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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Boolean Circuit Complexity and Two-Dimensional Cover Problems</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.14117v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS1" title="In 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Overview</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS1.SSS0.Px1" title="In 1.1 Overview ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title">Contributions.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS2" title="In 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Results</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS2.SSS0.Px1" title="In 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title">Notation.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS2.SSS0.Px2" title="In 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title">Organization.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS2.SSS0.Px3" title="In 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title">Acknowledgements.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Discrete Complexity</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS1" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Definitions and notation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Examples</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS1" title="In 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2.1 </span>Boolean circuit complexity</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS2" title="In 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2.2 </span>Bipartite graph complexity</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS3" title="In 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2.3 </span>Higher-dimensional generalizations of graph complexity</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS4" title="In 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2.4 </span>Combinatorial rectangles from communication complexity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS3" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Basic lemmas and other useful results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS4" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Transference of lower bounds</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS5" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5 </span>Cyclic Discrete Complexity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Characterizations of Discrete Complexity via Set-Theoretic Fusion</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Definitions and notation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1.SSS0.Px1" title="In 3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title">Cover Graph of <math alttext="A" class="ltx_Math" display="inline"><semantics><mi>A</mi><annotation-xml encoding="MathML-Content"><ci>𝐴</ci></annotation-xml><annotation encoding="application/x-tex">A</annotation><annotation encoding="application/x-llamapun">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline"><semantics><mi class="ltx_font_mathcaligraphic">ℬ</mi><annotation-xml encoding="MathML-Content"><ci>ℬ</ci></annotation-xml><annotation encoding="application/x-tex">\mathcal{B}</annotation><annotation encoding="application/x-llamapun">caligraphic_B</annotation></semantics></math>.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS2" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Discrete complexity lower bounds using the fusion method</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS3" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Set-theoretic fusion as a complete framework for lower bounds</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>An exact characterization via cyclic discrete complexity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Graph Complexity and Two-Dimensional Cover Problems</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS1" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Basic results and connections</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>A simple lower bound example</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS3" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Nondeterministic graph complexity</span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Boolean Circuit Complexity and Two-Dimensional Cover Problems</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Bruno P. Cavalar <br class="ltx_break"/><span class="ltx_text" id="id1.1.id1" style="font-size:90%;">Department of Computer Science <br class="ltx_break"/>University of Oxford</span> </span><span class="ltx_author_notes"><span class="ltx_text ltx_font_typewriter" id="id2.2.id1">E-mail: bruno.cavalar@cs.oxford.ac.uk</span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Igor C. Oliveira <br class="ltx_break"/><span class="ltx_text" id="id3.1.id1" style="font-size:90%;">Department of Computer Science <br class="ltx_break"/>University of Warwick</span> </span><span class="ltx_author_notes"><span class="ltx_text ltx_font_typewriter" id="id4.2.id1">E-mail: igor.oliveira@warwick.ac.uk</span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id5.id1">We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudlák, Rödl, and Savický (1988). For convenience, we formalize these ideas in the more general setting of “discrete complexity”, i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures.</p> <p class="ltx_p" id="id6.id2">We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity.</p> <p class="ltx_p" id="id7.id3">Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of “cyclic” Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS1" title="In 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Overview</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.SS2" title="In 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Results</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Discrete Complexity</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS1" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Definitions and notation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Examples</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS3" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Basic lemmas and other useful results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS4" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Transference of lower bounds</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS5" title="In 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5 </span>Cyclic Discrete Complexity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Characterizations of Discrete Complexity via Set-Theoretic Fusion</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Definitions and notation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS2" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Discrete complexity lower bounds using the fusion method</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS3" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Set-theoretic fusion as a complete framework for lower bounds</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="In 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>An exact characterization via cyclic discrete complexity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4" title="In Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Graph Complexity and Two-Dimensional Cover Problems</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS1" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Basic results and connections</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>A simple lower bound example</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS3" title="In 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Nondeterministic graph complexity</span></a></li> </ol> </li> </ol></nav> <section class="ltx_section ltx_indent_first" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <section class="ltx_subsection ltx_indent_first" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Overview</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.1">Obtaining circuit size lower bounds for explicit Boolean functions is a central research problem in theoretical computer science. While restricted classes of circuits such as constant-depth circuits and monotone circuits are reasonably well understood (see, e.g., <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib7" title="">7</a>]</cite>), understanding the power and limitations of general (unrestricted) Boolean circuits remains a major challenge.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.4">The strongest known lower bounds on the number of gates necessary to compute an explicit Boolean function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.4"><semantics id="S1.SS1.p2.1.m1.4a"><mrow id="S1.SS1.p2.1.m1.4.5" xref="S1.SS1.p2.1.m1.4.5.cmml"><mi id="S1.SS1.p2.1.m1.4.5.2" xref="S1.SS1.p2.1.m1.4.5.2.cmml">f</mi><mo id="S1.SS1.p2.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S1.SS1.p2.1.m1.4.5.1.cmml">:</mo><mrow id="S1.SS1.p2.1.m1.4.5.3" xref="S1.SS1.p2.1.m1.4.5.3.cmml"><msup id="S1.SS1.p2.1.m1.4.5.3.2" xref="S1.SS1.p2.1.m1.4.5.3.2.cmml"><mrow id="S1.SS1.p2.1.m1.4.5.3.2.2.2" xref="S1.SS1.p2.1.m1.4.5.3.2.2.1.cmml"><mo id="S1.SS1.p2.1.m1.4.5.3.2.2.2.1" stretchy="false" xref="S1.SS1.p2.1.m1.4.5.3.2.2.1.cmml">{</mo><mn id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml">0</mn><mo id="S1.SS1.p2.1.m1.4.5.3.2.2.2.2" xref="S1.SS1.p2.1.m1.4.5.3.2.2.1.cmml">,</mo><mn id="S1.SS1.p2.1.m1.2.2" xref="S1.SS1.p2.1.m1.2.2.cmml">1</mn><mo id="S1.SS1.p2.1.m1.4.5.3.2.2.2.3" stretchy="false" xref="S1.SS1.p2.1.m1.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S1.SS1.p2.1.m1.4.5.3.2.3" xref="S1.SS1.p2.1.m1.4.5.3.2.3.cmml">n</mi></msup><mo id="S1.SS1.p2.1.m1.4.5.3.1" stretchy="false" xref="S1.SS1.p2.1.m1.4.5.3.1.cmml">→</mo><mrow id="S1.SS1.p2.1.m1.4.5.3.3.2" xref="S1.SS1.p2.1.m1.4.5.3.3.1.cmml"><mo id="S1.SS1.p2.1.m1.4.5.3.3.2.1" stretchy="false" xref="S1.SS1.p2.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S1.SS1.p2.1.m1.3.3" xref="S1.SS1.p2.1.m1.3.3.cmml">0</mn><mo id="S1.SS1.p2.1.m1.4.5.3.3.2.2" xref="S1.SS1.p2.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S1.SS1.p2.1.m1.4.4" xref="S1.SS1.p2.1.m1.4.4.cmml">1</mn><mo id="S1.SS1.p2.1.m1.4.5.3.3.2.3" stretchy="false" xref="S1.SS1.p2.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.4b"><apply id="S1.SS1.p2.1.m1.4.5.cmml" xref="S1.SS1.p2.1.m1.4.5"><ci id="S1.SS1.p2.1.m1.4.5.1.cmml" xref="S1.SS1.p2.1.m1.4.5.1">:</ci><ci id="S1.SS1.p2.1.m1.4.5.2.cmml" xref="S1.SS1.p2.1.m1.4.5.2">𝑓</ci><apply id="S1.SS1.p2.1.m1.4.5.3.cmml" xref="S1.SS1.p2.1.m1.4.5.3"><ci id="S1.SS1.p2.1.m1.4.5.3.1.cmml" xref="S1.SS1.p2.1.m1.4.5.3.1">→</ci><apply id="S1.SS1.p2.1.m1.4.5.3.2.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="S1.SS1.p2.1.m1.4.5.3.2.1.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2">superscript</csymbol><set id="S1.SS1.p2.1.m1.4.5.3.2.2.1.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2.2.2"><cn id="S1.SS1.p2.1.m1.1.1.cmml" type="integer" xref="S1.SS1.p2.1.m1.1.1">0</cn><cn id="S1.SS1.p2.1.m1.2.2.cmml" type="integer" xref="S1.SS1.p2.1.m1.2.2">1</cn></set><ci id="S1.SS1.p2.1.m1.4.5.3.2.3.cmml" xref="S1.SS1.p2.1.m1.4.5.3.2.3">𝑛</ci></apply><set id="S1.SS1.p2.1.m1.4.5.3.3.1.cmml" xref="S1.SS1.p2.1.m1.4.5.3.3.2"><cn id="S1.SS1.p2.1.m1.3.3.cmml" type="integer" xref="S1.SS1.p2.1.m1.3.3">0</cn><cn id="S1.SS1.p2.1.m1.4.4.cmml" type="integer" xref="S1.SS1.p2.1.m1.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> are of the form <math alttext="C\cdot n" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.1"><semantics id="S1.SS1.p2.2.m2.1a"><mrow id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml"><mi id="S1.SS1.p2.2.m2.1.1.2" xref="S1.SS1.p2.2.m2.1.1.2.cmml">C</mi><mo id="S1.SS1.p2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.SS1.p2.2.m2.1.1.1.cmml">⋅</mo><mi id="S1.SS1.p2.2.m2.1.1.3" xref="S1.SS1.p2.2.m2.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.1b"><apply id="S1.SS1.p2.2.m2.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1"><ci id="S1.SS1.p2.2.m2.1.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1.1">⋅</ci><ci id="S1.SS1.p2.2.m2.1.1.2.cmml" xref="S1.SS1.p2.2.m2.1.1.2">𝐶</ci><ci id="S1.SS1.p2.2.m2.1.1.3.cmml" xref="S1.SS1.p2.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.1c">C\cdot n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.1d">italic_C ⋅ italic_n</annotation></semantics></math> for a constant <math alttext="C\leq 5" class="ltx_Math" display="inline" id="S1.SS1.p2.3.m3.1"><semantics id="S1.SS1.p2.3.m3.1a"><mrow id="S1.SS1.p2.3.m3.1.1" xref="S1.SS1.p2.3.m3.1.1.cmml"><mi id="S1.SS1.p2.3.m3.1.1.2" xref="S1.SS1.p2.3.m3.1.1.2.cmml">C</mi><mo id="S1.SS1.p2.3.m3.1.1.1" xref="S1.SS1.p2.3.m3.1.1.1.cmml">≤</mo><mn id="S1.SS1.p2.3.m3.1.1.3" xref="S1.SS1.p2.3.m3.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.3.m3.1b"><apply id="S1.SS1.p2.3.m3.1.1.cmml" xref="S1.SS1.p2.3.m3.1.1"><leq id="S1.SS1.p2.3.m3.1.1.1.cmml" xref="S1.SS1.p2.3.m3.1.1.1"></leq><ci id="S1.SS1.p2.3.m3.1.1.2.cmml" xref="S1.SS1.p2.3.m3.1.1.2">𝐶</ci><cn id="S1.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S1.SS1.p2.3.m3.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.3.m3.1c">C\leq 5</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.3.m3.1d">italic_C ≤ 5</annotation></semantics></math>. The largest known value of <math alttext="C" class="ltx_Math" display="inline" id="S1.SS1.p2.4.m4.1"><semantics id="S1.SS1.p2.4.m4.1a"><mi id="S1.SS1.p2.4.m4.1.1" xref="S1.SS1.p2.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.4.m4.1b"><ci id="S1.SS1.p2.4.m4.1.1.cmml" xref="S1.SS1.p2.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.4.m4.1d">italic_C</annotation></semantics></math> depends on the exact set of allowed operations (see <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib3" title="">3</a>]</cite> and references therein). To the best of our knowledge, the existing lower bounds on gate complexity for unrestricted Boolean circuits with a single output bit have all been obtained via the gate elimination method and its extensions. Unfortunately, it is not expected that this technique can lead to much better bounds <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib5" title="">5</a>]</cite>, let alone super-linear circuit size lower bounds.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.1">This paper revisits a classical approach to lower bounds known as the fusion method <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>. The latter reduces the analysis of the circuit complexity of a Boolean function to obtaining bounds on certain related combinatorial cover problems. The method can also be adapted to weaker circuit classes, where it has been successful in some contexts (see <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib19" title="">19</a>]</cite> for an overview of results).<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>The fusion method can be seen as an instantiation of the generalized approximation method. For a self-contained exposition of the connection between the fusion method and the approximation method, we refer the reader to <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib13" title="">13</a>]</cite>.</span></span></span></p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.1">An advantage of the fusion method over the gate elimination method is that it provides a tight characterization (up to a constant or polynomial factor, depending on the formulation) of the circuit complexity of a function. In particular, if a strong enough circuit lower bound holds, then in principle it can be established via the fusion method.</p> </div> <section class="ltx_paragraph ltx_indentfirst" id="S1.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Contributions.</h5> <div class="ltx_para" id="S1.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS1.SSS0.Px1.p1.1">We can informally summarize our contributions as follows:</p> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">We exhibit a new instantiation of the fusion method that reduces the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of “two-dimensional” combinatorial cover problems.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">To achieve this, we introduce a framework that combines the fusion method for lower bounds with the notion of graph complexity and its variants <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite>. In particular, we observe that cover complexity offers a particularly strong “transference” theorem between Boolean circuit complexity and graph complexity.</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">As a byproduct of our conceptual and technical contributions, we obtain a tight asymptotic bound on the cover complexity of a random graph, and introduce a useful notion of nondeterministic graph complexity.</p> </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S1.I1.i4.p1"> <p class="ltx_p" id="S1.I1.i4.p1.2">Finally, we describe an exact correspondence between cover complexity and circuit complexity. This is relevant for the investigation of state-of-the-art circuit lower bounds of the form <math alttext="C\cdot n" class="ltx_Math" display="inline" id="S1.I1.i4.p1.1.m1.1"><semantics id="S1.I1.i4.p1.1.m1.1a"><mrow id="S1.I1.i4.p1.1.m1.1.1" xref="S1.I1.i4.p1.1.m1.1.1.cmml"><mi id="S1.I1.i4.p1.1.m1.1.1.2" xref="S1.I1.i4.p1.1.m1.1.1.2.cmml">C</mi><mo id="S1.I1.i4.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.I1.i4.p1.1.m1.1.1.1.cmml">⋅</mo><mi id="S1.I1.i4.p1.1.m1.1.1.3" xref="S1.I1.i4.p1.1.m1.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.1.m1.1b"><apply id="S1.I1.i4.p1.1.m1.1.1.cmml" xref="S1.I1.i4.p1.1.m1.1.1"><ci id="S1.I1.i4.p1.1.m1.1.1.1.cmml" xref="S1.I1.i4.p1.1.m1.1.1.1">⋅</ci><ci id="S1.I1.i4.p1.1.m1.1.1.2.cmml" xref="S1.I1.i4.p1.1.m1.1.1.2">𝐶</ci><ci id="S1.I1.i4.p1.1.m1.1.1.3.cmml" xref="S1.I1.i4.p1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.1.m1.1c">C\cdot n</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.1.m1.1d">italic_C ⋅ italic_n</annotation></semantics></math>, where <math alttext="C" class="ltx_Math" display="inline" id="S1.I1.i4.p1.2.m2.1"><semantics id="S1.I1.i4.p1.2.m2.1a"><mi id="S1.I1.i4.p1.2.m2.1.1" xref="S1.I1.i4.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.2.m2.1b"><ci id="S1.I1.i4.p1.2.m2.1.1.cmml" xref="S1.I1.i4.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.2.m2.1d">italic_C</annotation></semantics></math> is constant.</p> </div> </li> </ol> </div> <div class="ltx_para ltx_noindent" id="S1.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS1.SSS0.Px1.p2.1">In the next section, we describe these results and their connections to previous work in more detail.</p> </div> </section> </section> <section class="ltx_subsection ltx_indent_first" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Results</h3> <section class="ltx_paragraph ltx_indentfirst" id="S1.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Notation.</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p1.13">Given a family <math alttext="\mathcal{B}=\{B_{1},\ldots,B_{m}\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.1.m1.3"><semantics id="S1.SS2.SSS0.Px1.p1.1.m1.3a"><mrow id="S1.SS2.SSS0.Px1.p1.1.m1.3.3" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.4" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.4.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.3" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.3.cmml">=</mo><mrow id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml">{</mo><msub id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.2.cmml">B</mi><mn id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.4" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p1.1.m1.1.1" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.5" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.2" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.2.cmml">B</mi><mi id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.3" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.6" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.1.m1.3b"><apply id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3"><eq id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.3.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.3"></eq><ci id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.4.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.4">ℬ</ci><set id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.3.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2"><apply id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.2">𝐵</ci><cn id="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1">…</ci><apply id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.2">𝐵</ci><ci id="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p1.1.m1.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.1.m1.3c">\mathcal{B}=\{B_{1},\ldots,B_{m}\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.1.m1.3d">caligraphic_B = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math>, where each set <math alttext="B_{i}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p1.2.m2.1a"><msub id="S1.SS2.SSS0.Px1.p1.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.2" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.2.cmml">B</mi><mi id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.3" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.2.m2.1b"><apply id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.2">𝐵</ci><ci id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.2.m2.1c">B_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is contained in a finite fixed ground set <math alttext="\Gamma" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p1.3.m3.1a"><mi id="S1.SS2.SSS0.Px1.p1.3.m3.1.1" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.3.m3.1b"><ci id="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.3.m3.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.3.m3.1d">roman_Γ</annotation></semantics></math>, and a target set <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p1.4.m4.1a"><mi id="S1.SS2.SSS0.Px1.p1.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.4.m4.1b"><ci id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.4.m4.1d">italic_A</annotation></semantics></math>, we let <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p1.5.m5.1a"><mrow id="S1.SS2.SSS0.Px1.p1.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.3" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.2" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.5.m5.1b"><apply id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1"><times id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.5.m5.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.5.m5.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> denote the minimum total number of pairwise unions and intersections needed to construct <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p1.6.m6.1a"><mi id="S1.SS2.SSS0.Px1.p1.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.6.m6.1b"><ci id="S1.SS2.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.6.m6.1d">italic_A</annotation></semantics></math> starting from <math alttext="B_{1},\ldots,B_{m}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.7.m7.3"><semantics id="S1.SS2.SSS0.Px1.p1.7.m7.3a"><mrow id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml"><msub id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2.cmml">B</mi><mn id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.3" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p1.7.m7.1.1" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p1.7.m7.1.1.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.4" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2.cmml">B</mi><mi id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.7.m7.3b"><list id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.3.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2"><apply id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.2">𝐵</ci><cn id="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.7.m7.2.2.1.1.3">1</cn></apply><ci id="S1.SS2.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.1.1">…</ci><apply id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.2">𝐵</ci><ci id="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.3.3.2.2.3">𝑚</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.7.m7.3c">B_{1},\ldots,B_{m}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.7.m7.3d">italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. We say that <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.8.m8.1"><semantics id="S1.SS2.SSS0.Px1.p1.8.m8.1a"><mrow id="S1.SS2.SSS0.Px1.p1.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.8.m8.1b"><apply id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1"><times id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.8.m8.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.8.m8.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p1.13.1">discrete complexity</em> of <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.9.m9.1"><semantics id="S1.SS2.SSS0.Px1.p1.9.m9.1a"><mi id="S1.SS2.SSS0.Px1.p1.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p1.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.9.m9.1b"><ci id="S1.SS2.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.9.m9.1d">italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.10.m10.1"><semantics id="S1.SS2.SSS0.Px1.p1.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p1.10.m10.1.1" xref="S1.SS2.SSS0.Px1.p1.10.m10.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.10.m10.1b"><ci id="S1.SS2.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.10.m10.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.10.m10.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.10.m10.1d">caligraphic_B</annotation></semantics></math> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS1" title="2.1 Definitions and notation ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> for a formal presentation). We will be interested in the discrete complexity of <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p1.13.2">non-trivial</em> sets <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.11.m11.1"><semantics id="S1.SS2.SSS0.Px1.p1.11.m11.1a"><mi id="S1.SS2.SSS0.Px1.p1.11.m11.1.1" xref="S1.SS2.SSS0.Px1.p1.11.m11.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.11.m11.1b"><ci id="S1.SS2.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.11.m11.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.11.m11.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.11.m11.1d">italic_A</annotation></semantics></math>, i.e., when <math alttext="A\neq\emptyset" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.12.m12.1"><semantics id="S1.SS2.SSS0.Px1.p1.12.m12.1a"><mrow id="S1.SS2.SSS0.Px1.p1.12.m12.1.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.cmml">≠</mo><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.12.m12.1b"><apply id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1"><neq id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1"></neq><ci id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2">𝐴</ci><emptyset id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.12.m12.1c">A\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.12.m12.1d">italic_A ≠ ∅</annotation></semantics></math> and <math alttext="A\neq\Gamma" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.13.m13.1"><semantics id="S1.SS2.SSS0.Px1.p1.13.m13.1a"><mrow id="S1.SS2.SSS0.Px1.p1.13.m13.1.1" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.2" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.1" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.1.cmml">≠</mo><mi id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.3" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.13.m13.1b"><apply id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1"><neq id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.1"></neq><ci id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.13.m13.1c">A\neq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.13.m13.1d">italic_A ≠ roman_Γ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p2.7">This general definition can be used to capture a variety of problems. For instance, the monotone circuit complexity of a function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p2.1.m1.4"><semantics id="S1.SS2.SSS0.Px1.p2.1.m1.4a"><mrow id="S1.SS2.SSS0.Px1.p2.1.m1.4.5" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.cmml"><mi id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.2.cmml">f</mi><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.1.cmml">:</mo><mrow id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.cmml"><msup id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.cmml"><mrow id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p2.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p2.1.m1.1.1.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.2.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p2.1.m1.2.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.3" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.3.cmml">n</mi></msup><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.1.cmml">→</mo><mrow id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p2.1.m1.3.3" xref="S1.SS2.SSS0.Px1.p2.1.m1.3.3.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.2.2" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p2.1.m1.4.4" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.4.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p2.1.m1.4b"><apply id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5"><ci id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.1.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.1">:</ci><ci id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.2.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.2">𝑓</ci><apply id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3"><ci id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.1">→</ci><apply id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2">superscript</csymbol><set id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.2.2"><cn id="S1.SS2.SSS0.Px1.p2.1.m1.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.1.m1.1.1">0</cn><cn id="S1.SS2.SSS0.Px1.p2.1.m1.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.1.m1.2.2">1</cn></set><ci id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.2.3">𝑛</ci></apply><set id="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.5.3.3.2"><cn id="S1.SS2.SSS0.Px1.p2.1.m1.3.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.1.m1.3.3">0</cn><cn id="S1.SS2.SSS0.Px1.p2.1.m1.4.4.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.1.m1.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p2.1.m1.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p2.1.m1.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> is simply <math alttext="D(f^{-1}(1)\mid\{x_{1},\ldots,x_{n},\emptyset,\bar{1}\})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p2.2.m2.5"><semantics id="S1.SS2.SSS0.Px1.p2.2.m2.5a"><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.cmml"><mi id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.3" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.cmml"><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.cmml"><msup id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.cmml"><mi id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.2.cmml">f</mi><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3a" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.cmml">−</mo><mn id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.2.cmml">1</mn></mrow></msup><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.3.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.cmml"><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.cmml">(</mo><mn id="S1.SS2.SSS0.Px1.p2.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.1.1.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.cmml">)</mo></mrow></mrow><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.3" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.3.cmml">∣</mo><mrow id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">{</mo><msub id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.2.cmml">x</mi><mn id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.4" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p2.2.m2.2.2" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p2.2.m2.2.2.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.5" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.2.cmml">x</mi><mi id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.3" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.3.cmml">n</mi></msub><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.6" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p2.2.m2.3.3" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p2.2.m2.3.3.cmml">∅</mi><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.7" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">,</mo><mover accent="true" id="S1.SS2.SSS0.Px1.p2.2.m2.4.4" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4.cmml"><mn id="S1.SS2.SSS0.Px1.p2.2.m2.4.4.2" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.2.m2.4.4.1" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4.1.cmml">¯</mo></mover><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.8" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml">}</mo></mrow></mrow><mo id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p2.2.m2.5b"><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5"><times id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.2"></times><ci id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.3">conditional</csymbol><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4"><times id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.1"></times><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2">superscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.2">𝑓</ci><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3"><minus id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3"></minus><cn id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.4.2.3.2">1</cn></apply></apply><cn id="S1.SS2.SSS0.Px1.p2.2.m2.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.2.m2.1.1">1</cn></apply><set id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2"><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.2">𝑥</ci><cn id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.1.1.1.3">1</cn></apply><ci id="S1.SS2.SSS0.Px1.p2.2.m2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.2.2">…</ci><apply id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.2">𝑥</ci><ci id="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.5.5.1.1.1.2.2.2.3">𝑛</ci></apply><emptyset id="S1.SS2.SSS0.Px1.p2.2.m2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.3.3"></emptyset><apply id="S1.SS2.SSS0.Px1.p2.2.m2.4.4.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4"><ci id="S1.SS2.SSS0.Px1.p2.2.m2.4.4.1.cmml" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4.1">¯</ci><cn id="S1.SS2.SSS0.Px1.p2.2.m2.4.4.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.2.m2.4.4.2">1</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p2.2.m2.5c">D(f^{-1}(1)\mid\{x_{1},\ldots,x_{n},\emptyset,\bar{1}\})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p2.2.m2.5d">italic_D ( italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) ∣ { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ∅ , over¯ start_ARG 1 end_ARG } )</annotation></semantics></math>, where each symbol from <math alttext="\{x_{1},\ldots,x_{n},\emptyset,\bar{1}\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p2.3.m3.5"><semantics id="S1.SS2.SSS0.Px1.p2.3.m3.5a"><mrow id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml"><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">{</mo><msub id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.2" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.2.cmml">x</mi><mn id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.3" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.4" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p2.3.m3.1.1" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p2.3.m3.1.1.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.5" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.2" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.2.cmml">x</mi><mi id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.3" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.3.cmml">n</mi></msub><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.6" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p2.3.m3.2.2" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p2.3.m3.2.2.cmml">∅</mi><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.7" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">,</mo><mover accent="true" id="S1.SS2.SSS0.Px1.p2.3.m3.3.3" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3.cmml"><mn id="S1.SS2.SSS0.Px1.p2.3.m3.3.3.2" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.3.m3.3.3.1" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3.1.cmml">¯</mo></mover><mo id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.8" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p2.3.m3.5b"><set id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.3.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2"><apply id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.2">𝑥</ci><cn id="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.3.m3.4.4.1.1.3">1</cn></apply><ci id="S1.SS2.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.1.1">…</ci><apply id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.2">𝑥</ci><ci id="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.5.5.2.2.3">𝑛</ci></apply><emptyset id="S1.SS2.SSS0.Px1.p2.3.m3.2.2.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.2.2"></emptyset><apply id="S1.SS2.SSS0.Px1.p2.3.m3.3.3.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3"><ci id="S1.SS2.SSS0.Px1.p2.3.m3.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3.1">¯</ci><cn id="S1.SS2.SSS0.Px1.p2.3.m3.3.3.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.3.m3.3.3.2">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p2.3.m3.5c">\{x_{1},\ldots,x_{n},\emptyset,\bar{1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p2.3.m3.5d">{ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ∅ , over¯ start_ARG 1 end_ARG }</annotation></semantics></math> represents the natural corresponding subset of <math alttext="\{0,1\}^{n}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p2.4.m4.2"><semantics id="S1.SS2.SSS0.Px1.p2.4.m4.2a"><msup id="S1.SS2.SSS0.Px1.p2.4.m4.2.3" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.cmml"><mrow id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.2" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p2.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p2.4.m4.1.1.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.2.2" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p2.4.m4.2.2" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.1.cmml">}</mo></mrow><mi id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.3" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p2.4.m4.2b"><apply id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.cmml" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3">superscript</csymbol><set id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.2.2"><cn id="S1.SS2.SSS0.Px1.p2.4.m4.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.4.m4.1.1">0</cn><cn id="S1.SS2.SSS0.Px1.p2.4.m4.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.2">1</cn></set><ci id="S1.SS2.SSS0.Px1.p2.4.m4.2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p2.4.m4.2.3.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p2.4.m4.2c">\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p2.4.m4.2d">{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Similarly, we can capture (non-monotone) Boolean circuit complexity by considering the family <math alttext="\mathcal{B}_{n}=\{x_{1},\ldots,x_{n},\overline{x_{1}},\ldots,\overline{x_{n}}\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p2.5.m5.6"><semantics id="S1.SS2.SSS0.Px1.p2.5.m5.6a"><mrow id="S1.SS2.SSS0.Px1.p2.5.m5.6.6" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.cmml"><msub id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4.2" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4.2.cmml">ℬ</mi><mi id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4.3" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.4.3.cmml">n</mi></msub><mo id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.3" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.3.cmml">=</mo><mrow id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.2.2" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p2.5.m5.6.6.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p2.5.m5.6.6.2.3.cmml">{</mo><msub 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id="S1.SS2.SSS0.Px1.p2.7.m7.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p2.7.m7.1.1">1</cn></apply><apply id="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.2">ℬ</ci><ci id="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p2.7.m7.2.2.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p2.7.m7.2c">D(f^{-1}(1)\mid\mathcal{B}_{n})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p2.7.m7.2d">italic_D ( italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) ∣ caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>This captures the <em class="ltx_emph ltx_font_italic" id="footnote2.1">DeMorgan circuit complexity</em>, where negations are at the bottom of the circuit.</span></span></span></p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p3"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p3.13">Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p3.1.m1.1a"><mrow id="S1.SS2.SSS0.Px1.p3.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.2" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.2.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.1" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.1.cmml">=</mo><msup id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.cmml"><mn id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.2" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.2.cmml">2</mn><mi id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.3" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.1.m1.1b"><apply id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1"><eq id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.2">𝑁</ci><apply id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3">superscript</csymbol><cn id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.2">2</cn><ci id="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p3.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.1.m1.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.1.m1.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p3.2.m2.1a"><mrow id="S1.SS2.SSS0.Px1.p3.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.2" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.2.cmml">n</mi><mo id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.1" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.1.cmml">∈</mo><mi id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.3" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.2.m2.1b"><apply id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1"><in id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.1"></in><ci id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.2">𝑛</ci><ci id="S1.SS2.SSS0.Px1.p3.2.m2.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p3.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.2.m2.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, and let <math alttext="[N]=\{1,2,\ldots,N\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.3.m3.5"><semantics id="S1.SS2.SSS0.Px1.p3.3.m3.5a"><mrow id="S1.SS2.SSS0.Px1.p3.3.m3.5.6" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.cmml"><mrow id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.2" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p3.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p3.3.m3.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.1.1.cmml">]</mo></mrow><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.1" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.1.cmml">=</mo><mrow id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p3.3.m3.2.2" xref="S1.SS2.SSS0.Px1.p3.3.m3.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2.2" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p3.3.m3.3.3" xref="S1.SS2.SSS0.Px1.p3.3.m3.3.3.cmml">2</mn><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2.3" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p3.3.m3.4.4" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p3.3.m3.4.4.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2.4" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p3.3.m3.5.5" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.5.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2.5" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.3.m3.5b"><apply id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6"><eq id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.1.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.1"></eq><apply id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.1.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.2.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.1.1">𝑁</ci></apply><set id="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.6.3.2"><cn id="S1.SS2.SSS0.Px1.p3.3.m3.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p3.3.m3.2.2">1</cn><cn id="S1.SS2.SSS0.Px1.p3.3.m3.3.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p3.3.m3.3.3">2</cn><ci id="S1.SS2.SSS0.Px1.p3.3.m3.4.4.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.4.4">…</ci><ci id="S1.SS2.SSS0.Px1.p3.3.m3.5.5.cmml" xref="S1.SS2.SSS0.Px1.p3.3.m3.5.5">𝑁</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.3.m3.5c">[N]=\{1,2,\ldots,N\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.3.m3.5d">[ italic_N ] = { 1 , 2 , … , italic_N }</annotation></semantics></math>. As another example in discrete complexity, we can consider subsets <math alttext="R_{1},\ldots,R_{N},C_{1},\ldots,C_{N}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.4.m4.6"><semantics id="S1.SS2.SSS0.Px1.p3.4.m4.6a"><mrow id="S1.SS2.SSS0.Px1.p3.4.m4.6.6.4" xref="S1.SS2.SSS0.Px1.p3.4.m4.6.6.5.cmml"><msub id="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1" xref="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1.2" xref="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1.2.cmml">R</mi><mn id="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1.3" xref="S1.SS2.SSS0.Px1.p3.4.m4.3.3.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.SSS0.Px1.p3.4.m4.6.6.4.5" xref="S1.SS2.SSS0.Px1.p3.4.m4.6.6.5.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p3.4.m4.1.1" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p3.4.m4.1.1.cmml">…</mi><mo id="S1.SS2.SSS0.Px1.p3.4.m4.6.6.4.6" xref="S1.SS2.SSS0.Px1.p3.4.m4.6.6.5.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p3.4.m4.4.4.2.2" xref="S1.SS2.SSS0.Px1.p3.4.m4.4.4.2.2.cmml"><mi 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xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.4.3">𝑖</ci></apply><apply id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.3.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.3">conditional-set</csymbol><interval closure="open" id="S1.SS2.SSS0.Px1.p3.6.m6.4.4.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.4.4.1.1.1.2"><ci id="S1.SS2.SSS0.Px1.p3.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.1.1">𝑖</ci><ci id="S1.SS2.SSS0.Px1.p3.6.m6.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.2.2">𝑗</ci></interval><apply id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2"><in id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.1"></in><ci id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.2">𝑗</ci><apply id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.3.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.5.5.2.2.2.3.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p3.6.m6.3.3.cmml" xref="S1.SS2.SSS0.Px1.p3.6.m6.3.3">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.6.m6.5c">R_{i}=\{(i,j)\mid j\in[N]\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.6.m6.5d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { ( italic_i , italic_j ) ∣ italic_j ∈ [ italic_N ] }</annotation></semantics></math> corresponds to the <math alttext="i" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.7.m7.1"><semantics id="S1.SS2.SSS0.Px1.p3.7.m7.1a"><mi id="S1.SS2.SSS0.Px1.p3.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p3.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.7.m7.1b"><ci id="S1.SS2.SSS0.Px1.p3.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.7.m7.1d">italic_i</annotation></semantics></math>-th “row”, and each set <math alttext="C_{j}=\{(i,j)\mid i\in[N]\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.8.m8.5"><semantics id="S1.SS2.SSS0.Px1.p3.8.m8.5a"><mrow id="S1.SS2.SSS0.Px1.p3.8.m8.5.5" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.cmml"><msub id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.cmml"><mi id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.2.cmml">C</mi><mi id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.3" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.3.cmml">j</mi></msub><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.3" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.3.cmml">=</mo><mrow id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.1.cmml">{</mo><mrow id="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.1.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p3.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p3.8.m8.1.1.cmml">i</mi><mo id="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.2.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.1.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p3.8.m8.2.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.2.2.cmml">j</mi><mo id="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.1.cmml">)</mo></mrow><mo fence="true" id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.4" lspace="0em" rspace="0em" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.1.cmml">∣</mo><mrow id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.2.cmml">i</mi><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.1" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.1.cmml">∈</mo><mrow id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.2" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p3.8.m8.3.3" xref="S1.SS2.SSS0.Px1.p3.8.m8.3.3.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.5" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.8.m8.5b"><apply id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5"><eq id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.3.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.3"></eq><apply id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.2.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.2">𝐶</ci><ci id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.3.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.4.3">𝑗</ci></apply><apply id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.3">conditional-set</csymbol><interval closure="open" id="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.4.4.1.1.1.2"><ci id="S1.SS2.SSS0.Px1.p3.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.1.1">𝑖</ci><ci id="S1.SS2.SSS0.Px1.p3.8.m8.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.2.2">𝑗</ci></interval><apply id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2"><in id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.1"></in><ci id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.2">𝑖</ci><apply id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.5.5.2.2.2.3.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p3.8.m8.3.3.cmml" xref="S1.SS2.SSS0.Px1.p3.8.m8.3.3">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.8.m8.5c">C_{j}=\{(i,j)\mid i\in[N]\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.8.m8.5d">italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = { ( italic_i , italic_j ) ∣ italic_i ∈ [ italic_N ] }</annotation></semantics></math> corresponds to the <math alttext="j" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.9.m9.1"><semantics id="S1.SS2.SSS0.Px1.p3.9.m9.1a"><mi id="S1.SS2.SSS0.Px1.p3.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p3.9.m9.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.9.m9.1b"><ci id="S1.SS2.SSS0.Px1.p3.9.m9.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.9.m9.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.9.m9.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.9.m9.1d">italic_j</annotation></semantics></math>-th “column”. Then, given a set <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.10.m10.2"><semantics id="S1.SS2.SSS0.Px1.p3.10.m10.2a"><mrow id="S1.SS2.SSS0.Px1.p3.10.m10.2.3" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.2" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.2.cmml">G</mi><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.1" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.1.cmml">⊆</mo><mrow id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.cmml"><mrow id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.2" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p3.10.m10.1.1" xref="S1.SS2.SSS0.Px1.p3.10.m10.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.1" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.1.cmml">×</mo><mrow id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.2" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p3.10.m10.2.2" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.2.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.10.m10.2b"><apply id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3"><subset id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.1"></subset><ci id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.2">𝐺</ci><apply id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3"><times id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.1"></times><apply id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p3.10.m10.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.1.1">𝑁</ci></apply><apply id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p3.10.m10.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.10.m10.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.10.m10.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.10.m10.2d">italic_G ⊆ [ italic_N ] × [ 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cd="ambiguous" id="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4.1.cmml" xref="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4.2.cmml" xref="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4.2">𝐶</ci><ci id="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4.3.cmml" xref="S1.SS2.SSS0.Px1.p3.11.m11.8.8.4.4.4.3">𝑁</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.11.m11.8c">\mathcal{G}_{N,N}=\{R_{1},\ldots,R_{N},C_{1},\ldots,C_{N}\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.11.m11.8d">caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT = { italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_C start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT }</annotation></semantics></math>, the quantity <math alttext="D(G\mid\mathcal{G}_{N,N})" 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xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.2">𝐺</ci><apply id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.3.3.1.1.1.3.2">𝒢</ci><list id="S1.SS2.SSS0.Px1.p3.12.m12.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.2.2.2.4"><ci id="S1.SS2.SSS0.Px1.p3.12.m12.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.1.1.1.1">𝑁</ci><ci id="S1.SS2.SSS0.Px1.p3.12.m12.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p3.12.m12.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.12.m12.3c">D(G\mid\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.12.m12.3d">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> is known as the <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p3.13.1">graph complexity</em> of <math alttext="G" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p3.13.m13.1"><semantics id="S1.SS2.SSS0.Px1.p3.13.m13.1a"><mi id="S1.SS2.SSS0.Px1.p3.13.m13.1.1" xref="S1.SS2.SSS0.Px1.p3.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p3.13.m13.1b"><ci id="S1.SS2.SSS0.Px1.p3.13.m13.1.1.cmml" xref="S1.SS2.SSS0.Px1.p3.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p3.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p3.13.m13.1d">italic_G</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite>).</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p4"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p4.11">For the discussion below, we will need another definition. We let <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p4.1.m1.1a"><mrow id="S1.SS2.SSS0.Px1.p4.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.cmml"><msub id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.2" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.2.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.3" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.3.cmml">∩</mo></msub><mo id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.2" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.1.m1.1b"><apply id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1"><times id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.2"></times><apply id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.2">𝐷</ci><intersect id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.3.3"></intersect></apply><apply id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.1.m1.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.1.m1.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.1.m1.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> denote the minimum number of pairwise intersections sufficient to construct <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p4.2.m2.1a"><mi id="S1.SS2.SSS0.Px1.p4.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p4.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.2.m2.1b"><ci id="S1.SS2.SSS0.Px1.p4.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.2.m2.1d">italic_A</annotation></semantics></math> from the sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p4.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p4.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.3.m3.1b"><ci id="S1.SS2.SSS0.Px1.p4.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.3.m3.1d">caligraphic_B</annotation></semantics></math>. We say that <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p4.4.m4.1a"><mrow id="S1.SS2.SSS0.Px1.p4.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.cmml"><msub id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.2" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.2.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.3" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.3.cmml">∩</mo></msub><mo id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.2" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.4.m4.1b"><apply id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1"><times id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.2"></times><apply id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.2">𝐷</ci><intersect id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.3.3"></intersect></apply><apply id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.4.m4.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.4.m4.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.4.m4.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p4.11.1">intersection complexity</em> of <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p4.5.m5.1a"><mi id="S1.SS2.SSS0.Px1.p4.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p4.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.5.m5.1b"><ci id="S1.SS2.SSS0.Px1.p4.5.m5.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.5.m5.1d">italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p4.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p4.6.m6.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.6.m6.1b"><ci id="S1.SS2.SSS0.Px1.p4.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.6.m6.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.6.m6.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.6.m6.1d">caligraphic_B</annotation></semantics></math>. When <math alttext="\mathcal{B}=\mathcal{B}_{n}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.7.m7.1"><semantics id="S1.SS2.SSS0.Px1.p4.7.m7.1a"><mrow id="S1.SS2.SSS0.Px1.p4.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.2" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.2.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.1" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.1.cmml">=</mo><msub id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.2" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.2.cmml">ℬ</mi><mi id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.3" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.7.m7.1b"><apply id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1"><eq id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.2">ℬ</ci><apply id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.2">ℬ</ci><ci id="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p4.7.m7.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.7.m7.1c">\mathcal{B}=\mathcal{B}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.7.m7.1d">caligraphic_B = caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, we may refer to intersection complexity with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.8.m8.1"><semantics id="S1.SS2.SSS0.Px1.p4.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p4.8.m8.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.8.m8.1b"><ci id="S1.SS2.SSS0.Px1.p4.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.8.m8.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.8.m8.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.8.m8.1d">caligraphic_B</annotation></semantics></math> as <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p4.11.2">AND complexity</em>. We refer to <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.F1" title="In Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a> for an example. It is possible to show that <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.9.m9.1"><semantics id="S1.SS2.SSS0.Px1.p4.9.m9.1a"><mrow id="S1.SS2.SSS0.Px1.p4.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.cmml"><msub id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.2" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.2.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.3" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.3.cmml">∩</mo></msub><mo id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.2" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.9.m9.1b"><apply id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1"><times id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.2"></times><apply id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.2">𝐷</ci><intersect id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.3.3"></intersect></apply><apply id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.9.m9.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.9.m9.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.9.m9.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> and <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.10.m10.1"><semantics id="S1.SS2.SSS0.Px1.p4.10.m10.1a"><mrow id="S1.SS2.SSS0.Px1.p4.10.m10.1.1" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.3" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.2" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.10.m10.1b"><apply id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1"><times id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p4.10.m10.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.10.m10.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.10.m10.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> are polynomially related, with a dependency on <math alttext="|\mathcal{B}|" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p4.11.m11.1"><semantics id="S1.SS2.SSS0.Px1.p4.11.m11.1a"><mrow id="S1.SS2.SSS0.Px1.p4.11.m11.1.2.2" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p4.11.m11.1.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p4.11.m11.1.1" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.1.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p4.11.m11.1.2.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p4.11.m11.1b"><apply id="S1.SS2.SSS0.Px1.p4.11.m11.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.2.2"><abs id="S1.SS2.SSS0.Px1.p4.11.m11.1.2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.2.2.1"></abs><ci id="S1.SS2.SSS0.Px1.p4.11.m11.1.1.cmml" xref="S1.SS2.SSS0.Px1.p4.11.m11.1.1">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p4.11.m11.1c">|\mathcal{B}|</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p4.11.m11.1d">| caligraphic_B |</annotation></semantics></math> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS3" title="2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.3</span></a> for more details).</p> </div> <figure class="ltx_figure" id="S1.F1"><svg class="ltx_picture ltx_centering" height="119.77" id="S1.F1.pic1" overflow="visible" version="1.1" width="119.77"><g fill="#000000" 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47.24 94.49 L 47.24 118.11 L 70.87 118.11 L 70.87 94.49 Z M 70.87 118.11" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 70.87 0 M 70.87 0 L 70.87 23.62 L 94.49 23.62 L 94.49 0 Z M 94.49 23.62" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 70.87 23.62 M 70.87 23.62 L 70.87 47.24 L 94.49 47.24 L 94.49 23.62 Z M 94.49 47.24" style="stroke:none"></path></g><g color="#FFFFFF" fill="#FFFFFF" stroke="#FFFFFF"><path d="M 70.87 23.62 M 70.87 23.62 L 70.87 47.24 L 94.49 47.24 L 94.49 23.62 Z M 94.49 47.24" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 70.87 47.24 M 70.87 47.24 L 70.87 70.87 L 94.49 70.87 L 94.49 47.24 Z M 94.49 70.87" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 70.87 70.87 M 70.87 70.87 L 70.87 94.49 L 94.49 94.49 L 94.49 70.87 Z M 94.49 94.49" style="stroke:none"></path></g><g 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47.24 Z M 118.11 70.87" style="stroke:none"></path></g><g color="#FFFFFF" fill="#FFFFFF" stroke="#FFFFFF"><path d="M 94.49 47.24 M 94.49 47.24 L 94.49 70.87 L 118.11 70.87 L 118.11 47.24 Z M 118.11 70.87" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 94.49 70.87 M 94.49 70.87 L 94.49 94.49 L 118.11 94.49 L 118.11 70.87 Z M 118.11 94.49" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 94.49 94.49 M 94.49 94.49 L 94.49 118.11 L 118.11 118.11 L 118.11 94.49 Z M 118.11 118.11" style="stroke:none"></path></g><g color="#FFFFFF" fill="#FFFFFF" stroke="#FFFFFF"><path d="M 94.49 94.49 M 94.49 94.49 L 94.49 118.11 L 118.11 118.11 L 118.11 94.49 Z M 118.11 118.11" style="stroke:none"></path></g></g><g color="#000000" fill="#000000" stroke="#000000" stroke-width="0.8pt"><path d="M 0 0 M 0 0 L 118.11 0 M 0 23.62 L 118.11 23.62 M 0 47.24 L 118.11 47.24 M 0 70.87 L 118.11 70.87 M 0 94.49 L 118.11 94.49 M 0 118.1 L 118.11 118.1 M 0 0 L 0 118.11 M 23.62 0 L 23.62 118.11 M 47.24 0 L 47.24 118.11 M 70.87 0 L 70.87 118.11 M 94.49 0 L 94.49 118.11 M 118.1 0 L 118.1 118.11 M 118.11 118.11" style="fill:none"></path></g><g stroke-width="1.2pt"><path d="M 0 0 M 0 0 L 0 118.11 L 118.11 118.11 L 118.11 0 Z M 118.11 118.11" style="fill:none"></path></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>A graphical representation of a set <math alttext="G\subseteq[5]\times[5]" class="ltx_Math" display="inline" id="S1.F1.4.m1.2"><semantics id="S1.F1.4.m1.2b"><mrow id="S1.F1.4.m1.2.3" xref="S1.F1.4.m1.2.3.cmml"><mi id="S1.F1.4.m1.2.3.2" xref="S1.F1.4.m1.2.3.2.cmml">G</mi><mo id="S1.F1.4.m1.2.3.1" xref="S1.F1.4.m1.2.3.1.cmml">⊆</mo><mrow id="S1.F1.4.m1.2.3.3" xref="S1.F1.4.m1.2.3.3.cmml"><mrow id="S1.F1.4.m1.2.3.3.2.2" xref="S1.F1.4.m1.2.3.3.2.1.cmml"><mo id="S1.F1.4.m1.2.3.3.2.2.1" stretchy="false" xref="S1.F1.4.m1.2.3.3.2.1.1.cmml">[</mo><mn id="S1.F1.4.m1.1.1" xref="S1.F1.4.m1.1.1.cmml">5</mn><mo id="S1.F1.4.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S1.F1.4.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S1.F1.4.m1.2.3.3.1" rspace="0.222em" xref="S1.F1.4.m1.2.3.3.1.cmml">×</mo><mrow id="S1.F1.4.m1.2.3.3.3.2" xref="S1.F1.4.m1.2.3.3.3.1.cmml"><mo id="S1.F1.4.m1.2.3.3.3.2.1" stretchy="false" xref="S1.F1.4.m1.2.3.3.3.1.1.cmml">[</mo><mn id="S1.F1.4.m1.2.2" xref="S1.F1.4.m1.2.2.cmml">5</mn><mo id="S1.F1.4.m1.2.3.3.3.2.2" stretchy="false" xref="S1.F1.4.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.4.m1.2c"><apply id="S1.F1.4.m1.2.3.cmml" xref="S1.F1.4.m1.2.3"><subset id="S1.F1.4.m1.2.3.1.cmml" xref="S1.F1.4.m1.2.3.1"></subset><ci id="S1.F1.4.m1.2.3.2.cmml" xref="S1.F1.4.m1.2.3.2">𝐺</ci><apply id="S1.F1.4.m1.2.3.3.cmml" xref="S1.F1.4.m1.2.3.3"><times id="S1.F1.4.m1.2.3.3.1.cmml" xref="S1.F1.4.m1.2.3.3.1"></times><apply id="S1.F1.4.m1.2.3.3.2.1.cmml" xref="S1.F1.4.m1.2.3.3.2.2"><csymbol cd="latexml" id="S1.F1.4.m1.2.3.3.2.1.1.cmml" xref="S1.F1.4.m1.2.3.3.2.2.1">delimited-[]</csymbol><cn id="S1.F1.4.m1.1.1.cmml" type="integer" xref="S1.F1.4.m1.1.1">5</cn></apply><apply id="S1.F1.4.m1.2.3.3.3.1.cmml" xref="S1.F1.4.m1.2.3.3.3.2"><csymbol cd="latexml" id="S1.F1.4.m1.2.3.3.3.1.1.cmml" xref="S1.F1.4.m1.2.3.3.3.2.1">delimited-[]</csymbol><cn id="S1.F1.4.m1.2.2.cmml" type="integer" xref="S1.F1.4.m1.2.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.4.m1.2d">G\subseteq[5]\times[5]</annotation><annotation encoding="application/x-llamapun" id="S1.F1.4.m1.2e">italic_G ⊆ [ 5 ] × [ 5 ]</annotation></semantics></math> of intersection complexity <math alttext="D_{\cap}(G\mid\mathcal{G}_{5,5})\leq 2" class="ltx_Math" display="inline" id="S1.F1.5.m2.3"><semantics id="S1.F1.5.m2.3b"><mrow id="S1.F1.5.m2.3.3" xref="S1.F1.5.m2.3.3.cmml"><mrow id="S1.F1.5.m2.3.3.1" xref="S1.F1.5.m2.3.3.1.cmml"><msub id="S1.F1.5.m2.3.3.1.3" xref="S1.F1.5.m2.3.3.1.3.cmml"><mi id="S1.F1.5.m2.3.3.1.3.2" xref="S1.F1.5.m2.3.3.1.3.2.cmml">D</mi><mo id="S1.F1.5.m2.3.3.1.3.3" xref="S1.F1.5.m2.3.3.1.3.3.cmml">∩</mo></msub><mo id="S1.F1.5.m2.3.3.1.2" xref="S1.F1.5.m2.3.3.1.2.cmml">⁢</mo><mrow id="S1.F1.5.m2.3.3.1.1.1" xref="S1.F1.5.m2.3.3.1.1.1.1.cmml"><mo id="S1.F1.5.m2.3.3.1.1.1.2" stretchy="false" xref="S1.F1.5.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="S1.F1.5.m2.3.3.1.1.1.1" xref="S1.F1.5.m2.3.3.1.1.1.1.cmml"><mi id="S1.F1.5.m2.3.3.1.1.1.1.2" xref="S1.F1.5.m2.3.3.1.1.1.1.2.cmml">G</mi><mo id="S1.F1.5.m2.3.3.1.1.1.1.1" xref="S1.F1.5.m2.3.3.1.1.1.1.1.cmml">∣</mo><msub id="S1.F1.5.m2.3.3.1.1.1.1.3" xref="S1.F1.5.m2.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.F1.5.m2.3.3.1.1.1.1.3.2" xref="S1.F1.5.m2.3.3.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S1.F1.5.m2.2.2.2.4" xref="S1.F1.5.m2.2.2.2.3.cmml"><mn id="S1.F1.5.m2.1.1.1.1" xref="S1.F1.5.m2.1.1.1.1.cmml">5</mn><mo id="S1.F1.5.m2.2.2.2.4.1" xref="S1.F1.5.m2.2.2.2.3.cmml">,</mo><mn id="S1.F1.5.m2.2.2.2.2" xref="S1.F1.5.m2.2.2.2.2.cmml">5</mn></mrow></msub></mrow><mo id="S1.F1.5.m2.3.3.1.1.1.3" stretchy="false" xref="S1.F1.5.m2.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.F1.5.m2.3.3.2" xref="S1.F1.5.m2.3.3.2.cmml">≤</mo><mn id="S1.F1.5.m2.3.3.3" xref="S1.F1.5.m2.3.3.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.5.m2.3c"><apply id="S1.F1.5.m2.3.3.cmml" xref="S1.F1.5.m2.3.3"><leq id="S1.F1.5.m2.3.3.2.cmml" xref="S1.F1.5.m2.3.3.2"></leq><apply id="S1.F1.5.m2.3.3.1.cmml" xref="S1.F1.5.m2.3.3.1"><times id="S1.F1.5.m2.3.3.1.2.cmml" xref="S1.F1.5.m2.3.3.1.2"></times><apply id="S1.F1.5.m2.3.3.1.3.cmml" xref="S1.F1.5.m2.3.3.1.3"><csymbol cd="ambiguous" id="S1.F1.5.m2.3.3.1.3.1.cmml" xref="S1.F1.5.m2.3.3.1.3">subscript</csymbol><ci id="S1.F1.5.m2.3.3.1.3.2.cmml" xref="S1.F1.5.m2.3.3.1.3.2">𝐷</ci><intersect id="S1.F1.5.m2.3.3.1.3.3.cmml" xref="S1.F1.5.m2.3.3.1.3.3"></intersect></apply><apply id="S1.F1.5.m2.3.3.1.1.1.1.cmml" xref="S1.F1.5.m2.3.3.1.1.1"><csymbol cd="latexml" id="S1.F1.5.m2.3.3.1.1.1.1.1.cmml" xref="S1.F1.5.m2.3.3.1.1.1.1.1">conditional</csymbol><ci id="S1.F1.5.m2.3.3.1.1.1.1.2.cmml" xref="S1.F1.5.m2.3.3.1.1.1.1.2">𝐺</ci><apply id="S1.F1.5.m2.3.3.1.1.1.1.3.cmml" xref="S1.F1.5.m2.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.F1.5.m2.3.3.1.1.1.1.3.1.cmml" xref="S1.F1.5.m2.3.3.1.1.1.1.3">subscript</csymbol><ci id="S1.F1.5.m2.3.3.1.1.1.1.3.2.cmml" xref="S1.F1.5.m2.3.3.1.1.1.1.3.2">𝒢</ci><list id="S1.F1.5.m2.2.2.2.3.cmml" xref="S1.F1.5.m2.2.2.2.4"><cn id="S1.F1.5.m2.1.1.1.1.cmml" type="integer" xref="S1.F1.5.m2.1.1.1.1">5</cn><cn id="S1.F1.5.m2.2.2.2.2.cmml" type="integer" xref="S1.F1.5.m2.2.2.2.2">5</cn></list></apply></apply></apply><cn id="S1.F1.5.m2.3.3.3.cmml" type="integer" xref="S1.F1.5.m2.3.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.5.m2.3d">D_{\cap}(G\mid\mathcal{G}_{5,5})\leq 2</annotation><annotation encoding="application/x-llamapun" id="S1.F1.5.m2.3e">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT 5 , 5 end_POSTSUBSCRIPT ) ≤ 2</annotation></semantics></math> via <math alttext="G=\big{(}(R_{2}\cup R_{4})\cap(C_{1}\cup C_{3}\cup C_{5})\big{)}\cup\big{(}(C_% {2}\cup C_{4})\cap(R_{1}\cup R_{3}\cup R_{5})\big{)}" class="ltx_Math" display="inline" id="S1.F1.6.m3.2"><semantics id="S1.F1.6.m3.2b"><mrow id="S1.F1.6.m3.2.2" xref="S1.F1.6.m3.2.2.cmml"><mi id="S1.F1.6.m3.2.2.4" xref="S1.F1.6.m3.2.2.4.cmml">G</mi><mo id="S1.F1.6.m3.2.2.3" xref="S1.F1.6.m3.2.2.3.cmml">=</mo><mrow id="S1.F1.6.m3.2.2.2" xref="S1.F1.6.m3.2.2.2.cmml"><mrow id="S1.F1.6.m3.1.1.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.cmml"><mo id="S1.F1.6.m3.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S1.F1.6.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.F1.6.m3.1.1.1.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.cmml"><mrow id="S1.F1.6.m3.1.1.1.1.1.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S1.F1.6.m3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.cmml"><msub id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2.2" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2.2.cmml">R</mi><mn id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2.3" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.1.cmml">∪</mo><msub id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3.2" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3.2.cmml">R</mi><mn id="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3.3" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.3.3.cmml">4</mn></msub></mrow><mo id="S1.F1.6.m3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.F1.6.m3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.F1.6.m3.1.1.1.1.1.1.3" xref="S1.F1.6.m3.1.1.1.1.1.1.3.cmml">∩</mo><mrow id="S1.F1.6.m3.1.1.1.1.1.1.2.1" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.cmml"><mo id="S1.F1.6.m3.1.1.1.1.1.1.2.1.2" stretchy="false" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.cmml"><msub id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2.cmml"><mi id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2.2" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2.2.cmml">C</mi><mn id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2.3" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.2.3.cmml">1</mn></msub><mo id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.1" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.1.cmml">∪</mo><msub id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3.cmml"><mi id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3.2" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3.2.cmml">C</mi><mn id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3.3" xref="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.3.3.cmml">3</mn></msub><mo id="S1.F1.6.m3.1.1.1.1.1.1.2.1.1.1b" 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xref="S1.F1.6.m3.2.2.2.2.1.1.2.1.1.4">subscript</csymbol><ci id="S1.F1.6.m3.2.2.2.2.1.1.2.1.1.4.2.cmml" xref="S1.F1.6.m3.2.2.2.2.1.1.2.1.1.4.2">𝑅</ci><cn id="S1.F1.6.m3.2.2.2.2.1.1.2.1.1.4.3.cmml" type="integer" xref="S1.F1.6.m3.2.2.2.2.1.1.2.1.1.4.3">5</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.6.m3.2d">G=\big{(}(R_{2}\cup R_{4})\cap(C_{1}\cup C_{3}\cup C_{5})\big{)}\cup\big{(}(C_% {2}\cup C_{4})\cap(R_{1}\cup R_{3}\cup R_{5})\big{)}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.6.m3.2e">italic_G = ( ( italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∪ italic_R start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∩ ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_C start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∪ italic_C start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ) ) ∪ ( ( italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∪ italic_C start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∩ ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_R start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∪ italic_R start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ) )</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p5"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p5.14">Given an arbitrary set <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p5.1.m1.1a"><mi id="S1.SS2.SSS0.Px1.p5.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p5.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.1.m1.1b"><ci id="S1.SS2.SSS0.Px1.p5.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.1.m1.1d">italic_A</annotation></semantics></math> and a family <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p5.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p5.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.2.m2.1b"><ci id="S1.SS2.SSS0.Px1.p5.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.2.m2.1d">caligraphic_B</annotation></semantics></math> as above, one can introduce a complexity measure <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.3.m3.2"><semantics id="S1.SS2.SSS0.Px1.p5.3.m3.2a"><mrow id="S1.SS2.SSS0.Px1.p5.3.m3.2.3" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.2" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.2.cmml">ρ</mi><mo id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.1" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.2" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.1.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p5.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p5.3.m3.1.1.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.2.2" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.3.m3.2.2" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.2.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.3.m3.2b"><apply id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3"><times id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.1"></times><ci id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.2">𝜌</ci><interval closure="open" id="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.3.3.2"><ci id="S1.SS2.SSS0.Px1.p5.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.1.1">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.3.m3.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.3.m3.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.3.m3.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.3.m3.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> that is closely related to <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p5.4.m4.1a"><mrow id="S1.SS2.SSS0.Px1.p5.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.3" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.2" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.4.m4.1b"><apply id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1"><times id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p5.4.m4.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.4.m4.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.4.m4.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math>. In more detail, we define an appropriate bipartite graph <math alttext="\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.5.m5.5"><semantics id="S1.SS2.SSS0.Px1.p5.5.m5.5a"><mrow id="S1.SS2.SSS0.Px1.p5.5.m5.5.5" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.cmml"><msub id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.cmml"><mi id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.2" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.2.cmml">Φ</mi><mrow id="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.4" xref="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.5.m5.1.1.1.1" xref="S1.SS2.SSS0.Px1.p5.5.m5.1.1.1.1.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.4.1" xref="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.2" xref="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.2.cmml">ℬ</mi></mrow></msub><mo id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.3" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.3.cmml">=</mo><mrow id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml">(</mo><msub id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.2" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.2.cmml">V</mi><mi id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.3" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><mo id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.4" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.2" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.2.cmml">V</mi><mi id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.3" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><mo id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.5" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.5.m5.3.3" xref="S1.SS2.SSS0.Px1.p5.5.m5.3.3.cmml">ℰ</mi><mo id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.6" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.5.m5.5b"><apply id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5"><eq id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.3"></eq><apply id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.1.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.2.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.4.2">Φ</ci><list id="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.4"><ci id="S1.SS2.SSS0.Px1.p5.5.m5.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.1.1.1.1">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.2.2.2.2">ℬ</ci></list></apply><vector id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2"><apply id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.2">𝑉</ci><ci id="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.4.4.1.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply><apply id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.2">𝑉</ci><ci id="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.5.5.2.2.2.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply><ci id="S1.SS2.SSS0.Px1.p5.5.m5.3.3.cmml" xref="S1.SS2.SSS0.Px1.p5.5.m5.3.3">ℰ</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.5.m5.5c">\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.5.m5.5d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT = ( italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT , caligraphic_E )</annotation></semantics></math>, called the <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p5.14.1">cover graph</em> of <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p5.6.m6.1a"><mi id="S1.SS2.SSS0.Px1.p5.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p5.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.6.m6.1b"><ci id="S1.SS2.SSS0.Px1.p5.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.6.m6.1d">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.7.m7.1"><semantics id="S1.SS2.SSS0.Px1.p5.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p5.7.m7.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.7.m7.1b"><ci id="S1.SS2.SSS0.Px1.p5.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.7.m7.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.7.m7.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.7.m7.1d">caligraphic_B</annotation></semantics></math>, and let <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.8.m8.2"><semantics id="S1.SS2.SSS0.Px1.p5.8.m8.2a"><mrow id="S1.SS2.SSS0.Px1.p5.8.m8.2.3" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.2" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.2.cmml">ρ</mi><mo id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.1" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.2" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.1.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p5.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p5.8.m8.1.1.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.2.2" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.8.m8.2.2" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.2.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.8.m8.2b"><apply id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3"><times id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.1"></times><ci id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.2">𝜌</ci><interval closure="open" id="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.3.3.2"><ci id="S1.SS2.SSS0.Px1.p5.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.1.1">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.8.m8.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.8.m8.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.8.m8.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.8.m8.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> denote the minimum number of vertices in <math alttext="V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.9.m9.1"><semantics id="S1.SS2.SSS0.Px1.p5.9.m9.1a"><msub id="S1.SS2.SSS0.Px1.p5.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.2" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1.2.cmml">V</mi><mi id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.3" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.9.m9.1b"><apply id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1.2">𝑉</ci><ci id="S1.SS2.SSS0.Px1.p5.9.m9.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p5.9.m9.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.9.m9.1c">V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.9.m9.1d">italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math> whose adjacent edges cover all the vertices in <math alttext="V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.10.m10.1"><semantics id="S1.SS2.SSS0.Px1.p5.10.m10.1a"><msub id="S1.SS2.SSS0.Px1.p5.10.m10.1.1" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.2" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1.2.cmml">V</mi><mi id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.3" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.10.m10.1b"><apply id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1.2">𝑉</ci><ci id="S1.SS2.SSS0.Px1.p5.10.m10.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p5.10.m10.1.1.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.10.m10.1c">V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.10.m10.1d">italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math>. (Since the definition of the graph <math alttext="\Phi_{A,\mathcal{B}}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.11.m11.2"><semantics id="S1.SS2.SSS0.Px1.p5.11.m11.2a"><msub id="S1.SS2.SSS0.Px1.p5.11.m11.2.3" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.11.m11.2.3.2" mathvariant="normal" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.3.2.cmml">Φ</mi><mrow id="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.4" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.11.m11.1.1.1.1" xref="S1.SS2.SSS0.Px1.p5.11.m11.1.1.1.1.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.4.1" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.2" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.2.cmml">ℬ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.11.m11.2b"><apply id="S1.SS2.SSS0.Px1.p5.11.m11.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p5.11.m11.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p5.11.m11.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.3.2">Φ</ci><list id="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.4"><ci id="S1.SS2.SSS0.Px1.p5.11.m11.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.1.1.1.1">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.11.m11.2.2.2.2">ℬ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.11.m11.2c">\Phi_{A,\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.11.m11.2d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> is somewhat technical and won’t be needed in the subsequent discussion, it is deferred to <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1" title="3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.1</span></a>). We say that <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.12.m12.2"><semantics id="S1.SS2.SSS0.Px1.p5.12.m12.2a"><mrow id="S1.SS2.SSS0.Px1.p5.12.m12.2.3" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.2" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.2.cmml">ρ</mi><mo id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.1" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.2" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.1.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p5.12.m12.1.1" xref="S1.SS2.SSS0.Px1.p5.12.m12.1.1.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.2.2" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.12.m12.2.2" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.2.cmml">ℬ</mi><mo id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.12.m12.2b"><apply id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3"><times id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.1"></times><ci id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.2">𝜌</ci><interval closure="open" id="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.3.3.2"><ci id="S1.SS2.SSS0.Px1.p5.12.m12.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.1.1">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p5.12.m12.2.2.cmml" xref="S1.SS2.SSS0.Px1.p5.12.m12.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.12.m12.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.12.m12.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p5.14.2">cover complexity</em> of <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.13.m13.1"><semantics id="S1.SS2.SSS0.Px1.p5.13.m13.1a"><mi id="S1.SS2.SSS0.Px1.p5.13.m13.1.1" xref="S1.SS2.SSS0.Px1.p5.13.m13.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.13.m13.1b"><ci id="S1.SS2.SSS0.Px1.p5.13.m13.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.13.m13.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.13.m13.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.13.m13.1d">italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p5.14.m14.1"><semantics id="S1.SS2.SSS0.Px1.p5.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p5.14.m14.1.1" xref="S1.SS2.SSS0.Px1.p5.14.m14.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p5.14.m14.1b"><ci id="S1.SS2.SSS0.Px1.p5.14.m14.1.1.cmml" xref="S1.SS2.SSS0.Px1.p5.14.m14.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p5.14.m14.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p5.14.m14.1d">caligraphic_B</annotation></semantics></math>. This measure of complexity generalises the cover problem introduced by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib19" title="">19</a>]</cite> to capture circuit complexity. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p6"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p6.10">Our first observation is that, by a straightforward adaptation of the fusion method for lower bounds <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib19" title="">19</a>]</cite> to our framework, the following relation holds:</p> <table class="ltx_equation ltx_eqn_table" id="S1.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(A,\mathcal{B})\;\leq\;D_{\cap}(A\mid\mathcal{B})\;\leq\;\rho(A,\mathcal{B% })^{2}." class="ltx_Math" display="block" id="S1.E1.m1.5"><semantics id="S1.E1.m1.5a"><mrow id="S1.E1.m1.5.5.1" xref="S1.E1.m1.5.5.1.1.cmml"><mrow id="S1.E1.m1.5.5.1.1" xref="S1.E1.m1.5.5.1.1.cmml"><mrow id="S1.E1.m1.5.5.1.1.3" xref="S1.E1.m1.5.5.1.1.3.cmml"><mi id="S1.E1.m1.5.5.1.1.3.2" xref="S1.E1.m1.5.5.1.1.3.2.cmml">ρ</mi><mo id="S1.E1.m1.5.5.1.1.3.1" xref="S1.E1.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S1.E1.m1.5.5.1.1.3.3.2" xref="S1.E1.m1.5.5.1.1.3.3.1.cmml"><mo id="S1.E1.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S1.E1.m1.5.5.1.1.3.3.1.cmml">(</mo><mi id="S1.E1.m1.1.1" xref="S1.E1.m1.1.1.cmml">A</mi><mo id="S1.E1.m1.5.5.1.1.3.3.2.2" xref="S1.E1.m1.5.5.1.1.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.E1.m1.2.2" xref="S1.E1.m1.2.2.cmml">ℬ</mi><mo id="S1.E1.m1.5.5.1.1.3.3.2.3" rspace="0.280em" stretchy="false" xref="S1.E1.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S1.E1.m1.5.5.1.1.4" rspace="0.558em" xref="S1.E1.m1.5.5.1.1.4.cmml">≤</mo><mrow id="S1.E1.m1.5.5.1.1.1" xref="S1.E1.m1.5.5.1.1.1.cmml"><msub id="S1.E1.m1.5.5.1.1.1.3" xref="S1.E1.m1.5.5.1.1.1.3.cmml"><mi id="S1.E1.m1.5.5.1.1.1.3.2" xref="S1.E1.m1.5.5.1.1.1.3.2.cmml">D</mi><mo id="S1.E1.m1.5.5.1.1.1.3.3" xref="S1.E1.m1.5.5.1.1.1.3.3.cmml">∩</mo></msub><mo id="S1.E1.m1.5.5.1.1.1.2" xref="S1.E1.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S1.E1.m1.5.5.1.1.1.1.1" xref="S1.E1.m1.5.5.1.1.1.1.1.1.cmml"><mo id="S1.E1.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S1.E1.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.E1.m1.5.5.1.1.1.1.1.1" xref="S1.E1.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S1.E1.m1.5.5.1.1.1.1.1.1.2" xref="S1.E1.m1.5.5.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.E1.m1.5.5.1.1.1.1.1.1.1" xref="S1.E1.m1.5.5.1.1.1.1.1.1.1.cmml">∣</mo><mi 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href="https://arxiv.org/html/2503.14117v1#S1.E1.m1.5.5.1.1.1.cmml" id="S1.E1.m1.5.5.1.1d.cmml" xref="S1.E1.m1.5.5.1"></share><apply id="S1.E1.m1.5.5.1.1.6.cmml" xref="S1.E1.m1.5.5.1.1.6"><times id="S1.E1.m1.5.5.1.1.6.1.cmml" xref="S1.E1.m1.5.5.1.1.6.1"></times><ci id="S1.E1.m1.5.5.1.1.6.2.cmml" xref="S1.E1.m1.5.5.1.1.6.2">𝜌</ci><apply id="S1.E1.m1.5.5.1.1.6.3.cmml" xref="S1.E1.m1.5.5.1.1.6.3"><csymbol cd="ambiguous" id="S1.E1.m1.5.5.1.1.6.3.1.cmml" xref="S1.E1.m1.5.5.1.1.6.3">superscript</csymbol><interval closure="open" id="S1.E1.m1.5.5.1.1.6.3.2.1.cmml" xref="S1.E1.m1.5.5.1.1.6.3.2.2"><ci id="S1.E1.m1.3.3.cmml" xref="S1.E1.m1.3.3">𝐴</ci><ci id="S1.E1.m1.4.4.cmml" xref="S1.E1.m1.4.4">ℬ</ci></interval><cn id="S1.E1.m1.5.5.1.1.6.3.3.cmml" type="integer" xref="S1.E1.m1.5.5.1.1.6.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.5c">\rho(A,\mathcal{B})\;\leq\;D_{\cap}(A\mid\mathcal{B})\;\leq\;\rho(A,\mathcal{B% })^{2}.</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.5d">italic_ρ ( italic_A , caligraphic_B ) ≤ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_ρ ( italic_A , caligraphic_B ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p6.9">In particular, cover complexity provides a lower bound on intersection complexity. We are particularly interested in applications of the inequalities above to graph complexity. There are two main reasons for this. Firstly, to each graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.1.m1.2"><semantics id="S1.SS2.SSS0.Px1.p6.1.m1.2a"><mrow id="S1.SS2.SSS0.Px1.p6.1.m1.2.3" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.2" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.2.cmml">G</mi><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.1" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.1.cmml">⊆</mo><mrow id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.cmml"><mrow id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.2" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p6.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p6.1.m1.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.1" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.1.cmml">×</mo><mrow id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.2" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p6.1.m1.2.2" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.2.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.1.m1.2b"><apply id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3"><subset id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.1"></subset><ci id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.2">𝐺</ci><apply id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3"><times id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.1"></times><apply id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p6.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.1.1">𝑁</ci></apply><apply id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p6.1.m1.2.2.cmml" xref="S1.SS2.SSS0.Px1.p6.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> one can associate a natural Boolean function <math 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xref="S1.SS2.SSS0.Px1.p6.2.m2.6.6">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.2.m2.6c">f_{G}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.2.m2.6d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS4" title="2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.4</span></a>), where <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p6.3.m3.1a"><mrow id="S1.SS2.SSS0.Px1.p6.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.2" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.2.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.1" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.1.cmml">=</mo><msup id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.cmml"><mn id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.2" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.2.cmml">2</mn><mi id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.3" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.3.m3.1b"><apply id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1"><eq id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.2">𝑁</ci><apply id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3">superscript</csymbol><cn id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.2">2</cn><ci id="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p6.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.3.m3.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.3.m3.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and it is known that lower bounds on the graph complexity of <math alttext="G" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p6.4.m4.1a"><mi id="S1.SS2.SSS0.Px1.p6.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p6.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.4.m4.1b"><ci id="S1.SS2.SSS0.Px1.p6.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.4.m4.1d">italic_G</annotation></semantics></math> yield lower bounds on the Boolean circuit complexity of <math alttext="f_{G}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p6.5.m5.1a"><msub id="S1.SS2.SSS0.Px1.p6.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.2" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1.2.cmml">f</mi><mi id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.3" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.5.m5.1b"><apply id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1.2">𝑓</ci><ci id="S1.SS2.SSS0.Px1.p6.5.m5.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p6.5.m5.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.5.m5.1c">f_{G}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib14" title="">14</a>]</cite>. (There can be a significant loss on the parameters of such transference results depending on the context. We refer to <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite> for more details. See also the discussion before <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem14" title="Remark 14 (Circuit lower bounds from graph complexity lower bounds). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Remark</span> <span class="ltx_text ltx_ref_tag">14</span></a> below.) Secondly, the cover problem defining <math alttext="\rho(G,\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.6.m6.4"><semantics id="S1.SS2.SSS0.Px1.p6.6.m6.4a"><mrow id="S1.SS2.SSS0.Px1.p6.6.m6.4.4" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.cmml"><mi id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.3" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.3.cmml">ρ</mi><mo id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.2" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.2.cmml"><mo id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.2.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p6.6.m6.3.3" xref="S1.SS2.SSS0.Px1.p6.6.m6.3.3.cmml">G</mi><mo id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.3" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.2.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.1" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.1.2" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.1.2.cmml">𝒢</mi><mrow id="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.4" xref="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p6.6.m6.1.1.1.1" xref="S1.SS2.SSS0.Px1.p6.6.m6.1.1.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.4.1" xref="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.2" xref="S1.SS2.SSS0.Px1.p6.6.m6.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1.4" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.6.m6.4b"><apply id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.cmml" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4"><times id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.2.cmml" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.2"></times><ci id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.3.cmml" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.3">𝜌</ci><interval closure="open" id="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.2.cmml" xref="S1.SS2.SSS0.Px1.p6.6.m6.4.4.1.1"><ci 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end_POSTSUBSCRIPT )</annotation></semantics></math> involves a two-dimensional ground set <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.7.m7.2"><semantics id="S1.SS2.SSS0.Px1.p6.7.m7.2a"><mrow id="S1.SS2.SSS0.Px1.p6.7.m7.2.3" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.cmml"><mrow id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.2" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.1.1.cmml">[</mo><mi id="S1.SS2.SSS0.Px1.p6.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p6.7.m7.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.2.1.1.cmml">]</mo></mrow><mo id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.1" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.1.cmml">×</mo><mrow id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.3.2" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p6.7.m7.2.3.3.2.1" stretchy="false" 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xref="S1.SS2.SSS0.Px1.p6.7.m7.2.3.3.2.1">delimited-[]</csymbol><ci id="S1.SS2.SSS0.Px1.p6.7.m7.2.2.cmml" xref="S1.SS2.SSS0.Px1.p6.7.m7.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.7.m7.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.7.m7.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math>, in contrast to the <math alttext="n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.8.m8.1"><semantics id="S1.SS2.SSS0.Px1.p6.8.m8.1a"><mi id="S1.SS2.SSS0.Px1.p6.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p6.8.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.8.m8.1b"><ci id="S1.SS2.SSS0.Px1.p6.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p6.8.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.8.m8.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.8.m8.1d">italic_n</annotation></semantics></math>-dimensional ground set <math alttext="\{0,1\}^{n}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p6.9.m9.2"><semantics id="S1.SS2.SSS0.Px1.p6.9.m9.2a"><msup id="S1.SS2.SSS0.Px1.p6.9.m9.2.3" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.cmml"><mrow id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.2" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p6.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p6.9.m9.1.1.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.2.2" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p6.9.m9.2.2" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.1.cmml">}</mo></mrow><mi id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.3" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p6.9.m9.2b"><apply id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.cmml" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3">superscript</csymbol><set id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.2.2"><cn id="S1.SS2.SSS0.Px1.p6.9.m9.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p6.9.m9.1.1">0</cn><cn id="S1.SS2.SSS0.Px1.p6.9.m9.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.2">1</cn></set><ci id="S1.SS2.SSS0.Px1.p6.9.m9.2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p6.9.m9.2.3.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p6.9.m9.2c">\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p6.9.m9.2d">{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> found in Boolean function complexity. We hope this perspective can inspire new techniques, and indeed we show how this perspective can be used to give a tight bound for a natural Boolean function in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p7"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p7.1">Our second observation is that a tight connection can be established between graph complexity and Boolean circuit complexity by focusing on intersection complexity and cover complexity.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem1.1.1.1">Lemma 1</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem1.2.2"> </span>(Transference of Lower Bounds)<span class="ltx_text ltx_font_bold" id="Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem1.p1"> <p class="ltx_p" id="Thmtheorem1.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem1.p1.2.2">For every non-trivial bipartite graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem1.p1.1.1.m1.2"><semantics id="Thmtheorem1.p1.1.1.m1.2a"><mrow id="Thmtheorem1.p1.1.1.m1.2.3" xref="Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem1.p1.1.1.m1.2.3.2" xref="Thmtheorem1.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="Thmtheorem1.p1.1.1.m1.2.3.1" xref="Thmtheorem1.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem1.p1.1.1.m1.2.3.3" xref="Thmtheorem1.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem1.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem1.p1.1.1.m1.2.3.3.2.2.1" 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id="Thmtheorem1.p1.1.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.1.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> and corresponding Boolean function <math alttext="f_{G}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.2.2.m2.6"><semantics id="Thmtheorem1.p1.2.2.m2.6a"><mrow id="Thmtheorem1.p1.2.2.m2.6.7" xref="Thmtheorem1.p1.2.2.m2.6.7.cmml"><msub id="Thmtheorem1.p1.2.2.m2.6.7.2" xref="Thmtheorem1.p1.2.2.m2.6.7.2.cmml"><mi id="Thmtheorem1.p1.2.2.m2.6.7.2.2" xref="Thmtheorem1.p1.2.2.m2.6.7.2.2.cmml">f</mi><mi id="Thmtheorem1.p1.2.2.m2.6.7.2.3" xref="Thmtheorem1.p1.2.2.m2.6.7.2.3.cmml">G</mi></msub><mo id="Thmtheorem1.p1.2.2.m2.6.7.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem1.p1.2.2.m2.6.7.1.cmml">:</mo><mrow id="Thmtheorem1.p1.2.2.m2.6.7.3" xref="Thmtheorem1.p1.2.2.m2.6.7.3.cmml"><mrow id="Thmtheorem1.p1.2.2.m2.6.7.3.2" 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xref="Thmtheorem1.p1.2.2.m2.6.7.3.2.2.3">𝑛</ci></apply><apply id="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.cmml" xref="Thmtheorem1.p1.2.2.m2.6.7.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.1.cmml" xref="Thmtheorem1.p1.2.2.m2.6.7.3.2.3">superscript</csymbol><set id="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.2.1.cmml" xref="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.2.2"><cn id="Thmtheorem1.p1.2.2.m2.3.3.cmml" type="integer" xref="Thmtheorem1.p1.2.2.m2.3.3">0</cn><cn id="Thmtheorem1.p1.2.2.m2.4.4.cmml" type="integer" xref="Thmtheorem1.p1.2.2.m2.4.4">1</cn></set><ci id="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.3.cmml" xref="Thmtheorem1.p1.2.2.m2.6.7.3.2.3.3">𝑛</ci></apply></apply><set id="Thmtheorem1.p1.2.2.m2.6.7.3.3.1.cmml" xref="Thmtheorem1.p1.2.2.m2.6.7.3.3.2"><cn id="Thmtheorem1.p1.2.2.m2.5.5.cmml" type="integer" xref="Thmtheorem1.p1.2.2.m2.5.5">0</cn><cn id="Thmtheorem1.p1.2.2.m2.6.6.cmml" type="integer" xref="Thmtheorem1.p1.2.2.m2.6.6">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.2.2.m2.6c">f_{G}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.2.2.m2.6d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>, we have</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx1"> <tbody id="S1.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\rho(f_{G}^{-1}(1),\mathcal{B}_{2n})" class="ltx_Math" display="inline" id="S1.E2.m1.3"><semantics id="S1.E2.m1.3a"><mrow id="S1.E2.m1.3.3" xref="S1.E2.m1.3.3.cmml"><mi id="S1.E2.m1.3.3.4" xref="S1.E2.m1.3.3.4.cmml">ρ</mi><mo id="S1.E2.m1.3.3.3" xref="S1.E2.m1.3.3.3.cmml">⁢</mo><mrow id="S1.E2.m1.3.3.2.2" xref="S1.E2.m1.3.3.2.3.cmml"><mo id="S1.E2.m1.3.3.2.2.3" stretchy="false" xref="S1.E2.m1.3.3.2.3.cmml">(</mo><mrow id="S1.E2.m1.2.2.1.1.1" xref="S1.E2.m1.2.2.1.1.1.cmml"><msubsup id="S1.E2.m1.2.2.1.1.1.2" xref="S1.E2.m1.2.2.1.1.1.2.cmml"><mi id="S1.E2.m1.2.2.1.1.1.2.2.2" xref="S1.E2.m1.2.2.1.1.1.2.2.2.cmml">f</mi><mi id="S1.E2.m1.2.2.1.1.1.2.2.3" xref="S1.E2.m1.2.2.1.1.1.2.2.3.cmml">G</mi><mrow id="S1.E2.m1.2.2.1.1.1.2.3" xref="S1.E2.m1.2.2.1.1.1.2.3.cmml"><mo id="S1.E2.m1.2.2.1.1.1.2.3a" xref="S1.E2.m1.2.2.1.1.1.2.3.cmml">−</mo><mn id="S1.E2.m1.2.2.1.1.1.2.3.2" xref="S1.E2.m1.2.2.1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S1.E2.m1.2.2.1.1.1.1" xref="S1.E2.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.E2.m1.2.2.1.1.1.3.2" xref="S1.E2.m1.2.2.1.1.1.cmml"><mo id="S1.E2.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="S1.E2.m1.2.2.1.1.1.cmml">(</mo><mn id="S1.E2.m1.1.1" xref="S1.E2.m1.1.1.cmml">1</mn><mo id="S1.E2.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="S1.E2.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.E2.m1.3.3.2.2.4" xref="S1.E2.m1.3.3.2.3.cmml">,</mo><msub id="S1.E2.m1.3.3.2.2.2" xref="S1.E2.m1.3.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E2.m1.3.3.2.2.2.2" xref="S1.E2.m1.3.3.2.2.2.2.cmml">ℬ</mi><mrow id="S1.E2.m1.3.3.2.2.2.3" xref="S1.E2.m1.3.3.2.2.2.3.cmml"><mn id="S1.E2.m1.3.3.2.2.2.3.2" xref="S1.E2.m1.3.3.2.2.2.3.2.cmml">2</mn><mo id="S1.E2.m1.3.3.2.2.2.3.1" xref="S1.E2.m1.3.3.2.2.2.3.1.cmml">⁢</mo><mi id="S1.E2.m1.3.3.2.2.2.3.3" xref="S1.E2.m1.3.3.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S1.E2.m1.3.3.2.2.5" stretchy="false" xref="S1.E2.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.E2.m1.3b"><apply id="S1.E2.m1.3.3.cmml" xref="S1.E2.m1.3.3"><times id="S1.E2.m1.3.3.3.cmml" xref="S1.E2.m1.3.3.3"></times><ci id="S1.E2.m1.3.3.4.cmml" xref="S1.E2.m1.3.3.4">𝜌</ci><interval closure="open" id="S1.E2.m1.3.3.2.3.cmml" xref="S1.E2.m1.3.3.2.2"><apply id="S1.E2.m1.2.2.1.1.1.cmml" xref="S1.E2.m1.2.2.1.1.1"><times id="S1.E2.m1.2.2.1.1.1.1.cmml" xref="S1.E2.m1.2.2.1.1.1.1"></times><apply id="S1.E2.m1.2.2.1.1.1.2.cmml" xref="S1.E2.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S1.E2.m1.2.2.1.1.1.2.1.cmml" xref="S1.E2.m1.2.2.1.1.1.2">superscript</csymbol><apply id="S1.E2.m1.2.2.1.1.1.2.2.cmml" xref="S1.E2.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S1.E2.m1.2.2.1.1.1.2.2.1.cmml" xref="S1.E2.m1.2.2.1.1.1.2">subscript</csymbol><ci id="S1.E2.m1.2.2.1.1.1.2.2.2.cmml" xref="S1.E2.m1.2.2.1.1.1.2.2.2">𝑓</ci><ci id="S1.E2.m1.2.2.1.1.1.2.2.3.cmml" xref="S1.E2.m1.2.2.1.1.1.2.2.3">𝐺</ci></apply><apply id="S1.E2.m1.2.2.1.1.1.2.3.cmml" xref="S1.E2.m1.2.2.1.1.1.2.3"><minus id="S1.E2.m1.2.2.1.1.1.2.3.1.cmml" xref="S1.E2.m1.2.2.1.1.1.2.3"></minus><cn id="S1.E2.m1.2.2.1.1.1.2.3.2.cmml" type="integer" xref="S1.E2.m1.2.2.1.1.1.2.3.2">1</cn></apply></apply><cn id="S1.E2.m1.1.1.cmml" type="integer" xref="S1.E2.m1.1.1">1</cn></apply><apply id="S1.E2.m1.3.3.2.2.2.cmml" xref="S1.E2.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.E2.m1.3.3.2.2.2.1.cmml" xref="S1.E2.m1.3.3.2.2.2">subscript</csymbol><ci id="S1.E2.m1.3.3.2.2.2.2.cmml" xref="S1.E2.m1.3.3.2.2.2.2">ℬ</ci><apply id="S1.E2.m1.3.3.2.2.2.3.cmml" xref="S1.E2.m1.3.3.2.2.2.3"><times id="S1.E2.m1.3.3.2.2.2.3.1.cmml" xref="S1.E2.m1.3.3.2.2.2.3.1"></times><cn id="S1.E2.m1.3.3.2.2.2.3.2.cmml" type="integer" xref="S1.E2.m1.3.3.2.2.2.3.2">2</cn><ci id="S1.E2.m1.3.3.2.2.2.3.3.cmml" xref="S1.E2.m1.3.3.2.2.2.3.3">𝑛</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E2.m1.3c">\displaystyle\rho(f_{G}^{-1}(1),\mathcal{B}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S1.E2.m1.3d">italic_ρ ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S1.E2.m2.1"><semantics id="S1.E2.m2.1a"><mo id="S1.E2.m2.1.1" xref="S1.E2.m2.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S1.E2.m2.1b"><geq id="S1.E2.m2.1.1.cmml" xref="S1.E2.m2.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S1.E2.m2.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S1.E2.m2.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rho(G,\mathcal{G}_{N,N}),~{}\text{and}" class="ltx_Math" display="inline" id="S1.E2.m3.5"><semantics id="S1.E2.m3.5a"><mrow id="S1.E2.m3.5.5.1" xref="S1.E2.m3.5.5.2.cmml"><mrow id="S1.E2.m3.5.5.1.1" xref="S1.E2.m3.5.5.1.1.cmml"><mi id="S1.E2.m3.5.5.1.1.3" xref="S1.E2.m3.5.5.1.1.3.cmml">ρ</mi><mo id="S1.E2.m3.5.5.1.1.2" xref="S1.E2.m3.5.5.1.1.2.cmml">⁢</mo><mrow id="S1.E2.m3.5.5.1.1.1.1" xref="S1.E2.m3.5.5.1.1.1.2.cmml"><mo id="S1.E2.m3.5.5.1.1.1.1.2" stretchy="false" xref="S1.E2.m3.5.5.1.1.1.2.cmml">(</mo><mi id="S1.E2.m3.3.3" xref="S1.E2.m3.3.3.cmml">G</mi><mo id="S1.E2.m3.5.5.1.1.1.1.3" xref="S1.E2.m3.5.5.1.1.1.2.cmml">,</mo><msub id="S1.E2.m3.5.5.1.1.1.1.1" xref="S1.E2.m3.5.5.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E2.m3.5.5.1.1.1.1.1.2" xref="S1.E2.m3.5.5.1.1.1.1.1.2.cmml">𝒢</mi><mrow id="S1.E2.m3.2.2.2.4" xref="S1.E2.m3.2.2.2.3.cmml"><mi id="S1.E2.m3.1.1.1.1" xref="S1.E2.m3.1.1.1.1.cmml">N</mi><mo id="S1.E2.m3.2.2.2.4.1" xref="S1.E2.m3.2.2.2.3.cmml">,</mo><mi id="S1.E2.m3.2.2.2.2" xref="S1.E2.m3.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S1.E2.m3.5.5.1.1.1.1.4" stretchy="false" xref="S1.E2.m3.5.5.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.E2.m3.5.5.1.2" rspace="0.497em" xref="S1.E2.m3.5.5.2.cmml">,</mo><mtext class="ltx_mathvariant_italic" id="S1.E2.m3.4.4" xref="S1.E2.m3.4.4a.cmml">and</mtext></mrow><annotation-xml encoding="MathML-Content" id="S1.E2.m3.5b"><list id="S1.E2.m3.5.5.2.cmml" xref="S1.E2.m3.5.5.1"><apply id="S1.E2.m3.5.5.1.1.cmml" xref="S1.E2.m3.5.5.1.1"><times id="S1.E2.m3.5.5.1.1.2.cmml" xref="S1.E2.m3.5.5.1.1.2"></times><ci id="S1.E2.m3.5.5.1.1.3.cmml" xref="S1.E2.m3.5.5.1.1.3">𝜌</ci><interval closure="open" id="S1.E2.m3.5.5.1.1.1.2.cmml" xref="S1.E2.m3.5.5.1.1.1.1"><ci id="S1.E2.m3.3.3.cmml" xref="S1.E2.m3.3.3">𝐺</ci><apply id="S1.E2.m3.5.5.1.1.1.1.1.cmml" xref="S1.E2.m3.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.E2.m3.5.5.1.1.1.1.1.1.cmml" xref="S1.E2.m3.5.5.1.1.1.1.1">subscript</csymbol><ci id="S1.E2.m3.5.5.1.1.1.1.1.2.cmml" xref="S1.E2.m3.5.5.1.1.1.1.1.2">𝒢</ci><list id="S1.E2.m3.2.2.2.3.cmml" xref="S1.E2.m3.2.2.2.4"><ci id="S1.E2.m3.1.1.1.1.cmml" xref="S1.E2.m3.1.1.1.1">𝑁</ci><ci id="S1.E2.m3.2.2.2.2.cmml" xref="S1.E2.m3.2.2.2.2">𝑁</ci></list></apply></interval></apply><ci id="S1.E2.m3.4.4a.cmml" xref="S1.E2.m3.4.4"><mtext class="ltx_mathvariant_italic" id="S1.E2.m3.4.4.cmml" xref="S1.E2.m3.4.4">and</mtext></ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.E2.m3.5c">\displaystyle\rho(G,\mathcal{G}_{N,N}),~{}\text{and}</annotation><annotation encoding="application/x-llamapun" id="S1.E2.m3.5d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) , and</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> <tbody id="S1.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D(f_{G}^{-1}(1)\mid\mathcal{B}_{2n})" class="ltx_Math" display="inline" id="S1.E3.m1.2"><semantics id="S1.E3.m1.2a"><mrow id="S1.E3.m1.2.2" xref="S1.E3.m1.2.2.cmml"><mi id="S1.E3.m1.2.2.3" xref="S1.E3.m1.2.2.3.cmml">D</mi><mo id="S1.E3.m1.2.2.2" xref="S1.E3.m1.2.2.2.cmml">⁢</mo><mrow id="S1.E3.m1.2.2.1.1" xref="S1.E3.m1.2.2.1.1.1.cmml"><mo id="S1.E3.m1.2.2.1.1.2" stretchy="false" xref="S1.E3.m1.2.2.1.1.1.cmml">(</mo><mrow id="S1.E3.m1.2.2.1.1.1" xref="S1.E3.m1.2.2.1.1.1.cmml"><mrow id="S1.E3.m1.2.2.1.1.1.2" xref="S1.E3.m1.2.2.1.1.1.2.cmml"><msubsup id="S1.E3.m1.2.2.1.1.1.2.2" xref="S1.E3.m1.2.2.1.1.1.2.2.cmml"><mi id="S1.E3.m1.2.2.1.1.1.2.2.2.2" xref="S1.E3.m1.2.2.1.1.1.2.2.2.2.cmml">f</mi><mi id="S1.E3.m1.2.2.1.1.1.2.2.2.3" xref="S1.E3.m1.2.2.1.1.1.2.2.2.3.cmml">G</mi><mrow id="S1.E3.m1.2.2.1.1.1.2.2.3" xref="S1.E3.m1.2.2.1.1.1.2.2.3.cmml"><mo id="S1.E3.m1.2.2.1.1.1.2.2.3a" xref="S1.E3.m1.2.2.1.1.1.2.2.3.cmml">−</mo><mn id="S1.E3.m1.2.2.1.1.1.2.2.3.2" xref="S1.E3.m1.2.2.1.1.1.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S1.E3.m1.2.2.1.1.1.2.1" xref="S1.E3.m1.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S1.E3.m1.2.2.1.1.1.2.3.2" 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id="S1.E3.m2.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle D_{\cap}(G\mid\mathcal{G}_{N,N})." class="ltx_Math" display="inline" id="S1.E3.m3.3"><semantics id="S1.E3.m3.3a"><mrow id="S1.E3.m3.3.3.1" xref="S1.E3.m3.3.3.1.1.cmml"><mrow id="S1.E3.m3.3.3.1.1" xref="S1.E3.m3.3.3.1.1.cmml"><msub id="S1.E3.m3.3.3.1.1.3" xref="S1.E3.m3.3.3.1.1.3.cmml"><mi id="S1.E3.m3.3.3.1.1.3.2" xref="S1.E3.m3.3.3.1.1.3.2.cmml">D</mi><mo id="S1.E3.m3.3.3.1.1.3.3" xref="S1.E3.m3.3.3.1.1.3.3.cmml">∩</mo></msub><mo id="S1.E3.m3.3.3.1.1.2" xref="S1.E3.m3.3.3.1.1.2.cmml">⁢</mo><mrow id="S1.E3.m3.3.3.1.1.1.1" xref="S1.E3.m3.3.3.1.1.1.1.1.cmml"><mo id="S1.E3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S1.E3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S1.E3.m3.3.3.1.1.1.1.1" xref="S1.E3.m3.3.3.1.1.1.1.1.cmml"><mi id="S1.E3.m3.3.3.1.1.1.1.1.2" xref="S1.E3.m3.3.3.1.1.1.1.1.2.cmml">G</mi><mo id="S1.E3.m3.3.3.1.1.1.1.1.1" xref="S1.E3.m3.3.3.1.1.1.1.1.1.cmml">∣</mo><msub id="S1.E3.m3.3.3.1.1.1.1.1.3" xref="S1.E3.m3.3.3.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E3.m3.3.3.1.1.1.1.1.3.2" xref="S1.E3.m3.3.3.1.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S1.E3.m3.2.2.2.4" xref="S1.E3.m3.2.2.2.3.cmml"><mi id="S1.E3.m3.1.1.1.1" xref="S1.E3.m3.1.1.1.1.cmml">N</mi><mo id="S1.E3.m3.2.2.2.4.1" xref="S1.E3.m3.2.2.2.3.cmml">,</mo><mi id="S1.E3.m3.2.2.2.2" xref="S1.E3.m3.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S1.E3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S1.E3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.E3.m3.3.3.1.2" lspace="0em" xref="S1.E3.m3.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E3.m3.3b"><apply id="S1.E3.m3.3.3.1.1.cmml" xref="S1.E3.m3.3.3.1"><times id="S1.E3.m3.3.3.1.1.2.cmml" xref="S1.E3.m3.3.3.1.1.2"></times><apply id="S1.E3.m3.3.3.1.1.3.cmml" xref="S1.E3.m3.3.3.1.1.3"><csymbol cd="ambiguous" id="S1.E3.m3.3.3.1.1.3.1.cmml" xref="S1.E3.m3.3.3.1.1.3">subscript</csymbol><ci id="S1.E3.m3.3.3.1.1.3.2.cmml" xref="S1.E3.m3.3.3.1.1.3.2">𝐷</ci><intersect id="S1.E3.m3.3.3.1.1.3.3.cmml" xref="S1.E3.m3.3.3.1.1.3.3"></intersect></apply><apply id="S1.E3.m3.3.3.1.1.1.1.1.cmml" xref="S1.E3.m3.3.3.1.1.1.1"><csymbol cd="latexml" id="S1.E3.m3.3.3.1.1.1.1.1.1.cmml" xref="S1.E3.m3.3.3.1.1.1.1.1.1">conditional</csymbol><ci id="S1.E3.m3.3.3.1.1.1.1.1.2.cmml" xref="S1.E3.m3.3.3.1.1.1.1.1.2">𝐺</ci><apply id="S1.E3.m3.3.3.1.1.1.1.1.3.cmml" xref="S1.E3.m3.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.E3.m3.3.3.1.1.1.1.1.3.1.cmml" xref="S1.E3.m3.3.3.1.1.1.1.1.3">subscript</csymbol><ci id="S1.E3.m3.3.3.1.1.1.1.1.3.2.cmml" xref="S1.E3.m3.3.3.1.1.1.1.1.3.2">𝒢</ci><list id="S1.E3.m3.2.2.2.3.cmml" xref="S1.E3.m3.2.2.2.4"><ci id="S1.E3.m3.1.1.1.1.cmml" xref="S1.E3.m3.1.1.1.1">𝑁</ci><ci id="S1.E3.m3.2.2.2.2.cmml" xref="S1.E3.m3.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E3.m3.3c">\displaystyle D_{\cap}(G\mid\mathcal{G}_{N,N}).</annotation><annotation encoding="application/x-llamapun" id="S1.E3.m3.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p8"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p8.7">The second inequality is implicit in the literature on graph complexity. We include it in the statement of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem1" title="Lemma 1 (Transference of Lower Bounds). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">1</span></a> for completeness. Using <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem1" title="Lemma 1 (Transference of Lower Bounds). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">1</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.E1" title="In Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">1</span></a>, and another idea, we note in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS4" title="2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.4</span></a> that a lower bound of the form <math alttext="C\cdot\log N" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p8.1.m1.1a"><mrow id="S1.SS2.SSS0.Px1.p8.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.2" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.2.cmml">C</mi><mo id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.1.cmml">⋅</mo><mrow id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.1" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.1.cmml">log</mi><mo id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3a" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.cmml">⁡</mo><mi id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.2" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.2.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.1.m1.1b"><apply id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1"><ci id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.1">⋅</ci><ci id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.2">𝐶</ci><apply id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3"><log id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.1"></log><ci id="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p8.1.m1.1.1.3.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.1.m1.1c">C\cdot\log N</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.1.m1.1d">italic_C ⋅ roman_log italic_N</annotation></semantics></math> on <math alttext="\rho(G,\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.2.m2.4"><semantics id="S1.SS2.SSS0.Px1.p8.2.m2.4a"><mrow id="S1.SS2.SSS0.Px1.p8.2.m2.4.4" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.cmml"><mi id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.3" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.3.cmml">ρ</mi><mo id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.2" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.2.cmml"><mo id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.2.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p8.2.m2.3.3" xref="S1.SS2.SSS0.Px1.p8.2.m2.3.3.cmml">G</mi><mo id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.3" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.2.cmml">,</mo><msub id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.2" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.2.cmml">𝒢</mi><mrow id="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.4" xref="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p8.2.m2.1.1.1.1" xref="S1.SS2.SSS0.Px1.p8.2.m2.1.1.1.1.cmml">N</mi><mo id="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.4.1" xref="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.3.cmml">,</mo><mi id="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.2" xref="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.4" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.2.m2.4b"><apply id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4"><times id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.2.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.2"></times><ci id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.3.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.3">𝜌</ci><interval closure="open" id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1"><ci id="S1.SS2.SSS0.Px1.p8.2.m2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.3.3">𝐺</ci><apply id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.4.4.1.1.1.2">𝒢</ci><list id="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.4"><ci id="S1.SS2.SSS0.Px1.p8.2.m2.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.1.1.1.1">𝑁</ci><ci id="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p8.2.m2.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.2.m2.4c">\rho(G,\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.2.m2.4d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> yields a lower bound of the form <math alttext="C\cdot m-O(1)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p8.3.m3.1a"><mrow id="S1.SS2.SSS0.Px1.p8.3.m3.1.2" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.cmml"><mrow id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.cmml"><mi id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.2" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.2.cmml">C</mi><mo id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.1.cmml">⋅</mo><mi id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.3" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.3.cmml">m</mi></mrow><mo id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.1" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.1.cmml">−</mo><mrow id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.2" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.2.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.1" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.3.2" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.cmml">(</mo><mn id="S1.SS2.SSS0.Px1.p8.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.1.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.3.m3.1b"><apply id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2"><minus id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.1"></minus><apply id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2"><ci id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.1">⋅</ci><ci id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.2.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.2">𝐶</ci><ci id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.3.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.2.3">𝑚</ci></apply><apply id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3"><times id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.1"></times><ci id="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.2.3.2">𝑂</ci><cn id="S1.SS2.SSS0.Px1.p8.3.m3.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.3.m3.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.3.m3.1c">C\cdot m-O(1)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.3.m3.1d">italic_C ⋅ italic_m - italic_O ( 1 )</annotation></semantics></math> on the AND complexity of a related function <math alttext="F\colon\{0,1\}^{m}\to\{0,1\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.4.m4.4"><semantics id="S1.SS2.SSS0.Px1.p8.4.m4.4a"><mrow id="S1.SS2.SSS0.Px1.p8.4.m4.4.5" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.cmml"><mi id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.2.cmml">F</mi><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.1" lspace="0.278em" rspace="0.278em" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.1.cmml">:</mo><mrow id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.cmml"><msup id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.cmml"><mrow id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.1.cmml"><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p8.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p8.4.m4.1.1.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.2.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p8.4.m4.2.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.3" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.3.cmml">m</mi></msup><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.1.cmml">→</mo><mrow id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p8.4.m4.3.3" xref="S1.SS2.SSS0.Px1.p8.4.m4.3.3.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.2.2" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p8.4.m4.4.4" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.4.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.4.m4.4b"><apply id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5"><ci id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.1.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.1">:</ci><ci id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.2.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.2">𝐹</ci><apply id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3"><ci id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.1.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.1">→</ci><apply id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.1.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2">superscript</csymbol><set id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.1.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.2.2"><cn id="S1.SS2.SSS0.Px1.p8.4.m4.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.4.m4.1.1">0</cn><cn id="S1.SS2.SSS0.Px1.p8.4.m4.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.4.m4.2.2">1</cn></set><ci id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.3.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.2.3">𝑚</ci></apply><set id="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.1.cmml" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.5.3.3.2"><cn id="S1.SS2.SSS0.Px1.p8.4.m4.3.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.4.m4.3.3">0</cn><cn id="S1.SS2.SSS0.Px1.p8.4.m4.4.4.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.4.m4.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.4.m4.4c">F\colon\{0,1\}^{m}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.4.m4.4d">italic_F : { 0 , 1 } start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>. It is worth noting that lower bounds of the form <math alttext="Cn" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p8.5.m5.1a"><mrow id="S1.SS2.SSS0.Px1.p8.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.2" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.2.cmml">C</mi><mo id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.1" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.3" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.5.m5.1b"><apply id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1"><times id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.1"></times><ci id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.2">𝐶</ci><ci id="S1.SS2.SSS0.Px1.p8.5.m5.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p8.5.m5.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.5.m5.1c">Cn</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.5.m5.1d">italic_C italic_n</annotation></semantics></math> for <math alttext="C>1" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p8.6.m6.1a"><mrow id="S1.SS2.SSS0.Px1.p8.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.2" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.2.cmml">C</mi><mo id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.1" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.1.cmml">></mo><mn id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.3" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.6.m6.1b"><apply id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1"><gt id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.1"></gt><ci id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.2">𝐶</ci><cn id="S1.SS2.SSS0.Px1.p8.6.m6.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p8.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.6.m6.1c">C>1</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.6.m6.1d">italic_C > 1</annotation></semantics></math> on the AND complexity of explicit Boolean functions can be obtained using gate-elimination techniques <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib6" title="">6</a>]</cite>, so the problem considered here does not suffer from a “barrier” at <math alttext="n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p8.7.m7.1"><semantics id="S1.SS2.SSS0.Px1.p8.7.m7.1a"><mi id="S1.SS2.SSS0.Px1.p8.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p8.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p8.7.m7.1b"><ci id="S1.SS2.SSS0.Px1.p8.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p8.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p8.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p8.7.m7.1d">italic_n</annotation></semantics></math> gates as in the setting of multiplicative complexity <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib18" title="">18</a>]</cite>. We leave open the problem of matching (or more ambitiously strengthening) existing Boolean circuit lower bounds obtained via gate elimination using our framework.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p9"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p9.1">Complementing the approach to non-trivial circuit lower bounds discussed above, we show the following result for non-explicit graphs.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem2.1.1.1">Theorem 2</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem2.2.2"> </span>(Cover complexity of a random graph)<span class="ltx_text ltx_font_bold" id="Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem2.p1"> <p class="ltx_p" id="Thmtheorem2.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem2.p1.2.2">Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.1.1.m1.1"><semantics id="Thmtheorem2.p1.1.1.m1.1a"><mrow id="Thmtheorem2.p1.1.1.m1.1.1" xref="Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem2.p1.1.1.m1.1.1.2" xref="Thmtheorem2.p1.1.1.m1.1.1.2.cmml">N</mi><mo id="Thmtheorem2.p1.1.1.m1.1.1.1" xref="Thmtheorem2.p1.1.1.m1.1.1.1.cmml">=</mo><msup id="Thmtheorem2.p1.1.1.m1.1.1.3" xref="Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mn id="Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">2</mn><mi id="Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.1.1.m1.1b"><apply id="Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1"><eq id="Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1.1"></eq><ci id="Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1.2">𝑁</ci><apply id="Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1.3">superscript</csymbol><cn id="Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" type="integer" xref="Thmtheorem2.p1.1.1.m1.1.1.3.2">2</cn><ci id="Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem2.p1.1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.1.1.m1.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.1.1.m1.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and let <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem2.p1.2.2.m2.2"><semantics id="Thmtheorem2.p1.2.2.m2.2a"><mrow id="Thmtheorem2.p1.2.2.m2.2.3" xref="Thmtheorem2.p1.2.2.m2.2.3.cmml"><mi id="Thmtheorem2.p1.2.2.m2.2.3.2" xref="Thmtheorem2.p1.2.2.m2.2.3.2.cmml">G</mi><mo id="Thmtheorem2.p1.2.2.m2.2.3.1" xref="Thmtheorem2.p1.2.2.m2.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem2.p1.2.2.m2.2.3.3" xref="Thmtheorem2.p1.2.2.m2.2.3.3.cmml"><mrow id="Thmtheorem2.p1.2.2.m2.2.3.3.2.2" xref="Thmtheorem2.p1.2.2.m2.2.3.3.2.1.cmml"><mo id="Thmtheorem2.p1.2.2.m2.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem2.p1.2.2.m2.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem2.p1.2.2.m2.1.1" xref="Thmtheorem2.p1.2.2.m2.1.1.cmml">N</mi><mo id="Thmtheorem2.p1.2.2.m2.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem2.p1.2.2.m2.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem2.p1.2.2.m2.2.3.3.1" rspace="0.222em" xref="Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem2.p1.2.2.m2.2.3.3.3.2" xref="Thmtheorem2.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="Thmtheorem2.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem2.p1.2.2.m2.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem2.p1.2.2.m2.2.2" xref="Thmtheorem2.p1.2.2.m2.2.2.cmml">N</mi><mo id="Thmtheorem2.p1.2.2.m2.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem2.p1.2.2.m2.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.2.2.m2.2b"><apply id="Thmtheorem2.p1.2.2.m2.2.3.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3"><subset id="Thmtheorem2.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.1"></subset><ci id="Thmtheorem2.p1.2.2.m2.2.3.2.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.2">𝐺</ci><apply id="Thmtheorem2.p1.2.2.m2.2.3.3.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3"><times id="Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3.1"></times><apply id="Thmtheorem2.p1.2.2.m2.2.3.3.2.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem2.p1.2.2.m2.2.3.3.2.1.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="Thmtheorem2.p1.2.2.m2.1.1">𝑁</ci></apply><apply id="Thmtheorem2.p1.2.2.m2.2.3.3.3.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem2.p1.2.2.m2.2.3.3.3.1.1.cmml" xref="Thmtheorem2.p1.2.2.m2.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem2.p1.2.2.m2.2.2.cmml" xref="Thmtheorem2.p1.2.2.m2.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.2.2.m2.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.2.2.m2.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> be a uniformly random bipartite graph. Then, asymptotically almost surely,</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(G,\mathcal{G}_{N,N})\;=\;\Theta(N)." class="ltx_Math" display="block" id="S1.Ex1.m1.5"><semantics id="S1.Ex1.m1.5a"><mrow id="S1.Ex1.m1.5.5.1" xref="S1.Ex1.m1.5.5.1.1.cmml"><mrow id="S1.Ex1.m1.5.5.1.1" xref="S1.Ex1.m1.5.5.1.1.cmml"><mrow id="S1.Ex1.m1.5.5.1.1.1" xref="S1.Ex1.m1.5.5.1.1.1.cmml"><mi id="S1.Ex1.m1.5.5.1.1.1.3" xref="S1.Ex1.m1.5.5.1.1.1.3.cmml">ρ</mi><mo id="S1.Ex1.m1.5.5.1.1.1.2" xref="S1.Ex1.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.5.5.1.1.1.1.1" xref="S1.Ex1.m1.5.5.1.1.1.1.2.cmml"><mo id="S1.Ex1.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S1.Ex1.m1.5.5.1.1.1.1.2.cmml">(</mo><mi id="S1.Ex1.m1.3.3" xref="S1.Ex1.m1.3.3.cmml">G</mi><mo id="S1.Ex1.m1.5.5.1.1.1.1.1.3" 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p10"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p10.6">Since the state of the art in Boolean circuit lower bounds is of the form <math alttext="C\cdot n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p10.1.m1.1a"><mrow id="S1.SS2.SSS0.Px1.p10.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.2" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.2.cmml">C</mi><mo id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.1.cmml">⋅</mo><mi id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.3" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.1.m1.1b"><apply id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1"><ci id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.1">⋅</ci><ci id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.2">𝐶</ci><ci id="S1.SS2.SSS0.Px1.p10.1.m1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p10.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.1.m1.1c">C\cdot n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.1.m1.1d">italic_C ⋅ italic_n</annotation></semantics></math> for a small constant <math alttext="C" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.2.m2.1"><semantics id="S1.SS2.SSS0.Px1.p10.2.m2.1a"><mi id="S1.SS2.SSS0.Px1.p10.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p10.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.2.m2.1b"><ci id="S1.SS2.SSS0.Px1.p10.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.2.m2.1d">italic_C</annotation></semantics></math>, the discussion above motivates the investigation of a tighter version of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.E1" title="In Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">1</span></a>. Next, we show that cover complexity can be <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p10.6.1">exactly</em> characterized using the complexity of <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p10.6.2">cyclic constructions</em>. Roughly speaking, <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p10.3.m3.1a"><mrow id="S1.SS2.SSS0.Px1.p10.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.3" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.3.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.2" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.3.m3.1b"><apply id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1"><times id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.3">𝐷</ci><apply id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p10.3.m3.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.3.m3.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.3.m3.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> denotes the minimum number of unions and intersections in a cyclic construction of <math alttext="A" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p10.4.m4.1a"><mi id="S1.SS2.SSS0.Px1.p10.4.m4.1.1" xref="S1.SS2.SSS0.Px1.p10.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.4.m4.1b"><ci id="S1.SS2.SSS0.Px1.p10.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.4.m4.1d">italic_A</annotation></semantics></math> from sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p10.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p10.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p10.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.5.m5.1b"><ci id="S1.SS2.SSS0.Px1.p10.5.m5.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.5.m5.1d">caligraphic_B</annotation></semantics></math>, where a cyclic construction can be seen as the analogue of a Boolean circuit allowed to contain cycles. We refer to <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS5" title="2.5 Cyclic Discrete Complexity ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.5</span></a> for the definition. Similarly, we can also consider <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p10.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p10.6.m6.1a"><mrow id="S1.SS2.SSS0.Px1.p10.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.cmml"><msub id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.2" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.2.cmml">D</mi><mo id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.3" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.3.cmml">∩</mo></msub><mo id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.2" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.2.cmml">A</mi><mo id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p10.6.m6.1b"><apply id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1"><times id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.2"></times><apply id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3">subscript</csymbol><ci id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.2">𝐷</ci><intersect id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.3.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.3.3"></intersect></apply><apply id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1"><csymbol cd="latexml" id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.1">conditional</csymbol><ci id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.2">𝐴</ci><ci id="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p10.6.m6.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p10.6.m6.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p10.6.m6.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math>, the intersection complexity of cyclic constructions.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem3.1.1.1">Theorem 3</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem3.2.2"> </span>(Exact characterization of cover complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem3.p1"> <p class="ltx_p" id="Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem3.p1.2.2">Let <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem3.p1.1.1.m1.1"><semantics id="Thmtheorem3.p1.1.1.m1.1a"><mrow id="Thmtheorem3.p1.1.1.m1.1.1" xref="Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem3.p1.1.1.m1.1.1.2" xref="Thmtheorem3.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem3.p1.1.1.m1.1.1.1" xref="Thmtheorem3.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem3.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem3.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.1.1.m1.1b"><apply id="Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="Thmtheorem3.p1.1.1.m1.1.1"><subset id="Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem3.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem3.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem3.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> be a non-trivial set, and let <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.2.2.m2.1"><semantics id="Thmtheorem3.p1.2.2.m2.1a"><mrow id="Thmtheorem3.p1.2.2.m2.1.2" xref="Thmtheorem3.p1.2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem3.p1.2.2.m2.1.2.2" xref="Thmtheorem3.p1.2.2.m2.1.2.2.cmml">ℬ</mi><mo id="Thmtheorem3.p1.2.2.m2.1.2.1" xref="Thmtheorem3.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem3.p1.2.2.m2.1.2.3" xref="Thmtheorem3.p1.2.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem3.p1.2.2.m2.1.2.3.2" xref="Thmtheorem3.p1.2.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem3.p1.2.2.m2.1.2.3.1" xref="Thmtheorem3.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.2.2.m2.1.2.3.3.2" xref="Thmtheorem3.p1.2.2.m2.1.2.3.cmml"><mo id="Thmtheorem3.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmtheorem3.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem3.p1.2.2.m2.1.1.cmml">Γ</mi><mo id="Thmtheorem3.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem3.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.2.2.m2.1b"><apply id="Thmtheorem3.p1.2.2.m2.1.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2"><subset id="Thmtheorem3.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2.1"></subset><ci id="Thmtheorem3.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2.2">ℬ</ci><apply id="Thmtheorem3.p1.2.2.m2.1.2.3.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2.3"><times id="Thmtheorem3.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2.3.1"></times><ci id="Thmtheorem3.p1.2.2.m2.1.2.3.2.cmml" xref="Thmtheorem3.p1.2.2.m2.1.2.3.2">𝒫</ci><ci id="Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="Thmtheorem3.p1.2.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.2.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.2.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math> be a non-empty family of sets. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(A,\mathcal{B})\;=\;D_{\cap}(A\mid\mathcal{B})." class="ltx_Math" display="block" id="S1.Ex2.m1.3"><semantics id="S1.Ex2.m1.3a"><mrow id="S1.Ex2.m1.3.3.1" xref="S1.Ex2.m1.3.3.1.1.cmml"><mrow id="S1.Ex2.m1.3.3.1.1" xref="S1.Ex2.m1.3.3.1.1.cmml"><mrow id="S1.Ex2.m1.3.3.1.1.3" xref="S1.Ex2.m1.3.3.1.1.3.cmml"><mi id="S1.Ex2.m1.3.3.1.1.3.2" xref="S1.Ex2.m1.3.3.1.1.3.2.cmml">ρ</mi><mo id="S1.Ex2.m1.3.3.1.1.3.1" xref="S1.Ex2.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.3.3.1.1.3.3.2" xref="S1.Ex2.m1.3.3.1.1.3.3.1.cmml"><mo id="S1.Ex2.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.3.3.1.cmml">(</mo><mi id="S1.Ex2.m1.1.1" xref="S1.Ex2.m1.1.1.cmml">A</mi><mo id="S1.Ex2.m1.3.3.1.1.3.3.2.2" xref="S1.Ex2.m1.3.3.1.1.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml">ℬ</mi><mo id="S1.Ex2.m1.3.3.1.1.3.3.2.3" rspace="0.280em" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.3.3.1.1.2" rspace="0.558em" xref="S1.Ex2.m1.3.3.1.1.2.cmml">=</mo><mrow id="S1.Ex2.m1.3.3.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.cmml"><msub id="S1.Ex2.m1.3.3.1.1.1.3" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml"><mi id="S1.Ex2.m1.3.3.1.1.1.3.2" xref="S1.Ex2.m1.3.3.1.1.1.3.2.cmml">D</mi><mo id="S1.Ex2.m1.3.3.1.1.1.3.3" xref="S1.Ex2.m1.3.3.1.1.1.3.3.cmml">∩</mo></msub><mo id="S1.Ex2.m1.3.3.1.1.1.2" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.3.3.1.1.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S1.Ex2.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex2.m1.3.3.1.1.1.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S1.Ex2.m1.3.3.1.1.1.1.1.1.2" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.Ex2.m1.3.3.1.1.1.1.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.3.3.1.1.1.1.1.1.3" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S1.Ex2.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Ex2.m1.3.3.1.2" lspace="0em" xref="S1.Ex2.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex2.m1.3b"><apply id="S1.Ex2.m1.3.3.1.1.cmml" xref="S1.Ex2.m1.3.3.1"><eq id="S1.Ex2.m1.3.3.1.1.2.cmml" xref="S1.Ex2.m1.3.3.1.1.2"></eq><apply id="S1.Ex2.m1.3.3.1.1.3.cmml" xref="S1.Ex2.m1.3.3.1.1.3"><times id="S1.Ex2.m1.3.3.1.1.3.1.cmml" xref="S1.Ex2.m1.3.3.1.1.3.1"></times><ci id="S1.Ex2.m1.3.3.1.1.3.2.cmml" xref="S1.Ex2.m1.3.3.1.1.3.2">𝜌</ci><interval closure="open" id="S1.Ex2.m1.3.3.1.1.3.3.1.cmml" xref="S1.Ex2.m1.3.3.1.1.3.3.2"><ci id="S1.Ex2.m1.1.1.cmml" xref="S1.Ex2.m1.1.1">𝐴</ci><ci id="S1.Ex2.m1.2.2.cmml" xref="S1.Ex2.m1.2.2">ℬ</ci></interval></apply><apply id="S1.Ex2.m1.3.3.1.1.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1"><times id="S1.Ex2.m1.3.3.1.1.1.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.2"></times><apply id="S1.Ex2.m1.3.3.1.1.1.3.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S1.Ex2.m1.3.3.1.1.1.3.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S1.Ex2.m1.3.3.1.1.1.3.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3.2">𝐷</ci><intersect id="S1.Ex2.m1.3.3.1.1.1.3.3.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3.3"></intersect></apply><apply id="S1.Ex2.m1.3.3.1.1.1.1.1.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S1.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.1">conditional</csymbol><ci id="S1.Ex2.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.2">𝐴</ci><ci id="S1.Ex2.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S1.Ex2.m1.3.3.1.1.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.3c">\rho(A,\mathcal{B})\;=\;D_{\cap}(A\mid\mathcal{B}).</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.3d">italic_ρ ( italic_A , caligraphic_B ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p11"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p11.1">This precise correspondence is obtained by refining an idea from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite>, which obtained a characterization of a variant of cover complexity up to a constant factor. There are some technical differences though. In contrast to their work, here we consider (monotone) semi-filters instead of a more general class of functionals <math alttext="\mathcal{F}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p11.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p11.1.m1.1a"><mrow id="S1.SS2.SSS0.Px1.p11.1.m1.1.2" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.2" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.2.cmml">ℱ</mi><mo id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.1" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.1.cmml">⊆</mo><mrow id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.2" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.2.cmml">𝒫</mi><mo id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.1" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.3.2" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.cmml"><mo id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.cmml">(</mo><mi id="S1.SS2.SSS0.Px1.p11.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.1.cmml">U</mi><mo id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.3.2.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p11.1.m1.1b"><apply id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2"><subset id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.1"></subset><ci id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.2.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.2">ℱ</ci><apply id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3"><times id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.1"></times><ci id="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.2.3.2">𝒫</ci><ci id="S1.SS2.SSS0.Px1.p11.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p11.1.m1.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p11.1.m1.1c">\mathcal{F}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p11.1.m1.1d">caligraphic_F ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> in the definition of cover complexity, and intersection complexity instead of Boolean circuit complexity. Additionally, the result is presented in the set-theoretic framework of the fusion method (which is closer to our notion of discrete complexity), while <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite> employed a formulation via legitimate models and the generalized approximation method.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p12"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p12.3">As an immediate consequence of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem3" title="Theorem 3 (Exact characterization of cover complexity). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3</span></a> and a cover complexity lower bound from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>, it follows that every monotone <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p12.3.1">cyclic</em> Boolean circuit that decides if an input graph on <math alttext="n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p12.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p12.1.m1.1a"><mi id="S1.SS2.SSS0.Px1.p12.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p12.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p12.1.m1.1b"><ci id="S1.SS2.SSS0.Px1.p12.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p12.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p12.1.m1.1d">italic_n</annotation></semantics></math> vertices contains a triangle contains at least <math alttext="\Omega(n^{3}/(\log n)^{4})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p12.2.m2.1"><semantics 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xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.3.cmml">4</mn></msup></mrow><mo id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p12.2.m2.1b"><apply id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1"><times id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.3">Ω</ci><apply id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1"><divide id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.2"></divide><apply id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3">superscript</csymbol><ci id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.2">𝑛</ci><cn id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.3.3">3</cn></apply><apply id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1">superscript</csymbol><apply id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1"><log id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.1"></log><ci id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.1.1.1.2">𝑛</ci></apply><cn id="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p12.2.m2.1.1.1.1.1.1.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p12.2.m2.1c">\Omega(n^{3}/(\log n)^{4})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p12.2.m2.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / ( roman_log italic_n ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math> fan-in two <math alttext="\mathsf{AND}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p12.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p12.3.m3.1a"><mi id="S1.SS2.SSS0.Px1.p12.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p12.3.m3.1.1.cmml">𝖠𝖭𝖣</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p12.3.m3.1b"><ci id="S1.SS2.SSS0.Px1.p12.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p12.3.m3.1.1">𝖠𝖭𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p12.3.m3.1c">\mathsf{AND}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p12.3.m3.1d">sansserif_AND</annotation></semantics></math> gates.<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>This consequence does not immediately follow from the work of <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite>, as their formulation is not consistent with the use of monotone functionals employed in the definition of <math alttext="\rho" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mi id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><ci id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_ρ</annotation></semantics></math> followed here and in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>.</span></span></span> We refer to <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.4</span></a> for more details.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p13"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p13.1">The tight bound in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem3" title="Theorem 3 (Exact characterization of cover complexity). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3</span></a> highlights a mathematical advantage of the investigation of cyclic constructions and cyclic Boolean circuits. Interestingly, the strongest known lower bounds against unrestricted (non-monotone) Boolean circuits obtained via the gate elimination method <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib3" title="">3</a>]</cite> also incorporate concepts related to cyclic computations.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p14"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p14.1">Our last contribution is of a conceptual nature. The fusion method offers a different yet equivalent formulation of circuit complexity. This allows us to port some of the abstractions and characterizations provided by different notions of cover complexity to the setting of discrete complexity. As an example, we introduce <em class="ltx_emph ltx_font_italic" id="S1.SS2.SSS0.Px1.p14.1.1">nondeterministic graph complexity</em> through a dual notion of “nondeterministic” cover complexity from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>, and show a simple application to nondeterministic Boolean circuit lower bounds via a transference lemma for nondeterministic complexity.<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>Observe that the definition of nondeterministic complexity for Boolean functions relies on Boolean circuits extended with extra input variables. It is not obvious how to introduce a natural analogue in the context of graph complexity, which relies on graph constructions.</span></span></span></p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p15"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p15.1">Going beyond the contrast between state-of-the-art lower bounds for monotone and non-monotone computations, it would also be interesting to obtain an improved understanding of which settings of discrete complexity are susceptible to strong unconditional lower bounds.</p> </div> </section> <section class="ltx_paragraph ltx_indentfirst" id="S1.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Organization.</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p1.1">The main definitions are given in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2" title="2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2</span></a>. To make the paper self-contained, we include a proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S1.E1" title="In Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">1</span></a> in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3" title="3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a>. The proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem1" title="Lemma 1 (Transference of Lower Bounds). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">1</span></a> appears in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS4" title="2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS1" title="4.1 Basic results and connections ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>. The proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem2" title="Theorem 2 (Cover complexity of a random graph). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2</span></a> is presented in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS1" title="4.1 Basic results and connections ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>, while the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem3" title="Theorem 3 (Exact characterization of cover complexity). ‣ Notation. ‣ 1.2 Results ‣ 1 Introduction ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3</span></a> is given in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.4</span></a>. Finally, a discussion on nondeterministic graph complexity and a simple application of this notion appear in <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS3" title="4.3 Nondeterministic graph complexity ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>.</p> </div> </section> <section class="ltx_paragraph ltx_indentfirst" id="S1.SS2.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">Acknowledgements.</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px3.p1.4">We would like to thank Sasha Golovnev and Rahul Santhanam for discussions about the AND complexity of Boolean functions. This work received support from the Royal Society University Research Fellowship URF<math alttext="\setminus" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px3.p1.1.m1.1"><semantics id="S1.SS2.SSS0.Px3.p1.1.m1.1a"><mo id="S1.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S1.SS2.SSS0.Px3.p1.1.m1.1.1.cmml">∖</mo><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px3.p1.1.m1.1b"><setdiff id="S1.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px3.p1.1.m1.1.1"></setdiff></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px3.p1.1.m1.1c">\setminus</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px3.p1.1.m1.1d">∖</annotation></semantics></math>R1<math alttext="\setminus" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px3.p1.2.m2.1"><semantics id="S1.SS2.SSS0.Px3.p1.2.m2.1a"><mo id="S1.SS2.SSS0.Px3.p1.2.m2.1.1" xref="S1.SS2.SSS0.Px3.p1.2.m2.1.1.cmml">∖</mo><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px3.p1.2.m2.1b"><setdiff 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id="S1.SS2.SSS0.Px3.p1.4.m4.1"><semantics id="S1.SS2.SSS0.Px3.p1.4.m4.1a"><mo id="S1.SS2.SSS0.Px3.p1.4.m4.1.1" xref="S1.SS2.SSS0.Px3.p1.4.m4.1.1.cmml">∖</mo><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px3.p1.4.m4.1b"><setdiff id="S1.SS2.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px3.p1.4.m4.1.1"></setdiff></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px3.p1.4.m4.1c">\setminus</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px3.p1.4.m4.1d">∖</annotation></semantics></math>211106; the UKRI Frontier Research Guarantee Grant EP/Y007999/1; and the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick.</p> </div> </section> </section> </section> <section class="ltx_section ltx_indent_first" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Discrete Complexity</h2> <section class="ltx_subsection ltx_indent_first" id="S2.SS1"> <h3 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id="S2.SS1.p1.3.m3.2.2.cmml" type="integer" xref="S2.SS1.p1.3.m3.2.2">1</cn><ci id="S2.SS1.p1.3.m3.3.3.cmml" xref="S2.SS1.p1.3.m3.3.3">…</ci><ci id="S2.SS1.p1.3.m3.4.4.cmml" xref="S2.SS1.p1.3.m3.4.4">𝑡</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.4c">[t]\stackrel{{\scriptstyle\rm def}}{{=}}\{1,\ldots,t\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.4d">[ italic_t ] start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { 1 , … , italic_t }</annotation></semantics></math>, where <math alttext="t\in\mathbb{N}^{+}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">t</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml"><mi id="S2.SS1.p1.4.m4.1.1.3.2" xref="S2.SS1.p1.4.m4.1.1.3.2.cmml">ℕ</mi><mo id="S2.SS1.p1.4.m4.1.1.3.3" xref="S2.SS1.p1.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><in id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></in><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑡</ci><apply id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m4.1.1.3.1.cmml" xref="S2.SS1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS1.p1.4.m4.1.1.3.2.cmml" xref="S2.SS1.p1.4.m4.1.1.3.2">ℕ</ci><plus id="S2.SS1.p1.4.m4.1.1.3.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">t\in\mathbb{N}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_t ∈ blackboard_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\mathcal{P}(\cdot)" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mrow id="S2.SS1.p1.5.m5.1.2" xref="S2.SS1.p1.5.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.5.m5.1.2.2" xref="S2.SS1.p1.5.m5.1.2.2.cmml">𝒫</mi><mo id="S2.SS1.p1.5.m5.1.2.1" xref="S2.SS1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.5.m5.1.2.3.2" xref="S2.SS1.p1.5.m5.1.2.cmml"><mo id="S2.SS1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.5.m5.1.2.cmml">(</mo><mo id="S2.SS1.p1.5.m5.1.1" lspace="0em" rspace="0em" xref="S2.SS1.p1.5.m5.1.1.cmml">⋅</mo><mo id="S2.SS1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.2.cmml" xref="S2.SS1.p1.5.m5.1.2"><times id="S2.SS1.p1.5.m5.1.2.1.cmml" xref="S2.SS1.p1.5.m5.1.2.1"></times><ci id="S2.SS1.p1.5.m5.1.2.2.cmml" xref="S2.SS1.p1.5.m5.1.2.2">𝒫</ci><ci id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">\mathcal{P}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">caligraphic_P ( ⋅ )</annotation></semantics></math> is the power-set construction.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.23">Let <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi id="S2.SS1.p2.1.m1.1.1" mathvariant="normal" xref="S2.SS1.p2.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">roman_Γ</annotation></semantics></math> be a nonempty finite set. We refer to this set as the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.1">ground set</em>, or the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.2">ambient space</em>. Let <math alttext="\mathcal{B}=\{B_{1},\ldots,B_{m}\}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.3"><semantics id="S2.SS1.p2.2.m2.3a"><mrow id="S2.SS1.p2.2.m2.3.3" xref="S2.SS1.p2.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.2.m2.3.3.4" xref="S2.SS1.p2.2.m2.3.3.4.cmml">ℬ</mi><mo id="S2.SS1.p2.2.m2.3.3.3" xref="S2.SS1.p2.2.m2.3.3.3.cmml">=</mo><mrow id="S2.SS1.p2.2.m2.3.3.2.2" xref="S2.SS1.p2.2.m2.3.3.2.3.cmml"><mo id="S2.SS1.p2.2.m2.3.3.2.2.3" stretchy="false" xref="S2.SS1.p2.2.m2.3.3.2.3.cmml">{</mo><msub id="S2.SS1.p2.2.m2.2.2.1.1.1" xref="S2.SS1.p2.2.m2.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.2.m2.2.2.1.1.1.2" xref="S2.SS1.p2.2.m2.2.2.1.1.1.2.cmml">B</mi><mn id="S2.SS1.p2.2.m2.2.2.1.1.1.3" xref="S2.SS1.p2.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p2.2.m2.3.3.2.2.4" xref="S2.SS1.p2.2.m2.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p2.2.m2.1.1" mathvariant="normal" xref="S2.SS1.p2.2.m2.1.1.cmml">…</mi><mo id="S2.SS1.p2.2.m2.3.3.2.2.5" xref="S2.SS1.p2.2.m2.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.2.m2.3.3.2.2.2" xref="S2.SS1.p2.2.m2.3.3.2.2.2.cmml"><mi id="S2.SS1.p2.2.m2.3.3.2.2.2.2" xref="S2.SS1.p2.2.m2.3.3.2.2.2.2.cmml">B</mi><mi id="S2.SS1.p2.2.m2.3.3.2.2.2.3" xref="S2.SS1.p2.2.m2.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S2.SS1.p2.2.m2.3.3.2.2.6" stretchy="false" xref="S2.SS1.p2.2.m2.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.3b"><apply id="S2.SS1.p2.2.m2.3.3.cmml" xref="S2.SS1.p2.2.m2.3.3"><eq id="S2.SS1.p2.2.m2.3.3.3.cmml" xref="S2.SS1.p2.2.m2.3.3.3"></eq><ci id="S2.SS1.p2.2.m2.3.3.4.cmml" xref="S2.SS1.p2.2.m2.3.3.4">ℬ</ci><set id="S2.SS1.p2.2.m2.3.3.2.3.cmml" xref="S2.SS1.p2.2.m2.3.3.2.2"><apply id="S2.SS1.p2.2.m2.2.2.1.1.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.2.m2.2.2.1.1.1.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.2">𝐵</ci><cn id="S2.SS1.p2.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">…</ci><apply id="S2.SS1.p2.2.m2.3.3.2.2.2.cmml" xref="S2.SS1.p2.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.3.3.2.2.2.1.cmml" xref="S2.SS1.p2.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.2.m2.3.3.2.2.2.2.cmml" xref="S2.SS1.p2.2.m2.3.3.2.2.2.2">𝐵</ci><ci id="S2.SS1.p2.2.m2.3.3.2.2.2.3.cmml" xref="S2.SS1.p2.2.m2.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.3c">\mathcal{B}=\{B_{1},\ldots,B_{m}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.3d">caligraphic_B = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> be a family of subsets of <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mi id="S2.SS1.p2.3.m3.1.1" mathvariant="normal" xref="S2.SS1.p2.3.m3.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">roman_Γ</annotation></semantics></math>. We say that a set <math alttext="B_{i}\in\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><msub id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2.2" xref="S2.SS1.p2.4.m4.1.1.2.2.cmml">B</mi><mi id="S2.SS1.p2.4.m4.1.1.2.3" xref="S2.SS1.p2.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.4.m4.1.1.1" xref="S2.SS1.p2.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><in id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1.1"></in><apply id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.1.2.1.cmml" xref="S2.SS1.p2.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.1.2.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2.2">𝐵</ci><ci id="S2.SS1.p2.4.m4.1.1.2.3.cmml" xref="S2.SS1.p2.4.m4.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">B_{i}\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math> is a <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.3">generator</em>. Given a set <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mrow id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml"><mi id="S2.SS1.p2.5.m5.1.1.2" xref="S2.SS1.p2.5.m5.1.1.2.cmml">A</mi><mo id="S2.SS1.p2.5.m5.1.1.1" xref="S2.SS1.p2.5.m5.1.1.1.cmml">⊆</mo><mi id="S2.SS1.p2.5.m5.1.1.3" mathvariant="normal" xref="S2.SS1.p2.5.m5.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><apply id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1"><subset id="S2.SS1.p2.5.m5.1.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1.1"></subset><ci id="S2.SS1.p2.5.m5.1.1.2.cmml" xref="S2.SS1.p2.5.m5.1.1.2">𝐴</ci><ci id="S2.SS1.p2.5.m5.1.1.3.cmml" xref="S2.SS1.p2.5.m5.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_A ⊆ roman_Γ</annotation></semantics></math>, we are interested in the minimum number of elementary set operations necessary to construct <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><mi id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.1b"><ci id="S2.SS1.p2.6.m6.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">italic_A</annotation></semantics></math> from the generator sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.1b"><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">caligraphic_B</annotation></semantics></math>. The allowed operations are <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.4">union</em> and <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.5">intersection</em>. Formally, we let <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><mrow id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml"><mi id="S2.SS1.p2.8.m8.1.1.3" xref="S2.SS1.p2.8.m8.1.1.3.cmml">D</mi><mo id="S2.SS1.p2.8.m8.1.1.2" xref="S2.SS1.p2.8.m8.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.8.m8.1.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.8.m8.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.8.m8.1.1.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.8.m8.1.1.1.1.1.2" xref="S2.SS1.p2.8.m8.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p2.8.m8.1.1.1.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.8.m8.1.1.1.1.1.3" xref="S2.SS1.p2.8.m8.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p2.8.m8.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><apply id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1"><times id="S2.SS1.p2.8.m8.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2"></times><ci id="S2.SS1.p2.8.m8.1.1.3.cmml" xref="S2.SS1.p2.8.m8.1.1.3">𝐷</ci><apply id="S2.SS1.p2.8.m8.1.1.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p2.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p2.8.m8.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.8.m8.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> be the minimum number <math alttext="t\geq 1" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><mrow id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml"><mi id="S2.SS1.p2.9.m9.1.1.2" xref="S2.SS1.p2.9.m9.1.1.2.cmml">t</mi><mo id="S2.SS1.p2.9.m9.1.1.1" xref="S2.SS1.p2.9.m9.1.1.1.cmml">≥</mo><mn id="S2.SS1.p2.9.m9.1.1.3" xref="S2.SS1.p2.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><apply id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1"><geq id="S2.SS1.p2.9.m9.1.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1.1"></geq><ci id="S2.SS1.p2.9.m9.1.1.2.cmml" xref="S2.SS1.p2.9.m9.1.1.2">𝑡</ci><cn id="S2.SS1.p2.9.m9.1.1.3.cmml" type="integer" xref="S2.SS1.p2.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">t\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">italic_t ≥ 1</annotation></semantics></math> such that there exists a <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.6">sequence</em> <math alttext="A_{1},\ldots,A_{t}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.3"><semantics id="S2.SS1.p2.10.m10.3a"><mrow id="S2.SS1.p2.10.m10.3.3.2" xref="S2.SS1.p2.10.m10.3.3.3.cmml"><msub id="S2.SS1.p2.10.m10.2.2.1.1" xref="S2.SS1.p2.10.m10.2.2.1.1.cmml"><mi id="S2.SS1.p2.10.m10.2.2.1.1.2" xref="S2.SS1.p2.10.m10.2.2.1.1.2.cmml">A</mi><mn id="S2.SS1.p2.10.m10.2.2.1.1.3" xref="S2.SS1.p2.10.m10.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p2.10.m10.3.3.2.3" xref="S2.SS1.p2.10.m10.3.3.3.cmml">,</mo><mi id="S2.SS1.p2.10.m10.1.1" mathvariant="normal" xref="S2.SS1.p2.10.m10.1.1.cmml">…</mi><mo id="S2.SS1.p2.10.m10.3.3.2.4" xref="S2.SS1.p2.10.m10.3.3.3.cmml">,</mo><msub id="S2.SS1.p2.10.m10.3.3.2.2" xref="S2.SS1.p2.10.m10.3.3.2.2.cmml"><mi id="S2.SS1.p2.10.m10.3.3.2.2.2" xref="S2.SS1.p2.10.m10.3.3.2.2.2.cmml">A</mi><mi id="S2.SS1.p2.10.m10.3.3.2.2.3" xref="S2.SS1.p2.10.m10.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.3b"><list id="S2.SS1.p2.10.m10.3.3.3.cmml" xref="S2.SS1.p2.10.m10.3.3.2"><apply id="S2.SS1.p2.10.m10.2.2.1.1.cmml" xref="S2.SS1.p2.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.10.m10.2.2.1.1.1.cmml" xref="S2.SS1.p2.10.m10.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p2.10.m10.2.2.1.1.2.cmml" xref="S2.SS1.p2.10.m10.2.2.1.1.2">𝐴</ci><cn id="S2.SS1.p2.10.m10.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p2.10.m10.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">…</ci><apply id="S2.SS1.p2.10.m10.3.3.2.2.cmml" xref="S2.SS1.p2.10.m10.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.10.m10.3.3.2.2.1.cmml" xref="S2.SS1.p2.10.m10.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p2.10.m10.3.3.2.2.2.cmml" xref="S2.SS1.p2.10.m10.3.3.2.2.2">𝐴</ci><ci id="S2.SS1.p2.10.m10.3.3.2.2.3.cmml" xref="S2.SS1.p2.10.m10.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.3c">A_{1},\ldots,A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> of sets contained in <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p2.11.m11.1"><semantics id="S2.SS1.p2.11.m11.1a"><mi id="S2.SS1.p2.11.m11.1.1" mathvariant="normal" xref="S2.SS1.p2.11.m11.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.11.m11.1b"><ci id="S2.SS1.p2.11.m11.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.11.m11.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.11.m11.1d">roman_Γ</annotation></semantics></math> for which the following holds: <math alttext="A_{t}=A" class="ltx_Math" display="inline" id="S2.SS1.p2.12.m12.1"><semantics id="S2.SS1.p2.12.m12.1a"><mrow id="S2.SS1.p2.12.m12.1.1" xref="S2.SS1.p2.12.m12.1.1.cmml"><msub id="S2.SS1.p2.12.m12.1.1.2" xref="S2.SS1.p2.12.m12.1.1.2.cmml"><mi id="S2.SS1.p2.12.m12.1.1.2.2" xref="S2.SS1.p2.12.m12.1.1.2.2.cmml">A</mi><mi id="S2.SS1.p2.12.m12.1.1.2.3" xref="S2.SS1.p2.12.m12.1.1.2.3.cmml">t</mi></msub><mo id="S2.SS1.p2.12.m12.1.1.1" xref="S2.SS1.p2.12.m12.1.1.1.cmml">=</mo><mi id="S2.SS1.p2.12.m12.1.1.3" xref="S2.SS1.p2.12.m12.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.12.m12.1b"><apply id="S2.SS1.p2.12.m12.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1"><eq id="S2.SS1.p2.12.m12.1.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1.1"></eq><apply id="S2.SS1.p2.12.m12.1.1.2.cmml" xref="S2.SS1.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.12.m12.1.1.2.1.cmml" xref="S2.SS1.p2.12.m12.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.12.m12.1.1.2.2.cmml" xref="S2.SS1.p2.12.m12.1.1.2.2">𝐴</ci><ci id="S2.SS1.p2.12.m12.1.1.2.3.cmml" xref="S2.SS1.p2.12.m12.1.1.2.3">𝑡</ci></apply><ci id="S2.SS1.p2.12.m12.1.1.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.12.m12.1c">A_{t}=A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.12.m12.1d">italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_A</annotation></semantics></math>, and for every <math alttext="i\in[t]" class="ltx_Math" display="inline" id="S2.SS1.p2.13.m13.1"><semantics id="S2.SS1.p2.13.m13.1a"><mrow id="S2.SS1.p2.13.m13.1.2" xref="S2.SS1.p2.13.m13.1.2.cmml"><mi id="S2.SS1.p2.13.m13.1.2.2" xref="S2.SS1.p2.13.m13.1.2.2.cmml">i</mi><mo id="S2.SS1.p2.13.m13.1.2.1" xref="S2.SS1.p2.13.m13.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p2.13.m13.1.2.3.2" xref="S2.SS1.p2.13.m13.1.2.3.1.cmml"><mo id="S2.SS1.p2.13.m13.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.13.m13.1.2.3.1.1.cmml">[</mo><mi id="S2.SS1.p2.13.m13.1.1" xref="S2.SS1.p2.13.m13.1.1.cmml">t</mi><mo id="S2.SS1.p2.13.m13.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.13.m13.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.13.m13.1b"><apply id="S2.SS1.p2.13.m13.1.2.cmml" xref="S2.SS1.p2.13.m13.1.2"><in id="S2.SS1.p2.13.m13.1.2.1.cmml" xref="S2.SS1.p2.13.m13.1.2.1"></in><ci id="S2.SS1.p2.13.m13.1.2.2.cmml" xref="S2.SS1.p2.13.m13.1.2.2">𝑖</ci><apply id="S2.SS1.p2.13.m13.1.2.3.1.cmml" xref="S2.SS1.p2.13.m13.1.2.3.2"><csymbol cd="latexml" id="S2.SS1.p2.13.m13.1.2.3.1.1.cmml" xref="S2.SS1.p2.13.m13.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS1.p2.13.m13.1.1.cmml" xref="S2.SS1.p2.13.m13.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.13.m13.1c">i\in[t]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.13.m13.1d">italic_i ∈ [ italic_t ]</annotation></semantics></math>, <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS1.p2.14.m14.1"><semantics id="S2.SS1.p2.14.m14.1a"><msub id="S2.SS1.p2.14.m14.1.1" xref="S2.SS1.p2.14.m14.1.1.cmml"><mi id="S2.SS1.p2.14.m14.1.1.2" xref="S2.SS1.p2.14.m14.1.1.2.cmml">A</mi><mi id="S2.SS1.p2.14.m14.1.1.3" xref="S2.SS1.p2.14.m14.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.14.m14.1b"><apply id="S2.SS1.p2.14.m14.1.1.cmml" xref="S2.SS1.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.14.m14.1.1.1.cmml" xref="S2.SS1.p2.14.m14.1.1">subscript</csymbol><ci id="S2.SS1.p2.14.m14.1.1.2.cmml" xref="S2.SS1.p2.14.m14.1.1.2">𝐴</ci><ci id="S2.SS1.p2.14.m14.1.1.3.cmml" xref="S2.SS1.p2.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.14.m14.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.14.m14.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is either the union or the intersection of two (not necessarily distinct) sets in <math alttext="\mathcal{B}\cup\{A_{1},\ldots,A_{i-1}\}" class="ltx_Math" display="inline" id="S2.SS1.p2.15.m15.3"><semantics id="S2.SS1.p2.15.m15.3a"><mrow id="S2.SS1.p2.15.m15.3.3" xref="S2.SS1.p2.15.m15.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.15.m15.3.3.4" xref="S2.SS1.p2.15.m15.3.3.4.cmml">ℬ</mi><mo id="S2.SS1.p2.15.m15.3.3.3" xref="S2.SS1.p2.15.m15.3.3.3.cmml">∪</mo><mrow id="S2.SS1.p2.15.m15.3.3.2.2" xref="S2.SS1.p2.15.m15.3.3.2.3.cmml"><mo id="S2.SS1.p2.15.m15.3.3.2.2.3" stretchy="false" xref="S2.SS1.p2.15.m15.3.3.2.3.cmml">{</mo><msub id="S2.SS1.p2.15.m15.2.2.1.1.1" xref="S2.SS1.p2.15.m15.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.15.m15.2.2.1.1.1.2" xref="S2.SS1.p2.15.m15.2.2.1.1.1.2.cmml">A</mi><mn id="S2.SS1.p2.15.m15.2.2.1.1.1.3" xref="S2.SS1.p2.15.m15.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p2.15.m15.3.3.2.2.4" xref="S2.SS1.p2.15.m15.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p2.15.m15.1.1" mathvariant="normal" xref="S2.SS1.p2.15.m15.1.1.cmml">…</mi><mo id="S2.SS1.p2.15.m15.3.3.2.2.5" xref="S2.SS1.p2.15.m15.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p2.15.m15.3.3.2.2.2" xref="S2.SS1.p2.15.m15.3.3.2.2.2.cmml"><mi id="S2.SS1.p2.15.m15.3.3.2.2.2.2" xref="S2.SS1.p2.15.m15.3.3.2.2.2.2.cmml">A</mi><mrow id="S2.SS1.p2.15.m15.3.3.2.2.2.3" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.cmml"><mi id="S2.SS1.p2.15.m15.3.3.2.2.2.3.2" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.2.cmml">i</mi><mo id="S2.SS1.p2.15.m15.3.3.2.2.2.3.1" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.1.cmml">−</mo><mn id="S2.SS1.p2.15.m15.3.3.2.2.2.3.3" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p2.15.m15.3.3.2.2.6" stretchy="false" xref="S2.SS1.p2.15.m15.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.15.m15.3b"><apply id="S2.SS1.p2.15.m15.3.3.cmml" xref="S2.SS1.p2.15.m15.3.3"><union id="S2.SS1.p2.15.m15.3.3.3.cmml" xref="S2.SS1.p2.15.m15.3.3.3"></union><ci id="S2.SS1.p2.15.m15.3.3.4.cmml" xref="S2.SS1.p2.15.m15.3.3.4">ℬ</ci><set id="S2.SS1.p2.15.m15.3.3.2.3.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2"><apply id="S2.SS1.p2.15.m15.2.2.1.1.1.cmml" xref="S2.SS1.p2.15.m15.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.15.m15.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.15.m15.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.15.m15.2.2.1.1.1.2.cmml" xref="S2.SS1.p2.15.m15.2.2.1.1.1.2">𝐴</ci><cn id="S2.SS1.p2.15.m15.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.15.m15.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p2.15.m15.1.1.cmml" xref="S2.SS1.p2.15.m15.1.1">…</ci><apply id="S2.SS1.p2.15.m15.3.3.2.2.2.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.15.m15.3.3.2.2.2.1.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.15.m15.3.3.2.2.2.2.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2.2">𝐴</ci><apply id="S2.SS1.p2.15.m15.3.3.2.2.2.3.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3"><minus id="S2.SS1.p2.15.m15.3.3.2.2.2.3.1.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.1"></minus><ci id="S2.SS1.p2.15.m15.3.3.2.2.2.3.2.cmml" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.2">𝑖</ci><cn id="S2.SS1.p2.15.m15.3.3.2.2.2.3.3.cmml" type="integer" xref="S2.SS1.p2.15.m15.3.3.2.2.2.3.3">1</cn></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.15.m15.3c">\mathcal{B}\cup\{A_{1},\ldots,A_{i-1}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.15.m15.3d">caligraphic_B ∪ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. We say that a sequence of this form <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.7">generates</em> <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p2.16.m16.1"><semantics id="S2.SS1.p2.16.m16.1a"><mi id="S2.SS1.p2.16.m16.1.1" xref="S2.SS1.p2.16.m16.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.16.m16.1b"><ci id="S2.SS1.p2.16.m16.1.1.cmml" xref="S2.SS1.p2.16.m16.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.16.m16.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.16.m16.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p2.17.m17.1"><semantics id="S2.SS1.p2.17.m17.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.17.m17.1.1" xref="S2.SS1.p2.17.m17.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.17.m17.1b"><ci id="S2.SS1.p2.17.m17.1.1.cmml" xref="S2.SS1.p2.17.m17.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.17.m17.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.17.m17.1d">caligraphic_B</annotation></semantics></math>. If there is no finite <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p2.18.m18.1"><semantics id="S2.SS1.p2.18.m18.1a"><mi id="S2.SS1.p2.18.m18.1.1" xref="S2.SS1.p2.18.m18.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.18.m18.1b"><ci id="S2.SS1.p2.18.m18.1.1.cmml" xref="S2.SS1.p2.18.m18.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.18.m18.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.18.m18.1d">italic_t</annotation></semantics></math> for which such a sequence exists, then <math alttext="D(A\mid\mathcal{B})\stackrel{{\scriptstyle\rm def}}{{=}}\infty" class="ltx_Math" display="inline" id="S2.SS1.p2.19.m19.1"><semantics id="S2.SS1.p2.19.m19.1a"><mrow id="S2.SS1.p2.19.m19.1.1" xref="S2.SS1.p2.19.m19.1.1.cmml"><mrow id="S2.SS1.p2.19.m19.1.1.1" xref="S2.SS1.p2.19.m19.1.1.1.cmml"><mi id="S2.SS1.p2.19.m19.1.1.1.3" xref="S2.SS1.p2.19.m19.1.1.1.3.cmml">D</mi><mo id="S2.SS1.p2.19.m19.1.1.1.2" xref="S2.SS1.p2.19.m19.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.19.m19.1.1.1.1.1" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.19.m19.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.19.m19.1.1.1.1.1.1" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.19.m19.1.1.1.1.1.1.2" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p2.19.m19.1.1.1.1.1.1.1" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.19.m19.1.1.1.1.1.1.3" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p2.19.m19.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mover id="S2.SS1.p2.19.m19.1.1.2" xref="S2.SS1.p2.19.m19.1.1.2.cmml"><mo id="S2.SS1.p2.19.m19.1.1.2.2" xref="S2.SS1.p2.19.m19.1.1.2.2.cmml">=</mo><mi id="S2.SS1.p2.19.m19.1.1.2.3" xref="S2.SS1.p2.19.m19.1.1.2.3.cmml">def</mi></mover><mi id="S2.SS1.p2.19.m19.1.1.3" mathvariant="normal" xref="S2.SS1.p2.19.m19.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.19.m19.1b"><apply id="S2.SS1.p2.19.m19.1.1.cmml" xref="S2.SS1.p2.19.m19.1.1"><apply id="S2.SS1.p2.19.m19.1.1.2.cmml" xref="S2.SS1.p2.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.19.m19.1.1.2.1.cmml" xref="S2.SS1.p2.19.m19.1.1.2">superscript</csymbol><eq id="S2.SS1.p2.19.m19.1.1.2.2.cmml" xref="S2.SS1.p2.19.m19.1.1.2.2"></eq><ci id="S2.SS1.p2.19.m19.1.1.2.3.cmml" xref="S2.SS1.p2.19.m19.1.1.2.3">def</ci></apply><apply id="S2.SS1.p2.19.m19.1.1.1.cmml" xref="S2.SS1.p2.19.m19.1.1.1"><times id="S2.SS1.p2.19.m19.1.1.1.2.cmml" xref="S2.SS1.p2.19.m19.1.1.1.2"></times><ci id="S2.SS1.p2.19.m19.1.1.1.3.cmml" xref="S2.SS1.p2.19.m19.1.1.1.3">𝐷</ci><apply id="S2.SS1.p2.19.m19.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.19.m19.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.19.m19.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p2.19.m19.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p2.19.m19.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.19.m19.1.1.1.1.1.1.3">ℬ</ci></apply></apply><infinity id="S2.SS1.p2.19.m19.1.1.3.cmml" xref="S2.SS1.p2.19.m19.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.19.m19.1c">D(A\mid\mathcal{B})\stackrel{{\scriptstyle\rm def}}{{=}}\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.19.m19.1d">italic_D ( italic_A ∣ caligraphic_B ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP ∞</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote5"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>A simple example is that of a non-monotone Boolean function represented by <math alttext="A\subseteq\{0,1\}^{n}" class="ltx_Math" display="inline" id="footnote5.m1.2"><semantics id="footnote5.m1.2b"><mrow id="footnote5.m1.2.3" xref="footnote5.m1.2.3.cmml"><mi id="footnote5.m1.2.3.2" xref="footnote5.m1.2.3.2.cmml">A</mi><mo id="footnote5.m1.2.3.1" xref="footnote5.m1.2.3.1.cmml">⊆</mo><msup id="footnote5.m1.2.3.3" xref="footnote5.m1.2.3.3.cmml"><mrow id="footnote5.m1.2.3.3.2.2" xref="footnote5.m1.2.3.3.2.1.cmml"><mo id="footnote5.m1.2.3.3.2.2.1" stretchy="false" xref="footnote5.m1.2.3.3.2.1.cmml">{</mo><mn id="footnote5.m1.1.1" xref="footnote5.m1.1.1.cmml">0</mn><mo id="footnote5.m1.2.3.3.2.2.2" xref="footnote5.m1.2.3.3.2.1.cmml">,</mo><mn id="footnote5.m1.2.2" xref="footnote5.m1.2.2.cmml">1</mn><mo id="footnote5.m1.2.3.3.2.2.3" stretchy="false" xref="footnote5.m1.2.3.3.2.1.cmml">}</mo></mrow><mi id="footnote5.m1.2.3.3.3" xref="footnote5.m1.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote5.m1.2c"><apply id="footnote5.m1.2.3.cmml" xref="footnote5.m1.2.3"><subset id="footnote5.m1.2.3.1.cmml" xref="footnote5.m1.2.3.1"></subset><ci id="footnote5.m1.2.3.2.cmml" xref="footnote5.m1.2.3.2">𝐴</ci><apply id="footnote5.m1.2.3.3.cmml" xref="footnote5.m1.2.3.3"><csymbol cd="ambiguous" id="footnote5.m1.2.3.3.1.cmml" xref="footnote5.m1.2.3.3">superscript</csymbol><set id="footnote5.m1.2.3.3.2.1.cmml" xref="footnote5.m1.2.3.3.2.2"><cn id="footnote5.m1.1.1.cmml" type="integer" xref="footnote5.m1.1.1">0</cn><cn id="footnote5.m1.2.2.cmml" type="integer" xref="footnote5.m1.2.2">1</cn></set><ci id="footnote5.m1.2.3.3.3.cmml" xref="footnote5.m1.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m1.2d">A\subseteq\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="footnote5.m1.2e">italic_A ⊆ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="footnote5.m2.1"><semantics id="footnote5.m2.1b"><mi class="ltx_font_mathcaligraphic" id="footnote5.m2.1.1" xref="footnote5.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="footnote5.m2.1c"><ci id="footnote5.m2.1.1.cmml" xref="footnote5.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m2.1d">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="footnote5.m2.1e">caligraphic_B</annotation></semantics></math> as the family of generators in monotone circuit complexity.</span></span></span> Consequently, <math alttext="D\colon\mathcal{P}(\Gamma)\times\mathcal{P}(\mathcal{P}(\Gamma))\to\mathbb{N}^% {+}\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.SS1.p2.20.m20.4"><semantics id="S2.SS1.p2.20.m20.4a"><mrow id="S2.SS1.p2.20.m20.4.4" xref="S2.SS1.p2.20.m20.4.4.cmml"><mi id="S2.SS1.p2.20.m20.4.4.3" xref="S2.SS1.p2.20.m20.4.4.3.cmml">D</mi><mo id="S2.SS1.p2.20.m20.4.4.2" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.20.m20.4.4.2.cmml">:</mo><mrow id="S2.SS1.p2.20.m20.4.4.1" xref="S2.SS1.p2.20.m20.4.4.1.cmml"><mrow id="S2.SS1.p2.20.m20.4.4.1.1" xref="S2.SS1.p2.20.m20.4.4.1.1.cmml"><mrow id="S2.SS1.p2.20.m20.4.4.1.1.3" xref="S2.SS1.p2.20.m20.4.4.1.1.3.cmml"><mrow id="S2.SS1.p2.20.m20.4.4.1.1.3.2" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.20.m20.4.4.1.1.3.2.2" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.2.cmml">𝒫</mi><mo id="S2.SS1.p2.20.m20.4.4.1.1.3.2.1" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.1.3.2.3.2" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.cmml"><mo id="S2.SS1.p2.20.m20.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.cmml">(</mo><mi id="S2.SS1.p2.20.m20.1.1" mathvariant="normal" xref="S2.SS1.p2.20.m20.1.1.cmml">Γ</mi><mo id="S2.SS1.p2.20.m20.4.4.1.1.3.2.3.2.2" rspace="0.055em" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.20.m20.4.4.1.1.3.1" rspace="0.222em" xref="S2.SS1.p2.20.m20.4.4.1.1.3.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.20.m20.4.4.1.1.3.3" xref="S2.SS1.p2.20.m20.4.4.1.1.3.3.cmml">𝒫</mi></mrow><mo id="S2.SS1.p2.20.m20.4.4.1.1.2" xref="S2.SS1.p2.20.m20.4.4.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.1.1.1" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.20.m20.4.4.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.2" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.2.cmml">𝒫</mi><mo id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.1" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.3.2" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml">(</mo><mi id="S2.SS1.p2.20.m20.2.2" mathvariant="normal" xref="S2.SS1.p2.20.m20.2.2.cmml">Γ</mi><mo id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.20.m20.4.4.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.20.m20.4.4.1.2" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.2.cmml">→</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.3" xref="S2.SS1.p2.20.m20.4.4.1.3.cmml"><msup id="S2.SS1.p2.20.m20.4.4.1.3.2" xref="S2.SS1.p2.20.m20.4.4.1.3.2.cmml"><mi id="S2.SS1.p2.20.m20.4.4.1.3.2.2" xref="S2.SS1.p2.20.m20.4.4.1.3.2.2.cmml">ℕ</mi><mo id="S2.SS1.p2.20.m20.4.4.1.3.2.3" xref="S2.SS1.p2.20.m20.4.4.1.3.2.3.cmml">+</mo></msup><mo id="S2.SS1.p2.20.m20.4.4.1.3.1" xref="S2.SS1.p2.20.m20.4.4.1.3.1.cmml">∪</mo><mrow id="S2.SS1.p2.20.m20.4.4.1.3.3.2" xref="S2.SS1.p2.20.m20.4.4.1.3.3.1.cmml"><mo id="S2.SS1.p2.20.m20.4.4.1.3.3.2.1" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.3.3.1.cmml">{</mo><mi id="S2.SS1.p2.20.m20.3.3" mathvariant="normal" xref="S2.SS1.p2.20.m20.3.3.cmml">∞</mi><mo id="S2.SS1.p2.20.m20.4.4.1.3.3.2.2" stretchy="false" xref="S2.SS1.p2.20.m20.4.4.1.3.3.1.cmml">}</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.20.m20.4b"><apply id="S2.SS1.p2.20.m20.4.4.cmml" xref="S2.SS1.p2.20.m20.4.4"><ci id="S2.SS1.p2.20.m20.4.4.2.cmml" xref="S2.SS1.p2.20.m20.4.4.2">:</ci><ci id="S2.SS1.p2.20.m20.4.4.3.cmml" xref="S2.SS1.p2.20.m20.4.4.3">𝐷</ci><apply id="S2.SS1.p2.20.m20.4.4.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1"><ci id="S2.SS1.p2.20.m20.4.4.1.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.2">→</ci><apply id="S2.SS1.p2.20.m20.4.4.1.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1"><times id="S2.SS1.p2.20.m20.4.4.1.1.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.2"></times><apply id="S2.SS1.p2.20.m20.4.4.1.1.3.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3"><times id="S2.SS1.p2.20.m20.4.4.1.1.3.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3.1"></times><apply id="S2.SS1.p2.20.m20.4.4.1.1.3.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2"><times id="S2.SS1.p2.20.m20.4.4.1.1.3.2.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.1"></times><ci id="S2.SS1.p2.20.m20.4.4.1.1.3.2.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3.2.2">𝒫</ci><ci id="S2.SS1.p2.20.m20.1.1.cmml" xref="S2.SS1.p2.20.m20.1.1">Γ</ci></apply><ci id="S2.SS1.p2.20.m20.4.4.1.1.3.3.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.3.3">𝒫</ci></apply><apply id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1"><times id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.1"></times><ci id="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.1.1.1.1.2">𝒫</ci><ci id="S2.SS1.p2.20.m20.2.2.cmml" xref="S2.SS1.p2.20.m20.2.2">Γ</ci></apply></apply><apply id="S2.SS1.p2.20.m20.4.4.1.3.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3"><union id="S2.SS1.p2.20.m20.4.4.1.3.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.1"></union><apply id="S2.SS1.p2.20.m20.4.4.1.3.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p2.20.m20.4.4.1.3.2.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.2">superscript</csymbol><ci id="S2.SS1.p2.20.m20.4.4.1.3.2.2.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.2.2">ℕ</ci><plus id="S2.SS1.p2.20.m20.4.4.1.3.2.3.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.2.3"></plus></apply><set id="S2.SS1.p2.20.m20.4.4.1.3.3.1.cmml" xref="S2.SS1.p2.20.m20.4.4.1.3.3.2"><infinity id="S2.SS1.p2.20.m20.3.3.cmml" xref="S2.SS1.p2.20.m20.3.3"></infinity></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.20.m20.4c">D\colon\mathcal{P}(\Gamma)\times\mathcal{P}(\mathcal{P}(\Gamma))\to\mathbb{N}^% {+}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.20.m20.4d">italic_D : caligraphic_P ( roman_Γ ) × caligraphic_P ( caligraphic_P ( roman_Γ ) ) → blackboard_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ { ∞ }</annotation></semantics></math>. We say that <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p2.21.m21.1"><semantics id="S2.SS1.p2.21.m21.1a"><mrow id="S2.SS1.p2.21.m21.1.1" xref="S2.SS1.p2.21.m21.1.1.cmml"><mi id="S2.SS1.p2.21.m21.1.1.3" xref="S2.SS1.p2.21.m21.1.1.3.cmml">D</mi><mo id="S2.SS1.p2.21.m21.1.1.2" xref="S2.SS1.p2.21.m21.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.21.m21.1.1.1.1" xref="S2.SS1.p2.21.m21.1.1.1.1.1.cmml"><mo id="S2.SS1.p2.21.m21.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.21.m21.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.21.m21.1.1.1.1.1" xref="S2.SS1.p2.21.m21.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.21.m21.1.1.1.1.1.2" xref="S2.SS1.p2.21.m21.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p2.21.m21.1.1.1.1.1.1" xref="S2.SS1.p2.21.m21.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.21.m21.1.1.1.1.1.3" xref="S2.SS1.p2.21.m21.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p2.21.m21.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.21.m21.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.21.m21.1b"><apply id="S2.SS1.p2.21.m21.1.1.cmml" xref="S2.SS1.p2.21.m21.1.1"><times id="S2.SS1.p2.21.m21.1.1.2.cmml" xref="S2.SS1.p2.21.m21.1.1.2"></times><ci id="S2.SS1.p2.21.m21.1.1.3.cmml" xref="S2.SS1.p2.21.m21.1.1.3">𝐷</ci><apply id="S2.SS1.p2.21.m21.1.1.1.1.1.cmml" xref="S2.SS1.p2.21.m21.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.21.m21.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.21.m21.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p2.21.m21.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.21.m21.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p2.21.m21.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.21.m21.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.21.m21.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.21.m21.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.23.8">discrete complexity</em> of <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p2.22.m22.1"><semantics id="S2.SS1.p2.22.m22.1a"><mi id="S2.SS1.p2.22.m22.1.1" xref="S2.SS1.p2.22.m22.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.22.m22.1b"><ci id="S2.SS1.p2.22.m22.1.1.cmml" xref="S2.SS1.p2.22.m22.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.22.m22.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.22.m22.1d">italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p2.23.m23.1"><semantics id="S2.SS1.p2.23.m23.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.23.m23.1.1" xref="S2.SS1.p2.23.m23.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.23.m23.1b"><ci id="S2.SS1.p2.23.m23.1.1.cmml" xref="S2.SS1.p2.23.m23.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.23.m23.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.23.m23.1d">caligraphic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.8">We use <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><msub id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">D</mi><mo id="S2.SS1.p3.1.m1.1.1.3.3" xref="S2.SS1.p3.1.m1.1.1.3.3.cmml">∩</mo></msub><mo id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.1.m1.1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.1.m1.1.1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p3.1.m1.1.1.1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.1.m1.1.1.1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><times id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2"></times><apply id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2">𝐷</ci><intersect id="S2.SS1.p3.1.m1.1.1.3.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3.3"></intersect></apply><apply id="S2.SS1.p3.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p3.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p3.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p3.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> to denote the minimum number of <em class="ltx_emph ltx_font_italic" id="S2.SS1.p3.8.1">intersections</em> in any sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">caligraphic_B</annotation></semantics></math>. The value <math alttext="D_{\cup}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><mrow id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><msub id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml"><mi id="S2.SS1.p3.4.m4.1.1.3.2" xref="S2.SS1.p3.4.m4.1.1.3.2.cmml">D</mi><mo id="S2.SS1.p3.4.m4.1.1.3.3" xref="S2.SS1.p3.4.m4.1.1.3.3.cmml">∪</mo></msub><mo id="S2.SS1.p3.4.m4.1.1.2" xref="S2.SS1.p3.4.m4.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.4.m4.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.4.m4.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.4.m4.1.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.4.m4.1.1.1.1.1.2" xref="S2.SS1.p3.4.m4.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p3.4.m4.1.1.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.4.m4.1.1.1.1.1.3" xref="S2.SS1.p3.4.m4.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p3.4.m4.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><apply id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1"><times id="S2.SS1.p3.4.m4.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.2"></times><apply id="S2.SS1.p3.4.m4.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.3.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.3.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2">𝐷</ci><union id="S2.SS1.p3.4.m4.1.1.3.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.3"></union></apply><apply id="S2.SS1.p3.4.m4.1.1.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p3.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p3.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p3.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">D_{\cup}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is defined analogously. We will often refer to these measures as <em class="ltx_emph ltx_font_italic" id="S2.SS1.p3.8.2">intersection complexity</em> and <em class="ltx_emph ltx_font_italic" id="S2.SS1.p3.8.3">union complexity</em>, respectively. Given sets <math alttext="A_{1},\dots,A_{s}\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.3"><semantics id="S2.SS1.p3.5.m5.3a"><mrow id="S2.SS1.p3.5.m5.3.3" xref="S2.SS1.p3.5.m5.3.3.cmml"><mrow id="S2.SS1.p3.5.m5.3.3.2.2" xref="S2.SS1.p3.5.m5.3.3.2.3.cmml"><msub id="S2.SS1.p3.5.m5.2.2.1.1.1" xref="S2.SS1.p3.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS1.p3.5.m5.2.2.1.1.1.2" xref="S2.SS1.p3.5.m5.2.2.1.1.1.2.cmml">A</mi><mn id="S2.SS1.p3.5.m5.2.2.1.1.1.3" xref="S2.SS1.p3.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p3.5.m5.3.3.2.2.3" xref="S2.SS1.p3.5.m5.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p3.5.m5.1.1" mathvariant="normal" xref="S2.SS1.p3.5.m5.1.1.cmml">…</mi><mo id="S2.SS1.p3.5.m5.3.3.2.2.4" xref="S2.SS1.p3.5.m5.3.3.2.3.cmml">,</mo><msub id="S2.SS1.p3.5.m5.3.3.2.2.2" xref="S2.SS1.p3.5.m5.3.3.2.2.2.cmml"><mi id="S2.SS1.p3.5.m5.3.3.2.2.2.2" xref="S2.SS1.p3.5.m5.3.3.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p3.5.m5.3.3.2.2.2.3" xref="S2.SS1.p3.5.m5.3.3.2.2.2.3.cmml">s</mi></msub></mrow><mo id="S2.SS1.p3.5.m5.3.3.3" xref="S2.SS1.p3.5.m5.3.3.3.cmml">⊆</mo><mi id="S2.SS1.p3.5.m5.3.3.4" mathvariant="normal" xref="S2.SS1.p3.5.m5.3.3.4.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.3b"><apply id="S2.SS1.p3.5.m5.3.3.cmml" xref="S2.SS1.p3.5.m5.3.3"><subset id="S2.SS1.p3.5.m5.3.3.3.cmml" xref="S2.SS1.p3.5.m5.3.3.3"></subset><list id="S2.SS1.p3.5.m5.3.3.2.3.cmml" xref="S2.SS1.p3.5.m5.3.3.2.2"><apply id="S2.SS1.p3.5.m5.2.2.1.1.1.cmml" xref="S2.SS1.p3.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS1.p3.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS1.p3.5.m5.2.2.1.1.1.2">𝐴</ci><cn id="S2.SS1.p3.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.5.m5.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">…</ci><apply id="S2.SS1.p3.5.m5.3.3.2.2.2.cmml" xref="S2.SS1.p3.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.3.3.2.2.2.1.cmml" xref="S2.SS1.p3.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S2.SS1.p3.5.m5.3.3.2.2.2.2.cmml" xref="S2.SS1.p3.5.m5.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS1.p3.5.m5.3.3.2.2.2.3.cmml" xref="S2.SS1.p3.5.m5.3.3.2.2.2.3">𝑠</ci></apply></list><ci id="S2.SS1.p3.5.m5.3.3.4.cmml" xref="S2.SS1.p3.5.m5.3.3.4">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.3c">A_{1},\dots,A_{s}\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⊆ roman_Γ</annotation></semantics></math>, we will write <math alttext="D(A_{1},\dots,A_{s}\mid{\mathcal{B}})" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.3"><semantics id="S2.SS1.p3.6.m6.3a"><mrow id="S2.SS1.p3.6.m6.3.3" xref="S2.SS1.p3.6.m6.3.3.cmml"><mi id="S2.SS1.p3.6.m6.3.3.4" xref="S2.SS1.p3.6.m6.3.3.4.cmml">D</mi><mo id="S2.SS1.p3.6.m6.3.3.3" xref="S2.SS1.p3.6.m6.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p3.6.m6.3.3.2.2" xref="S2.SS1.p3.6.m6.3.3.2.3.cmml"><mo id="S2.SS1.p3.6.m6.3.3.2.2.3" stretchy="false" xref="S2.SS1.p3.6.m6.3.3.2.3.cmml">(</mo><msub id="S2.SS1.p3.6.m6.2.2.1.1.1" xref="S2.SS1.p3.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS1.p3.6.m6.2.2.1.1.1.2" xref="S2.SS1.p3.6.m6.2.2.1.1.1.2.cmml">A</mi><mn id="S2.SS1.p3.6.m6.2.2.1.1.1.3" xref="S2.SS1.p3.6.m6.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p3.6.m6.3.3.2.2.4" xref="S2.SS1.p3.6.m6.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p3.6.m6.1.1" mathvariant="normal" xref="S2.SS1.p3.6.m6.1.1.cmml">…</mi><mo id="S2.SS1.p3.6.m6.3.3.2.2.5" xref="S2.SS1.p3.6.m6.3.3.2.3.cmml">,</mo><mrow id="S2.SS1.p3.6.m6.3.3.2.2.2" xref="S2.SS1.p3.6.m6.3.3.2.2.2.cmml"><msub id="S2.SS1.p3.6.m6.3.3.2.2.2.2" xref="S2.SS1.p3.6.m6.3.3.2.2.2.2.cmml"><mi id="S2.SS1.p3.6.m6.3.3.2.2.2.2.2" xref="S2.SS1.p3.6.m6.3.3.2.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p3.6.m6.3.3.2.2.2.2.3" xref="S2.SS1.p3.6.m6.3.3.2.2.2.2.3.cmml">s</mi></msub><mo id="S2.SS1.p3.6.m6.3.3.2.2.2.1" xref="S2.SS1.p3.6.m6.3.3.2.2.2.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.6.m6.3.3.2.2.2.3" xref="S2.SS1.p3.6.m6.3.3.2.2.2.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p3.6.m6.3.3.2.2.6" stretchy="false" xref="S2.SS1.p3.6.m6.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.3b"><apply id="S2.SS1.p3.6.m6.3.3.cmml" xref="S2.SS1.p3.6.m6.3.3"><times id="S2.SS1.p3.6.m6.3.3.3.cmml" xref="S2.SS1.p3.6.m6.3.3.3"></times><ci id="S2.SS1.p3.6.m6.3.3.4.cmml" xref="S2.SS1.p3.6.m6.3.3.4">𝐷</ci><vector id="S2.SS1.p3.6.m6.3.3.2.3.cmml" xref="S2.SS1.p3.6.m6.3.3.2.2"><apply id="S2.SS1.p3.6.m6.2.2.1.1.1.cmml" xref="S2.SS1.p3.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS1.p3.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.6.m6.2.2.1.1.1.2.cmml" 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id="S2.SS1.p3.6.m6.3c">D(A_{1},\dots,A_{s}\mid{\mathcal{B}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.3d">italic_D ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ∣ caligraphic_B )</annotation></semantics></math> to refer to the minimum length of a sequence that generates all the sets <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m7.1"><semantics id="S2.SS1.p3.7.m7.1a"><msub id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml"><mi id="S2.SS1.p3.7.m7.1.1.2" xref="S2.SS1.p3.7.m7.1.1.2.cmml">A</mi><mi id="S2.SS1.p3.7.m7.1.1.3" xref="S2.SS1.p3.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.1b"><apply id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m7.1.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p3.7.m7.1.1.2.cmml" xref="S2.SS1.p3.7.m7.1.1.2">𝐴</ci><ci id="S2.SS1.p3.7.m7.1.1.3.cmml" xref="S2.SS1.p3.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m7.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; the measure <math alttext="D_{\cap}(A_{1},\dots,A_{s}\mid{\mathcal{B}})" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m8.3"><semantics id="S2.SS1.p3.8.m8.3a"><mrow id="S2.SS1.p3.8.m8.3.3" xref="S2.SS1.p3.8.m8.3.3.cmml"><msub id="S2.SS1.p3.8.m8.3.3.4" xref="S2.SS1.p3.8.m8.3.3.4.cmml"><mi id="S2.SS1.p3.8.m8.3.3.4.2" xref="S2.SS1.p3.8.m8.3.3.4.2.cmml">D</mi><mo id="S2.SS1.p3.8.m8.3.3.4.3" xref="S2.SS1.p3.8.m8.3.3.4.3.cmml">∩</mo></msub><mo id="S2.SS1.p3.8.m8.3.3.3" xref="S2.SS1.p3.8.m8.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p3.8.m8.3.3.2.2" xref="S2.SS1.p3.8.m8.3.3.2.3.cmml"><mo id="S2.SS1.p3.8.m8.3.3.2.2.3" stretchy="false" 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xref="S2.SS1.p3.8.m8.3.3.2.2.2.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p3.8.m8.3.3.2.2.6" stretchy="false" xref="S2.SS1.p3.8.m8.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.3b"><apply id="S2.SS1.p3.8.m8.3.3.cmml" xref="S2.SS1.p3.8.m8.3.3"><times id="S2.SS1.p3.8.m8.3.3.3.cmml" xref="S2.SS1.p3.8.m8.3.3.3"></times><apply id="S2.SS1.p3.8.m8.3.3.4.cmml" xref="S2.SS1.p3.8.m8.3.3.4"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.3.3.4.1.cmml" xref="S2.SS1.p3.8.m8.3.3.4">subscript</csymbol><ci id="S2.SS1.p3.8.m8.3.3.4.2.cmml" xref="S2.SS1.p3.8.m8.3.3.4.2">𝐷</ci><intersect id="S2.SS1.p3.8.m8.3.3.4.3.cmml" xref="S2.SS1.p3.8.m8.3.3.4.3"></intersect></apply><vector id="S2.SS1.p3.8.m8.3.3.2.3.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2"><apply id="S2.SS1.p3.8.m8.2.2.1.1.1.cmml" xref="S2.SS1.p3.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.2.2.1.1.1.1.cmml" xref="S2.SS1.p3.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.8.m8.2.2.1.1.1.2.cmml" xref="S2.SS1.p3.8.m8.2.2.1.1.1.2">𝐴</ci><cn id="S2.SS1.p3.8.m8.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.8.m8.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p3.8.m8.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1">…</ci><apply id="S2.SS1.p3.8.m8.3.3.2.2.2.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2"><csymbol cd="latexml" id="S2.SS1.p3.8.m8.3.3.2.2.2.1.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.1">conditional</csymbol><apply id="S2.SS1.p3.8.m8.3.3.2.2.2.2.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.3.3.2.2.2.2.1.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p3.8.m8.3.3.2.2.2.2.2.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.2.2">𝐴</ci><ci id="S2.SS1.p3.8.m8.3.3.2.2.2.2.3.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.2.3">𝑠</ci></apply><ci id="S2.SS1.p3.8.m8.3.3.2.2.2.3.cmml" xref="S2.SS1.p3.8.m8.3.3.2.2.2.3">ℬ</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.3c">D_{\cap}(A_{1},\dots,A_{s}\mid{\mathcal{B}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m8.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ∣ caligraphic_B )</annotation></semantics></math> is defined analogously.</p> </div> <div class="ltx_theorem ltx_theorem_fact" id="Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem4.1.1.1">Fact 4</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem4.p1"> <p class="ltx_p" id="Thmtheorem4.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem4.p1.3.3">If <math alttext="A\in\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem4.p1.1.1.m1.1"><semantics id="Thmtheorem4.p1.1.1.m1.1a"><mrow id="Thmtheorem4.p1.1.1.m1.1.1" xref="Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem4.p1.1.1.m1.1.1.2" xref="Thmtheorem4.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem4.p1.1.1.m1.1.1.1" xref="Thmtheorem4.p1.1.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem4.p1.1.1.m1.1.1.3" xref="Thmtheorem4.p1.1.1.m1.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.1.1.m1.1b"><apply id="Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="Thmtheorem4.p1.1.1.m1.1.1"><in id="Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem4.p1.1.1.m1.1.1.1"></in><ci id="Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem4.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem4.p1.1.1.m1.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.1.1.m1.1c">A\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.1.1.m1.1d">italic_A ∈ caligraphic_B</annotation></semantics></math>, then <math alttext="D(A\mid\mathcal{B})=1" class="ltx_Math" display="inline" id="Thmtheorem4.p1.2.2.m2.1"><semantics id="Thmtheorem4.p1.2.2.m2.1a"><mrow id="Thmtheorem4.p1.2.2.m2.1.1" xref="Thmtheorem4.p1.2.2.m2.1.1.cmml"><mrow id="Thmtheorem4.p1.2.2.m2.1.1.1" xref="Thmtheorem4.p1.2.2.m2.1.1.1.cmml"><mi id="Thmtheorem4.p1.2.2.m2.1.1.1.3" xref="Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml">D</mi><mo id="Thmtheorem4.p1.2.2.m2.1.1.1.2" xref="Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.1" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem4.p1.2.2.m2.1.1.2" xref="Thmtheorem4.p1.2.2.m2.1.1.2.cmml">=</mo><mn id="Thmtheorem4.p1.2.2.m2.1.1.3" xref="Thmtheorem4.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.2.2.m2.1b"><apply id="Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1"><eq id="Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.2"></eq><apply id="Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1"><times id="Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.2"></times><ci id="Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.3">𝐷</ci><apply id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3">ℬ</ci></apply></apply><cn id="Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="integer" xref="Thmtheorem4.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.2.2.m2.1c">D(A\mid\mathcal{B})=1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.2.2.m2.1d">italic_D ( italic_A ∣ caligraphic_B ) = 1</annotation></semantics></math> and <math alttext="D_{\cap}(A\mid\mathcal{B})=D_{\cup}(A\mid\mathcal{B})=0" class="ltx_Math" display="inline" id="Thmtheorem4.p1.3.3.m3.2"><semantics id="Thmtheorem4.p1.3.3.m3.2a"><mrow id="Thmtheorem4.p1.3.3.m3.2.2" xref="Thmtheorem4.p1.3.3.m3.2.2.cmml"><mrow id="Thmtheorem4.p1.3.3.m3.1.1.1" xref="Thmtheorem4.p1.3.3.m3.1.1.1.cmml"><msub id="Thmtheorem4.p1.3.3.m3.1.1.1.3" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3.cmml"><mi id="Thmtheorem4.p1.3.3.m3.1.1.1.3.2" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3.2.cmml">D</mi><mo id="Thmtheorem4.p1.3.3.m3.1.1.1.3.3" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem4.p1.3.3.m3.1.1.1.2" xref="Thmtheorem4.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.2" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.1" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.3" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem4.p1.3.3.m3.2.2.4" xref="Thmtheorem4.p1.3.3.m3.2.2.4.cmml">=</mo><mrow id="Thmtheorem4.p1.3.3.m3.2.2.2" xref="Thmtheorem4.p1.3.3.m3.2.2.2.cmml"><msub id="Thmtheorem4.p1.3.3.m3.2.2.2.3" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3.cmml"><mi id="Thmtheorem4.p1.3.3.m3.2.2.2.3.2" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3.2.cmml">D</mi><mo id="Thmtheorem4.p1.3.3.m3.2.2.2.3.3" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3.3.cmml">∪</mo></msub><mo id="Thmtheorem4.p1.3.3.m3.2.2.2.2" xref="Thmtheorem4.p1.3.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.cmml"><mo id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.2" stretchy="false" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow 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xref="Thmtheorem4.p1.3.3.m3.2.2"><eq id="Thmtheorem4.p1.3.3.m3.2.2.4.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.4"></eq><apply id="Thmtheorem4.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1"><times id="Thmtheorem4.p1.3.3.m3.1.1.1.2.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.2"></times><apply id="Thmtheorem4.p1.3.3.m3.1.1.1.3.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem4.p1.3.3.m3.1.1.1.3.1.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3">subscript</csymbol><ci id="Thmtheorem4.p1.3.3.m3.1.1.1.3.2.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3.2">𝐷</ci><intersect id="Thmtheorem4.p1.3.3.m3.1.1.1.3.3.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.3.3"></intersect></apply><apply id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="Thmtheorem4.p1.3.3.m3.1.1.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem4.p1.3.3.m3.2.2.2.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2"><times id="Thmtheorem4.p1.3.3.m3.2.2.2.2.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.2"></times><apply id="Thmtheorem4.p1.3.3.m3.2.2.2.3.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3"><csymbol cd="ambiguous" id="Thmtheorem4.p1.3.3.m3.2.2.2.3.1.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3">subscript</csymbol><ci id="Thmtheorem4.p1.3.3.m3.2.2.2.3.2.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3.2">𝐷</ci><union id="Thmtheorem4.p1.3.3.m3.2.2.2.3.3.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.3.3"></union></apply><apply id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1"><csymbol cd="latexml" id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.1.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.2.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.2">𝐴</ci><ci id="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.3.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.2.1.1.1.3">ℬ</ci></apply></apply></apply><apply id="Thmtheorem4.p1.3.3.m3.2.2c.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2"><eq id="Thmtheorem4.p1.3.3.m3.2.2.5.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2.5"></eq><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem4.p1.3.3.m3.2.2.2.cmml" id="Thmtheorem4.p1.3.3.m3.2.2d.cmml" xref="Thmtheorem4.p1.3.3.m3.2.2"></share><cn id="Thmtheorem4.p1.3.3.m3.2.2.6.cmml" type="integer" xref="Thmtheorem4.p1.3.3.m3.2.2.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.3.3.m3.2c">D_{\cap}(A\mid\mathcal{B})=D_{\cup}(A\mid\mathcal{B})=0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.3.3.m3.2d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.1">We have the following obvious inequality, which in general does not need to be tight (Fact <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem4" title="Fact 4. ‣ 2.1 Definitions and notation ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4</span></a> offers a trivial example).</p> </div> <div class="ltx_theorem ltx_theorem_fact" id="Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem5.1.1.1">Fact 5</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem5.p1"> <p class="ltx_p" id="Thmtheorem5.p1.1"><math alttext="D(A\mid\mathcal{B})\geq D_{\cap}(A\mid\mathcal{B})+D_{\cup}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="Thmtheorem5.p1.1.m1.3"><semantics id="Thmtheorem5.p1.1.m1.3a"><mrow id="Thmtheorem5.p1.1.m1.3.3" xref="Thmtheorem5.p1.1.m1.3.3.cmml"><mrow id="Thmtheorem5.p1.1.m1.1.1.1" xref="Thmtheorem5.p1.1.m1.1.1.1.cmml"><mi id="Thmtheorem5.p1.1.m1.1.1.1.3" xref="Thmtheorem5.p1.1.m1.1.1.1.3.cmml">D</mi><mo id="Thmtheorem5.p1.1.m1.1.1.1.2" xref="Thmtheorem5.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem5.p1.1.m1.1.1.1.1.1" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmtheorem5.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.2" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.1" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.3" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem5.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.1.m1.3.3.4" xref="Thmtheorem5.p1.1.m1.3.3.4.cmml">≥</mo><mrow id="Thmtheorem5.p1.1.m1.3.3.3" xref="Thmtheorem5.p1.1.m1.3.3.3.cmml"><mrow id="Thmtheorem5.p1.1.m1.2.2.2.1" xref="Thmtheorem5.p1.1.m1.2.2.2.1.cmml"><msub id="Thmtheorem5.p1.1.m1.2.2.2.1.3" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3.cmml"><mi id="Thmtheorem5.p1.1.m1.2.2.2.1.3.2" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3.2.cmml">D</mi><mo id="Thmtheorem5.p1.1.m1.2.2.2.1.3.3" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem5.p1.1.m1.2.2.2.1.2" xref="Thmtheorem5.p1.1.m1.2.2.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.cmml"><mo id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.2" stretchy="false" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.cmml"><mi id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.2" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.1" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.3" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.1.m1.3.3.3.3" xref="Thmtheorem5.p1.1.m1.3.3.3.3.cmml">+</mo><mrow id="Thmtheorem5.p1.1.m1.3.3.3.2" xref="Thmtheorem5.p1.1.m1.3.3.3.2.cmml"><msub id="Thmtheorem5.p1.1.m1.3.3.3.2.3" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3.cmml"><mi id="Thmtheorem5.p1.1.m1.3.3.3.2.3.2" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3.2.cmml">D</mi><mo id="Thmtheorem5.p1.1.m1.3.3.3.2.3.3" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3.3.cmml">∪</mo></msub><mo id="Thmtheorem5.p1.1.m1.3.3.3.2.2" xref="Thmtheorem5.p1.1.m1.3.3.3.2.2.cmml">⁢</mo><mrow id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.cmml"><mo id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.2" stretchy="false" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.cmml">(</mo><mrow id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.cmml"><mi id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.2" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.2.cmml">A</mi><mo id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.1" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.3" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.3" stretchy="false" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.1.m1.3b"><apply id="Thmtheorem5.p1.1.m1.3.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3"><geq id="Thmtheorem5.p1.1.m1.3.3.4.cmml" xref="Thmtheorem5.p1.1.m1.3.3.4"></geq><apply id="Thmtheorem5.p1.1.m1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1"><times id="Thmtheorem5.p1.1.m1.1.1.1.2.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.2"></times><ci id="Thmtheorem5.p1.1.m1.1.1.1.3.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.3">𝐷</ci><apply id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmtheorem5.p1.1.m1.1.1.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem5.p1.1.m1.3.3.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3"><plus id="Thmtheorem5.p1.1.m1.3.3.3.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.3"></plus><apply id="Thmtheorem5.p1.1.m1.2.2.2.1.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1"><times id="Thmtheorem5.p1.1.m1.2.2.2.1.2.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.2"></times><apply id="Thmtheorem5.p1.1.m1.2.2.2.1.3.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="Thmtheorem5.p1.1.m1.2.2.2.1.3.1.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3">subscript</csymbol><ci id="Thmtheorem5.p1.1.m1.2.2.2.1.3.2.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3.2">𝐷</ci><intersect id="Thmtheorem5.p1.1.m1.2.2.2.1.3.3.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.3.3"></intersect></apply><apply id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1"><csymbol cd="latexml" id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.2.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.3.cmml" xref="Thmtheorem5.p1.1.m1.2.2.2.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem5.p1.1.m1.3.3.3.2.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2"><times id="Thmtheorem5.p1.1.m1.3.3.3.2.2.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.2"></times><apply id="Thmtheorem5.p1.1.m1.3.3.3.2.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem5.p1.1.m1.3.3.3.2.3.1.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3">subscript</csymbol><ci id="Thmtheorem5.p1.1.m1.3.3.3.2.3.2.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3.2">𝐷</ci><union id="Thmtheorem5.p1.1.m1.3.3.3.2.3.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.3.3"></union></apply><apply id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1"><csymbol cd="latexml" id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.2.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.2">𝐴</ci><ci id="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.3.cmml" xref="Thmtheorem5.p1.1.m1.3.3.3.2.1.1.1.3">ℬ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.1.m1.3c">D(A\mid\mathcal{B})\geq D_{\cap}(A\mid\mathcal{B})+D_{\cup}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.1.m1.3d">italic_D ( italic_A ∣ caligraphic_B ) ≥ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) + italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmtheorem5.p1.1.1">.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.8">When the ambient space <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><mi id="S2.SS1.p5.1.m1.1.1" mathvariant="normal" xref="S2.SS1.p5.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><ci id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">roman_Γ</annotation></semantics></math> is clear from the context, we let <math alttext="E^{c}\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><mrow id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml"><msup id="S2.SS1.p5.2.m2.1.1.2" xref="S2.SS1.p5.2.m2.1.1.2.cmml"><mi id="S2.SS1.p5.2.m2.1.1.2.2" xref="S2.SS1.p5.2.m2.1.1.2.2.cmml">E</mi><mi id="S2.SS1.p5.2.m2.1.1.2.3" xref="S2.SS1.p5.2.m2.1.1.2.3.cmml">c</mi></msup><mo id="S2.SS1.p5.2.m2.1.1.1" xref="S2.SS1.p5.2.m2.1.1.1.cmml">⊆</mo><mi id="S2.SS1.p5.2.m2.1.1.3" mathvariant="normal" xref="S2.SS1.p5.2.m2.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><apply id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1"><subset id="S2.SS1.p5.2.m2.1.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1.1"></subset><apply id="S2.SS1.p5.2.m2.1.1.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.2.1.cmml" xref="S2.SS1.p5.2.m2.1.1.2">superscript</csymbol><ci id="S2.SS1.p5.2.m2.1.1.2.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2.2">𝐸</ci><ci id="S2.SS1.p5.2.m2.1.1.2.3.cmml" xref="S2.SS1.p5.2.m2.1.1.2.3">𝑐</ci></apply><ci id="S2.SS1.p5.2.m2.1.1.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">E^{c}\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">italic_E start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ⊆ roman_Γ</annotation></semantics></math> denote the complement of a set <math alttext="E\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><mrow id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml"><mi id="S2.SS1.p5.3.m3.1.1.2" xref="S2.SS1.p5.3.m3.1.1.2.cmml">E</mi><mo id="S2.SS1.p5.3.m3.1.1.1" xref="S2.SS1.p5.3.m3.1.1.1.cmml">⊆</mo><mi id="S2.SS1.p5.3.m3.1.1.3" mathvariant="normal" xref="S2.SS1.p5.3.m3.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><apply id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1"><subset id="S2.SS1.p5.3.m3.1.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1.1"></subset><ci id="S2.SS1.p5.3.m3.1.1.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2">𝐸</ci><ci id="S2.SS1.p5.3.m3.1.1.3.cmml" xref="S2.SS1.p5.3.m3.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">E\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_E ⊆ roman_Γ</annotation></semantics></math>. For convenience, for a set <math alttext="U\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><mrow id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml"><mi id="S2.SS1.p5.4.m4.1.1.2" xref="S2.SS1.p5.4.m4.1.1.2.cmml">U</mi><mo id="S2.SS1.p5.4.m4.1.1.1" xref="S2.SS1.p5.4.m4.1.1.1.cmml">⊆</mo><mi id="S2.SS1.p5.4.m4.1.1.3" mathvariant="normal" xref="S2.SS1.p5.4.m4.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><apply id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1"><subset id="S2.SS1.p5.4.m4.1.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1.1"></subset><ci id="S2.SS1.p5.4.m4.1.1.2.cmml" xref="S2.SS1.p5.4.m4.1.1.2">𝑈</ci><ci id="S2.SS1.p5.4.m4.1.1.3.cmml" xref="S2.SS1.p5.4.m4.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">U\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">italic_U ⊆ roman_Γ</annotation></semantics></math>, we use <math alttext="B_{U}" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m5.1"><semantics id="S2.SS1.p5.5.m5.1a"><msub id="S2.SS1.p5.5.m5.1.1" xref="S2.SS1.p5.5.m5.1.1.cmml"><mi id="S2.SS1.p5.5.m5.1.1.2" xref="S2.SS1.p5.5.m5.1.1.2.cmml">B</mi><mi id="S2.SS1.p5.5.m5.1.1.3" xref="S2.SS1.p5.5.m5.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m5.1b"><apply id="S2.SS1.p5.5.m5.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p5.5.m5.1.1.2.cmml" xref="S2.SS1.p5.5.m5.1.1.2">𝐵</ci><ci id="S2.SS1.p5.5.m5.1.1.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m5.1c">B_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m5.1d">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> as a shorthand for <math alttext="B\cap U" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m6.1"><semantics id="S2.SS1.p5.6.m6.1a"><mrow id="S2.SS1.p5.6.m6.1.1" xref="S2.SS1.p5.6.m6.1.1.cmml"><mi id="S2.SS1.p5.6.m6.1.1.2" xref="S2.SS1.p5.6.m6.1.1.2.cmml">B</mi><mo id="S2.SS1.p5.6.m6.1.1.1" xref="S2.SS1.p5.6.m6.1.1.1.cmml">∩</mo><mi id="S2.SS1.p5.6.m6.1.1.3" xref="S2.SS1.p5.6.m6.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.6.m6.1b"><apply id="S2.SS1.p5.6.m6.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1"><intersect id="S2.SS1.p5.6.m6.1.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1.1"></intersect><ci id="S2.SS1.p5.6.m6.1.1.2.cmml" xref="S2.SS1.p5.6.m6.1.1.2">𝐵</ci><ci id="S2.SS1.p5.6.m6.1.1.3.cmml" xref="S2.SS1.p5.6.m6.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.6.m6.1c">B\cap U</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.6.m6.1d">italic_B ∩ italic_U</annotation></semantics></math>. For a family of sets <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p5.7.m7.1"><semantics id="S2.SS1.p5.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.7.m7.1.1" xref="S2.SS1.p5.7.m7.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.7.m7.1b"><ci id="S2.SS1.p5.7.m7.1.1.cmml" xref="S2.SS1.p5.7.m7.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.7.m7.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.7.m7.1d">caligraphic_B</annotation></semantics></math>, we let <math alttext="\mathcal{B}_{U}\stackrel{{\scriptstyle\rm def}}{{=}}\{B_{U}\mid B\in\mathcal{B}\}" class="ltx_Math" display="inline" id="S2.SS1.p5.8.m8.2"><semantics id="S2.SS1.p5.8.m8.2a"><mrow id="S2.SS1.p5.8.m8.2.2" xref="S2.SS1.p5.8.m8.2.2.cmml"><msub id="S2.SS1.p5.8.m8.2.2.4" xref="S2.SS1.p5.8.m8.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.8.m8.2.2.4.2" xref="S2.SS1.p5.8.m8.2.2.4.2.cmml">ℬ</mi><mi id="S2.SS1.p5.8.m8.2.2.4.3" xref="S2.SS1.p5.8.m8.2.2.4.3.cmml">U</mi></msub><mover id="S2.SS1.p5.8.m8.2.2.3" xref="S2.SS1.p5.8.m8.2.2.3.cmml"><mo id="S2.SS1.p5.8.m8.2.2.3.2" xref="S2.SS1.p5.8.m8.2.2.3.2.cmml">=</mo><mi id="S2.SS1.p5.8.m8.2.2.3.3" xref="S2.SS1.p5.8.m8.2.2.3.3.cmml">def</mi></mover><mrow id="S2.SS1.p5.8.m8.2.2.2.2" xref="S2.SS1.p5.8.m8.2.2.2.3.cmml"><mo id="S2.SS1.p5.8.m8.2.2.2.2.3" stretchy="false" xref="S2.SS1.p5.8.m8.2.2.2.3.1.cmml">{</mo><msub id="S2.SS1.p5.8.m8.1.1.1.1.1" xref="S2.SS1.p5.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.8.m8.1.1.1.1.1.2" xref="S2.SS1.p5.8.m8.1.1.1.1.1.2.cmml">B</mi><mi id="S2.SS1.p5.8.m8.1.1.1.1.1.3" xref="S2.SS1.p5.8.m8.1.1.1.1.1.3.cmml">U</mi></msub><mo fence="true" id="S2.SS1.p5.8.m8.2.2.2.2.4" lspace="0em" rspace="0em" xref="S2.SS1.p5.8.m8.2.2.2.3.1.cmml">∣</mo><mrow id="S2.SS1.p5.8.m8.2.2.2.2.2" xref="S2.SS1.p5.8.m8.2.2.2.2.2.cmml"><mi id="S2.SS1.p5.8.m8.2.2.2.2.2.2" xref="S2.SS1.p5.8.m8.2.2.2.2.2.2.cmml">B</mi><mo id="S2.SS1.p5.8.m8.2.2.2.2.2.1" xref="S2.SS1.p5.8.m8.2.2.2.2.2.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.8.m8.2.2.2.2.2.3" xref="S2.SS1.p5.8.m8.2.2.2.2.2.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p5.8.m8.2.2.2.2.5" stretchy="false" xref="S2.SS1.p5.8.m8.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.8.m8.2b"><apply id="S2.SS1.p5.8.m8.2.2.cmml" xref="S2.SS1.p5.8.m8.2.2"><apply id="S2.SS1.p5.8.m8.2.2.3.cmml" xref="S2.SS1.p5.8.m8.2.2.3"><csymbol cd="ambiguous" id="S2.SS1.p5.8.m8.2.2.3.1.cmml" xref="S2.SS1.p5.8.m8.2.2.3">superscript</csymbol><eq id="S2.SS1.p5.8.m8.2.2.3.2.cmml" xref="S2.SS1.p5.8.m8.2.2.3.2"></eq><ci id="S2.SS1.p5.8.m8.2.2.3.3.cmml" xref="S2.SS1.p5.8.m8.2.2.3.3">def</ci></apply><apply id="S2.SS1.p5.8.m8.2.2.4.cmml" xref="S2.SS1.p5.8.m8.2.2.4"><csymbol cd="ambiguous" id="S2.SS1.p5.8.m8.2.2.4.1.cmml" xref="S2.SS1.p5.8.m8.2.2.4">subscript</csymbol><ci id="S2.SS1.p5.8.m8.2.2.4.2.cmml" xref="S2.SS1.p5.8.m8.2.2.4.2">ℬ</ci><ci id="S2.SS1.p5.8.m8.2.2.4.3.cmml" xref="S2.SS1.p5.8.m8.2.2.4.3">𝑈</ci></apply><apply id="S2.SS1.p5.8.m8.2.2.2.3.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2"><csymbol cd="latexml" id="S2.SS1.p5.8.m8.2.2.2.3.1.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2.3">conditional-set</csymbol><apply id="S2.SS1.p5.8.m8.1.1.1.1.1.cmml" xref="S2.SS1.p5.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS1.p5.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p5.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS1.p5.8.m8.1.1.1.1.1.2">𝐵</ci><ci id="S2.SS1.p5.8.m8.1.1.1.1.1.3.cmml" xref="S2.SS1.p5.8.m8.1.1.1.1.1.3">𝑈</ci></apply><apply id="S2.SS1.p5.8.m8.2.2.2.2.2.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2.2"><in id="S2.SS1.p5.8.m8.2.2.2.2.2.1.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2.2.1"></in><ci id="S2.SS1.p5.8.m8.2.2.2.2.2.2.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2.2.2">𝐵</ci><ci id="S2.SS1.p5.8.m8.2.2.2.2.2.3.cmml" xref="S2.SS1.p5.8.m8.2.2.2.2.2.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.8.m8.2c">\mathcal{B}_{U}\stackrel{{\scriptstyle\rm def}}{{=}}\{B_{U}\mid B\in\mathcal{B}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.8.m8.2d">caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∣ italic_B ∈ caligraphic_B }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.11">Let <math alttext="A_{1},\ldots,A_{t}" class="ltx_Math" display="inline" id="S2.SS1.p6.1.m1.3"><semantics id="S2.SS1.p6.1.m1.3a"><mrow id="S2.SS1.p6.1.m1.3.3.2" xref="S2.SS1.p6.1.m1.3.3.3.cmml"><msub id="S2.SS1.p6.1.m1.2.2.1.1" xref="S2.SS1.p6.1.m1.2.2.1.1.cmml"><mi id="S2.SS1.p6.1.m1.2.2.1.1.2" xref="S2.SS1.p6.1.m1.2.2.1.1.2.cmml">A</mi><mn id="S2.SS1.p6.1.m1.2.2.1.1.3" xref="S2.SS1.p6.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p6.1.m1.3.3.2.3" xref="S2.SS1.p6.1.m1.3.3.3.cmml">,</mo><mi id="S2.SS1.p6.1.m1.1.1" mathvariant="normal" xref="S2.SS1.p6.1.m1.1.1.cmml">…</mi><mo id="S2.SS1.p6.1.m1.3.3.2.4" xref="S2.SS1.p6.1.m1.3.3.3.cmml">,</mo><msub id="S2.SS1.p6.1.m1.3.3.2.2" xref="S2.SS1.p6.1.m1.3.3.2.2.cmml"><mi id="S2.SS1.p6.1.m1.3.3.2.2.2" xref="S2.SS1.p6.1.m1.3.3.2.2.2.cmml">A</mi><mi id="S2.SS1.p6.1.m1.3.3.2.2.3" xref="S2.SS1.p6.1.m1.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.1.m1.3b"><list id="S2.SS1.p6.1.m1.3.3.3.cmml" xref="S2.SS1.p6.1.m1.3.3.2"><apply id="S2.SS1.p6.1.m1.2.2.1.1.cmml" xref="S2.SS1.p6.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.1.m1.2.2.1.1.1.cmml" xref="S2.SS1.p6.1.m1.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p6.1.m1.2.2.1.1.2.cmml" xref="S2.SS1.p6.1.m1.2.2.1.1.2">𝐴</ci><cn id="S2.SS1.p6.1.m1.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p6.1.m1.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p6.1.m1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1">…</ci><apply id="S2.SS1.p6.1.m1.3.3.2.2.cmml" xref="S2.SS1.p6.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p6.1.m1.3.3.2.2.1.cmml" xref="S2.SS1.p6.1.m1.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p6.1.m1.3.3.2.2.2.cmml" xref="S2.SS1.p6.1.m1.3.3.2.2.2">𝐴</ci><ci id="S2.SS1.p6.1.m1.3.3.2.2.3.cmml" xref="S2.SS1.p6.1.m1.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.1.m1.3c">A_{1},\ldots,A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.1.m1.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> be a sequence of sets that generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p6.2.m2.1"><semantics id="S2.SS1.p6.2.m2.1a"><mi id="S2.SS1.p6.2.m2.1.1" xref="S2.SS1.p6.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.2.m2.1b"><ci id="S2.SS1.p6.2.m2.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.2.m2.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p6.3.m3.1"><semantics id="S2.SS1.p6.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.3.m3.1.1" xref="S2.SS1.p6.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.3.m3.1b"><ci id="S2.SS1.p6.3.m3.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.3.m3.1d">caligraphic_B</annotation></semantics></math>, where <math alttext="|\mathcal{B}|=m" class="ltx_Math" display="inline" id="S2.SS1.p6.4.m4.1"><semantics id="S2.SS1.p6.4.m4.1a"><mrow id="S2.SS1.p6.4.m4.1.2" xref="S2.SS1.p6.4.m4.1.2.cmml"><mrow id="S2.SS1.p6.4.m4.1.2.2.2" xref="S2.SS1.p6.4.m4.1.2.2.1.cmml"><mo id="S2.SS1.p6.4.m4.1.2.2.2.1" stretchy="false" xref="S2.SS1.p6.4.m4.1.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.4.m4.1.1" xref="S2.SS1.p6.4.m4.1.1.cmml">ℬ</mi><mo id="S2.SS1.p6.4.m4.1.2.2.2.2" stretchy="false" xref="S2.SS1.p6.4.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS1.p6.4.m4.1.2.1" xref="S2.SS1.p6.4.m4.1.2.1.cmml">=</mo><mi id="S2.SS1.p6.4.m4.1.2.3" xref="S2.SS1.p6.4.m4.1.2.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.4.m4.1b"><apply id="S2.SS1.p6.4.m4.1.2.cmml" xref="S2.SS1.p6.4.m4.1.2"><eq id="S2.SS1.p6.4.m4.1.2.1.cmml" xref="S2.SS1.p6.4.m4.1.2.1"></eq><apply id="S2.SS1.p6.4.m4.1.2.2.1.cmml" xref="S2.SS1.p6.4.m4.1.2.2.2"><abs id="S2.SS1.p6.4.m4.1.2.2.1.1.cmml" xref="S2.SS1.p6.4.m4.1.2.2.2.1"></abs><ci id="S2.SS1.p6.4.m4.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1">ℬ</ci></apply><ci id="S2.SS1.p6.4.m4.1.2.3.cmml" xref="S2.SS1.p6.4.m4.1.2.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.4.m4.1c">|\mathcal{B}|=m</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.4.m4.1d">| caligraphic_B | = italic_m</annotation></semantics></math>. It will be convenient in some inductive proofs to consider the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p6.11.1">extended sequence</em> <math alttext="B_{1},\ldots,B_{m},A_{1},\ldots,A_{t}" class="ltx_Math" display="inline" id="S2.SS1.p6.5.m5.6"><semantics id="S2.SS1.p6.5.m5.6a"><mrow id="S2.SS1.p6.5.m5.6.6.4" xref="S2.SS1.p6.5.m5.6.6.5.cmml"><msub id="S2.SS1.p6.5.m5.3.3.1.1" xref="S2.SS1.p6.5.m5.3.3.1.1.cmml"><mi id="S2.SS1.p6.5.m5.3.3.1.1.2" xref="S2.SS1.p6.5.m5.3.3.1.1.2.cmml">B</mi><mn id="S2.SS1.p6.5.m5.3.3.1.1.3" xref="S2.SS1.p6.5.m5.3.3.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p6.5.m5.6.6.4.5" xref="S2.SS1.p6.5.m5.6.6.5.cmml">,</mo><mi id="S2.SS1.p6.5.m5.1.1" mathvariant="normal" xref="S2.SS1.p6.5.m5.1.1.cmml">…</mi><mo id="S2.SS1.p6.5.m5.6.6.4.6" xref="S2.SS1.p6.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS1.p6.5.m5.4.4.2.2" xref="S2.SS1.p6.5.m5.4.4.2.2.cmml"><mi id="S2.SS1.p6.5.m5.4.4.2.2.2" xref="S2.SS1.p6.5.m5.4.4.2.2.2.cmml">B</mi><mi id="S2.SS1.p6.5.m5.4.4.2.2.3" xref="S2.SS1.p6.5.m5.4.4.2.2.3.cmml">m</mi></msub><mo id="S2.SS1.p6.5.m5.6.6.4.7" xref="S2.SS1.p6.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS1.p6.5.m5.5.5.3.3" xref="S2.SS1.p6.5.m5.5.5.3.3.cmml"><mi id="S2.SS1.p6.5.m5.5.5.3.3.2" xref="S2.SS1.p6.5.m5.5.5.3.3.2.cmml">A</mi><mn id="S2.SS1.p6.5.m5.5.5.3.3.3" xref="S2.SS1.p6.5.m5.5.5.3.3.3.cmml">1</mn></msub><mo id="S2.SS1.p6.5.m5.6.6.4.8" xref="S2.SS1.p6.5.m5.6.6.5.cmml">,</mo><mi id="S2.SS1.p6.5.m5.2.2" mathvariant="normal" xref="S2.SS1.p6.5.m5.2.2.cmml">…</mi><mo id="S2.SS1.p6.5.m5.6.6.4.9" xref="S2.SS1.p6.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS1.p6.5.m5.6.6.4.4" xref="S2.SS1.p6.5.m5.6.6.4.4.cmml"><mi id="S2.SS1.p6.5.m5.6.6.4.4.2" xref="S2.SS1.p6.5.m5.6.6.4.4.2.cmml">A</mi><mi id="S2.SS1.p6.5.m5.6.6.4.4.3" xref="S2.SS1.p6.5.m5.6.6.4.4.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.5.m5.6b"><list id="S2.SS1.p6.5.m5.6.6.5.cmml" xref="S2.SS1.p6.5.m5.6.6.4"><apply id="S2.SS1.p6.5.m5.3.3.1.1.cmml" xref="S2.SS1.p6.5.m5.3.3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.3.3.1.1.1.cmml" xref="S2.SS1.p6.5.m5.3.3.1.1">subscript</csymbol><ci id="S2.SS1.p6.5.m5.3.3.1.1.2.cmml" xref="S2.SS1.p6.5.m5.3.3.1.1.2">𝐵</ci><cn id="S2.SS1.p6.5.m5.3.3.1.1.3.cmml" type="integer" xref="S2.SS1.p6.5.m5.3.3.1.1.3">1</cn></apply><ci id="S2.SS1.p6.5.m5.1.1.cmml" xref="S2.SS1.p6.5.m5.1.1">…</ci><apply id="S2.SS1.p6.5.m5.4.4.2.2.cmml" xref="S2.SS1.p6.5.m5.4.4.2.2"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.4.4.2.2.1.cmml" xref="S2.SS1.p6.5.m5.4.4.2.2">subscript</csymbol><ci id="S2.SS1.p6.5.m5.4.4.2.2.2.cmml" xref="S2.SS1.p6.5.m5.4.4.2.2.2">𝐵</ci><ci id="S2.SS1.p6.5.m5.4.4.2.2.3.cmml" xref="S2.SS1.p6.5.m5.4.4.2.2.3">𝑚</ci></apply><apply id="S2.SS1.p6.5.m5.5.5.3.3.cmml" xref="S2.SS1.p6.5.m5.5.5.3.3"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.5.5.3.3.1.cmml" xref="S2.SS1.p6.5.m5.5.5.3.3">subscript</csymbol><ci id="S2.SS1.p6.5.m5.5.5.3.3.2.cmml" xref="S2.SS1.p6.5.m5.5.5.3.3.2">𝐴</ci><cn id="S2.SS1.p6.5.m5.5.5.3.3.3.cmml" type="integer" xref="S2.SS1.p6.5.m5.5.5.3.3.3">1</cn></apply><ci id="S2.SS1.p6.5.m5.2.2.cmml" xref="S2.SS1.p6.5.m5.2.2">…</ci><apply id="S2.SS1.p6.5.m5.6.6.4.4.cmml" xref="S2.SS1.p6.5.m5.6.6.4.4"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.6.6.4.4.1.cmml" xref="S2.SS1.p6.5.m5.6.6.4.4">subscript</csymbol><ci id="S2.SS1.p6.5.m5.6.6.4.4.2.cmml" xref="S2.SS1.p6.5.m5.6.6.4.4.2">𝐴</ci><ci id="S2.SS1.p6.5.m5.6.6.4.4.3.cmml" xref="S2.SS1.p6.5.m5.6.6.4.4.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.5.m5.6c">B_{1},\ldots,B_{m},A_{1},\ldots,A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.5.m5.6d">italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> that includes as a prefix the generators from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p6.6.m6.1"><semantics id="S2.SS1.p6.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.6.m6.1.1" xref="S2.SS1.p6.6.m6.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.6.m6.1b"><ci id="S2.SS1.p6.6.m6.1.1.cmml" xref="S2.SS1.p6.6.m6.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.6.m6.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.6.m6.1d">caligraphic_B</annotation></semantics></math>. The particular order of the sets <math alttext="B_{i}" class="ltx_Math" display="inline" id="S2.SS1.p6.7.m7.1"><semantics id="S2.SS1.p6.7.m7.1a"><msub id="S2.SS1.p6.7.m7.1.1" xref="S2.SS1.p6.7.m7.1.1.cmml"><mi id="S2.SS1.p6.7.m7.1.1.2" xref="S2.SS1.p6.7.m7.1.1.2.cmml">B</mi><mi id="S2.SS1.p6.7.m7.1.1.3" xref="S2.SS1.p6.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.7.m7.1b"><apply id="S2.SS1.p6.7.m7.1.1.cmml" xref="S2.SS1.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.7.m7.1.1.1.cmml" xref="S2.SS1.p6.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p6.7.m7.1.1.2.cmml" xref="S2.SS1.p6.7.m7.1.1.2">𝐵</ci><ci id="S2.SS1.p6.7.m7.1.1.3.cmml" xref="S2.SS1.p6.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.7.m7.1c">B_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.7.m7.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is not relevant. While the extended sequence has length <math alttext="m+t" class="ltx_Math" display="inline" id="S2.SS1.p6.8.m8.1"><semantics id="S2.SS1.p6.8.m8.1a"><mrow id="S2.SS1.p6.8.m8.1.1" xref="S2.SS1.p6.8.m8.1.1.cmml"><mi id="S2.SS1.p6.8.m8.1.1.2" xref="S2.SS1.p6.8.m8.1.1.2.cmml">m</mi><mo id="S2.SS1.p6.8.m8.1.1.1" xref="S2.SS1.p6.8.m8.1.1.1.cmml">+</mo><mi id="S2.SS1.p6.8.m8.1.1.3" xref="S2.SS1.p6.8.m8.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.8.m8.1b"><apply id="S2.SS1.p6.8.m8.1.1.cmml" xref="S2.SS1.p6.8.m8.1.1"><plus id="S2.SS1.p6.8.m8.1.1.1.cmml" xref="S2.SS1.p6.8.m8.1.1.1"></plus><ci id="S2.SS1.p6.8.m8.1.1.2.cmml" xref="S2.SS1.p6.8.m8.1.1.2">𝑚</ci><ci id="S2.SS1.p6.8.m8.1.1.3.cmml" xref="S2.SS1.p6.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.8.m8.1c">m+t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.8.m8.1d">italic_m + italic_t</annotation></semantics></math>, we will refer to it as a sequence of complexity <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p6.9.m9.1"><semantics id="S2.SS1.p6.9.m9.1a"><mi id="S2.SS1.p6.9.m9.1.1" xref="S2.SS1.p6.9.m9.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.9.m9.1b"><ci id="S2.SS1.p6.9.m9.1.1.cmml" xref="S2.SS1.p6.9.m9.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.9.m9.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.9.m9.1d">italic_t</annotation></semantics></math>. Similarly, if the number of intersections employed in the definition of the sequence is <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p6.10.m10.1"><semantics id="S2.SS1.p6.10.m10.1a"><mi id="S2.SS1.p6.10.m10.1.1" xref="S2.SS1.p6.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.10.m10.1b"><ci id="S2.SS1.p6.10.m10.1.1.cmml" xref="S2.SS1.p6.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.10.m10.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.10.m10.1d">italic_k</annotation></semantics></math>, we say it has intersection complexity <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p6.11.m11.1"><semantics id="S2.SS1.p6.11.m11.1a"><mi id="S2.SS1.p6.11.m11.1.1" xref="S2.SS1.p6.11.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.11.m11.1b"><ci id="S2.SS1.p6.11.m11.1.1.cmml" xref="S2.SS1.p6.11.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.11.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.11.m11.1d">italic_k</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p7"> <p class="ltx_p" id="S2.SS1.p7.6">Given a construction of <math alttext="A" class="ltx_Math" display="inline" id="S2.SS1.p7.1.m1.1"><semantics id="S2.SS1.p7.1.m1.1a"><mi id="S2.SS1.p7.1.m1.1.1" xref="S2.SS1.p7.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.1.m1.1b"><ci id="S2.SS1.p7.1.m1.1.1.cmml" xref="S2.SS1.p7.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.1.m1.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p7.2.m2.1"><semantics id="S2.SS1.p7.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.2.m2.1.1" xref="S2.SS1.p7.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.2.m2.1b"><ci id="S2.SS1.p7.2.m2.1.1.cmml" xref="S2.SS1.p7.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.2.m2.1d">caligraphic_B</annotation></semantics></math> specified by a sequence <math alttext="A_{1},\ldots,A_{t}" class="ltx_Math" display="inline" id="S2.SS1.p7.3.m3.3"><semantics id="S2.SS1.p7.3.m3.3a"><mrow id="S2.SS1.p7.3.m3.3.3.2" xref="S2.SS1.p7.3.m3.3.3.3.cmml"><msub id="S2.SS1.p7.3.m3.2.2.1.1" xref="S2.SS1.p7.3.m3.2.2.1.1.cmml"><mi id="S2.SS1.p7.3.m3.2.2.1.1.2" xref="S2.SS1.p7.3.m3.2.2.1.1.2.cmml">A</mi><mn id="S2.SS1.p7.3.m3.2.2.1.1.3" xref="S2.SS1.p7.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p7.3.m3.3.3.2.3" xref="S2.SS1.p7.3.m3.3.3.3.cmml">,</mo><mi id="S2.SS1.p7.3.m3.1.1" mathvariant="normal" xref="S2.SS1.p7.3.m3.1.1.cmml">…</mi><mo id="S2.SS1.p7.3.m3.3.3.2.4" xref="S2.SS1.p7.3.m3.3.3.3.cmml">,</mo><msub id="S2.SS1.p7.3.m3.3.3.2.2" xref="S2.SS1.p7.3.m3.3.3.2.2.cmml"><mi id="S2.SS1.p7.3.m3.3.3.2.2.2" xref="S2.SS1.p7.3.m3.3.3.2.2.2.cmml">A</mi><mi id="S2.SS1.p7.3.m3.3.3.2.2.3" xref="S2.SS1.p7.3.m3.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.3.m3.3b"><list id="S2.SS1.p7.3.m3.3.3.3.cmml" xref="S2.SS1.p7.3.m3.3.3.2"><apply id="S2.SS1.p7.3.m3.2.2.1.1.cmml" xref="S2.SS1.p7.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.3.m3.2.2.1.1.1.cmml" xref="S2.SS1.p7.3.m3.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p7.3.m3.2.2.1.1.2.cmml" xref="S2.SS1.p7.3.m3.2.2.1.1.2">𝐴</ci><cn id="S2.SS1.p7.3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p7.3.m3.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p7.3.m3.1.1.cmml" xref="S2.SS1.p7.3.m3.1.1">…</ci><apply id="S2.SS1.p7.3.m3.3.3.2.2.cmml" xref="S2.SS1.p7.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p7.3.m3.3.3.2.2.1.cmml" xref="S2.SS1.p7.3.m3.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p7.3.m3.3.3.2.2.2.cmml" xref="S2.SS1.p7.3.m3.3.3.2.2.2">𝐴</ci><ci id="S2.SS1.p7.3.m3.3.3.2.2.3.cmml" xref="S2.SS1.p7.3.m3.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.3.m3.3c">A_{1},\ldots,A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.3.m3.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and its corresponding union and intersection operations, we let <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.SS1.p7.4.m4.1"><semantics id="S2.SS1.p7.4.m4.1a"><mi id="S2.SS1.p7.4.m4.1.1" mathvariant="normal" xref="S2.SS1.p7.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.4.m4.1b"><ci id="S2.SS1.p7.4.m4.1.1.cmml" xref="S2.SS1.p7.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.4.m4.1d">roman_Λ</annotation></semantics></math> be the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p7.6.1">set of intersections</em> in the sequence, where we represent an intersection operation <math alttext="A_{\ell}=A_{i}\cap A_{j}" class="ltx_Math" display="inline" id="S2.SS1.p7.5.m5.1"><semantics id="S2.SS1.p7.5.m5.1a"><mrow id="S2.SS1.p7.5.m5.1.1" xref="S2.SS1.p7.5.m5.1.1.cmml"><msub id="S2.SS1.p7.5.m5.1.1.2" xref="S2.SS1.p7.5.m5.1.1.2.cmml"><mi id="S2.SS1.p7.5.m5.1.1.2.2" xref="S2.SS1.p7.5.m5.1.1.2.2.cmml">A</mi><mi id="S2.SS1.p7.5.m5.1.1.2.3" mathvariant="normal" xref="S2.SS1.p7.5.m5.1.1.2.3.cmml">ℓ</mi></msub><mo id="S2.SS1.p7.5.m5.1.1.1" xref="S2.SS1.p7.5.m5.1.1.1.cmml">=</mo><mrow id="S2.SS1.p7.5.m5.1.1.3" xref="S2.SS1.p7.5.m5.1.1.3.cmml"><msub id="S2.SS1.p7.5.m5.1.1.3.2" xref="S2.SS1.p7.5.m5.1.1.3.2.cmml"><mi id="S2.SS1.p7.5.m5.1.1.3.2.2" xref="S2.SS1.p7.5.m5.1.1.3.2.2.cmml">A</mi><mi id="S2.SS1.p7.5.m5.1.1.3.2.3" xref="S2.SS1.p7.5.m5.1.1.3.2.3.cmml">i</mi></msub><mo id="S2.SS1.p7.5.m5.1.1.3.1" xref="S2.SS1.p7.5.m5.1.1.3.1.cmml">∩</mo><msub id="S2.SS1.p7.5.m5.1.1.3.3" xref="S2.SS1.p7.5.m5.1.1.3.3.cmml"><mi id="S2.SS1.p7.5.m5.1.1.3.3.2" xref="S2.SS1.p7.5.m5.1.1.3.3.2.cmml">A</mi><mi id="S2.SS1.p7.5.m5.1.1.3.3.3" xref="S2.SS1.p7.5.m5.1.1.3.3.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.5.m5.1b"><apply id="S2.SS1.p7.5.m5.1.1.cmml" xref="S2.SS1.p7.5.m5.1.1"><eq id="S2.SS1.p7.5.m5.1.1.1.cmml" xref="S2.SS1.p7.5.m5.1.1.1"></eq><apply id="S2.SS1.p7.5.m5.1.1.2.cmml" xref="S2.SS1.p7.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p7.5.m5.1.1.2.1.cmml" xref="S2.SS1.p7.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS1.p7.5.m5.1.1.2.2.cmml" xref="S2.SS1.p7.5.m5.1.1.2.2">𝐴</ci><ci id="S2.SS1.p7.5.m5.1.1.2.3.cmml" xref="S2.SS1.p7.5.m5.1.1.2.3">ℓ</ci></apply><apply id="S2.SS1.p7.5.m5.1.1.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3"><intersect id="S2.SS1.p7.5.m5.1.1.3.1.cmml" xref="S2.SS1.p7.5.m5.1.1.3.1"></intersect><apply id="S2.SS1.p7.5.m5.1.1.3.2.cmml" xref="S2.SS1.p7.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p7.5.m5.1.1.3.2.1.cmml" xref="S2.SS1.p7.5.m5.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p7.5.m5.1.1.3.2.2.cmml" xref="S2.SS1.p7.5.m5.1.1.3.2.2">𝐴</ci><ci id="S2.SS1.p7.5.m5.1.1.3.2.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3.2.3">𝑖</ci></apply><apply id="S2.SS1.p7.5.m5.1.1.3.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p7.5.m5.1.1.3.3.1.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p7.5.m5.1.1.3.3.2.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3.2">𝐴</ci><ci id="S2.SS1.p7.5.m5.1.1.3.3.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.5.m5.1c">A_{\ell}=A_{i}\cap A_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.5.m5.1d">italic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> by the pair <math alttext="(A_{i},A_{j})" class="ltx_Math" display="inline" id="S2.SS1.p7.6.m6.2"><semantics id="S2.SS1.p7.6.m6.2a"><mrow id="S2.SS1.p7.6.m6.2.2.2" xref="S2.SS1.p7.6.m6.2.2.3.cmml"><mo id="S2.SS1.p7.6.m6.2.2.2.3" stretchy="false" xref="S2.SS1.p7.6.m6.2.2.3.cmml">(</mo><msub id="S2.SS1.p7.6.m6.1.1.1.1" xref="S2.SS1.p7.6.m6.1.1.1.1.cmml"><mi id="S2.SS1.p7.6.m6.1.1.1.1.2" xref="S2.SS1.p7.6.m6.1.1.1.1.2.cmml">A</mi><mi id="S2.SS1.p7.6.m6.1.1.1.1.3" xref="S2.SS1.p7.6.m6.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p7.6.m6.2.2.2.4" xref="S2.SS1.p7.6.m6.2.2.3.cmml">,</mo><msub id="S2.SS1.p7.6.m6.2.2.2.2" xref="S2.SS1.p7.6.m6.2.2.2.2.cmml"><mi id="S2.SS1.p7.6.m6.2.2.2.2.2" xref="S2.SS1.p7.6.m6.2.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p7.6.m6.2.2.2.2.3" xref="S2.SS1.p7.6.m6.2.2.2.2.3.cmml">j</mi></msub><mo id="S2.SS1.p7.6.m6.2.2.2.5" stretchy="false" xref="S2.SS1.p7.6.m6.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.6.m6.2b"><interval closure="open" id="S2.SS1.p7.6.m6.2.2.3.cmml" xref="S2.SS1.p7.6.m6.2.2.2"><apply id="S2.SS1.p7.6.m6.1.1.1.1.cmml" xref="S2.SS1.p7.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.6.m6.1.1.1.1.1.cmml" xref="S2.SS1.p7.6.m6.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p7.6.m6.1.1.1.1.2.cmml" xref="S2.SS1.p7.6.m6.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p7.6.m6.1.1.1.1.3.cmml" xref="S2.SS1.p7.6.m6.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS1.p7.6.m6.2.2.2.2.cmml" xref="S2.SS1.p7.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p7.6.m6.2.2.2.2.1.cmml" xref="S2.SS1.p7.6.m6.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p7.6.m6.2.2.2.2.2.cmml" xref="S2.SS1.p7.6.m6.2.2.2.2.2">𝐴</ci><ci id="S2.SS1.p7.6.m6.2.2.2.2.3.cmml" xref="S2.SS1.p7.6.m6.2.2.2.2.3">𝑗</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.6.m6.2c">(A_{i},A_{j})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.6.m6.2d">( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p8"> <p class="ltx_p" id="S2.SS1.p8.13">For an ambient space <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p8.1.m1.1"><semantics id="S2.SS1.p8.1.m1.1a"><mi id="S2.SS1.p8.1.m1.1.1" mathvariant="normal" xref="S2.SS1.p8.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.1.m1.1b"><ci id="S2.SS1.p8.1.m1.1.1.cmml" xref="S2.SS1.p8.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.1.m1.1d">roman_Γ</annotation></semantics></math> and <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="S2.SS1.p8.2.m2.1"><semantics id="S2.SS1.p8.2.m2.1a"><mrow id="S2.SS1.p8.2.m2.1.2" xref="S2.SS1.p8.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.2.m2.1.2.2" xref="S2.SS1.p8.2.m2.1.2.2.cmml">ℬ</mi><mo id="S2.SS1.p8.2.m2.1.2.1" xref="S2.SS1.p8.2.m2.1.2.1.cmml">⊆</mo><mrow id="S2.SS1.p8.2.m2.1.2.3" xref="S2.SS1.p8.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.2.m2.1.2.3.2" xref="S2.SS1.p8.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="S2.SS1.p8.2.m2.1.2.3.1" xref="S2.SS1.p8.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p8.2.m2.1.2.3.3.2" xref="S2.SS1.p8.2.m2.1.2.3.cmml"><mo id="S2.SS1.p8.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p8.2.m2.1.2.3.cmml">(</mo><mi id="S2.SS1.p8.2.m2.1.1" mathvariant="normal" xref="S2.SS1.p8.2.m2.1.1.cmml">Γ</mi><mo id="S2.SS1.p8.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p8.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.2.m2.1b"><apply id="S2.SS1.p8.2.m2.1.2.cmml" xref="S2.SS1.p8.2.m2.1.2"><subset id="S2.SS1.p8.2.m2.1.2.1.cmml" xref="S2.SS1.p8.2.m2.1.2.1"></subset><ci id="S2.SS1.p8.2.m2.1.2.2.cmml" xref="S2.SS1.p8.2.m2.1.2.2">ℬ</ci><apply id="S2.SS1.p8.2.m2.1.2.3.cmml" xref="S2.SS1.p8.2.m2.1.2.3"><times id="S2.SS1.p8.2.m2.1.2.3.1.cmml" xref="S2.SS1.p8.2.m2.1.2.3.1"></times><ci id="S2.SS1.p8.2.m2.1.2.3.2.cmml" xref="S2.SS1.p8.2.m2.1.2.3.2">𝒫</ci><ci id="S2.SS1.p8.2.m2.1.1.cmml" xref="S2.SS1.p8.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math>, we use <math alttext="\langle\Gamma,\mathcal{B}\rangle" class="ltx_Math" display="inline" id="S2.SS1.p8.3.m3.2"><semantics id="S2.SS1.p8.3.m3.2a"><mrow id="S2.SS1.p8.3.m3.2.3.2" xref="S2.SS1.p8.3.m3.2.3.1.cmml"><mo id="S2.SS1.p8.3.m3.2.3.2.1" stretchy="false" xref="S2.SS1.p8.3.m3.2.3.1.cmml">⟨</mo><mi id="S2.SS1.p8.3.m3.1.1" mathvariant="normal" xref="S2.SS1.p8.3.m3.1.1.cmml">Γ</mi><mo id="S2.SS1.p8.3.m3.2.3.2.2" xref="S2.SS1.p8.3.m3.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.3.m3.2.2" xref="S2.SS1.p8.3.m3.2.2.cmml">ℬ</mi><mo id="S2.SS1.p8.3.m3.2.3.2.3" stretchy="false" xref="S2.SS1.p8.3.m3.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.3.m3.2b"><list id="S2.SS1.p8.3.m3.2.3.1.cmml" xref="S2.SS1.p8.3.m3.2.3.2"><ci id="S2.SS1.p8.3.m3.1.1.cmml" xref="S2.SS1.p8.3.m3.1.1">Γ</ci><ci id="S2.SS1.p8.3.m3.2.2.cmml" xref="S2.SS1.p8.3.m3.2.2">ℬ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.3.m3.2c">\langle\Gamma,\mathcal{B}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.3.m3.2d">⟨ roman_Γ , caligraphic_B ⟩</annotation></semantics></math> to represent the corresponding <em class="ltx_emph ltx_font_italic" id="S2.SS1.p8.13.1">discrete space</em>. We assume for simplicity that <math alttext="\Gamma=\bigcup_{B\in\mathcal{B}}B" class="ltx_Math" display="inline" id="S2.SS1.p8.4.m4.1"><semantics id="S2.SS1.p8.4.m4.1a"><mrow id="S2.SS1.p8.4.m4.1.1" xref="S2.SS1.p8.4.m4.1.1.cmml"><mi id="S2.SS1.p8.4.m4.1.1.2" mathvariant="normal" xref="S2.SS1.p8.4.m4.1.1.2.cmml">Γ</mi><mo id="S2.SS1.p8.4.m4.1.1.1" rspace="0.111em" xref="S2.SS1.p8.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS1.p8.4.m4.1.1.3" xref="S2.SS1.p8.4.m4.1.1.3.cmml"><msub id="S2.SS1.p8.4.m4.1.1.3.1" xref="S2.SS1.p8.4.m4.1.1.3.1.cmml"><mo id="S2.SS1.p8.4.m4.1.1.3.1.2" xref="S2.SS1.p8.4.m4.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.SS1.p8.4.m4.1.1.3.1.3" xref="S2.SS1.p8.4.m4.1.1.3.1.3.cmml"><mi id="S2.SS1.p8.4.m4.1.1.3.1.3.2" xref="S2.SS1.p8.4.m4.1.1.3.1.3.2.cmml">B</mi><mo id="S2.SS1.p8.4.m4.1.1.3.1.3.1" xref="S2.SS1.p8.4.m4.1.1.3.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.4.m4.1.1.3.1.3.3" xref="S2.SS1.p8.4.m4.1.1.3.1.3.3.cmml">ℬ</mi></mrow></msub><mi id="S2.SS1.p8.4.m4.1.1.3.2" xref="S2.SS1.p8.4.m4.1.1.3.2.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.4.m4.1b"><apply id="S2.SS1.p8.4.m4.1.1.cmml" xref="S2.SS1.p8.4.m4.1.1"><eq id="S2.SS1.p8.4.m4.1.1.1.cmml" xref="S2.SS1.p8.4.m4.1.1.1"></eq><ci id="S2.SS1.p8.4.m4.1.1.2.cmml" xref="S2.SS1.p8.4.m4.1.1.2">Γ</ci><apply id="S2.SS1.p8.4.m4.1.1.3.cmml" xref="S2.SS1.p8.4.m4.1.1.3"><apply id="S2.SS1.p8.4.m4.1.1.3.1.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS1.p8.4.m4.1.1.3.1.1.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1">subscript</csymbol><union id="S2.SS1.p8.4.m4.1.1.3.1.2.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1.2"></union><apply id="S2.SS1.p8.4.m4.1.1.3.1.3.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1.3"><in id="S2.SS1.p8.4.m4.1.1.3.1.3.1.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1.3.1"></in><ci id="S2.SS1.p8.4.m4.1.1.3.1.3.2.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1.3.2">𝐵</ci><ci id="S2.SS1.p8.4.m4.1.1.3.1.3.3.cmml" xref="S2.SS1.p8.4.m4.1.1.3.1.3.3">ℬ</ci></apply></apply><ci id="S2.SS1.p8.4.m4.1.1.3.2.cmml" xref="S2.SS1.p8.4.m4.1.1.3.2">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.4.m4.1c">\Gamma=\bigcup_{B\in\mathcal{B}}B</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.4.m4.1d">roman_Γ = ⋃ start_POSTSUBSCRIPT italic_B ∈ caligraphic_B end_POSTSUBSCRIPT italic_B</annotation></semantics></math>. We extend the notation introduced above, and use <math alttext="D(A_{1},\ldots,A_{\ell}\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p8.5.m5.3"><semantics id="S2.SS1.p8.5.m5.3a"><mrow id="S2.SS1.p8.5.m5.3.3" xref="S2.SS1.p8.5.m5.3.3.cmml"><mi id="S2.SS1.p8.5.m5.3.3.4" xref="S2.SS1.p8.5.m5.3.3.4.cmml">D</mi><mo id="S2.SS1.p8.5.m5.3.3.3" xref="S2.SS1.p8.5.m5.3.3.3.cmml">⁢</mo><mrow id="S2.SS1.p8.5.m5.3.3.2.2" xref="S2.SS1.p8.5.m5.3.3.2.3.cmml"><mo id="S2.SS1.p8.5.m5.3.3.2.2.3" stretchy="false" xref="S2.SS1.p8.5.m5.3.3.2.3.cmml">(</mo><msub id="S2.SS1.p8.5.m5.2.2.1.1.1" xref="S2.SS1.p8.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS1.p8.5.m5.2.2.1.1.1.2" xref="S2.SS1.p8.5.m5.2.2.1.1.1.2.cmml">A</mi><mn id="S2.SS1.p8.5.m5.2.2.1.1.1.3" xref="S2.SS1.p8.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p8.5.m5.3.3.2.2.4" xref="S2.SS1.p8.5.m5.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p8.5.m5.1.1" mathvariant="normal" xref="S2.SS1.p8.5.m5.1.1.cmml">…</mi><mo id="S2.SS1.p8.5.m5.3.3.2.2.5" xref="S2.SS1.p8.5.m5.3.3.2.3.cmml">,</mo><mrow id="S2.SS1.p8.5.m5.3.3.2.2.2" xref="S2.SS1.p8.5.m5.3.3.2.2.2.cmml"><msub id="S2.SS1.p8.5.m5.3.3.2.2.2.2" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2.cmml"><mi id="S2.SS1.p8.5.m5.3.3.2.2.2.2.2" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2.2.cmml">A</mi><mi id="S2.SS1.p8.5.m5.3.3.2.2.2.2.3" mathvariant="normal" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2.3.cmml">ℓ</mi></msub><mo id="S2.SS1.p8.5.m5.3.3.2.2.2.1" xref="S2.SS1.p8.5.m5.3.3.2.2.2.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.5.m5.3.3.2.2.2.3" xref="S2.SS1.p8.5.m5.3.3.2.2.2.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p8.5.m5.3.3.2.2.6" stretchy="false" xref="S2.SS1.p8.5.m5.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.5.m5.3b"><apply id="S2.SS1.p8.5.m5.3.3.cmml" xref="S2.SS1.p8.5.m5.3.3"><times id="S2.SS1.p8.5.m5.3.3.3.cmml" xref="S2.SS1.p8.5.m5.3.3.3"></times><ci id="S2.SS1.p8.5.m5.3.3.4.cmml" xref="S2.SS1.p8.5.m5.3.3.4">𝐷</ci><vector id="S2.SS1.p8.5.m5.3.3.2.3.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2"><apply id="S2.SS1.p8.5.m5.2.2.1.1.1.cmml" xref="S2.SS1.p8.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS1.p8.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p8.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS1.p8.5.m5.2.2.1.1.1.2">𝐴</ci><cn id="S2.SS1.p8.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p8.5.m5.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS1.p8.5.m5.1.1.cmml" xref="S2.SS1.p8.5.m5.1.1">…</ci><apply id="S2.SS1.p8.5.m5.3.3.2.2.2.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2"><csymbol cd="latexml" id="S2.SS1.p8.5.m5.3.3.2.2.2.1.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.1">conditional</csymbol><apply id="S2.SS1.p8.5.m5.3.3.2.2.2.2.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p8.5.m5.3.3.2.2.2.2.1.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p8.5.m5.3.3.2.2.2.2.2.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2.2">𝐴</ci><ci id="S2.SS1.p8.5.m5.3.3.2.2.2.2.3.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.2.3">ℓ</ci></apply><ci id="S2.SS1.p8.5.m5.3.3.2.2.2.3.cmml" xref="S2.SS1.p8.5.m5.3.3.2.2.2.3">ℬ</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.5.m5.3c">D(A_{1},\ldots,A_{\ell}\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.5.m5.3d">italic_D ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∣ caligraphic_B )</annotation></semantics></math> to denote the discrete complexity of simultaneously generating <math alttext="A_{1},\ldots,A_{\ell}" class="ltx_Math" display="inline" id="S2.SS1.p8.6.m6.3"><semantics id="S2.SS1.p8.6.m6.3a"><mrow id="S2.SS1.p8.6.m6.3.3.2" xref="S2.SS1.p8.6.m6.3.3.3.cmml"><msub id="S2.SS1.p8.6.m6.2.2.1.1" xref="S2.SS1.p8.6.m6.2.2.1.1.cmml"><mi id="S2.SS1.p8.6.m6.2.2.1.1.2" xref="S2.SS1.p8.6.m6.2.2.1.1.2.cmml">A</mi><mn id="S2.SS1.p8.6.m6.2.2.1.1.3" xref="S2.SS1.p8.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p8.6.m6.3.3.2.3" xref="S2.SS1.p8.6.m6.3.3.3.cmml">,</mo><mi id="S2.SS1.p8.6.m6.1.1" mathvariant="normal" xref="S2.SS1.p8.6.m6.1.1.cmml">…</mi><mo id="S2.SS1.p8.6.m6.3.3.2.4" xref="S2.SS1.p8.6.m6.3.3.3.cmml">,</mo><msub id="S2.SS1.p8.6.m6.3.3.2.2" xref="S2.SS1.p8.6.m6.3.3.2.2.cmml"><mi id="S2.SS1.p8.6.m6.3.3.2.2.2" xref="S2.SS1.p8.6.m6.3.3.2.2.2.cmml">A</mi><mi id="S2.SS1.p8.6.m6.3.3.2.2.3" mathvariant="normal" xref="S2.SS1.p8.6.m6.3.3.2.2.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.6.m6.3b"><list id="S2.SS1.p8.6.m6.3.3.3.cmml" xref="S2.SS1.p8.6.m6.3.3.2"><apply id="S2.SS1.p8.6.m6.2.2.1.1.cmml" xref="S2.SS1.p8.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.6.m6.2.2.1.1.1.cmml" xref="S2.SS1.p8.6.m6.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p8.6.m6.2.2.1.1.2.cmml" xref="S2.SS1.p8.6.m6.2.2.1.1.2">𝐴</ci><cn id="S2.SS1.p8.6.m6.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p8.6.m6.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p8.6.m6.1.1.cmml" xref="S2.SS1.p8.6.m6.1.1">…</ci><apply id="S2.SS1.p8.6.m6.3.3.2.2.cmml" xref="S2.SS1.p8.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p8.6.m6.3.3.2.2.1.cmml" xref="S2.SS1.p8.6.m6.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p8.6.m6.3.3.2.2.2.cmml" xref="S2.SS1.p8.6.m6.3.3.2.2.2">𝐴</ci><ci id="S2.SS1.p8.6.m6.3.3.2.2.3.cmml" xref="S2.SS1.p8.6.m6.3.3.2.2.3">ℓ</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.6.m6.3c">A_{1},\ldots,A_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.6.m6.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p8.7.m7.1"><semantics id="S2.SS1.p8.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.7.m7.1.1" xref="S2.SS1.p8.7.m7.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.7.m7.1b"><ci id="S2.SS1.p8.7.m7.1.1.cmml" xref="S2.SS1.p8.7.m7.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.7.m7.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.7.m7.1d">caligraphic_B</annotation></semantics></math>. In other words, this is the minimum number <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p8.8.m8.1"><semantics id="S2.SS1.p8.8.m8.1a"><mi id="S2.SS1.p8.8.m8.1.1" xref="S2.SS1.p8.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.8.m8.1b"><ci id="S2.SS1.p8.8.m8.1.1.cmml" xref="S2.SS1.p8.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.8.m8.1d">italic_t</annotation></semantics></math> such that there exists a sequence <math alttext="E_{1},\ldots,E_{t}" class="ltx_Math" display="inline" id="S2.SS1.p8.9.m9.3"><semantics id="S2.SS1.p8.9.m9.3a"><mrow id="S2.SS1.p8.9.m9.3.3.2" xref="S2.SS1.p8.9.m9.3.3.3.cmml"><msub id="S2.SS1.p8.9.m9.2.2.1.1" xref="S2.SS1.p8.9.m9.2.2.1.1.cmml"><mi id="S2.SS1.p8.9.m9.2.2.1.1.2" xref="S2.SS1.p8.9.m9.2.2.1.1.2.cmml">E</mi><mn id="S2.SS1.p8.9.m9.2.2.1.1.3" xref="S2.SS1.p8.9.m9.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p8.9.m9.3.3.2.3" xref="S2.SS1.p8.9.m9.3.3.3.cmml">,</mo><mi id="S2.SS1.p8.9.m9.1.1" mathvariant="normal" xref="S2.SS1.p8.9.m9.1.1.cmml">…</mi><mo id="S2.SS1.p8.9.m9.3.3.2.4" xref="S2.SS1.p8.9.m9.3.3.3.cmml">,</mo><msub id="S2.SS1.p8.9.m9.3.3.2.2" xref="S2.SS1.p8.9.m9.3.3.2.2.cmml"><mi id="S2.SS1.p8.9.m9.3.3.2.2.2" xref="S2.SS1.p8.9.m9.3.3.2.2.2.cmml">E</mi><mi id="S2.SS1.p8.9.m9.3.3.2.2.3" xref="S2.SS1.p8.9.m9.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.9.m9.3b"><list id="S2.SS1.p8.9.m9.3.3.3.cmml" xref="S2.SS1.p8.9.m9.3.3.2"><apply id="S2.SS1.p8.9.m9.2.2.1.1.cmml" xref="S2.SS1.p8.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.9.m9.2.2.1.1.1.cmml" xref="S2.SS1.p8.9.m9.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p8.9.m9.2.2.1.1.2.cmml" xref="S2.SS1.p8.9.m9.2.2.1.1.2">𝐸</ci><cn id="S2.SS1.p8.9.m9.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p8.9.m9.2.2.1.1.3">1</cn></apply><ci id="S2.SS1.p8.9.m9.1.1.cmml" xref="S2.SS1.p8.9.m9.1.1">…</ci><apply id="S2.SS1.p8.9.m9.3.3.2.2.cmml" xref="S2.SS1.p8.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p8.9.m9.3.3.2.2.1.cmml" xref="S2.SS1.p8.9.m9.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p8.9.m9.3.3.2.2.2.cmml" xref="S2.SS1.p8.9.m9.3.3.2.2.2">𝐸</ci><ci id="S2.SS1.p8.9.m9.3.3.2.2.3.cmml" xref="S2.SS1.p8.9.m9.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.9.m9.3c">E_{1},\ldots,E_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.9.m9.3d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> of sets contained in <math alttext="\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p8.10.m10.1"><semantics id="S2.SS1.p8.10.m10.1a"><mi id="S2.SS1.p8.10.m10.1.1" mathvariant="normal" xref="S2.SS1.p8.10.m10.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.10.m10.1b"><ci id="S2.SS1.p8.10.m10.1.1.cmml" xref="S2.SS1.p8.10.m10.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.10.m10.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.10.m10.1d">roman_Γ</annotation></semantics></math> such that every set <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS1.p8.11.m11.1"><semantics id="S2.SS1.p8.11.m11.1a"><msub id="S2.SS1.p8.11.m11.1.1" xref="S2.SS1.p8.11.m11.1.1.cmml"><mi id="S2.SS1.p8.11.m11.1.1.2" xref="S2.SS1.p8.11.m11.1.1.2.cmml">A</mi><mi id="S2.SS1.p8.11.m11.1.1.3" xref="S2.SS1.p8.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.11.m11.1b"><apply id="S2.SS1.p8.11.m11.1.1.cmml" xref="S2.SS1.p8.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.11.m11.1.1.1.cmml" xref="S2.SS1.p8.11.m11.1.1">subscript</csymbol><ci id="S2.SS1.p8.11.m11.1.1.2.cmml" xref="S2.SS1.p8.11.m11.1.1.2">𝐴</ci><ci id="S2.SS1.p8.11.m11.1.1.3.cmml" xref="S2.SS1.p8.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.11.m11.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.11.m11.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> appears in the sequence at least once, and each <math alttext="E_{j}" class="ltx_Math" display="inline" id="S2.SS1.p8.12.m12.1"><semantics id="S2.SS1.p8.12.m12.1a"><msub id="S2.SS1.p8.12.m12.1.1" xref="S2.SS1.p8.12.m12.1.1.cmml"><mi id="S2.SS1.p8.12.m12.1.1.2" xref="S2.SS1.p8.12.m12.1.1.2.cmml">E</mi><mi id="S2.SS1.p8.12.m12.1.1.3" xref="S2.SS1.p8.12.m12.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.12.m12.1b"><apply id="S2.SS1.p8.12.m12.1.1.cmml" xref="S2.SS1.p8.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.12.m12.1.1.1.cmml" xref="S2.SS1.p8.12.m12.1.1">subscript</csymbol><ci id="S2.SS1.p8.12.m12.1.1.2.cmml" xref="S2.SS1.p8.12.m12.1.1.2">𝐸</ci><ci id="S2.SS1.p8.12.m12.1.1.3.cmml" xref="S2.SS1.p8.12.m12.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.12.m12.1c">E_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.12.m12.1d">italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is obtained from the preceding sets in the sequence and the sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS1.p8.13.m13.1"><semantics id="S2.SS1.p8.13.m13.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.13.m13.1.1" xref="S2.SS1.p8.13.m13.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.13.m13.1b"><ci id="S2.SS1.p8.13.m13.1.1.cmml" xref="S2.SS1.p8.13.m13.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.13.m13.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.13.m13.1d">caligraphic_B</annotation></semantics></math> either by a union or by an intersection operation.</p> </div> <div class="ltx_para" id="S2.SS1.p9"> <p class="ltx_p" id="S2.SS1.p9.2">Finally, note that we tacitly assume in most proofs presented in this section that <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS1.p9.1.m1.1"><semantics id="S2.SS1.p9.1.m1.1a"><mrow id="S2.SS1.p9.1.m1.1.1" xref="S2.SS1.p9.1.m1.1.1.cmml"><mi id="S2.SS1.p9.1.m1.1.1.3" xref="S2.SS1.p9.1.m1.1.1.3.cmml">D</mi><mo id="S2.SS1.p9.1.m1.1.1.2" xref="S2.SS1.p9.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p9.1.m1.1.1.1.1" xref="S2.SS1.p9.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS1.p9.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p9.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p9.1.m1.1.1.1.1.1" xref="S2.SS1.p9.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p9.1.m1.1.1.1.1.1.2" xref="S2.SS1.p9.1.m1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS1.p9.1.m1.1.1.1.1.1.1" xref="S2.SS1.p9.1.m1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p9.1.m1.1.1.1.1.1.3" xref="S2.SS1.p9.1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS1.p9.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p9.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.1.m1.1b"><apply id="S2.SS1.p9.1.m1.1.1.cmml" xref="S2.SS1.p9.1.m1.1.1"><times id="S2.SS1.p9.1.m1.1.1.2.cmml" xref="S2.SS1.p9.1.m1.1.1.2"></times><ci id="S2.SS1.p9.1.m1.1.1.3.cmml" xref="S2.SS1.p9.1.m1.1.1.3">𝐷</ci><apply id="S2.SS1.p9.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.p9.1.m1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p9.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS1.p9.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p9.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p9.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS1.p9.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS1.p9.1.m1.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.1.m1.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.1.m1.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is finite, as otherwise the corresponding statements are trivially true. We will also assume in these statements that <math alttext="A\subseteq\bigcup_{B\in\mathcal{B}}B=\Gamma" class="ltx_Math" display="inline" id="S2.SS1.p9.2.m2.1"><semantics id="S2.SS1.p9.2.m2.1a"><mrow id="S2.SS1.p9.2.m2.1.1" xref="S2.SS1.p9.2.m2.1.1.cmml"><mi id="S2.SS1.p9.2.m2.1.1.2" xref="S2.SS1.p9.2.m2.1.1.2.cmml">A</mi><mo id="S2.SS1.p9.2.m2.1.1.3" rspace="0.111em" xref="S2.SS1.p9.2.m2.1.1.3.cmml">⊆</mo><mrow id="S2.SS1.p9.2.m2.1.1.4" xref="S2.SS1.p9.2.m2.1.1.4.cmml"><msub id="S2.SS1.p9.2.m2.1.1.4.1" xref="S2.SS1.p9.2.m2.1.1.4.1.cmml"><mo id="S2.SS1.p9.2.m2.1.1.4.1.2" xref="S2.SS1.p9.2.m2.1.1.4.1.2.cmml">⋃</mo><mrow id="S2.SS1.p9.2.m2.1.1.4.1.3" xref="S2.SS1.p9.2.m2.1.1.4.1.3.cmml"><mi id="S2.SS1.p9.2.m2.1.1.4.1.3.2" xref="S2.SS1.p9.2.m2.1.1.4.1.3.2.cmml">B</mi><mo id="S2.SS1.p9.2.m2.1.1.4.1.3.1" xref="S2.SS1.p9.2.m2.1.1.4.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p9.2.m2.1.1.4.1.3.3" xref="S2.SS1.p9.2.m2.1.1.4.1.3.3.cmml">ℬ</mi></mrow></msub><mi 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start_POSTSUBSCRIPT italic_B ∈ caligraphic_B end_POSTSUBSCRIPT italic_B = roman_Γ</annotation></semantics></math> in order to avoid trivial considerations.</p> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Examples</h3> <section class="ltx_subsubsection ltx_indent_first" id="S2.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.2.1 </span>Boolean circuit complexity</h4> <div class="ltx_para" id="S2.SS2.SSS1.p1"> <p class="ltx_p" id="S2.SS2.SSS1.p1.14">This is the classical setting where for each <math alttext="n\in\mathbb{N}^{+}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.1.m1.1"><semantics id="S2.SS2.SSS1.p1.1.m1.1a"><mrow id="S2.SS2.SSS1.p1.1.m1.1.1" xref="S2.SS2.SSS1.p1.1.m1.1.1.cmml"><mi id="S2.SS2.SSS1.p1.1.m1.1.1.2" xref="S2.SS2.SSS1.p1.1.m1.1.1.2.cmml">n</mi><mo id="S2.SS2.SSS1.p1.1.m1.1.1.1" xref="S2.SS2.SSS1.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S2.SS2.SSS1.p1.1.m1.1.1.3" xref="S2.SS2.SSS1.p1.1.m1.1.1.3.cmml"><mi id="S2.SS2.SSS1.p1.1.m1.1.1.3.2" xref="S2.SS2.SSS1.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mo id="S2.SS2.SSS1.p1.1.m1.1.1.3.3" xref="S2.SS2.SSS1.p1.1.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.1.m1.1b"><apply id="S2.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1"><in id="S2.SS2.SSS1.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.1"></in><ci id="S2.SS2.SSS1.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.2">𝑛</ci><apply id="S2.SS2.SSS1.p1.1.m1.1.1.3.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.3.2">ℕ</ci><plus id="S2.SS2.SSS1.p1.1.m1.1.1.3.3.cmml" xref="S2.SS2.SSS1.p1.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.1.m1.1c">n\in\mathbb{N}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.1.m1.1d">italic_n ∈ blackboard_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\Gamma=\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.2.m2.2"><semantics id="S2.SS2.SSS1.p1.2.m2.2a"><mrow id="S2.SS2.SSS1.p1.2.m2.2.3" xref="S2.SS2.SSS1.p1.2.m2.2.3.cmml"><mi id="S2.SS2.SSS1.p1.2.m2.2.3.2" mathvariant="normal" xref="S2.SS2.SSS1.p1.2.m2.2.3.2.cmml">Γ</mi><mo id="S2.SS2.SSS1.p1.2.m2.2.3.1" xref="S2.SS2.SSS1.p1.2.m2.2.3.1.cmml">=</mo><msup id="S2.SS2.SSS1.p1.2.m2.2.3.3" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.cmml"><mrow id="S2.SS2.SSS1.p1.2.m2.2.3.3.2.2" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS1.p1.2.m2.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.2.1.cmml">{</mo><mn id="S2.SS2.SSS1.p1.2.m2.1.1" xref="S2.SS2.SSS1.p1.2.m2.1.1.cmml">0</mn><mo id="S2.SS2.SSS1.p1.2.m2.2.3.3.2.2.2" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.2.1.cmml">,</mo><mn id="S2.SS2.SSS1.p1.2.m2.2.2" xref="S2.SS2.SSS1.p1.2.m2.2.2.cmml">1</mn><mo id="S2.SS2.SSS1.p1.2.m2.2.3.3.2.2.3" stretchy="false" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS2.SSS1.p1.2.m2.2.3.3.3" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.2.m2.2b"><apply id="S2.SS2.SSS1.p1.2.m2.2.3.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3"><eq id="S2.SS2.SSS1.p1.2.m2.2.3.1.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.1"></eq><ci id="S2.SS2.SSS1.p1.2.m2.2.3.2.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.2">Γ</ci><apply id="S2.SS2.SSS1.p1.2.m2.2.3.3.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.2.m2.2.3.3.1.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.3">superscript</csymbol><set id="S2.SS2.SSS1.p1.2.m2.2.3.3.2.1.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.2.2"><cn id="S2.SS2.SSS1.p1.2.m2.1.1.cmml" type="integer" xref="S2.SS2.SSS1.p1.2.m2.1.1">0</cn><cn id="S2.SS2.SSS1.p1.2.m2.2.2.cmml" type="integer" xref="S2.SS2.SSS1.p1.2.m2.2.2">1</cn></set><ci id="S2.SS2.SSS1.p1.2.m2.2.3.3.3.cmml" xref="S2.SS2.SSS1.p1.2.m2.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.2.m2.2c">\Gamma=\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.2.m2.2d">roman_Γ = { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is the set of vertices of the <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.3.m3.1"><semantics id="S2.SS2.SSS1.p1.3.m3.1a"><mi id="S2.SS2.SSS1.p1.3.m3.1.1" xref="S2.SS2.SSS1.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.3.m3.1b"><ci id="S2.SS2.SSS1.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS1.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.3.m3.1d">italic_n</annotation></semantics></math>-dimensional hypercube, <math alttext="A" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.4.m4.1"><semantics id="S2.SS2.SSS1.p1.4.m4.1a"><mi id="S2.SS2.SSS1.p1.4.m4.1.1" xref="S2.SS2.SSS1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.4.m4.1b"><ci id="S2.SS2.SSS1.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.4.m4.1d">italic_A</annotation></semantics></math> corresponds to <math alttext="f^{-1}(1)" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.5.m5.1"><semantics id="S2.SS2.SSS1.p1.5.m5.1a"><mrow id="S2.SS2.SSS1.p1.5.m5.1.2" xref="S2.SS2.SSS1.p1.5.m5.1.2.cmml"><msup id="S2.SS2.SSS1.p1.5.m5.1.2.2" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.cmml"><mi id="S2.SS2.SSS1.p1.5.m5.1.2.2.2" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.2.cmml">f</mi><mrow id="S2.SS2.SSS1.p1.5.m5.1.2.2.3" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3.cmml"><mo id="S2.SS2.SSS1.p1.5.m5.1.2.2.3a" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3.cmml">−</mo><mn id="S2.SS2.SSS1.p1.5.m5.1.2.2.3.2" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS2.SSS1.p1.5.m5.1.2.1" xref="S2.SS2.SSS1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS1.p1.5.m5.1.2.3.2" xref="S2.SS2.SSS1.p1.5.m5.1.2.cmml"><mo id="S2.SS2.SSS1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS1.p1.5.m5.1.2.cmml">(</mo><mn id="S2.SS2.SSS1.p1.5.m5.1.1" xref="S2.SS2.SSS1.p1.5.m5.1.1.cmml">1</mn><mo id="S2.SS2.SSS1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.5.m5.1b"><apply id="S2.SS2.SSS1.p1.5.m5.1.2.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2"><times id="S2.SS2.SSS1.p1.5.m5.1.2.1.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.1"></times><apply id="S2.SS2.SSS1.p1.5.m5.1.2.2.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.5.m5.1.2.2.1.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.2">superscript</csymbol><ci id="S2.SS2.SSS1.p1.5.m5.1.2.2.2.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.2">𝑓</ci><apply id="S2.SS2.SSS1.p1.5.m5.1.2.2.3.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3"><minus id="S2.SS2.SSS1.p1.5.m5.1.2.2.3.1.cmml" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3"></minus><cn id="S2.SS2.SSS1.p1.5.m5.1.2.2.3.2.cmml" type="integer" xref="S2.SS2.SSS1.p1.5.m5.1.2.2.3.2">1</cn></apply></apply><cn id="S2.SS2.SSS1.p1.5.m5.1.1.cmml" type="integer" xref="S2.SS2.SSS1.p1.5.m5.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.5.m5.1c">f^{-1}(1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.5.m5.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 )</annotation></semantics></math> for a Boolean function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.6.m6.4"><semantics id="S2.SS2.SSS1.p1.6.m6.4a"><mrow id="S2.SS2.SSS1.p1.6.m6.4.5" xref="S2.SS2.SSS1.p1.6.m6.4.5.cmml"><mi id="S2.SS2.SSS1.p1.6.m6.4.5.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.2.cmml">f</mi><mo id="S2.SS2.SSS1.p1.6.m6.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.SSS1.p1.6.m6.4.5.1.cmml">:</mo><mrow id="S2.SS2.SSS1.p1.6.m6.4.5.3" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.cmml"><msup id="S2.SS2.SSS1.p1.6.m6.4.5.3.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.cmml"><mrow id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.1.cmml"><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.2.1" stretchy="false" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.1.cmml">{</mo><mn id="S2.SS2.SSS1.p1.6.m6.1.1" xref="S2.SS2.SSS1.p1.6.m6.1.1.cmml">0</mn><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.2.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.1.cmml">,</mo><mn id="S2.SS2.SSS1.p1.6.m6.2.2" xref="S2.SS2.SSS1.p1.6.m6.2.2.cmml">1</mn><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.2.3" stretchy="false" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.3" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.3.cmml">n</mi></msup><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.1" stretchy="false" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.1.cmml">→</mo><mrow id="S2.SS2.SSS1.p1.6.m6.4.5.3.3.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.3.1.cmml"><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.3.2.1" stretchy="false" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.3.1.cmml">{</mo><mn id="S2.SS2.SSS1.p1.6.m6.3.3" xref="S2.SS2.SSS1.p1.6.m6.3.3.cmml">0</mn><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.3.2.2" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.3.1.cmml">,</mo><mn id="S2.SS2.SSS1.p1.6.m6.4.4" xref="S2.SS2.SSS1.p1.6.m6.4.4.cmml">1</mn><mo id="S2.SS2.SSS1.p1.6.m6.4.5.3.3.2.3" stretchy="false" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.6.m6.4b"><apply id="S2.SS2.SSS1.p1.6.m6.4.5.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5"><ci id="S2.SS2.SSS1.p1.6.m6.4.5.1.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.1">:</ci><ci id="S2.SS2.SSS1.p1.6.m6.4.5.2.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.2">𝑓</ci><apply id="S2.SS2.SSS1.p1.6.m6.4.5.3.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3"><ci id="S2.SS2.SSS1.p1.6.m6.4.5.3.1.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.1">→</ci><apply id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.1.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2">superscript</csymbol><set id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.1.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.2.2"><cn id="S2.SS2.SSS1.p1.6.m6.1.1.cmml" type="integer" xref="S2.SS2.SSS1.p1.6.m6.1.1">0</cn><cn id="S2.SS2.SSS1.p1.6.m6.2.2.cmml" type="integer" xref="S2.SS2.SSS1.p1.6.m6.2.2">1</cn></set><ci id="S2.SS2.SSS1.p1.6.m6.4.5.3.2.3.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.2.3">𝑛</ci></apply><set id="S2.SS2.SSS1.p1.6.m6.4.5.3.3.1.cmml" xref="S2.SS2.SSS1.p1.6.m6.4.5.3.3.2"><cn id="S2.SS2.SSS1.p1.6.m6.3.3.cmml" type="integer" xref="S2.SS2.SSS1.p1.6.m6.3.3">0</cn><cn id="S2.SS2.SSS1.p1.6.m6.4.4.cmml" type="integer" xref="S2.SS2.SSS1.p1.6.m6.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.6.m6.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.6.m6.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>, and <math alttext="\mathcal{B}=\{B_{1},\ldots,B_{n},B_{1}^{c},\ldots,B_{n}^{c}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.7.m7.6"><semantics id="S2.SS2.SSS1.p1.7.m7.6a"><mrow id="S2.SS2.SSS1.p1.7.m7.6.6" xref="S2.SS2.SSS1.p1.7.m7.6.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS1.p1.7.m7.6.6.6" xref="S2.SS2.SSS1.p1.7.m7.6.6.6.cmml">ℬ</mi><mo id="S2.SS2.SSS1.p1.7.m7.6.6.5" xref="S2.SS2.SSS1.p1.7.m7.6.6.5.cmml">=</mo><mrow id="S2.SS2.SSS1.p1.7.m7.6.6.4.4" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.5.cmml"><mo id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.5" stretchy="false" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.5.cmml">{</mo><msub id="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1" xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.cmml"><mi id="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.2" xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.2.cmml">B</mi><mn id="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.3" xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.6" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.5.cmml">,</mo><mi id="S2.SS2.SSS1.p1.7.m7.1.1" mathvariant="normal" xref="S2.SS2.SSS1.p1.7.m7.1.1.cmml">…</mi><mo id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.7" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.5.cmml">,</mo><msub id="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2" xref="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.cmml"><mi 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xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.2">𝐵</ci><cn id="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS1.p1.7.m7.3.3.1.1.1.3">1</cn></apply><ci id="S2.SS2.SSS1.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.1.1">…</ci><apply id="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.2">𝐵</ci><ci id="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.3.cmml" xref="S2.SS2.SSS1.p1.7.m7.4.4.2.2.2.3">𝑛</ci></apply><apply id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3">superscript</csymbol><apply id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3">subscript</csymbol><ci id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.2">𝐵</ci><cn id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.3.cmml" type="integer" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.2.3">1</cn></apply><ci id="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.3.cmml" xref="S2.SS2.SSS1.p1.7.m7.5.5.3.3.3.3">𝑐</ci></apply><ci id="S2.SS2.SSS1.p1.7.m7.2.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.2.2">…</ci><apply id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4">superscript</csymbol><apply id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.1.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4">subscript</csymbol><ci id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.2.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.2">𝐵</ci><ci id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.3.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.2.3">𝑛</ci></apply><ci id="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.3.cmml" xref="S2.SS2.SSS1.p1.7.m7.6.6.4.4.4.3">𝑐</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.7.m7.6c">\mathcal{B}=\{B_{1},\ldots,B_{n},B_{1}^{c},\ldots,B_{n}^{c}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.7.m7.6d">caligraphic_B = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT }</annotation></semantics></math>, where <math alttext="B_{i}=\{v\in\Gamma\mid v_{i}=1\}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.8.m8.2"><semantics id="S2.SS2.SSS1.p1.8.m8.2a"><mrow id="S2.SS2.SSS1.p1.8.m8.2.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.cmml"><msub id="S2.SS2.SSS1.p1.8.m8.2.2.4" xref="S2.SS2.SSS1.p1.8.m8.2.2.4.cmml"><mi id="S2.SS2.SSS1.p1.8.m8.2.2.4.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.4.2.cmml">B</mi><mi id="S2.SS2.SSS1.p1.8.m8.2.2.4.3" xref="S2.SS2.SSS1.p1.8.m8.2.2.4.3.cmml">i</mi></msub><mo id="S2.SS2.SSS1.p1.8.m8.2.2.3" xref="S2.SS2.SSS1.p1.8.m8.2.2.3.cmml">=</mo><mrow id="S2.SS2.SSS1.p1.8.m8.2.2.2.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.3.cmml"><mo id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.3" stretchy="false" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.3.1.cmml">{</mo><mrow id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.2" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.2.cmml">v</mi><mo id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.1" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.3" mathvariant="normal" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.3.cmml">Γ</mi></mrow><mo fence="true" id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.4" lspace="0em" rspace="0em" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.3.1.cmml">∣</mo><mrow id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.cmml"><msub id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.cmml"><mi id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.2" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.2.cmml">v</mi><mi id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.3" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.1" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.1.cmml">=</mo><mn id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.3" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.5" stretchy="false" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.8.m8.2b"><apply id="S2.SS2.SSS1.p1.8.m8.2.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2"><eq id="S2.SS2.SSS1.p1.8.m8.2.2.3.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.3"></eq><apply id="S2.SS2.SSS1.p1.8.m8.2.2.4.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.8.m8.2.2.4.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.4">subscript</csymbol><ci id="S2.SS2.SSS1.p1.8.m8.2.2.4.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.4.2">𝐵</ci><ci id="S2.SS2.SSS1.p1.8.m8.2.2.4.3.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.4.3">𝑖</ci></apply><apply id="S2.SS2.SSS1.p1.8.m8.2.2.2.3.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2"><csymbol cd="latexml" id="S2.SS2.SSS1.p1.8.m8.2.2.2.3.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.3">conditional-set</csymbol><apply id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1"><in id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.1"></in><ci id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.2">𝑣</ci><ci id="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS1.p1.8.m8.1.1.1.1.1.3">Γ</ci></apply><apply id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2"><eq id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.1"></eq><apply id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.1.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.2.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.2">𝑣</ci><ci id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.3.cmml" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.2.3">𝑖</ci></apply><cn id="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.SSS1.p1.8.m8.2.2.2.2.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.8.m8.2c">B_{i}=\{v\in\Gamma\mid v_{i}=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.8.m8.2d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_v ∈ roman_Γ ∣ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 }</annotation></semantics></math>. By definition, <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.9.m9.1"><semantics id="S2.SS2.SSS1.p1.9.m9.1a"><mrow id="S2.SS2.SSS1.p1.9.m9.1.1" xref="S2.SS2.SSS1.p1.9.m9.1.1.cmml"><mi id="S2.SS2.SSS1.p1.9.m9.1.1.3" xref="S2.SS2.SSS1.p1.9.m9.1.1.3.cmml">D</mi><mo id="S2.SS2.SSS1.p1.9.m9.1.1.2" xref="S2.SS2.SSS1.p1.9.m9.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS1.p1.9.m9.1.1.1.1" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.2" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.1" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.3" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.9.m9.1b"><apply id="S2.SS2.SSS1.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1"><times id="S2.SS2.SSS1.p1.9.m9.1.1.2.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.2"></times><ci id="S2.SS2.SSS1.p1.9.m9.1.1.3.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.3">𝐷</ci><apply id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS1.p1.9.m9.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.9.m9.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.9.m9.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> captures the <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS1.p1.14.1">circuit complexity</em> of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.10.m10.1"><semantics id="S2.SS2.SSS1.p1.10.m10.1a"><mi id="S2.SS2.SSS1.p1.10.m10.1.1" xref="S2.SS2.SSS1.p1.10.m10.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.10.m10.1b"><ci id="S2.SS2.SSS1.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS1.p1.10.m10.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.10.m10.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.10.m10.1d">italic_f</annotation></semantics></math>. If we drop the generators <math alttext="B_{i}^{c}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.11.m11.1"><semantics id="S2.SS2.SSS1.p1.11.m11.1a"><msubsup id="S2.SS2.SSS1.p1.11.m11.1.1" xref="S2.SS2.SSS1.p1.11.m11.1.1.cmml"><mi id="S2.SS2.SSS1.p1.11.m11.1.1.2.2" xref="S2.SS2.SSS1.p1.11.m11.1.1.2.2.cmml">B</mi><mi id="S2.SS2.SSS1.p1.11.m11.1.1.2.3" xref="S2.SS2.SSS1.p1.11.m11.1.1.2.3.cmml">i</mi><mi id="S2.SS2.SSS1.p1.11.m11.1.1.3" xref="S2.SS2.SSS1.p1.11.m11.1.1.3.cmml">c</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.11.m11.1b"><apply id="S2.SS2.SSS1.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.11.m11.1.1.1.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1">superscript</csymbol><apply id="S2.SS2.SSS1.p1.11.m11.1.1.2.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.11.m11.1.1.2.1.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1">subscript</csymbol><ci id="S2.SS2.SSS1.p1.11.m11.1.1.2.2.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1.2.2">𝐵</ci><ci id="S2.SS2.SSS1.p1.11.m11.1.1.2.3.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1.2.3">𝑖</ci></apply><ci id="S2.SS2.SSS1.p1.11.m11.1.1.3.cmml" xref="S2.SS2.SSS1.p1.11.m11.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.11.m11.1c">B_{i}^{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.11.m11.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math> from the family <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.12.m12.1"><semantics id="S2.SS2.SSS1.p1.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS1.p1.12.m12.1.1" xref="S2.SS2.SSS1.p1.12.m12.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.12.m12.1b"><ci id="S2.SS2.SSS1.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS1.p1.12.m12.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.12.m12.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.12.m12.1d">caligraphic_B</annotation></semantics></math>, and add the sets <math alttext="\emptyset" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.13.m13.1"><semantics id="S2.SS2.SSS1.p1.13.m13.1a"><mi id="S2.SS2.SSS1.p1.13.m13.1.1" mathvariant="normal" xref="S2.SS2.SSS1.p1.13.m13.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.13.m13.1b"><emptyset id="S2.SS2.SSS1.p1.13.m13.1.1.cmml" xref="S2.SS2.SSS1.p1.13.m13.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.13.m13.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.13.m13.1d">∅</annotation></semantics></math> and <math alttext="\bar{1}\stackrel{{\scriptstyle\rm def}}{{=}}\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS2.SSS1.p1.14.m14.2"><semantics id="S2.SS2.SSS1.p1.14.m14.2a"><mrow id="S2.SS2.SSS1.p1.14.m14.2.3" xref="S2.SS2.SSS1.p1.14.m14.2.3.cmml"><mover accent="true" id="S2.SS2.SSS1.p1.14.m14.2.3.2" xref="S2.SS2.SSS1.p1.14.m14.2.3.2.cmml"><mn id="S2.SS2.SSS1.p1.14.m14.2.3.2.2" xref="S2.SS2.SSS1.p1.14.m14.2.3.2.2.cmml">1</mn><mo id="S2.SS2.SSS1.p1.14.m14.2.3.2.1" xref="S2.SS2.SSS1.p1.14.m14.2.3.2.1.cmml">¯</mo></mover><mover id="S2.SS2.SSS1.p1.14.m14.2.3.1" xref="S2.SS2.SSS1.p1.14.m14.2.3.1.cmml"><mo id="S2.SS2.SSS1.p1.14.m14.2.3.1.2" xref="S2.SS2.SSS1.p1.14.m14.2.3.1.2.cmml">=</mo><mi id="S2.SS2.SSS1.p1.14.m14.2.3.1.3" xref="S2.SS2.SSS1.p1.14.m14.2.3.1.3.cmml">def</mi></mover><msup id="S2.SS2.SSS1.p1.14.m14.2.3.3" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.cmml"><mrow id="S2.SS2.SSS1.p1.14.m14.2.3.3.2.2" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS1.p1.14.m14.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.2.1.cmml">{</mo><mn id="S2.SS2.SSS1.p1.14.m14.1.1" xref="S2.SS2.SSS1.p1.14.m14.1.1.cmml">0</mn><mo id="S2.SS2.SSS1.p1.14.m14.2.3.3.2.2.2" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.2.1.cmml">,</mo><mn id="S2.SS2.SSS1.p1.14.m14.2.2" xref="S2.SS2.SSS1.p1.14.m14.2.2.cmml">1</mn><mo id="S2.SS2.SSS1.p1.14.m14.2.3.3.2.2.3" stretchy="false" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS2.SSS1.p1.14.m14.2.3.3.3" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS1.p1.14.m14.2b"><apply id="S2.SS2.SSS1.p1.14.m14.2.3.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3"><apply id="S2.SS2.SSS1.p1.14.m14.2.3.1.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.14.m14.2.3.1.1.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.1">superscript</csymbol><eq id="S2.SS2.SSS1.p1.14.m14.2.3.1.2.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.1.2"></eq><ci id="S2.SS2.SSS1.p1.14.m14.2.3.1.3.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.1.3">def</ci></apply><apply id="S2.SS2.SSS1.p1.14.m14.2.3.2.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.2"><ci id="S2.SS2.SSS1.p1.14.m14.2.3.2.1.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.2.1">¯</ci><cn id="S2.SS2.SSS1.p1.14.m14.2.3.2.2.cmml" type="integer" xref="S2.SS2.SSS1.p1.14.m14.2.3.2.2">1</cn></apply><apply id="S2.SS2.SSS1.p1.14.m14.2.3.3.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS1.p1.14.m14.2.3.3.1.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.3">superscript</csymbol><set id="S2.SS2.SSS1.p1.14.m14.2.3.3.2.1.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.2.2"><cn id="S2.SS2.SSS1.p1.14.m14.1.1.cmml" type="integer" xref="S2.SS2.SSS1.p1.14.m14.1.1">0</cn><cn id="S2.SS2.SSS1.p1.14.m14.2.2.cmml" type="integer" xref="S2.SS2.SSS1.p1.14.m14.2.2">1</cn></set><ci id="S2.SS2.SSS1.p1.14.m14.2.3.3.3.cmml" xref="S2.SS2.SSS1.p1.14.m14.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS1.p1.14.m14.2c">\bar{1}\stackrel{{\scriptstyle\rm def}}{{=}}\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS1.p1.14.m14.2d">over¯ start_ARG 1 end_ARG start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> to it, we get <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS1.p1.14.2">monotone circuit complexity</em> instead of circuit complexity.</p> </div> </section> <section class="ltx_subsubsection ltx_indent_first" id="S2.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.2.2 </span>Bipartite graph complexity</h4> <div class="ltx_para" id="S2.SS2.SSS2.p1"> <p class="ltx_p" id="S2.SS2.SSS2.p1.22">Let <math alttext="\Gamma=[N]\times[M]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.1.m1.2"><semantics id="S2.SS2.SSS2.p1.1.m1.2a"><mrow id="S2.SS2.SSS2.p1.1.m1.2.3" xref="S2.SS2.SSS2.p1.1.m1.2.3.cmml"><mi id="S2.SS2.SSS2.p1.1.m1.2.3.2" mathvariant="normal" xref="S2.SS2.SSS2.p1.1.m1.2.3.2.cmml">Γ</mi><mo id="S2.SS2.SSS2.p1.1.m1.2.3.1" xref="S2.SS2.SSS2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS2.SSS2.p1.1.m1.2.3.3" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.cmml"><mrow id="S2.SS2.SSS2.p1.1.m1.2.3.3.2.2" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS2.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.1.m1.1.1" xref="S2.SS2.SSS2.p1.1.m1.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS2.p1.1.m1.2.3.3.1" rspace="0.222em" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">×</mo><mrow id="S2.SS2.SSS2.p1.1.m1.2.3.3.3.2" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.3.1.cmml"><mo id="S2.SS2.SSS2.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.1.m1.2.2" xref="S2.SS2.SSS2.p1.1.m1.2.2.cmml">M</mi><mo id="S2.SS2.SSS2.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.1.m1.2b"><apply id="S2.SS2.SSS2.p1.1.m1.2.3.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3"><eq id="S2.SS2.SSS2.p1.1.m1.2.3.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.1"></eq><ci id="S2.SS2.SSS2.p1.1.m1.2.3.2.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.2">Γ</ci><apply id="S2.SS2.SSS2.p1.1.m1.2.3.3.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3"><times id="S2.SS2.SSS2.p1.1.m1.2.3.3.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.1"></times><apply id="S2.SS2.SSS2.p1.1.m1.2.3.3.2.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.1.m1.2.3.3.2.1.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS2.p1.1.m1.2.3.3.3.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.1.m1.2.3.3.3.1.1.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.1.m1.2.2.cmml" xref="S2.SS2.SSS2.p1.1.m1.2.2">𝑀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.1.m1.2c">\Gamma=[N]\times[M]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.1.m1.2d">roman_Γ = [ italic_N ] × [ italic_M ]</annotation></semantics></math>, where <math alttext="N,M\in\mathbb{N}^{+}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.2.m2.2"><semantics id="S2.SS2.SSS2.p1.2.m2.2a"><mrow id="S2.SS2.SSS2.p1.2.m2.2.3" xref="S2.SS2.SSS2.p1.2.m2.2.3.cmml"><mrow id="S2.SS2.SSS2.p1.2.m2.2.3.2.2" xref="S2.SS2.SSS2.p1.2.m2.2.3.2.1.cmml"><mi id="S2.SS2.SSS2.p1.2.m2.1.1" xref="S2.SS2.SSS2.p1.2.m2.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.2.m2.2.3.2.2.1" xref="S2.SS2.SSS2.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S2.SS2.SSS2.p1.2.m2.2.2" xref="S2.SS2.SSS2.p1.2.m2.2.2.cmml">M</mi></mrow><mo id="S2.SS2.SSS2.p1.2.m2.2.3.1" xref="S2.SS2.SSS2.p1.2.m2.2.3.1.cmml">∈</mo><msup id="S2.SS2.SSS2.p1.2.m2.2.3.3" xref="S2.SS2.SSS2.p1.2.m2.2.3.3.cmml"><mi id="S2.SS2.SSS2.p1.2.m2.2.3.3.2" xref="S2.SS2.SSS2.p1.2.m2.2.3.3.2.cmml">ℕ</mi><mo id="S2.SS2.SSS2.p1.2.m2.2.3.3.3" xref="S2.SS2.SSS2.p1.2.m2.2.3.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.2.m2.2b"><apply id="S2.SS2.SSS2.p1.2.m2.2.3.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3"><in id="S2.SS2.SSS2.p1.2.m2.2.3.1.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.1"></in><list id="S2.SS2.SSS2.p1.2.m2.2.3.2.1.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.2.2"><ci id="S2.SS2.SSS2.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS2.p1.2.m2.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p1.2.m2.2.2.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.2">𝑀</ci></list><apply id="S2.SS2.SSS2.p1.2.m2.2.3.3.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.2.m2.2.3.3.1.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.3">superscript</csymbol><ci id="S2.SS2.SSS2.p1.2.m2.2.3.3.2.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.3.2">ℕ</ci><plus id="S2.SS2.SSS2.p1.2.m2.2.3.3.3.cmml" xref="S2.SS2.SSS2.p1.2.m2.2.3.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.2.m2.2c">N,M\in\mathbb{N}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.2.m2.2d">italic_N , italic_M ∈ blackboard_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. A set <math alttext="G\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.3.m3.1"><semantics id="S2.SS2.SSS2.p1.3.m3.1a"><mrow id="S2.SS2.SSS2.p1.3.m3.1.1" xref="S2.SS2.SSS2.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS2.p1.3.m3.1.1.2" xref="S2.SS2.SSS2.p1.3.m3.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS2.p1.3.m3.1.1.1" xref="S2.SS2.SSS2.p1.3.m3.1.1.1.cmml">⊆</mo><mi id="S2.SS2.SSS2.p1.3.m3.1.1.3" mathvariant="normal" xref="S2.SS2.SSS2.p1.3.m3.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.3.m3.1b"><apply id="S2.SS2.SSS2.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS2.p1.3.m3.1.1"><subset id="S2.SS2.SSS2.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS2.p1.3.m3.1.1.1"></subset><ci id="S2.SS2.SSS2.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS2.p1.3.m3.1.1.2">𝐺</ci><ci id="S2.SS2.SSS2.p1.3.m3.1.1.3.cmml" xref="S2.SS2.SSS2.p1.3.m3.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.3.m3.1c">G\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.3.m3.1d">italic_G ⊆ roman_Γ</annotation></semantics></math> can be viewed either as a bipartite graph with parts <math alttext="L=[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.4.m4.1"><semantics id="S2.SS2.SSS2.p1.4.m4.1a"><mrow id="S2.SS2.SSS2.p1.4.m4.1.2" xref="S2.SS2.SSS2.p1.4.m4.1.2.cmml"><mi id="S2.SS2.SSS2.p1.4.m4.1.2.2" xref="S2.SS2.SSS2.p1.4.m4.1.2.2.cmml">L</mi><mo id="S2.SS2.SSS2.p1.4.m4.1.2.1" xref="S2.SS2.SSS2.p1.4.m4.1.2.1.cmml">=</mo><mrow id="S2.SS2.SSS2.p1.4.m4.1.2.3.2" xref="S2.SS2.SSS2.p1.4.m4.1.2.3.1.cmml"><mo id="S2.SS2.SSS2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.4.m4.1.1" xref="S2.SS2.SSS2.p1.4.m4.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p1.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.4.m4.1b"><apply id="S2.SS2.SSS2.p1.4.m4.1.2.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.2"><eq id="S2.SS2.SSS2.p1.4.m4.1.2.1.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.2.1"></eq><ci id="S2.SS2.SSS2.p1.4.m4.1.2.2.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.2.2">𝐿</ci><apply id="S2.SS2.SSS2.p1.4.m4.1.2.3.1.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.4.m4.1.2.3.1.1.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS2.p1.4.m4.1.1">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.4.m4.1c">L=[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.4.m4.1d">italic_L = [ italic_N ]</annotation></semantics></math> and <math alttext="R=[M]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.5.m5.1"><semantics id="S2.SS2.SSS2.p1.5.m5.1a"><mrow id="S2.SS2.SSS2.p1.5.m5.1.2" xref="S2.SS2.SSS2.p1.5.m5.1.2.cmml"><mi id="S2.SS2.SSS2.p1.5.m5.1.2.2" xref="S2.SS2.SSS2.p1.5.m5.1.2.2.cmml">R</mi><mo id="S2.SS2.SSS2.p1.5.m5.1.2.1" xref="S2.SS2.SSS2.p1.5.m5.1.2.1.cmml">=</mo><mrow id="S2.SS2.SSS2.p1.5.m5.1.2.3.2" xref="S2.SS2.SSS2.p1.5.m5.1.2.3.1.cmml"><mo id="S2.SS2.SSS2.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.5.m5.1.1" xref="S2.SS2.SSS2.p1.5.m5.1.1.cmml">M</mi><mo id="S2.SS2.SSS2.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p1.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.5.m5.1b"><apply id="S2.SS2.SSS2.p1.5.m5.1.2.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.2"><eq id="S2.SS2.SSS2.p1.5.m5.1.2.1.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.2.1"></eq><ci id="S2.SS2.SSS2.p1.5.m5.1.2.2.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.2.2">𝑅</ci><apply id="S2.SS2.SSS2.p1.5.m5.1.2.3.1.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.5.m5.1.2.3.1.1.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS2.p1.5.m5.1.1">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.5.m5.1c">R=[M]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.5.m5.1d">italic_R = [ italic_M ]</annotation></semantics></math>, or as an <math alttext="N\times M" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.6.m6.1"><semantics id="S2.SS2.SSS2.p1.6.m6.1a"><mrow id="S2.SS2.SSS2.p1.6.m6.1.1" xref="S2.SS2.SSS2.p1.6.m6.1.1.cmml"><mi id="S2.SS2.SSS2.p1.6.m6.1.1.2" xref="S2.SS2.SSS2.p1.6.m6.1.1.2.cmml">N</mi><mo id="S2.SS2.SSS2.p1.6.m6.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.SSS2.p1.6.m6.1.1.1.cmml">×</mo><mi id="S2.SS2.SSS2.p1.6.m6.1.1.3" xref="S2.SS2.SSS2.p1.6.m6.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.6.m6.1b"><apply id="S2.SS2.SSS2.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS2.p1.6.m6.1.1"><times id="S2.SS2.SSS2.p1.6.m6.1.1.1.cmml" xref="S2.SS2.SSS2.p1.6.m6.1.1.1"></times><ci id="S2.SS2.SSS2.p1.6.m6.1.1.2.cmml" xref="S2.SS2.SSS2.p1.6.m6.1.1.2">𝑁</ci><ci id="S2.SS2.SSS2.p1.6.m6.1.1.3.cmml" xref="S2.SS2.SSS2.p1.6.m6.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.6.m6.1c">N\times M</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.6.m6.1d">italic_N × italic_M</annotation></semantics></math> <math alttext="\{0,1\}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.7.m7.2"><semantics id="S2.SS2.SSS2.p1.7.m7.2a"><mrow id="S2.SS2.SSS2.p1.7.m7.2.3.2" xref="S2.SS2.SSS2.p1.7.m7.2.3.1.cmml"><mo id="S2.SS2.SSS2.p1.7.m7.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.7.m7.2.3.1.cmml">{</mo><mn id="S2.SS2.SSS2.p1.7.m7.1.1" xref="S2.SS2.SSS2.p1.7.m7.1.1.cmml">0</mn><mo id="S2.SS2.SSS2.p1.7.m7.2.3.2.2" xref="S2.SS2.SSS2.p1.7.m7.2.3.1.cmml">,</mo><mn id="S2.SS2.SSS2.p1.7.m7.2.2" xref="S2.SS2.SSS2.p1.7.m7.2.2.cmml">1</mn><mo id="S2.SS2.SSS2.p1.7.m7.2.3.2.3" stretchy="false" xref="S2.SS2.SSS2.p1.7.m7.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.7.m7.2b"><set id="S2.SS2.SSS2.p1.7.m7.2.3.1.cmml" xref="S2.SS2.SSS2.p1.7.m7.2.3.2"><cn id="S2.SS2.SSS2.p1.7.m7.1.1.cmml" type="integer" xref="S2.SS2.SSS2.p1.7.m7.1.1">0</cn><cn id="S2.SS2.SSS2.p1.7.m7.2.2.cmml" type="integer" xref="S2.SS2.SSS2.p1.7.m7.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.7.m7.2c">\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.7.m7.2d">{ 0 , 1 }</annotation></semantics></math>-valued matrix. We let <math alttext="R_{i}\subseteq[N]\times[M]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.8.m8.2"><semantics id="S2.SS2.SSS2.p1.8.m8.2a"><mrow id="S2.SS2.SSS2.p1.8.m8.2.3" xref="S2.SS2.SSS2.p1.8.m8.2.3.cmml"><msub id="S2.SS2.SSS2.p1.8.m8.2.3.2" xref="S2.SS2.SSS2.p1.8.m8.2.3.2.cmml"><mi id="S2.SS2.SSS2.p1.8.m8.2.3.2.2" xref="S2.SS2.SSS2.p1.8.m8.2.3.2.2.cmml">R</mi><mi id="S2.SS2.SSS2.p1.8.m8.2.3.2.3" xref="S2.SS2.SSS2.p1.8.m8.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS2.SSS2.p1.8.m8.2.3.1" xref="S2.SS2.SSS2.p1.8.m8.2.3.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p1.8.m8.2.3.3" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.cmml"><mrow id="S2.SS2.SSS2.p1.8.m8.2.3.3.2.2" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS2.p1.8.m8.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.8.m8.1.1" xref="S2.SS2.SSS2.p1.8.m8.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.8.m8.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS2.p1.8.m8.2.3.3.1" rspace="0.222em" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.1.cmml">×</mo><mrow id="S2.SS2.SSS2.p1.8.m8.2.3.3.3.2" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.3.1.cmml"><mo id="S2.SS2.SSS2.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.8.m8.2.2" xref="S2.SS2.SSS2.p1.8.m8.2.2.cmml">M</mi><mo id="S2.SS2.SSS2.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.8.m8.2b"><apply id="S2.SS2.SSS2.p1.8.m8.2.3.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3"><subset id="S2.SS2.SSS2.p1.8.m8.2.3.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.1"></subset><apply id="S2.SS2.SSS2.p1.8.m8.2.3.2.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.8.m8.2.3.2.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.2">subscript</csymbol><ci id="S2.SS2.SSS2.p1.8.m8.2.3.2.2.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.2.2">𝑅</ci><ci id="S2.SS2.SSS2.p1.8.m8.2.3.2.3.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.2.3">𝑖</ci></apply><apply id="S2.SS2.SSS2.p1.8.m8.2.3.3.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3"><times id="S2.SS2.SSS2.p1.8.m8.2.3.3.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.1"></times><apply id="S2.SS2.SSS2.p1.8.m8.2.3.3.2.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.8.m8.2.3.3.2.1.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.8.m8.1.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS2.p1.8.m8.2.3.3.3.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.8.m8.2.3.3.3.1.1.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.8.m8.2.2.cmml" xref="S2.SS2.SSS2.p1.8.m8.2.2">𝑀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.8.m8.2c">R_{i}\subseteq[N]\times[M]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.8.m8.2d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ [ italic_N ] × [ italic_M ]</annotation></semantics></math> denote the matrix with <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.9.m9.1"><semantics id="S2.SS2.SSS2.p1.9.m9.1a"><mn id="S2.SS2.SSS2.p1.9.m9.1.1" xref="S2.SS2.SSS2.p1.9.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.9.m9.1b"><cn id="S2.SS2.SSS2.p1.9.m9.1.1.cmml" type="integer" xref="S2.SS2.SSS2.p1.9.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.9.m9.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.9.m9.1d">1</annotation></semantics></math>’s in the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.10.m10.1"><semantics id="S2.SS2.SSS2.p1.10.m10.1a"><mi id="S2.SS2.SSS2.p1.10.m10.1.1" xref="S2.SS2.SSS2.p1.10.m10.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.10.m10.1b"><ci id="S2.SS2.SSS2.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS2.p1.10.m10.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.10.m10.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.10.m10.1d">italic_i</annotation></semantics></math>-th row, and <math alttext="0" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.11.m11.1"><semantics id="S2.SS2.SSS2.p1.11.m11.1a"><mn id="S2.SS2.SSS2.p1.11.m11.1.1" xref="S2.SS2.SSS2.p1.11.m11.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.11.m11.1b"><cn id="S2.SS2.SSS2.p1.11.m11.1.1.cmml" type="integer" xref="S2.SS2.SSS2.p1.11.m11.1.1">0</cn></annotation-xml></semantics></math>’s elsewhere. Similarly, <math alttext="C_{j}\subseteq[N]\times[M]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.12.m12.2"><semantics id="S2.SS2.SSS2.p1.12.m12.2a"><mrow id="S2.SS2.SSS2.p1.12.m12.2.3" xref="S2.SS2.SSS2.p1.12.m12.2.3.cmml"><msub id="S2.SS2.SSS2.p1.12.m12.2.3.2" xref="S2.SS2.SSS2.p1.12.m12.2.3.2.cmml"><mi id="S2.SS2.SSS2.p1.12.m12.2.3.2.2" xref="S2.SS2.SSS2.p1.12.m12.2.3.2.2.cmml">C</mi><mi id="S2.SS2.SSS2.p1.12.m12.2.3.2.3" xref="S2.SS2.SSS2.p1.12.m12.2.3.2.3.cmml">j</mi></msub><mo id="S2.SS2.SSS2.p1.12.m12.2.3.1" xref="S2.SS2.SSS2.p1.12.m12.2.3.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p1.12.m12.2.3.3" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.cmml"><mrow id="S2.SS2.SSS2.p1.12.m12.2.3.3.2.2" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS2.p1.12.m12.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.12.m12.1.1" xref="S2.SS2.SSS2.p1.12.m12.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.12.m12.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS2.p1.12.m12.2.3.3.1" rspace="0.222em" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.1.cmml">×</mo><mrow id="S2.SS2.SSS2.p1.12.m12.2.3.3.3.2" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.3.1.cmml"><mo id="S2.SS2.SSS2.p1.12.m12.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p1.12.m12.2.2" xref="S2.SS2.SSS2.p1.12.m12.2.2.cmml">M</mi><mo id="S2.SS2.SSS2.p1.12.m12.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.12.m12.2b"><apply id="S2.SS2.SSS2.p1.12.m12.2.3.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3"><subset id="S2.SS2.SSS2.p1.12.m12.2.3.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.1"></subset><apply id="S2.SS2.SSS2.p1.12.m12.2.3.2.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.12.m12.2.3.2.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.2">subscript</csymbol><ci id="S2.SS2.SSS2.p1.12.m12.2.3.2.2.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.2.2">𝐶</ci><ci id="S2.SS2.SSS2.p1.12.m12.2.3.2.3.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.2.3">𝑗</ci></apply><apply id="S2.SS2.SSS2.p1.12.m12.2.3.3.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3"><times id="S2.SS2.SSS2.p1.12.m12.2.3.3.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.1"></times><apply id="S2.SS2.SSS2.p1.12.m12.2.3.3.2.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.12.m12.2.3.3.2.1.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS2.p1.12.m12.2.3.3.3.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.12.m12.2.3.3.3.1.1.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p1.12.m12.2.2.cmml" xref="S2.SS2.SSS2.p1.12.m12.2.2">𝑀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.12.m12.2c">C_{j}\subseteq[N]\times[M]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.12.m12.2d">italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊆ [ italic_N ] × [ italic_M ]</annotation></semantics></math> denotes the matrix with <math alttext="1" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.13.m13.1"><semantics id="S2.SS2.SSS2.p1.13.m13.1a"><mn id="S2.SS2.SSS2.p1.13.m13.1.1" xref="S2.SS2.SSS2.p1.13.m13.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.13.m13.1b"><cn id="S2.SS2.SSS2.p1.13.m13.1.1.cmml" type="integer" xref="S2.SS2.SSS2.p1.13.m13.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.13.m13.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.13.m13.1d">1</annotation></semantics></math>’s in the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.14.m14.1"><semantics id="S2.SS2.SSS2.p1.14.m14.1a"><mi id="S2.SS2.SSS2.p1.14.m14.1.1" xref="S2.SS2.SSS2.p1.14.m14.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.14.m14.1b"><ci id="S2.SS2.SSS2.p1.14.m14.1.1.cmml" xref="S2.SS2.SSS2.p1.14.m14.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.14.m14.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.14.m14.1d">italic_j</annotation></semantics></math>-th column, and <math alttext="0" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.15.m15.1"><semantics id="S2.SS2.SSS2.p1.15.m15.1a"><mn id="S2.SS2.SSS2.p1.15.m15.1.1" xref="S2.SS2.SSS2.p1.15.m15.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.15.m15.1b"><cn id="S2.SS2.SSS2.p1.15.m15.1.1.cmml" type="integer" xref="S2.SS2.SSS2.p1.15.m15.1.1">0</cn></annotation-xml></semantics></math>’s elsewhere. (Each <math alttext="R_{i}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.16.m16.1"><semantics id="S2.SS2.SSS2.p1.16.m16.1a"><msub id="S2.SS2.SSS2.p1.16.m16.1.1" xref="S2.SS2.SSS2.p1.16.m16.1.1.cmml"><mi id="S2.SS2.SSS2.p1.16.m16.1.1.2" xref="S2.SS2.SSS2.p1.16.m16.1.1.2.cmml">R</mi><mi id="S2.SS2.SSS2.p1.16.m16.1.1.3" xref="S2.SS2.SSS2.p1.16.m16.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.16.m16.1b"><apply id="S2.SS2.SSS2.p1.16.m16.1.1.cmml" xref="S2.SS2.SSS2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.16.m16.1.1.1.cmml" xref="S2.SS2.SSS2.p1.16.m16.1.1">subscript</csymbol><ci id="S2.SS2.SSS2.p1.16.m16.1.1.2.cmml" xref="S2.SS2.SSS2.p1.16.m16.1.1.2">𝑅</ci><ci id="S2.SS2.SSS2.p1.16.m16.1.1.3.cmml" xref="S2.SS2.SSS2.p1.16.m16.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.16.m16.1c">R_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.16.m16.1d">italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="C_{j}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.17.m17.1"><semantics id="S2.SS2.SSS2.p1.17.m17.1a"><msub id="S2.SS2.SSS2.p1.17.m17.1.1" xref="S2.SS2.SSS2.p1.17.m17.1.1.cmml"><mi id="S2.SS2.SSS2.p1.17.m17.1.1.2" xref="S2.SS2.SSS2.p1.17.m17.1.1.2.cmml">C</mi><mi id="S2.SS2.SSS2.p1.17.m17.1.1.3" xref="S2.SS2.SSS2.p1.17.m17.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.17.m17.1b"><apply id="S2.SS2.SSS2.p1.17.m17.1.1.cmml" xref="S2.SS2.SSS2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.17.m17.1.1.1.cmml" xref="S2.SS2.SSS2.p1.17.m17.1.1">subscript</csymbol><ci id="S2.SS2.SSS2.p1.17.m17.1.1.2.cmml" xref="S2.SS2.SSS2.p1.17.m17.1.1.2">𝐶</ci><ci id="S2.SS2.SSS2.p1.17.m17.1.1.3.cmml" xref="S2.SS2.SSS2.p1.17.m17.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.17.m17.1c">C_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.17.m17.1d">italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is called a <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p1.22.1">star</em> in graph terminology). We let <math alttext="\mathcal{G}_{N,M}=\{R_{1},\ldots,R_{N},C_{1},\ldots,C_{M}\}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.18.m18.8"><semantics id="S2.SS2.SSS2.p1.18.m18.8a"><mrow id="S2.SS2.SSS2.p1.18.m18.8.8" xref="S2.SS2.SSS2.p1.18.m18.8.8.cmml"><msub id="S2.SS2.SSS2.p1.18.m18.8.8.6" xref="S2.SS2.SSS2.p1.18.m18.8.8.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS2.p1.18.m18.8.8.6.2" xref="S2.SS2.SSS2.p1.18.m18.8.8.6.2.cmml">𝒢</mi><mrow id="S2.SS2.SSS2.p1.18.m18.2.2.2.4" xref="S2.SS2.SSS2.p1.18.m18.2.2.2.3.cmml"><mi id="S2.SS2.SSS2.p1.18.m18.1.1.1.1" xref="S2.SS2.SSS2.p1.18.m18.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.18.m18.2.2.2.4.1" xref="S2.SS2.SSS2.p1.18.m18.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS2.p1.18.m18.2.2.2.2" xref="S2.SS2.SSS2.p1.18.m18.2.2.2.2.cmml">M</mi></mrow></msub><mo id="S2.SS2.SSS2.p1.18.m18.8.8.5" xref="S2.SS2.SSS2.p1.18.m18.8.8.5.cmml">=</mo><mrow id="S2.SS2.SSS2.p1.18.m18.8.8.4.4" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml"><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.5" stretchy="false" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">{</mo><msub id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.cmml"><mi id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.2" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.2.cmml">R</mi><mn id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.3" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.6" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">,</mo><mi id="S2.SS2.SSS2.p1.18.m18.3.3" mathvariant="normal" xref="S2.SS2.SSS2.p1.18.m18.3.3.cmml">…</mi><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.7" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">,</mo><msub id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.cmml"><mi id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.2" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.2.cmml">R</mi><mi id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.3" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.3.cmml">N</mi></msub><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.8" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">,</mo><msub id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.cmml"><mi id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.2" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.2.cmml">C</mi><mn id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.3" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.3.cmml">1</mn></msub><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.9" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">,</mo><mi id="S2.SS2.SSS2.p1.18.m18.4.4" mathvariant="normal" xref="S2.SS2.SSS2.p1.18.m18.4.4.cmml">…</mi><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.10" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">,</mo><msub id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.cmml"><mi id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.2" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.2.cmml">C</mi><mi id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.3" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.3.cmml">M</mi></msub><mo id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.11" stretchy="false" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.18.m18.8b"><apply id="S2.SS2.SSS2.p1.18.m18.8.8.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8"><eq id="S2.SS2.SSS2.p1.18.m18.8.8.5.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.5"></eq><apply id="S2.SS2.SSS2.p1.18.m18.8.8.6.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.6"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.18.m18.8.8.6.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.6">subscript</csymbol><ci id="S2.SS2.SSS2.p1.18.m18.8.8.6.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.6.2">𝒢</ci><list id="S2.SS2.SSS2.p1.18.m18.2.2.2.3.cmml" xref="S2.SS2.SSS2.p1.18.m18.2.2.2.4"><ci id="S2.SS2.SSS2.p1.18.m18.1.1.1.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p1.18.m18.2.2.2.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.2.2.2.2">𝑀</ci></list></apply><set id="S2.SS2.SSS2.p1.18.m18.8.8.4.5.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4"><apply id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.2">𝑅</ci><cn id="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS2.p1.18.m18.5.5.1.1.1.3">1</cn></apply><ci id="S2.SS2.SSS2.p1.18.m18.3.3.cmml" xref="S2.SS2.SSS2.p1.18.m18.3.3">…</ci><apply id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2">subscript</csymbol><ci id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.2">𝑅</ci><ci id="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.3.cmml" xref="S2.SS2.SSS2.p1.18.m18.6.6.2.2.2.3">𝑁</ci></apply><apply id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.cmml" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3">subscript</csymbol><ci id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.2">𝐶</ci><cn id="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.3.cmml" type="integer" xref="S2.SS2.SSS2.p1.18.m18.7.7.3.3.3.3">1</cn></apply><ci id="S2.SS2.SSS2.p1.18.m18.4.4.cmml" xref="S2.SS2.SSS2.p1.18.m18.4.4">…</ci><apply id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.1.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4">subscript</csymbol><ci id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.2.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.2">𝐶</ci><ci id="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.3.cmml" xref="S2.SS2.SSS2.p1.18.m18.8.8.4.4.4.3">𝑀</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.18.m18.8c">\mathcal{G}_{N,M}=\{R_{1},\ldots,R_{N},C_{1},\ldots,C_{M}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.18.m18.8d">caligraphic_G start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT = { italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_C start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT }</annotation></semantics></math>. The value <math alttext="D(G\mid\mathcal{G}_{N,M})" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.19.m19.3"><semantics id="S2.SS2.SSS2.p1.19.m19.3a"><mrow id="S2.SS2.SSS2.p1.19.m19.3.3" xref="S2.SS2.SSS2.p1.19.m19.3.3.cmml"><mi id="S2.SS2.SSS2.p1.19.m19.3.3.3" xref="S2.SS2.SSS2.p1.19.m19.3.3.3.cmml">D</mi><mo id="S2.SS2.SSS2.p1.19.m19.3.3.2" xref="S2.SS2.SSS2.p1.19.m19.3.3.2.cmml">⁢</mo><mrow id="S2.SS2.SSS2.p1.19.m19.3.3.1.1" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.cmml"><mo id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.2" stretchy="false" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.cmml"><mi id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.2" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.1" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.1.cmml">∣</mo><msub id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.2" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS2.SSS2.p1.19.m19.2.2.2.4" xref="S2.SS2.SSS2.p1.19.m19.2.2.2.3.cmml"><mi id="S2.SS2.SSS2.p1.19.m19.1.1.1.1" xref="S2.SS2.SSS2.p1.19.m19.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.19.m19.2.2.2.4.1" xref="S2.SS2.SSS2.p1.19.m19.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS2.p1.19.m19.2.2.2.2" xref="S2.SS2.SSS2.p1.19.m19.2.2.2.2.cmml">M</mi></mrow></msub></mrow><mo id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.3" stretchy="false" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.19.m19.3b"><apply id="S2.SS2.SSS2.p1.19.m19.3.3.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3"><times id="S2.SS2.SSS2.p1.19.m19.3.3.2.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.2"></times><ci id="S2.SS2.SSS2.p1.19.m19.3.3.3.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.3">𝐷</ci><apply id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1"><csymbol cd="latexml" id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.1.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.1">conditional</csymbol><ci id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.2.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.2">𝐺</ci><apply id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.1.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.2.cmml" xref="S2.SS2.SSS2.p1.19.m19.3.3.1.1.1.3.2">𝒢</ci><list id="S2.SS2.SSS2.p1.19.m19.2.2.2.3.cmml" xref="S2.SS2.SSS2.p1.19.m19.2.2.2.4"><ci id="S2.SS2.SSS2.p1.19.m19.1.1.1.1.cmml" xref="S2.SS2.SSS2.p1.19.m19.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p1.19.m19.2.2.2.2.cmml" xref="S2.SS2.SSS2.p1.19.m19.2.2.2.2">𝑀</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.19.m19.3c">D(G\mid\mathcal{G}_{N,M})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.19.m19.3d">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT )</annotation></semantics></math> is known as the <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p1.22.2">star complexity</em> of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.20.m20.1"><semantics id="S2.SS2.SSS2.p1.20.m20.1a"><mi id="S2.SS2.SSS2.p1.20.m20.1.1" xref="S2.SS2.SSS2.p1.20.m20.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.20.m20.1b"><ci id="S2.SS2.SSS2.p1.20.m20.1.1.cmml" xref="S2.SS2.SSS2.p1.20.m20.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.20.m20.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.20.m20.1d">italic_G</annotation></semantics></math> (<cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib14" title="">14</a>]</cite>, see also <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite> and references therein). We will refer to it simply as <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p1.22.3">graph complexity</em>. Notice that, for every non-empty graph <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.21.m21.1"><semantics id="S2.SS2.SSS2.p1.21.m21.1a"><mi id="S2.SS2.SSS2.p1.21.m21.1.1" xref="S2.SS2.SSS2.p1.21.m21.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.21.m21.1b"><ci id="S2.SS2.SSS2.p1.21.m21.1.1.cmml" xref="S2.SS2.SSS2.p1.21.m21.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.21.m21.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.21.m21.1d">italic_G</annotation></semantics></math>, <math alttext="D_{\cap}(G\mid\mathcal{G}_{N,M})\leq\min\{N,M\}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p1.22.m22.6"><semantics id="S2.SS2.SSS2.p1.22.m22.6a"><mrow id="S2.SS2.SSS2.p1.22.m22.6.6" xref="S2.SS2.SSS2.p1.22.m22.6.6.cmml"><mrow id="S2.SS2.SSS2.p1.22.m22.6.6.1" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.cmml"><msub id="S2.SS2.SSS2.p1.22.m22.6.6.1.3" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.3.cmml"><mi id="S2.SS2.SSS2.p1.22.m22.6.6.1.3.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.3.2.cmml">D</mi><mo id="S2.SS2.SSS2.p1.22.m22.6.6.1.3.3" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.3.3.cmml">∩</mo></msub><mo id="S2.SS2.SSS2.p1.22.m22.6.6.1.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.cmml"><mo id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.2" stretchy="false" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.cmml"><mi id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.1" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.3" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.3.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS2.SSS2.p1.22.m22.2.2.2.4" xref="S2.SS2.SSS2.p1.22.m22.2.2.2.3.cmml"><mi id="S2.SS2.SSS2.p1.22.m22.1.1.1.1" xref="S2.SS2.SSS2.p1.22.m22.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p1.22.m22.2.2.2.4.1" xref="S2.SS2.SSS2.p1.22.m22.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS2.p1.22.m22.2.2.2.2" xref="S2.SS2.SSS2.p1.22.m22.2.2.2.2.cmml">M</mi></mrow></msub></mrow><mo id="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.3" stretchy="false" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS2.p1.22.m22.6.6.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.2.cmml">≤</mo><mrow id="S2.SS2.SSS2.p1.22.m22.6.6.3.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml"><mi id="S2.SS2.SSS2.p1.22.m22.3.3" xref="S2.SS2.SSS2.p1.22.m22.3.3.cmml">min</mi><mo id="S2.SS2.SSS2.p1.22.m22.6.6.3.2a" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml">⁡</mo><mrow id="S2.SS2.SSS2.p1.22.m22.6.6.3.2.1" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml"><mo id="S2.SS2.SSS2.p1.22.m22.6.6.3.2.1.1" stretchy="false" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml">{</mo><mi id="S2.SS2.SSS2.p1.22.m22.4.4" xref="S2.SS2.SSS2.p1.22.m22.4.4.cmml">N</mi><mo id="S2.SS2.SSS2.p1.22.m22.6.6.3.2.1.2" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml">,</mo><mi id="S2.SS2.SSS2.p1.22.m22.5.5" xref="S2.SS2.SSS2.p1.22.m22.5.5.cmml">M</mi><mo id="S2.SS2.SSS2.p1.22.m22.6.6.3.2.1.3" stretchy="false" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p1.22.m22.6b"><apply id="S2.SS2.SSS2.p1.22.m22.6.6.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6"><leq id="S2.SS2.SSS2.p1.22.m22.6.6.2.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6.2"></leq><apply id="S2.SS2.SSS2.p1.22.m22.6.6.1.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6.1"><times id="S2.SS2.SSS2.p1.22.m22.6.6.1.2.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.2"></times><apply id="S2.SS2.SSS2.p1.22.m22.6.6.1.3.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6.1.3"><csymbol cd="ambiguous" 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xref="S2.SS2.SSS2.p1.22.m22.2.2.2.4"><ci id="S2.SS2.SSS2.p1.22.m22.1.1.1.1.cmml" xref="S2.SS2.SSS2.p1.22.m22.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p1.22.m22.2.2.2.2.cmml" xref="S2.SS2.SSS2.p1.22.m22.2.2.2.2">𝑀</ci></list></apply></apply></apply><apply id="S2.SS2.SSS2.p1.22.m22.6.6.3.1.cmml" xref="S2.SS2.SSS2.p1.22.m22.6.6.3.2"><min id="S2.SS2.SSS2.p1.22.m22.3.3.cmml" xref="S2.SS2.SSS2.p1.22.m22.3.3"></min><ci id="S2.SS2.SSS2.p1.22.m22.4.4.cmml" xref="S2.SS2.SSS2.p1.22.m22.4.4">𝑁</ci><ci id="S2.SS2.SSS2.p1.22.m22.5.5.cmml" xref="S2.SS2.SSS2.p1.22.m22.5.5">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p1.22.m22.6c">D_{\cap}(G\mid\mathcal{G}_{N,M})\leq\min\{N,M\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p1.22.m22.6d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT ) ≤ roman_min { italic_N , italic_M }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS2.p2"> <p class="ltx_p" id="S2.SS2.SSS2.p2.7">We remark that a related notion of <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p2.7.1">clique complexity</em> is discussed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib7" title="">7</a>]</cite>. In this notion, the generators are sets of the form <math alttext="W_{S}:=\bigcup_{i\in S}R_{i}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.1.m1.1"><semantics id="S2.SS2.SSS2.p2.1.m1.1a"><mrow id="S2.SS2.SSS2.p2.1.m1.1.1" xref="S2.SS2.SSS2.p2.1.m1.1.1.cmml"><msub id="S2.SS2.SSS2.p2.1.m1.1.1.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.2.cmml"><mi id="S2.SS2.SSS2.p2.1.m1.1.1.2.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.2.2.cmml">W</mi><mi id="S2.SS2.SSS2.p2.1.m1.1.1.2.3" xref="S2.SS2.SSS2.p2.1.m1.1.1.2.3.cmml">S</mi></msub><mo id="S2.SS2.SSS2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.111em" xref="S2.SS2.SSS2.p2.1.m1.1.1.1.cmml">:=</mo><mrow id="S2.SS2.SSS2.p2.1.m1.1.1.3" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.cmml"><msub id="S2.SS2.SSS2.p2.1.m1.1.1.3.1" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.cmml"><mo id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.cmml"><mi id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.2.cmml">i</mi><mo id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.1" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.1.cmml">∈</mo><mi id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.3" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.3.cmml">S</mi></mrow></msub><msub id="S2.SS2.SSS2.p2.1.m1.1.1.3.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2.cmml"><mi id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.2" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2.2.cmml">R</mi><mi id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.3" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.1.m1.1b"><apply id="S2.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.1">assign</csymbol><apply id="S2.SS2.SSS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.1.m1.1.1.2.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS2.p2.1.m1.1.1.2.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.2.2">𝑊</ci><ci id="S2.SS2.SSS2.p2.1.m1.1.1.2.3.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.2.3">𝑆</ci></apply><apply id="S2.SS2.SSS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3"><apply id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1">subscript</csymbol><union id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.2"></union><apply id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3"><in id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.1"></in><ci id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.2">𝑖</ci><ci id="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.3.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.1.3.3">𝑆</ci></apply></apply><apply id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.1.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2">subscript</csymbol><ci id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.2.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2.2">𝑅</ci><ci id="S2.SS2.SSS2.p2.1.m1.1.1.3.2.3.cmml" xref="S2.SS2.SSS2.p2.1.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.1.m1.1c">W_{S}:=\bigcup_{i\in S}R_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.1.m1.1d">italic_W start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT := ⋃ start_POSTSUBSCRIPT italic_i ∈ italic_S end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Z_{T}:=\bigcup_{j\in T}C_{j}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.2.m2.1"><semantics id="S2.SS2.SSS2.p2.2.m2.1a"><mrow id="S2.SS2.SSS2.p2.2.m2.1.1" xref="S2.SS2.SSS2.p2.2.m2.1.1.cmml"><msub id="S2.SS2.SSS2.p2.2.m2.1.1.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS2.SSS2.p2.2.m2.1.1.2.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.2.2.cmml">Z</mi><mi id="S2.SS2.SSS2.p2.2.m2.1.1.2.3" xref="S2.SS2.SSS2.p2.2.m2.1.1.2.3.cmml">T</mi></msub><mo id="S2.SS2.SSS2.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.111em" xref="S2.SS2.SSS2.p2.2.m2.1.1.1.cmml">:=</mo><mrow id="S2.SS2.SSS2.p2.2.m2.1.1.3" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.cmml"><msub id="S2.SS2.SSS2.p2.2.m2.1.1.3.1" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.cmml"><mo id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.cmml"><mi id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.2.cmml">j</mi><mo id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.1" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.1.cmml">∈</mo><mi id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.3" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.3.cmml">T</mi></mrow></msub><msub id="S2.SS2.SSS2.p2.2.m2.1.1.3.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.2" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2.2.cmml">C</mi><mi id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.3" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.2.m2.1b"><apply id="S2.SS2.SSS2.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.1">assign</csymbol><apply id="S2.SS2.SSS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.2.m2.1.1.2.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.SSS2.p2.2.m2.1.1.2.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.2.2">𝑍</ci><ci id="S2.SS2.SSS2.p2.2.m2.1.1.2.3.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.2.3">𝑇</ci></apply><apply id="S2.SS2.SSS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3"><apply id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1">subscript</csymbol><union id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.2"></union><apply id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3"><in id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.1"></in><ci id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.2">𝑗</ci><ci id="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.3.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.1.3.3">𝑇</ci></apply></apply><apply id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.1.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2">subscript</csymbol><ci id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.2.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2.2">𝐶</ci><ci id="S2.SS2.SSS2.p2.2.m2.1.1.3.2.3.cmml" xref="S2.SS2.SSS2.p2.2.m2.1.1.3.2.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.2.m2.1c">Z_{T}:=\bigcup_{j\in T}C_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.2.m2.1d">italic_Z start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT := ⋃ start_POSTSUBSCRIPT italic_j ∈ italic_T end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, for some <math alttext="S\subseteq[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.3.m3.1"><semantics id="S2.SS2.SSS2.p2.3.m3.1a"><mrow id="S2.SS2.SSS2.p2.3.m3.1.2" xref="S2.SS2.SSS2.p2.3.m3.1.2.cmml"><mi id="S2.SS2.SSS2.p2.3.m3.1.2.2" xref="S2.SS2.SSS2.p2.3.m3.1.2.2.cmml">S</mi><mo id="S2.SS2.SSS2.p2.3.m3.1.2.1" xref="S2.SS2.SSS2.p2.3.m3.1.2.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p2.3.m3.1.2.3.2" xref="S2.SS2.SSS2.p2.3.m3.1.2.3.1.cmml"><mo id="S2.SS2.SSS2.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p2.3.m3.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p2.3.m3.1.1" xref="S2.SS2.SSS2.p2.3.m3.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p2.3.m3.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.3.m3.1b"><apply id="S2.SS2.SSS2.p2.3.m3.1.2.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.2"><subset id="S2.SS2.SSS2.p2.3.m3.1.2.1.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.2.1"></subset><ci id="S2.SS2.SSS2.p2.3.m3.1.2.2.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.2.2">𝑆</ci><apply id="S2.SS2.SSS2.p2.3.m3.1.2.3.1.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.3.m3.1.2.3.1.1.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS2.p2.3.m3.1.1">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.3.m3.1c">S\subseteq[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.3.m3.1d">italic_S ⊆ [ italic_N ]</annotation></semantics></math> and <math alttext="T\subseteq[M]" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.4.m4.1"><semantics id="S2.SS2.SSS2.p2.4.m4.1a"><mrow id="S2.SS2.SSS2.p2.4.m4.1.2" xref="S2.SS2.SSS2.p2.4.m4.1.2.cmml"><mi id="S2.SS2.SSS2.p2.4.m4.1.2.2" xref="S2.SS2.SSS2.p2.4.m4.1.2.2.cmml">T</mi><mo id="S2.SS2.SSS2.p2.4.m4.1.2.1" xref="S2.SS2.SSS2.p2.4.m4.1.2.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p2.4.m4.1.2.3.2" xref="S2.SS2.SSS2.p2.4.m4.1.2.3.1.cmml"><mo id="S2.SS2.SSS2.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p2.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p2.4.m4.1.1" xref="S2.SS2.SSS2.p2.4.m4.1.1.cmml">M</mi><mo id="S2.SS2.SSS2.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p2.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.4.m4.1b"><apply id="S2.SS2.SSS2.p2.4.m4.1.2.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.2"><subset id="S2.SS2.SSS2.p2.4.m4.1.2.1.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.2.1"></subset><ci id="S2.SS2.SSS2.p2.4.m4.1.2.2.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.2.2">𝑇</ci><apply id="S2.SS2.SSS2.p2.4.m4.1.2.3.1.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.4.m4.1.2.3.1.1.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS2.p2.4.m4.1.1">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.4.m4.1c">T\subseteq[M]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.4.m4.1d">italic_T ⊆ [ italic_M ]</annotation></semantics></math>. Let <math alttext="{\mathcal{K}}_{N,M}=\left\{W_{S}:S\subseteq[N]\right\}\cup\left\{Z_{T}:T% \subseteq[M]\right\}" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.5.m5.8"><semantics id="S2.SS2.SSS2.p2.5.m5.8a"><mrow id="S2.SS2.SSS2.p2.5.m5.8.8" xref="S2.SS2.SSS2.p2.5.m5.8.8.cmml"><msub id="S2.SS2.SSS2.p2.5.m5.8.8.6" xref="S2.SS2.SSS2.p2.5.m5.8.8.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS2.p2.5.m5.8.8.6.2" xref="S2.SS2.SSS2.p2.5.m5.8.8.6.2.cmml">𝒦</mi><mrow id="S2.SS2.SSS2.p2.5.m5.2.2.2.4" xref="S2.SS2.SSS2.p2.5.m5.2.2.2.3.cmml"><mi id="S2.SS2.SSS2.p2.5.m5.1.1.1.1" xref="S2.SS2.SSS2.p2.5.m5.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS2.p2.5.m5.2.2.2.4.1" xref="S2.SS2.SSS2.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS2.p2.5.m5.2.2.2.2" xref="S2.SS2.SSS2.p2.5.m5.2.2.2.2.cmml">M</mi></mrow></msub><mo id="S2.SS2.SSS2.p2.5.m5.8.8.5" xref="S2.SS2.SSS2.p2.5.m5.8.8.5.cmml">=</mo><mrow id="S2.SS2.SSS2.p2.5.m5.8.8.4" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.cmml"><mrow id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.cmml"><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.3" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.1.cmml">{</mo><msub id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.cmml"><mi id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.2" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.2.cmml">W</mi><mi id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.3" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.3.cmml">S</mi></msub><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.1.cmml">:</mo><mrow id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.cmml"><mi id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.2" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.2.cmml">S</mi><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.1" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.2" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.1.cmml"><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p2.5.m5.3.3" xref="S2.SS2.SSS2.p2.5.m5.3.3.cmml">N</mi><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.5" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.1.cmml">}</mo></mrow><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.5" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.5.cmml">∪</mo><mrow id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.cmml"><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.3" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.1.cmml">{</mo><msub id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.cmml"><mi id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.2" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.2.cmml">Z</mi><mi id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.3" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.3.cmml">T</mi></msub><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.4" lspace="0.278em" rspace="0.278em" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.1.cmml">:</mo><mrow id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.cmml"><mi id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.2" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.2.cmml">T</mi><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.1" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.1.cmml">⊆</mo><mrow id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.2" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.1.cmml"><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS2.p2.5.m5.4.4" xref="S2.SS2.SSS2.p2.5.m5.4.4.cmml">M</mi><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.2.2" stretchy="false" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.5" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.5.m5.8b"><apply id="S2.SS2.SSS2.p2.5.m5.8.8.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8"><eq id="S2.SS2.SSS2.p2.5.m5.8.8.5.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.5"></eq><apply id="S2.SS2.SSS2.p2.5.m5.8.8.6.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.6"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.5.m5.8.8.6.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.6">subscript</csymbol><ci id="S2.SS2.SSS2.p2.5.m5.8.8.6.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.6.2">𝒦</ci><list id="S2.SS2.SSS2.p2.5.m5.2.2.2.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.2.2.2.4"><ci id="S2.SS2.SSS2.p2.5.m5.1.1.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p2.5.m5.2.2.2.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.2.2.2.2">𝑀</ci></list></apply><apply id="S2.SS2.SSS2.p2.5.m5.8.8.4.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4"><union id="S2.SS2.SSS2.p2.5.m5.8.8.4.5.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.5"></union><apply id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.3.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.3">conditional-set</csymbol><apply id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.2">𝑊</ci><ci id="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.5.5.1.1.1.1.3">𝑆</ci></apply><apply id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2"><subset id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.1"></subset><ci id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.2">𝑆</ci><apply id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.6.6.2.2.2.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p2.5.m5.3.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.3.3">𝑁</ci></apply></apply></apply><apply id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.3.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.3">conditional-set</csymbol><apply id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1">subscript</csymbol><ci id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.2">𝑍</ci><ci id="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.3.cmml" xref="S2.SS2.SSS2.p2.5.m5.7.7.3.3.1.1.3">𝑇</ci></apply><apply id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2"><subset id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.1"></subset><ci id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.2.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.2">𝑇</ci><apply id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.1.1.cmml" xref="S2.SS2.SSS2.p2.5.m5.8.8.4.4.2.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS2.p2.5.m5.4.4.cmml" xref="S2.SS2.SSS2.p2.5.m5.4.4">𝑀</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.5.m5.8c">{\mathcal{K}}_{N,M}=\left\{W_{S}:S\subseteq[N]\right\}\cup\left\{Z_{T}:T% \subseteq[M]\right\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.5.m5.8d">caligraphic_K start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT = { italic_W start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT : italic_S ⊆ [ italic_N ] } ∪ { italic_Z start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT : italic_T ⊆ [ italic_M ] }</annotation></semantics></math>. Note that the intersection clique complexity of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.6.m6.1"><semantics id="S2.SS2.SSS2.p2.6.m6.1a"><mi id="S2.SS2.SSS2.p2.6.m6.1.1" xref="S2.SS2.SSS2.p2.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS2.p2.6.m6.1b"><ci id="S2.SS2.SSS2.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS2.p2.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.6.m6.1d">italic_G</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p2.7.2">equal</em> to its intersection graph complexity (i.e., <math alttext="D_{\cap}(G\mid{\mathcal{K}}_{N,M})=D_{\cap}(G\mid{\mathcal{G}}_{N,M})" class="ltx_Math" display="inline" id="S2.SS2.SSS2.p2.7.m7.6"><semantics id="S2.SS2.SSS2.p2.7.m7.6a"><mrow id="S2.SS2.SSS2.p2.7.m7.6.6" xref="S2.SS2.SSS2.p2.7.m7.6.6.cmml"><mrow 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xref="S2.SS2.SSS2.p2.7.m7.6.6.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS2.p2.7.m7.6.6.2.1.1.1.3.1.cmml" xref="S2.SS2.SSS2.p2.7.m7.6.6.2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS2.p2.7.m7.6.6.2.1.1.1.3.2.cmml" xref="S2.SS2.SSS2.p2.7.m7.6.6.2.1.1.1.3.2">𝒢</ci><list id="S2.SS2.SSS2.p2.7.m7.4.4.2.3.cmml" xref="S2.SS2.SSS2.p2.7.m7.4.4.2.4"><ci id="S2.SS2.SSS2.p2.7.m7.3.3.1.1.cmml" xref="S2.SS2.SSS2.p2.7.m7.3.3.1.1">𝑁</ci><ci id="S2.SS2.SSS2.p2.7.m7.4.4.2.2.cmml" xref="S2.SS2.SSS2.p2.7.m7.4.4.2.2">𝑀</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS2.p2.7.m7.6c">D_{\cap}(G\mid{\mathcal{K}}_{N,M})=D_{\cap}(G\mid{\mathcal{G}}_{N,M})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS2.p2.7.m7.6d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_K start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT )</annotation></semantics></math>).<span class="ltx_note ltx_role_footnote" id="footnote6"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span> We also remark that the <em class="ltx_emph ltx_font_italic" id="footnote6.1">decision tree clique complexity</em> of a graph <math alttext="G" class="ltx_Math" display="inline" id="footnote6.m1.1"><semantics id="footnote6.m1.1b"><mi id="footnote6.m1.1.1" xref="footnote6.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="footnote6.m1.1c"><ci id="footnote6.m1.1.1.cmml" xref="footnote6.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="footnote6.m1.1e">italic_G</annotation></semantics></math> (in which we are allowed to query an arbitrary generator from <math alttext="{\mathcal{K}}_{N,M}" class="ltx_Math" display="inline" id="footnote6.m2.2"><semantics id="footnote6.m2.2b"><msub id="footnote6.m2.2.3" xref="footnote6.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote6.m2.2.3.2" xref="footnote6.m2.2.3.2.cmml">𝒦</mi><mrow id="footnote6.m2.2.2.2.4" xref="footnote6.m2.2.2.2.3.cmml"><mi id="footnote6.m2.1.1.1.1" xref="footnote6.m2.1.1.1.1.cmml">N</mi><mo id="footnote6.m2.2.2.2.4.1" xref="footnote6.m2.2.2.2.3.cmml">,</mo><mi id="footnote6.m2.2.2.2.2" xref="footnote6.m2.2.2.2.2.cmml">M</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote6.m2.2c"><apply id="footnote6.m2.2.3.cmml" xref="footnote6.m2.2.3"><csymbol cd="ambiguous" id="footnote6.m2.2.3.1.cmml" xref="footnote6.m2.2.3">subscript</csymbol><ci id="footnote6.m2.2.3.2.cmml" xref="footnote6.m2.2.3.2">𝒦</ci><list id="footnote6.m2.2.2.2.3.cmml" xref="footnote6.m2.2.2.2.4"><ci id="footnote6.m2.1.1.1.1.cmml" xref="footnote6.m2.1.1.1.1">𝑁</ci><ci id="footnote6.m2.2.2.2.2.cmml" xref="footnote6.m2.2.2.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m2.2d">{\mathcal{K}}_{N,M}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m2.2e">caligraphic_K start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT</annotation></semantics></math>) is known to capture <em class="ltx_emph ltx_font_italic" id="footnote6.2">exactly</em> the communication complexity of an associated function <math alttext="f_{G}" class="ltx_Math" display="inline" id="footnote6.m3.1"><semantics id="footnote6.m3.1b"><msub id="footnote6.m3.1.1" xref="footnote6.m3.1.1.cmml"><mi id="footnote6.m3.1.1.2" xref="footnote6.m3.1.1.2.cmml">f</mi><mi id="footnote6.m3.1.1.3" xref="footnote6.m3.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="footnote6.m3.1c"><apply id="footnote6.m3.1.1.cmml" xref="footnote6.m3.1.1"><csymbol cd="ambiguous" id="footnote6.m3.1.1.1.cmml" xref="footnote6.m3.1.1">subscript</csymbol><ci id="footnote6.m3.1.1.2.cmml" xref="footnote6.m3.1.1.2">𝑓</ci><ci id="footnote6.m3.1.1.3.cmml" xref="footnote6.m3.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m3.1d">f_{G}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m3.1e">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib14" title="">14</a>, Section 3]</cite>. </span></span></span></p> </div> <div class="ltx_para" id="S2.SS2.SSS2.p3"> <p class="ltx_p" id="S2.SS2.SSS2.p3.1">One can also consider the graph complexity of <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS2.p3.1.1">non-bipartite</em> graphs via an appropriate choice of generators (as in, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite>), though we will not be concerned with this variant in this work.</p> </div> </section> <section class="ltx_subsubsection ltx_indent_first" id="S2.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.2.3 </span>Higher-dimensional generalizations of graph complexity</h4> <div class="ltx_para" id="S2.SS2.SSS3.p1"> <p class="ltx_p" id="S2.SS2.SSS3.p1.12">This is the natural extension of the ambient space <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.1.m1.2"><semantics id="S2.SS2.SSS3.p1.1.m1.2a"><mrow id="S2.SS2.SSS3.p1.1.m1.2.3" xref="S2.SS2.SSS3.p1.1.m1.2.3.cmml"><mrow id="S2.SS2.SSS3.p1.1.m1.2.3.2.2" xref="S2.SS2.SSS3.p1.1.m1.2.3.2.1.cmml"><mo id="S2.SS2.SSS3.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.1.m1.2.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.1.m1.1.1" xref="S2.SS2.SSS3.p1.1.m1.1.1.cmml">N</mi><mo id="S2.SS2.SSS3.p1.1.m1.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS3.p1.1.m1.2.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS3.p1.1.m1.2.3.1" rspace="0.222em" xref="S2.SS2.SSS3.p1.1.m1.2.3.1.cmml">×</mo><mrow id="S2.SS2.SSS3.p1.1.m1.2.3.3.2" xref="S2.SS2.SSS3.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS2.SSS3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.1.m1.2.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.1.m1.2.2" xref="S2.SS2.SSS3.p1.1.m1.2.2.cmml">N</mi><mo id="S2.SS2.SSS3.p1.1.m1.2.3.3.2.2" stretchy="false" xref="S2.SS2.SSS3.p1.1.m1.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.1.m1.2b"><apply id="S2.SS2.SSS3.p1.1.m1.2.3.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3"><times id="S2.SS2.SSS3.p1.1.m1.2.3.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3.1"></times><apply id="S2.SS2.SSS3.p1.1.m1.2.3.2.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.1.m1.2.3.2.1.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS3.p1.1.m1.2.3.3.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.1.m1.2.3.3.1.1.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.1.m1.2.2.cmml" xref="S2.SS2.SSS3.p1.1.m1.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.1.m1.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.1.m1.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math> to <math alttext="[N]^{d}" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.2.m2.1"><semantics id="S2.SS2.SSS3.p1.2.m2.1a"><msup id="S2.SS2.SSS3.p1.2.m2.1.2" xref="S2.SS2.SSS3.p1.2.m2.1.2.cmml"><mrow id="S2.SS2.SSS3.p1.2.m2.1.2.2.2" xref="S2.SS2.SSS3.p1.2.m2.1.2.2.1.cmml"><mo id="S2.SS2.SSS3.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.2.m2.1.2.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.2.m2.1.1" xref="S2.SS2.SSS3.p1.2.m2.1.1.cmml">N</mi><mo id="S2.SS2.SSS3.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S2.SS2.SSS3.p1.2.m2.1.2.2.1.1.cmml">]</mo></mrow><mi id="S2.SS2.SSS3.p1.2.m2.1.2.3" xref="S2.SS2.SSS3.p1.2.m2.1.2.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.2.m2.1b"><apply id="S2.SS2.SSS3.p1.2.m2.1.2.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.2.m2.1.2.1.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.2">superscript</csymbol><apply id="S2.SS2.SSS3.p1.2.m2.1.2.2.1.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.2.2.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.2.m2.1.2.2.1.1.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.2.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.1">𝑁</ci></apply><ci id="S2.SS2.SSS3.p1.2.m2.1.2.3.cmml" xref="S2.SS2.SSS3.p1.2.m2.1.2.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.2.m2.1c">[N]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.2.m2.1d">[ italic_N ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="d\in\mathbb{N}^{+}" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.3.m3.1"><semantics id="S2.SS2.SSS3.p1.3.m3.1a"><mrow id="S2.SS2.SSS3.p1.3.m3.1.1" xref="S2.SS2.SSS3.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS3.p1.3.m3.1.1.2" xref="S2.SS2.SSS3.p1.3.m3.1.1.2.cmml">d</mi><mo id="S2.SS2.SSS3.p1.3.m3.1.1.1" xref="S2.SS2.SSS3.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S2.SS2.SSS3.p1.3.m3.1.1.3" xref="S2.SS2.SSS3.p1.3.m3.1.1.3.cmml"><mi id="S2.SS2.SSS3.p1.3.m3.1.1.3.2" xref="S2.SS2.SSS3.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mo id="S2.SS2.SSS3.p1.3.m3.1.1.3.3" xref="S2.SS2.SSS3.p1.3.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.3.m3.1b"><apply id="S2.SS2.SSS3.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1"><in id="S2.SS2.SSS3.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.1"></in><ci id="S2.SS2.SSS3.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.2">𝑑</ci><apply id="S2.SS2.SSS3.p1.3.m3.1.1.3.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS3.p1.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.3.2">ℕ</ci><plus id="S2.SS2.SSS3.p1.3.m3.1.1.3.3.cmml" xref="S2.SS2.SSS3.p1.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.3.m3.1c">d\in\mathbb{N}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.3.m3.1d">italic_d ∈ blackboard_N start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is a fixed dimension. Every generator contained in <math alttext="[N]^{d}" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.4.m4.1"><semantics id="S2.SS2.SSS3.p1.4.m4.1a"><msup id="S2.SS2.SSS3.p1.4.m4.1.2" xref="S2.SS2.SSS3.p1.4.m4.1.2.cmml"><mrow id="S2.SS2.SSS3.p1.4.m4.1.2.2.2" xref="S2.SS2.SSS3.p1.4.m4.1.2.2.1.cmml"><mo id="S2.SS2.SSS3.p1.4.m4.1.2.2.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.4.m4.1.2.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.4.m4.1.1" xref="S2.SS2.SSS3.p1.4.m4.1.1.cmml">N</mi><mo id="S2.SS2.SSS3.p1.4.m4.1.2.2.2.2" stretchy="false" xref="S2.SS2.SSS3.p1.4.m4.1.2.2.1.1.cmml">]</mo></mrow><mi id="S2.SS2.SSS3.p1.4.m4.1.2.3" xref="S2.SS2.SSS3.p1.4.m4.1.2.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.4.m4.1b"><apply id="S2.SS2.SSS3.p1.4.m4.1.2.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.4.m4.1.2.1.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.2">superscript</csymbol><apply id="S2.SS2.SSS3.p1.4.m4.1.2.2.1.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.2.2.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.4.m4.1.2.2.1.1.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.2.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.1">𝑁</ci></apply><ci id="S2.SS2.SSS3.p1.4.m4.1.2.3.cmml" xref="S2.SS2.SSS3.p1.4.m4.1.2.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.4.m4.1c">[N]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.4.m4.1d">[ italic_N ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is a set of elements described by a sequence of the form <math alttext="(\star,\ldots,\star,a,\star,\ldots,\star)" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.5.m5.7"><semantics id="S2.SS2.SSS3.p1.5.m5.7a"><mrow id="S2.SS2.SSS3.p1.5.m5.7.8.2" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml"><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">(</mo><mo id="S2.SS2.SSS3.p1.5.m5.1.1" lspace="0em" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.1.1.cmml">⋆</mo><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.2" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mi id="S2.SS2.SSS3.p1.5.m5.2.2" mathvariant="normal" xref="S2.SS2.SSS3.p1.5.m5.2.2.cmml">…</mi><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.3" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mo id="S2.SS2.SSS3.p1.5.m5.3.3" lspace="0em" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.3.3.cmml">⋆</mo><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.4" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mi id="S2.SS2.SSS3.p1.5.m5.4.4" xref="S2.SS2.SSS3.p1.5.m5.4.4.cmml">a</mi><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.5" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mo id="S2.SS2.SSS3.p1.5.m5.5.5" lspace="0em" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.5.5.cmml">⋆</mo><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.6" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mi id="S2.SS2.SSS3.p1.5.m5.6.6" mathvariant="normal" xref="S2.SS2.SSS3.p1.5.m5.6.6.cmml">…</mi><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.7" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">,</mo><mo id="S2.SS2.SSS3.p1.5.m5.7.7" lspace="0em" rspace="0em" xref="S2.SS2.SSS3.p1.5.m5.7.7.cmml">⋆</mo><mo id="S2.SS2.SSS3.p1.5.m5.7.8.2.8" stretchy="false" xref="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.5.m5.7b"><vector id="S2.SS2.SSS3.p1.5.m5.7.8.1.cmml" xref="S2.SS2.SSS3.p1.5.m5.7.8.2"><ci id="S2.SS2.SSS3.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS3.p1.5.m5.1.1">⋆</ci><ci id="S2.SS2.SSS3.p1.5.m5.2.2.cmml" xref="S2.SS2.SSS3.p1.5.m5.2.2">…</ci><ci id="S2.SS2.SSS3.p1.5.m5.3.3.cmml" xref="S2.SS2.SSS3.p1.5.m5.3.3">⋆</ci><ci id="S2.SS2.SSS3.p1.5.m5.4.4.cmml" xref="S2.SS2.SSS3.p1.5.m5.4.4">𝑎</ci><ci id="S2.SS2.SSS3.p1.5.m5.5.5.cmml" xref="S2.SS2.SSS3.p1.5.m5.5.5">⋆</ci><ci id="S2.SS2.SSS3.p1.5.m5.6.6.cmml" xref="S2.SS2.SSS3.p1.5.m5.6.6">…</ci><ci id="S2.SS2.SSS3.p1.5.m5.7.7.cmml" xref="S2.SS2.SSS3.p1.5.m5.7.7">⋆</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.5.m5.7c">(\star,\ldots,\star,a,\star,\ldots,\star)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.5.m5.7d">( ⋆ , … , ⋆ , italic_a , ⋆ , … , ⋆ )</annotation></semantics></math>, where an element <math alttext="a\in[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.6.m6.1"><semantics id="S2.SS2.SSS3.p1.6.m6.1a"><mrow id="S2.SS2.SSS3.p1.6.m6.1.2" xref="S2.SS2.SSS3.p1.6.m6.1.2.cmml"><mi id="S2.SS2.SSS3.p1.6.m6.1.2.2" xref="S2.SS2.SSS3.p1.6.m6.1.2.2.cmml">a</mi><mo id="S2.SS2.SSS3.p1.6.m6.1.2.1" xref="S2.SS2.SSS3.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.SS2.SSS3.p1.6.m6.1.2.3.2" xref="S2.SS2.SSS3.p1.6.m6.1.2.3.1.cmml"><mo id="S2.SS2.SSS3.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.6.m6.1.1" xref="S2.SS2.SSS3.p1.6.m6.1.1.cmml">N</mi><mo id="S2.SS2.SSS3.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS3.p1.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.6.m6.1b"><apply id="S2.SS2.SSS3.p1.6.m6.1.2.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.2"><in id="S2.SS2.SSS3.p1.6.m6.1.2.1.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.2.1"></in><ci id="S2.SS2.SSS3.p1.6.m6.1.2.2.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.2.2">𝑎</ci><apply id="S2.SS2.SSS3.p1.6.m6.1.2.3.1.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.2.3.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.6.m6.1.2.3.1.1.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS3.p1.6.m6.1.1">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.6.m6.1c">a\in[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.6.m6.1d">italic_a ∈ [ italic_N ]</annotation></semantics></math> is fixed in exactly one coordinate. We let <math alttext="\mathcal{G}_{N}^{(d)}" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.7.m7.1"><semantics id="S2.SS2.SSS3.p1.7.m7.1a"><msubsup id="S2.SS2.SSS3.p1.7.m7.1.2" xref="S2.SS2.SSS3.p1.7.m7.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS3.p1.7.m7.1.2.2.2" xref="S2.SS2.SSS3.p1.7.m7.1.2.2.2.cmml">𝒢</mi><mi id="S2.SS2.SSS3.p1.7.m7.1.2.2.3" xref="S2.SS2.SSS3.p1.7.m7.1.2.2.3.cmml">N</mi><mrow id="S2.SS2.SSS3.p1.7.m7.1.1.1.3" xref="S2.SS2.SSS3.p1.7.m7.1.2.cmml"><mo id="S2.SS2.SSS3.p1.7.m7.1.1.1.3.1" stretchy="false" xref="S2.SS2.SSS3.p1.7.m7.1.2.cmml">(</mo><mi id="S2.SS2.SSS3.p1.7.m7.1.1.1.1" xref="S2.SS2.SSS3.p1.7.m7.1.1.1.1.cmml">d</mi><mo id="S2.SS2.SSS3.p1.7.m7.1.1.1.3.2" stretchy="false" xref="S2.SS2.SSS3.p1.7.m7.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.7.m7.1b"><apply id="S2.SS2.SSS3.p1.7.m7.1.2.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.7.m7.1.2.1.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2">superscript</csymbol><apply id="S2.SS2.SSS3.p1.7.m7.1.2.2.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.7.m7.1.2.2.1.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2">subscript</csymbol><ci id="S2.SS2.SSS3.p1.7.m7.1.2.2.2.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2.2.2">𝒢</ci><ci id="S2.SS2.SSS3.p1.7.m7.1.2.2.3.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.2.2.3">𝑁</ci></apply><ci id="S2.SS2.SSS3.p1.7.m7.1.1.1.1.cmml" xref="S2.SS2.SSS3.p1.7.m7.1.1.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.7.m7.1c">\mathcal{G}_{N}^{(d)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.7.m7.1d">caligraphic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT</annotation></semantics></math> be the corresponding family of generators. Notice that <math alttext="|\mathcal{G}^{(d)}_{N}|=dN" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.8.m8.2"><semantics id="S2.SS2.SSS3.p1.8.m8.2a"><mrow id="S2.SS2.SSS3.p1.8.m8.2.2" xref="S2.SS2.SSS3.p1.8.m8.2.2.cmml"><mrow id="S2.SS2.SSS3.p1.8.m8.2.2.1.1" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.2.cmml"><mo id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.2.1.cmml">|</mo><msubsup id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.2" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.2.cmml">𝒢</mi><mi id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.3" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.3.cmml">N</mi><mrow id="S2.SS2.SSS3.p1.8.m8.1.1.1.3" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.cmml"><mo id="S2.SS2.SSS3.p1.8.m8.1.1.1.3.1" stretchy="false" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.cmml">(</mo><mi id="S2.SS2.SSS3.p1.8.m8.1.1.1.1" xref="S2.SS2.SSS3.p1.8.m8.1.1.1.1.cmml">d</mi><mo id="S2.SS2.SSS3.p1.8.m8.1.1.1.3.2" stretchy="false" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.3" stretchy="false" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.2.1.cmml">|</mo></mrow><mo id="S2.SS2.SSS3.p1.8.m8.2.2.2" xref="S2.SS2.SSS3.p1.8.m8.2.2.2.cmml">=</mo><mrow id="S2.SS2.SSS3.p1.8.m8.2.2.3" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.cmml"><mi id="S2.SS2.SSS3.p1.8.m8.2.2.3.2" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.2.cmml">d</mi><mo id="S2.SS2.SSS3.p1.8.m8.2.2.3.1" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.1.cmml">⁢</mo><mi id="S2.SS2.SSS3.p1.8.m8.2.2.3.3" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.8.m8.2b"><apply id="S2.SS2.SSS3.p1.8.m8.2.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2"><eq id="S2.SS2.SSS3.p1.8.m8.2.2.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.2"></eq><apply id="S2.SS2.SSS3.p1.8.m8.2.2.1.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1"><abs id="S2.SS2.SSS3.p1.8.m8.2.2.1.2.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.2"></abs><apply id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1">subscript</csymbol><apply id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1">superscript</csymbol><ci id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.2.2">𝒢</ci><ci id="S2.SS2.SSS3.p1.8.m8.1.1.1.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.1.1.1.1">𝑑</ci></apply><ci id="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.3.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.1.1.1.3">𝑁</ci></apply></apply><apply id="S2.SS2.SSS3.p1.8.m8.2.2.3.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.3"><times id="S2.SS2.SSS3.p1.8.m8.2.2.3.1.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.1"></times><ci id="S2.SS2.SSS3.p1.8.m8.2.2.3.2.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.2">𝑑</ci><ci id="S2.SS2.SSS3.p1.8.m8.2.2.3.3.cmml" xref="S2.SS2.SSS3.p1.8.m8.2.2.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.8.m8.2c">|\mathcal{G}^{(d)}_{N}|=dN</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.8.m8.2d">| caligraphic_G start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT | = italic_d italic_N</annotation></semantics></math>. Given a <math alttext="d" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.9.m9.1"><semantics id="S2.SS2.SSS3.p1.9.m9.1a"><mi id="S2.SS2.SSS3.p1.9.m9.1.1" xref="S2.SS2.SSS3.p1.9.m9.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.9.m9.1b"><ci id="S2.SS2.SSS3.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS3.p1.9.m9.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.9.m9.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.9.m9.1d">italic_d</annotation></semantics></math>-dimensional tensor <math alttext="A\subseteq[N]^{d}" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.10.m10.1"><semantics id="S2.SS2.SSS3.p1.10.m10.1a"><mrow id="S2.SS2.SSS3.p1.10.m10.1.2" xref="S2.SS2.SSS3.p1.10.m10.1.2.cmml"><mi id="S2.SS2.SSS3.p1.10.m10.1.2.2" xref="S2.SS2.SSS3.p1.10.m10.1.2.2.cmml">A</mi><mo id="S2.SS2.SSS3.p1.10.m10.1.2.1" xref="S2.SS2.SSS3.p1.10.m10.1.2.1.cmml">⊆</mo><msup id="S2.SS2.SSS3.p1.10.m10.1.2.3" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.cmml"><mrow id="S2.SS2.SSS3.p1.10.m10.1.2.3.2.2" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.2.1.cmml"><mo id="S2.SS2.SSS3.p1.10.m10.1.2.3.2.2.1" stretchy="false" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS3.p1.10.m10.1.1" xref="S2.SS2.SSS3.p1.10.m10.1.1.cmml">N</mi><mo id="S2.SS2.SSS3.p1.10.m10.1.2.3.2.2.2" stretchy="false" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.2.1.1.cmml">]</mo></mrow><mi id="S2.SS2.SSS3.p1.10.m10.1.2.3.3" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.10.m10.1b"><apply id="S2.SS2.SSS3.p1.10.m10.1.2.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2"><subset id="S2.SS2.SSS3.p1.10.m10.1.2.1.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.1"></subset><ci id="S2.SS2.SSS3.p1.10.m10.1.2.2.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.2">𝐴</ci><apply id="S2.SS2.SSS3.p1.10.m10.1.2.3.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS3.p1.10.m10.1.2.3.1.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.3">superscript</csymbol><apply id="S2.SS2.SSS3.p1.10.m10.1.2.3.2.1.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS3.p1.10.m10.1.2.3.2.1.1.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS3.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.1">𝑁</ci></apply><ci id="S2.SS2.SSS3.p1.10.m10.1.2.3.3.cmml" xref="S2.SS2.SSS3.p1.10.m10.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.10.m10.1c">A\subseteq[N]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.10.m10.1d">italic_A ⊆ [ italic_N ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, we denote its <math alttext="d" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.11.m11.1"><semantics id="S2.SS2.SSS3.p1.11.m11.1a"><mi id="S2.SS2.SSS3.p1.11.m11.1.1" xref="S2.SS2.SSS3.p1.11.m11.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS3.p1.11.m11.1b"><ci id="S2.SS2.SSS3.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS3.p1.11.m11.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS3.p1.11.m11.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS3.p1.11.m11.1d">italic_d</annotation></semantics></math><em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS3.p1.12.1">-dimensional graph complexity</em> by <math alttext="D(A\mid\mathcal{G}_{N}^{(d)})" class="ltx_Math" display="inline" id="S2.SS2.SSS3.p1.12.m12.2"><semantics id="S2.SS2.SSS3.p1.12.m12.2a"><mrow id="S2.SS2.SSS3.p1.12.m12.2.2" xref="S2.SS2.SSS3.p1.12.m12.2.2.cmml"><mi id="S2.SS2.SSS3.p1.12.m12.2.2.3" xref="S2.SS2.SSS3.p1.12.m12.2.2.3.cmml">D</mi><mo id="S2.SS2.SSS3.p1.12.m12.2.2.2" xref="S2.SS2.SSS3.p1.12.m12.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.SSS3.p1.12.m12.2.2.1.1" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.cmml"><mo id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.2" stretchy="false" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.cmml"><mi id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.2" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.2.cmml">A</mi><mo id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.1" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.1.cmml">∣</mo><msubsup id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.2.2" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.2.2.cmml">𝒢</mi><mi id="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.2.3" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.2.3.cmml">N</mi><mrow id="S2.SS2.SSS3.p1.12.m12.1.1.1.3" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.cmml"><mo id="S2.SS2.SSS3.p1.12.m12.1.1.1.3.1" stretchy="false" xref="S2.SS2.SSS3.p1.12.m12.2.2.1.1.1.3.cmml">(</mo><mi id="S2.SS2.SSS3.p1.12.m12.1.1.1.1" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS3.p2"> <p class="ltx_p" id="S2.SS2.SSS3.p2.1">To some extent, graph complexity and Boolean circuit complexity are extremal examples of non-trivial discrete spaces, in the sense that the former minimizes the number of dimensions and maximizes the possible values in each coordinate, while the latter does the opposite. The higher dimensional graphs generalize both cases.</p> </div> </section> <section class="ltx_subsubsection ltx_indent_first" id="S2.SS2.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.2.4 </span>Combinatorial rectangles from communication complexity</h4> <div class="ltx_para" id="S2.SS2.SSS4.p1"> <p class="ltx_p" id="S2.SS2.SSS4.p1.11">The domain is <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.1.m1.2"><semantics id="S2.SS2.SSS4.p1.1.m1.2a"><mrow id="S2.SS2.SSS4.p1.1.m1.2.3" xref="S2.SS2.SSS4.p1.1.m1.2.3.cmml"><mrow id="S2.SS2.SSS4.p1.1.m1.2.3.2.2" xref="S2.SS2.SSS4.p1.1.m1.2.3.2.1.cmml"><mo id="S2.SS2.SSS4.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.1.m1.2.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.1.m1.1.1" xref="S2.SS2.SSS4.p1.1.m1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.1.m1.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS4.p1.1.m1.2.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS4.p1.1.m1.2.3.1" rspace="0.222em" xref="S2.SS2.SSS4.p1.1.m1.2.3.1.cmml">×</mo><mrow id="S2.SS2.SSS4.p1.1.m1.2.3.3.2" xref="S2.SS2.SSS4.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS2.SSS4.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.1.m1.2.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.1.m1.2.2" xref="S2.SS2.SSS4.p1.1.m1.2.2.cmml">N</mi><mo id="S2.SS2.SSS4.p1.1.m1.2.3.3.2.2" stretchy="false" xref="S2.SS2.SSS4.p1.1.m1.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.1.m1.2b"><apply id="S2.SS2.SSS4.p1.1.m1.2.3.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3"><times id="S2.SS2.SSS4.p1.1.m1.2.3.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3.1"></times><apply id="S2.SS2.SSS4.p1.1.m1.2.3.2.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.1.m1.2.3.2.1.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS4.p1.1.m1.2.3.3.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.1.m1.2.3.3.1.1.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.1.m1.2.2.cmml" xref="S2.SS2.SSS4.p1.1.m1.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.1.m1.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.1.m1.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math>, and its associated family <math alttext="\mathcal{R}_{N,N}" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.2.m2.2"><semantics id="S2.SS2.SSS4.p1.2.m2.2a"><msub id="S2.SS2.SSS4.p1.2.m2.2.3" xref="S2.SS2.SSS4.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.2.m2.2.3.2" xref="S2.SS2.SSS4.p1.2.m2.2.3.2.cmml">ℛ</mi><mrow id="S2.SS2.SSS4.p1.2.m2.2.2.2.4" xref="S2.SS2.SSS4.p1.2.m2.2.2.2.3.cmml"><mi id="S2.SS2.SSS4.p1.2.m2.1.1.1.1" xref="S2.SS2.SSS4.p1.2.m2.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.2.m2.2.2.2.4.1" xref="S2.SS2.SSS4.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.2.m2.2.2.2.2" xref="S2.SS2.SSS4.p1.2.m2.2.2.2.2.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.2.m2.2b"><apply id="S2.SS2.SSS4.p1.2.m2.2.3.cmml" xref="S2.SS2.SSS4.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.2.m2.2.3.1.cmml" xref="S2.SS2.SSS4.p1.2.m2.2.3">subscript</csymbol><ci id="S2.SS2.SSS4.p1.2.m2.2.3.2.cmml" xref="S2.SS2.SSS4.p1.2.m2.2.3.2">ℛ</ci><list id="S2.SS2.SSS4.p1.2.m2.2.2.2.3.cmml" xref="S2.SS2.SSS4.p1.2.m2.2.2.2.4"><ci id="S2.SS2.SSS4.p1.2.m2.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.2.m2.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS4.p1.2.m2.2.2.2.2.cmml" xref="S2.SS2.SSS4.p1.2.m2.2.2.2.2">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.2.m2.2c">\mathcal{R}_{N,N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.2.m2.2d">caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT</annotation></semantics></math> of generators contains every <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS4.p1.11.1">combinatorial rectangle</em> <math alttext="R=U\times V" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.3.m3.1"><semantics id="S2.SS2.SSS4.p1.3.m3.1a"><mrow id="S2.SS2.SSS4.p1.3.m3.1.1" xref="S2.SS2.SSS4.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS4.p1.3.m3.1.1.2" xref="S2.SS2.SSS4.p1.3.m3.1.1.2.cmml">R</mi><mo id="S2.SS2.SSS4.p1.3.m3.1.1.1" xref="S2.SS2.SSS4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS2.SSS4.p1.3.m3.1.1.3" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.cmml"><mi id="S2.SS2.SSS4.p1.3.m3.1.1.3.2" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.2.cmml">U</mi><mo id="S2.SS2.SSS4.p1.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.1.cmml">×</mo><mi id="S2.SS2.SSS4.p1.3.m3.1.1.3.3" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.3.m3.1b"><apply id="S2.SS2.SSS4.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1"><eq id="S2.SS2.SSS4.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.1"></eq><ci id="S2.SS2.SSS4.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.2">𝑅</ci><apply id="S2.SS2.SSS4.p1.3.m3.1.1.3.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.3"><times id="S2.SS2.SSS4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.1"></times><ci id="S2.SS2.SSS4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.2">𝑈</ci><ci id="S2.SS2.SSS4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS2.SSS4.p1.3.m3.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.3.m3.1c">R=U\times V</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.3.m3.1d">italic_R = italic_U × italic_V</annotation></semantics></math>, where <math alttext="U,V\subseteq[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.4.m4.3"><semantics id="S2.SS2.SSS4.p1.4.m4.3a"><mrow id="S2.SS2.SSS4.p1.4.m4.3.4" xref="S2.SS2.SSS4.p1.4.m4.3.4.cmml"><mrow id="S2.SS2.SSS4.p1.4.m4.3.4.2.2" xref="S2.SS2.SSS4.p1.4.m4.3.4.2.1.cmml"><mi id="S2.SS2.SSS4.p1.4.m4.2.2" xref="S2.SS2.SSS4.p1.4.m4.2.2.cmml">U</mi><mo id="S2.SS2.SSS4.p1.4.m4.3.4.2.2.1" xref="S2.SS2.SSS4.p1.4.m4.3.4.2.1.cmml">,</mo><mi id="S2.SS2.SSS4.p1.4.m4.3.3" xref="S2.SS2.SSS4.p1.4.m4.3.3.cmml">V</mi></mrow><mo id="S2.SS2.SSS4.p1.4.m4.3.4.1" xref="S2.SS2.SSS4.p1.4.m4.3.4.1.cmml">⊆</mo><mrow id="S2.SS2.SSS4.p1.4.m4.3.4.3.2" xref="S2.SS2.SSS4.p1.4.m4.3.4.3.1.cmml"><mo id="S2.SS2.SSS4.p1.4.m4.3.4.3.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.4.m4.3.4.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.4.m4.1.1" xref="S2.SS2.SSS4.p1.4.m4.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.4.m4.3.4.3.2.2" stretchy="false" xref="S2.SS2.SSS4.p1.4.m4.3.4.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.4.m4.3b"><apply id="S2.SS2.SSS4.p1.4.m4.3.4.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.4"><subset id="S2.SS2.SSS4.p1.4.m4.3.4.1.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.4.1"></subset><list id="S2.SS2.SSS4.p1.4.m4.3.4.2.1.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.4.2.2"><ci id="S2.SS2.SSS4.p1.4.m4.2.2.cmml" xref="S2.SS2.SSS4.p1.4.m4.2.2">𝑈</ci><ci id="S2.SS2.SSS4.p1.4.m4.3.3.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.3">𝑉</ci></list><apply id="S2.SS2.SSS4.p1.4.m4.3.4.3.1.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.4.3.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.4.m4.3.4.3.1.1.cmml" xref="S2.SS2.SSS4.p1.4.m4.3.4.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS4.p1.4.m4.1.1">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.4.m4.3c">U,V\subseteq[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.4.m4.3d">italic_U , italic_V ⊆ [ italic_N ]</annotation></semantics></math> are arbitrary subsets. In particular, <math alttext="|\mathcal{R}_{N,N}|=2^{2N}" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.5.m5.3"><semantics id="S2.SS2.SSS4.p1.5.m5.3a"><mrow id="S2.SS2.SSS4.p1.5.m5.3.3" xref="S2.SS2.SSS4.p1.5.m5.3.3.cmml"><mrow id="S2.SS2.SSS4.p1.5.m5.3.3.1.1" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.2.cmml"><mo id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.2.1.cmml">|</mo><msub id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.2" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.2.cmml">ℛ</mi><mrow id="S2.SS2.SSS4.p1.5.m5.2.2.2.4" xref="S2.SS2.SSS4.p1.5.m5.2.2.2.3.cmml"><mi id="S2.SS2.SSS4.p1.5.m5.1.1.1.1" xref="S2.SS2.SSS4.p1.5.m5.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.5.m5.2.2.2.4.1" xref="S2.SS2.SSS4.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.5.m5.2.2.2.2" xref="S2.SS2.SSS4.p1.5.m5.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.3" stretchy="false" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.2.1.cmml">|</mo></mrow><mo id="S2.SS2.SSS4.p1.5.m5.3.3.2" xref="S2.SS2.SSS4.p1.5.m5.3.3.2.cmml">=</mo><msup id="S2.SS2.SSS4.p1.5.m5.3.3.3" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.cmml"><mn id="S2.SS2.SSS4.p1.5.m5.3.3.3.2" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.2.cmml">2</mn><mrow id="S2.SS2.SSS4.p1.5.m5.3.3.3.3" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.cmml"><mn id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.2" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.2.cmml">2</mn><mo id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.1" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.1.cmml">⁢</mo><mi id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.3" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.3.cmml">N</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.5.m5.3b"><apply id="S2.SS2.SSS4.p1.5.m5.3.3.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3"><eq id="S2.SS2.SSS4.p1.5.m5.3.3.2.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.2"></eq><apply id="S2.SS2.SSS4.p1.5.m5.3.3.1.2.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1"><abs id="S2.SS2.SSS4.p1.5.m5.3.3.1.2.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.2"></abs><apply id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.2.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.1.1.1.2">ℛ</ci><list id="S2.SS2.SSS4.p1.5.m5.2.2.2.3.cmml" xref="S2.SS2.SSS4.p1.5.m5.2.2.2.4"><ci id="S2.SS2.SSS4.p1.5.m5.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS4.p1.5.m5.2.2.2.2.cmml" xref="S2.SS2.SSS4.p1.5.m5.2.2.2.2">𝑁</ci></list></apply></apply><apply id="S2.SS2.SSS4.p1.5.m5.3.3.3.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.5.m5.3.3.3.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.3">superscript</csymbol><cn id="S2.SS2.SSS4.p1.5.m5.3.3.3.2.cmml" type="integer" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.2">2</cn><apply id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3"><times id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.1.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.1"></times><cn id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.2.cmml" type="integer" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.2">2</cn><ci id="S2.SS2.SSS4.p1.5.m5.3.3.3.3.3.cmml" xref="S2.SS2.SSS4.p1.5.m5.3.3.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.5.m5.3c">|\mathcal{R}_{N,N}|=2^{2N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.5.m5.3d">| caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT | = 2 start_POSTSUPERSCRIPT 2 italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>, while the number of subsets of <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.6.m6.2"><semantics id="S2.SS2.SSS4.p1.6.m6.2a"><mrow id="S2.SS2.SSS4.p1.6.m6.2.3" xref="S2.SS2.SSS4.p1.6.m6.2.3.cmml"><mrow id="S2.SS2.SSS4.p1.6.m6.2.3.2.2" xref="S2.SS2.SSS4.p1.6.m6.2.3.2.1.cmml"><mo id="S2.SS2.SSS4.p1.6.m6.2.3.2.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.6.m6.2.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.6.m6.1.1" xref="S2.SS2.SSS4.p1.6.m6.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.6.m6.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS4.p1.6.m6.2.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS4.p1.6.m6.2.3.1" rspace="0.222em" xref="S2.SS2.SSS4.p1.6.m6.2.3.1.cmml">×</mo><mrow id="S2.SS2.SSS4.p1.6.m6.2.3.3.2" xref="S2.SS2.SSS4.p1.6.m6.2.3.3.1.cmml"><mo id="S2.SS2.SSS4.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.6.m6.2.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.6.m6.2.2" xref="S2.SS2.SSS4.p1.6.m6.2.2.cmml">N</mi><mo id="S2.SS2.SSS4.p1.6.m6.2.3.3.2.2" stretchy="false" xref="S2.SS2.SSS4.p1.6.m6.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.6.m6.2b"><apply id="S2.SS2.SSS4.p1.6.m6.2.3.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3"><times id="S2.SS2.SSS4.p1.6.m6.2.3.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3.1"></times><apply id="S2.SS2.SSS4.p1.6.m6.2.3.2.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.6.m6.2.3.2.1.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS4.p1.6.m6.2.3.3.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.6.m6.2.3.3.1.1.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.6.m6.2.2.cmml" xref="S2.SS2.SSS4.p1.6.m6.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.6.m6.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.6.m6.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math> is <math alttext="2^{N^{2}}" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.7.m7.1"><semantics id="S2.SS2.SSS4.p1.7.m7.1a"><msup id="S2.SS2.SSS4.p1.7.m7.1.1" xref="S2.SS2.SSS4.p1.7.m7.1.1.cmml"><mn id="S2.SS2.SSS4.p1.7.m7.1.1.2" xref="S2.SS2.SSS4.p1.7.m7.1.1.2.cmml">2</mn><msup id="S2.SS2.SSS4.p1.7.m7.1.1.3" xref="S2.SS2.SSS4.p1.7.m7.1.1.3.cmml"><mi id="S2.SS2.SSS4.p1.7.m7.1.1.3.2" xref="S2.SS2.SSS4.p1.7.m7.1.1.3.2.cmml">N</mi><mn id="S2.SS2.SSS4.p1.7.m7.1.1.3.3" xref="S2.SS2.SSS4.p1.7.m7.1.1.3.3.cmml">2</mn></msup></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.7.m7.1b"><apply id="S2.SS2.SSS4.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.7.m7.1.1.1.cmml" xref="S2.SS2.SSS4.p1.7.m7.1.1">superscript</csymbol><cn id="S2.SS2.SSS4.p1.7.m7.1.1.2.cmml" type="integer" xref="S2.SS2.SSS4.p1.7.m7.1.1.2">2</cn><apply id="S2.SS2.SSS4.p1.7.m7.1.1.3.cmml" xref="S2.SS2.SSS4.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.7.m7.1.1.3.1.cmml" xref="S2.SS2.SSS4.p1.7.m7.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS4.p1.7.m7.1.1.3.2.cmml" xref="S2.SS2.SSS4.p1.7.m7.1.1.3.2">𝑁</ci><cn id="S2.SS2.SSS4.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS2.SSS4.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.7.m7.1c">2^{N^{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.7.m7.1d">2 start_POSTSUPERSCRIPT italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Observe that <math alttext="\mathcal{R}_{N,N}" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.8.m8.2"><semantics id="S2.SS2.SSS4.p1.8.m8.2a"><msub id="S2.SS2.SSS4.p1.8.m8.2.3" xref="S2.SS2.SSS4.p1.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.8.m8.2.3.2" xref="S2.SS2.SSS4.p1.8.m8.2.3.2.cmml">ℛ</mi><mrow id="S2.SS2.SSS4.p1.8.m8.2.2.2.4" xref="S2.SS2.SSS4.p1.8.m8.2.2.2.3.cmml"><mi id="S2.SS2.SSS4.p1.8.m8.1.1.1.1" xref="S2.SS2.SSS4.p1.8.m8.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.8.m8.2.2.2.4.1" xref="S2.SS2.SSS4.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.8.m8.2.2.2.2" xref="S2.SS2.SSS4.p1.8.m8.2.2.2.2.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.8.m8.2b"><apply id="S2.SS2.SSS4.p1.8.m8.2.3.cmml" xref="S2.SS2.SSS4.p1.8.m8.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.8.m8.2.3.1.cmml" xref="S2.SS2.SSS4.p1.8.m8.2.3">subscript</csymbol><ci id="S2.SS2.SSS4.p1.8.m8.2.3.2.cmml" xref="S2.SS2.SSS4.p1.8.m8.2.3.2">ℛ</ci><list id="S2.SS2.SSS4.p1.8.m8.2.2.2.3.cmml" xref="S2.SS2.SSS4.p1.8.m8.2.2.2.4"><ci id="S2.SS2.SSS4.p1.8.m8.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.8.m8.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS4.p1.8.m8.2.2.2.2.cmml" xref="S2.SS2.SSS4.p1.8.m8.2.2.2.2">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.8.m8.2c">\mathcal{R}_{N,N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.8.m8.2d">caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT</annotation></semantics></math> extends the set of generators employed in graph complexity. Consequently, for <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.9.m9.2"><semantics id="S2.SS2.SSS4.p1.9.m9.2a"><mrow id="S2.SS2.SSS4.p1.9.m9.2.3" xref="S2.SS2.SSS4.p1.9.m9.2.3.cmml"><mi id="S2.SS2.SSS4.p1.9.m9.2.3.2" xref="S2.SS2.SSS4.p1.9.m9.2.3.2.cmml">G</mi><mo id="S2.SS2.SSS4.p1.9.m9.2.3.1" xref="S2.SS2.SSS4.p1.9.m9.2.3.1.cmml">⊆</mo><mrow id="S2.SS2.SSS4.p1.9.m9.2.3.3" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.cmml"><mrow id="S2.SS2.SSS4.p1.9.m9.2.3.3.2.2" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.2.1.cmml"><mo id="S2.SS2.SSS4.p1.9.m9.2.3.3.2.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.9.m9.1.1" xref="S2.SS2.SSS4.p1.9.m9.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.9.m9.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS2.SSS4.p1.9.m9.2.3.3.1" rspace="0.222em" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.1.cmml">×</mo><mrow id="S2.SS2.SSS4.p1.9.m9.2.3.3.3.2" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.3.1.cmml"><mo id="S2.SS2.SSS4.p1.9.m9.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS2.SSS4.p1.9.m9.2.2" xref="S2.SS2.SSS4.p1.9.m9.2.2.cmml">N</mi><mo id="S2.SS2.SSS4.p1.9.m9.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.9.m9.2b"><apply id="S2.SS2.SSS4.p1.9.m9.2.3.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3"><subset id="S2.SS2.SSS4.p1.9.m9.2.3.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.1"></subset><ci id="S2.SS2.SSS4.p1.9.m9.2.3.2.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.2">𝐺</ci><apply id="S2.SS2.SSS4.p1.9.m9.2.3.3.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3"><times id="S2.SS2.SSS4.p1.9.m9.2.3.3.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.1"></times><apply id="S2.SS2.SSS4.p1.9.m9.2.3.3.2.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.9.m9.2.3.3.2.1.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.1.1">𝑁</ci></apply><apply id="S2.SS2.SSS4.p1.9.m9.2.3.3.3.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.9.m9.2.3.3.3.1.1.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS2.SSS4.p1.9.m9.2.2.cmml" xref="S2.SS2.SSS4.p1.9.m9.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.9.m9.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.9.m9.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>, <math alttext="D(G\mid\mathcal{R}_{N,N})\leq D(G\mid\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.10.m10.6"><semantics id="S2.SS2.SSS4.p1.10.m10.6a"><mrow id="S2.SS2.SSS4.p1.10.m10.6.6" xref="S2.SS2.SSS4.p1.10.m10.6.6.cmml"><mrow id="S2.SS2.SSS4.p1.10.m10.5.5.1" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.5.5.1.3" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.3.cmml">D</mi><mo id="S2.SS2.SSS4.p1.10.m10.5.5.1.2" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.cmml"><mo id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.2" stretchy="false" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.2" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3.2" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3.2.cmml">ℛ</mi><mrow id="S2.SS2.SSS4.p1.10.m10.2.2.2.4" xref="S2.SS2.SSS4.p1.10.m10.2.2.2.3.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.1.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.10.m10.2.2.2.4.1" xref="S2.SS2.SSS4.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.10.m10.2.2.2.2" xref="S2.SS2.SSS4.p1.10.m10.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.3" stretchy="false" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS4.p1.10.m10.6.6.3" xref="S2.SS2.SSS4.p1.10.m10.6.6.3.cmml">≤</mo><mrow id="S2.SS2.SSS4.p1.10.m10.6.6.2" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.6.6.2.3" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.3.cmml">D</mi><mo id="S2.SS2.SSS4.p1.10.m10.6.6.2.2" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.2.cmml">⁢</mo><mrow id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.cmml"><mo id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.2" stretchy="false" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.2" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.1" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.1.cmml">∣</mo><msub id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.2" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS2.SSS4.p1.10.m10.4.4.2.4" xref="S2.SS2.SSS4.p1.10.m10.4.4.2.3.cmml"><mi id="S2.SS2.SSS4.p1.10.m10.3.3.1.1" xref="S2.SS2.SSS4.p1.10.m10.3.3.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.10.m10.4.4.2.4.1" xref="S2.SS2.SSS4.p1.10.m10.4.4.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.10.m10.4.4.2.2" xref="S2.SS2.SSS4.p1.10.m10.4.4.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.3" stretchy="false" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.10.m10.6b"><apply id="S2.SS2.SSS4.p1.10.m10.6.6.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6"><leq id="S2.SS2.SSS4.p1.10.m10.6.6.3.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6.3"></leq><apply id="S2.SS2.SSS4.p1.10.m10.5.5.1.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1"><times id="S2.SS2.SSS4.p1.10.m10.5.5.1.2.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.2"></times><ci id="S2.SS2.SSS4.p1.10.m10.5.5.1.3.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.3">𝐷</ci><apply id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.1">conditional</csymbol><ci id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.2.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.2">𝐺</ci><apply id="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3.cmml" xref="S2.SS2.SSS4.p1.10.m10.5.5.1.1.1.1.3"><csymbol 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id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.2.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.2">𝐺</ci><apply id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.1.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.2.cmml" xref="S2.SS2.SSS4.p1.10.m10.6.6.2.1.1.1.3.2">𝒢</ci><list id="S2.SS2.SSS4.p1.10.m10.4.4.2.3.cmml" xref="S2.SS2.SSS4.p1.10.m10.4.4.2.4"><ci id="S2.SS2.SSS4.p1.10.m10.3.3.1.1.cmml" xref="S2.SS2.SSS4.p1.10.m10.3.3.1.1">𝑁</ci><ci id="S2.SS2.SSS4.p1.10.m10.4.4.2.2.cmml" xref="S2.SS2.SSS4.p1.10.m10.4.4.2.2">𝑁</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.10.m10.6c">D(G\mid\mathcal{R}_{N,N})\leq D(G\mid\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.10.m10.6d">italic_D ( italic_G ∣ caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>. Moreover, <math alttext="D_{\cap}(G\mid\mathcal{R}_{N,N})=0" class="ltx_Math" display="inline" id="S2.SS2.SSS4.p1.11.m11.3"><semantics id="S2.SS2.SSS4.p1.11.m11.3a"><mrow id="S2.SS2.SSS4.p1.11.m11.3.3" xref="S2.SS2.SSS4.p1.11.m11.3.3.cmml"><mrow id="S2.SS2.SSS4.p1.11.m11.3.3.1" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.cmml"><msub id="S2.SS2.SSS4.p1.11.m11.3.3.1.3" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3.cmml"><mi id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.2" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3.2.cmml">D</mi><mo id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.3" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3.3.cmml">∩</mo></msub><mo id="S2.SS2.SSS4.p1.11.m11.3.3.1.2" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.cmml"><mo id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.2" stretchy="false" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.cmml"><mi id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.2" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.2.cmml">G</mi><mo id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.1" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.2" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.2.cmml">ℛ</mi><mrow id="S2.SS2.SSS4.p1.11.m11.2.2.2.4" xref="S2.SS2.SSS4.p1.11.m11.2.2.2.3.cmml"><mi id="S2.SS2.SSS4.p1.11.m11.1.1.1.1" xref="S2.SS2.SSS4.p1.11.m11.1.1.1.1.cmml">N</mi><mo id="S2.SS2.SSS4.p1.11.m11.2.2.2.4.1" xref="S2.SS2.SSS4.p1.11.m11.2.2.2.3.cmml">,</mo><mi id="S2.SS2.SSS4.p1.11.m11.2.2.2.2" xref="S2.SS2.SSS4.p1.11.m11.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.3" stretchy="false" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS4.p1.11.m11.3.3.2" xref="S2.SS2.SSS4.p1.11.m11.3.3.2.cmml">=</mo><mn id="S2.SS2.SSS4.p1.11.m11.3.3.3" xref="S2.SS2.SSS4.p1.11.m11.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS4.p1.11.m11.3b"><apply id="S2.SS2.SSS4.p1.11.m11.3.3.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3"><eq id="S2.SS2.SSS4.p1.11.m11.3.3.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.2"></eq><apply id="S2.SS2.SSS4.p1.11.m11.3.3.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1"><times id="S2.SS2.SSS4.p1.11.m11.3.3.1.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.2"></times><apply id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3">subscript</csymbol><ci id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3.2">𝐷</ci><intersect id="S2.SS2.SSS4.p1.11.m11.3.3.1.3.3.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.3.3"></intersect></apply><apply id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1"><csymbol cd="latexml" id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.1">conditional</csymbol><ci id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.2">𝐺</ci><apply id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.3.3.1.1.1.1.3.2">ℛ</ci><list id="S2.SS2.SSS4.p1.11.m11.2.2.2.3.cmml" xref="S2.SS2.SSS4.p1.11.m11.2.2.2.4"><ci id="S2.SS2.SSS4.p1.11.m11.1.1.1.1.cmml" xref="S2.SS2.SSS4.p1.11.m11.1.1.1.1">𝑁</ci><ci id="S2.SS2.SSS4.p1.11.m11.2.2.2.2.cmml" xref="S2.SS2.SSS4.p1.11.m11.2.2.2.2">𝑁</ci></list></apply></apply></apply><cn id="S2.SS2.SSS4.p1.11.m11.3.3.3.cmml" type="integer" xref="S2.SS2.SSS4.p1.11.m11.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS4.p1.11.m11.3c">D_{\cap}(G\mid\mathcal{R}_{N,N})=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS4.p1.11.m11.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> for every graph. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S2.SS2.SSS4.p2"> <p class="ltx_p" id="S2.SS2.SSS4.p2.1">Observe that there is an interesting contrast among all these different spaces: the ratio between the <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS4.p2.1.1">size of the ambient space</em> and <em class="ltx_emph ltx_font_italic" id="S2.SS2.SSS4.p2.1.2">the number of generators</em>. For instance, in graph complexity the two are polynomially related, in Boolean circuits the ambient space is exponentially larger, and in the discrete space involving combinatorial rectangles the opposite happens. These natural discrete spaces exhibit three important regimes of parameters in discrete complexity.</p> </div> </section> </section> <section class="ltx_subsection ltx_indent_first" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Basic lemmas and other useful results</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">By combining sequences, we have the following trivial inequality.</p> </div> <div class="ltx_theorem ltx_theorem_fact" id="Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem6.1.1.1">Fact 6</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem6.p1"> <p class="ltx_p" id="Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem6.p1.3.3">For every set <math alttext="E\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem6.p1.1.1.m1.1"><semantics id="Thmtheorem6.p1.1.1.m1.1a"><mrow id="Thmtheorem6.p1.1.1.m1.1.1" xref="Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem6.p1.1.1.m1.1.1.2" xref="Thmtheorem6.p1.1.1.m1.1.1.2.cmml">E</mi><mo id="Thmtheorem6.p1.1.1.m1.1.1.1" xref="Thmtheorem6.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem6.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem6.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.1.1.m1.1b"><apply id="Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="Thmtheorem6.p1.1.1.m1.1.1"><subset id="Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem6.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem6.p1.1.1.m1.1.1.2">𝐸</ci><ci id="Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem6.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.1.1.m1.1c">E\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.1.1.m1.1d">italic_E ⊆ roman_Γ</annotation></semantics></math> and <math alttext="\diamond\in\{\cap,\cup\}" class="ltx_math_unparsed" display="inline" id="Thmtheorem6.p1.2.2.m2.2"><semantics id="Thmtheorem6.p1.2.2.m2.2a"><mrow id="Thmtheorem6.p1.2.2.m2.2b"><mo id="Thmtheorem6.p1.2.2.m2.1.1" rspace="0em">⋄</mo><mo id="Thmtheorem6.p1.2.2.m2.2.2" lspace="0em">∈</mo><mrow id="Thmtheorem6.p1.2.2.m2.2.3"><mo id="Thmtheorem6.p1.2.2.m2.2.3.1" stretchy="false">{</mo><mo id="Thmtheorem6.p1.2.2.m2.2.3.2" lspace="0em" rspace="0em">∩</mo><mo id="Thmtheorem6.p1.2.2.m2.2.3.3" rspace="0em">,</mo><mo id="Thmtheorem6.p1.2.2.m2.2.3.4" lspace="0em" rspace="0em">∪</mo><mo id="Thmtheorem6.p1.2.2.m2.2.3.5" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmtheorem6.p1.2.2.m2.2c">\diamond\in\{\cap,\cup\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.2.2.m2.2d">⋄ ∈ { ∩ , ∪ }</annotation></semantics></math>, <math alttext="D_{\diamond}(A\mid\mathcal{B})\leq D_{\diamond}(A\mid E,\mathcal{B})+D_{% \diamond}(E\mid\mathcal{B})" class="ltx_Math" display="inline" id="Thmtheorem6.p1.3.3.m3.5"><semantics id="Thmtheorem6.p1.3.3.m3.5a"><mrow id="Thmtheorem6.p1.3.3.m3.5.5" xref="Thmtheorem6.p1.3.3.m3.5.5.cmml"><mrow id="Thmtheorem6.p1.3.3.m3.3.3.1" xref="Thmtheorem6.p1.3.3.m3.3.3.1.cmml"><msub id="Thmtheorem6.p1.3.3.m3.3.3.1.3" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3.cmml"><mi id="Thmtheorem6.p1.3.3.m3.3.3.1.3.2" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3.2.cmml">D</mi><mo id="Thmtheorem6.p1.3.3.m3.3.3.1.3.3" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3.3.cmml">⋄</mo></msub><mo id="Thmtheorem6.p1.3.3.m3.3.3.1.2" xref="Thmtheorem6.p1.3.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.cmml"><mo id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.2" stretchy="false" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.cmml"><mi id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.2" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.1" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.3" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.3" stretchy="false" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem6.p1.3.3.m3.5.5.4" xref="Thmtheorem6.p1.3.3.m3.5.5.4.cmml">≤</mo><mrow id="Thmtheorem6.p1.3.3.m3.5.5.3" xref="Thmtheorem6.p1.3.3.m3.5.5.3.cmml"><mrow id="Thmtheorem6.p1.3.3.m3.4.4.2.1" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.cmml"><msub id="Thmtheorem6.p1.3.3.m3.4.4.2.1.3" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.3.cmml"><mi id="Thmtheorem6.p1.3.3.m3.4.4.2.1.3.2" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.3.2.cmml">D</mi><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.3.3" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.3.3.cmml">⋄</mo></msub><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.2" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.cmml"><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.2" stretchy="false" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.cmml"><mi id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.2" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.1" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.1.cmml">∣</mo><mrow id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.2" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.1.cmml"><mi id="Thmtheorem6.p1.3.3.m3.1.1" xref="Thmtheorem6.p1.3.3.m3.1.1.cmml">E</mi><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.2.1" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem6.p1.3.3.m3.2.2" xref="Thmtheorem6.p1.3.3.m3.2.2.cmml">ℬ</mi></mrow></mrow><mo id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.3" stretchy="false" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem6.p1.3.3.m3.5.5.3.3" xref="Thmtheorem6.p1.3.3.m3.5.5.3.3.cmml">+</mo><mrow id="Thmtheorem6.p1.3.3.m3.5.5.3.2" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.cmml"><msub id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.cmml"><mi id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.2" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.2.cmml">D</mi><mo id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.3" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.3.cmml">⋄</mo></msub><mo id="Thmtheorem6.p1.3.3.m3.5.5.3.2.2" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.2.cmml">⁢</mo><mrow id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.cmml"><mo id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.2" stretchy="false" 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xref="Thmtheorem6.p1.3.3.m3.3.3.1.2"></times><apply id="Thmtheorem6.p1.3.3.m3.3.3.1.3.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3"><csymbol cd="ambiguous" id="Thmtheorem6.p1.3.3.m3.3.3.1.3.1.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3">subscript</csymbol><ci id="Thmtheorem6.p1.3.3.m3.3.3.1.3.2.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3.2">𝐷</ci><ci id="Thmtheorem6.p1.3.3.m3.3.3.1.3.3.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.3.3">⋄</ci></apply><apply id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1"><csymbol cd="latexml" id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.1.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.2.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.3.cmml" xref="Thmtheorem6.p1.3.3.m3.3.3.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem6.p1.3.3.m3.5.5.3.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3"><plus 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xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.2">𝐴</ci><list id="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.1.cmml" xref="Thmtheorem6.p1.3.3.m3.4.4.2.1.1.1.1.3.2"><ci id="Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="Thmtheorem6.p1.3.3.m3.1.1">𝐸</ci><ci id="Thmtheorem6.p1.3.3.m3.2.2.cmml" xref="Thmtheorem6.p1.3.3.m3.2.2">ℬ</ci></list></apply></apply><apply id="Thmtheorem6.p1.3.3.m3.5.5.3.2.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2"><times id="Thmtheorem6.p1.3.3.m3.5.5.3.2.2.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.2"></times><apply id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.1.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3">subscript</csymbol><ci id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.2.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.2">𝐷</ci><ci id="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.3.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.3.3">⋄</ci></apply><apply id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1"><csymbol cd="latexml" id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.1.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.2.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.2">𝐸</ci><ci id="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.3.cmml" xref="Thmtheorem6.p1.3.3.m3.5.5.3.2.1.1.1.3">ℬ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.3.3.m3.5c">D_{\diamond}(A\mid\mathcal{B})\leq D_{\diamond}(A\mid E,\mathcal{B})+D_{% \diamond}(E\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.3.3.m3.5d">italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_A ∣ italic_E , caligraphic_B ) + italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_E ∣ caligraphic_B )</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote7"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note"><span class="ltx_text ltx_font_upright" id="footnote7.1.1.1">7</span></span><span class="ltx_text ltx_font_upright" id="footnote7.9">We often abuse notation and write </span><math alttext="D(A\mid E,\mathcal{B})" class="ltx_Math" display="inline" id="footnote7.m1.3"><semantics id="footnote7.m1.3b"><mrow id="footnote7.m1.3.3" xref="footnote7.m1.3.3.cmml"><mi id="footnote7.m1.3.3.3" xref="footnote7.m1.3.3.3.cmml">D</mi><mo id="footnote7.m1.3.3.2" xref="footnote7.m1.3.3.2.cmml">⁢</mo><mrow id="footnote7.m1.3.3.1.1" xref="footnote7.m1.3.3.1.1.1.cmml"><mo id="footnote7.m1.3.3.1.1.2" stretchy="false" xref="footnote7.m1.3.3.1.1.1.cmml">(</mo><mrow id="footnote7.m1.3.3.1.1.1" xref="footnote7.m1.3.3.1.1.1.cmml"><mi id="footnote7.m1.3.3.1.1.1.2" xref="footnote7.m1.3.3.1.1.1.2.cmml">A</mi><mo id="footnote7.m1.3.3.1.1.1.1" xref="footnote7.m1.3.3.1.1.1.1.cmml">∣</mo><mrow id="footnote7.m1.3.3.1.1.1.3.2" xref="footnote7.m1.3.3.1.1.1.3.1.cmml"><mi id="footnote7.m1.1.1" xref="footnote7.m1.1.1.cmml">E</mi><mo id="footnote7.m1.3.3.1.1.1.3.2.1" xref="footnote7.m1.3.3.1.1.1.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="footnote7.m1.2.2" xref="footnote7.m1.2.2.cmml">ℬ</mi></mrow></mrow><mo id="footnote7.m1.3.3.1.1.3" stretchy="false" xref="footnote7.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m1.3c"><apply id="footnote7.m1.3.3.cmml" xref="footnote7.m1.3.3"><times id="footnote7.m1.3.3.2.cmml" xref="footnote7.m1.3.3.2"></times><ci id="footnote7.m1.3.3.3.cmml" xref="footnote7.m1.3.3.3">𝐷</ci><apply id="footnote7.m1.3.3.1.1.1.cmml" xref="footnote7.m1.3.3.1.1"><csymbol cd="latexml" id="footnote7.m1.3.3.1.1.1.1.cmml" xref="footnote7.m1.3.3.1.1.1.1">conditional</csymbol><ci id="footnote7.m1.3.3.1.1.1.2.cmml" xref="footnote7.m1.3.3.1.1.1.2">𝐴</ci><list id="footnote7.m1.3.3.1.1.1.3.1.cmml" xref="footnote7.m1.3.3.1.1.1.3.2"><ci id="footnote7.m1.1.1.cmml" xref="footnote7.m1.1.1">𝐸</ci><ci id="footnote7.m1.2.2.cmml" xref="footnote7.m1.2.2">ℬ</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m1.3d">D(A\mid E,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="footnote7.m1.3e">italic_D ( italic_A ∣ italic_E , caligraphic_B )</annotation></semantics></math><span class="ltx_text ltx_font_upright" id="footnote7.10"> instead of </span><math alttext="D(A\mid\{E\}\cup\mathcal{B})" class="ltx_Math" display="inline" id="footnote7.m2.2"><semantics id="footnote7.m2.2b"><mrow id="footnote7.m2.2.2" xref="footnote7.m2.2.2.cmml"><mi id="footnote7.m2.2.2.3" xref="footnote7.m2.2.2.3.cmml">D</mi><mo id="footnote7.m2.2.2.2" xref="footnote7.m2.2.2.2.cmml">⁢</mo><mrow id="footnote7.m2.2.2.1.1" xref="footnote7.m2.2.2.1.1.1.cmml"><mo id="footnote7.m2.2.2.1.1.2" stretchy="false" xref="footnote7.m2.2.2.1.1.1.cmml">(</mo><mrow id="footnote7.m2.2.2.1.1.1" xref="footnote7.m2.2.2.1.1.1.cmml"><mi id="footnote7.m2.2.2.1.1.1.2" xref="footnote7.m2.2.2.1.1.1.2.cmml">A</mi><mo id="footnote7.m2.2.2.1.1.1.1" xref="footnote7.m2.2.2.1.1.1.1.cmml">∣</mo><mrow id="footnote7.m2.2.2.1.1.1.3" xref="footnote7.m2.2.2.1.1.1.3.cmml"><mrow id="footnote7.m2.2.2.1.1.1.3.2.2" xref="footnote7.m2.2.2.1.1.1.3.2.1.cmml"><mo id="footnote7.m2.2.2.1.1.1.3.2.2.1" stretchy="false" xref="footnote7.m2.2.2.1.1.1.3.2.1.cmml">{</mo><mi id="footnote7.m2.1.1" xref="footnote7.m2.1.1.cmml">E</mi><mo id="footnote7.m2.2.2.1.1.1.3.2.2.2" stretchy="false" xref="footnote7.m2.2.2.1.1.1.3.2.1.cmml">}</mo></mrow><mo id="footnote7.m2.2.2.1.1.1.3.1" xref="footnote7.m2.2.2.1.1.1.3.1.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="footnote7.m2.2.2.1.1.1.3.3" xref="footnote7.m2.2.2.1.1.1.3.3.cmml">ℬ</mi></mrow></mrow><mo id="footnote7.m2.2.2.1.1.3" stretchy="false" xref="footnote7.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m2.2c"><apply id="footnote7.m2.2.2.cmml" xref="footnote7.m2.2.2"><times id="footnote7.m2.2.2.2.cmml" xref="footnote7.m2.2.2.2"></times><ci id="footnote7.m2.2.2.3.cmml" xref="footnote7.m2.2.2.3">𝐷</ci><apply id="footnote7.m2.2.2.1.1.1.cmml" xref="footnote7.m2.2.2.1.1"><csymbol cd="latexml" id="footnote7.m2.2.2.1.1.1.1.cmml" xref="footnote7.m2.2.2.1.1.1.1">conditional</csymbol><ci id="footnote7.m2.2.2.1.1.1.2.cmml" xref="footnote7.m2.2.2.1.1.1.2">𝐴</ci><apply id="footnote7.m2.2.2.1.1.1.3.cmml" xref="footnote7.m2.2.2.1.1.1.3"><union id="footnote7.m2.2.2.1.1.1.3.1.cmml" xref="footnote7.m2.2.2.1.1.1.3.1"></union><set id="footnote7.m2.2.2.1.1.1.3.2.1.cmml" xref="footnote7.m2.2.2.1.1.1.3.2.2"><ci id="footnote7.m2.1.1.cmml" xref="footnote7.m2.1.1">𝐸</ci></set><ci id="footnote7.m2.2.2.1.1.1.3.3.cmml" xref="footnote7.m2.2.2.1.1.1.3.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m2.2d">D(A\mid\{E\}\cup\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="footnote7.m2.2e">italic_D ( italic_A ∣ { italic_E } ∪ caligraphic_B )</annotation></semantics></math><span class="ltx_text ltx_font_upright" id="footnote7.11">.</span></span></span></span></span></p> </div> </div> <div class="ltx_proof" id="S2.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS3.1.p1"> <p class="ltx_p" id="S2.SS3.1.p1.7">Let <math alttext="t_{1}=D_{\diamond}(A\mid E,\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS3.1.p1.1.m1.3"><semantics id="S2.SS3.1.p1.1.m1.3a"><mrow id="S2.SS3.1.p1.1.m1.3.3" xref="S2.SS3.1.p1.1.m1.3.3.cmml"><msub id="S2.SS3.1.p1.1.m1.3.3.3" xref="S2.SS3.1.p1.1.m1.3.3.3.cmml"><mi id="S2.SS3.1.p1.1.m1.3.3.3.2" xref="S2.SS3.1.p1.1.m1.3.3.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.1.m1.3.3.3.3" xref="S2.SS3.1.p1.1.m1.3.3.3.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.1.m1.3.3.2" xref="S2.SS3.1.p1.1.m1.3.3.2.cmml">=</mo><mrow id="S2.SS3.1.p1.1.m1.3.3.1" xref="S2.SS3.1.p1.1.m1.3.3.1.cmml"><msub id="S2.SS3.1.p1.1.m1.3.3.1.3" xref="S2.SS3.1.p1.1.m1.3.3.1.3.cmml"><mi id="S2.SS3.1.p1.1.m1.3.3.1.3.2" xref="S2.SS3.1.p1.1.m1.3.3.1.3.2.cmml">D</mi><mo id="S2.SS3.1.p1.1.m1.3.3.1.3.3" xref="S2.SS3.1.p1.1.m1.3.3.1.3.3.cmml">⋄</mo></msub><mo id="S2.SS3.1.p1.1.m1.3.3.1.2" xref="S2.SS3.1.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS3.1.p1.1.m1.3.3.1.1.1" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.cmml"><mo id="S2.SS3.1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.2" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.2.cmml">A</mi><mo id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.1" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.1.cmml">∣</mo><mrow id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.2" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.1.cmml"><mi id="S2.SS3.1.p1.1.m1.1.1" xref="S2.SS3.1.p1.1.m1.1.1.cmml">E</mi><mo id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.2.1" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.1.p1.1.m1.2.2" xref="S2.SS3.1.p1.1.m1.2.2.cmml">ℬ</mi></mrow></mrow><mo id="S2.SS3.1.p1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.1.m1.3b"><apply id="S2.SS3.1.p1.1.m1.3.3.cmml" xref="S2.SS3.1.p1.1.m1.3.3"><eq id="S2.SS3.1.p1.1.m1.3.3.2.cmml" xref="S2.SS3.1.p1.1.m1.3.3.2"></eq><apply id="S2.SS3.1.p1.1.m1.3.3.3.cmml" xref="S2.SS3.1.p1.1.m1.3.3.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.1.m1.3.3.3.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.3">subscript</csymbol><ci id="S2.SS3.1.p1.1.m1.3.3.3.2.cmml" xref="S2.SS3.1.p1.1.m1.3.3.3.2">𝑡</ci><cn id="S2.SS3.1.p1.1.m1.3.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.1.m1.3.3.3.3">1</cn></apply><apply id="S2.SS3.1.p1.1.m1.3.3.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1"><times id="S2.SS3.1.p1.1.m1.3.3.1.2.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.2"></times><apply id="S2.SS3.1.p1.1.m1.3.3.1.3.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.1.m1.3.3.1.3.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.3">subscript</csymbol><ci id="S2.SS3.1.p1.1.m1.3.3.1.3.2.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.3.2">𝐷</ci><ci id="S2.SS3.1.p1.1.m1.3.3.1.3.3.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.3.3">⋄</ci></apply><apply id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1"><csymbol cd="latexml" id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.2">𝐴</ci><list id="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.1.cmml" xref="S2.SS3.1.p1.1.m1.3.3.1.1.1.1.3.2"><ci id="S2.SS3.1.p1.1.m1.1.1.cmml" xref="S2.SS3.1.p1.1.m1.1.1">𝐸</ci><ci id="S2.SS3.1.p1.1.m1.2.2.cmml" xref="S2.SS3.1.p1.1.m1.2.2">ℬ</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.1.m1.3c">t_{1}=D_{\diamond}(A\mid E,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.1.m1.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_A ∣ italic_E , caligraphic_B )</annotation></semantics></math>, witnessed by the sequence <math alttext="A_{1},\ldots,A_{t_{1}}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.2.m2.3"><semantics id="S2.SS3.1.p1.2.m2.3a"><mrow id="S2.SS3.1.p1.2.m2.3.3.2" xref="S2.SS3.1.p1.2.m2.3.3.3.cmml"><msub id="S2.SS3.1.p1.2.m2.2.2.1.1" xref="S2.SS3.1.p1.2.m2.2.2.1.1.cmml"><mi id="S2.SS3.1.p1.2.m2.2.2.1.1.2" xref="S2.SS3.1.p1.2.m2.2.2.1.1.2.cmml">A</mi><mn id="S2.SS3.1.p1.2.m2.2.2.1.1.3" xref="S2.SS3.1.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.2.m2.3.3.2.3" xref="S2.SS3.1.p1.2.m2.3.3.3.cmml">,</mo><mi id="S2.SS3.1.p1.2.m2.1.1" mathvariant="normal" xref="S2.SS3.1.p1.2.m2.1.1.cmml">…</mi><mo id="S2.SS3.1.p1.2.m2.3.3.2.4" xref="S2.SS3.1.p1.2.m2.3.3.3.cmml">,</mo><msub id="S2.SS3.1.p1.2.m2.3.3.2.2" xref="S2.SS3.1.p1.2.m2.3.3.2.2.cmml"><mi id="S2.SS3.1.p1.2.m2.3.3.2.2.2" xref="S2.SS3.1.p1.2.m2.3.3.2.2.2.cmml">A</mi><msub id="S2.SS3.1.p1.2.m2.3.3.2.2.3" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3.cmml"><mi id="S2.SS3.1.p1.2.m2.3.3.2.2.3.2" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.2.m2.3.3.2.2.3.3" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3.3.cmml">1</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.2.m2.3b"><list id="S2.SS3.1.p1.2.m2.3.3.3.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2"><apply id="S2.SS3.1.p1.2.m2.2.2.1.1.cmml" xref="S2.SS3.1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.1.p1.2.m2.2.2.1.1.1.cmml" xref="S2.SS3.1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S2.SS3.1.p1.2.m2.2.2.1.1.2.cmml" xref="S2.SS3.1.p1.2.m2.2.2.1.1.2">𝐴</ci><cn id="S2.SS3.1.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S2.SS3.1.p1.2.m2.2.2.1.1.3">1</cn></apply><ci id="S2.SS3.1.p1.2.m2.1.1.cmml" xref="S2.SS3.1.p1.2.m2.1.1">…</ci><apply id="S2.SS3.1.p1.2.m2.3.3.2.2.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.1.p1.2.m2.3.3.2.2.1.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2">subscript</csymbol><ci id="S2.SS3.1.p1.2.m2.3.3.2.2.2.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2.2">𝐴</ci><apply id="S2.SS3.1.p1.2.m2.3.3.2.2.3.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.2.m2.3.3.2.2.3.1.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3">subscript</csymbol><ci id="S2.SS3.1.p1.2.m2.3.3.2.2.3.2.cmml" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3.2">𝑡</ci><cn id="S2.SS3.1.p1.2.m2.3.3.2.2.3.3.cmml" type="integer" xref="S2.SS3.1.p1.2.m2.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.2.m2.3c">A_{1},\ldots,A_{t_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.2.m2.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Also, let <math alttext="t_{2}=D_{\diamond}(E\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS3.1.p1.3.m3.1"><semantics id="S2.SS3.1.p1.3.m3.1a"><mrow id="S2.SS3.1.p1.3.m3.1.1" xref="S2.SS3.1.p1.3.m3.1.1.cmml"><msub id="S2.SS3.1.p1.3.m3.1.1.3" xref="S2.SS3.1.p1.3.m3.1.1.3.cmml"><mi id="S2.SS3.1.p1.3.m3.1.1.3.2" xref="S2.SS3.1.p1.3.m3.1.1.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.3.m3.1.1.3.3" xref="S2.SS3.1.p1.3.m3.1.1.3.3.cmml">2</mn></msub><mo id="S2.SS3.1.p1.3.m3.1.1.2" xref="S2.SS3.1.p1.3.m3.1.1.2.cmml">=</mo><mrow id="S2.SS3.1.p1.3.m3.1.1.1" xref="S2.SS3.1.p1.3.m3.1.1.1.cmml"><msub id="S2.SS3.1.p1.3.m3.1.1.1.3" xref="S2.SS3.1.p1.3.m3.1.1.1.3.cmml"><mi id="S2.SS3.1.p1.3.m3.1.1.1.3.2" xref="S2.SS3.1.p1.3.m3.1.1.1.3.2.cmml">D</mi><mo id="S2.SS3.1.p1.3.m3.1.1.1.3.3" xref="S2.SS3.1.p1.3.m3.1.1.1.3.3.cmml">⋄</mo></msub><mo id="S2.SS3.1.p1.3.m3.1.1.1.2" xref="S2.SS3.1.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.1.p1.3.m3.1.1.1.1.1" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS3.1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.2" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.2.cmml">E</mi><mo id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.1" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.3" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS3.1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.3.m3.1b"><apply id="S2.SS3.1.p1.3.m3.1.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1"><eq id="S2.SS3.1.p1.3.m3.1.1.2.cmml" xref="S2.SS3.1.p1.3.m3.1.1.2"></eq><apply id="S2.SS3.1.p1.3.m3.1.1.3.cmml" xref="S2.SS3.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.3.m3.1.1.3.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS3.1.p1.3.m3.1.1.3.2.cmml" xref="S2.SS3.1.p1.3.m3.1.1.3.2">𝑡</ci><cn id="S2.SS3.1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS3.1.p1.3.m3.1.1.3.3">2</cn></apply><apply id="S2.SS3.1.p1.3.m3.1.1.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1"><times id="S2.SS3.1.p1.3.m3.1.1.1.2.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.2"></times><apply id="S2.SS3.1.p1.3.m3.1.1.1.3.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.3.m3.1.1.1.3.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="S2.SS3.1.p1.3.m3.1.1.1.3.2.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.3.2">𝐷</ci><ci id="S2.SS3.1.p1.3.m3.1.1.1.3.3.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.3.3">⋄</ci></apply><apply id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.2">𝐸</ci><ci id="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S2.SS3.1.p1.3.m3.1.1.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.3.m3.1c">t_{2}=D_{\diamond}(E\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.3.m3.1d">italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_E ∣ caligraphic_B )</annotation></semantics></math>, with a corresponding sequence <math alttext="E_{1},\ldots,E_{t_{2}}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.4.m4.3"><semantics id="S2.SS3.1.p1.4.m4.3a"><mrow id="S2.SS3.1.p1.4.m4.3.3.2" xref="S2.SS3.1.p1.4.m4.3.3.3.cmml"><msub id="S2.SS3.1.p1.4.m4.2.2.1.1" xref="S2.SS3.1.p1.4.m4.2.2.1.1.cmml"><mi id="S2.SS3.1.p1.4.m4.2.2.1.1.2" xref="S2.SS3.1.p1.4.m4.2.2.1.1.2.cmml">E</mi><mn id="S2.SS3.1.p1.4.m4.2.2.1.1.3" xref="S2.SS3.1.p1.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.4.m4.3.3.2.3" xref="S2.SS3.1.p1.4.m4.3.3.3.cmml">,</mo><mi id="S2.SS3.1.p1.4.m4.1.1" mathvariant="normal" xref="S2.SS3.1.p1.4.m4.1.1.cmml">…</mi><mo id="S2.SS3.1.p1.4.m4.3.3.2.4" xref="S2.SS3.1.p1.4.m4.3.3.3.cmml">,</mo><msub id="S2.SS3.1.p1.4.m4.3.3.2.2" xref="S2.SS3.1.p1.4.m4.3.3.2.2.cmml"><mi id="S2.SS3.1.p1.4.m4.3.3.2.2.2" xref="S2.SS3.1.p1.4.m4.3.3.2.2.2.cmml">E</mi><msub id="S2.SS3.1.p1.4.m4.3.3.2.2.3" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3.cmml"><mi id="S2.SS3.1.p1.4.m4.3.3.2.2.3.2" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.4.m4.3.3.2.2.3.3" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3.3.cmml">2</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.4.m4.3b"><list id="S2.SS3.1.p1.4.m4.3.3.3.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2"><apply id="S2.SS3.1.p1.4.m4.2.2.1.1.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.1.p1.4.m4.2.2.1.1.1.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1">subscript</csymbol><ci id="S2.SS3.1.p1.4.m4.2.2.1.1.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.2">𝐸</ci><cn id="S2.SS3.1.p1.4.m4.2.2.1.1.3.cmml" type="integer" xref="S2.SS3.1.p1.4.m4.2.2.1.1.3">1</cn></apply><ci id="S2.SS3.1.p1.4.m4.1.1.cmml" xref="S2.SS3.1.p1.4.m4.1.1">…</ci><apply id="S2.SS3.1.p1.4.m4.3.3.2.2.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.1.p1.4.m4.3.3.2.2.1.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2">subscript</csymbol><ci id="S2.SS3.1.p1.4.m4.3.3.2.2.2.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2.2">𝐸</ci><apply id="S2.SS3.1.p1.4.m4.3.3.2.2.3.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.4.m4.3.3.2.2.3.1.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3">subscript</csymbol><ci id="S2.SS3.1.p1.4.m4.3.3.2.2.3.2.cmml" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3.2">𝑡</ci><cn id="S2.SS3.1.p1.4.m4.3.3.2.2.3.3.cmml" type="integer" xref="S2.SS3.1.p1.4.m4.3.3.2.2.3.3">2</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.4.m4.3c">E_{1},\ldots,E_{t_{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.4.m4.3d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_E start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="E_{1},\ldots,E_{t_{2}},A_{1},\ldots,A_{t_{1}}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.5.m5.6"><semantics id="S2.SS3.1.p1.5.m5.6a"><mrow id="S2.SS3.1.p1.5.m5.6.6.4" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml"><msub id="S2.SS3.1.p1.5.m5.3.3.1.1" xref="S2.SS3.1.p1.5.m5.3.3.1.1.cmml"><mi id="S2.SS3.1.p1.5.m5.3.3.1.1.2" xref="S2.SS3.1.p1.5.m5.3.3.1.1.2.cmml">E</mi><mn id="S2.SS3.1.p1.5.m5.3.3.1.1.3" xref="S2.SS3.1.p1.5.m5.3.3.1.1.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.5.m5.6.6.4.5" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml">,</mo><mi id="S2.SS3.1.p1.5.m5.1.1" mathvariant="normal" xref="S2.SS3.1.p1.5.m5.1.1.cmml">…</mi><mo id="S2.SS3.1.p1.5.m5.6.6.4.6" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS3.1.p1.5.m5.4.4.2.2" xref="S2.SS3.1.p1.5.m5.4.4.2.2.cmml"><mi id="S2.SS3.1.p1.5.m5.4.4.2.2.2" xref="S2.SS3.1.p1.5.m5.4.4.2.2.2.cmml">E</mi><msub id="S2.SS3.1.p1.5.m5.4.4.2.2.3" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3.cmml"><mi id="S2.SS3.1.p1.5.m5.4.4.2.2.3.2" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.5.m5.4.4.2.2.3.3" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3.3.cmml">2</mn></msub></msub><mo id="S2.SS3.1.p1.5.m5.6.6.4.7" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS3.1.p1.5.m5.5.5.3.3" xref="S2.SS3.1.p1.5.m5.5.5.3.3.cmml"><mi id="S2.SS3.1.p1.5.m5.5.5.3.3.2" xref="S2.SS3.1.p1.5.m5.5.5.3.3.2.cmml">A</mi><mn id="S2.SS3.1.p1.5.m5.5.5.3.3.3" xref="S2.SS3.1.p1.5.m5.5.5.3.3.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.5.m5.6.6.4.8" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml">,</mo><mi id="S2.SS3.1.p1.5.m5.2.2" mathvariant="normal" xref="S2.SS3.1.p1.5.m5.2.2.cmml">…</mi><mo id="S2.SS3.1.p1.5.m5.6.6.4.9" xref="S2.SS3.1.p1.5.m5.6.6.5.cmml">,</mo><msub id="S2.SS3.1.p1.5.m5.6.6.4.4" xref="S2.SS3.1.p1.5.m5.6.6.4.4.cmml"><mi id="S2.SS3.1.p1.5.m5.6.6.4.4.2" xref="S2.SS3.1.p1.5.m5.6.6.4.4.2.cmml">A</mi><msub id="S2.SS3.1.p1.5.m5.6.6.4.4.3" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3.cmml"><mi id="S2.SS3.1.p1.5.m5.6.6.4.4.3.2" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.5.m5.6.6.4.4.3.3" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3.3.cmml">1</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.5.m5.6b"><list id="S2.SS3.1.p1.5.m5.6.6.5.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4"><apply id="S2.SS3.1.p1.5.m5.3.3.1.1.cmml" xref="S2.SS3.1.p1.5.m5.3.3.1.1"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.3.3.1.1.1.cmml" xref="S2.SS3.1.p1.5.m5.3.3.1.1">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.3.3.1.1.2.cmml" xref="S2.SS3.1.p1.5.m5.3.3.1.1.2">𝐸</ci><cn id="S2.SS3.1.p1.5.m5.3.3.1.1.3.cmml" type="integer" xref="S2.SS3.1.p1.5.m5.3.3.1.1.3">1</cn></apply><ci id="S2.SS3.1.p1.5.m5.1.1.cmml" xref="S2.SS3.1.p1.5.m5.1.1">…</ci><apply id="S2.SS3.1.p1.5.m5.4.4.2.2.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.4.4.2.2.1.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.4.4.2.2.2.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2.2">𝐸</ci><apply id="S2.SS3.1.p1.5.m5.4.4.2.2.3.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.4.4.2.2.3.1.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.4.4.2.2.3.2.cmml" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3.2">𝑡</ci><cn id="S2.SS3.1.p1.5.m5.4.4.2.2.3.3.cmml" type="integer" xref="S2.SS3.1.p1.5.m5.4.4.2.2.3.3">2</cn></apply></apply><apply id="S2.SS3.1.p1.5.m5.5.5.3.3.cmml" xref="S2.SS3.1.p1.5.m5.5.5.3.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.5.5.3.3.1.cmml" xref="S2.SS3.1.p1.5.m5.5.5.3.3">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.5.5.3.3.2.cmml" xref="S2.SS3.1.p1.5.m5.5.5.3.3.2">𝐴</ci><cn id="S2.SS3.1.p1.5.m5.5.5.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.5.m5.5.5.3.3.3">1</cn></apply><ci id="S2.SS3.1.p1.5.m5.2.2.cmml" xref="S2.SS3.1.p1.5.m5.2.2">…</ci><apply id="S2.SS3.1.p1.5.m5.6.6.4.4.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.6.6.4.4.1.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.6.6.4.4.2.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4.2">𝐴</ci><apply id="S2.SS3.1.p1.5.m5.6.6.4.4.3.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.5.m5.6.6.4.4.3.1.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3">subscript</csymbol><ci id="S2.SS3.1.p1.5.m5.6.6.4.4.3.2.cmml" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3.2">𝑡</ci><cn id="S2.SS3.1.p1.5.m5.6.6.4.4.3.3.cmml" type="integer" xref="S2.SS3.1.p1.5.m5.6.6.4.4.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.5.m5.6c">E_{1},\ldots,E_{t_{2}},A_{1},\ldots,A_{t_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.5.m5.6d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_E start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a sequence of length <math alttext="t_{1}+t_{2}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.6.m6.1"><semantics id="S2.SS3.1.p1.6.m6.1a"><mrow id="S2.SS3.1.p1.6.m6.1.1" xref="S2.SS3.1.p1.6.m6.1.1.cmml"><msub id="S2.SS3.1.p1.6.m6.1.1.2" xref="S2.SS3.1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS3.1.p1.6.m6.1.1.2.2" xref="S2.SS3.1.p1.6.m6.1.1.2.2.cmml">t</mi><mn id="S2.SS3.1.p1.6.m6.1.1.2.3" xref="S2.SS3.1.p1.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.6.m6.1.1.1" xref="S2.SS3.1.p1.6.m6.1.1.1.cmml">+</mo><msub id="S2.SS3.1.p1.6.m6.1.1.3" xref="S2.SS3.1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS3.1.p1.6.m6.1.1.3.2" xref="S2.SS3.1.p1.6.m6.1.1.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.6.m6.1.1.3.3" xref="S2.SS3.1.p1.6.m6.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.6.m6.1b"><apply id="S2.SS3.1.p1.6.m6.1.1.cmml" xref="S2.SS3.1.p1.6.m6.1.1"><plus id="S2.SS3.1.p1.6.m6.1.1.1.cmml" xref="S2.SS3.1.p1.6.m6.1.1.1"></plus><apply id="S2.SS3.1.p1.6.m6.1.1.2.cmml" xref="S2.SS3.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS3.1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS3.1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS3.1.p1.6.m6.1.1.2.2">𝑡</ci><cn id="S2.SS3.1.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S2.SS3.1.p1.6.m6.1.1.2.3">1</cn></apply><apply id="S2.SS3.1.p1.6.m6.1.1.3.cmml" xref="S2.SS3.1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS3.1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS3.1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS3.1.p1.6.m6.1.1.3.2">𝑡</ci><cn id="S2.SS3.1.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS3.1.p1.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.6.m6.1c">t_{1}+t_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.6.m6.1d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> showing that <math alttext="D_{\diamond}(A\mid\mathcal{B})\leq t_{1}+t_{2}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.7.m7.1"><semantics id="S2.SS3.1.p1.7.m7.1a"><mrow id="S2.SS3.1.p1.7.m7.1.1" xref="S2.SS3.1.p1.7.m7.1.1.cmml"><mrow id="S2.SS3.1.p1.7.m7.1.1.1" xref="S2.SS3.1.p1.7.m7.1.1.1.cmml"><msub id="S2.SS3.1.p1.7.m7.1.1.1.3" xref="S2.SS3.1.p1.7.m7.1.1.1.3.cmml"><mi id="S2.SS3.1.p1.7.m7.1.1.1.3.2" xref="S2.SS3.1.p1.7.m7.1.1.1.3.2.cmml">D</mi><mo id="S2.SS3.1.p1.7.m7.1.1.1.3.3" xref="S2.SS3.1.p1.7.m7.1.1.1.3.3.cmml">⋄</mo></msub><mo id="S2.SS3.1.p1.7.m7.1.1.1.2" xref="S2.SS3.1.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.1.p1.7.m7.1.1.1.1.1" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS3.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.2" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.1" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.3" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS3.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.1.p1.7.m7.1.1.2" xref="S2.SS3.1.p1.7.m7.1.1.2.cmml">≤</mo><mrow id="S2.SS3.1.p1.7.m7.1.1.3" xref="S2.SS3.1.p1.7.m7.1.1.3.cmml"><msub id="S2.SS3.1.p1.7.m7.1.1.3.2" xref="S2.SS3.1.p1.7.m7.1.1.3.2.cmml"><mi id="S2.SS3.1.p1.7.m7.1.1.3.2.2" xref="S2.SS3.1.p1.7.m7.1.1.3.2.2.cmml">t</mi><mn id="S2.SS3.1.p1.7.m7.1.1.3.2.3" xref="S2.SS3.1.p1.7.m7.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS3.1.p1.7.m7.1.1.3.1" xref="S2.SS3.1.p1.7.m7.1.1.3.1.cmml">+</mo><msub id="S2.SS3.1.p1.7.m7.1.1.3.3" xref="S2.SS3.1.p1.7.m7.1.1.3.3.cmml"><mi id="S2.SS3.1.p1.7.m7.1.1.3.3.2" xref="S2.SS3.1.p1.7.m7.1.1.3.3.2.cmml">t</mi><mn id="S2.SS3.1.p1.7.m7.1.1.3.3.3" xref="S2.SS3.1.p1.7.m7.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.7.m7.1b"><apply id="S2.SS3.1.p1.7.m7.1.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1"><leq id="S2.SS3.1.p1.7.m7.1.1.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.2"></leq><apply id="S2.SS3.1.p1.7.m7.1.1.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1"><times id="S2.SS3.1.p1.7.m7.1.1.1.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.2"></times><apply id="S2.SS3.1.p1.7.m7.1.1.1.3.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.7.m7.1.1.1.3.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.3">subscript</csymbol><ci id="S2.SS3.1.p1.7.m7.1.1.1.3.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.3.2">𝐷</ci><ci id="S2.SS3.1.p1.7.m7.1.1.1.3.3.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.3.3">⋄</ci></apply><apply id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S2.SS3.1.p1.7.m7.1.1.1.1.1.1.3">ℬ</ci></apply></apply><apply id="S2.SS3.1.p1.7.m7.1.1.3.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3"><plus id="S2.SS3.1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.1"></plus><apply id="S2.SS3.1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS3.1.p1.7.m7.1.1.3.2.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S2.SS3.1.p1.7.m7.1.1.3.2.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.2.2">𝑡</ci><cn id="S2.SS3.1.p1.7.m7.1.1.3.2.3.cmml" type="integer" xref="S2.SS3.1.p1.7.m7.1.1.3.2.3">1</cn></apply><apply id="S2.SS3.1.p1.7.m7.1.1.3.3.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.1.p1.7.m7.1.1.3.3.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S2.SS3.1.p1.7.m7.1.1.3.3.2.cmml" xref="S2.SS3.1.p1.7.m7.1.1.3.3.2">𝑡</ci><cn id="S2.SS3.1.p1.7.m7.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.7.m7.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.7.m7.1c">D_{\diamond}(A\mid\mathcal{B})\leq t_{1}+t_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.7.m7.1d">italic_D start_POSTSUBSCRIPT ⋄ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.11">Observe that a construction of an arbitrary set <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">caligraphic_B</annotation></semantics></math> provides a construction of <math alttext="A_{U}" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><msub id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml">A</mi><mi id="S2.SS3.p2.3.m3.1.1.3" xref="S2.SS3.p2.3.m3.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS3.p2.3.m3.1.1.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2">𝐴</ci><ci id="S2.SS3.p2.3.m3.1.1.3.cmml" xref="S2.SS3.p2.3.m3.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">A_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> from the sets in <math alttext="\mathcal{B}_{U}" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><msub id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.4.m4.1.1.2" xref="S2.SS3.p2.4.m4.1.1.2.cmml">ℬ</mi><mi id="S2.SS3.p2.4.m4.1.1.3" xref="S2.SS3.p2.4.m4.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><apply id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2">ℬ</ci><ci id="S2.SS3.p2.4.m4.1.1.3.cmml" xref="S2.SS3.p2.4.m4.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">\mathcal{B}_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> (recall that <math alttext="A_{U}\stackrel{{\scriptstyle\rm def}}{{=}}A\cap U" class="ltx_Math" display="inline" id="S2.SS3.p2.5.m5.1"><semantics id="S2.SS3.p2.5.m5.1a"><mrow id="S2.SS3.p2.5.m5.1.1" xref="S2.SS3.p2.5.m5.1.1.cmml"><msub id="S2.SS3.p2.5.m5.1.1.2" xref="S2.SS3.p2.5.m5.1.1.2.cmml"><mi id="S2.SS3.p2.5.m5.1.1.2.2" xref="S2.SS3.p2.5.m5.1.1.2.2.cmml">A</mi><mi id="S2.SS3.p2.5.m5.1.1.2.3" xref="S2.SS3.p2.5.m5.1.1.2.3.cmml">U</mi></msub><mover id="S2.SS3.p2.5.m5.1.1.1" xref="S2.SS3.p2.5.m5.1.1.1.cmml"><mo id="S2.SS3.p2.5.m5.1.1.1.2" xref="S2.SS3.p2.5.m5.1.1.1.2.cmml">=</mo><mi id="S2.SS3.p2.5.m5.1.1.1.3" xref="S2.SS3.p2.5.m5.1.1.1.3.cmml">def</mi></mover><mrow id="S2.SS3.p2.5.m5.1.1.3" xref="S2.SS3.p2.5.m5.1.1.3.cmml"><mi id="S2.SS3.p2.5.m5.1.1.3.2" xref="S2.SS3.p2.5.m5.1.1.3.2.cmml">A</mi><mo id="S2.SS3.p2.5.m5.1.1.3.1" xref="S2.SS3.p2.5.m5.1.1.3.1.cmml">∩</mo><mi id="S2.SS3.p2.5.m5.1.1.3.3" xref="S2.SS3.p2.5.m5.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.5.m5.1b"><apply id="S2.SS3.p2.5.m5.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1"><apply id="S2.SS3.p2.5.m5.1.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.5.m5.1.1.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1.1">superscript</csymbol><eq id="S2.SS3.p2.5.m5.1.1.1.2.cmml" xref="S2.SS3.p2.5.m5.1.1.1.2"></eq><ci id="S2.SS3.p2.5.m5.1.1.1.3.cmml" xref="S2.SS3.p2.5.m5.1.1.1.3">def</ci></apply><apply id="S2.SS3.p2.5.m5.1.1.2.cmml" xref="S2.SS3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.5.m5.1.1.2.1.cmml" xref="S2.SS3.p2.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.5.m5.1.1.2.2.cmml" xref="S2.SS3.p2.5.m5.1.1.2.2">𝐴</ci><ci id="S2.SS3.p2.5.m5.1.1.2.3.cmml" xref="S2.SS3.p2.5.m5.1.1.2.3">𝑈</ci></apply><apply id="S2.SS3.p2.5.m5.1.1.3.cmml" xref="S2.SS3.p2.5.m5.1.1.3"><intersect id="S2.SS3.p2.5.m5.1.1.3.1.cmml" xref="S2.SS3.p2.5.m5.1.1.3.1"></intersect><ci id="S2.SS3.p2.5.m5.1.1.3.2.cmml" xref="S2.SS3.p2.5.m5.1.1.3.2">𝐴</ci><ci id="S2.SS3.p2.5.m5.1.1.3.3.cmml" xref="S2.SS3.p2.5.m5.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.5.m5.1c">A_{U}\stackrel{{\scriptstyle\rm def}}{{=}}A\cap U</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.5.m5.1d">italic_A start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP italic_A ∩ italic_U</annotation></semantics></math>, etc.). Indeed, it is easy to see that if <math alttext="A^{1},\ldots,A^{t}" class="ltx_Math" display="inline" id="S2.SS3.p2.6.m6.3"><semantics id="S2.SS3.p2.6.m6.3a"><mrow id="S2.SS3.p2.6.m6.3.3.2" xref="S2.SS3.p2.6.m6.3.3.3.cmml"><msup id="S2.SS3.p2.6.m6.2.2.1.1" xref="S2.SS3.p2.6.m6.2.2.1.1.cmml"><mi id="S2.SS3.p2.6.m6.2.2.1.1.2" xref="S2.SS3.p2.6.m6.2.2.1.1.2.cmml">A</mi><mn id="S2.SS3.p2.6.m6.2.2.1.1.3" xref="S2.SS3.p2.6.m6.2.2.1.1.3.cmml">1</mn></msup><mo id="S2.SS3.p2.6.m6.3.3.2.3" xref="S2.SS3.p2.6.m6.3.3.3.cmml">,</mo><mi id="S2.SS3.p2.6.m6.1.1" mathvariant="normal" xref="S2.SS3.p2.6.m6.1.1.cmml">…</mi><mo id="S2.SS3.p2.6.m6.3.3.2.4" xref="S2.SS3.p2.6.m6.3.3.3.cmml">,</mo><msup id="S2.SS3.p2.6.m6.3.3.2.2" xref="S2.SS3.p2.6.m6.3.3.2.2.cmml"><mi id="S2.SS3.p2.6.m6.3.3.2.2.2" xref="S2.SS3.p2.6.m6.3.3.2.2.2.cmml">A</mi><mi id="S2.SS3.p2.6.m6.3.3.2.2.3" xref="S2.SS3.p2.6.m6.3.3.2.2.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.6.m6.3b"><list id="S2.SS3.p2.6.m6.3.3.3.cmml" xref="S2.SS3.p2.6.m6.3.3.2"><apply id="S2.SS3.p2.6.m6.2.2.1.1.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.6.m6.2.2.1.1.1.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1">superscript</csymbol><ci id="S2.SS3.p2.6.m6.2.2.1.1.2.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1.2">𝐴</ci><cn id="S2.SS3.p2.6.m6.2.2.1.1.3.cmml" type="integer" xref="S2.SS3.p2.6.m6.2.2.1.1.3">1</cn></apply><ci id="S2.SS3.p2.6.m6.1.1.cmml" xref="S2.SS3.p2.6.m6.1.1">…</ci><apply id="S2.SS3.p2.6.m6.3.3.2.2.cmml" xref="S2.SS3.p2.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p2.6.m6.3.3.2.2.1.cmml" xref="S2.SS3.p2.6.m6.3.3.2.2">superscript</csymbol><ci id="S2.SS3.p2.6.m6.3.3.2.2.2.cmml" xref="S2.SS3.p2.6.m6.3.3.2.2.2">𝐴</ci><ci id="S2.SS3.p2.6.m6.3.3.2.2.3.cmml" xref="S2.SS3.p2.6.m6.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.6.m6.3c">A^{1},\ldots,A^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.6.m6.3d">italic_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_A start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.p2.7.m7.1"><semantics id="S2.SS3.p2.7.m7.1a"><mi id="S2.SS3.p2.7.m7.1.1" xref="S2.SS3.p2.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.7.m7.1b"><ci id="S2.SS3.p2.7.m7.1.1.cmml" xref="S2.SS3.p2.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.7.m7.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS3.p2.8.m8.1"><semantics id="S2.SS3.p2.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.8.m8.1.1" xref="S2.SS3.p2.8.m8.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.8.m8.1b"><ci id="S2.SS3.p2.8.m8.1.1.cmml" xref="S2.SS3.p2.8.m8.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.8.m8.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.8.m8.1d">caligraphic_B</annotation></semantics></math>, then <math alttext="A^{1}_{U},\ldots,A^{t}_{U}" class="ltx_Math" display="inline" id="S2.SS3.p2.9.m9.3"><semantics id="S2.SS3.p2.9.m9.3a"><mrow id="S2.SS3.p2.9.m9.3.3.2" xref="S2.SS3.p2.9.m9.3.3.3.cmml"><msubsup id="S2.SS3.p2.9.m9.2.2.1.1" xref="S2.SS3.p2.9.m9.2.2.1.1.cmml"><mi id="S2.SS3.p2.9.m9.2.2.1.1.2.2" xref="S2.SS3.p2.9.m9.2.2.1.1.2.2.cmml">A</mi><mi id="S2.SS3.p2.9.m9.2.2.1.1.3" xref="S2.SS3.p2.9.m9.2.2.1.1.3.cmml">U</mi><mn id="S2.SS3.p2.9.m9.2.2.1.1.2.3" xref="S2.SS3.p2.9.m9.2.2.1.1.2.3.cmml">1</mn></msubsup><mo id="S2.SS3.p2.9.m9.3.3.2.3" xref="S2.SS3.p2.9.m9.3.3.3.cmml">,</mo><mi id="S2.SS3.p2.9.m9.1.1" mathvariant="normal" xref="S2.SS3.p2.9.m9.1.1.cmml">…</mi><mo id="S2.SS3.p2.9.m9.3.3.2.4" xref="S2.SS3.p2.9.m9.3.3.3.cmml">,</mo><msubsup id="S2.SS3.p2.9.m9.3.3.2.2" xref="S2.SS3.p2.9.m9.3.3.2.2.cmml"><mi id="S2.SS3.p2.9.m9.3.3.2.2.2.2" xref="S2.SS3.p2.9.m9.3.3.2.2.2.2.cmml">A</mi><mi id="S2.SS3.p2.9.m9.3.3.2.2.3" xref="S2.SS3.p2.9.m9.3.3.2.2.3.cmml">U</mi><mi id="S2.SS3.p2.9.m9.3.3.2.2.2.3" xref="S2.SS3.p2.9.m9.3.3.2.2.2.3.cmml">t</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.9.m9.3b"><list id="S2.SS3.p2.9.m9.3.3.3.cmml" xref="S2.SS3.p2.9.m9.3.3.2"><apply id="S2.SS3.p2.9.m9.2.2.1.1.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.9.m9.2.2.1.1.1.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1">subscript</csymbol><apply id="S2.SS3.p2.9.m9.2.2.1.1.2.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.9.m9.2.2.1.1.2.1.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1">superscript</csymbol><ci id="S2.SS3.p2.9.m9.2.2.1.1.2.2.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1.2.2">𝐴</ci><cn id="S2.SS3.p2.9.m9.2.2.1.1.2.3.cmml" type="integer" xref="S2.SS3.p2.9.m9.2.2.1.1.2.3">1</cn></apply><ci id="S2.SS3.p2.9.m9.2.2.1.1.3.cmml" xref="S2.SS3.p2.9.m9.2.2.1.1.3">𝑈</ci></apply><ci id="S2.SS3.p2.9.m9.1.1.cmml" xref="S2.SS3.p2.9.m9.1.1">…</ci><apply id="S2.SS3.p2.9.m9.3.3.2.2.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p2.9.m9.3.3.2.2.1.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2">subscript</csymbol><apply id="S2.SS3.p2.9.m9.3.3.2.2.2.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p2.9.m9.3.3.2.2.2.1.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2">superscript</csymbol><ci id="S2.SS3.p2.9.m9.3.3.2.2.2.2.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS3.p2.9.m9.3.3.2.2.2.3.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2.2.3">𝑡</ci></apply><ci id="S2.SS3.p2.9.m9.3.3.2.2.3.cmml" xref="S2.SS3.p2.9.m9.3.3.2.2.3">𝑈</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.9.m9.3c">A^{1}_{U},\ldots,A^{t}_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.9.m9.3d">italic_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , … , italic_A start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> generates <math alttext="A_{U}" class="ltx_Math" display="inline" id="S2.SS3.p2.10.m10.1"><semantics id="S2.SS3.p2.10.m10.1a"><msub id="S2.SS3.p2.10.m10.1.1" xref="S2.SS3.p2.10.m10.1.1.cmml"><mi id="S2.SS3.p2.10.m10.1.1.2" xref="S2.SS3.p2.10.m10.1.1.2.cmml">A</mi><mi id="S2.SS3.p2.10.m10.1.1.3" xref="S2.SS3.p2.10.m10.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.10.m10.1b"><apply id="S2.SS3.p2.10.m10.1.1.cmml" xref="S2.SS3.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.10.m10.1.1.1.cmml" xref="S2.SS3.p2.10.m10.1.1">subscript</csymbol><ci id="S2.SS3.p2.10.m10.1.1.2.cmml" xref="S2.SS3.p2.10.m10.1.1.2">𝐴</ci><ci id="S2.SS3.p2.10.m10.1.1.3.cmml" xref="S2.SS3.p2.10.m10.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.10.m10.1c">A_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.10.m10.1d">italic_A start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\mathcal{B}_{U}" class="ltx_Math" display="inline" id="S2.SS3.p2.11.m11.1"><semantics id="S2.SS3.p2.11.m11.1a"><msub id="S2.SS3.p2.11.m11.1.1" xref="S2.SS3.p2.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.11.m11.1.1.2" xref="S2.SS3.p2.11.m11.1.1.2.cmml">ℬ</mi><mi id="S2.SS3.p2.11.m11.1.1.3" xref="S2.SS3.p2.11.m11.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.11.m11.1b"><apply id="S2.SS3.p2.11.m11.1.1.cmml" xref="S2.SS3.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.11.m11.1.1.1.cmml" xref="S2.SS3.p2.11.m11.1.1">subscript</csymbol><ci id="S2.SS3.p2.11.m11.1.1.2.cmml" xref="S2.SS3.p2.11.m11.1.1.2">ℬ</ci><ci id="S2.SS3.p2.11.m11.1.1.3.cmml" xref="S2.SS3.p2.11.m11.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.11.m11.1c">\mathcal{B}_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.11.m11.1d">caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_fact" id="Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem7.1.1.1">Fact 7</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem7.p1"> <p class="ltx_p" id="Thmtheorem7.p1.1"><math alttext="D(A_{U}\mid\mathcal{B}_{U})\leq D(A\mid\mathcal{B})." class="ltx_Math" display="inline" id="Thmtheorem7.p1.1.m1.1"><semantics id="Thmtheorem7.p1.1.m1.1a"><mrow id="Thmtheorem7.p1.1.m1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.cmml"><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.cmml"><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.cmml"><mi id="Thmtheorem7.p1.1.m1.1.1.1.1.1.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.3.cmml">D</mi><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.1.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.cmml"><msub id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.3.cmml">U</mi></msub><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">∣</mo><msub id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.3.cmml">U</mi></msub></mrow><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.3.cmml">≤</mo><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.cmml"><mi id="Thmtheorem7.p1.1.m1.1.1.1.1.2.3" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.3.cmml">D</mi><mo id="Thmtheorem7.p1.1.m1.1.1.1.1.2.2" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1" 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xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.2.3">𝑈</ci></apply><apply id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.2">ℬ</ci><ci id="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.1.1.1.1.3.3">𝑈</ci></apply></apply></apply><apply id="Thmtheorem7.p1.1.m1.1.1.1.1.2.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2"><times id="Thmtheorem7.p1.1.m1.1.1.1.1.2.2.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.2"></times><ci id="Thmtheorem7.p1.1.m1.1.1.1.1.2.3.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.3">𝐷</ci><apply id="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1"><csymbol cd="latexml" id="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.1.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.2.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.2">𝐴</ci><ci id="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.3.cmml" xref="Thmtheorem7.p1.1.m1.1.1.1.1.2.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem7.p1.1.m1.1c">D(A_{U}\mid\mathcal{B}_{U})\leq D(A\mid\mathcal{B}).</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem7.p1.1.m1.1d">italic_D ( italic_A start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ) ≤ italic_D ( italic_A ∣ caligraphic_B ) .</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmtheorem7.p1.1.1"></span></p> </div> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.3">For convenience, we say that <math alttext="A^{1}_{U},\ldots,A^{t}_{U}" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.3"><semantics id="S2.SS3.p3.1.m1.3a"><mrow id="S2.SS3.p3.1.m1.3.3.2" xref="S2.SS3.p3.1.m1.3.3.3.cmml"><msubsup id="S2.SS3.p3.1.m1.2.2.1.1" xref="S2.SS3.p3.1.m1.2.2.1.1.cmml"><mi id="S2.SS3.p3.1.m1.2.2.1.1.2.2" xref="S2.SS3.p3.1.m1.2.2.1.1.2.2.cmml">A</mi><mi id="S2.SS3.p3.1.m1.2.2.1.1.3" xref="S2.SS3.p3.1.m1.2.2.1.1.3.cmml">U</mi><mn id="S2.SS3.p3.1.m1.2.2.1.1.2.3" xref="S2.SS3.p3.1.m1.2.2.1.1.2.3.cmml">1</mn></msubsup><mo id="S2.SS3.p3.1.m1.3.3.2.3" xref="S2.SS3.p3.1.m1.3.3.3.cmml">,</mo><mi id="S2.SS3.p3.1.m1.1.1" mathvariant="normal" xref="S2.SS3.p3.1.m1.1.1.cmml">…</mi><mo id="S2.SS3.p3.1.m1.3.3.2.4" xref="S2.SS3.p3.1.m1.3.3.3.cmml">,</mo><msubsup id="S2.SS3.p3.1.m1.3.3.2.2" xref="S2.SS3.p3.1.m1.3.3.2.2.cmml"><mi id="S2.SS3.p3.1.m1.3.3.2.2.2.2" xref="S2.SS3.p3.1.m1.3.3.2.2.2.2.cmml">A</mi><mi id="S2.SS3.p3.1.m1.3.3.2.2.3" xref="S2.SS3.p3.1.m1.3.3.2.2.3.cmml">U</mi><mi id="S2.SS3.p3.1.m1.3.3.2.2.2.3" xref="S2.SS3.p3.1.m1.3.3.2.2.2.3.cmml">t</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.3b"><list id="S2.SS3.p3.1.m1.3.3.3.cmml" xref="S2.SS3.p3.1.m1.3.3.2"><apply id="S2.SS3.p3.1.m1.2.2.1.1.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.2.2.1.1.1.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1">subscript</csymbol><apply id="S2.SS3.p3.1.m1.2.2.1.1.2.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.2.2.1.1.2.1.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1">superscript</csymbol><ci id="S2.SS3.p3.1.m1.2.2.1.1.2.2.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.2.2">𝐴</ci><cn id="S2.SS3.p3.1.m1.2.2.1.1.2.3.cmml" type="integer" xref="S2.SS3.p3.1.m1.2.2.1.1.2.3">1</cn></apply><ci id="S2.SS3.p3.1.m1.2.2.1.1.3.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.3">𝑈</ci></apply><ci id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1">…</ci><apply id="S2.SS3.p3.1.m1.3.3.2.2.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.3.3.2.2.1.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2">subscript</csymbol><apply id="S2.SS3.p3.1.m1.3.3.2.2.2.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.3.3.2.2.2.1.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2">superscript</csymbol><ci id="S2.SS3.p3.1.m1.3.3.2.2.2.2.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2.2.2">𝐴</ci><ci id="S2.SS3.p3.1.m1.3.3.2.2.2.3.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2.2.3">𝑡</ci></apply><ci id="S2.SS3.p3.1.m1.3.3.2.2.3.cmml" xref="S2.SS3.p3.1.m1.3.3.2.2.3">𝑈</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.3c">A^{1}_{U},\ldots,A^{t}_{U}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.3d">italic_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , … , italic_A start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S2.SS3.p3.3.1">relativization</em> of the sequence <math alttext="A^{1},\ldots,A^{t}" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.3"><semantics id="S2.SS3.p3.2.m2.3a"><mrow id="S2.SS3.p3.2.m2.3.3.2" xref="S2.SS3.p3.2.m2.3.3.3.cmml"><msup id="S2.SS3.p3.2.m2.2.2.1.1" xref="S2.SS3.p3.2.m2.2.2.1.1.cmml"><mi id="S2.SS3.p3.2.m2.2.2.1.1.2" xref="S2.SS3.p3.2.m2.2.2.1.1.2.cmml">A</mi><mn id="S2.SS3.p3.2.m2.2.2.1.1.3" xref="S2.SS3.p3.2.m2.2.2.1.1.3.cmml">1</mn></msup><mo id="S2.SS3.p3.2.m2.3.3.2.3" xref="S2.SS3.p3.2.m2.3.3.3.cmml">,</mo><mi id="S2.SS3.p3.2.m2.1.1" mathvariant="normal" xref="S2.SS3.p3.2.m2.1.1.cmml">…</mi><mo id="S2.SS3.p3.2.m2.3.3.2.4" xref="S2.SS3.p3.2.m2.3.3.3.cmml">,</mo><msup id="S2.SS3.p3.2.m2.3.3.2.2" xref="S2.SS3.p3.2.m2.3.3.2.2.cmml"><mi id="S2.SS3.p3.2.m2.3.3.2.2.2" xref="S2.SS3.p3.2.m2.3.3.2.2.2.cmml">A</mi><mi id="S2.SS3.p3.2.m2.3.3.2.2.3" xref="S2.SS3.p3.2.m2.3.3.2.2.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.3b"><list id="S2.SS3.p3.2.m2.3.3.3.cmml" xref="S2.SS3.p3.2.m2.3.3.2"><apply id="S2.SS3.p3.2.m2.2.2.1.1.cmml" xref="S2.SS3.p3.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p3.2.m2.2.2.1.1.1.cmml" xref="S2.SS3.p3.2.m2.2.2.1.1">superscript</csymbol><ci id="S2.SS3.p3.2.m2.2.2.1.1.2.cmml" xref="S2.SS3.p3.2.m2.2.2.1.1.2">𝐴</ci><cn id="S2.SS3.p3.2.m2.2.2.1.1.3.cmml" type="integer" xref="S2.SS3.p3.2.m2.2.2.1.1.3">1</cn></apply><ci id="S2.SS3.p3.2.m2.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1">…</ci><apply id="S2.SS3.p3.2.m2.3.3.2.2.cmml" xref="S2.SS3.p3.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.2.m2.3.3.2.2.1.cmml" xref="S2.SS3.p3.2.m2.3.3.2.2">superscript</csymbol><ci id="S2.SS3.p3.2.m2.3.3.2.2.2.cmml" xref="S2.SS3.p3.2.m2.3.3.2.2.2">𝐴</ci><ci id="S2.SS3.p3.2.m2.3.3.2.2.3.cmml" xref="S2.SS3.p3.2.m2.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.3c">A^{1},\ldots,A^{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.3d">italic_A start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_A start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> with respect to <math alttext="U" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.1"><semantics id="S2.SS3.p3.3.m3.1a"><mi id="S2.SS3.p3.3.m3.1.1" xref="S2.SS3.p3.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.3.m3.1b"><ci id="S2.SS3.p3.3.m3.1.1.cmml" xref="S2.SS3.p3.3.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.3.m3.1d">italic_U</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.1">The following simple technical fact will be useful. The proof is an easy induction via extended sequences.</p> </div> <div class="ltx_theorem ltx_theorem_fact" id="Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem8.1.1.1">Fact 8</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem8.p1"> <p class="ltx_p" id="Thmtheorem8.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem8.p1.2.2">If <math alttext="A" class="ltx_Math" display="inline" id="Thmtheorem8.p1.1.1.m1.1"><semantics id="Thmtheorem8.p1.1.1.m1.1a"><mi id="Thmtheorem8.p1.1.1.m1.1.1" xref="Thmtheorem8.p1.1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.1.1.m1.1b"><ci id="Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="Thmtheorem8.p1.1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.1.1.m1.1d">italic_A</annotation></semantics></math> is non-empty, then <math alttext="D_{\cap}(A\mid\mathcal{B})=D_{\cap}(A\mid\mathcal{B}\cup\{\emptyset\})" class="ltx_Math" display="inline" id="Thmtheorem8.p1.2.2.m2.3"><semantics id="Thmtheorem8.p1.2.2.m2.3a"><mrow id="Thmtheorem8.p1.2.2.m2.3.3" xref="Thmtheorem8.p1.2.2.m2.3.3.cmml"><mrow id="Thmtheorem8.p1.2.2.m2.2.2.1" xref="Thmtheorem8.p1.2.2.m2.2.2.1.cmml"><msub id="Thmtheorem8.p1.2.2.m2.2.2.1.3" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3.cmml"><mi id="Thmtheorem8.p1.2.2.m2.2.2.1.3.2" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3.2.cmml">D</mi><mo id="Thmtheorem8.p1.2.2.m2.2.2.1.3.3" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem8.p1.2.2.m2.2.2.1.2" xref="Thmtheorem8.p1.2.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.cmml"><mo id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.2" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.cmml"><mi id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.2" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.1" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.3" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem8.p1.2.2.m2.3.3.3" xref="Thmtheorem8.p1.2.2.m2.3.3.3.cmml">=</mo><mrow id="Thmtheorem8.p1.2.2.m2.3.3.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.cmml"><msub id="Thmtheorem8.p1.2.2.m2.3.3.2.3" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3.cmml"><mi id="Thmtheorem8.p1.2.2.m2.3.3.2.3.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3.2.cmml">D</mi><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.3.3" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3.3.cmml">∩</mo></msub><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.2.cmml">⁢</mo><mrow id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.cmml"><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.2" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.cmml">(</mo><mrow id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.cmml"><mi id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.2.cmml">A</mi><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.1" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.1.cmml">∣</mo><mrow id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.2.cmml">ℬ</mi><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.1" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.1.cmml">∪</mo><mrow id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.2" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.1.cmml"><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.2.1" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.1.cmml">{</mo><mi id="Thmtheorem8.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem8.p1.2.2.m2.1.1.cmml">∅</mi><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.2.2" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.3" stretchy="false" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.2.2.m2.3b"><apply id="Thmtheorem8.p1.2.2.m2.3.3.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3"><eq id="Thmtheorem8.p1.2.2.m2.3.3.3.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.3"></eq><apply id="Thmtheorem8.p1.2.2.m2.2.2.1.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1"><times id="Thmtheorem8.p1.2.2.m2.2.2.1.2.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.2"></times><apply id="Thmtheorem8.p1.2.2.m2.2.2.1.3.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3"><csymbol cd="ambiguous" id="Thmtheorem8.p1.2.2.m2.2.2.1.3.1.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3">subscript</csymbol><ci id="Thmtheorem8.p1.2.2.m2.2.2.1.3.2.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3.2">𝐷</ci><intersect id="Thmtheorem8.p1.2.2.m2.2.2.1.3.3.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.3.3"></intersect></apply><apply id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1"><csymbol cd="latexml" id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.1.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.2.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.3.cmml" xref="Thmtheorem8.p1.2.2.m2.2.2.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem8.p1.2.2.m2.3.3.2.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2"><times id="Thmtheorem8.p1.2.2.m2.3.3.2.2.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.2"></times><apply id="Thmtheorem8.p1.2.2.m2.3.3.2.3.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem8.p1.2.2.m2.3.3.2.3.1.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3">subscript</csymbol><ci id="Thmtheorem8.p1.2.2.m2.3.3.2.3.2.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3.2">𝐷</ci><intersect id="Thmtheorem8.p1.2.2.m2.3.3.2.3.3.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.3.3"></intersect></apply><apply id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1"><csymbol cd="latexml" id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.1.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.2.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.2">𝐴</ci><apply id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3"><union id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.1.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.1"></union><ci id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.2.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.2">ℬ</ci><set id="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.1.cmml" xref="Thmtheorem8.p1.2.2.m2.3.3.2.1.1.1.3.3.2"><emptyset id="Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="Thmtheorem8.p1.2.2.m2.1.1"></emptyset></set></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.2.2.m2.3c">D_{\cap}(A\mid\mathcal{B})=D_{\cap}(A\mid\mathcal{B}\cup\{\emptyset\})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.2.2.m2.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ∪ { ∅ } )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS3.p5"> <p class="ltx_p" id="S2.SS3.p5.1">The next lemma shows that intersection complexity and discrete complexity are polynomally related, with a dependency on <math alttext="|\mathcal{B}|" class="ltx_Math" display="inline" id="S2.SS3.p5.1.m1.1"><semantics id="S2.SS3.p5.1.m1.1a"><mrow id="S2.SS3.p5.1.m1.1.2.2" xref="S2.SS3.p5.1.m1.1.2.1.cmml"><mo id="S2.SS3.p5.1.m1.1.2.2.1" stretchy="false" xref="S2.SS3.p5.1.m1.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p5.1.m1.1.1" xref="S2.SS3.p5.1.m1.1.1.cmml">ℬ</mi><mo id="S2.SS3.p5.1.m1.1.2.2.2" stretchy="false" xref="S2.SS3.p5.1.m1.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.1.m1.1b"><apply id="S2.SS3.p5.1.m1.1.2.1.cmml" xref="S2.SS3.p5.1.m1.1.2.2"><abs id="S2.SS3.p5.1.m1.1.2.1.1.cmml" xref="S2.SS3.p5.1.m1.1.2.2.1"></abs><ci id="S2.SS3.p5.1.m1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.1.m1.1c">|\mathcal{B}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.1.m1.1d">| caligraphic_B |</annotation></semantics></math>. This was first observed for monotone circuits in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib1" title="">1</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem9.1.1.1">Lemma 9</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem9.2.2"> </span>(Immediate from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib21" title="">21</a>]</cite>)<span class="ltx_text ltx_font_bold" id="Thmtheorem9.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem9.p1"> <p class="ltx_p" id="Thmtheorem9.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem9.p1.1.1">If <math alttext="1<D_{\cap}(A\mid\mathcal{B})=k<\infty" class="ltx_Math" display="inline" id="Thmtheorem9.p1.1.1.m1.1"><semantics id="Thmtheorem9.p1.1.1.m1.1a"><mrow id="Thmtheorem9.p1.1.1.m1.1.1" xref="Thmtheorem9.p1.1.1.m1.1.1.cmml"><mn id="Thmtheorem9.p1.1.1.m1.1.1.3" xref="Thmtheorem9.p1.1.1.m1.1.1.3.cmml">1</mn><mo id="Thmtheorem9.p1.1.1.m1.1.1.4" xref="Thmtheorem9.p1.1.1.m1.1.1.4.cmml"><</mo><mrow id="Thmtheorem9.p1.1.1.m1.1.1.1" xref="Thmtheorem9.p1.1.1.m1.1.1.1.cmml"><msub id="Thmtheorem9.p1.1.1.m1.1.1.1.3" xref="Thmtheorem9.p1.1.1.m1.1.1.1.3.cmml"><mi id="Thmtheorem9.p1.1.1.m1.1.1.1.3.2" xref="Thmtheorem9.p1.1.1.m1.1.1.1.3.2.cmml">D</mi><mo id="Thmtheorem9.p1.1.1.m1.1.1.1.3.3" xref="Thmtheorem9.p1.1.1.m1.1.1.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem9.p1.1.1.m1.1.1.1.2" xref="Thmtheorem9.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.2" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.1" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.3" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem9.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem9.p1.1.1.m1.1.1.5" xref="Thmtheorem9.p1.1.1.m1.1.1.5.cmml">=</mo><mi id="Thmtheorem9.p1.1.1.m1.1.1.6" xref="Thmtheorem9.p1.1.1.m1.1.1.6.cmml">k</mi><mo id="Thmtheorem9.p1.1.1.m1.1.1.7" xref="Thmtheorem9.p1.1.1.m1.1.1.7.cmml"><</mo><mi id="Thmtheorem9.p1.1.1.m1.1.1.8" mathvariant="normal" xref="Thmtheorem9.p1.1.1.m1.1.1.8.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem9.p1.1.1.m1.1b"><apply id="Thmtheorem9.p1.1.1.m1.1.1.cmml" 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xref="Thmtheorem9.p1.1.1.m1.1.1.7"></lt><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem9.p1.1.1.m1.1.1.6.cmml" id="Thmtheorem9.p1.1.1.m1.1.1f.cmml" xref="Thmtheorem9.p1.1.1.m1.1.1"></share><infinity id="Thmtheorem9.p1.1.1.m1.1.1.8.cmml" xref="Thmtheorem9.p1.1.1.m1.1.1.8"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem9.p1.1.1.m1.1c">1<D_{\cap}(A\mid\mathcal{B})=k<\infty</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem9.p1.1.1.m1.1d">1 < italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_k < ∞</annotation></semantics></math>, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D(A\mid\mathcal{B})=O(k(|\mathcal{B}|+k)/\log k)." class="ltx_Math" display="block" 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id="S2.Ex3.m1.2c">D(A\mid\mathcal{B})=O(k(|\mathcal{B}|+k)/\log k).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.2d">italic_D ( italic_A ∣ caligraphic_B ) = italic_O ( italic_k ( | caligraphic_B | + italic_k ) / roman_log italic_k ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.1">We describe a self-contained, indirect proof of a weaker form of this lemma in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS3" title="3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.3</span></a> (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem28" title="Corollary 28 (Intersection complexity versus discrete complexity). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">28</span></a>).</p> </div> <div class="ltx_para" id="S2.SS3.p7"> <p class="ltx_p" id="S2.SS3.p7.16">Given <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.p7.1.m1.1"><semantics id="S2.SS3.p7.1.m1.1a"><mi id="S2.SS3.p7.1.m1.1.1" xref="S2.SS3.p7.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.1.m1.1b"><ci id="S2.SS3.p7.1.m1.1.1.cmml" xref="S2.SS3.p7.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS3.p7.2.m2.1"><semantics id="S2.SS3.p7.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p7.2.m2.1.1" xref="S2.SS3.p7.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.2.m2.1b"><ci id="S2.SS3.p7.2.m2.1.1.cmml" xref="S2.SS3.p7.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.2.m2.1d">caligraphic_B</annotation></semantics></math>, there is a simple test to decide if <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS3.p7.3.m3.1"><semantics id="S2.SS3.p7.3.m3.1a"><mrow id="S2.SS3.p7.3.m3.1.1" xref="S2.SS3.p7.3.m3.1.1.cmml"><mi id="S2.SS3.p7.3.m3.1.1.3" xref="S2.SS3.p7.3.m3.1.1.3.cmml">D</mi><mo id="S2.SS3.p7.3.m3.1.1.2" xref="S2.SS3.p7.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.p7.3.m3.1.1.1.1" xref="S2.SS3.p7.3.m3.1.1.1.1.1.cmml"><mo id="S2.SS3.p7.3.m3.1.1.1.1.2" stretchy="false" xref="S2.SS3.p7.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p7.3.m3.1.1.1.1.1" xref="S2.SS3.p7.3.m3.1.1.1.1.1.cmml"><mi id="S2.SS3.p7.3.m3.1.1.1.1.1.2" xref="S2.SS3.p7.3.m3.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS3.p7.3.m3.1.1.1.1.1.1" xref="S2.SS3.p7.3.m3.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p7.3.m3.1.1.1.1.1.3" xref="S2.SS3.p7.3.m3.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS3.p7.3.m3.1.1.1.1.3" stretchy="false" xref="S2.SS3.p7.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.3.m3.1b"><apply id="S2.SS3.p7.3.m3.1.1.cmml" xref="S2.SS3.p7.3.m3.1.1"><times id="S2.SS3.p7.3.m3.1.1.2.cmml" xref="S2.SS3.p7.3.m3.1.1.2"></times><ci id="S2.SS3.p7.3.m3.1.1.3.cmml" xref="S2.SS3.p7.3.m3.1.1.3">𝐷</ci><apply id="S2.SS3.p7.3.m3.1.1.1.1.1.cmml" xref="S2.SS3.p7.3.m3.1.1.1.1"><csymbol cd="latexml" id="S2.SS3.p7.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS3.p7.3.m3.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.p7.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS3.p7.3.m3.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS3.p7.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS3.p7.3.m3.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.3.m3.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.3.m3.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> is finite, i.e., if there exists a finite sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.p7.4.m4.1"><semantics id="S2.SS3.p7.4.m4.1a"><mi id="S2.SS3.p7.4.m4.1.1" xref="S2.SS3.p7.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.4.m4.1b"><ci id="S2.SS3.p7.4.m4.1.1.cmml" xref="S2.SS3.p7.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.4.m4.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS3.p7.5.m5.1"><semantics id="S2.SS3.p7.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p7.5.m5.1.1" xref="S2.SS3.p7.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.5.m5.1b"><ci id="S2.SS3.p7.5.m5.1.1.cmml" xref="S2.SS3.p7.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.5.m5.1d">caligraphic_B</annotation></semantics></math>. Let <math alttext="\mathcal{B}=\{B_{1},\ldots,B_{m}\}" class="ltx_Math" display="inline" id="S2.SS3.p7.6.m6.3"><semantics id="S2.SS3.p7.6.m6.3a"><mrow id="S2.SS3.p7.6.m6.3.3" xref="S2.SS3.p7.6.m6.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p7.6.m6.3.3.4" xref="S2.SS3.p7.6.m6.3.3.4.cmml">ℬ</mi><mo id="S2.SS3.p7.6.m6.3.3.3" xref="S2.SS3.p7.6.m6.3.3.3.cmml">=</mo><mrow id="S2.SS3.p7.6.m6.3.3.2.2" xref="S2.SS3.p7.6.m6.3.3.2.3.cmml"><mo id="S2.SS3.p7.6.m6.3.3.2.2.3" stretchy="false" xref="S2.SS3.p7.6.m6.3.3.2.3.cmml">{</mo><msub id="S2.SS3.p7.6.m6.2.2.1.1.1" xref="S2.SS3.p7.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS3.p7.6.m6.2.2.1.1.1.2" xref="S2.SS3.p7.6.m6.2.2.1.1.1.2.cmml">B</mi><mn id="S2.SS3.p7.6.m6.2.2.1.1.1.3" xref="S2.SS3.p7.6.m6.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS3.p7.6.m6.3.3.2.2.4" xref="S2.SS3.p7.6.m6.3.3.2.3.cmml">,</mo><mi id="S2.SS3.p7.6.m6.1.1" mathvariant="normal" xref="S2.SS3.p7.6.m6.1.1.cmml">…</mi><mo id="S2.SS3.p7.6.m6.3.3.2.2.5" xref="S2.SS3.p7.6.m6.3.3.2.3.cmml">,</mo><msub id="S2.SS3.p7.6.m6.3.3.2.2.2" xref="S2.SS3.p7.6.m6.3.3.2.2.2.cmml"><mi id="S2.SS3.p7.6.m6.3.3.2.2.2.2" xref="S2.SS3.p7.6.m6.3.3.2.2.2.2.cmml">B</mi><mi id="S2.SS3.p7.6.m6.3.3.2.2.2.3" xref="S2.SS3.p7.6.m6.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S2.SS3.p7.6.m6.3.3.2.2.6" stretchy="false" xref="S2.SS3.p7.6.m6.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.6.m6.3b"><apply id="S2.SS3.p7.6.m6.3.3.cmml" xref="S2.SS3.p7.6.m6.3.3"><eq id="S2.SS3.p7.6.m6.3.3.3.cmml" xref="S2.SS3.p7.6.m6.3.3.3"></eq><ci id="S2.SS3.p7.6.m6.3.3.4.cmml" xref="S2.SS3.p7.6.m6.3.3.4">ℬ</ci><set id="S2.SS3.p7.6.m6.3.3.2.3.cmml" xref="S2.SS3.p7.6.m6.3.3.2.2"><apply id="S2.SS3.p7.6.m6.2.2.1.1.1.cmml" xref="S2.SS3.p7.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p7.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS3.p7.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.SS3.p7.6.m6.2.2.1.1.1.2.cmml" xref="S2.SS3.p7.6.m6.2.2.1.1.1.2">𝐵</ci><cn id="S2.SS3.p7.6.m6.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS3.p7.6.m6.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS3.p7.6.m6.1.1.cmml" xref="S2.SS3.p7.6.m6.1.1">…</ci><apply id="S2.SS3.p7.6.m6.3.3.2.2.2.cmml" xref="S2.SS3.p7.6.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p7.6.m6.3.3.2.2.2.1.cmml" xref="S2.SS3.p7.6.m6.3.3.2.2.2">subscript</csymbol><ci id="S2.SS3.p7.6.m6.3.3.2.2.2.2.cmml" xref="S2.SS3.p7.6.m6.3.3.2.2.2.2">𝐵</ci><ci id="S2.SS3.p7.6.m6.3.3.2.2.2.3.cmml" xref="S2.SS3.p7.6.m6.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.6.m6.3c">\mathcal{B}=\{B_{1},\ldots,B_{m}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.6.m6.3d">caligraphic_B = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math>. Given <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="S2.SS3.p7.7.m7.1"><semantics id="S2.SS3.p7.7.m7.1a"><mrow id="S2.SS3.p7.7.m7.1.1" xref="S2.SS3.p7.7.m7.1.1.cmml"><mi id="S2.SS3.p7.7.m7.1.1.2" xref="S2.SS3.p7.7.m7.1.1.2.cmml">w</mi><mo id="S2.SS3.p7.7.m7.1.1.1" xref="S2.SS3.p7.7.m7.1.1.1.cmml">∈</mo><mi id="S2.SS3.p7.7.m7.1.1.3" mathvariant="normal" xref="S2.SS3.p7.7.m7.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.7.m7.1b"><apply id="S2.SS3.p7.7.m7.1.1.cmml" xref="S2.SS3.p7.7.m7.1.1"><in id="S2.SS3.p7.7.m7.1.1.1.cmml" xref="S2.SS3.p7.7.m7.1.1.1"></in><ci id="S2.SS3.p7.7.m7.1.1.2.cmml" xref="S2.SS3.p7.7.m7.1.1.2">𝑤</ci><ci id="S2.SS3.p7.7.m7.1.1.3.cmml" xref="S2.SS3.p7.7.m7.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.7.m7.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.7.m7.1d">italic_w ∈ roman_Γ</annotation></semantics></math>, we let <math alttext="\mathsf{vec}(w)\in\{0,1\}^{m}" class="ltx_Math" display="inline" id="S2.SS3.p7.8.m8.3"><semantics id="S2.SS3.p7.8.m8.3a"><mrow id="S2.SS3.p7.8.m8.3.4" xref="S2.SS3.p7.8.m8.3.4.cmml"><mrow id="S2.SS3.p7.8.m8.3.4.2" xref="S2.SS3.p7.8.m8.3.4.2.cmml"><mi id="S2.SS3.p7.8.m8.3.4.2.2" xref="S2.SS3.p7.8.m8.3.4.2.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.p7.8.m8.3.4.2.1" xref="S2.SS3.p7.8.m8.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS3.p7.8.m8.3.4.2.3.2" xref="S2.SS3.p7.8.m8.3.4.2.cmml"><mo id="S2.SS3.p7.8.m8.3.4.2.3.2.1" stretchy="false" xref="S2.SS3.p7.8.m8.3.4.2.cmml">(</mo><mi id="S2.SS3.p7.8.m8.1.1" xref="S2.SS3.p7.8.m8.1.1.cmml">w</mi><mo id="S2.SS3.p7.8.m8.3.4.2.3.2.2" stretchy="false" xref="S2.SS3.p7.8.m8.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p7.8.m8.3.4.1" xref="S2.SS3.p7.8.m8.3.4.1.cmml">∈</mo><msup id="S2.SS3.p7.8.m8.3.4.3" xref="S2.SS3.p7.8.m8.3.4.3.cmml"><mrow id="S2.SS3.p7.8.m8.3.4.3.2.2" xref="S2.SS3.p7.8.m8.3.4.3.2.1.cmml"><mo id="S2.SS3.p7.8.m8.3.4.3.2.2.1" stretchy="false" xref="S2.SS3.p7.8.m8.3.4.3.2.1.cmml">{</mo><mn id="S2.SS3.p7.8.m8.2.2" xref="S2.SS3.p7.8.m8.2.2.cmml">0</mn><mo id="S2.SS3.p7.8.m8.3.4.3.2.2.2" xref="S2.SS3.p7.8.m8.3.4.3.2.1.cmml">,</mo><mn id="S2.SS3.p7.8.m8.3.3" xref="S2.SS3.p7.8.m8.3.3.cmml">1</mn><mo id="S2.SS3.p7.8.m8.3.4.3.2.2.3" stretchy="false" xref="S2.SS3.p7.8.m8.3.4.3.2.1.cmml">}</mo></mrow><mi id="S2.SS3.p7.8.m8.3.4.3.3" xref="S2.SS3.p7.8.m8.3.4.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.8.m8.3b"><apply id="S2.SS3.p7.8.m8.3.4.cmml" xref="S2.SS3.p7.8.m8.3.4"><in id="S2.SS3.p7.8.m8.3.4.1.cmml" xref="S2.SS3.p7.8.m8.3.4.1"></in><apply id="S2.SS3.p7.8.m8.3.4.2.cmml" xref="S2.SS3.p7.8.m8.3.4.2"><times id="S2.SS3.p7.8.m8.3.4.2.1.cmml" xref="S2.SS3.p7.8.m8.3.4.2.1"></times><ci id="S2.SS3.p7.8.m8.3.4.2.2.cmml" xref="S2.SS3.p7.8.m8.3.4.2.2">𝗏𝖾𝖼</ci><ci id="S2.SS3.p7.8.m8.1.1.cmml" xref="S2.SS3.p7.8.m8.1.1">𝑤</ci></apply><apply id="S2.SS3.p7.8.m8.3.4.3.cmml" xref="S2.SS3.p7.8.m8.3.4.3"><csymbol cd="ambiguous" id="S2.SS3.p7.8.m8.3.4.3.1.cmml" xref="S2.SS3.p7.8.m8.3.4.3">superscript</csymbol><set id="S2.SS3.p7.8.m8.3.4.3.2.1.cmml" xref="S2.SS3.p7.8.m8.3.4.3.2.2"><cn id="S2.SS3.p7.8.m8.2.2.cmml" type="integer" xref="S2.SS3.p7.8.m8.2.2">0</cn><cn id="S2.SS3.p7.8.m8.3.3.cmml" type="integer" xref="S2.SS3.p7.8.m8.3.3">1</cn></set><ci id="S2.SS3.p7.8.m8.3.4.3.3.cmml" xref="S2.SS3.p7.8.m8.3.4.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.8.m8.3c">\mathsf{vec}(w)\in\{0,1\}^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.8.m8.3d">sansserif_vec ( italic_w ) ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> be the vector with <math alttext="\mathsf{vec}(w)_{i}=1" class="ltx_Math" display="inline" id="S2.SS3.p7.9.m9.1"><semantics id="S2.SS3.p7.9.m9.1a"><mrow id="S2.SS3.p7.9.m9.1.2" xref="S2.SS3.p7.9.m9.1.2.cmml"><mrow id="S2.SS3.p7.9.m9.1.2.2" xref="S2.SS3.p7.9.m9.1.2.2.cmml"><mi id="S2.SS3.p7.9.m9.1.2.2.2" xref="S2.SS3.p7.9.m9.1.2.2.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.p7.9.m9.1.2.2.1" xref="S2.SS3.p7.9.m9.1.2.2.1.cmml">⁢</mo><msub id="S2.SS3.p7.9.m9.1.2.2.3" xref="S2.SS3.p7.9.m9.1.2.2.3.cmml"><mrow id="S2.SS3.p7.9.m9.1.2.2.3.2.2" xref="S2.SS3.p7.9.m9.1.2.2.3.cmml"><mo id="S2.SS3.p7.9.m9.1.2.2.3.2.2.1" stretchy="false" xref="S2.SS3.p7.9.m9.1.2.2.3.cmml">(</mo><mi id="S2.SS3.p7.9.m9.1.1" xref="S2.SS3.p7.9.m9.1.1.cmml">w</mi><mo id="S2.SS3.p7.9.m9.1.2.2.3.2.2.2" stretchy="false" xref="S2.SS3.p7.9.m9.1.2.2.3.cmml">)</mo></mrow><mi id="S2.SS3.p7.9.m9.1.2.2.3.3" xref="S2.SS3.p7.9.m9.1.2.2.3.3.cmml">i</mi></msub></mrow><mo id="S2.SS3.p7.9.m9.1.2.1" xref="S2.SS3.p7.9.m9.1.2.1.cmml">=</mo><mn id="S2.SS3.p7.9.m9.1.2.3" xref="S2.SS3.p7.9.m9.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.9.m9.1b"><apply id="S2.SS3.p7.9.m9.1.2.cmml" xref="S2.SS3.p7.9.m9.1.2"><eq id="S2.SS3.p7.9.m9.1.2.1.cmml" xref="S2.SS3.p7.9.m9.1.2.1"></eq><apply id="S2.SS3.p7.9.m9.1.2.2.cmml" xref="S2.SS3.p7.9.m9.1.2.2"><times id="S2.SS3.p7.9.m9.1.2.2.1.cmml" xref="S2.SS3.p7.9.m9.1.2.2.1"></times><ci id="S2.SS3.p7.9.m9.1.2.2.2.cmml" xref="S2.SS3.p7.9.m9.1.2.2.2">𝗏𝖾𝖼</ci><apply id="S2.SS3.p7.9.m9.1.2.2.3.cmml" xref="S2.SS3.p7.9.m9.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p7.9.m9.1.2.2.3.1.cmml" xref="S2.SS3.p7.9.m9.1.2.2.3">subscript</csymbol><ci id="S2.SS3.p7.9.m9.1.1.cmml" xref="S2.SS3.p7.9.m9.1.1">𝑤</ci><ci id="S2.SS3.p7.9.m9.1.2.2.3.3.cmml" xref="S2.SS3.p7.9.m9.1.2.2.3.3">𝑖</ci></apply></apply><cn id="S2.SS3.p7.9.m9.1.2.3.cmml" type="integer" xref="S2.SS3.p7.9.m9.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.9.m9.1c">\mathsf{vec}(w)_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.9.m9.1d">sansserif_vec ( italic_w ) start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> if and only if <math alttext="w\in B_{i}" class="ltx_Math" display="inline" id="S2.SS3.p7.10.m10.1"><semantics id="S2.SS3.p7.10.m10.1a"><mrow id="S2.SS3.p7.10.m10.1.1" xref="S2.SS3.p7.10.m10.1.1.cmml"><mi id="S2.SS3.p7.10.m10.1.1.2" xref="S2.SS3.p7.10.m10.1.1.2.cmml">w</mi><mo id="S2.SS3.p7.10.m10.1.1.1" xref="S2.SS3.p7.10.m10.1.1.1.cmml">∈</mo><msub id="S2.SS3.p7.10.m10.1.1.3" xref="S2.SS3.p7.10.m10.1.1.3.cmml"><mi id="S2.SS3.p7.10.m10.1.1.3.2" xref="S2.SS3.p7.10.m10.1.1.3.2.cmml">B</mi><mi id="S2.SS3.p7.10.m10.1.1.3.3" xref="S2.SS3.p7.10.m10.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.10.m10.1b"><apply id="S2.SS3.p7.10.m10.1.1.cmml" xref="S2.SS3.p7.10.m10.1.1"><in id="S2.SS3.p7.10.m10.1.1.1.cmml" xref="S2.SS3.p7.10.m10.1.1.1"></in><ci id="S2.SS3.p7.10.m10.1.1.2.cmml" xref="S2.SS3.p7.10.m10.1.1.2">𝑤</ci><apply id="S2.SS3.p7.10.m10.1.1.3.cmml" xref="S2.SS3.p7.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p7.10.m10.1.1.3.1.cmml" xref="S2.SS3.p7.10.m10.1.1.3">subscript</csymbol><ci id="S2.SS3.p7.10.m10.1.1.3.2.cmml" xref="S2.SS3.p7.10.m10.1.1.3.2">𝐵</ci><ci id="S2.SS3.p7.10.m10.1.1.3.3.cmml" xref="S2.SS3.p7.10.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.10.m10.1c">w\in B_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.10.m10.1d">italic_w ∈ italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. For a set <math alttext="C\subseteq\Gamma" class="ltx_Math" display="inline" id="S2.SS3.p7.11.m11.1"><semantics id="S2.SS3.p7.11.m11.1a"><mrow id="S2.SS3.p7.11.m11.1.1" xref="S2.SS3.p7.11.m11.1.1.cmml"><mi id="S2.SS3.p7.11.m11.1.1.2" xref="S2.SS3.p7.11.m11.1.1.2.cmml">C</mi><mo id="S2.SS3.p7.11.m11.1.1.1" xref="S2.SS3.p7.11.m11.1.1.1.cmml">⊆</mo><mi id="S2.SS3.p7.11.m11.1.1.3" mathvariant="normal" xref="S2.SS3.p7.11.m11.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.11.m11.1b"><apply id="S2.SS3.p7.11.m11.1.1.cmml" xref="S2.SS3.p7.11.m11.1.1"><subset id="S2.SS3.p7.11.m11.1.1.1.cmml" xref="S2.SS3.p7.11.m11.1.1.1"></subset><ci id="S2.SS3.p7.11.m11.1.1.2.cmml" xref="S2.SS3.p7.11.m11.1.1.2">𝐶</ci><ci id="S2.SS3.p7.11.m11.1.1.3.cmml" xref="S2.SS3.p7.11.m11.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.11.m11.1c">C\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.11.m11.1d">italic_C ⊆ roman_Γ</annotation></semantics></math>, let <math alttext="\mathsf{vec}(C)=\{\mathsf{vec}(c)\mid c\in C\}" class="ltx_Math" display="inline" id="S2.SS3.p7.12.m12.4"><semantics id="S2.SS3.p7.12.m12.4a"><mrow id="S2.SS3.p7.12.m12.4.4" xref="S2.SS3.p7.12.m12.4.4.cmml"><mrow id="S2.SS3.p7.12.m12.4.4.4" xref="S2.SS3.p7.12.m12.4.4.4.cmml"><mi id="S2.SS3.p7.12.m12.4.4.4.2" xref="S2.SS3.p7.12.m12.4.4.4.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.p7.12.m12.4.4.4.1" xref="S2.SS3.p7.12.m12.4.4.4.1.cmml">⁢</mo><mrow id="S2.SS3.p7.12.m12.4.4.4.3.2" xref="S2.SS3.p7.12.m12.4.4.4.cmml"><mo id="S2.SS3.p7.12.m12.4.4.4.3.2.1" stretchy="false" xref="S2.SS3.p7.12.m12.4.4.4.cmml">(</mo><mi id="S2.SS3.p7.12.m12.1.1" xref="S2.SS3.p7.12.m12.1.1.cmml">C</mi><mo id="S2.SS3.p7.12.m12.4.4.4.3.2.2" stretchy="false" xref="S2.SS3.p7.12.m12.4.4.4.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p7.12.m12.4.4.3" xref="S2.SS3.p7.12.m12.4.4.3.cmml">=</mo><mrow id="S2.SS3.p7.12.m12.4.4.2.2" xref="S2.SS3.p7.12.m12.4.4.2.3.cmml"><mo id="S2.SS3.p7.12.m12.4.4.2.2.3" stretchy="false" xref="S2.SS3.p7.12.m12.4.4.2.3.1.cmml">{</mo><mrow id="S2.SS3.p7.12.m12.3.3.1.1.1" xref="S2.SS3.p7.12.m12.3.3.1.1.1.cmml"><mi id="S2.SS3.p7.12.m12.3.3.1.1.1.2" xref="S2.SS3.p7.12.m12.3.3.1.1.1.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.p7.12.m12.3.3.1.1.1.1" xref="S2.SS3.p7.12.m12.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS3.p7.12.m12.3.3.1.1.1.3.2" xref="S2.SS3.p7.12.m12.3.3.1.1.1.cmml"><mo id="S2.SS3.p7.12.m12.3.3.1.1.1.3.2.1" stretchy="false" xref="S2.SS3.p7.12.m12.3.3.1.1.1.cmml">(</mo><mi id="S2.SS3.p7.12.m12.2.2" xref="S2.SS3.p7.12.m12.2.2.cmml">c</mi><mo id="S2.SS3.p7.12.m12.3.3.1.1.1.3.2.2" stretchy="false" xref="S2.SS3.p7.12.m12.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="S2.SS3.p7.12.m12.4.4.2.2.4" lspace="0em" rspace="0em" xref="S2.SS3.p7.12.m12.4.4.2.3.1.cmml">∣</mo><mrow id="S2.SS3.p7.12.m12.4.4.2.2.2" xref="S2.SS3.p7.12.m12.4.4.2.2.2.cmml"><mi id="S2.SS3.p7.12.m12.4.4.2.2.2.2" xref="S2.SS3.p7.12.m12.4.4.2.2.2.2.cmml">c</mi><mo id="S2.SS3.p7.12.m12.4.4.2.2.2.1" xref="S2.SS3.p7.12.m12.4.4.2.2.2.1.cmml">∈</mo><mi id="S2.SS3.p7.12.m12.4.4.2.2.2.3" xref="S2.SS3.p7.12.m12.4.4.2.2.2.3.cmml">C</mi></mrow><mo id="S2.SS3.p7.12.m12.4.4.2.2.5" stretchy="false" xref="S2.SS3.p7.12.m12.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.12.m12.4b"><apply id="S2.SS3.p7.12.m12.4.4.cmml" xref="S2.SS3.p7.12.m12.4.4"><eq id="S2.SS3.p7.12.m12.4.4.3.cmml" xref="S2.SS3.p7.12.m12.4.4.3"></eq><apply id="S2.SS3.p7.12.m12.4.4.4.cmml" xref="S2.SS3.p7.12.m12.4.4.4"><times id="S2.SS3.p7.12.m12.4.4.4.1.cmml" xref="S2.SS3.p7.12.m12.4.4.4.1"></times><ci id="S2.SS3.p7.12.m12.4.4.4.2.cmml" xref="S2.SS3.p7.12.m12.4.4.4.2">𝗏𝖾𝖼</ci><ci id="S2.SS3.p7.12.m12.1.1.cmml" xref="S2.SS3.p7.12.m12.1.1">𝐶</ci></apply><apply id="S2.SS3.p7.12.m12.4.4.2.3.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2"><csymbol cd="latexml" id="S2.SS3.p7.12.m12.4.4.2.3.1.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2.3">conditional-set</csymbol><apply id="S2.SS3.p7.12.m12.3.3.1.1.1.cmml" xref="S2.SS3.p7.12.m12.3.3.1.1.1"><times id="S2.SS3.p7.12.m12.3.3.1.1.1.1.cmml" xref="S2.SS3.p7.12.m12.3.3.1.1.1.1"></times><ci id="S2.SS3.p7.12.m12.3.3.1.1.1.2.cmml" xref="S2.SS3.p7.12.m12.3.3.1.1.1.2">𝗏𝖾𝖼</ci><ci id="S2.SS3.p7.12.m12.2.2.cmml" xref="S2.SS3.p7.12.m12.2.2">𝑐</ci></apply><apply id="S2.SS3.p7.12.m12.4.4.2.2.2.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2.2"><in id="S2.SS3.p7.12.m12.4.4.2.2.2.1.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2.2.1"></in><ci id="S2.SS3.p7.12.m12.4.4.2.2.2.2.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2.2.2">𝑐</ci><ci id="S2.SS3.p7.12.m12.4.4.2.2.2.3.cmml" xref="S2.SS3.p7.12.m12.4.4.2.2.2.3">𝐶</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.12.m12.4c">\mathsf{vec}(C)=\{\mathsf{vec}(c)\mid c\in C\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.12.m12.4d">sansserif_vec ( italic_C ) = { sansserif_vec ( italic_c ) ∣ italic_c ∈ italic_C }</annotation></semantics></math>. For vectors <math alttext="u,v\in\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS3.p7.13.m13.4"><semantics id="S2.SS3.p7.13.m13.4a"><mrow id="S2.SS3.p7.13.m13.4.5" xref="S2.SS3.p7.13.m13.4.5.cmml"><mrow id="S2.SS3.p7.13.m13.4.5.2.2" xref="S2.SS3.p7.13.m13.4.5.2.1.cmml"><mi id="S2.SS3.p7.13.m13.3.3" xref="S2.SS3.p7.13.m13.3.3.cmml">u</mi><mo id="S2.SS3.p7.13.m13.4.5.2.2.1" xref="S2.SS3.p7.13.m13.4.5.2.1.cmml">,</mo><mi id="S2.SS3.p7.13.m13.4.4" xref="S2.SS3.p7.13.m13.4.4.cmml">v</mi></mrow><mo id="S2.SS3.p7.13.m13.4.5.1" xref="S2.SS3.p7.13.m13.4.5.1.cmml">∈</mo><msup id="S2.SS3.p7.13.m13.4.5.3" xref="S2.SS3.p7.13.m13.4.5.3.cmml"><mrow id="S2.SS3.p7.13.m13.4.5.3.2.2" xref="S2.SS3.p7.13.m13.4.5.3.2.1.cmml"><mo id="S2.SS3.p7.13.m13.4.5.3.2.2.1" stretchy="false" xref="S2.SS3.p7.13.m13.4.5.3.2.1.cmml">{</mo><mn id="S2.SS3.p7.13.m13.1.1" xref="S2.SS3.p7.13.m13.1.1.cmml">0</mn><mo id="S2.SS3.p7.13.m13.4.5.3.2.2.2" xref="S2.SS3.p7.13.m13.4.5.3.2.1.cmml">,</mo><mn id="S2.SS3.p7.13.m13.2.2" xref="S2.SS3.p7.13.m13.2.2.cmml">1</mn><mo id="S2.SS3.p7.13.m13.4.5.3.2.2.3" stretchy="false" xref="S2.SS3.p7.13.m13.4.5.3.2.1.cmml">}</mo></mrow><mi id="S2.SS3.p7.13.m13.4.5.3.3" xref="S2.SS3.p7.13.m13.4.5.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.13.m13.4b"><apply id="S2.SS3.p7.13.m13.4.5.cmml" xref="S2.SS3.p7.13.m13.4.5"><in id="S2.SS3.p7.13.m13.4.5.1.cmml" xref="S2.SS3.p7.13.m13.4.5.1"></in><list id="S2.SS3.p7.13.m13.4.5.2.1.cmml" xref="S2.SS3.p7.13.m13.4.5.2.2"><ci id="S2.SS3.p7.13.m13.3.3.cmml" xref="S2.SS3.p7.13.m13.3.3">𝑢</ci><ci id="S2.SS3.p7.13.m13.4.4.cmml" xref="S2.SS3.p7.13.m13.4.4">𝑣</ci></list><apply id="S2.SS3.p7.13.m13.4.5.3.cmml" xref="S2.SS3.p7.13.m13.4.5.3"><csymbol cd="ambiguous" id="S2.SS3.p7.13.m13.4.5.3.1.cmml" xref="S2.SS3.p7.13.m13.4.5.3">superscript</csymbol><set id="S2.SS3.p7.13.m13.4.5.3.2.1.cmml" xref="S2.SS3.p7.13.m13.4.5.3.2.2"><cn id="S2.SS3.p7.13.m13.1.1.cmml" type="integer" xref="S2.SS3.p7.13.m13.1.1">0</cn><cn id="S2.SS3.p7.13.m13.2.2.cmml" type="integer" xref="S2.SS3.p7.13.m13.2.2">1</cn></set><ci id="S2.SS3.p7.13.m13.4.5.3.3.cmml" xref="S2.SS3.p7.13.m13.4.5.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.13.m13.4c">u,v\in\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.13.m13.4d">italic_u , italic_v ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, we write <math alttext="u\preceq v" class="ltx_Math" display="inline" id="S2.SS3.p7.14.m14.1"><semantics id="S2.SS3.p7.14.m14.1a"><mrow id="S2.SS3.p7.14.m14.1.1" xref="S2.SS3.p7.14.m14.1.1.cmml"><mi id="S2.SS3.p7.14.m14.1.1.2" xref="S2.SS3.p7.14.m14.1.1.2.cmml">u</mi><mo id="S2.SS3.p7.14.m14.1.1.1" xref="S2.SS3.p7.14.m14.1.1.1.cmml">⪯</mo><mi id="S2.SS3.p7.14.m14.1.1.3" xref="S2.SS3.p7.14.m14.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.14.m14.1b"><apply id="S2.SS3.p7.14.m14.1.1.cmml" xref="S2.SS3.p7.14.m14.1.1"><csymbol cd="latexml" id="S2.SS3.p7.14.m14.1.1.1.cmml" xref="S2.SS3.p7.14.m14.1.1.1">precedes-or-equals</csymbol><ci id="S2.SS3.p7.14.m14.1.1.2.cmml" xref="S2.SS3.p7.14.m14.1.1.2">𝑢</ci><ci id="S2.SS3.p7.14.m14.1.1.3.cmml" xref="S2.SS3.p7.14.m14.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.14.m14.1c">u\preceq v</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.14.m14.1d">italic_u ⪯ italic_v</annotation></semantics></math> if <math alttext="u_{i}\leq v_{i}" class="ltx_Math" display="inline" id="S2.SS3.p7.15.m15.1"><semantics id="S2.SS3.p7.15.m15.1a"><mrow id="S2.SS3.p7.15.m15.1.1" xref="S2.SS3.p7.15.m15.1.1.cmml"><msub id="S2.SS3.p7.15.m15.1.1.2" xref="S2.SS3.p7.15.m15.1.1.2.cmml"><mi id="S2.SS3.p7.15.m15.1.1.2.2" xref="S2.SS3.p7.15.m15.1.1.2.2.cmml">u</mi><mi id="S2.SS3.p7.15.m15.1.1.2.3" xref="S2.SS3.p7.15.m15.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS3.p7.15.m15.1.1.1" xref="S2.SS3.p7.15.m15.1.1.1.cmml">≤</mo><msub id="S2.SS3.p7.15.m15.1.1.3" xref="S2.SS3.p7.15.m15.1.1.3.cmml"><mi id="S2.SS3.p7.15.m15.1.1.3.2" xref="S2.SS3.p7.15.m15.1.1.3.2.cmml">v</mi><mi id="S2.SS3.p7.15.m15.1.1.3.3" xref="S2.SS3.p7.15.m15.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.15.m15.1b"><apply id="S2.SS3.p7.15.m15.1.1.cmml" xref="S2.SS3.p7.15.m15.1.1"><leq id="S2.SS3.p7.15.m15.1.1.1.cmml" xref="S2.SS3.p7.15.m15.1.1.1"></leq><apply id="S2.SS3.p7.15.m15.1.1.2.cmml" xref="S2.SS3.p7.15.m15.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p7.15.m15.1.1.2.1.cmml" xref="S2.SS3.p7.15.m15.1.1.2">subscript</csymbol><ci id="S2.SS3.p7.15.m15.1.1.2.2.cmml" xref="S2.SS3.p7.15.m15.1.1.2.2">𝑢</ci><ci id="S2.SS3.p7.15.m15.1.1.2.3.cmml" xref="S2.SS3.p7.15.m15.1.1.2.3">𝑖</ci></apply><apply id="S2.SS3.p7.15.m15.1.1.3.cmml" xref="S2.SS3.p7.15.m15.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p7.15.m15.1.1.3.1.cmml" xref="S2.SS3.p7.15.m15.1.1.3">subscript</csymbol><ci id="S2.SS3.p7.15.m15.1.1.3.2.cmml" xref="S2.SS3.p7.15.m15.1.1.3.2">𝑣</ci><ci id="S2.SS3.p7.15.m15.1.1.3.3.cmml" xref="S2.SS3.p7.15.m15.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.15.m15.1c">u_{i}\leq v_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.15.m15.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="i\in[n]" class="ltx_Math" display="inline" id="S2.SS3.p7.16.m16.1"><semantics id="S2.SS3.p7.16.m16.1a"><mrow id="S2.SS3.p7.16.m16.1.2" xref="S2.SS3.p7.16.m16.1.2.cmml"><mi id="S2.SS3.p7.16.m16.1.2.2" xref="S2.SS3.p7.16.m16.1.2.2.cmml">i</mi><mo id="S2.SS3.p7.16.m16.1.2.1" xref="S2.SS3.p7.16.m16.1.2.1.cmml">∈</mo><mrow id="S2.SS3.p7.16.m16.1.2.3.2" xref="S2.SS3.p7.16.m16.1.2.3.1.cmml"><mo id="S2.SS3.p7.16.m16.1.2.3.2.1" stretchy="false" xref="S2.SS3.p7.16.m16.1.2.3.1.1.cmml">[</mo><mi id="S2.SS3.p7.16.m16.1.1" xref="S2.SS3.p7.16.m16.1.1.cmml">n</mi><mo id="S2.SS3.p7.16.m16.1.2.3.2.2" stretchy="false" xref="S2.SS3.p7.16.m16.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.16.m16.1b"><apply id="S2.SS3.p7.16.m16.1.2.cmml" xref="S2.SS3.p7.16.m16.1.2"><in id="S2.SS3.p7.16.m16.1.2.1.cmml" xref="S2.SS3.p7.16.m16.1.2.1"></in><ci id="S2.SS3.p7.16.m16.1.2.2.cmml" xref="S2.SS3.p7.16.m16.1.2.2">𝑖</ci><apply id="S2.SS3.p7.16.m16.1.2.3.1.cmml" xref="S2.SS3.p7.16.m16.1.2.3.2"><csymbol cd="latexml" id="S2.SS3.p7.16.m16.1.2.3.1.1.cmml" xref="S2.SS3.p7.16.m16.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS3.p7.16.m16.1.1.cmml" xref="S2.SS3.p7.16.m16.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.16.m16.1c">i\in[n]</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.16.m16.1d">italic_i ∈ [ italic_n ]</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote8"><sup class="ltx_note_mark">8</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">8</sup><span class="ltx_tag ltx_tag_note">8</span> We note that <math alttext="\mathsf{vec}(w)" class="ltx_Math" display="inline" id="footnote8.m1.1"><semantics id="footnote8.m1.1b"><mrow id="footnote8.m1.1.2" xref="footnote8.m1.1.2.cmml"><mi id="footnote8.m1.1.2.2" xref="footnote8.m1.1.2.2.cmml">𝗏𝖾𝖼</mi><mo id="footnote8.m1.1.2.1" xref="footnote8.m1.1.2.1.cmml">⁢</mo><mrow id="footnote8.m1.1.2.3.2" xref="footnote8.m1.1.2.cmml"><mo id="footnote8.m1.1.2.3.2.1" stretchy="false" xref="footnote8.m1.1.2.cmml">(</mo><mi id="footnote8.m1.1.1" xref="footnote8.m1.1.1.cmml">w</mi><mo id="footnote8.m1.1.2.3.2.2" stretchy="false" xref="footnote8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote8.m1.1c"><apply id="footnote8.m1.1.2.cmml" xref="footnote8.m1.1.2"><times id="footnote8.m1.1.2.1.cmml" xref="footnote8.m1.1.2.1"></times><ci id="footnote8.m1.1.2.2.cmml" xref="footnote8.m1.1.2.2">𝗏𝖾𝖼</ci><ci id="footnote8.m1.1.1.cmml" xref="footnote8.m1.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m1.1d">\mathsf{vec}(w)</annotation><annotation encoding="application/x-llamapun" id="footnote8.m1.1e">sansserif_vec ( italic_w )</annotation></semantics></math> always has Hamming weight exactly 2 when <math alttext="{\mathcal{B}}={\mathcal{G}}_{N,M}" class="ltx_Math" display="inline" id="footnote8.m2.2"><semantics id="footnote8.m2.2b"><mrow id="footnote8.m2.2.3" xref="footnote8.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote8.m2.2.3.2" xref="footnote8.m2.2.3.2.cmml">ℬ</mi><mo id="footnote8.m2.2.3.1" xref="footnote8.m2.2.3.1.cmml">=</mo><msub id="footnote8.m2.2.3.3" xref="footnote8.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote8.m2.2.3.3.2" xref="footnote8.m2.2.3.3.2.cmml">𝒢</mi><mrow id="footnote8.m2.2.2.2.4" xref="footnote8.m2.2.2.2.3.cmml"><mi id="footnote8.m2.1.1.1.1" xref="footnote8.m2.1.1.1.1.cmml">N</mi><mo id="footnote8.m2.2.2.2.4.1" xref="footnote8.m2.2.2.2.3.cmml">,</mo><mi id="footnote8.m2.2.2.2.2" xref="footnote8.m2.2.2.2.2.cmml">M</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote8.m2.2c"><apply id="footnote8.m2.2.3.cmml" xref="footnote8.m2.2.3"><eq id="footnote8.m2.2.3.1.cmml" xref="footnote8.m2.2.3.1"></eq><ci id="footnote8.m2.2.3.2.cmml" xref="footnote8.m2.2.3.2">ℬ</ci><apply id="footnote8.m2.2.3.3.cmml" xref="footnote8.m2.2.3.3"><csymbol cd="ambiguous" id="footnote8.m2.2.3.3.1.cmml" xref="footnote8.m2.2.3.3">subscript</csymbol><ci id="footnote8.m2.2.3.3.2.cmml" xref="footnote8.m2.2.3.3.2">𝒢</ci><list id="footnote8.m2.2.2.2.3.cmml" xref="footnote8.m2.2.2.2.4"><ci id="footnote8.m2.1.1.1.1.cmml" xref="footnote8.m2.1.1.1.1">𝑁</ci><ci id="footnote8.m2.2.2.2.2.cmml" xref="footnote8.m2.2.2.2.2">𝑀</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m2.2d">{\mathcal{B}}={\mathcal{G}}_{N,M}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m2.2e">caligraphic_B = caligraphic_G start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w\in[N]\times[M]" class="ltx_Math" display="inline" id="footnote8.m3.2"><semantics id="footnote8.m3.2b"><mrow id="footnote8.m3.2.3" xref="footnote8.m3.2.3.cmml"><mi id="footnote8.m3.2.3.2" xref="footnote8.m3.2.3.2.cmml">w</mi><mo id="footnote8.m3.2.3.1" xref="footnote8.m3.2.3.1.cmml">∈</mo><mrow id="footnote8.m3.2.3.3" xref="footnote8.m3.2.3.3.cmml"><mrow id="footnote8.m3.2.3.3.2.2" xref="footnote8.m3.2.3.3.2.1.cmml"><mo id="footnote8.m3.2.3.3.2.2.1" stretchy="false" xref="footnote8.m3.2.3.3.2.1.1.cmml">[</mo><mi id="footnote8.m3.1.1" xref="footnote8.m3.1.1.cmml">N</mi><mo id="footnote8.m3.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="footnote8.m3.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="footnote8.m3.2.3.3.1" rspace="0.222em" xref="footnote8.m3.2.3.3.1.cmml">×</mo><mrow id="footnote8.m3.2.3.3.3.2" xref="footnote8.m3.2.3.3.3.1.cmml"><mo id="footnote8.m3.2.3.3.3.2.1" stretchy="false" xref="footnote8.m3.2.3.3.3.1.1.cmml">[</mo><mi id="footnote8.m3.2.2" xref="footnote8.m3.2.2.cmml">M</mi><mo id="footnote8.m3.2.3.3.3.2.2" stretchy="false" xref="footnote8.m3.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote8.m3.2c"><apply id="footnote8.m3.2.3.cmml" xref="footnote8.m3.2.3"><in id="footnote8.m3.2.3.1.cmml" xref="footnote8.m3.2.3.1"></in><ci id="footnote8.m3.2.3.2.cmml" xref="footnote8.m3.2.3.2">𝑤</ci><apply id="footnote8.m3.2.3.3.cmml" xref="footnote8.m3.2.3.3"><times id="footnote8.m3.2.3.3.1.cmml" xref="footnote8.m3.2.3.3.1"></times><apply id="footnote8.m3.2.3.3.2.1.cmml" xref="footnote8.m3.2.3.3.2.2"><csymbol cd="latexml" id="footnote8.m3.2.3.3.2.1.1.cmml" xref="footnote8.m3.2.3.3.2.2.1">delimited-[]</csymbol><ci id="footnote8.m3.1.1.cmml" xref="footnote8.m3.1.1">𝑁</ci></apply><apply id="footnote8.m3.2.3.3.3.1.cmml" xref="footnote8.m3.2.3.3.3.2"><csymbol cd="latexml" id="footnote8.m3.2.3.3.3.1.1.cmml" xref="footnote8.m3.2.3.3.3.2.1">delimited-[]</csymbol><ci id="footnote8.m3.2.2.cmml" xref="footnote8.m3.2.2">𝑀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m3.2d">w\in[N]\times[M]</annotation><annotation encoding="application/x-llamapun" id="footnote8.m3.2e">italic_w ∈ [ italic_N ] × [ italic_M ]</annotation></semantics></math>. There is a well-known connection between slice functions and graph complexity (see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib10" title="">10</a>]</cite>). </span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem10.1.1.1">Proposition 10</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem10.2.2"> </span>(Finiteness test)<span class="ltx_text ltx_font_bold" id="Thmtheorem10.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem10.p1"> <p class="ltx_p" id="Thmtheorem10.p1.4"><math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="Thmtheorem10.p1.1.m1.1"><semantics id="Thmtheorem10.p1.1.m1.1a"><mrow id="Thmtheorem10.p1.1.m1.1.1" xref="Thmtheorem10.p1.1.m1.1.1.cmml"><mi id="Thmtheorem10.p1.1.m1.1.1.3" xref="Thmtheorem10.p1.1.m1.1.1.3.cmml">D</mi><mo id="Thmtheorem10.p1.1.m1.1.1.2" xref="Thmtheorem10.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem10.p1.1.m1.1.1.1.1" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml"><mo id="Thmtheorem10.p1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem10.p1.1.m1.1.1.1.1.1" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml"><mi id="Thmtheorem10.p1.1.m1.1.1.1.1.1.2" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem10.p1.1.m1.1.1.1.1.1.1" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem10.p1.1.m1.1.1.1.1.1.3" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem10.p1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem10.p1.1.m1.1b"><apply id="Thmtheorem10.p1.1.m1.1.1.cmml" xref="Thmtheorem10.p1.1.m1.1.1"><times id="Thmtheorem10.p1.1.m1.1.1.2.cmml" xref="Thmtheorem10.p1.1.m1.1.1.2"></times><ci id="Thmtheorem10.p1.1.m1.1.1.3.cmml" xref="Thmtheorem10.p1.1.m1.1.1.3">𝐷</ci><apply id="Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml" xref="Thmtheorem10.p1.1.m1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem10.p1.1.m1.1.1.1.1.1.2.cmml" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem10.p1.1.m1.1.1.1.1.1.3.cmml" xref="Thmtheorem10.p1.1.m1.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem10.p1.1.m1.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem10.p1.1.m1.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="Thmtheorem10.p1.4.3"> is finite if and only if there are no vectors <math alttext="u\in\mathsf{vec}(A)" class="ltx_Math" display="inline" id="Thmtheorem10.p1.2.1.m1.1"><semantics id="Thmtheorem10.p1.2.1.m1.1a"><mrow id="Thmtheorem10.p1.2.1.m1.1.2" xref="Thmtheorem10.p1.2.1.m1.1.2.cmml"><mi id="Thmtheorem10.p1.2.1.m1.1.2.2" xref="Thmtheorem10.p1.2.1.m1.1.2.2.cmml">u</mi><mo id="Thmtheorem10.p1.2.1.m1.1.2.1" xref="Thmtheorem10.p1.2.1.m1.1.2.1.cmml">∈</mo><mrow id="Thmtheorem10.p1.2.1.m1.1.2.3" xref="Thmtheorem10.p1.2.1.m1.1.2.3.cmml"><mi id="Thmtheorem10.p1.2.1.m1.1.2.3.2" xref="Thmtheorem10.p1.2.1.m1.1.2.3.2.cmml">𝗏𝖾𝖼</mi><mo id="Thmtheorem10.p1.2.1.m1.1.2.3.1" xref="Thmtheorem10.p1.2.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem10.p1.2.1.m1.1.2.3.3.2" xref="Thmtheorem10.p1.2.1.m1.1.2.3.cmml"><mo id="Thmtheorem10.p1.2.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem10.p1.2.1.m1.1.2.3.cmml">(</mo><mi id="Thmtheorem10.p1.2.1.m1.1.1" xref="Thmtheorem10.p1.2.1.m1.1.1.cmml">A</mi><mo id="Thmtheorem10.p1.2.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem10.p1.2.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem10.p1.2.1.m1.1b"><apply id="Thmtheorem10.p1.2.1.m1.1.2.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2"><in id="Thmtheorem10.p1.2.1.m1.1.2.1.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2.1"></in><ci id="Thmtheorem10.p1.2.1.m1.1.2.2.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2.2">𝑢</ci><apply id="Thmtheorem10.p1.2.1.m1.1.2.3.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2.3"><times id="Thmtheorem10.p1.2.1.m1.1.2.3.1.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2.3.1"></times><ci id="Thmtheorem10.p1.2.1.m1.1.2.3.2.cmml" xref="Thmtheorem10.p1.2.1.m1.1.2.3.2">𝗏𝖾𝖼</ci><ci id="Thmtheorem10.p1.2.1.m1.1.1.cmml" xref="Thmtheorem10.p1.2.1.m1.1.1">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem10.p1.2.1.m1.1c">u\in\mathsf{vec}(A)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem10.p1.2.1.m1.1d">italic_u ∈ sansserif_vec ( italic_A )</annotation></semantics></math> and <math alttext="v\in\mathsf{vec}(A^{c})" class="ltx_Math" display="inline" id="Thmtheorem10.p1.3.2.m2.1"><semantics id="Thmtheorem10.p1.3.2.m2.1a"><mrow id="Thmtheorem10.p1.3.2.m2.1.1" xref="Thmtheorem10.p1.3.2.m2.1.1.cmml"><mi id="Thmtheorem10.p1.3.2.m2.1.1.3" xref="Thmtheorem10.p1.3.2.m2.1.1.3.cmml">v</mi><mo id="Thmtheorem10.p1.3.2.m2.1.1.2" xref="Thmtheorem10.p1.3.2.m2.1.1.2.cmml">∈</mo><mrow id="Thmtheorem10.p1.3.2.m2.1.1.1" xref="Thmtheorem10.p1.3.2.m2.1.1.1.cmml"><mi id="Thmtheorem10.p1.3.2.m2.1.1.1.3" xref="Thmtheorem10.p1.3.2.m2.1.1.1.3.cmml">𝗏𝖾𝖼</mi><mo id="Thmtheorem10.p1.3.2.m2.1.1.1.2" xref="Thmtheorem10.p1.3.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.cmml"><mo id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.2" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mi id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.3" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.3.cmml">c</mi></msup><mo id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem10.p1.3.2.m2.1b"><apply id="Thmtheorem10.p1.3.2.m2.1.1.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1"><in id="Thmtheorem10.p1.3.2.m2.1.1.2.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.2"></in><ci id="Thmtheorem10.p1.3.2.m2.1.1.3.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.3">𝑣</ci><apply id="Thmtheorem10.p1.3.2.m2.1.1.1.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1"><times id="Thmtheorem10.p1.3.2.m2.1.1.1.2.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.2"></times><ci id="Thmtheorem10.p1.3.2.m2.1.1.1.3.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.3">𝗏𝖾𝖼</ci><apply id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1">superscript</csymbol><ci id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.3.cmml" xref="Thmtheorem10.p1.3.2.m2.1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem10.p1.3.2.m2.1c">v\in\mathsf{vec}(A^{c})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem10.p1.3.2.m2.1d">italic_v ∈ sansserif_vec ( italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="u\preceq v" class="ltx_Math" display="inline" id="Thmtheorem10.p1.4.3.m3.1"><semantics id="Thmtheorem10.p1.4.3.m3.1a"><mrow id="Thmtheorem10.p1.4.3.m3.1.1" xref="Thmtheorem10.p1.4.3.m3.1.1.cmml"><mi id="Thmtheorem10.p1.4.3.m3.1.1.2" xref="Thmtheorem10.p1.4.3.m3.1.1.2.cmml">u</mi><mo id="Thmtheorem10.p1.4.3.m3.1.1.1" xref="Thmtheorem10.p1.4.3.m3.1.1.1.cmml">⪯</mo><mi id="Thmtheorem10.p1.4.3.m3.1.1.3" xref="Thmtheorem10.p1.4.3.m3.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem10.p1.4.3.m3.1b"><apply id="Thmtheorem10.p1.4.3.m3.1.1.cmml" xref="Thmtheorem10.p1.4.3.m3.1.1"><csymbol cd="latexml" id="Thmtheorem10.p1.4.3.m3.1.1.1.cmml" xref="Thmtheorem10.p1.4.3.m3.1.1.1">precedes-or-equals</csymbol><ci id="Thmtheorem10.p1.4.3.m3.1.1.2.cmml" xref="Thmtheorem10.p1.4.3.m3.1.1.2">𝑢</ci><ci id="Thmtheorem10.p1.4.3.m3.1.1.3.cmml" xref="Thmtheorem10.p1.4.3.m3.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem10.p1.4.3.m3.1c">u\preceq v</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem10.p1.4.3.m3.1d">italic_u ⪯ italic_v</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS3.2.p1"> <p class="ltx_p" id="S2.SS3.2.p1.10">Let <math alttext="a\in A" class="ltx_Math" display="inline" id="S2.SS3.2.p1.1.m1.1"><semantics id="S2.SS3.2.p1.1.m1.1a"><mrow id="S2.SS3.2.p1.1.m1.1.1" xref="S2.SS3.2.p1.1.m1.1.1.cmml"><mi id="S2.SS3.2.p1.1.m1.1.1.2" xref="S2.SS3.2.p1.1.m1.1.1.2.cmml">a</mi><mo id="S2.SS3.2.p1.1.m1.1.1.1" xref="S2.SS3.2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S2.SS3.2.p1.1.m1.1.1.3" xref="S2.SS3.2.p1.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.1.m1.1b"><apply id="S2.SS3.2.p1.1.m1.1.1.cmml" xref="S2.SS3.2.p1.1.m1.1.1"><in id="S2.SS3.2.p1.1.m1.1.1.1.cmml" xref="S2.SS3.2.p1.1.m1.1.1.1"></in><ci id="S2.SS3.2.p1.1.m1.1.1.2.cmml" xref="S2.SS3.2.p1.1.m1.1.1.2">𝑎</ci><ci id="S2.SS3.2.p1.1.m1.1.1.3.cmml" xref="S2.SS3.2.p1.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.1.m1.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.1.m1.1d">italic_a ∈ italic_A</annotation></semantics></math> and <math alttext="b\in A^{c}" class="ltx_Math" display="inline" id="S2.SS3.2.p1.2.m2.1"><semantics id="S2.SS3.2.p1.2.m2.1a"><mrow id="S2.SS3.2.p1.2.m2.1.1" xref="S2.SS3.2.p1.2.m2.1.1.cmml"><mi id="S2.SS3.2.p1.2.m2.1.1.2" xref="S2.SS3.2.p1.2.m2.1.1.2.cmml">b</mi><mo id="S2.SS3.2.p1.2.m2.1.1.1" xref="S2.SS3.2.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S2.SS3.2.p1.2.m2.1.1.3" xref="S2.SS3.2.p1.2.m2.1.1.3.cmml"><mi id="S2.SS3.2.p1.2.m2.1.1.3.2" xref="S2.SS3.2.p1.2.m2.1.1.3.2.cmml">A</mi><mi id="S2.SS3.2.p1.2.m2.1.1.3.3" xref="S2.SS3.2.p1.2.m2.1.1.3.3.cmml">c</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.2.m2.1b"><apply id="S2.SS3.2.p1.2.m2.1.1.cmml" xref="S2.SS3.2.p1.2.m2.1.1"><in id="S2.SS3.2.p1.2.m2.1.1.1.cmml" xref="S2.SS3.2.p1.2.m2.1.1.1"></in><ci id="S2.SS3.2.p1.2.m2.1.1.2.cmml" xref="S2.SS3.2.p1.2.m2.1.1.2">𝑏</ci><apply id="S2.SS3.2.p1.2.m2.1.1.3.cmml" xref="S2.SS3.2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.2.p1.2.m2.1.1.3.1.cmml" xref="S2.SS3.2.p1.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS3.2.p1.2.m2.1.1.3.2.cmml" xref="S2.SS3.2.p1.2.m2.1.1.3.2">𝐴</ci><ci id="S2.SS3.2.p1.2.m2.1.1.3.3.cmml" xref="S2.SS3.2.p1.2.m2.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.2.m2.1c">b\in A^{c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.2.m2.1d">italic_b ∈ italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math> be elements such that <math alttext="u=\mathsf{vec}(a)\preceq\mathsf{vec}(b)=v" class="ltx_Math" display="inline" id="S2.SS3.2.p1.3.m3.2"><semantics id="S2.SS3.2.p1.3.m3.2a"><mrow id="S2.SS3.2.p1.3.m3.2.3" xref="S2.SS3.2.p1.3.m3.2.3.cmml"><mi id="S2.SS3.2.p1.3.m3.2.3.2" xref="S2.SS3.2.p1.3.m3.2.3.2.cmml">u</mi><mo id="S2.SS3.2.p1.3.m3.2.3.3" xref="S2.SS3.2.p1.3.m3.2.3.3.cmml">=</mo><mrow id="S2.SS3.2.p1.3.m3.2.3.4" xref="S2.SS3.2.p1.3.m3.2.3.4.cmml"><mi id="S2.SS3.2.p1.3.m3.2.3.4.2" xref="S2.SS3.2.p1.3.m3.2.3.4.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.2.p1.3.m3.2.3.4.1" xref="S2.SS3.2.p1.3.m3.2.3.4.1.cmml">⁢</mo><mrow id="S2.SS3.2.p1.3.m3.2.3.4.3.2" xref="S2.SS3.2.p1.3.m3.2.3.4.cmml"><mo id="S2.SS3.2.p1.3.m3.2.3.4.3.2.1" stretchy="false" xref="S2.SS3.2.p1.3.m3.2.3.4.cmml">(</mo><mi id="S2.SS3.2.p1.3.m3.1.1" xref="S2.SS3.2.p1.3.m3.1.1.cmml">a</mi><mo id="S2.SS3.2.p1.3.m3.2.3.4.3.2.2" stretchy="false" xref="S2.SS3.2.p1.3.m3.2.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS3.2.p1.3.m3.2.3.5" xref="S2.SS3.2.p1.3.m3.2.3.5.cmml">⪯</mo><mrow id="S2.SS3.2.p1.3.m3.2.3.6" xref="S2.SS3.2.p1.3.m3.2.3.6.cmml"><mi id="S2.SS3.2.p1.3.m3.2.3.6.2" xref="S2.SS3.2.p1.3.m3.2.3.6.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.2.p1.3.m3.2.3.6.1" xref="S2.SS3.2.p1.3.m3.2.3.6.1.cmml">⁢</mo><mrow id="S2.SS3.2.p1.3.m3.2.3.6.3.2" xref="S2.SS3.2.p1.3.m3.2.3.6.cmml"><mo id="S2.SS3.2.p1.3.m3.2.3.6.3.2.1" stretchy="false" xref="S2.SS3.2.p1.3.m3.2.3.6.cmml">(</mo><mi id="S2.SS3.2.p1.3.m3.2.2" xref="S2.SS3.2.p1.3.m3.2.2.cmml">b</mi><mo id="S2.SS3.2.p1.3.m3.2.3.6.3.2.2" stretchy="false" xref="S2.SS3.2.p1.3.m3.2.3.6.cmml">)</mo></mrow></mrow><mo id="S2.SS3.2.p1.3.m3.2.3.7" xref="S2.SS3.2.p1.3.m3.2.3.7.cmml">=</mo><mi id="S2.SS3.2.p1.3.m3.2.3.8" xref="S2.SS3.2.p1.3.m3.2.3.8.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.3.m3.2b"><apply id="S2.SS3.2.p1.3.m3.2.3.cmml" xref="S2.SS3.2.p1.3.m3.2.3"><and id="S2.SS3.2.p1.3.m3.2.3a.cmml" xref="S2.SS3.2.p1.3.m3.2.3"></and><apply id="S2.SS3.2.p1.3.m3.2.3b.cmml" xref="S2.SS3.2.p1.3.m3.2.3"><eq id="S2.SS3.2.p1.3.m3.2.3.3.cmml" xref="S2.SS3.2.p1.3.m3.2.3.3"></eq><ci id="S2.SS3.2.p1.3.m3.2.3.2.cmml" xref="S2.SS3.2.p1.3.m3.2.3.2">𝑢</ci><apply id="S2.SS3.2.p1.3.m3.2.3.4.cmml" xref="S2.SS3.2.p1.3.m3.2.3.4"><times id="S2.SS3.2.p1.3.m3.2.3.4.1.cmml" xref="S2.SS3.2.p1.3.m3.2.3.4.1"></times><ci id="S2.SS3.2.p1.3.m3.2.3.4.2.cmml" xref="S2.SS3.2.p1.3.m3.2.3.4.2">𝗏𝖾𝖼</ci><ci id="S2.SS3.2.p1.3.m3.1.1.cmml" xref="S2.SS3.2.p1.3.m3.1.1">𝑎</ci></apply></apply><apply id="S2.SS3.2.p1.3.m3.2.3c.cmml" xref="S2.SS3.2.p1.3.m3.2.3"><csymbol cd="latexml" id="S2.SS3.2.p1.3.m3.2.3.5.cmml" xref="S2.SS3.2.p1.3.m3.2.3.5">precedes-or-equals</csymbol><share href="https://arxiv.org/html/2503.14117v1#S2.SS3.2.p1.3.m3.2.3.4.cmml" id="S2.SS3.2.p1.3.m3.2.3d.cmml" xref="S2.SS3.2.p1.3.m3.2.3"></share><apply id="S2.SS3.2.p1.3.m3.2.3.6.cmml" xref="S2.SS3.2.p1.3.m3.2.3.6"><times id="S2.SS3.2.p1.3.m3.2.3.6.1.cmml" xref="S2.SS3.2.p1.3.m3.2.3.6.1"></times><ci id="S2.SS3.2.p1.3.m3.2.3.6.2.cmml" xref="S2.SS3.2.p1.3.m3.2.3.6.2">𝗏𝖾𝖼</ci><ci id="S2.SS3.2.p1.3.m3.2.2.cmml" xref="S2.SS3.2.p1.3.m3.2.2">𝑏</ci></apply></apply><apply id="S2.SS3.2.p1.3.m3.2.3e.cmml" xref="S2.SS3.2.p1.3.m3.2.3"><eq id="S2.SS3.2.p1.3.m3.2.3.7.cmml" xref="S2.SS3.2.p1.3.m3.2.3.7"></eq><share href="https://arxiv.org/html/2503.14117v1#S2.SS3.2.p1.3.m3.2.3.6.cmml" id="S2.SS3.2.p1.3.m3.2.3f.cmml" xref="S2.SS3.2.p1.3.m3.2.3"></share><ci id="S2.SS3.2.p1.3.m3.2.3.8.cmml" xref="S2.SS3.2.p1.3.m3.2.3.8">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.3.m3.2c">u=\mathsf{vec}(a)\preceq\mathsf{vec}(b)=v</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.3.m3.2d">italic_u = sansserif_vec ( italic_a ) ⪯ sansserif_vec ( italic_b ) = italic_v</annotation></semantics></math>. Suppose there is a construction <math alttext="A_{1},\ldots,A_{t}" class="ltx_Math" display="inline" id="S2.SS3.2.p1.4.m4.3"><semantics id="S2.SS3.2.p1.4.m4.3a"><mrow id="S2.SS3.2.p1.4.m4.3.3.2" xref="S2.SS3.2.p1.4.m4.3.3.3.cmml"><msub id="S2.SS3.2.p1.4.m4.2.2.1.1" xref="S2.SS3.2.p1.4.m4.2.2.1.1.cmml"><mi id="S2.SS3.2.p1.4.m4.2.2.1.1.2" xref="S2.SS3.2.p1.4.m4.2.2.1.1.2.cmml">A</mi><mn id="S2.SS3.2.p1.4.m4.2.2.1.1.3" xref="S2.SS3.2.p1.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS3.2.p1.4.m4.3.3.2.3" xref="S2.SS3.2.p1.4.m4.3.3.3.cmml">,</mo><mi id="S2.SS3.2.p1.4.m4.1.1" mathvariant="normal" xref="S2.SS3.2.p1.4.m4.1.1.cmml">…</mi><mo id="S2.SS3.2.p1.4.m4.3.3.2.4" xref="S2.SS3.2.p1.4.m4.3.3.3.cmml">,</mo><msub id="S2.SS3.2.p1.4.m4.3.3.2.2" xref="S2.SS3.2.p1.4.m4.3.3.2.2.cmml"><mi id="S2.SS3.2.p1.4.m4.3.3.2.2.2" xref="S2.SS3.2.p1.4.m4.3.3.2.2.2.cmml">A</mi><mi id="S2.SS3.2.p1.4.m4.3.3.2.2.3" xref="S2.SS3.2.p1.4.m4.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.4.m4.3b"><list id="S2.SS3.2.p1.4.m4.3.3.3.cmml" xref="S2.SS3.2.p1.4.m4.3.3.2"><apply id="S2.SS3.2.p1.4.m4.2.2.1.1.cmml" xref="S2.SS3.2.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS3.2.p1.4.m4.2.2.1.1.1.cmml" xref="S2.SS3.2.p1.4.m4.2.2.1.1">subscript</csymbol><ci id="S2.SS3.2.p1.4.m4.2.2.1.1.2.cmml" xref="S2.SS3.2.p1.4.m4.2.2.1.1.2">𝐴</ci><cn id="S2.SS3.2.p1.4.m4.2.2.1.1.3.cmml" type="integer" xref="S2.SS3.2.p1.4.m4.2.2.1.1.3">1</cn></apply><ci id="S2.SS3.2.p1.4.m4.1.1.cmml" xref="S2.SS3.2.p1.4.m4.1.1">…</ci><apply id="S2.SS3.2.p1.4.m4.3.3.2.2.cmml" xref="S2.SS3.2.p1.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.2.p1.4.m4.3.3.2.2.1.cmml" xref="S2.SS3.2.p1.4.m4.3.3.2.2">subscript</csymbol><ci id="S2.SS3.2.p1.4.m4.3.3.2.2.2.cmml" xref="S2.SS3.2.p1.4.m4.3.3.2.2.2">𝐴</ci><ci id="S2.SS3.2.p1.4.m4.3.3.2.2.3.cmml" xref="S2.SS3.2.p1.4.m4.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.4.m4.3c">A_{1},\ldots,A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.4.m4.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.2.p1.5.m5.1"><semantics id="S2.SS3.2.p1.5.m5.1a"><mi id="S2.SS3.2.p1.5.m5.1.1" xref="S2.SS3.2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.5.m5.1b"><ci id="S2.SS3.2.p1.5.m5.1.1.cmml" xref="S2.SS3.2.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.5.m5.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS3.2.p1.6.m6.1"><semantics id="S2.SS3.2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.2.p1.6.m6.1.1" xref="S2.SS3.2.p1.6.m6.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.6.m6.1b"><ci id="S2.SS3.2.p1.6.m6.1.1.cmml" xref="S2.SS3.2.p1.6.m6.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.6.m6.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.6.m6.1d">caligraphic_B</annotation></semantics></math>. It follows easily by induction that <math alttext="b\in A_{t}" class="ltx_Math" display="inline" id="S2.SS3.2.p1.7.m7.1"><semantics id="S2.SS3.2.p1.7.m7.1a"><mrow id="S2.SS3.2.p1.7.m7.1.1" xref="S2.SS3.2.p1.7.m7.1.1.cmml"><mi id="S2.SS3.2.p1.7.m7.1.1.2" xref="S2.SS3.2.p1.7.m7.1.1.2.cmml">b</mi><mo id="S2.SS3.2.p1.7.m7.1.1.1" xref="S2.SS3.2.p1.7.m7.1.1.1.cmml">∈</mo><msub id="S2.SS3.2.p1.7.m7.1.1.3" xref="S2.SS3.2.p1.7.m7.1.1.3.cmml"><mi id="S2.SS3.2.p1.7.m7.1.1.3.2" xref="S2.SS3.2.p1.7.m7.1.1.3.2.cmml">A</mi><mi id="S2.SS3.2.p1.7.m7.1.1.3.3" xref="S2.SS3.2.p1.7.m7.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.7.m7.1b"><apply id="S2.SS3.2.p1.7.m7.1.1.cmml" xref="S2.SS3.2.p1.7.m7.1.1"><in id="S2.SS3.2.p1.7.m7.1.1.1.cmml" xref="S2.SS3.2.p1.7.m7.1.1.1"></in><ci id="S2.SS3.2.p1.7.m7.1.1.2.cmml" xref="S2.SS3.2.p1.7.m7.1.1.2">𝑏</ci><apply id="S2.SS3.2.p1.7.m7.1.1.3.cmml" xref="S2.SS3.2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.2.p1.7.m7.1.1.3.1.cmml" xref="S2.SS3.2.p1.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS3.2.p1.7.m7.1.1.3.2.cmml" xref="S2.SS3.2.p1.7.m7.1.1.3.2">𝐴</ci><ci id="S2.SS3.2.p1.7.m7.1.1.3.3.cmml" xref="S2.SS3.2.p1.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.7.m7.1c">b\in A_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.7.m7.1d">italic_b ∈ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, which is contradictory. On the other hand, if there is no element <math alttext="b" class="ltx_Math" display="inline" id="S2.SS3.2.p1.8.m8.1"><semantics id="S2.SS3.2.p1.8.m8.1a"><mi id="S2.SS3.2.p1.8.m8.1.1" xref="S2.SS3.2.p1.8.m8.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.8.m8.1b"><ci id="S2.SS3.2.p1.8.m8.1.1.cmml" xref="S2.SS3.2.p1.8.m8.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.8.m8.1c">b</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.8.m8.1d">italic_b</annotation></semantics></math> and vector <math alttext="v" class="ltx_Math" display="inline" id="S2.SS3.2.p1.9.m9.1"><semantics id="S2.SS3.2.p1.9.m9.1a"><mi id="S2.SS3.2.p1.9.m9.1.1" xref="S2.SS3.2.p1.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.9.m9.1b"><ci id="S2.SS3.2.p1.9.m9.1.1.cmml" xref="S2.SS3.2.p1.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.9.m9.1d">italic_v</annotation></semantics></math> with this property, it is not hard to see that <math alttext="A=\bigcup_{u\in\mathsf{vec}(A)}\bigcap_{i:u_{i}=1}B_{i}" class="ltx_Math" display="inline" id="S2.SS3.2.p1.10.m10.1"><semantics id="S2.SS3.2.p1.10.m10.1a"><mrow id="S2.SS3.2.p1.10.m10.1.2" xref="S2.SS3.2.p1.10.m10.1.2.cmml"><mi id="S2.SS3.2.p1.10.m10.1.2.2" xref="S2.SS3.2.p1.10.m10.1.2.2.cmml">A</mi><mo id="S2.SS3.2.p1.10.m10.1.2.1" rspace="0.111em" xref="S2.SS3.2.p1.10.m10.1.2.1.cmml">=</mo><mrow id="S2.SS3.2.p1.10.m10.1.2.3" xref="S2.SS3.2.p1.10.m10.1.2.3.cmml"><msub id="S2.SS3.2.p1.10.m10.1.2.3.1" xref="S2.SS3.2.p1.10.m10.1.2.3.1.cmml"><mo id="S2.SS3.2.p1.10.m10.1.2.3.1.2" rspace="0em" xref="S2.SS3.2.p1.10.m10.1.2.3.1.2.cmml">⋃</mo><mrow id="S2.SS3.2.p1.10.m10.1.1.1" xref="S2.SS3.2.p1.10.m10.1.1.1.cmml"><mi id="S2.SS3.2.p1.10.m10.1.1.1.3" xref="S2.SS3.2.p1.10.m10.1.1.1.3.cmml">u</mi><mo id="S2.SS3.2.p1.10.m10.1.1.1.2" xref="S2.SS3.2.p1.10.m10.1.1.1.2.cmml">∈</mo><mrow id="S2.SS3.2.p1.10.m10.1.1.1.4" xref="S2.SS3.2.p1.10.m10.1.1.1.4.cmml"><mi id="S2.SS3.2.p1.10.m10.1.1.1.4.2" xref="S2.SS3.2.p1.10.m10.1.1.1.4.2.cmml">𝗏𝖾𝖼</mi><mo id="S2.SS3.2.p1.10.m10.1.1.1.4.1" xref="S2.SS3.2.p1.10.m10.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.SS3.2.p1.10.m10.1.1.1.4.3.2" xref="S2.SS3.2.p1.10.m10.1.1.1.4.cmml"><mo id="S2.SS3.2.p1.10.m10.1.1.1.4.3.2.1" stretchy="false" xref="S2.SS3.2.p1.10.m10.1.1.1.4.cmml">(</mo><mi id="S2.SS3.2.p1.10.m10.1.1.1.1" xref="S2.SS3.2.p1.10.m10.1.1.1.1.cmml">A</mi><mo id="S2.SS3.2.p1.10.m10.1.1.1.4.3.2.2" stretchy="false" xref="S2.SS3.2.p1.10.m10.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S2.SS3.2.p1.10.m10.1.2.3.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.cmml"><msub id="S2.SS3.2.p1.10.m10.1.2.3.2.1" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.cmml"><mo id="S2.SS3.2.p1.10.m10.1.2.3.2.1.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.2.cmml">⋂</mo><mrow id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.cmml"><mi id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.2.cmml">i</mi><mo id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.1.cmml">:</mo><mrow id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.cmml"><msub id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.cmml"><mi id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.2.cmml">u</mi><mi id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.3" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.3.cmml">i</mi></msub><mo id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.1" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.1.cmml">=</mo><mn id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.3" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.3.cmml">1</mn></mrow></mrow></msub><msub id="S2.SS3.2.p1.10.m10.1.2.3.2.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2.cmml"><mi id="S2.SS3.2.p1.10.m10.1.2.3.2.2.2" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2.2.cmml">B</mi><mi id="S2.SS3.2.p1.10.m10.1.2.3.2.2.3" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.10.m10.1b"><apply id="S2.SS3.2.p1.10.m10.1.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2"><eq id="S2.SS3.2.p1.10.m10.1.2.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.1"></eq><ci id="S2.SS3.2.p1.10.m10.1.2.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.2">𝐴</ci><apply id="S2.SS3.2.p1.10.m10.1.2.3.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3"><apply id="S2.SS3.2.p1.10.m10.1.2.3.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.1"><csymbol cd="ambiguous" id="S2.SS3.2.p1.10.m10.1.2.3.1.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.1">subscript</csymbol><union id="S2.SS3.2.p1.10.m10.1.2.3.1.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.1.2"></union><apply id="S2.SS3.2.p1.10.m10.1.1.1.cmml" xref="S2.SS3.2.p1.10.m10.1.1.1"><in 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id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.1">:</ci><ci id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.2">𝑖</ci><apply id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3"><eq id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.1"></eq><apply id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2">subscript</csymbol><ci id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.2">𝑢</ci><ci id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.3.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.2.3">𝑖</ci></apply><cn id="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.3.cmml" type="integer" xref="S2.SS3.2.p1.10.m10.1.2.3.2.1.3.3.3">1</cn></apply></apply></apply><apply id="S2.SS3.2.p1.10.m10.1.2.3.2.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.2.p1.10.m10.1.2.3.2.2.1.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2">subscript</csymbol><ci id="S2.SS3.2.p1.10.m10.1.2.3.2.2.2.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2.2">𝐵</ci><ci id="S2.SS3.2.p1.10.m10.1.2.3.2.2.3.cmml" xref="S2.SS3.2.p1.10.m10.1.2.3.2.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.10.m10.1c">A=\bigcup_{u\in\mathsf{vec}(A)}\bigcap_{i:u_{i}=1}B_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.10.m10.1d">italic_A = ⋃ start_POSTSUBSCRIPT italic_u ∈ sansserif_vec ( italic_A ) end_POSTSUBSCRIPT ⋂ start_POSTSUBSCRIPT italic_i : italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. This completes the proof of the proposition. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS3.p8"> <p class="ltx_p" id="S2.SS3.p8.1">Finally, observe that standard counting arguments yield the existence of sets of high discrete complexity.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem11.1.1.1">Lemma 11</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem11.2.2"> </span>(Complex sets)<span class="ltx_text ltx_font_bold" id="Thmtheorem11.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem11.p1"> <p class="ltx_p" id="Thmtheorem11.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem11.p1.5.5">Let <math alttext="k=|\Gamma|" class="ltx_Math" display="inline" id="Thmtheorem11.p1.1.1.m1.1"><semantics id="Thmtheorem11.p1.1.1.m1.1a"><mrow id="Thmtheorem11.p1.1.1.m1.1.2" xref="Thmtheorem11.p1.1.1.m1.1.2.cmml"><mi id="Thmtheorem11.p1.1.1.m1.1.2.2" xref="Thmtheorem11.p1.1.1.m1.1.2.2.cmml">k</mi><mo id="Thmtheorem11.p1.1.1.m1.1.2.1" xref="Thmtheorem11.p1.1.1.m1.1.2.1.cmml">=</mo><mrow id="Thmtheorem11.p1.1.1.m1.1.2.3.2" xref="Thmtheorem11.p1.1.1.m1.1.2.3.1.cmml"><mo id="Thmtheorem11.p1.1.1.m1.1.2.3.2.1" stretchy="false" xref="Thmtheorem11.p1.1.1.m1.1.2.3.1.1.cmml">|</mo><mi id="Thmtheorem11.p1.1.1.m1.1.1" mathvariant="normal" xref="Thmtheorem11.p1.1.1.m1.1.1.cmml">Γ</mi><mo id="Thmtheorem11.p1.1.1.m1.1.2.3.2.2" stretchy="false" xref="Thmtheorem11.p1.1.1.m1.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem11.p1.1.1.m1.1b"><apply id="Thmtheorem11.p1.1.1.m1.1.2.cmml" xref="Thmtheorem11.p1.1.1.m1.1.2"><eq id="Thmtheorem11.p1.1.1.m1.1.2.1.cmml" xref="Thmtheorem11.p1.1.1.m1.1.2.1"></eq><ci id="Thmtheorem11.p1.1.1.m1.1.2.2.cmml" xref="Thmtheorem11.p1.1.1.m1.1.2.2">𝑘</ci><apply id="Thmtheorem11.p1.1.1.m1.1.2.3.1.cmml" xref="Thmtheorem11.p1.1.1.m1.1.2.3.2"><abs id="Thmtheorem11.p1.1.1.m1.1.2.3.1.1.cmml" xref="Thmtheorem11.p1.1.1.m1.1.2.3.2.1"></abs><ci id="Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="Thmtheorem11.p1.1.1.m1.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem11.p1.1.1.m1.1c">k=|\Gamma|</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem11.p1.1.1.m1.1d">italic_k = | roman_Γ |</annotation></semantics></math> and <math alttext="m=|\mathcal{B}|" class="ltx_Math" display="inline" id="Thmtheorem11.p1.2.2.m2.1"><semantics id="Thmtheorem11.p1.2.2.m2.1a"><mrow id="Thmtheorem11.p1.2.2.m2.1.2" xref="Thmtheorem11.p1.2.2.m2.1.2.cmml"><mi id="Thmtheorem11.p1.2.2.m2.1.2.2" xref="Thmtheorem11.p1.2.2.m2.1.2.2.cmml">m</mi><mo id="Thmtheorem11.p1.2.2.m2.1.2.1" xref="Thmtheorem11.p1.2.2.m2.1.2.1.cmml">=</mo><mrow id="Thmtheorem11.p1.2.2.m2.1.2.3.2" xref="Thmtheorem11.p1.2.2.m2.1.2.3.1.cmml"><mo id="Thmtheorem11.p1.2.2.m2.1.2.3.2.1" stretchy="false" xref="Thmtheorem11.p1.2.2.m2.1.2.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem11.p1.2.2.m2.1.1" xref="Thmtheorem11.p1.2.2.m2.1.1.cmml">ℬ</mi><mo id="Thmtheorem11.p1.2.2.m2.1.2.3.2.2" stretchy="false" xref="Thmtheorem11.p1.2.2.m2.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem11.p1.2.2.m2.1b"><apply id="Thmtheorem11.p1.2.2.m2.1.2.cmml" xref="Thmtheorem11.p1.2.2.m2.1.2"><eq id="Thmtheorem11.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem11.p1.2.2.m2.1.2.1"></eq><ci id="Thmtheorem11.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem11.p1.2.2.m2.1.2.2">𝑚</ci><apply id="Thmtheorem11.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem11.p1.2.2.m2.1.2.3.2"><abs id="Thmtheorem11.p1.2.2.m2.1.2.3.1.1.cmml" xref="Thmtheorem11.p1.2.2.m2.1.2.3.2.1"></abs><ci id="Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="Thmtheorem11.p1.2.2.m2.1.1">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem11.p1.2.2.m2.1c">m=|\mathcal{B}|</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem11.p1.2.2.m2.1d">italic_m = | caligraphic_B |</annotation></semantics></math>. 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id="Thmtheorem11.p1.3.3.m3.1.1" xref="Thmtheorem11.p1.3.3.m3.1.1.cmml">log</mi><mo id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1a" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.2.cmml"><mo id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.2.cmml">(</mo><mrow id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.cmml"><mi id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.cmml">m</mi><mo id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.cmml">s</mi></mrow><mo id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="Thmtheorem11.p1.3.3.m3.2.2.2" xref="Thmtheorem11.p1.3.3.m3.2.2.2.cmml"><</mo><mi id="Thmtheorem11.p1.3.3.m3.2.2.3" xref="Thmtheorem11.p1.3.3.m3.2.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem11.p1.3.3.m3.2b"><apply id="Thmtheorem11.p1.3.3.m3.2.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2"><lt id="Thmtheorem11.p1.3.3.m3.2.2.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.2"></lt><apply id="Thmtheorem11.p1.3.3.m3.2.2.1.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1"><times id="Thmtheorem11.p1.3.3.m3.2.2.1.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.2"></times><cn id="Thmtheorem11.p1.3.3.m3.2.2.1.3.cmml" type="integer" xref="Thmtheorem11.p1.3.3.m3.2.2.1.3">3</cn><ci id="Thmtheorem11.p1.3.3.m3.2.2.1.4.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.4">𝑠</ci><apply id="Thmtheorem11.p1.3.3.m3.2.2.1.1.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1"><ceiling id="Thmtheorem11.p1.3.3.m3.2.2.1.1.2.1.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.2"></ceiling><apply id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1"><log id="Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="Thmtheorem11.p1.3.3.m3.1.1"></log><apply id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1"><plus id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.1"></plus><ci id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2">𝑚</ci><ci id="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3">𝑠</ci></apply></apply></apply></apply><ci id="Thmtheorem11.p1.3.3.m3.2.2.3.cmml" xref="Thmtheorem11.p1.3.3.m3.2.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem11.p1.3.3.m3.2c">\;3s\lceil\log(m+s)\rceil<k</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem11.p1.3.3.m3.2d">3 italic_s ⌈ roman_log ( italic_m + italic_s ) ⌉ < italic_k</annotation></semantics></math>, there exists a set <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem11.p1.4.4.m4.1"><semantics id="Thmtheorem11.p1.4.4.m4.1a"><mrow id="Thmtheorem11.p1.4.4.m4.1.1" xref="Thmtheorem11.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem11.p1.4.4.m4.1.1.2" xref="Thmtheorem11.p1.4.4.m4.1.1.2.cmml">A</mi><mo id="Thmtheorem11.p1.4.4.m4.1.1.1" xref="Thmtheorem11.p1.4.4.m4.1.1.1.cmml">⊆</mo><mi id="Thmtheorem11.p1.4.4.m4.1.1.3" mathvariant="normal" xref="Thmtheorem11.p1.4.4.m4.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem11.p1.4.4.m4.1b"><apply id="Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="Thmtheorem11.p1.4.4.m4.1.1"><subset id="Thmtheorem11.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem11.p1.4.4.m4.1.1.1"></subset><ci id="Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem11.p1.4.4.m4.1.1.2">𝐴</ci><ci id="Thmtheorem11.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem11.p1.4.4.m4.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem11.p1.4.4.m4.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem11.p1.4.4.m4.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> such that <math alttext="D(A\mid\mathcal{B})\geq s" class="ltx_Math" display="inline" id="Thmtheorem11.p1.5.5.m5.1"><semantics id="Thmtheorem11.p1.5.5.m5.1a"><mrow id="Thmtheorem11.p1.5.5.m5.1.1" xref="Thmtheorem11.p1.5.5.m5.1.1.cmml"><mrow id="Thmtheorem11.p1.5.5.m5.1.1.1" xref="Thmtheorem11.p1.5.5.m5.1.1.1.cmml"><mi id="Thmtheorem11.p1.5.5.m5.1.1.1.3" xref="Thmtheorem11.p1.5.5.m5.1.1.1.3.cmml">D</mi><mo id="Thmtheorem11.p1.5.5.m5.1.1.1.2" xref="Thmtheorem11.p1.5.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.cmml"><mo id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.cmml"><mi id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.2" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.1" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.3" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem11.p1.5.5.m5.1.1.2" xref="Thmtheorem11.p1.5.5.m5.1.1.2.cmml">≥</mo><mi id="Thmtheorem11.p1.5.5.m5.1.1.3" xref="Thmtheorem11.p1.5.5.m5.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem11.p1.5.5.m5.1b"><apply id="Thmtheorem11.p1.5.5.m5.1.1.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1"><geq id="Thmtheorem11.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.2"></geq><apply id="Thmtheorem11.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1"><times id="Thmtheorem11.p1.5.5.m5.1.1.1.2.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.2"></times><ci id="Thmtheorem11.p1.5.5.m5.1.1.1.3.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.3">𝐷</ci><apply id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.1.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.2.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.3.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.1.1.1.1.3">ℬ</ci></apply></apply><ci id="Thmtheorem11.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem11.p1.5.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem11.p1.5.5.m5.1c">D(A\mid\mathcal{B})\geq s</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem11.p1.5.5.m5.1d">italic_D ( italic_A ∣ caligraphic_B ) ≥ italic_s</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS3.p9"> <p class="ltx_p" id="S2.SS3.p9.4">For instance, a random matrix <math alttext="M\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS3.p9.1.m1.2"><semantics id="S2.SS3.p9.1.m1.2a"><mrow id="S2.SS3.p9.1.m1.2.3" xref="S2.SS3.p9.1.m1.2.3.cmml"><mi id="S2.SS3.p9.1.m1.2.3.2" xref="S2.SS3.p9.1.m1.2.3.2.cmml">M</mi><mo id="S2.SS3.p9.1.m1.2.3.1" xref="S2.SS3.p9.1.m1.2.3.1.cmml">⊆</mo><mrow id="S2.SS3.p9.1.m1.2.3.3" xref="S2.SS3.p9.1.m1.2.3.3.cmml"><mrow id="S2.SS3.p9.1.m1.2.3.3.2.2" xref="S2.SS3.p9.1.m1.2.3.3.2.1.cmml"><mo id="S2.SS3.p9.1.m1.2.3.3.2.2.1" stretchy="false" xref="S2.SS3.p9.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS3.p9.1.m1.1.1" xref="S2.SS3.p9.1.m1.1.1.cmml">N</mi><mo id="S2.SS3.p9.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS3.p9.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS3.p9.1.m1.2.3.3.1" rspace="0.222em" xref="S2.SS3.p9.1.m1.2.3.3.1.cmml">×</mo><mrow id="S2.SS3.p9.1.m1.2.3.3.3.2" xref="S2.SS3.p9.1.m1.2.3.3.3.1.cmml"><mo id="S2.SS3.p9.1.m1.2.3.3.3.2.1" stretchy="false" xref="S2.SS3.p9.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS3.p9.1.m1.2.2" xref="S2.SS3.p9.1.m1.2.2.cmml">N</mi><mo id="S2.SS3.p9.1.m1.2.3.3.3.2.2" stretchy="false" xref="S2.SS3.p9.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.1.m1.2b"><apply id="S2.SS3.p9.1.m1.2.3.cmml" xref="S2.SS3.p9.1.m1.2.3"><subset id="S2.SS3.p9.1.m1.2.3.1.cmml" xref="S2.SS3.p9.1.m1.2.3.1"></subset><ci id="S2.SS3.p9.1.m1.2.3.2.cmml" xref="S2.SS3.p9.1.m1.2.3.2">𝑀</ci><apply id="S2.SS3.p9.1.m1.2.3.3.cmml" xref="S2.SS3.p9.1.m1.2.3.3"><times id="S2.SS3.p9.1.m1.2.3.3.1.cmml" xref="S2.SS3.p9.1.m1.2.3.3.1"></times><apply id="S2.SS3.p9.1.m1.2.3.3.2.1.cmml" xref="S2.SS3.p9.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS3.p9.1.m1.2.3.3.2.1.1.cmml" xref="S2.SS3.p9.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS3.p9.1.m1.1.1.cmml" xref="S2.SS3.p9.1.m1.1.1">𝑁</ci></apply><apply id="S2.SS3.p9.1.m1.2.3.3.3.1.cmml" xref="S2.SS3.p9.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS3.p9.1.m1.2.3.3.3.1.1.cmml" xref="S2.SS3.p9.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS3.p9.1.m1.2.2.cmml" xref="S2.SS3.p9.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.1.m1.2c">M\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.1.m1.2d">italic_M ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> satisfies <math alttext="D(M\mid\mathcal{R}_{N,N})=\Omega(N)" class="ltx_Math" display="inline" id="S2.SS3.p9.2.m2.4"><semantics id="S2.SS3.p9.2.m2.4a"><mrow id="S2.SS3.p9.2.m2.4.4" xref="S2.SS3.p9.2.m2.4.4.cmml"><mrow id="S2.SS3.p9.2.m2.4.4.1" xref="S2.SS3.p9.2.m2.4.4.1.cmml"><mi id="S2.SS3.p9.2.m2.4.4.1.3" xref="S2.SS3.p9.2.m2.4.4.1.3.cmml">D</mi><mo id="S2.SS3.p9.2.m2.4.4.1.2" xref="S2.SS3.p9.2.m2.4.4.1.2.cmml">⁢</mo><mrow id="S2.SS3.p9.2.m2.4.4.1.1.1" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.cmml"><mo id="S2.SS3.p9.2.m2.4.4.1.1.1.2" stretchy="false" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p9.2.m2.4.4.1.1.1.1" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.cmml"><mi id="S2.SS3.p9.2.m2.4.4.1.1.1.1.2" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.2.cmml">M</mi><mo id="S2.SS3.p9.2.m2.4.4.1.1.1.1.1" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS3.p9.2.m2.4.4.1.1.1.1.3" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.2" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.2.cmml">ℛ</mi><mrow id="S2.SS3.p9.2.m2.2.2.2.4" xref="S2.SS3.p9.2.m2.2.2.2.3.cmml"><mi id="S2.SS3.p9.2.m2.1.1.1.1" xref="S2.SS3.p9.2.m2.1.1.1.1.cmml">N</mi><mo id="S2.SS3.p9.2.m2.2.2.2.4.1" xref="S2.SS3.p9.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS3.p9.2.m2.2.2.2.2" xref="S2.SS3.p9.2.m2.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS3.p9.2.m2.4.4.1.1.1.3" stretchy="false" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p9.2.m2.4.4.2" xref="S2.SS3.p9.2.m2.4.4.2.cmml">=</mo><mrow id="S2.SS3.p9.2.m2.4.4.3" xref="S2.SS3.p9.2.m2.4.4.3.cmml"><mi id="S2.SS3.p9.2.m2.4.4.3.2" mathvariant="normal" xref="S2.SS3.p9.2.m2.4.4.3.2.cmml">Ω</mi><mo id="S2.SS3.p9.2.m2.4.4.3.1" xref="S2.SS3.p9.2.m2.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS3.p9.2.m2.4.4.3.3.2" xref="S2.SS3.p9.2.m2.4.4.3.cmml"><mo id="S2.SS3.p9.2.m2.4.4.3.3.2.1" stretchy="false" xref="S2.SS3.p9.2.m2.4.4.3.cmml">(</mo><mi id="S2.SS3.p9.2.m2.3.3" xref="S2.SS3.p9.2.m2.3.3.cmml">N</mi><mo id="S2.SS3.p9.2.m2.4.4.3.3.2.2" stretchy="false" xref="S2.SS3.p9.2.m2.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.2.m2.4b"><apply id="S2.SS3.p9.2.m2.4.4.cmml" xref="S2.SS3.p9.2.m2.4.4"><eq id="S2.SS3.p9.2.m2.4.4.2.cmml" xref="S2.SS3.p9.2.m2.4.4.2"></eq><apply id="S2.SS3.p9.2.m2.4.4.1.cmml" xref="S2.SS3.p9.2.m2.4.4.1"><times id="S2.SS3.p9.2.m2.4.4.1.2.cmml" xref="S2.SS3.p9.2.m2.4.4.1.2"></times><ci id="S2.SS3.p9.2.m2.4.4.1.3.cmml" xref="S2.SS3.p9.2.m2.4.4.1.3">𝐷</ci><apply id="S2.SS3.p9.2.m2.4.4.1.1.1.1.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1"><csymbol cd="latexml" id="S2.SS3.p9.2.m2.4.4.1.1.1.1.1.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.p9.2.m2.4.4.1.1.1.1.2.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.2">𝑀</ci><apply id="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.1.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.2.cmml" xref="S2.SS3.p9.2.m2.4.4.1.1.1.1.3.2">ℛ</ci><list id="S2.SS3.p9.2.m2.2.2.2.3.cmml" xref="S2.SS3.p9.2.m2.2.2.2.4"><ci id="S2.SS3.p9.2.m2.1.1.1.1.cmml" xref="S2.SS3.p9.2.m2.1.1.1.1">𝑁</ci><ci id="S2.SS3.p9.2.m2.2.2.2.2.cmml" xref="S2.SS3.p9.2.m2.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="S2.SS3.p9.2.m2.4.4.3.cmml" xref="S2.SS3.p9.2.m2.4.4.3"><times id="S2.SS3.p9.2.m2.4.4.3.1.cmml" xref="S2.SS3.p9.2.m2.4.4.3.1"></times><ci id="S2.SS3.p9.2.m2.4.4.3.2.cmml" xref="S2.SS3.p9.2.m2.4.4.3.2">Ω</ci><ci id="S2.SS3.p9.2.m2.3.3.cmml" xref="S2.SS3.p9.2.m2.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.2.m2.4c">D(M\mid\mathcal{R}_{N,N})=\Omega(N)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.2.m2.4d">italic_D ( italic_M ∣ caligraphic_R start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Ω ( italic_N )</annotation></semantics></math>, while a random graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S2.SS3.p9.3.m3.2"><semantics id="S2.SS3.p9.3.m3.2a"><mrow id="S2.SS3.p9.3.m3.2.3" xref="S2.SS3.p9.3.m3.2.3.cmml"><mi id="S2.SS3.p9.3.m3.2.3.2" xref="S2.SS3.p9.3.m3.2.3.2.cmml">G</mi><mo id="S2.SS3.p9.3.m3.2.3.1" xref="S2.SS3.p9.3.m3.2.3.1.cmml">⊆</mo><mrow id="S2.SS3.p9.3.m3.2.3.3" xref="S2.SS3.p9.3.m3.2.3.3.cmml"><mrow id="S2.SS3.p9.3.m3.2.3.3.2.2" xref="S2.SS3.p9.3.m3.2.3.3.2.1.cmml"><mo id="S2.SS3.p9.3.m3.2.3.3.2.2.1" stretchy="false" xref="S2.SS3.p9.3.m3.2.3.3.2.1.1.cmml">[</mo><mi id="S2.SS3.p9.3.m3.1.1" xref="S2.SS3.p9.3.m3.1.1.cmml">N</mi><mo id="S2.SS3.p9.3.m3.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.SS3.p9.3.m3.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS3.p9.3.m3.2.3.3.1" rspace="0.222em" xref="S2.SS3.p9.3.m3.2.3.3.1.cmml">×</mo><mrow id="S2.SS3.p9.3.m3.2.3.3.3.2" xref="S2.SS3.p9.3.m3.2.3.3.3.1.cmml"><mo id="S2.SS3.p9.3.m3.2.3.3.3.2.1" stretchy="false" xref="S2.SS3.p9.3.m3.2.3.3.3.1.1.cmml">[</mo><mi id="S2.SS3.p9.3.m3.2.2" xref="S2.SS3.p9.3.m3.2.2.cmml">N</mi><mo id="S2.SS3.p9.3.m3.2.3.3.3.2.2" stretchy="false" xref="S2.SS3.p9.3.m3.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.3.m3.2b"><apply id="S2.SS3.p9.3.m3.2.3.cmml" xref="S2.SS3.p9.3.m3.2.3"><subset id="S2.SS3.p9.3.m3.2.3.1.cmml" xref="S2.SS3.p9.3.m3.2.3.1"></subset><ci id="S2.SS3.p9.3.m3.2.3.2.cmml" xref="S2.SS3.p9.3.m3.2.3.2">𝐺</ci><apply id="S2.SS3.p9.3.m3.2.3.3.cmml" xref="S2.SS3.p9.3.m3.2.3.3"><times id="S2.SS3.p9.3.m3.2.3.3.1.cmml" xref="S2.SS3.p9.3.m3.2.3.3.1"></times><apply id="S2.SS3.p9.3.m3.2.3.3.2.1.cmml" xref="S2.SS3.p9.3.m3.2.3.3.2.2"><csymbol cd="latexml" id="S2.SS3.p9.3.m3.2.3.3.2.1.1.cmml" xref="S2.SS3.p9.3.m3.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS3.p9.3.m3.1.1.cmml" xref="S2.SS3.p9.3.m3.1.1">𝑁</ci></apply><apply id="S2.SS3.p9.3.m3.2.3.3.3.1.cmml" xref="S2.SS3.p9.3.m3.2.3.3.3.2"><csymbol cd="latexml" id="S2.SS3.p9.3.m3.2.3.3.3.1.1.cmml" xref="S2.SS3.p9.3.m3.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S2.SS3.p9.3.m3.2.2.cmml" xref="S2.SS3.p9.3.m3.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.3.m3.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.3.m3.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> has <math alttext="D(G\mid\mathcal{G}_{N,N})=\Omega(N^{2}/\log N)" class="ltx_Math" display="inline" id="S2.SS3.p9.4.m4.4"><semantics id="S2.SS3.p9.4.m4.4a"><mrow id="S2.SS3.p9.4.m4.4.4" xref="S2.SS3.p9.4.m4.4.4.cmml"><mrow id="S2.SS3.p9.4.m4.3.3.1" xref="S2.SS3.p9.4.m4.3.3.1.cmml"><mi id="S2.SS3.p9.4.m4.3.3.1.3" xref="S2.SS3.p9.4.m4.3.3.1.3.cmml">D</mi><mo id="S2.SS3.p9.4.m4.3.3.1.2" xref="S2.SS3.p9.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS3.p9.4.m4.3.3.1.1.1" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.cmml"><mo id="S2.SS3.p9.4.m4.3.3.1.1.1.2" stretchy="false" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p9.4.m4.3.3.1.1.1.1" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.cmml"><mi id="S2.SS3.p9.4.m4.3.3.1.1.1.1.2" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.2.cmml">G</mi><mo id="S2.SS3.p9.4.m4.3.3.1.1.1.1.1" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS3.p9.4.m4.3.3.1.1.1.1.3" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.2" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS3.p9.4.m4.2.2.2.4" xref="S2.SS3.p9.4.m4.2.2.2.3.cmml"><mi id="S2.SS3.p9.4.m4.1.1.1.1" xref="S2.SS3.p9.4.m4.1.1.1.1.cmml">N</mi><mo id="S2.SS3.p9.4.m4.2.2.2.4.1" xref="S2.SS3.p9.4.m4.2.2.2.3.cmml">,</mo><mi id="S2.SS3.p9.4.m4.2.2.2.2" xref="S2.SS3.p9.4.m4.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS3.p9.4.m4.3.3.1.1.1.3" stretchy="false" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p9.4.m4.4.4.3" xref="S2.SS3.p9.4.m4.4.4.3.cmml">=</mo><mrow id="S2.SS3.p9.4.m4.4.4.2" xref="S2.SS3.p9.4.m4.4.4.2.cmml"><mi id="S2.SS3.p9.4.m4.4.4.2.3" mathvariant="normal" xref="S2.SS3.p9.4.m4.4.4.2.3.cmml">Ω</mi><mo id="S2.SS3.p9.4.m4.4.4.2.2" xref="S2.SS3.p9.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="S2.SS3.p9.4.m4.4.4.2.1.1" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.cmml"><mo id="S2.SS3.p9.4.m4.4.4.2.1.1.2" stretchy="false" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.cmml">(</mo><mrow id="S2.SS3.p9.4.m4.4.4.2.1.1.1" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.cmml"><msup id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.cmml"><mi id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.2" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.2.cmml">N</mi><mn id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.3" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.SS3.p9.4.m4.4.4.2.1.1.1.1" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.1.cmml">/</mo><mrow id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.cmml"><mi id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.1" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.1.cmml">log</mi><mo id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3a" lspace="0.167em" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.cmml">⁡</mo><mi id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.2" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.2.cmml">N</mi></mrow></mrow><mo id="S2.SS3.p9.4.m4.4.4.2.1.1.3" stretchy="false" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.4.m4.4b"><apply id="S2.SS3.p9.4.m4.4.4.cmml" xref="S2.SS3.p9.4.m4.4.4"><eq id="S2.SS3.p9.4.m4.4.4.3.cmml" xref="S2.SS3.p9.4.m4.4.4.3"></eq><apply id="S2.SS3.p9.4.m4.3.3.1.cmml" xref="S2.SS3.p9.4.m4.3.3.1"><times id="S2.SS3.p9.4.m4.3.3.1.2.cmml" xref="S2.SS3.p9.4.m4.3.3.1.2"></times><ci id="S2.SS3.p9.4.m4.3.3.1.3.cmml" xref="S2.SS3.p9.4.m4.3.3.1.3">𝐷</ci><apply id="S2.SS3.p9.4.m4.3.3.1.1.1.1.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1"><csymbol cd="latexml" id="S2.SS3.p9.4.m4.3.3.1.1.1.1.1.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.1">conditional</csymbol><ci id="S2.SS3.p9.4.m4.3.3.1.1.1.1.2.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.2">𝐺</ci><apply id="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.1.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.3">subscript</csymbol><ci id="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.2.cmml" xref="S2.SS3.p9.4.m4.3.3.1.1.1.1.3.2">𝒢</ci><list id="S2.SS3.p9.4.m4.2.2.2.3.cmml" xref="S2.SS3.p9.4.m4.2.2.2.4"><ci id="S2.SS3.p9.4.m4.1.1.1.1.cmml" xref="S2.SS3.p9.4.m4.1.1.1.1">𝑁</ci><ci id="S2.SS3.p9.4.m4.2.2.2.2.cmml" xref="S2.SS3.p9.4.m4.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="S2.SS3.p9.4.m4.4.4.2.cmml" xref="S2.SS3.p9.4.m4.4.4.2"><times id="S2.SS3.p9.4.m4.4.4.2.2.cmml" xref="S2.SS3.p9.4.m4.4.4.2.2"></times><ci id="S2.SS3.p9.4.m4.4.4.2.3.cmml" xref="S2.SS3.p9.4.m4.4.4.2.3">Ω</ci><apply id="S2.SS3.p9.4.m4.4.4.2.1.1.1.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1"><divide id="S2.SS3.p9.4.m4.4.4.2.1.1.1.1.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.1"></divide><apply id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.1.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2">superscript</csymbol><ci id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.2.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.2">𝑁</ci><cn id="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.2.3">2</cn></apply><apply id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3"><log id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.1.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.1"></log><ci id="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.2.cmml" xref="S2.SS3.p9.4.m4.4.4.2.1.1.1.3.2">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.4.m4.4c">D(G\mid\mathcal{G}_{N,N})=\Omega(N^{2}/\log N)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.4.m4.4d">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Ω ( italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / roman_log italic_N )</annotation></semantics></math>. It is easy to see that the former lower bound is asymptotically tight. The tightness of the graph complexity bound is also known (cf. <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>, Theorem 1.7]</cite>).</p> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4 </span>Transference of lower bounds</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.1">The following lemma generalizes a similar reduction from graph complexity (see, e.g., <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>, Section 1.3]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem12.1.1.1">Lemma 12</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem12.p1"> <p class="ltx_p" id="Thmtheorem12.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem12.p1.5.5">Let <math alttext="\langle\Gamma_{1},\mathcal{B}_{1}\rangle" class="ltx_Math" display="inline" id="Thmtheorem12.p1.1.1.m1.2"><semantics id="Thmtheorem12.p1.1.1.m1.2a"><mrow id="Thmtheorem12.p1.1.1.m1.2.2.2" xref="Thmtheorem12.p1.1.1.m1.2.2.3.cmml"><mo id="Thmtheorem12.p1.1.1.m1.2.2.2.3" stretchy="false" xref="Thmtheorem12.p1.1.1.m1.2.2.3.cmml">⟨</mo><msub id="Thmtheorem12.p1.1.1.m1.1.1.1.1" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1.cmml"><mi id="Thmtheorem12.p1.1.1.m1.1.1.1.1.2" mathvariant="normal" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1.2.cmml">Γ</mi><mn id="Thmtheorem12.p1.1.1.m1.1.1.1.1.3" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmtheorem12.p1.1.1.m1.2.2.2.4" xref="Thmtheorem12.p1.1.1.m1.2.2.3.cmml">,</mo><msub id="Thmtheorem12.p1.1.1.m1.2.2.2.2" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem12.p1.1.1.m1.2.2.2.2.2" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2.2.cmml">ℬ</mi><mn id="Thmtheorem12.p1.1.1.m1.2.2.2.2.3" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2.3.cmml">1</mn></msub><mo id="Thmtheorem12.p1.1.1.m1.2.2.2.5" stretchy="false" xref="Thmtheorem12.p1.1.1.m1.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.1.1.m1.2b"><list id="Thmtheorem12.p1.1.1.m1.2.2.3.cmml" xref="Thmtheorem12.p1.1.1.m1.2.2.2"><apply id="Thmtheorem12.p1.1.1.m1.1.1.1.1.cmml" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1">subscript</csymbol><ci id="Thmtheorem12.p1.1.1.m1.1.1.1.1.2.cmml" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1.2">Γ</ci><cn id="Thmtheorem12.p1.1.1.m1.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem12.p1.1.1.m1.1.1.1.1.3">1</cn></apply><apply id="Thmtheorem12.p1.1.1.m1.2.2.2.2.cmml" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.1.1.m1.2.2.2.2.1.cmml" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2">subscript</csymbol><ci id="Thmtheorem12.p1.1.1.m1.2.2.2.2.2.cmml" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2.2">ℬ</ci><cn id="Thmtheorem12.p1.1.1.m1.2.2.2.2.3.cmml" type="integer" xref="Thmtheorem12.p1.1.1.m1.2.2.2.2.3">1</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.1.1.m1.2c">\langle\Gamma_{1},\mathcal{B}_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.1.1.m1.2d">⟨ roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> and <math alttext="\langle\Gamma_{2},\mathcal{B}_{2}\rangle" class="ltx_Math" display="inline" id="Thmtheorem12.p1.2.2.m2.2"><semantics id="Thmtheorem12.p1.2.2.m2.2a"><mrow id="Thmtheorem12.p1.2.2.m2.2.2.2" xref="Thmtheorem12.p1.2.2.m2.2.2.3.cmml"><mo id="Thmtheorem12.p1.2.2.m2.2.2.2.3" stretchy="false" xref="Thmtheorem12.p1.2.2.m2.2.2.3.cmml">⟨</mo><msub id="Thmtheorem12.p1.2.2.m2.1.1.1.1" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1.cmml"><mi id="Thmtheorem12.p1.2.2.m2.1.1.1.1.2" mathvariant="normal" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1.2.cmml">Γ</mi><mn id="Thmtheorem12.p1.2.2.m2.1.1.1.1.3" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1.3.cmml">2</mn></msub><mo id="Thmtheorem12.p1.2.2.m2.2.2.2.4" xref="Thmtheorem12.p1.2.2.m2.2.2.3.cmml">,</mo><msub id="Thmtheorem12.p1.2.2.m2.2.2.2.2" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem12.p1.2.2.m2.2.2.2.2.2" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2.2.cmml">ℬ</mi><mn id="Thmtheorem12.p1.2.2.m2.2.2.2.2.3" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2.3.cmml">2</mn></msub><mo id="Thmtheorem12.p1.2.2.m2.2.2.2.5" stretchy="false" xref="Thmtheorem12.p1.2.2.m2.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.2.2.m2.2b"><list id="Thmtheorem12.p1.2.2.m2.2.2.3.cmml" xref="Thmtheorem12.p1.2.2.m2.2.2.2"><apply id="Thmtheorem12.p1.2.2.m2.1.1.1.1.cmml" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.2.2.m2.1.1.1.1.1.cmml" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1">subscript</csymbol><ci id="Thmtheorem12.p1.2.2.m2.1.1.1.1.2.cmml" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1.2">Γ</ci><cn id="Thmtheorem12.p1.2.2.m2.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem12.p1.2.2.m2.1.1.1.1.3">2</cn></apply><apply id="Thmtheorem12.p1.2.2.m2.2.2.2.2.cmml" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.2.2.m2.2.2.2.2.1.cmml" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2">subscript</csymbol><ci id="Thmtheorem12.p1.2.2.m2.2.2.2.2.2.cmml" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2.2">ℬ</ci><cn id="Thmtheorem12.p1.2.2.m2.2.2.2.2.3.cmml" type="integer" xref="Thmtheorem12.p1.2.2.m2.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.2.2.m2.2c">\langle\Gamma_{2},\mathcal{B}_{2}\rangle</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.2.2.m2.2d">⟨ roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be discrete spaces, and <math alttext="\phi\colon\Gamma_{1}\to\Gamma_{2}" class="ltx_Math" display="inline" id="Thmtheorem12.p1.3.3.m3.1"><semantics id="Thmtheorem12.p1.3.3.m3.1a"><mrow id="Thmtheorem12.p1.3.3.m3.1.1" xref="Thmtheorem12.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem12.p1.3.3.m3.1.1.2" xref="Thmtheorem12.p1.3.3.m3.1.1.2.cmml">ϕ</mi><mo id="Thmtheorem12.p1.3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem12.p1.3.3.m3.1.1.1.cmml">:</mo><mrow id="Thmtheorem12.p1.3.3.m3.1.1.3" xref="Thmtheorem12.p1.3.3.m3.1.1.3.cmml"><msub id="Thmtheorem12.p1.3.3.m3.1.1.3.2" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2.cmml"><mi id="Thmtheorem12.p1.3.3.m3.1.1.3.2.2" mathvariant="normal" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2.2.cmml">Γ</mi><mn id="Thmtheorem12.p1.3.3.m3.1.1.3.2.3" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2.3.cmml">1</mn></msub><mo id="Thmtheorem12.p1.3.3.m3.1.1.3.1" stretchy="false" xref="Thmtheorem12.p1.3.3.m3.1.1.3.1.cmml">→</mo><msub id="Thmtheorem12.p1.3.3.m3.1.1.3.3" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3.cmml"><mi id="Thmtheorem12.p1.3.3.m3.1.1.3.3.2" mathvariant="normal" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3.2.cmml">Γ</mi><mn id="Thmtheorem12.p1.3.3.m3.1.1.3.3.3" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.3.3.m3.1b"><apply id="Thmtheorem12.p1.3.3.m3.1.1.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1"><ci id="Thmtheorem12.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.1">:</ci><ci id="Thmtheorem12.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.2">italic-ϕ</ci><apply id="Thmtheorem12.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3"><ci id="Thmtheorem12.p1.3.3.m3.1.1.3.1.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.1">→</ci><apply id="Thmtheorem12.p1.3.3.m3.1.1.3.2.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.3.3.m3.1.1.3.2.1.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2">subscript</csymbol><ci id="Thmtheorem12.p1.3.3.m3.1.1.3.2.2.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2.2">Γ</ci><cn id="Thmtheorem12.p1.3.3.m3.1.1.3.2.3.cmml" type="integer" xref="Thmtheorem12.p1.3.3.m3.1.1.3.2.3">1</cn></apply><apply id="Thmtheorem12.p1.3.3.m3.1.1.3.3.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem12.p1.3.3.m3.1.1.3.3.1.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3">subscript</csymbol><ci id="Thmtheorem12.p1.3.3.m3.1.1.3.3.2.cmml" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3.2">Γ</ci><cn id="Thmtheorem12.p1.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="Thmtheorem12.p1.3.3.m3.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.3.3.m3.1c">\phi\colon\Gamma_{1}\to\Gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.3.3.m3.1d">italic_ϕ : roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be an injective function. Assume that <math alttext="\mathcal{B}_{2}=\{B^{2}_{1},\ldots,B^{2}_{m}\}" class="ltx_Math" display="inline" id="Thmtheorem12.p1.4.4.m4.3"><semantics id="Thmtheorem12.p1.4.4.m4.3a"><mrow id="Thmtheorem12.p1.4.4.m4.3.3" xref="Thmtheorem12.p1.4.4.m4.3.3.cmml"><msub id="Thmtheorem12.p1.4.4.m4.3.3.4" xref="Thmtheorem12.p1.4.4.m4.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem12.p1.4.4.m4.3.3.4.2" xref="Thmtheorem12.p1.4.4.m4.3.3.4.2.cmml">ℬ</mi><mn id="Thmtheorem12.p1.4.4.m4.3.3.4.3" xref="Thmtheorem12.p1.4.4.m4.3.3.4.3.cmml">2</mn></msub><mo id="Thmtheorem12.p1.4.4.m4.3.3.3" xref="Thmtheorem12.p1.4.4.m4.3.3.3.cmml">=</mo><mrow id="Thmtheorem12.p1.4.4.m4.3.3.2.2" xref="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml"><mo id="Thmtheorem12.p1.4.4.m4.3.3.2.2.3" stretchy="false" xref="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml">{</mo><msubsup id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.cmml"><mi id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.2" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.2.cmml">B</mi><mn id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.3" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.3.cmml">1</mn><mn id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.3" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.3.cmml">2</mn></msubsup><mo id="Thmtheorem12.p1.4.4.m4.3.3.2.2.4" xref="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml">,</mo><mi id="Thmtheorem12.p1.4.4.m4.1.1" mathvariant="normal" xref="Thmtheorem12.p1.4.4.m4.1.1.cmml">…</mi><mo id="Thmtheorem12.p1.4.4.m4.3.3.2.2.5" xref="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml">,</mo><msubsup id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.cmml"><mi id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.2" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.2.cmml">B</mi><mi id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.3" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.3.cmml">m</mi><mn id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.3" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.3.cmml">2</mn></msubsup><mo id="Thmtheorem12.p1.4.4.m4.3.3.2.2.6" stretchy="false" xref="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.4.4.m4.3b"><apply id="Thmtheorem12.p1.4.4.m4.3.3.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3"><eq id="Thmtheorem12.p1.4.4.m4.3.3.3.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.3"></eq><apply id="Thmtheorem12.p1.4.4.m4.3.3.4.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.4"><csymbol cd="ambiguous" id="Thmtheorem12.p1.4.4.m4.3.3.4.1.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.4">subscript</csymbol><ci id="Thmtheorem12.p1.4.4.m4.3.3.4.2.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.4.2">ℬ</ci><cn id="Thmtheorem12.p1.4.4.m4.3.3.4.3.cmml" type="integer" xref="Thmtheorem12.p1.4.4.m4.3.3.4.3">2</cn></apply><set id="Thmtheorem12.p1.4.4.m4.3.3.2.3.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2"><apply id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.cmml" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1">subscript</csymbol><apply id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.cmml" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.1.cmml" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1">superscript</csymbol><ci id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.2.cmml" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.2">𝐵</ci><cn id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.3.cmml" type="integer" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.2.3">2</cn></apply><cn id="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="Thmtheorem12.p1.4.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="Thmtheorem12.p1.4.4.m4.1.1.cmml" xref="Thmtheorem12.p1.4.4.m4.1.1">…</ci><apply id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.1.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2">subscript</csymbol><apply id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.1.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2">superscript</csymbol><ci id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.2.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.2">𝐵</ci><cn id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.3.cmml" type="integer" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.2.3">2</cn></apply><ci id="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.3.cmml" xref="Thmtheorem12.p1.4.4.m4.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.4.4.m4.3c">\mathcal{B}_{2}=\{B^{2}_{1},\ldots,B^{2}_{m}\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.4.4.m4.3d">caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math>. Then, for every <math alttext="A_{1}\subseteq\Gamma_{1}" class="ltx_Math" display="inline" id="Thmtheorem12.p1.5.5.m5.1"><semantics id="Thmtheorem12.p1.5.5.m5.1a"><mrow id="Thmtheorem12.p1.5.5.m5.1.1" xref="Thmtheorem12.p1.5.5.m5.1.1.cmml"><msub id="Thmtheorem12.p1.5.5.m5.1.1.2" xref="Thmtheorem12.p1.5.5.m5.1.1.2.cmml"><mi id="Thmtheorem12.p1.5.5.m5.1.1.2.2" xref="Thmtheorem12.p1.5.5.m5.1.1.2.2.cmml">A</mi><mn id="Thmtheorem12.p1.5.5.m5.1.1.2.3" xref="Thmtheorem12.p1.5.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="Thmtheorem12.p1.5.5.m5.1.1.1" xref="Thmtheorem12.p1.5.5.m5.1.1.1.cmml">⊆</mo><msub id="Thmtheorem12.p1.5.5.m5.1.1.3" xref="Thmtheorem12.p1.5.5.m5.1.1.3.cmml"><mi id="Thmtheorem12.p1.5.5.m5.1.1.3.2" mathvariant="normal" xref="Thmtheorem12.p1.5.5.m5.1.1.3.2.cmml">Γ</mi><mn id="Thmtheorem12.p1.5.5.m5.1.1.3.3" xref="Thmtheorem12.p1.5.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.5.5.m5.1b"><apply id="Thmtheorem12.p1.5.5.m5.1.1.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1"><subset id="Thmtheorem12.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.1"></subset><apply id="Thmtheorem12.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem12.p1.5.5.m5.1.1.2.1.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="Thmtheorem12.p1.5.5.m5.1.1.2.2.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.2.2">𝐴</ci><cn id="Thmtheorem12.p1.5.5.m5.1.1.2.3.cmml" type="integer" xref="Thmtheorem12.p1.5.5.m5.1.1.2.3">1</cn></apply><apply id="Thmtheorem12.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem12.p1.5.5.m5.1.1.3.1.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="Thmtheorem12.p1.5.5.m5.1.1.3.2.cmml" xref="Thmtheorem12.p1.5.5.m5.1.1.3.2">Γ</ci><cn id="Thmtheorem12.p1.5.5.m5.1.1.3.3.cmml" type="integer" xref="Thmtheorem12.p1.5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.5.5.m5.1c">A_{1}\subseteq\Gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.5.5.m5.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>,</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx2"> <tbody id="S2.Ex5"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D(\phi(A_{1})\mid\mathcal{B}_{2})" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.3" xref="S2.Ex4.m1.1.1.3.cmml">D</mi><mo id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex4.m1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.cmml"><mrow id="S2.Ex4.m1.1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.1.1.3.cmml">ϕ</mi><mo id="S2.Ex4.m1.1.1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.1.1.1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex4.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.1.2.cmml">∣</mo><msub id="S2.Ex4.m1.1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex4.m1.1.1.1.1.1.3.2" xref="S2.Ex4.m1.1.1.1.1.1.3.2.cmml">ℬ</mi><mn id="S2.Ex4.m1.1.1.1.1.1.3.3" xref="S2.Ex4.m1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S2.Ex4.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"><times id="S2.Ex4.m1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.2"></times><ci id="S2.Ex4.m1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.3">𝐷</ci><apply id="S2.Ex4.m1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex4.m1.1.1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.1.1.2">conditional</csymbol><apply id="S2.Ex4.m1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1"><times id="S2.Ex4.m1.1.1.1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1.2"></times><ci id="S2.Ex4.m1.1.1.1.1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1.3">italic-ϕ</ci><apply id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Ex4.m1.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.Ex4.m1.1.1.1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.1.1.3.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex4.m1.1.1.1.1.1.3.2.cmml" xref="S2.Ex4.m1.1.1.1.1.1.3.2">ℬ</ci><cn id="S2.Ex4.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex4.m1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle D(\phi(A_{1})\mid\mathcal{B}_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">italic_D ( italic_ϕ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∣ caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S2.Ex4.m2.1"><semantics id="S2.Ex4.m2.1a"><mo id="S2.Ex4.m2.1.1" xref="S2.Ex4.m2.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex4.m2.1b"><geq id="S2.Ex4.m2.1.1.cmml" xref="S2.Ex4.m2.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle D(A_{1}\mid\mathcal{B}_{1})-D(\phi^{-1}(B^{2}_{1}),\ldots,\phi^{% -1}(B^{2}_{m})\mid\mathcal{B}_{1})" class="ltx_Math" display="inline" id="S2.Ex4.m3.4"><semantics id="S2.Ex4.m3.4a"><mrow 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xref="S2.Ex4.m3.2.2.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S2.Ex4.m3.2.2.1.1.1.3" stretchy="false" xref="S2.Ex4.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m3.4.4.4" xref="S2.Ex4.m3.4.4.4.cmml">−</mo><mrow id="S2.Ex4.m3.4.4.3" xref="S2.Ex4.m3.4.4.3.cmml"><mi id="S2.Ex4.m3.4.4.3.4" xref="S2.Ex4.m3.4.4.3.4.cmml">D</mi><mo id="S2.Ex4.m3.4.4.3.3" xref="S2.Ex4.m3.4.4.3.3.cmml">⁢</mo><mrow id="S2.Ex4.m3.4.4.3.2.2" xref="S2.Ex4.m3.4.4.3.2.3.cmml"><mo id="S2.Ex4.m3.4.4.3.2.2.3" stretchy="false" xref="S2.Ex4.m3.4.4.3.2.3.cmml">(</mo><mrow id="S2.Ex4.m3.3.3.2.1.1.1" xref="S2.Ex4.m3.3.3.2.1.1.1.cmml"><msup id="S2.Ex4.m3.3.3.2.1.1.1.3" xref="S2.Ex4.m3.3.3.2.1.1.1.3.cmml"><mi id="S2.Ex4.m3.3.3.2.1.1.1.3.2" xref="S2.Ex4.m3.3.3.2.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.Ex4.m3.3.3.2.1.1.1.3.3" xref="S2.Ex4.m3.3.3.2.1.1.1.3.3.cmml"><mo id="S2.Ex4.m3.3.3.2.1.1.1.3.3a" xref="S2.Ex4.m3.3.3.2.1.1.1.3.3.cmml">−</mo><mn id="S2.Ex4.m3.3.3.2.1.1.1.3.3.2" xref="S2.Ex4.m3.3.3.2.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.Ex4.m3.3.3.2.1.1.1.2" xref="S2.Ex4.m3.3.3.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m3.3.3.2.1.1.1.1.1" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m3.3.3.2.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.Ex4.m3.3.3.2.1.1.1.1.1.1" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.2.2" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.2.2.cmml">B</mi><mn id="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.3" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.3.cmml">1</mn><mn id="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.2.3" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.Ex4.m3.3.3.2.1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m3.3.3.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m3.4.4.3.2.2.4" xref="S2.Ex4.m3.4.4.3.2.3.cmml">,</mo><mi id="S2.Ex4.m3.1.1" mathvariant="normal" xref="S2.Ex4.m3.1.1.cmml">…</mi><mo id="S2.Ex4.m3.4.4.3.2.2.5" 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xref="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.2.2.cmml">B</mi><mi id="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.3" xref="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.3.cmml">m</mi><mn id="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.2.3" xref="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.3" stretchy="false" xref="S2.Ex4.m3.4.4.3.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m3.4.4.3.2.2.2.2" xref="S2.Ex4.m3.4.4.3.2.2.2.2.cmml">∣</mo><msub id="S2.Ex4.m3.4.4.3.2.2.2.3" xref="S2.Ex4.m3.4.4.3.2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex4.m3.4.4.3.2.2.2.3.2" xref="S2.Ex4.m3.4.4.3.2.2.2.3.2.cmml">ℬ</mi><mn id="S2.Ex4.m3.4.4.3.2.2.2.3.3" xref="S2.Ex4.m3.4.4.3.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.Ex4.m3.4.4.3.2.2.6" stretchy="false" xref="S2.Ex4.m3.4.4.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m3.4b"><apply id="S2.Ex4.m3.4.4.cmml" xref="S2.Ex4.m3.4.4"><minus id="S2.Ex4.m3.4.4.4.cmml" 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xref="S2.Ex4.m3.4.4.3.2.2.2.3.3">1</cn></apply></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m3.4c">\displaystyle D(A_{1}\mid\mathcal{B}_{1})-D(\phi^{-1}(B^{2}_{1}),\ldots,\phi^{% -1}(B^{2}_{m})\mid\mathcal{B}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m3.4d">italic_D ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_D ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td 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xref="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.2.2.3.2">1</cn></apply></apply><ci id="S2.Ex5.m2.1.1.cmml" xref="S2.Ex5.m2.1.1">𝐵</ci></apply><apply id="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.cmml" xref="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.1.cmml" xref="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.2.cmml" xref="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.2">ℬ</ci><cn id="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex5.m2.2.2.1.1.2.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m2.2c">\displaystyle D(A_{1}\mid\mathcal{B}_{1})-\sum_{B\in\mathcal{B}_{2}}D(\phi^{-1% }(B)\mid\mathcal{B}_{1}).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m2.2d">italic_D ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_B ∈ caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ) ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="Thmtheorem12.p1.7"><span class="ltx_text ltx_font_italic" id="Thmtheorem12.p1.7.2">The result also holds with respect to the discrete complexity measures <math alttext="D_{\cap}" class="ltx_Math" display="inline" id="Thmtheorem12.p1.6.1.m1.1"><semantics id="Thmtheorem12.p1.6.1.m1.1a"><msub id="Thmtheorem12.p1.6.1.m1.1.1" xref="Thmtheorem12.p1.6.1.m1.1.1.cmml"><mi id="Thmtheorem12.p1.6.1.m1.1.1.2" xref="Thmtheorem12.p1.6.1.m1.1.1.2.cmml">D</mi><mo id="Thmtheorem12.p1.6.1.m1.1.1.3" xref="Thmtheorem12.p1.6.1.m1.1.1.3.cmml">∩</mo></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.6.1.m1.1b"><apply id="Thmtheorem12.p1.6.1.m1.1.1.cmml" xref="Thmtheorem12.p1.6.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.6.1.m1.1.1.1.cmml" xref="Thmtheorem12.p1.6.1.m1.1.1">subscript</csymbol><ci id="Thmtheorem12.p1.6.1.m1.1.1.2.cmml" xref="Thmtheorem12.p1.6.1.m1.1.1.2">𝐷</ci><intersect id="Thmtheorem12.p1.6.1.m1.1.1.3.cmml" xref="Thmtheorem12.p1.6.1.m1.1.1.3"></intersect></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.6.1.m1.1c">D_{\cap}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.6.1.m1.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="D_{\cup}" class="ltx_Math" display="inline" id="Thmtheorem12.p1.7.2.m2.1"><semantics id="Thmtheorem12.p1.7.2.m2.1a"><msub id="Thmtheorem12.p1.7.2.m2.1.1" xref="Thmtheorem12.p1.7.2.m2.1.1.cmml"><mi id="Thmtheorem12.p1.7.2.m2.1.1.2" xref="Thmtheorem12.p1.7.2.m2.1.1.2.cmml">D</mi><mo id="Thmtheorem12.p1.7.2.m2.1.1.3" xref="Thmtheorem12.p1.7.2.m2.1.1.3.cmml">∪</mo></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem12.p1.7.2.m2.1b"><apply id="Thmtheorem12.p1.7.2.m2.1.1.cmml" xref="Thmtheorem12.p1.7.2.m2.1.1"><csymbol cd="ambiguous" id="Thmtheorem12.p1.7.2.m2.1.1.1.cmml" xref="Thmtheorem12.p1.7.2.m2.1.1">subscript</csymbol><ci id="Thmtheorem12.p1.7.2.m2.1.1.2.cmml" xref="Thmtheorem12.p1.7.2.m2.1.1.2">𝐷</ci><union id="Thmtheorem12.p1.7.2.m2.1.1.3.cmml" xref="Thmtheorem12.p1.7.2.m2.1.1.3"></union></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem12.p1.7.2.m2.1c">D_{\cup}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem12.p1.7.2.m2.1d">italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS4.1.p1"> <p class="ltx_p" id="S2.SS4.1.p1.7">Let <math 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id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.2" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.3" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.1.m1.1b"><apply id="S2.SS4.1.p1.1.m1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1"><eq id="S2.SS4.1.p1.1.m1.1.1.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.2"></eq><apply id="S2.SS4.1.p1.1.m1.1.1.3.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3.2">𝐴</ci><cn id="S2.SS4.1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.1.m1.1.1.3.3">2</cn></apply><apply id="S2.SS4.1.p1.1.m1.1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1"><times id="S2.SS4.1.p1.1.m1.1.1.1.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1.2"></times><ci id="S2.SS4.1.p1.1.m1.1.1.1.3.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1.3">italic-ϕ</ci><apply id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.1.m1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.1.m1.1c">A_{2}=\phi(A_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.1.m1.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Since <math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS4.1.p1.2.m2.1"><semantics id="S2.SS4.1.p1.2.m2.1a"><mi id="S2.SS4.1.p1.2.m2.1.1" xref="S2.SS4.1.p1.2.m2.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.2.m2.1b"><ci id="S2.SS4.1.p1.2.m2.1.1.cmml" xref="S2.SS4.1.p1.2.m2.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.2.m2.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.2.m2.1d">italic_ϕ</annotation></semantics></math> is injective, <math alttext="\phi^{-1}(A_{2})=A_{1}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.3.m3.1"><semantics id="S2.SS4.1.p1.3.m3.1a"><mrow id="S2.SS4.1.p1.3.m3.1.1" xref="S2.SS4.1.p1.3.m3.1.1.cmml"><mrow id="S2.SS4.1.p1.3.m3.1.1.1" xref="S2.SS4.1.p1.3.m3.1.1.1.cmml"><msup id="S2.SS4.1.p1.3.m3.1.1.1.3" xref="S2.SS4.1.p1.3.m3.1.1.1.3.cmml"><mi id="S2.SS4.1.p1.3.m3.1.1.1.3.2" xref="S2.SS4.1.p1.3.m3.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.1.p1.3.m3.1.1.1.3.3" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3.cmml"><mo id="S2.SS4.1.p1.3.m3.1.1.1.3.3a" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3.cmml">−</mo><mn id="S2.SS4.1.p1.3.m3.1.1.1.3.3.2" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.1.p1.3.m3.1.1.1.2" xref="S2.SS4.1.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.3.m3.1.1.1.1.1" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.2" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.3" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS4.1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.3.m3.1.1.2" xref="S2.SS4.1.p1.3.m3.1.1.2.cmml">=</mo><msub id="S2.SS4.1.p1.3.m3.1.1.3" xref="S2.SS4.1.p1.3.m3.1.1.3.cmml"><mi id="S2.SS4.1.p1.3.m3.1.1.3.2" xref="S2.SS4.1.p1.3.m3.1.1.3.2.cmml">A</mi><mn id="S2.SS4.1.p1.3.m3.1.1.3.3" xref="S2.SS4.1.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.3.m3.1b"><apply id="S2.SS4.1.p1.3.m3.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1"><eq id="S2.SS4.1.p1.3.m3.1.1.2.cmml" xref="S2.SS4.1.p1.3.m3.1.1.2"></eq><apply id="S2.SS4.1.p1.3.m3.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1"><times id="S2.SS4.1.p1.3.m3.1.1.1.2.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.2"></times><apply id="S2.SS4.1.p1.3.m3.1.1.1.3.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.3.m3.1.1.1.3.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.3.m3.1.1.1.3.2.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.3.m3.1.1.1.3.3.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3"><minus id="S2.SS4.1.p1.3.m3.1.1.1.3.3.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3"></minus><cn id="S2.SS4.1.p1.3.m3.1.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.3.m3.1.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.3.m3.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S2.SS4.1.p1.3.m3.1.1.3.cmml" xref="S2.SS4.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.3.m3.1.1.3.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.3.m3.1.1.3.2.cmml" xref="S2.SS4.1.p1.3.m3.1.1.3.2">𝐴</ci><cn id="S2.SS4.1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.3.m3.1c">\phi^{-1}(A_{2})=A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.3.m3.1d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="B^{2}_{1},\ldots,B^{2}_{m},C_{1},\ldots,C_{t}=A_{2}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.4.m4.6"><semantics id="S2.SS4.1.p1.4.m4.6a"><mrow id="S2.SS4.1.p1.4.m4.6.6" xref="S2.SS4.1.p1.4.m4.6.6.cmml"><mrow id="S2.SS4.1.p1.4.m4.6.6.4.4" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml"><msubsup id="S2.SS4.1.p1.4.m4.3.3.1.1.1" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.cmml"><mi id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.2" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.2.cmml">B</mi><mn id="S2.SS4.1.p1.4.m4.3.3.1.1.1.3" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.3.cmml">1</mn><mn id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.3" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.SS4.1.p1.4.m4.6.6.4.4.5" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml">,</mo><mi id="S2.SS4.1.p1.4.m4.1.1" mathvariant="normal" xref="S2.SS4.1.p1.4.m4.1.1.cmml">…</mi><mo id="S2.SS4.1.p1.4.m4.6.6.4.4.6" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml">,</mo><msubsup id="S2.SS4.1.p1.4.m4.4.4.2.2.2" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.cmml"><mi id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.2" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.2.cmml">B</mi><mi id="S2.SS4.1.p1.4.m4.4.4.2.2.2.3" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.3.cmml">m</mi><mn id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.3" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.3.cmml">2</mn></msubsup><mo id="S2.SS4.1.p1.4.m4.6.6.4.4.7" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml">,</mo><msub id="S2.SS4.1.p1.4.m4.5.5.3.3.3" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3.cmml"><mi id="S2.SS4.1.p1.4.m4.5.5.3.3.3.2" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3.2.cmml">C</mi><mn id="S2.SS4.1.p1.4.m4.5.5.3.3.3.3" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.4.m4.6.6.4.4.8" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml">,</mo><mi id="S2.SS4.1.p1.4.m4.2.2" mathvariant="normal" xref="S2.SS4.1.p1.4.m4.2.2.cmml">…</mi><mo id="S2.SS4.1.p1.4.m4.6.6.4.4.9" xref="S2.SS4.1.p1.4.m4.6.6.4.5.cmml">,</mo><msub id="S2.SS4.1.p1.4.m4.6.6.4.4.4" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4.cmml"><mi id="S2.SS4.1.p1.4.m4.6.6.4.4.4.2" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4.2.cmml">C</mi><mi id="S2.SS4.1.p1.4.m4.6.6.4.4.4.3" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4.3.cmml">t</mi></msub></mrow><mo id="S2.SS4.1.p1.4.m4.6.6.5" xref="S2.SS4.1.p1.4.m4.6.6.5.cmml">=</mo><msub id="S2.SS4.1.p1.4.m4.6.6.6" xref="S2.SS4.1.p1.4.m4.6.6.6.cmml"><mi id="S2.SS4.1.p1.4.m4.6.6.6.2" xref="S2.SS4.1.p1.4.m4.6.6.6.2.cmml">A</mi><mn id="S2.SS4.1.p1.4.m4.6.6.6.3" xref="S2.SS4.1.p1.4.m4.6.6.6.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.4.m4.6b"><apply id="S2.SS4.1.p1.4.m4.6.6.cmml" xref="S2.SS4.1.p1.4.m4.6.6"><eq id="S2.SS4.1.p1.4.m4.6.6.5.cmml" xref="S2.SS4.1.p1.4.m4.6.6.5"></eq><list id="S2.SS4.1.p1.4.m4.6.6.4.5.cmml" xref="S2.SS4.1.p1.4.m4.6.6.4.4"><apply id="S2.SS4.1.p1.4.m4.3.3.1.1.1.cmml" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.3.3.1.1.1.1.cmml" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1">subscript</csymbol><apply id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.cmml" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1">superscript</csymbol><ci id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.2">𝐵</ci><cn id="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.2.3">2</cn></apply><cn id="S2.SS4.1.p1.4.m4.3.3.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.3.3.1.1.1.3">1</cn></apply><ci id="S2.SS4.1.p1.4.m4.1.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1">…</ci><apply id="S2.SS4.1.p1.4.m4.4.4.2.2.2.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.4.4.2.2.2.1.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2">subscript</csymbol><apply id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.1.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2">superscript</csymbol><ci id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.2.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.2">𝐵</ci><cn id="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.2.3">2</cn></apply><ci id="S2.SS4.1.p1.4.m4.4.4.2.2.2.3.cmml" xref="S2.SS4.1.p1.4.m4.4.4.2.2.2.3">𝑚</ci></apply><apply id="S2.SS4.1.p1.4.m4.5.5.3.3.3.cmml" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.5.5.3.3.3.1.cmml" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3">subscript</csymbol><ci id="S2.SS4.1.p1.4.m4.5.5.3.3.3.2.cmml" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3.2">𝐶</ci><cn id="S2.SS4.1.p1.4.m4.5.5.3.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.5.5.3.3.3.3">1</cn></apply><ci id="S2.SS4.1.p1.4.m4.2.2.cmml" xref="S2.SS4.1.p1.4.m4.2.2">…</ci><apply id="S2.SS4.1.p1.4.m4.6.6.4.4.4.cmml" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.6.6.4.4.4.1.cmml" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4">subscript</csymbol><ci id="S2.SS4.1.p1.4.m4.6.6.4.4.4.2.cmml" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4.2">𝐶</ci><ci id="S2.SS4.1.p1.4.m4.6.6.4.4.4.3.cmml" xref="S2.SS4.1.p1.4.m4.6.6.4.4.4.3">𝑡</ci></apply></list><apply id="S2.SS4.1.p1.4.m4.6.6.6.cmml" xref="S2.SS4.1.p1.4.m4.6.6.6"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.6.6.6.1.cmml" xref="S2.SS4.1.p1.4.m4.6.6.6">subscript</csymbol><ci id="S2.SS4.1.p1.4.m4.6.6.6.2.cmml" xref="S2.SS4.1.p1.4.m4.6.6.6.2">𝐴</ci><cn id="S2.SS4.1.p1.4.m4.6.6.6.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.6.6.6.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.4.m4.6c">B^{2}_{1},\ldots,B^{2}_{m},C_{1},\ldots,C_{t}=A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.4.m4.6d">italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be an extended sequence that describes a construction of <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.5.m5.1"><semantics id="S2.SS4.1.p1.5.m5.1a"><msub id="S2.SS4.1.p1.5.m5.1.1" xref="S2.SS4.1.p1.5.m5.1.1.cmml"><mi id="S2.SS4.1.p1.5.m5.1.1.2" xref="S2.SS4.1.p1.5.m5.1.1.2.cmml">A</mi><mn id="S2.SS4.1.p1.5.m5.1.1.3" xref="S2.SS4.1.p1.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.5.m5.1b"><apply id="S2.SS4.1.p1.5.m5.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.5.m5.1.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.5.m5.1.1.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2">𝐴</ci><cn id="S2.SS4.1.p1.5.m5.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.5.m5.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.5.m5.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\mathcal{B}_{2}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.6.m6.1"><semantics id="S2.SS4.1.p1.6.m6.1a"><msub id="S2.SS4.1.p1.6.m6.1.1" xref="S2.SS4.1.p1.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.6.m6.1.1.2" xref="S2.SS4.1.p1.6.m6.1.1.2.cmml">ℬ</mi><mn id="S2.SS4.1.p1.6.m6.1.1.3" xref="S2.SS4.1.p1.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.6.m6.1b"><apply id="S2.SS4.1.p1.6.m6.1.1.cmml" xref="S2.SS4.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.6.m6.1.1.1.cmml" xref="S2.SS4.1.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.6.m6.1.1.2.cmml" xref="S2.SS4.1.p1.6.m6.1.1.2">ℬ</ci><cn id="S2.SS4.1.p1.6.m6.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.6.m6.1c">\mathcal{B}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.6.m6.1d">caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="t=D(A_{2}\mid\mathcal{B}_{2})" class="ltx_Math" display="inline" id="S2.SS4.1.p1.7.m7.1"><semantics id="S2.SS4.1.p1.7.m7.1a"><mrow id="S2.SS4.1.p1.7.m7.1.1" xref="S2.SS4.1.p1.7.m7.1.1.cmml"><mi id="S2.SS4.1.p1.7.m7.1.1.3" xref="S2.SS4.1.p1.7.m7.1.1.3.cmml">t</mi><mo id="S2.SS4.1.p1.7.m7.1.1.2" xref="S2.SS4.1.p1.7.m7.1.1.2.cmml">=</mo><mrow id="S2.SS4.1.p1.7.m7.1.1.1" xref="S2.SS4.1.p1.7.m7.1.1.1.cmml"><mi id="S2.SS4.1.p1.7.m7.1.1.1.3" xref="S2.SS4.1.p1.7.m7.1.1.1.3.cmml">D</mi><mo id="S2.SS4.1.p1.7.m7.1.1.1.2" xref="S2.SS4.1.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.7.m7.1.1.1.1.1" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.cmml"><msub id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.cmml"><mi id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.2" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.3" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.1" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.2" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.2.cmml">ℬ</mi><mn id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.3" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S2.SS4.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.7.m7.1b"><apply id="S2.SS4.1.p1.7.m7.1.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1"><eq id="S2.SS4.1.p1.7.m7.1.1.2.cmml" xref="S2.SS4.1.p1.7.m7.1.1.2"></eq><ci id="S2.SS4.1.p1.7.m7.1.1.3.cmml" xref="S2.SS4.1.p1.7.m7.1.1.3">𝑡</ci><apply id="S2.SS4.1.p1.7.m7.1.1.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1"><times id="S2.SS4.1.p1.7.m7.1.1.1.2.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.2"></times><ci id="S2.SS4.1.p1.7.m7.1.1.1.3.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.3">𝐷</ci><apply id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.1">conditional</csymbol><apply id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.2">𝐴</ci><cn id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.2.3">2</cn></apply><apply id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.2.cmml" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.2">ℬ</ci><cn id="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.7.m7.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.7.m7.1c">t=D(A_{2}\mid\mathcal{B}_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.7.m7.1d">italic_t = italic_D ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. We claim that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m}),\phi^{-1}(C_{1}),\ldots,\phi^% {-1}(C_{t})=A_{1}" class="ltx_Math" display="block" id="S2.Ex6.m1.6"><semantics id="S2.Ex6.m1.6a"><mrow id="S2.Ex6.m1.6.6" xref="S2.Ex6.m1.6.6.cmml"><mrow id="S2.Ex6.m1.6.6.4.4" xref="S2.Ex6.m1.6.6.4.5.cmml"><mrow id="S2.Ex6.m1.3.3.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.cmml"><msup id="S2.Ex6.m1.3.3.1.1.1.3" xref="S2.Ex6.m1.3.3.1.1.1.3.cmml"><mi id="S2.Ex6.m1.3.3.1.1.1.3.2" xref="S2.Ex6.m1.3.3.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.Ex6.m1.3.3.1.1.1.3.3" xref="S2.Ex6.m1.3.3.1.1.1.3.3.cmml"><mo id="S2.Ex6.m1.3.3.1.1.1.3.3a" xref="S2.Ex6.m1.3.3.1.1.1.3.3.cmml">−</mo><mn id="S2.Ex6.m1.3.3.1.1.1.3.3.2" xref="S2.Ex6.m1.3.3.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo 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cd="ambiguous" id="S2.Ex6.m1.6.6.6.1.cmml" xref="S2.Ex6.m1.6.6.6">subscript</csymbol><ci id="S2.Ex6.m1.6.6.6.2.cmml" xref="S2.Ex6.m1.6.6.6.2">𝐴</ci><cn id="S2.Ex6.m1.6.6.6.3.cmml" type="integer" xref="S2.Ex6.m1.6.6.6.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.6c">\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m}),\phi^{-1}(C_{1}),\ldots,\phi^% {-1}(C_{t})=A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.6d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) = italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.1.p1.12">is an extended sequence that describes a construction of <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.8.m1.1"><semantics id="S2.SS4.1.p1.8.m1.1a"><msub id="S2.SS4.1.p1.8.m1.1.1" xref="S2.SS4.1.p1.8.m1.1.1.cmml"><mi id="S2.SS4.1.p1.8.m1.1.1.2" xref="S2.SS4.1.p1.8.m1.1.1.2.cmml">A</mi><mn id="S2.SS4.1.p1.8.m1.1.1.3" xref="S2.SS4.1.p1.8.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.8.m1.1b"><apply id="S2.SS4.1.p1.8.m1.1.1.cmml" xref="S2.SS4.1.p1.8.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.8.m1.1.1.1.cmml" xref="S2.SS4.1.p1.8.m1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.8.m1.1.1.2.cmml" xref="S2.SS4.1.p1.8.m1.1.1.2">𝐴</ci><cn id="S2.SS4.1.p1.8.m1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.8.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.8.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.8.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\{\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m})\}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.9.m2.3"><semantics id="S2.SS4.1.p1.9.m2.3a"><mrow id="S2.SS4.1.p1.9.m2.3.3.2" xref="S2.SS4.1.p1.9.m2.3.3.3.cmml"><mo id="S2.SS4.1.p1.9.m2.3.3.2.3" stretchy="false" xref="S2.SS4.1.p1.9.m2.3.3.3.cmml">{</mo><mrow id="S2.SS4.1.p1.9.m2.2.2.1.1" xref="S2.SS4.1.p1.9.m2.2.2.1.1.cmml"><msup id="S2.SS4.1.p1.9.m2.2.2.1.1.3" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.cmml"><mi id="S2.SS4.1.p1.9.m2.2.2.1.1.3.2" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.cmml"><mo id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3a" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.cmml">−</mo><mn id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.2" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.1.p1.9.m2.2.2.1.1.2" xref="S2.SS4.1.p1.9.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.cmml"><mi id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.2" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.2.cmml">B</mi><mn id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.3" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.3.cmml">1</mn><mn id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.3" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.3" stretchy="false" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.9.m2.3.3.2.4" xref="S2.SS4.1.p1.9.m2.3.3.3.cmml">,</mo><mi id="S2.SS4.1.p1.9.m2.1.1" mathvariant="normal" xref="S2.SS4.1.p1.9.m2.1.1.cmml">…</mi><mo id="S2.SS4.1.p1.9.m2.3.3.2.5" xref="S2.SS4.1.p1.9.m2.3.3.3.cmml">,</mo><mrow id="S2.SS4.1.p1.9.m2.3.3.2.2" xref="S2.SS4.1.p1.9.m2.3.3.2.2.cmml"><msup id="S2.SS4.1.p1.9.m2.3.3.2.2.3" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.cmml"><mi id="S2.SS4.1.p1.9.m2.3.3.2.2.3.2" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.2.cmml">ϕ</mi><mrow id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.cmml"><mo id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3a" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.cmml">−</mo><mn id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.2" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.1.p1.9.m2.3.3.2.2.2" xref="S2.SS4.1.p1.9.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.cmml"><mo id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.2" stretchy="false" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.cmml">(</mo><msubsup id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.cmml"><mi id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.2" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.2.cmml">B</mi><mi id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.3" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.3.cmml">m</mi><mn id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.3" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.3" stretchy="false" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.9.m2.3.3.2.6" stretchy="false" xref="S2.SS4.1.p1.9.m2.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.9.m2.3b"><set id="S2.SS4.1.p1.9.m2.3.3.3.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2"><apply id="S2.SS4.1.p1.9.m2.2.2.1.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1"><times id="S2.SS4.1.p1.9.m2.2.2.1.1.2.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.2"></times><apply id="S2.SS4.1.p1.9.m2.2.2.1.1.3.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.2.2.1.1.3.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.9.m2.2.2.1.1.3.2.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3"><minus id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3"></minus><cn id="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.9.m2.2.2.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1">subscript</csymbol><apply id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.2">𝐵</ci><cn id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.2.3">2</cn></apply><cn id="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.9.m2.2.2.1.1.1.1.1.3">1</cn></apply></apply><ci id="S2.SS4.1.p1.9.m2.1.1.cmml" xref="S2.SS4.1.p1.9.m2.1.1">…</ci><apply id="S2.SS4.1.p1.9.m2.3.3.2.2.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2"><times id="S2.SS4.1.p1.9.m2.3.3.2.2.2.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.2"></times><apply id="S2.SS4.1.p1.9.m2.3.3.2.2.3.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.3.3.2.2.3.1.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3">superscript</csymbol><ci id="S2.SS4.1.p1.9.m2.3.3.2.2.3.2.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3"><minus id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.1.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3"></minus><cn id="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.9.m2.3.3.2.2.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1">subscript</csymbol><apply id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1">superscript</csymbol><ci id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.2">𝐵</ci><cn id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.2.3">2</cn></apply><ci id="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.3.cmml" xref="S2.SS4.1.p1.9.m2.3.3.2.2.1.1.1.3">𝑚</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.9.m2.3c">\{\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m})\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.9.m2.3d">{ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) }</annotation></semantics></math>. Indeed, this can be easily verified by induction using that <math alttext="\phi^{-1}(C_{1}\cap C_{2})=\phi^{-1}(C_{1})\cap\phi^{-1}(C_{2})" class="ltx_Math" display="inline" id="S2.SS4.1.p1.10.m3.3"><semantics id="S2.SS4.1.p1.10.m3.3a"><mrow id="S2.SS4.1.p1.10.m3.3.3" xref="S2.SS4.1.p1.10.m3.3.3.cmml"><mrow id="S2.SS4.1.p1.10.m3.1.1.1" xref="S2.SS4.1.p1.10.m3.1.1.1.cmml"><msup id="S2.SS4.1.p1.10.m3.1.1.1.3" xref="S2.SS4.1.p1.10.m3.1.1.1.3.cmml"><mi id="S2.SS4.1.p1.10.m3.1.1.1.3.2" xref="S2.SS4.1.p1.10.m3.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.1.p1.10.m3.1.1.1.3.3" xref="S2.SS4.1.p1.10.m3.1.1.1.3.3.cmml"><mo id="S2.SS4.1.p1.10.m3.1.1.1.3.3a" xref="S2.SS4.1.p1.10.m3.1.1.1.3.3.cmml">−</mo><mn id="S2.SS4.1.p1.10.m3.1.1.1.3.3.2" xref="S2.SS4.1.p1.10.m3.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.1.p1.10.m3.1.1.1.2" xref="S2.SS4.1.p1.10.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.10.m3.1.1.1.1.1" xref="S2.SS4.1.p1.10.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.10.m3.1.1.1.1.1.2" 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xref="S2.SS4.1.p1.10.m3.3.3.3.2.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.10.m3.3.3.3.2.1.1.1.2.cmml" xref="S2.SS4.1.p1.10.m3.3.3.3.2.1.1.1.2">𝐶</ci><cn id="S2.SS4.1.p1.10.m3.3.3.3.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.10.m3.3.3.3.2.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.10.m3.3c">\phi^{-1}(C_{1}\cap C_{2})=\phi^{-1}(C_{1})\cap\phi^{-1}(C_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.10.m3.3d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\phi^{-1}(C_{1}\cup 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xref="S2.SS4.1.p1.11.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.1.1.1.3.1.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.11.m4.1.1.1.3.2.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.11.m4.1.1.1.3.3.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.3.3"><minus id="S2.SS4.1.p1.11.m4.1.1.1.3.3.1.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.3.3"></minus><cn id="S2.SS4.1.p1.11.m4.1.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.11.m4.1.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.1.1"><union id="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.1"></union><apply id="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS4.1.p1.11.m4.1.1.1.1.1.1.2.2.cmml" 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xref="S2.SS4.1.p1.11.m4.2.2.2.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.2.2.2.1.3.1.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.11.m4.2.2.2.1.3.2.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.11.m4.2.2.2.1.3.3.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.3.3"><minus id="S2.SS4.1.p1.11.m4.2.2.2.1.3.3.1.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.3.3"></minus><cn id="S2.SS4.1.p1.11.m4.2.2.2.1.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.11.m4.2.2.2.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.2">𝐶</ci><cn id="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.11.m4.2.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="S2.SS4.1.p1.11.m4.3.3.3.2.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2"><times id="S2.SS4.1.p1.11.m4.3.3.3.2.2.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.2"></times><apply id="S2.SS4.1.p1.11.m4.3.3.3.2.3.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.3.3.3.2.3.1.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3">superscript</csymbol><ci id="S2.SS4.1.p1.11.m4.3.3.3.2.3.2.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3.2">italic-ϕ</ci><apply id="S2.SS4.1.p1.11.m4.3.3.3.2.3.3.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3.3"><minus id="S2.SS4.1.p1.11.m4.3.3.3.2.3.3.1.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3.3"></minus><cn id="S2.SS4.1.p1.11.m4.3.3.3.2.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.11.m4.3.3.3.2.3.3.2">1</cn></apply></apply><apply id="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.2.cmml" xref="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.2">𝐶</ci><cn id="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.11.m4.3.3.3.2.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.11.m4.3c">\phi^{-1}(C_{1}\cup C_{2})=\phi^{-1}(C_{1})\cup\phi^{-1}(C_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.11.m4.3d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∪ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. The result immediately follows by replacing the initial sets in the construction above by a sequence that realizes <math alttext="D(\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m})\mid\mathcal{B}_{1})" class="ltx_Math" display="inline" id="S2.SS4.1.p1.12.m5.3"><semantics id="S2.SS4.1.p1.12.m5.3a"><mrow id="S2.SS4.1.p1.12.m5.3.3" xref="S2.SS4.1.p1.12.m5.3.3.cmml"><mi id="S2.SS4.1.p1.12.m5.3.3.4" xref="S2.SS4.1.p1.12.m5.3.3.4.cmml">D</mi><mo id="S2.SS4.1.p1.12.m5.3.3.3" xref="S2.SS4.1.p1.12.m5.3.3.3.cmml">⁢</mo><mrow id="S2.SS4.1.p1.12.m5.3.3.2.2" xref="S2.SS4.1.p1.12.m5.3.3.2.3.cmml"><mo id="S2.SS4.1.p1.12.m5.3.3.2.2.3" stretchy="false" xref="S2.SS4.1.p1.12.m5.3.3.2.3.cmml">(</mo><mrow id="S2.SS4.1.p1.12.m5.2.2.1.1.1" xref="S2.SS4.1.p1.12.m5.2.2.1.1.1.cmml"><msup id="S2.SS4.1.p1.12.m5.2.2.1.1.1.3" xref="S2.SS4.1.p1.12.m5.2.2.1.1.1.3.cmml"><mi id="S2.SS4.1.p1.12.m5.2.2.1.1.1.3.2" xref="S2.SS4.1.p1.12.m5.2.2.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.1.p1.12.m5.2.2.1.1.1.3.3" 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id="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.2.2">𝐵</ci><cn id="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.2.3">2</cn></apply><ci id="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.3.cmml" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.1.1.1.1.3">𝑚</ci></apply></apply><apply id="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.cmml" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.1.cmml" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.3">subscript</csymbol><ci id="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.2.cmml" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.2">ℬ</ci><cn id="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.3.cmml" type="integer" xref="S2.SS4.1.p1.12.m5.3.3.2.2.2.3.3">1</cn></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.12.m5.3c">D(\phi^{-1}(B^{2}_{1}),\ldots,\phi^{-1}(B^{2}_{m})\mid\mathcal{B}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.12.m5.3d">italic_D ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.8">In particular, if we have a strong enough lower bound with respect to <math alttext="\langle\Gamma_{1},\mathcal{B}_{1}\rangle" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.2"><semantics id="S2.SS4.p2.1.m1.2a"><mrow id="S2.SS4.p2.1.m1.2.2.2" xref="S2.SS4.p2.1.m1.2.2.3.cmml"><mo id="S2.SS4.p2.1.m1.2.2.2.3" stretchy="false" xref="S2.SS4.p2.1.m1.2.2.3.cmml">⟨</mo><msub id="S2.SS4.p2.1.m1.1.1.1.1" xref="S2.SS4.p2.1.m1.1.1.1.1.cmml"><mi id="S2.SS4.p2.1.m1.1.1.1.1.2" mathvariant="normal" xref="S2.SS4.p2.1.m1.1.1.1.1.2.cmml">Γ</mi><mn id="S2.SS4.p2.1.m1.1.1.1.1.3" xref="S2.SS4.p2.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS4.p2.1.m1.2.2.2.4" xref="S2.SS4.p2.1.m1.2.2.3.cmml">,</mo><msub id="S2.SS4.p2.1.m1.2.2.2.2" xref="S2.SS4.p2.1.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p2.1.m1.2.2.2.2.2" xref="S2.SS4.p2.1.m1.2.2.2.2.2.cmml">ℬ</mi><mn id="S2.SS4.p2.1.m1.2.2.2.2.3" xref="S2.SS4.p2.1.m1.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS4.p2.1.m1.2.2.2.5" stretchy="false" xref="S2.SS4.p2.1.m1.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.2b"><list id="S2.SS4.p2.1.m1.2.2.3.cmml" xref="S2.SS4.p2.1.m1.2.2.2"><apply id="S2.SS4.p2.1.m1.1.1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.1.m1.1.1.1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.p2.1.m1.1.1.1.1.2.cmml" xref="S2.SS4.p2.1.m1.1.1.1.1.2">Γ</ci><cn id="S2.SS4.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p2.1.m1.1.1.1.1.3">1</cn></apply><apply id="S2.SS4.p2.1.m1.2.2.2.2.cmml" xref="S2.SS4.p2.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.1.m1.2.2.2.2.1.cmml" xref="S2.SS4.p2.1.m1.2.2.2.2">subscript</csymbol><ci id="S2.SS4.p2.1.m1.2.2.2.2.2.cmml" xref="S2.SS4.p2.1.m1.2.2.2.2.2">ℬ</ci><cn id="S2.SS4.p2.1.m1.2.2.2.2.3.cmml" type="integer" xref="S2.SS4.p2.1.m1.2.2.2.2.3">1</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.2c">\langle\Gamma_{1},\mathcal{B}_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.2d">⟨ roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, and can construct an injective map <math alttext="\phi\colon\Gamma_{1}\to\Gamma_{2}" class="ltx_Math" display="inline" id="S2.SS4.p2.2.m2.1"><semantics id="S2.SS4.p2.2.m2.1a"><mrow id="S2.SS4.p2.2.m2.1.1" xref="S2.SS4.p2.2.m2.1.1.cmml"><mi id="S2.SS4.p2.2.m2.1.1.2" xref="S2.SS4.p2.2.m2.1.1.2.cmml">ϕ</mi><mo id="S2.SS4.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS4.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S2.SS4.p2.2.m2.1.1.3" xref="S2.SS4.p2.2.m2.1.1.3.cmml"><msub id="S2.SS4.p2.2.m2.1.1.3.2" xref="S2.SS4.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS4.p2.2.m2.1.1.3.2.2" mathvariant="normal" xref="S2.SS4.p2.2.m2.1.1.3.2.2.cmml">Γ</mi><mn id="S2.SS4.p2.2.m2.1.1.3.2.3" xref="S2.SS4.p2.2.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS4.p2.2.m2.1.1.3.1" stretchy="false" xref="S2.SS4.p2.2.m2.1.1.3.1.cmml">→</mo><msub id="S2.SS4.p2.2.m2.1.1.3.3" xref="S2.SS4.p2.2.m2.1.1.3.3.cmml"><mi id="S2.SS4.p2.2.m2.1.1.3.3.2" mathvariant="normal" xref="S2.SS4.p2.2.m2.1.1.3.3.2.cmml">Γ</mi><mn id="S2.SS4.p2.2.m2.1.1.3.3.3" xref="S2.SS4.p2.2.m2.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.1b"><apply id="S2.SS4.p2.2.m2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1"><ci id="S2.SS4.p2.2.m2.1.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1.1">:</ci><ci id="S2.SS4.p2.2.m2.1.1.2.cmml" xref="S2.SS4.p2.2.m2.1.1.2">italic-ϕ</ci><apply id="S2.SS4.p2.2.m2.1.1.3.cmml" xref="S2.SS4.p2.2.m2.1.1.3"><ci id="S2.SS4.p2.2.m2.1.1.3.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.1">→</ci><apply id="S2.SS4.p2.2.m2.1.1.3.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.3.2.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2">subscript</csymbol><ci id="S2.SS4.p2.2.m2.1.1.3.2.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2.2">Γ</ci><cn id="S2.SS4.p2.2.m2.1.1.3.2.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3.2.3">1</cn></apply><apply id="S2.SS4.p2.2.m2.1.1.3.3.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.3.3.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS4.p2.2.m2.1.1.3.3.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.2">Γ</ci><cn id="S2.SS4.p2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.1c">\phi\colon\Gamma_{1}\to\Gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.1d">italic_ϕ : roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> such that for each <math alttext="B\in\mathcal{B}_{2}" class="ltx_Math" display="inline" id="S2.SS4.p2.3.m3.1"><semantics id="S2.SS4.p2.3.m3.1a"><mrow id="S2.SS4.p2.3.m3.1.1" xref="S2.SS4.p2.3.m3.1.1.cmml"><mi id="S2.SS4.p2.3.m3.1.1.2" xref="S2.SS4.p2.3.m3.1.1.2.cmml">B</mi><mo id="S2.SS4.p2.3.m3.1.1.1" xref="S2.SS4.p2.3.m3.1.1.1.cmml">∈</mo><msub id="S2.SS4.p2.3.m3.1.1.3" xref="S2.SS4.p2.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p2.3.m3.1.1.3.2" xref="S2.SS4.p2.3.m3.1.1.3.2.cmml">ℬ</mi><mn id="S2.SS4.p2.3.m3.1.1.3.3" xref="S2.SS4.p2.3.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.3.m3.1b"><apply id="S2.SS4.p2.3.m3.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1"><in id="S2.SS4.p2.3.m3.1.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1.1"></in><ci id="S2.SS4.p2.3.m3.1.1.2.cmml" xref="S2.SS4.p2.3.m3.1.1.2">𝐵</ci><apply id="S2.SS4.p2.3.m3.1.1.3.cmml" xref="S2.SS4.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.3.m3.1.1.3.1.cmml" xref="S2.SS4.p2.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS4.p2.3.m3.1.1.3.2.cmml" xref="S2.SS4.p2.3.m3.1.1.3.2">ℬ</ci><cn id="S2.SS4.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS4.p2.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.3.m3.1c">B\in\mathcal{B}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.3.m3.1d">italic_B ∈ caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> the value <math alttext="D(\phi^{-1}(B)\mid\mathcal{B}_{1})" class="ltx_Math" display="inline" id="S2.SS4.p2.4.m4.2"><semantics id="S2.SS4.p2.4.m4.2a"><mrow id="S2.SS4.p2.4.m4.2.2" xref="S2.SS4.p2.4.m4.2.2.cmml"><mi id="S2.SS4.p2.4.m4.2.2.3" xref="S2.SS4.p2.4.m4.2.2.3.cmml">D</mi><mo id="S2.SS4.p2.4.m4.2.2.2" xref="S2.SS4.p2.4.m4.2.2.2.cmml">⁢</mo><mrow id="S2.SS4.p2.4.m4.2.2.1.1" xref="S2.SS4.p2.4.m4.2.2.1.1.1.cmml"><mo id="S2.SS4.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S2.SS4.p2.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS4.p2.4.m4.2.2.1.1.1" xref="S2.SS4.p2.4.m4.2.2.1.1.1.cmml"><mrow id="S2.SS4.p2.4.m4.2.2.1.1.1.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.cmml"><msup id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.cmml"><mi id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.2.cmml">ϕ</mi><mrow id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.cmml"><mo id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3a" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.cmml">−</mo><mn id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.p2.4.m4.2.2.1.1.1.2.1" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p2.4.m4.2.2.1.1.1.2.3.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.cmml"><mo id="S2.SS4.p2.4.m4.2.2.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.cmml">(</mo><mi id="S2.SS4.p2.4.m4.1.1" xref="S2.SS4.p2.4.m4.1.1.cmml">B</mi><mo id="S2.SS4.p2.4.m4.2.2.1.1.1.2.3.2.2" stretchy="false" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p2.4.m4.2.2.1.1.1.1" xref="S2.SS4.p2.4.m4.2.2.1.1.1.1.cmml">∣</mo><msub id="S2.SS4.p2.4.m4.2.2.1.1.1.3" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p2.4.m4.2.2.1.1.1.3.2" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3.2.cmml">ℬ</mi><mn id="S2.SS4.p2.4.m4.2.2.1.1.1.3.3" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S2.SS4.p2.4.m4.2.2.1.1.3" stretchy="false" xref="S2.SS4.p2.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.4.m4.2b"><apply id="S2.SS4.p2.4.m4.2.2.cmml" xref="S2.SS4.p2.4.m4.2.2"><times id="S2.SS4.p2.4.m4.2.2.2.cmml" xref="S2.SS4.p2.4.m4.2.2.2"></times><ci id="S2.SS4.p2.4.m4.2.2.3.cmml" xref="S2.SS4.p2.4.m4.2.2.3">𝐷</ci><apply id="S2.SS4.p2.4.m4.2.2.1.1.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1"><csymbol cd="latexml" id="S2.SS4.p2.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.1">conditional</csymbol><apply id="S2.SS4.p2.4.m4.2.2.1.1.1.2.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2"><times id="S2.SS4.p2.4.m4.2.2.1.1.1.2.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.1"></times><apply id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2">superscript</csymbol><ci id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.2.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.2">italic-ϕ</ci><apply id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3"><minus id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3"></minus><cn id="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.2.cmml" type="integer" xref="S2.SS4.p2.4.m4.2.2.1.1.1.2.2.3.2">1</cn></apply></apply><ci id="S2.SS4.p2.4.m4.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1">𝐵</ci></apply><apply id="S2.SS4.p2.4.m4.2.2.1.1.1.3.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.4.m4.2.2.1.1.1.3.1.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3">subscript</csymbol><ci id="S2.SS4.p2.4.m4.2.2.1.1.1.3.2.cmml" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3.2">ℬ</ci><cn id="S2.SS4.p2.4.m4.2.2.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.p2.4.m4.2.2.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.4.m4.2c">D(\phi^{-1}(B)\mid\mathcal{B}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.4.m4.2d">italic_D ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ) ∣ caligraphic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is small, we get a lower bound in <math alttext="\langle\Gamma_{2},\mathcal{B}_{2}\rangle" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m5.2"><semantics id="S2.SS4.p2.5.m5.2a"><mrow id="S2.SS4.p2.5.m5.2.2.2" xref="S2.SS4.p2.5.m5.2.2.3.cmml"><mo id="S2.SS4.p2.5.m5.2.2.2.3" stretchy="false" xref="S2.SS4.p2.5.m5.2.2.3.cmml">⟨</mo><msub id="S2.SS4.p2.5.m5.1.1.1.1" xref="S2.SS4.p2.5.m5.1.1.1.1.cmml"><mi id="S2.SS4.p2.5.m5.1.1.1.1.2" mathvariant="normal" xref="S2.SS4.p2.5.m5.1.1.1.1.2.cmml">Γ</mi><mn id="S2.SS4.p2.5.m5.1.1.1.1.3" xref="S2.SS4.p2.5.m5.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS4.p2.5.m5.2.2.2.4" xref="S2.SS4.p2.5.m5.2.2.3.cmml">,</mo><msub id="S2.SS4.p2.5.m5.2.2.2.2" xref="S2.SS4.p2.5.m5.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p2.5.m5.2.2.2.2.2" xref="S2.SS4.p2.5.m5.2.2.2.2.2.cmml">ℬ</mi><mn id="S2.SS4.p2.5.m5.2.2.2.2.3" xref="S2.SS4.p2.5.m5.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS4.p2.5.m5.2.2.2.5" stretchy="false" xref="S2.SS4.p2.5.m5.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.5.m5.2b"><list id="S2.SS4.p2.5.m5.2.2.3.cmml" xref="S2.SS4.p2.5.m5.2.2.2"><apply id="S2.SS4.p2.5.m5.1.1.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.5.m5.1.1.1.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1.1.1">subscript</csymbol><ci id="S2.SS4.p2.5.m5.1.1.1.1.2.cmml" xref="S2.SS4.p2.5.m5.1.1.1.1.2">Γ</ci><cn id="S2.SS4.p2.5.m5.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p2.5.m5.1.1.1.1.3">2</cn></apply><apply id="S2.SS4.p2.5.m5.2.2.2.2.cmml" xref="S2.SS4.p2.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.5.m5.2.2.2.2.1.cmml" xref="S2.SS4.p2.5.m5.2.2.2.2">subscript</csymbol><ci id="S2.SS4.p2.5.m5.2.2.2.2.2.cmml" xref="S2.SS4.p2.5.m5.2.2.2.2.2">ℬ</ci><cn id="S2.SS4.p2.5.m5.2.2.2.2.3.cmml" type="integer" xref="S2.SS4.p2.5.m5.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m5.2c">\langle\Gamma_{2},\mathcal{B}_{2}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m5.2d">⟨ roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩</annotation></semantics></math>. Moreover, if the original set <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.SS4.p2.6.m6.1"><semantics id="S2.SS4.p2.6.m6.1a"><msub id="S2.SS4.p2.6.m6.1.1" xref="S2.SS4.p2.6.m6.1.1.cmml"><mi id="S2.SS4.p2.6.m6.1.1.2" xref="S2.SS4.p2.6.m6.1.1.2.cmml">A</mi><mn id="S2.SS4.p2.6.m6.1.1.3" xref="S2.SS4.p2.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.6.m6.1b"><apply id="S2.SS4.p2.6.m6.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.6.m6.1.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1">subscript</csymbol><ci id="S2.SS4.p2.6.m6.1.1.2.cmml" xref="S2.SS4.p2.6.m6.1.1.2">𝐴</ci><cn id="S2.SS4.p2.6.m6.1.1.3.cmml" type="integer" xref="S2.SS4.p2.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.6.m6.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.6.m6.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and the map <math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS4.p2.7.m7.1"><semantics id="S2.SS4.p2.7.m7.1a"><mi id="S2.SS4.p2.7.m7.1.1" xref="S2.SS4.p2.7.m7.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.7.m7.1b"><ci id="S2.SS4.p2.7.m7.1.1.cmml" xref="S2.SS4.p2.7.m7.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.7.m7.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.7.m7.1d">italic_ϕ</annotation></semantics></math> are “explicit”, <math alttext="A_{2}=\phi(A_{1})" class="ltx_Math" display="inline" id="S2.SS4.p2.8.m8.1"><semantics id="S2.SS4.p2.8.m8.1a"><mrow id="S2.SS4.p2.8.m8.1.1" xref="S2.SS4.p2.8.m8.1.1.cmml"><msub id="S2.SS4.p2.8.m8.1.1.3" xref="S2.SS4.p2.8.m8.1.1.3.cmml"><mi id="S2.SS4.p2.8.m8.1.1.3.2" xref="S2.SS4.p2.8.m8.1.1.3.2.cmml">A</mi><mn id="S2.SS4.p2.8.m8.1.1.3.3" xref="S2.SS4.p2.8.m8.1.1.3.3.cmml">2</mn></msub><mo id="S2.SS4.p2.8.m8.1.1.2" xref="S2.SS4.p2.8.m8.1.1.2.cmml">=</mo><mrow id="S2.SS4.p2.8.m8.1.1.1" xref="S2.SS4.p2.8.m8.1.1.1.cmml"><mi id="S2.SS4.p2.8.m8.1.1.1.3" xref="S2.SS4.p2.8.m8.1.1.1.3.cmml">ϕ</mi><mo id="S2.SS4.p2.8.m8.1.1.1.2" xref="S2.SS4.p2.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.p2.8.m8.1.1.1.1.1" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.cmml"><mo id="S2.SS4.p2.8.m8.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS4.p2.8.m8.1.1.1.1.1.1" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p2.8.m8.1.1.1.1.1.1.2" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.SS4.p2.8.m8.1.1.1.1.1.1.3" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS4.p2.8.m8.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.8.m8.1b"><apply id="S2.SS4.p2.8.m8.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1"><eq id="S2.SS4.p2.8.m8.1.1.2.cmml" xref="S2.SS4.p2.8.m8.1.1.2"></eq><apply id="S2.SS4.p2.8.m8.1.1.3.cmml" xref="S2.SS4.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.8.m8.1.1.3.1.cmml" xref="S2.SS4.p2.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS4.p2.8.m8.1.1.3.2.cmml" xref="S2.SS4.p2.8.m8.1.1.3.2">𝐴</ci><cn id="S2.SS4.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S2.SS4.p2.8.m8.1.1.3.3">2</cn></apply><apply id="S2.SS4.p2.8.m8.1.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1.1"><times id="S2.SS4.p2.8.m8.1.1.1.2.cmml" xref="S2.SS4.p2.8.m8.1.1.1.2"></times><ci id="S2.SS4.p2.8.m8.1.1.1.3.cmml" xref="S2.SS4.p2.8.m8.1.1.1.3">italic-ϕ</ci><apply id="S2.SS4.p2.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.SS4.p2.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p2.8.m8.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.8.m8.1c">A_{2}=\phi(A_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.8.m8.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is explicit as well.</p> </div> <div class="ltx_para" id="S2.SS4.p3"> <p class="ltx_p" id="S2.SS4.p3.9">We provide next a simple example that will be useful later in the text. Given a binary string <math alttext="w\in\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS4.p3.1.m1.2"><semantics id="S2.SS4.p3.1.m1.2a"><mrow id="S2.SS4.p3.1.m1.2.3" xref="S2.SS4.p3.1.m1.2.3.cmml"><mi id="S2.SS4.p3.1.m1.2.3.2" xref="S2.SS4.p3.1.m1.2.3.2.cmml">w</mi><mo id="S2.SS4.p3.1.m1.2.3.1" xref="S2.SS4.p3.1.m1.2.3.1.cmml">∈</mo><msup id="S2.SS4.p3.1.m1.2.3.3" xref="S2.SS4.p3.1.m1.2.3.3.cmml"><mrow id="S2.SS4.p3.1.m1.2.3.3.2.2" xref="S2.SS4.p3.1.m1.2.3.3.2.1.cmml"><mo id="S2.SS4.p3.1.m1.2.3.3.2.2.1" stretchy="false" xref="S2.SS4.p3.1.m1.2.3.3.2.1.cmml">{</mo><mn id="S2.SS4.p3.1.m1.1.1" xref="S2.SS4.p3.1.m1.1.1.cmml">0</mn><mo id="S2.SS4.p3.1.m1.2.3.3.2.2.2" xref="S2.SS4.p3.1.m1.2.3.3.2.1.cmml">,</mo><mn id="S2.SS4.p3.1.m1.2.2" xref="S2.SS4.p3.1.m1.2.2.cmml">1</mn><mo id="S2.SS4.p3.1.m1.2.3.3.2.2.3" stretchy="false" xref="S2.SS4.p3.1.m1.2.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS4.p3.1.m1.2.3.3.3" xref="S2.SS4.p3.1.m1.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.1.m1.2b"><apply id="S2.SS4.p3.1.m1.2.3.cmml" xref="S2.SS4.p3.1.m1.2.3"><in id="S2.SS4.p3.1.m1.2.3.1.cmml" xref="S2.SS4.p3.1.m1.2.3.1"></in><ci id="S2.SS4.p3.1.m1.2.3.2.cmml" xref="S2.SS4.p3.1.m1.2.3.2">𝑤</ci><apply id="S2.SS4.p3.1.m1.2.3.3.cmml" xref="S2.SS4.p3.1.m1.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.p3.1.m1.2.3.3.1.cmml" xref="S2.SS4.p3.1.m1.2.3.3">superscript</csymbol><set id="S2.SS4.p3.1.m1.2.3.3.2.1.cmml" xref="S2.SS4.p3.1.m1.2.3.3.2.2"><cn id="S2.SS4.p3.1.m1.1.1.cmml" type="integer" xref="S2.SS4.p3.1.m1.1.1">0</cn><cn id="S2.SS4.p3.1.m1.2.2.cmml" type="integer" xref="S2.SS4.p3.1.m1.2.2">1</cn></set><ci id="S2.SS4.p3.1.m1.2.3.3.3.cmml" xref="S2.SS4.p3.1.m1.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.1.m1.2c">w\in\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.1.m1.2d">italic_w ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, which we represent as <math alttext="w=w_{1}\ldots w_{n}" class="ltx_Math" display="inline" id="S2.SS4.p3.2.m2.1"><semantics id="S2.SS4.p3.2.m2.1a"><mrow id="S2.SS4.p3.2.m2.1.1" xref="S2.SS4.p3.2.m2.1.1.cmml"><mi id="S2.SS4.p3.2.m2.1.1.2" xref="S2.SS4.p3.2.m2.1.1.2.cmml">w</mi><mo id="S2.SS4.p3.2.m2.1.1.1" xref="S2.SS4.p3.2.m2.1.1.1.cmml">=</mo><mrow id="S2.SS4.p3.2.m2.1.1.3" xref="S2.SS4.p3.2.m2.1.1.3.cmml"><msub id="S2.SS4.p3.2.m2.1.1.3.2" xref="S2.SS4.p3.2.m2.1.1.3.2.cmml"><mi id="S2.SS4.p3.2.m2.1.1.3.2.2" xref="S2.SS4.p3.2.m2.1.1.3.2.2.cmml">w</mi><mn id="S2.SS4.p3.2.m2.1.1.3.2.3" xref="S2.SS4.p3.2.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS4.p3.2.m2.1.1.3.1" xref="S2.SS4.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS4.p3.2.m2.1.1.3.3" mathvariant="normal" xref="S2.SS4.p3.2.m2.1.1.3.3.cmml">…</mi><mo id="S2.SS4.p3.2.m2.1.1.3.1a" xref="S2.SS4.p3.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S2.SS4.p3.2.m2.1.1.3.4" xref="S2.SS4.p3.2.m2.1.1.3.4.cmml"><mi id="S2.SS4.p3.2.m2.1.1.3.4.2" xref="S2.SS4.p3.2.m2.1.1.3.4.2.cmml">w</mi><mi id="S2.SS4.p3.2.m2.1.1.3.4.3" xref="S2.SS4.p3.2.m2.1.1.3.4.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.2.m2.1b"><apply id="S2.SS4.p3.2.m2.1.1.cmml" xref="S2.SS4.p3.2.m2.1.1"><eq id="S2.SS4.p3.2.m2.1.1.1.cmml" xref="S2.SS4.p3.2.m2.1.1.1"></eq><ci id="S2.SS4.p3.2.m2.1.1.2.cmml" xref="S2.SS4.p3.2.m2.1.1.2">𝑤</ci><apply id="S2.SS4.p3.2.m2.1.1.3.cmml" xref="S2.SS4.p3.2.m2.1.1.3"><times id="S2.SS4.p3.2.m2.1.1.3.1.cmml" xref="S2.SS4.p3.2.m2.1.1.3.1"></times><apply id="S2.SS4.p3.2.m2.1.1.3.2.cmml" xref="S2.SS4.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS4.p3.2.m2.1.1.3.2.1.cmml" xref="S2.SS4.p3.2.m2.1.1.3.2">subscript</csymbol><ci id="S2.SS4.p3.2.m2.1.1.3.2.2.cmml" xref="S2.SS4.p3.2.m2.1.1.3.2.2">𝑤</ci><cn id="S2.SS4.p3.2.m2.1.1.3.2.3.cmml" type="integer" xref="S2.SS4.p3.2.m2.1.1.3.2.3">1</cn></apply><ci id="S2.SS4.p3.2.m2.1.1.3.3.cmml" xref="S2.SS4.p3.2.m2.1.1.3.3">…</ci><apply id="S2.SS4.p3.2.m2.1.1.3.4.cmml" xref="S2.SS4.p3.2.m2.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS4.p3.2.m2.1.1.3.4.1.cmml" xref="S2.SS4.p3.2.m2.1.1.3.4">subscript</csymbol><ci id="S2.SS4.p3.2.m2.1.1.3.4.2.cmml" xref="S2.SS4.p3.2.m2.1.1.3.4.2">𝑤</ci><ci id="S2.SS4.p3.2.m2.1.1.3.4.3.cmml" xref="S2.SS4.p3.2.m2.1.1.3.4.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.2.m2.1c">w=w_{1}\ldots w_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.2.m2.1d">italic_w = italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="\mathsf{number}(w)=\sum_{i=0}^{n-1}2^{i}\cdot w_{n-i}" class="ltx_Math" display="inline" id="S2.SS4.p3.3.m3.1"><semantics id="S2.SS4.p3.3.m3.1a"><mrow id="S2.SS4.p3.3.m3.1.2" xref="S2.SS4.p3.3.m3.1.2.cmml"><mrow id="S2.SS4.p3.3.m3.1.2.2" xref="S2.SS4.p3.3.m3.1.2.2.cmml"><mi id="S2.SS4.p3.3.m3.1.2.2.2" xref="S2.SS4.p3.3.m3.1.2.2.2.cmml">𝗇𝗎𝗆𝖻𝖾𝗋</mi><mo id="S2.SS4.p3.3.m3.1.2.2.1" xref="S2.SS4.p3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.p3.3.m3.1.2.2.3.2" xref="S2.SS4.p3.3.m3.1.2.2.cmml"><mo id="S2.SS4.p3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.p3.3.m3.1.2.2.cmml">(</mo><mi id="S2.SS4.p3.3.m3.1.1" xref="S2.SS4.p3.3.m3.1.1.cmml">w</mi><mo id="S2.SS4.p3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S2.SS4.p3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p3.3.m3.1.2.1" rspace="0.111em" xref="S2.SS4.p3.3.m3.1.2.1.cmml">=</mo><mrow id="S2.SS4.p3.3.m3.1.2.3" xref="S2.SS4.p3.3.m3.1.2.3.cmml"><msubsup id="S2.SS4.p3.3.m3.1.2.3.1" xref="S2.SS4.p3.3.m3.1.2.3.1.cmml"><mo id="S2.SS4.p3.3.m3.1.2.3.1.2.2" xref="S2.SS4.p3.3.m3.1.2.3.1.2.2.cmml">∑</mo><mrow id="S2.SS4.p3.3.m3.1.2.3.1.2.3" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.cmml"><mi id="S2.SS4.p3.3.m3.1.2.3.1.2.3.2" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.2.cmml">i</mi><mo id="S2.SS4.p3.3.m3.1.2.3.1.2.3.1" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.1.cmml">=</mo><mn id="S2.SS4.p3.3.m3.1.2.3.1.2.3.3" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.3.cmml">0</mn></mrow><mrow id="S2.SS4.p3.3.m3.1.2.3.1.3" xref="S2.SS4.p3.3.m3.1.2.3.1.3.cmml"><mi id="S2.SS4.p3.3.m3.1.2.3.1.3.2" xref="S2.SS4.p3.3.m3.1.2.3.1.3.2.cmml">n</mi><mo id="S2.SS4.p3.3.m3.1.2.3.1.3.1" xref="S2.SS4.p3.3.m3.1.2.3.1.3.1.cmml">−</mo><mn id="S2.SS4.p3.3.m3.1.2.3.1.3.3" xref="S2.SS4.p3.3.m3.1.2.3.1.3.3.cmml">1</mn></mrow></msubsup><mrow id="S2.SS4.p3.3.m3.1.2.3.2" xref="S2.SS4.p3.3.m3.1.2.3.2.cmml"><msup id="S2.SS4.p3.3.m3.1.2.3.2.2" xref="S2.SS4.p3.3.m3.1.2.3.2.2.cmml"><mn id="S2.SS4.p3.3.m3.1.2.3.2.2.2" xref="S2.SS4.p3.3.m3.1.2.3.2.2.2.cmml">2</mn><mi id="S2.SS4.p3.3.m3.1.2.3.2.2.3" xref="S2.SS4.p3.3.m3.1.2.3.2.2.3.cmml">i</mi></msup><mo id="S2.SS4.p3.3.m3.1.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS4.p3.3.m3.1.2.3.2.1.cmml">⋅</mo><msub id="S2.SS4.p3.3.m3.1.2.3.2.3" xref="S2.SS4.p3.3.m3.1.2.3.2.3.cmml"><mi id="S2.SS4.p3.3.m3.1.2.3.2.3.2" xref="S2.SS4.p3.3.m3.1.2.3.2.3.2.cmml">w</mi><mrow id="S2.SS4.p3.3.m3.1.2.3.2.3.3" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.cmml"><mi id="S2.SS4.p3.3.m3.1.2.3.2.3.3.2" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.2.cmml">n</mi><mo id="S2.SS4.p3.3.m3.1.2.3.2.3.3.1" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.1.cmml">−</mo><mi id="S2.SS4.p3.3.m3.1.2.3.2.3.3.3" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.3.cmml">i</mi></mrow></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.3.m3.1b"><apply id="S2.SS4.p3.3.m3.1.2.cmml" xref="S2.SS4.p3.3.m3.1.2"><eq id="S2.SS4.p3.3.m3.1.2.1.cmml" xref="S2.SS4.p3.3.m3.1.2.1"></eq><apply id="S2.SS4.p3.3.m3.1.2.2.cmml" xref="S2.SS4.p3.3.m3.1.2.2"><times id="S2.SS4.p3.3.m3.1.2.2.1.cmml" xref="S2.SS4.p3.3.m3.1.2.2.1"></times><ci id="S2.SS4.p3.3.m3.1.2.2.2.cmml" xref="S2.SS4.p3.3.m3.1.2.2.2">𝗇𝗎𝗆𝖻𝖾𝗋</ci><ci id="S2.SS4.p3.3.m3.1.1.cmml" xref="S2.SS4.p3.3.m3.1.1">𝑤</ci></apply><apply id="S2.SS4.p3.3.m3.1.2.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3"><apply id="S2.SS4.p3.3.m3.1.2.3.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1"><csymbol cd="ambiguous" id="S2.SS4.p3.3.m3.1.2.3.1.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1">superscript</csymbol><apply id="S2.SS4.p3.3.m3.1.2.3.1.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1"><csymbol cd="ambiguous" id="S2.SS4.p3.3.m3.1.2.3.1.2.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1">subscript</csymbol><sum id="S2.SS4.p3.3.m3.1.2.3.1.2.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.2.2"></sum><apply id="S2.SS4.p3.3.m3.1.2.3.1.2.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3"><eq id="S2.SS4.p3.3.m3.1.2.3.1.2.3.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.1"></eq><ci id="S2.SS4.p3.3.m3.1.2.3.1.2.3.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.2">𝑖</ci><cn id="S2.SS4.p3.3.m3.1.2.3.1.2.3.3.cmml" type="integer" xref="S2.SS4.p3.3.m3.1.2.3.1.2.3.3">0</cn></apply></apply><apply id="S2.SS4.p3.3.m3.1.2.3.1.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.3"><minus id="S2.SS4.p3.3.m3.1.2.3.1.3.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.3.1"></minus><ci id="S2.SS4.p3.3.m3.1.2.3.1.3.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.1.3.2">𝑛</ci><cn id="S2.SS4.p3.3.m3.1.2.3.1.3.3.cmml" type="integer" xref="S2.SS4.p3.3.m3.1.2.3.1.3.3">1</cn></apply></apply><apply id="S2.SS4.p3.3.m3.1.2.3.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2"><ci id="S2.SS4.p3.3.m3.1.2.3.2.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.1">⋅</ci><apply id="S2.SS4.p3.3.m3.1.2.3.2.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS4.p3.3.m3.1.2.3.2.2.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.2">superscript</csymbol><cn id="S2.SS4.p3.3.m3.1.2.3.2.2.2.cmml" type="integer" xref="S2.SS4.p3.3.m3.1.2.3.2.2.2">2</cn><ci id="S2.SS4.p3.3.m3.1.2.3.2.2.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.2.3">𝑖</ci></apply><apply id="S2.SS4.p3.3.m3.1.2.3.2.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3"><csymbol cd="ambiguous" id="S2.SS4.p3.3.m3.1.2.3.2.3.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3">subscript</csymbol><ci id="S2.SS4.p3.3.m3.1.2.3.2.3.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3.2">𝑤</ci><apply id="S2.SS4.p3.3.m3.1.2.3.2.3.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3"><minus id="S2.SS4.p3.3.m3.1.2.3.2.3.3.1.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.1"></minus><ci id="S2.SS4.p3.3.m3.1.2.3.2.3.3.2.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.2">𝑛</ci><ci id="S2.SS4.p3.3.m3.1.2.3.2.3.3.3.cmml" xref="S2.SS4.p3.3.m3.1.2.3.2.3.3.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.3.m3.1c">\mathsf{number}(w)=\sum_{i=0}^{n-1}2^{i}\cdot w_{n-i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.3.m3.1d">sansserif_number ( italic_w ) = ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ⋅ italic_w start_POSTSUBSCRIPT italic_n - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the number in <math alttext="\{0,\ldots,2^{n}-1\}" class="ltx_Math" display="inline" id="S2.SS4.p3.4.m4.3"><semantics id="S2.SS4.p3.4.m4.3a"><mrow id="S2.SS4.p3.4.m4.3.3.1" xref="S2.SS4.p3.4.m4.3.3.2.cmml"><mo id="S2.SS4.p3.4.m4.3.3.1.2" stretchy="false" xref="S2.SS4.p3.4.m4.3.3.2.cmml">{</mo><mn id="S2.SS4.p3.4.m4.1.1" xref="S2.SS4.p3.4.m4.1.1.cmml">0</mn><mo id="S2.SS4.p3.4.m4.3.3.1.3" xref="S2.SS4.p3.4.m4.3.3.2.cmml">,</mo><mi id="S2.SS4.p3.4.m4.2.2" mathvariant="normal" xref="S2.SS4.p3.4.m4.2.2.cmml">…</mi><mo id="S2.SS4.p3.4.m4.3.3.1.4" xref="S2.SS4.p3.4.m4.3.3.2.cmml">,</mo><mrow id="S2.SS4.p3.4.m4.3.3.1.1" xref="S2.SS4.p3.4.m4.3.3.1.1.cmml"><msup id="S2.SS4.p3.4.m4.3.3.1.1.2" xref="S2.SS4.p3.4.m4.3.3.1.1.2.cmml"><mn id="S2.SS4.p3.4.m4.3.3.1.1.2.2" xref="S2.SS4.p3.4.m4.3.3.1.1.2.2.cmml">2</mn><mi id="S2.SS4.p3.4.m4.3.3.1.1.2.3" xref="S2.SS4.p3.4.m4.3.3.1.1.2.3.cmml">n</mi></msup><mo id="S2.SS4.p3.4.m4.3.3.1.1.1" xref="S2.SS4.p3.4.m4.3.3.1.1.1.cmml">−</mo><mn id="S2.SS4.p3.4.m4.3.3.1.1.3" xref="S2.SS4.p3.4.m4.3.3.1.1.3.cmml">1</mn></mrow><mo id="S2.SS4.p3.4.m4.3.3.1.5" stretchy="false" xref="S2.SS4.p3.4.m4.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.4.m4.3b"><set id="S2.SS4.p3.4.m4.3.3.2.cmml" xref="S2.SS4.p3.4.m4.3.3.1"><cn id="S2.SS4.p3.4.m4.1.1.cmml" type="integer" xref="S2.SS4.p3.4.m4.1.1">0</cn><ci id="S2.SS4.p3.4.m4.2.2.cmml" xref="S2.SS4.p3.4.m4.2.2">…</ci><apply id="S2.SS4.p3.4.m4.3.3.1.1.cmml" xref="S2.SS4.p3.4.m4.3.3.1.1"><minus id="S2.SS4.p3.4.m4.3.3.1.1.1.cmml" xref="S2.SS4.p3.4.m4.3.3.1.1.1"></minus><apply id="S2.SS4.p3.4.m4.3.3.1.1.2.cmml" xref="S2.SS4.p3.4.m4.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.p3.4.m4.3.3.1.1.2.1.cmml" xref="S2.SS4.p3.4.m4.3.3.1.1.2">superscript</csymbol><cn id="S2.SS4.p3.4.m4.3.3.1.1.2.2.cmml" type="integer" xref="S2.SS4.p3.4.m4.3.3.1.1.2.2">2</cn><ci id="S2.SS4.p3.4.m4.3.3.1.1.2.3.cmml" xref="S2.SS4.p3.4.m4.3.3.1.1.2.3">𝑛</ci></apply><cn id="S2.SS4.p3.4.m4.3.3.1.1.3.cmml" type="integer" xref="S2.SS4.p3.4.m4.3.3.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.4.m4.3c">\{0,\ldots,2^{n}-1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.4.m4.3d">{ 0 , … , 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT - 1 }</annotation></semantics></math> encoded by <math alttext="w" class="ltx_Math" display="inline" id="S2.SS4.p3.5.m5.1"><semantics id="S2.SS4.p3.5.m5.1a"><mi id="S2.SS4.p3.5.m5.1.1" xref="S2.SS4.p3.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.5.m5.1b"><ci id="S2.SS4.p3.5.m5.1.1.cmml" xref="S2.SS4.p3.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.5.m5.1d">italic_w</annotation></semantics></math>. Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="S2.SS4.p3.6.m6.1"><semantics id="S2.SS4.p3.6.m6.1a"><mrow id="S2.SS4.p3.6.m6.1.1" xref="S2.SS4.p3.6.m6.1.1.cmml"><mi id="S2.SS4.p3.6.m6.1.1.2" xref="S2.SS4.p3.6.m6.1.1.2.cmml">N</mi><mo id="S2.SS4.p3.6.m6.1.1.1" xref="S2.SS4.p3.6.m6.1.1.1.cmml">=</mo><msup id="S2.SS4.p3.6.m6.1.1.3" xref="S2.SS4.p3.6.m6.1.1.3.cmml"><mn id="S2.SS4.p3.6.m6.1.1.3.2" xref="S2.SS4.p3.6.m6.1.1.3.2.cmml">2</mn><mi id="S2.SS4.p3.6.m6.1.1.3.3" xref="S2.SS4.p3.6.m6.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.6.m6.1b"><apply id="S2.SS4.p3.6.m6.1.1.cmml" xref="S2.SS4.p3.6.m6.1.1"><eq id="S2.SS4.p3.6.m6.1.1.1.cmml" xref="S2.SS4.p3.6.m6.1.1.1"></eq><ci id="S2.SS4.p3.6.m6.1.1.2.cmml" xref="S2.SS4.p3.6.m6.1.1.2">𝑁</ci><apply id="S2.SS4.p3.6.m6.1.1.3.cmml" xref="S2.SS4.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p3.6.m6.1.1.3.1.cmml" xref="S2.SS4.p3.6.m6.1.1.3">superscript</csymbol><cn id="S2.SS4.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S2.SS4.p3.6.m6.1.1.3.2">2</cn><ci id="S2.SS4.p3.6.m6.1.1.3.3.cmml" xref="S2.SS4.p3.6.m6.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.6.m6.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.6.m6.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and let <math alttext="\mathsf{binary}\colon[N]\to\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS4.p3.7.m7.3"><semantics id="S2.SS4.p3.7.m7.3a"><mrow id="S2.SS4.p3.7.m7.3.4" xref="S2.SS4.p3.7.m7.3.4.cmml"><mi id="S2.SS4.p3.7.m7.3.4.2" xref="S2.SS4.p3.7.m7.3.4.2.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo id="S2.SS4.p3.7.m7.3.4.1" lspace="0.278em" rspace="0.278em" xref="S2.SS4.p3.7.m7.3.4.1.cmml">:</mo><mrow id="S2.SS4.p3.7.m7.3.4.3" xref="S2.SS4.p3.7.m7.3.4.3.cmml"><mrow id="S2.SS4.p3.7.m7.3.4.3.2.2" xref="S2.SS4.p3.7.m7.3.4.3.2.1.cmml"><mo id="S2.SS4.p3.7.m7.3.4.3.2.2.1" stretchy="false" xref="S2.SS4.p3.7.m7.3.4.3.2.1.1.cmml">[</mo><mi id="S2.SS4.p3.7.m7.1.1" xref="S2.SS4.p3.7.m7.1.1.cmml">N</mi><mo id="S2.SS4.p3.7.m7.3.4.3.2.2.2" stretchy="false" xref="S2.SS4.p3.7.m7.3.4.3.2.1.1.cmml">]</mo></mrow><mo id="S2.SS4.p3.7.m7.3.4.3.1" stretchy="false" xref="S2.SS4.p3.7.m7.3.4.3.1.cmml">→</mo><msup id="S2.SS4.p3.7.m7.3.4.3.3" xref="S2.SS4.p3.7.m7.3.4.3.3.cmml"><mrow id="S2.SS4.p3.7.m7.3.4.3.3.2.2" xref="S2.SS4.p3.7.m7.3.4.3.3.2.1.cmml"><mo id="S2.SS4.p3.7.m7.3.4.3.3.2.2.1" stretchy="false" xref="S2.SS4.p3.7.m7.3.4.3.3.2.1.cmml">{</mo><mn id="S2.SS4.p3.7.m7.2.2" xref="S2.SS4.p3.7.m7.2.2.cmml">0</mn><mo id="S2.SS4.p3.7.m7.3.4.3.3.2.2.2" xref="S2.SS4.p3.7.m7.3.4.3.3.2.1.cmml">,</mo><mn id="S2.SS4.p3.7.m7.3.3" xref="S2.SS4.p3.7.m7.3.3.cmml">1</mn><mo id="S2.SS4.p3.7.m7.3.4.3.3.2.2.3" stretchy="false" xref="S2.SS4.p3.7.m7.3.4.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS4.p3.7.m7.3.4.3.3.3" xref="S2.SS4.p3.7.m7.3.4.3.3.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.7.m7.3b"><apply id="S2.SS4.p3.7.m7.3.4.cmml" xref="S2.SS4.p3.7.m7.3.4"><ci id="S2.SS4.p3.7.m7.3.4.1.cmml" xref="S2.SS4.p3.7.m7.3.4.1">:</ci><ci id="S2.SS4.p3.7.m7.3.4.2.cmml" xref="S2.SS4.p3.7.m7.3.4.2">𝖻𝗂𝗇𝖺𝗋𝗒</ci><apply id="S2.SS4.p3.7.m7.3.4.3.cmml" xref="S2.SS4.p3.7.m7.3.4.3"><ci id="S2.SS4.p3.7.m7.3.4.3.1.cmml" xref="S2.SS4.p3.7.m7.3.4.3.1">→</ci><apply id="S2.SS4.p3.7.m7.3.4.3.2.1.cmml" xref="S2.SS4.p3.7.m7.3.4.3.2.2"><csymbol cd="latexml" id="S2.SS4.p3.7.m7.3.4.3.2.1.1.cmml" xref="S2.SS4.p3.7.m7.3.4.3.2.2.1">delimited-[]</csymbol><ci id="S2.SS4.p3.7.m7.1.1.cmml" xref="S2.SS4.p3.7.m7.1.1">𝑁</ci></apply><apply id="S2.SS4.p3.7.m7.3.4.3.3.cmml" xref="S2.SS4.p3.7.m7.3.4.3.3"><csymbol cd="ambiguous" id="S2.SS4.p3.7.m7.3.4.3.3.1.cmml" xref="S2.SS4.p3.7.m7.3.4.3.3">superscript</csymbol><set id="S2.SS4.p3.7.m7.3.4.3.3.2.1.cmml" xref="S2.SS4.p3.7.m7.3.4.3.3.2.2"><cn id="S2.SS4.p3.7.m7.2.2.cmml" type="integer" xref="S2.SS4.p3.7.m7.2.2">0</cn><cn id="S2.SS4.p3.7.m7.3.3.cmml" type="integer" xref="S2.SS4.p3.7.m7.3.3">1</cn></set><ci id="S2.SS4.p3.7.m7.3.4.3.3.3.cmml" xref="S2.SS4.p3.7.m7.3.4.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.7.m7.3c">\mathsf{binary}\colon[N]\to\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.7.m7.3d">sansserif_binary : [ italic_N ] → { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be the <em class="ltx_emph ltx_font_italic" id="S2.SS4.p3.9.1">bijection</em> that maps the integer <math alttext="\mathsf{number}(w)+1" class="ltx_Math" display="inline" id="S2.SS4.p3.8.m8.1"><semantics id="S2.SS4.p3.8.m8.1a"><mrow id="S2.SS4.p3.8.m8.1.2" xref="S2.SS4.p3.8.m8.1.2.cmml"><mrow id="S2.SS4.p3.8.m8.1.2.2" xref="S2.SS4.p3.8.m8.1.2.2.cmml"><mi id="S2.SS4.p3.8.m8.1.2.2.2" xref="S2.SS4.p3.8.m8.1.2.2.2.cmml">𝗇𝗎𝗆𝖻𝖾𝗋</mi><mo id="S2.SS4.p3.8.m8.1.2.2.1" xref="S2.SS4.p3.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.p3.8.m8.1.2.2.3.2" xref="S2.SS4.p3.8.m8.1.2.2.cmml"><mo id="S2.SS4.p3.8.m8.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.p3.8.m8.1.2.2.cmml">(</mo><mi id="S2.SS4.p3.8.m8.1.1" xref="S2.SS4.p3.8.m8.1.1.cmml">w</mi><mo id="S2.SS4.p3.8.m8.1.2.2.3.2.2" stretchy="false" xref="S2.SS4.p3.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p3.8.m8.1.2.1" xref="S2.SS4.p3.8.m8.1.2.1.cmml">+</mo><mn id="S2.SS4.p3.8.m8.1.2.3" xref="S2.SS4.p3.8.m8.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.8.m8.1b"><apply id="S2.SS4.p3.8.m8.1.2.cmml" xref="S2.SS4.p3.8.m8.1.2"><plus id="S2.SS4.p3.8.m8.1.2.1.cmml" xref="S2.SS4.p3.8.m8.1.2.1"></plus><apply id="S2.SS4.p3.8.m8.1.2.2.cmml" xref="S2.SS4.p3.8.m8.1.2.2"><times id="S2.SS4.p3.8.m8.1.2.2.1.cmml" xref="S2.SS4.p3.8.m8.1.2.2.1"></times><ci id="S2.SS4.p3.8.m8.1.2.2.2.cmml" xref="S2.SS4.p3.8.m8.1.2.2.2">𝗇𝗎𝗆𝖻𝖾𝗋</ci><ci id="S2.SS4.p3.8.m8.1.1.cmml" xref="S2.SS4.p3.8.m8.1.1">𝑤</ci></apply><cn id="S2.SS4.p3.8.m8.1.2.3.cmml" type="integer" xref="S2.SS4.p3.8.m8.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.8.m8.1c">\mathsf{number}(w)+1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.8.m8.1d">sansserif_number ( italic_w ) + 1</annotation></semantics></math> to the corresponding string <math alttext="w\in\{0,1\}^{n}" class="ltx_Math" display="inline" id="S2.SS4.p3.9.m9.2"><semantics id="S2.SS4.p3.9.m9.2a"><mrow id="S2.SS4.p3.9.m9.2.3" xref="S2.SS4.p3.9.m9.2.3.cmml"><mi id="S2.SS4.p3.9.m9.2.3.2" xref="S2.SS4.p3.9.m9.2.3.2.cmml">w</mi><mo id="S2.SS4.p3.9.m9.2.3.1" xref="S2.SS4.p3.9.m9.2.3.1.cmml">∈</mo><msup id="S2.SS4.p3.9.m9.2.3.3" xref="S2.SS4.p3.9.m9.2.3.3.cmml"><mrow id="S2.SS4.p3.9.m9.2.3.3.2.2" xref="S2.SS4.p3.9.m9.2.3.3.2.1.cmml"><mo id="S2.SS4.p3.9.m9.2.3.3.2.2.1" stretchy="false" xref="S2.SS4.p3.9.m9.2.3.3.2.1.cmml">{</mo><mn id="S2.SS4.p3.9.m9.1.1" xref="S2.SS4.p3.9.m9.1.1.cmml">0</mn><mo id="S2.SS4.p3.9.m9.2.3.3.2.2.2" xref="S2.SS4.p3.9.m9.2.3.3.2.1.cmml">,</mo><mn id="S2.SS4.p3.9.m9.2.2" xref="S2.SS4.p3.9.m9.2.2.cmml">1</mn><mo id="S2.SS4.p3.9.m9.2.3.3.2.2.3" stretchy="false" xref="S2.SS4.p3.9.m9.2.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS4.p3.9.m9.2.3.3.3" xref="S2.SS4.p3.9.m9.2.3.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.9.m9.2b"><apply id="S2.SS4.p3.9.m9.2.3.cmml" xref="S2.SS4.p3.9.m9.2.3"><in id="S2.SS4.p3.9.m9.2.3.1.cmml" xref="S2.SS4.p3.9.m9.2.3.1"></in><ci id="S2.SS4.p3.9.m9.2.3.2.cmml" xref="S2.SS4.p3.9.m9.2.3.2">𝑤</ci><apply id="S2.SS4.p3.9.m9.2.3.3.cmml" xref="S2.SS4.p3.9.m9.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.p3.9.m9.2.3.3.1.cmml" xref="S2.SS4.p3.9.m9.2.3.3">superscript</csymbol><set id="S2.SS4.p3.9.m9.2.3.3.2.1.cmml" xref="S2.SS4.p3.9.m9.2.3.3.2.2"><cn id="S2.SS4.p3.9.m9.1.1.cmml" type="integer" xref="S2.SS4.p3.9.m9.1.1">0</cn><cn id="S2.SS4.p3.9.m9.2.2.cmml" type="integer" xref="S2.SS4.p3.9.m9.2.2">1</cn></set><ci id="S2.SS4.p3.9.m9.2.3.3.3.cmml" xref="S2.SS4.p3.9.m9.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.9.m9.2c">w\in\{0,1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.9.m9.2d">italic_w ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem13.1.1.1">Lemma 13</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem13.2.2"> </span>(Tight transference from graph complexity to circuit complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem13.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem13.p1"> <p class="ltx_p" id="Thmtheorem13.p1.8"><span class="ltx_text ltx_font_italic" id="Thmtheorem13.p1.8.8">Let <math alttext="\langle[N]\times[N],\mathcal{G}_{N,N}\rangle" class="ltx_Math" display="inline" id="Thmtheorem13.p1.1.1.m1.6"><semantics id="Thmtheorem13.p1.1.1.m1.6a"><mrow id="Thmtheorem13.p1.1.1.m1.6.6.2" xref="Thmtheorem13.p1.1.1.m1.6.6.3.cmml"><mo id="Thmtheorem13.p1.1.1.m1.6.6.2.3" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.6.6.3.cmml">⟨</mo><mrow id="Thmtheorem13.p1.1.1.m1.5.5.1.1" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.cmml"><mrow id="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.2" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.1.cmml"><mo id="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.2.1" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.1.1.m1.3.3" xref="Thmtheorem13.p1.1.1.m1.3.3.cmml">N</mi><mo id="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem13.p1.1.1.m1.5.5.1.1.1" rspace="0.222em" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.1.cmml">×</mo><mrow id="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.2" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.1.cmml"><mo id="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.2.1" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.1.1.m1.4.4" xref="Thmtheorem13.p1.1.1.m1.4.4.cmml">N</mi><mo id="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.2.2" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="Thmtheorem13.p1.1.1.m1.6.6.2.4" xref="Thmtheorem13.p1.1.1.m1.6.6.3.cmml">,</mo><msub id="Thmtheorem13.p1.1.1.m1.6.6.2.2" xref="Thmtheorem13.p1.1.1.m1.6.6.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem13.p1.1.1.m1.6.6.2.2.2" xref="Thmtheorem13.p1.1.1.m1.6.6.2.2.2.cmml">𝒢</mi><mrow id="Thmtheorem13.p1.1.1.m1.2.2.2.4" xref="Thmtheorem13.p1.1.1.m1.2.2.2.3.cmml"><mi id="Thmtheorem13.p1.1.1.m1.1.1.1.1" xref="Thmtheorem13.p1.1.1.m1.1.1.1.1.cmml">N</mi><mo id="Thmtheorem13.p1.1.1.m1.2.2.2.4.1" xref="Thmtheorem13.p1.1.1.m1.2.2.2.3.cmml">,</mo><mi id="Thmtheorem13.p1.1.1.m1.2.2.2.2" xref="Thmtheorem13.p1.1.1.m1.2.2.2.2.cmml">N</mi></mrow></msub><mo id="Thmtheorem13.p1.1.1.m1.6.6.2.5" stretchy="false" xref="Thmtheorem13.p1.1.1.m1.6.6.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.1.1.m1.6b"><list id="Thmtheorem13.p1.1.1.m1.6.6.3.cmml" xref="Thmtheorem13.p1.1.1.m1.6.6.2"><apply id="Thmtheorem13.p1.1.1.m1.5.5.1.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1"><times id="Thmtheorem13.p1.1.1.m1.5.5.1.1.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.1"></times><apply id="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.2"><csymbol cd="latexml" id="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.1.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.1.1.m1.3.3.cmml" xref="Thmtheorem13.p1.1.1.m1.3.3">𝑁</ci></apply><apply id="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.2"><csymbol cd="latexml" id="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.1.1.cmml" xref="Thmtheorem13.p1.1.1.m1.5.5.1.1.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.1.1.m1.4.4.cmml" xref="Thmtheorem13.p1.1.1.m1.4.4">𝑁</ci></apply></apply><apply id="Thmtheorem13.p1.1.1.m1.6.6.2.2.cmml" xref="Thmtheorem13.p1.1.1.m1.6.6.2.2"><csymbol cd="ambiguous" id="Thmtheorem13.p1.1.1.m1.6.6.2.2.1.cmml" xref="Thmtheorem13.p1.1.1.m1.6.6.2.2">subscript</csymbol><ci id="Thmtheorem13.p1.1.1.m1.6.6.2.2.2.cmml" xref="Thmtheorem13.p1.1.1.m1.6.6.2.2.2">𝒢</ci><list id="Thmtheorem13.p1.1.1.m1.2.2.2.3.cmml" xref="Thmtheorem13.p1.1.1.m1.2.2.2.4"><ci id="Thmtheorem13.p1.1.1.m1.1.1.1.1.cmml" xref="Thmtheorem13.p1.1.1.m1.1.1.1.1">𝑁</ci><ci id="Thmtheorem13.p1.1.1.m1.2.2.2.2.cmml" xref="Thmtheorem13.p1.1.1.m1.2.2.2.2">𝑁</ci></list></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.1.1.m1.6c">\langle[N]\times[N],\mathcal{G}_{N,N}\rangle</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.1.1.m1.6d">⟨ [ italic_N ] × [ italic_N ] , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ⟩</annotation></semantics></math> and <math alttext="\langle\{0,1\}^{2n},\mathcal{B}_{2n}\rangle" class="ltx_Math" display="inline" id="Thmtheorem13.p1.2.2.m2.4"><semantics id="Thmtheorem13.p1.2.2.m2.4a"><mrow id="Thmtheorem13.p1.2.2.m2.4.4.2" xref="Thmtheorem13.p1.2.2.m2.4.4.3.cmml"><mo id="Thmtheorem13.p1.2.2.m2.4.4.2.3" stretchy="false" xref="Thmtheorem13.p1.2.2.m2.4.4.3.cmml">⟨</mo><msup id="Thmtheorem13.p1.2.2.m2.3.3.1.1" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.cmml"><mrow id="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.2" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.1.cmml"><mo id="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.2.1" stretchy="false" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.1.cmml">{</mo><mn id="Thmtheorem13.p1.2.2.m2.1.1" xref="Thmtheorem13.p1.2.2.m2.1.1.cmml">0</mn><mo id="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.2.2" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.1.cmml">,</mo><mn id="Thmtheorem13.p1.2.2.m2.2.2" xref="Thmtheorem13.p1.2.2.m2.2.2.cmml">1</mn><mo id="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.2.3" stretchy="false" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.cmml"><mn id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.2" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.2.cmml">2</mn><mo id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.1" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.1.cmml">⁢</mo><mi id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.3" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.3.cmml">n</mi></mrow></msup><mo id="Thmtheorem13.p1.2.2.m2.4.4.2.4" xref="Thmtheorem13.p1.2.2.m2.4.4.3.cmml">,</mo><msub id="Thmtheorem13.p1.2.2.m2.4.4.2.2" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem13.p1.2.2.m2.4.4.2.2.2" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.2.cmml">ℬ</mi><mrow id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.cmml"><mn id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.2" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.2.cmml">2</mn><mo id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.1" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.1.cmml">⁢</mo><mi id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.3" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.3.cmml">n</mi></mrow></msub><mo id="Thmtheorem13.p1.2.2.m2.4.4.2.5" stretchy="false" xref="Thmtheorem13.p1.2.2.m2.4.4.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.2.2.m2.4b"><list id="Thmtheorem13.p1.2.2.m2.4.4.3.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2"><apply id="Thmtheorem13.p1.2.2.m2.3.3.1.1.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="Thmtheorem13.p1.2.2.m2.3.3.1.1.1.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1">superscript</csymbol><set id="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.1.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.2.2"><cn id="Thmtheorem13.p1.2.2.m2.1.1.cmml" type="integer" xref="Thmtheorem13.p1.2.2.m2.1.1">0</cn><cn id="Thmtheorem13.p1.2.2.m2.2.2.cmml" type="integer" xref="Thmtheorem13.p1.2.2.m2.2.2">1</cn></set><apply id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3"><times id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.1.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.1"></times><cn id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.2.cmml" type="integer" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.2">2</cn><ci id="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.3.cmml" xref="Thmtheorem13.p1.2.2.m2.3.3.1.1.3.3">𝑛</ci></apply></apply><apply id="Thmtheorem13.p1.2.2.m2.4.4.2.2.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2"><csymbol cd="ambiguous" id="Thmtheorem13.p1.2.2.m2.4.4.2.2.1.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2">subscript</csymbol><ci id="Thmtheorem13.p1.2.2.m2.4.4.2.2.2.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.2">ℬ</ci><apply id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3"><times id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.1.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.1"></times><cn id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.2.cmml" type="integer" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.2">2</cn><ci id="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.3.cmml" xref="Thmtheorem13.p1.2.2.m2.4.4.2.2.3.3">𝑛</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.2.2.m2.4c">\langle\{0,1\}^{2n},\mathcal{B}_{2n}\rangle</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.2.2.m2.4d">⟨ { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be the discrete spaces corresponding to <math alttext="N\times N" class="ltx_Math" display="inline" id="Thmtheorem13.p1.3.3.m3.1"><semantics id="Thmtheorem13.p1.3.3.m3.1a"><mrow id="Thmtheorem13.p1.3.3.m3.1.1" xref="Thmtheorem13.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem13.p1.3.3.m3.1.1.2" xref="Thmtheorem13.p1.3.3.m3.1.1.2.cmml">N</mi><mo id="Thmtheorem13.p1.3.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem13.p1.3.3.m3.1.1.1.cmml">×</mo><mi id="Thmtheorem13.p1.3.3.m3.1.1.3" xref="Thmtheorem13.p1.3.3.m3.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.3.3.m3.1b"><apply id="Thmtheorem13.p1.3.3.m3.1.1.cmml" xref="Thmtheorem13.p1.3.3.m3.1.1"><times id="Thmtheorem13.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem13.p1.3.3.m3.1.1.1"></times><ci id="Thmtheorem13.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem13.p1.3.3.m3.1.1.2">𝑁</ci><ci id="Thmtheorem13.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem13.p1.3.3.m3.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.3.3.m3.1c">N\times N</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.3.3.m3.1d">italic_N × italic_N</annotation></semantics></math> graph complexity and <math alttext="2n" class="ltx_Math" display="inline" id="Thmtheorem13.p1.4.4.m4.1"><semantics id="Thmtheorem13.p1.4.4.m4.1a"><mrow id="Thmtheorem13.p1.4.4.m4.1.1" xref="Thmtheorem13.p1.4.4.m4.1.1.cmml"><mn id="Thmtheorem13.p1.4.4.m4.1.1.2" xref="Thmtheorem13.p1.4.4.m4.1.1.2.cmml">2</mn><mo id="Thmtheorem13.p1.4.4.m4.1.1.1" xref="Thmtheorem13.p1.4.4.m4.1.1.1.cmml">⁢</mo><mi id="Thmtheorem13.p1.4.4.m4.1.1.3" xref="Thmtheorem13.p1.4.4.m4.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.4.4.m4.1b"><apply id="Thmtheorem13.p1.4.4.m4.1.1.cmml" xref="Thmtheorem13.p1.4.4.m4.1.1"><times id="Thmtheorem13.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem13.p1.4.4.m4.1.1.1"></times><cn id="Thmtheorem13.p1.4.4.m4.1.1.2.cmml" type="integer" xref="Thmtheorem13.p1.4.4.m4.1.1.2">2</cn><ci id="Thmtheorem13.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem13.p1.4.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.4.4.m4.1c">2n</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.4.4.m4.1d">2 italic_n</annotation></semantics></math>-bit circuit complexity, respectively, where <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="Thmtheorem13.p1.5.5.m5.1"><semantics id="Thmtheorem13.p1.5.5.m5.1a"><mrow id="Thmtheorem13.p1.5.5.m5.1.1" xref="Thmtheorem13.p1.5.5.m5.1.1.cmml"><mi id="Thmtheorem13.p1.5.5.m5.1.1.2" xref="Thmtheorem13.p1.5.5.m5.1.1.2.cmml">N</mi><mo id="Thmtheorem13.p1.5.5.m5.1.1.1" xref="Thmtheorem13.p1.5.5.m5.1.1.1.cmml">=</mo><msup id="Thmtheorem13.p1.5.5.m5.1.1.3" xref="Thmtheorem13.p1.5.5.m5.1.1.3.cmml"><mn id="Thmtheorem13.p1.5.5.m5.1.1.3.2" xref="Thmtheorem13.p1.5.5.m5.1.1.3.2.cmml">2</mn><mi id="Thmtheorem13.p1.5.5.m5.1.1.3.3" xref="Thmtheorem13.p1.5.5.m5.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.5.5.m5.1b"><apply id="Thmtheorem13.p1.5.5.m5.1.1.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1"><eq id="Thmtheorem13.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1.1"></eq><ci id="Thmtheorem13.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1.2">𝑁</ci><apply id="Thmtheorem13.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem13.p1.5.5.m5.1.1.3.1.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1.3">superscript</csymbol><cn id="Thmtheorem13.p1.5.5.m5.1.1.3.2.cmml" type="integer" xref="Thmtheorem13.p1.5.5.m5.1.1.3.2">2</cn><ci id="Thmtheorem13.p1.5.5.m5.1.1.3.3.cmml" xref="Thmtheorem13.p1.5.5.m5.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.5.5.m5.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.5.5.m5.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Moreover, let <math alttext="\phi\colon[N]\times[N]\to\{0,1\}^{2n}" class="ltx_Math" display="inline" id="Thmtheorem13.p1.6.6.m6.4"><semantics id="Thmtheorem13.p1.6.6.m6.4a"><mrow id="Thmtheorem13.p1.6.6.m6.4.5" xref="Thmtheorem13.p1.6.6.m6.4.5.cmml"><mi id="Thmtheorem13.p1.6.6.m6.4.5.2" xref="Thmtheorem13.p1.6.6.m6.4.5.2.cmml">ϕ</mi><mo id="Thmtheorem13.p1.6.6.m6.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem13.p1.6.6.m6.4.5.1.cmml">:</mo><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3" xref="Thmtheorem13.p1.6.6.m6.4.5.3.cmml"><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.cmml"><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.1.cmml"><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.6.6.m6.1.1" xref="Thmtheorem13.p1.6.6.m6.1.1.cmml">N</mi><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.2.1" rspace="0.222em" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.1.cmml">×</mo><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.1.cmml"><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.2.1" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.6.6.m6.2.2" xref="Thmtheorem13.p1.6.6.m6.2.2.cmml">N</mi><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.2.2" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.1" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.1.cmml">→</mo><msup id="Thmtheorem13.p1.6.6.m6.4.5.3.3" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.cmml"><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.1.cmml"><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.2.1" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.1.cmml">{</mo><mn id="Thmtheorem13.p1.6.6.m6.3.3" xref="Thmtheorem13.p1.6.6.m6.3.3.cmml">0</mn><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.2.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.1.cmml">,</mo><mn id="Thmtheorem13.p1.6.6.m6.4.4" xref="Thmtheorem13.p1.6.6.m6.4.4.cmml">1</mn><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.2.3" stretchy="false" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.cmml"><mn id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.2" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.2.cmml">2</mn><mo id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.1" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.1.cmml">⁢</mo><mi id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.3" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.3.cmml">n</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.6.6.m6.4b"><apply id="Thmtheorem13.p1.6.6.m6.4.5.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5"><ci id="Thmtheorem13.p1.6.6.m6.4.5.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.1">:</ci><ci id="Thmtheorem13.p1.6.6.m6.4.5.2.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.2">italic-ϕ</ci><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3"><ci id="Thmtheorem13.p1.6.6.m6.4.5.3.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.1">→</ci><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.2.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2"><times id="Thmtheorem13.p1.6.6.m6.4.5.3.2.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.1"></times><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.2"><csymbol cd="latexml" id="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.1.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.6.6.m6.1.1.cmml" xref="Thmtheorem13.p1.6.6.m6.1.1">𝑁</ci></apply><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.2"><csymbol cd="latexml" id="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.1.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.2.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.6.6.m6.2.2.cmml" xref="Thmtheorem13.p1.6.6.m6.2.2">𝑁</ci></apply></apply><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.3.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3"><csymbol cd="ambiguous" id="Thmtheorem13.p1.6.6.m6.4.5.3.3.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3">superscript</csymbol><set id="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.2.2"><cn id="Thmtheorem13.p1.6.6.m6.3.3.cmml" type="integer" xref="Thmtheorem13.p1.6.6.m6.3.3">0</cn><cn id="Thmtheorem13.p1.6.6.m6.4.4.cmml" type="integer" xref="Thmtheorem13.p1.6.6.m6.4.4">1</cn></set><apply id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3"><times id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.1.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.1"></times><cn id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.2.cmml" type="integer" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.2">2</cn><ci id="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.3.cmml" xref="Thmtheorem13.p1.6.6.m6.4.5.3.3.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.6.6.m6.4c">\phi\colon[N]\times[N]\to\{0,1\}^{2n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.6.6.m6.4d">italic_ϕ : [ italic_N ] × [ italic_N ] → { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be the bijective map defined by <math alttext="\phi(u,v)\stackrel{{\scriptstyle\rm def}}{{=}}\mathsf{binary}(u)\mathsf{binary% }(v)" class="ltx_Math" display="inline" id="Thmtheorem13.p1.7.7.m7.4"><semantics id="Thmtheorem13.p1.7.7.m7.4a"><mrow id="Thmtheorem13.p1.7.7.m7.4.5" xref="Thmtheorem13.p1.7.7.m7.4.5.cmml"><mrow id="Thmtheorem13.p1.7.7.m7.4.5.2" xref="Thmtheorem13.p1.7.7.m7.4.5.2.cmml"><mi id="Thmtheorem13.p1.7.7.m7.4.5.2.2" xref="Thmtheorem13.p1.7.7.m7.4.5.2.2.cmml">ϕ</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.2.1" xref="Thmtheorem13.p1.7.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="Thmtheorem13.p1.7.7.m7.4.5.2.3.2" xref="Thmtheorem13.p1.7.7.m7.4.5.2.3.1.cmml"><mo id="Thmtheorem13.p1.7.7.m7.4.5.2.3.2.1" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.2.3.1.cmml">(</mo><mi id="Thmtheorem13.p1.7.7.m7.1.1" xref="Thmtheorem13.p1.7.7.m7.1.1.cmml">u</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.2.3.2.2" xref="Thmtheorem13.p1.7.7.m7.4.5.2.3.1.cmml">,</mo><mi id="Thmtheorem13.p1.7.7.m7.2.2" xref="Thmtheorem13.p1.7.7.m7.2.2.cmml">v</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.2.3.2.3" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.2.3.1.cmml">)</mo></mrow></mrow><mover id="Thmtheorem13.p1.7.7.m7.4.5.1" xref="Thmtheorem13.p1.7.7.m7.4.5.1.cmml"><mo id="Thmtheorem13.p1.7.7.m7.4.5.1.2" xref="Thmtheorem13.p1.7.7.m7.4.5.1.2.cmml">=</mo><mi id="Thmtheorem13.p1.7.7.m7.4.5.1.3" xref="Thmtheorem13.p1.7.7.m7.4.5.1.3.cmml">def</mi></mover><mrow id="Thmtheorem13.p1.7.7.m7.4.5.3" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml"><mi id="Thmtheorem13.p1.7.7.m7.4.5.3.2" xref="Thmtheorem13.p1.7.7.m7.4.5.3.2.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.1" xref="Thmtheorem13.p1.7.7.m7.4.5.3.1.cmml">⁢</mo><mrow id="Thmtheorem13.p1.7.7.m7.4.5.3.3.2" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml"><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml">(</mo><mi id="Thmtheorem13.p1.7.7.m7.3.3" xref="Thmtheorem13.p1.7.7.m7.3.3.cmml">u</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.3.2.2" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml">)</mo></mrow><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.1a" xref="Thmtheorem13.p1.7.7.m7.4.5.3.1.cmml">⁢</mo><mi id="Thmtheorem13.p1.7.7.m7.4.5.3.4" xref="Thmtheorem13.p1.7.7.m7.4.5.3.4.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.1b" xref="Thmtheorem13.p1.7.7.m7.4.5.3.1.cmml">⁢</mo><mrow id="Thmtheorem13.p1.7.7.m7.4.5.3.5.2" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml"><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.5.2.1" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml">(</mo><mi id="Thmtheorem13.p1.7.7.m7.4.4" xref="Thmtheorem13.p1.7.7.m7.4.4.cmml">v</mi><mo id="Thmtheorem13.p1.7.7.m7.4.5.3.5.2.2" stretchy="false" xref="Thmtheorem13.p1.7.7.m7.4.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.7.7.m7.4b"><apply id="Thmtheorem13.p1.7.7.m7.4.5.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5"><apply id="Thmtheorem13.p1.7.7.m7.4.5.1.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.1"><csymbol cd="ambiguous" id="Thmtheorem13.p1.7.7.m7.4.5.1.1.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.1">superscript</csymbol><eq id="Thmtheorem13.p1.7.7.m7.4.5.1.2.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.1.2"></eq><ci id="Thmtheorem13.p1.7.7.m7.4.5.1.3.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.1.3">def</ci></apply><apply id="Thmtheorem13.p1.7.7.m7.4.5.2.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.2"><times id="Thmtheorem13.p1.7.7.m7.4.5.2.1.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.2.1"></times><ci id="Thmtheorem13.p1.7.7.m7.4.5.2.2.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.2.2">italic-ϕ</ci><interval closure="open" id="Thmtheorem13.p1.7.7.m7.4.5.2.3.1.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.2.3.2"><ci id="Thmtheorem13.p1.7.7.m7.1.1.cmml" xref="Thmtheorem13.p1.7.7.m7.1.1">𝑢</ci><ci id="Thmtheorem13.p1.7.7.m7.2.2.cmml" xref="Thmtheorem13.p1.7.7.m7.2.2">𝑣</ci></interval></apply><apply id="Thmtheorem13.p1.7.7.m7.4.5.3.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.3"><times id="Thmtheorem13.p1.7.7.m7.4.5.3.1.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.3.1"></times><ci id="Thmtheorem13.p1.7.7.m7.4.5.3.2.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.3.2">𝖻𝗂𝗇𝖺𝗋𝗒</ci><ci id="Thmtheorem13.p1.7.7.m7.3.3.cmml" xref="Thmtheorem13.p1.7.7.m7.3.3">𝑢</ci><ci id="Thmtheorem13.p1.7.7.m7.4.5.3.4.cmml" xref="Thmtheorem13.p1.7.7.m7.4.5.3.4">𝖻𝗂𝗇𝖺𝗋𝗒</ci><ci id="Thmtheorem13.p1.7.7.m7.4.4.cmml" xref="Thmtheorem13.p1.7.7.m7.4.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.7.7.m7.4c">\phi(u,v)\stackrel{{\scriptstyle\rm def}}{{=}}\mathsf{binary}(u)\mathsf{binary% }(v)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.7.7.m7.4d">italic_ϕ ( italic_u , italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP sansserif_binary ( italic_u ) sansserif_binary ( italic_v )</annotation></semantics></math>. For every <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem13.p1.8.8.m8.2"><semantics id="Thmtheorem13.p1.8.8.m8.2a"><mrow id="Thmtheorem13.p1.8.8.m8.2.3" xref="Thmtheorem13.p1.8.8.m8.2.3.cmml"><mi id="Thmtheorem13.p1.8.8.m8.2.3.2" xref="Thmtheorem13.p1.8.8.m8.2.3.2.cmml">G</mi><mo id="Thmtheorem13.p1.8.8.m8.2.3.1" xref="Thmtheorem13.p1.8.8.m8.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem13.p1.8.8.m8.2.3.3" xref="Thmtheorem13.p1.8.8.m8.2.3.3.cmml"><mrow id="Thmtheorem13.p1.8.8.m8.2.3.3.2.2" xref="Thmtheorem13.p1.8.8.m8.2.3.3.2.1.cmml"><mo id="Thmtheorem13.p1.8.8.m8.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem13.p1.8.8.m8.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.8.8.m8.1.1" xref="Thmtheorem13.p1.8.8.m8.1.1.cmml">N</mi><mo id="Thmtheorem13.p1.8.8.m8.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem13.p1.8.8.m8.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem13.p1.8.8.m8.2.3.3.1" rspace="0.222em" xref="Thmtheorem13.p1.8.8.m8.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem13.p1.8.8.m8.2.3.3.3.2" xref="Thmtheorem13.p1.8.8.m8.2.3.3.3.1.cmml"><mo id="Thmtheorem13.p1.8.8.m8.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem13.p1.8.8.m8.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem13.p1.8.8.m8.2.2" xref="Thmtheorem13.p1.8.8.m8.2.2.cmml">N</mi><mo id="Thmtheorem13.p1.8.8.m8.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem13.p1.8.8.m8.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem13.p1.8.8.m8.2b"><apply id="Thmtheorem13.p1.8.8.m8.2.3.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3"><subset id="Thmtheorem13.p1.8.8.m8.2.3.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.1"></subset><ci id="Thmtheorem13.p1.8.8.m8.2.3.2.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.2">𝐺</ci><apply id="Thmtheorem13.p1.8.8.m8.2.3.3.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3"><times id="Thmtheorem13.p1.8.8.m8.2.3.3.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3.1"></times><apply id="Thmtheorem13.p1.8.8.m8.2.3.3.2.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem13.p1.8.8.m8.2.3.3.2.1.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.8.8.m8.1.1.cmml" xref="Thmtheorem13.p1.8.8.m8.1.1">𝑁</ci></apply><apply id="Thmtheorem13.p1.8.8.m8.2.3.3.3.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem13.p1.8.8.m8.2.3.3.3.1.1.cmml" xref="Thmtheorem13.p1.8.8.m8.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem13.p1.8.8.m8.2.2.cmml" xref="Thmtheorem13.p1.8.8.m8.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem13.p1.8.8.m8.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem13.p1.8.8.m8.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex7"> <tbody><tr class="ltx_equation 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xref="S2.Ex7.m1.4.4.1.1.2.1.1.1.3.2">𝒢</ci><list id="S2.Ex7.m1.2.2.2.3.cmml" xref="S2.Ex7.m1.2.2.2.4"><ci id="S2.Ex7.m1.1.1.1.1.cmml" xref="S2.Ex7.m1.1.1.1.1">𝑁</ci><ci id="S2.Ex7.m1.2.2.2.2.cmml" xref="S2.Ex7.m1.2.2.2.2">𝑁</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7.m1.4c">D_{\cap}(\phi(G)\mid\mathcal{B}_{2n})\;\geq\;D_{\cap}(G\mid\mathcal{G}_{N,N}).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.4d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_ϕ ( italic_G ) ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) ≥ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem13.p1.9"><span class="ltx_text ltx_font_italic" id="Thmtheorem13.p1.9.1">In particular, graph intersection complexity lower bounds yield circuit complexity lower bounds.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS4.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS4.2.p1"> <p class="ltx_p" id="S2.SS4.2.p1.12">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem12" title="Lemma 12. ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">12</span></a>, it is enough to verify that for each <math alttext="B\in\mathcal{B}_{2n}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.1.m1.1"><semantics id="S2.SS4.2.p1.1.m1.1a"><mrow id="S2.SS4.2.p1.1.m1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.cmml"><mi id="S2.SS4.2.p1.1.m1.1.1.2" xref="S2.SS4.2.p1.1.m1.1.1.2.cmml">B</mi><mo id="S2.SS4.2.p1.1.m1.1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S2.SS4.2.p1.1.m1.1.1.3" xref="S2.SS4.2.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.1.m1.1.1.3.2" xref="S2.SS4.2.p1.1.m1.1.1.3.2.cmml">ℬ</mi><mrow id="S2.SS4.2.p1.1.m1.1.1.3.3" xref="S2.SS4.2.p1.1.m1.1.1.3.3.cmml"><mn id="S2.SS4.2.p1.1.m1.1.1.3.3.2" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.cmml">2</mn><mo id="S2.SS4.2.p1.1.m1.1.1.3.3.1" xref="S2.SS4.2.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.1.m1.1.1.3.3.3" xref="S2.SS4.2.p1.1.m1.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.1.m1.1b"><apply id="S2.SS4.2.p1.1.m1.1.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1"><in id="S2.SS4.2.p1.1.m1.1.1.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.1"></in><ci id="S2.SS4.2.p1.1.m1.1.1.2.cmml" xref="S2.SS4.2.p1.1.m1.1.1.2">𝐵</ci><apply id="S2.SS4.2.p1.1.m1.1.1.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.2">ℬ</ci><apply id="S2.SS4.2.p1.1.m1.1.1.3.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3"><times id="S2.SS4.2.p1.1.m1.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3.1"></times><cn id="S2.SS4.2.p1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2">2</cn><ci id="S2.SS4.2.p1.1.m1.1.1.3.3.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.1.m1.1c">B\in\mathcal{B}_{2n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.1.m1.1d">italic_B ∈ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="D_{\cap}(\phi^{-1}(B)\mid\mathcal{G}_{N,N})=0" class="ltx_Math" display="inline" id="S2.SS4.2.p1.2.m2.4"><semantics id="S2.SS4.2.p1.2.m2.4a"><mrow id="S2.SS4.2.p1.2.m2.4.4" xref="S2.SS4.2.p1.2.m2.4.4.cmml"><mrow id="S2.SS4.2.p1.2.m2.4.4.1" xref="S2.SS4.2.p1.2.m2.4.4.1.cmml"><msub 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xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.cmml">−</mo><mn id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.2" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.1" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.3.2" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.cmml"><mo id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.cmml">(</mo><mi id="S2.SS4.2.p1.2.m2.3.3" xref="S2.SS4.2.p1.2.m2.3.3.cmml">B</mi><mo id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.1" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.1.cmml">∣</mo><msub id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.2" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS4.2.p1.2.m2.2.2.2.4" xref="S2.SS4.2.p1.2.m2.2.2.2.3.cmml"><mi id="S2.SS4.2.p1.2.m2.1.1.1.1" xref="S2.SS4.2.p1.2.m2.1.1.1.1.cmml">N</mi><mo id="S2.SS4.2.p1.2.m2.2.2.2.4.1" xref="S2.SS4.2.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS4.2.p1.2.m2.2.2.2.2" xref="S2.SS4.2.p1.2.m2.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS4.2.p1.2.m2.4.4.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.2.m2.4.4.2" xref="S2.SS4.2.p1.2.m2.4.4.2.cmml">=</mo><mn id="S2.SS4.2.p1.2.m2.4.4.3" xref="S2.SS4.2.p1.2.m2.4.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.2.m2.4b"><apply id="S2.SS4.2.p1.2.m2.4.4.cmml" xref="S2.SS4.2.p1.2.m2.4.4"><eq id="S2.SS4.2.p1.2.m2.4.4.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.2"></eq><apply id="S2.SS4.2.p1.2.m2.4.4.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1"><times id="S2.SS4.2.p1.2.m2.4.4.1.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.2"></times><apply id="S2.SS4.2.p1.2.m2.4.4.1.3.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.2.m2.4.4.1.3.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.2.m2.4.4.1.3.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.3.2">𝐷</ci><intersect id="S2.SS4.2.p1.2.m2.4.4.1.3.3.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.3.3"></intersect></apply><apply id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1"><csymbol cd="latexml" id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.1">conditional</csymbol><apply id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2"><times id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.1"></times><apply id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2">superscript</csymbol><ci id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.2">italic-ϕ</ci><apply id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3"><minus id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3"></minus><cn id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.2.cmml" type="integer" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.2.2.3.2">1</cn></apply></apply><ci id="S2.SS4.2.p1.2.m2.3.3.cmml" xref="S2.SS4.2.p1.2.m2.3.3">𝐵</ci></apply><apply id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p1.2.m2.4.4.1.1.1.1.3.2">𝒢</ci><list id="S2.SS4.2.p1.2.m2.2.2.2.3.cmml" xref="S2.SS4.2.p1.2.m2.2.2.2.4"><ci id="S2.SS4.2.p1.2.m2.1.1.1.1.cmml" xref="S2.SS4.2.p1.2.m2.1.1.1.1">𝑁</ci><ci id="S2.SS4.2.p1.2.m2.2.2.2.2.cmml" xref="S2.SS4.2.p1.2.m2.2.2.2.2">𝑁</ci></list></apply></apply></apply><cn id="S2.SS4.2.p1.2.m2.4.4.3.cmml" type="integer" xref="S2.SS4.2.p1.2.m2.4.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.2.m2.4c">D_{\cap}(\phi^{-1}(B)\mid\mathcal{G}_{N,N})=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.2.m2.4d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ) ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>. Recall from Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS1" title="2.2.1 Boolean circuit complexity ‣ 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.2.1</span></a> that <math alttext="\mathcal{B}_{2n}=\{B_{1},\ldots,B_{2n},B_{1}^{c},\ldots,B_{2n}^{c}\}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.3.m3.6"><semantics id="S2.SS4.2.p1.3.m3.6a"><mrow id="S2.SS4.2.p1.3.m3.6.6" xref="S2.SS4.2.p1.3.m3.6.6.cmml"><msub id="S2.SS4.2.p1.3.m3.6.6.6" xref="S2.SS4.2.p1.3.m3.6.6.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.3.m3.6.6.6.2" xref="S2.SS4.2.p1.3.m3.6.6.6.2.cmml">ℬ</mi><mrow id="S2.SS4.2.p1.3.m3.6.6.6.3" xref="S2.SS4.2.p1.3.m3.6.6.6.3.cmml"><mn id="S2.SS4.2.p1.3.m3.6.6.6.3.2" xref="S2.SS4.2.p1.3.m3.6.6.6.3.2.cmml">2</mn><mo id="S2.SS4.2.p1.3.m3.6.6.6.3.1" xref="S2.SS4.2.p1.3.m3.6.6.6.3.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.3.m3.6.6.6.3.3" 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xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.2">𝐵</ci><apply id="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.cmml" xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3"><times id="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.1.cmml" xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.1"></times><cn id="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.2.cmml" type="integer" xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.2">2</cn><ci id="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.3.cmml" xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.2.3.3">𝑛</ci></apply></apply><ci id="S2.SS4.2.p1.3.m3.6.6.4.4.4.3.cmml" xref="S2.SS4.2.p1.3.m3.6.6.4.4.4.3">𝑐</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.3.m3.6c">\mathcal{B}_{2n}=\{B_{1},\ldots,B_{2n},B_{1}^{c},\ldots,B_{2n}^{c}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.3.m3.6d">caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 1 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id="S2.SS4.2.p1.4.m4.4b"><apply id="S2.SS4.2.p1.4.m4.4.4.cmml" xref="S2.SS4.2.p1.4.m4.4.4"><eq id="S2.SS4.2.p1.4.m4.4.4.3.cmml" xref="S2.SS4.2.p1.4.m4.4.4.3"></eq><apply id="S2.SS4.2.p1.4.m4.4.4.4.cmml" xref="S2.SS4.2.p1.4.m4.4.4.4"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.4.4.4.1.cmml" xref="S2.SS4.2.p1.4.m4.4.4.4">subscript</csymbol><ci id="S2.SS4.2.p1.4.m4.4.4.4.2.cmml" xref="S2.SS4.2.p1.4.m4.4.4.4.2">𝐵</ci><ci id="S2.SS4.2.p1.4.m4.4.4.4.3.cmml" xref="S2.SS4.2.p1.4.m4.4.4.4.3">𝑖</ci></apply><apply id="S2.SS4.2.p1.4.m4.4.4.2.3.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2"><csymbol cd="latexml" id="S2.SS4.2.p1.4.m4.4.4.2.3.1.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.3">conditional-set</csymbol><apply id="S2.SS4.2.p1.4.m4.3.3.1.1.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1"><in id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1"></in><ci id="S2.SS4.2.p1.4.m4.3.3.1.1.1.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.2">𝑣</ci><apply id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3">superscript</csymbol><set id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.2.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.2.2"><cn id="S2.SS4.2.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS4.2.p1.4.m4.1.1">0</cn><cn id="S2.SS4.2.p1.4.m4.2.2.cmml" type="integer" xref="S2.SS4.2.p1.4.m4.2.2">1</cn></set><apply id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3"><times id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.1"></times><cn id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.2">2</cn><ci id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.3.3.3">𝑛</ci></apply></apply></apply><apply id="S2.SS4.2.p1.4.m4.4.4.2.2.2.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2"><eq id="S2.SS4.2.p1.4.m4.4.4.2.2.2.1.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.1"></eq><apply id="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.1.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.2">subscript</csymbol><ci id="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.2.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.2">𝑣</ci><ci id="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.3.cmml" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.2.3">𝑖</ci></apply><cn id="S2.SS4.2.p1.4.m4.4.4.2.2.2.3.cmml" type="integer" xref="S2.SS4.2.p1.4.m4.4.4.2.2.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.4.m4.4c">B_{i}=\{v\in\{0,1\}^{2n}\mid v_{i}=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.4.m4.4d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_v ∈ { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT ∣ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 }</annotation></semantics></math>. If <math alttext="B_{i}\in\mathcal{B}_{2n}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.5.m5.1"><semantics id="S2.SS4.2.p1.5.m5.1a"><mrow id="S2.SS4.2.p1.5.m5.1.1" xref="S2.SS4.2.p1.5.m5.1.1.cmml"><msub id="S2.SS4.2.p1.5.m5.1.1.2" xref="S2.SS4.2.p1.5.m5.1.1.2.cmml"><mi id="S2.SS4.2.p1.5.m5.1.1.2.2" xref="S2.SS4.2.p1.5.m5.1.1.2.2.cmml">B</mi><mi id="S2.SS4.2.p1.5.m5.1.1.2.3" xref="S2.SS4.2.p1.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS4.2.p1.5.m5.1.1.1" xref="S2.SS4.2.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S2.SS4.2.p1.5.m5.1.1.3" xref="S2.SS4.2.p1.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.5.m5.1.1.3.2" xref="S2.SS4.2.p1.5.m5.1.1.3.2.cmml">ℬ</mi><mrow id="S2.SS4.2.p1.5.m5.1.1.3.3" xref="S2.SS4.2.p1.5.m5.1.1.3.3.cmml"><mn id="S2.SS4.2.p1.5.m5.1.1.3.3.2" xref="S2.SS4.2.p1.5.m5.1.1.3.3.2.cmml">2</mn><mo id="S2.SS4.2.p1.5.m5.1.1.3.3.1" xref="S2.SS4.2.p1.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.5.m5.1.1.3.3.3" xref="S2.SS4.2.p1.5.m5.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.5.m5.1b"><apply id="S2.SS4.2.p1.5.m5.1.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1"><in id="S2.SS4.2.p1.5.m5.1.1.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1.1"></in><apply id="S2.SS4.2.p1.5.m5.1.1.2.cmml" xref="S2.SS4.2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.5.m5.1.1.2.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS4.2.p1.5.m5.1.1.2.2.cmml" xref="S2.SS4.2.p1.5.m5.1.1.2.2">𝐵</ci><ci id="S2.SS4.2.p1.5.m5.1.1.2.3.cmml" xref="S2.SS4.2.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S2.SS4.2.p1.5.m5.1.1.3.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.5.m5.1.1.3.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.5.m5.1.1.3.2.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3.2">ℬ</ci><apply id="S2.SS4.2.p1.5.m5.1.1.3.3.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3.3"><times id="S2.SS4.2.p1.5.m5.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3.3.1"></times><cn id="S2.SS4.2.p1.5.m5.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.5.m5.1.1.3.3.2">2</cn><ci id="S2.SS4.2.p1.5.m5.1.1.3.3.3.cmml" xref="S2.SS4.2.p1.5.m5.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.5.m5.1c">B_{i}\in\mathcal{B}_{2n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.5.m5.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math> corresponds to the positive literal <math alttext="x_{i}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.6.m6.1"><semantics id="S2.SS4.2.p1.6.m6.1a"><msub id="S2.SS4.2.p1.6.m6.1.1" xref="S2.SS4.2.p1.6.m6.1.1.cmml"><mi id="S2.SS4.2.p1.6.m6.1.1.2" xref="S2.SS4.2.p1.6.m6.1.1.2.cmml">x</mi><mi id="S2.SS4.2.p1.6.m6.1.1.3" xref="S2.SS4.2.p1.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.6.m6.1b"><apply id="S2.SS4.2.p1.6.m6.1.1.cmml" xref="S2.SS4.2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p1.6.m6.1.1.1.cmml" xref="S2.SS4.2.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS4.2.p1.6.m6.1.1.2.cmml" xref="S2.SS4.2.p1.6.m6.1.1.2">𝑥</ci><ci id="S2.SS4.2.p1.6.m6.1.1.3.cmml" xref="S2.SS4.2.p1.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.6.m6.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="\phi^{-1}(B_{i})" class="ltx_Math" display="inline" id="S2.SS4.2.p1.7.m7.1"><semantics id="S2.SS4.2.p1.7.m7.1a"><mrow id="S2.SS4.2.p1.7.m7.1.1" xref="S2.SS4.2.p1.7.m7.1.1.cmml"><msup id="S2.SS4.2.p1.7.m7.1.1.3" xref="S2.SS4.2.p1.7.m7.1.1.3.cmml"><mi id="S2.SS4.2.p1.7.m7.1.1.3.2" xref="S2.SS4.2.p1.7.m7.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.2.p1.7.m7.1.1.3.3" xref="S2.SS4.2.p1.7.m7.1.1.3.3.cmml"><mo id="S2.SS4.2.p1.7.m7.1.1.3.3a" xref="S2.SS4.2.p1.7.m7.1.1.3.3.cmml">−</mo><mn id="S2.SS4.2.p1.7.m7.1.1.3.3.2" xref="S2.SS4.2.p1.7.m7.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.2.p1.7.m7.1.1.2" xref="S2.SS4.2.p1.7.m7.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.7.m7.1.1.1.1" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S2.SS4.2.p1.7.m7.1.1.1.1.1" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS4.2.p1.7.m7.1.1.1.1.1.2" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.2.cmml">B</mi><mi id="S2.SS4.2.p1.7.m7.1.1.1.1.1.3" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS4.2.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.7.m7.1b"><apply id="S2.SS4.2.p1.7.m7.1.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1"><times id="S2.SS4.2.p1.7.m7.1.1.2.cmml" xref="S2.SS4.2.p1.7.m7.1.1.2"></times><apply id="S2.SS4.2.p1.7.m7.1.1.3.cmml" xref="S2.SS4.2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.7.m7.1.1.3.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1.3">superscript</csymbol><ci id="S2.SS4.2.p1.7.m7.1.1.3.2.cmml" xref="S2.SS4.2.p1.7.m7.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.2.p1.7.m7.1.1.3.3.cmml" xref="S2.SS4.2.p1.7.m7.1.1.3.3"><minus id="S2.SS4.2.p1.7.m7.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1.3.3"></minus><cn id="S2.SS4.2.p1.7.m7.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.7.m7.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.2.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p1.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.2">𝐵</ci><ci id="S2.SS4.2.p1.7.m7.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.7.m7.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.7.m7.1c">\phi^{-1}(B_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.7.m7.1d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is either a union of columns (when <math alttext="i>n" class="ltx_Math" display="inline" id="S2.SS4.2.p1.8.m8.1"><semantics id="S2.SS4.2.p1.8.m8.1a"><mrow id="S2.SS4.2.p1.8.m8.1.1" xref="S2.SS4.2.p1.8.m8.1.1.cmml"><mi id="S2.SS4.2.p1.8.m8.1.1.2" xref="S2.SS4.2.p1.8.m8.1.1.2.cmml">i</mi><mo id="S2.SS4.2.p1.8.m8.1.1.1" xref="S2.SS4.2.p1.8.m8.1.1.1.cmml">></mo><mi id="S2.SS4.2.p1.8.m8.1.1.3" xref="S2.SS4.2.p1.8.m8.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.8.m8.1b"><apply id="S2.SS4.2.p1.8.m8.1.1.cmml" xref="S2.SS4.2.p1.8.m8.1.1"><gt id="S2.SS4.2.p1.8.m8.1.1.1.cmml" xref="S2.SS4.2.p1.8.m8.1.1.1"></gt><ci id="S2.SS4.2.p1.8.m8.1.1.2.cmml" xref="S2.SS4.2.p1.8.m8.1.1.2">𝑖</ci><ci id="S2.SS4.2.p1.8.m8.1.1.3.cmml" xref="S2.SS4.2.p1.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.8.m8.1c">i>n</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.8.m8.1d">italic_i > italic_n</annotation></semantics></math>) or a union of rows (when <math alttext="i\leq n" class="ltx_Math" display="inline" id="S2.SS4.2.p1.9.m9.1"><semantics id="S2.SS4.2.p1.9.m9.1a"><mrow id="S2.SS4.2.p1.9.m9.1.1" xref="S2.SS4.2.p1.9.m9.1.1.cmml"><mi id="S2.SS4.2.p1.9.m9.1.1.2" xref="S2.SS4.2.p1.9.m9.1.1.2.cmml">i</mi><mo id="S2.SS4.2.p1.9.m9.1.1.1" xref="S2.SS4.2.p1.9.m9.1.1.1.cmml">≤</mo><mi id="S2.SS4.2.p1.9.m9.1.1.3" xref="S2.SS4.2.p1.9.m9.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.9.m9.1b"><apply id="S2.SS4.2.p1.9.m9.1.1.cmml" xref="S2.SS4.2.p1.9.m9.1.1"><leq id="S2.SS4.2.p1.9.m9.1.1.1.cmml" xref="S2.SS4.2.p1.9.m9.1.1.1"></leq><ci id="S2.SS4.2.p1.9.m9.1.1.2.cmml" xref="S2.SS4.2.p1.9.m9.1.1.2">𝑖</ci><ci id="S2.SS4.2.p1.9.m9.1.1.3.cmml" xref="S2.SS4.2.p1.9.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.9.m9.1c">i\leq n</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.9.m9.1d">italic_i ≤ italic_n</annotation></semantics></math>) in graph complexity (cf. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS2" title="2.2.2 Bipartite graph complexity ‣ 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.2.2</span></a>). Consequently, in this case <math alttext="D_{\cap}(\phi^{-1}(B_{i})\mid\mathcal{G}_{N,N})=0" class="ltx_Math" display="inline" id="S2.SS4.2.p1.10.m10.3"><semantics id="S2.SS4.2.p1.10.m10.3a"><mrow id="S2.SS4.2.p1.10.m10.3.3" xref="S2.SS4.2.p1.10.m10.3.3.cmml"><mrow id="S2.SS4.2.p1.10.m10.3.3.1" xref="S2.SS4.2.p1.10.m10.3.3.1.cmml"><msub id="S2.SS4.2.p1.10.m10.3.3.1.3" xref="S2.SS4.2.p1.10.m10.3.3.1.3.cmml"><mi id="S2.SS4.2.p1.10.m10.3.3.1.3.2" xref="S2.SS4.2.p1.10.m10.3.3.1.3.2.cmml">D</mi><mo id="S2.SS4.2.p1.10.m10.3.3.1.3.3" xref="S2.SS4.2.p1.10.m10.3.3.1.3.3.cmml">∩</mo></msub><mo id="S2.SS4.2.p1.10.m10.3.3.1.2" xref="S2.SS4.2.p1.10.m10.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.10.m10.3.3.1.1.1" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.cmml"><mrow id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.cmml"><msup id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.cmml"><mi id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.cmml"><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3a" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.cmml">−</mo><mn id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.2.cmml">B</mi><mi id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.3" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.2.cmml">∣</mo><msub id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.2" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.SS4.2.p1.10.m10.2.2.2.4" xref="S2.SS4.2.p1.10.m10.2.2.2.3.cmml"><mi id="S2.SS4.2.p1.10.m10.1.1.1.1" xref="S2.SS4.2.p1.10.m10.1.1.1.1.cmml">N</mi><mo id="S2.SS4.2.p1.10.m10.2.2.2.4.1" xref="S2.SS4.2.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S2.SS4.2.p1.10.m10.2.2.2.2" xref="S2.SS4.2.p1.10.m10.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.SS4.2.p1.10.m10.3.3.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.10.m10.3.3.2" xref="S2.SS4.2.p1.10.m10.3.3.2.cmml">=</mo><mn id="S2.SS4.2.p1.10.m10.3.3.3" xref="S2.SS4.2.p1.10.m10.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.10.m10.3b"><apply id="S2.SS4.2.p1.10.m10.3.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3"><eq id="S2.SS4.2.p1.10.m10.3.3.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.2"></eq><apply id="S2.SS4.2.p1.10.m10.3.3.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1"><times id="S2.SS4.2.p1.10.m10.3.3.1.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.2"></times><apply id="S2.SS4.2.p1.10.m10.3.3.1.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.10.m10.3.3.1.3.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.10.m10.3.3.1.3.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.3.2">𝐷</ci><intersect id="S2.SS4.2.p1.10.m10.3.3.1.3.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.3.3"></intersect></apply><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1"><csymbol cd="latexml" id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.2">conditional</csymbol><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1"><times id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.2"></times><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3"><minus id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3"></minus><cn id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.2">𝐵</ci><ci id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p1.10.m10.3.3.1.1.1.1.3.2">𝒢</ci><list id="S2.SS4.2.p1.10.m10.2.2.2.3.cmml" xref="S2.SS4.2.p1.10.m10.2.2.2.4"><ci id="S2.SS4.2.p1.10.m10.1.1.1.1.cmml" xref="S2.SS4.2.p1.10.m10.1.1.1.1">𝑁</ci><ci id="S2.SS4.2.p1.10.m10.2.2.2.2.cmml" xref="S2.SS4.2.p1.10.m10.2.2.2.2">𝑁</ci></list></apply></apply></apply><cn id="S2.SS4.2.p1.10.m10.3.3.3.cmml" type="integer" xref="S2.SS4.2.p1.10.m10.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.10.m10.3c">D_{\cap}(\phi^{-1}(B_{i})\mid\mathcal{G}_{N,N})=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.10.m10.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> by Facts <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem4" title="Fact 4. ‣ 2.1 Definitions and notation ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem6" title="Fact 6. ‣ 2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">6</span></a>. On the other hand, for a <math alttext="B_{i}^{c}\in\mathcal{B}_{2n}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.11.m11.1"><semantics id="S2.SS4.2.p1.11.m11.1a"><mrow id="S2.SS4.2.p1.11.m11.1.1" xref="S2.SS4.2.p1.11.m11.1.1.cmml"><msubsup id="S2.SS4.2.p1.11.m11.1.1.2" xref="S2.SS4.2.p1.11.m11.1.1.2.cmml"><mi id="S2.SS4.2.p1.11.m11.1.1.2.2.2" xref="S2.SS4.2.p1.11.m11.1.1.2.2.2.cmml">B</mi><mi id="S2.SS4.2.p1.11.m11.1.1.2.2.3" xref="S2.SS4.2.p1.11.m11.1.1.2.2.3.cmml">i</mi><mi id="S2.SS4.2.p1.11.m11.1.1.2.3" xref="S2.SS4.2.p1.11.m11.1.1.2.3.cmml">c</mi></msubsup><mo id="S2.SS4.2.p1.11.m11.1.1.1" xref="S2.SS4.2.p1.11.m11.1.1.1.cmml">∈</mo><msub id="S2.SS4.2.p1.11.m11.1.1.3" xref="S2.SS4.2.p1.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.2.p1.11.m11.1.1.3.2" xref="S2.SS4.2.p1.11.m11.1.1.3.2.cmml">ℬ</mi><mrow id="S2.SS4.2.p1.11.m11.1.1.3.3" xref="S2.SS4.2.p1.11.m11.1.1.3.3.cmml"><mn id="S2.SS4.2.p1.11.m11.1.1.3.3.2" xref="S2.SS4.2.p1.11.m11.1.1.3.3.2.cmml">2</mn><mo id="S2.SS4.2.p1.11.m11.1.1.3.3.1" xref="S2.SS4.2.p1.11.m11.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.11.m11.1.1.3.3.3" xref="S2.SS4.2.p1.11.m11.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.11.m11.1b"><apply id="S2.SS4.2.p1.11.m11.1.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1"><in id="S2.SS4.2.p1.11.m11.1.1.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1.1"></in><apply id="S2.SS4.2.p1.11.m11.1.1.2.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.11.m11.1.1.2.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2">superscript</csymbol><apply id="S2.SS4.2.p1.11.m11.1.1.2.2.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.11.m11.1.1.2.2.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2">subscript</csymbol><ci id="S2.SS4.2.p1.11.m11.1.1.2.2.2.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2.2.2">𝐵</ci><ci id="S2.SS4.2.p1.11.m11.1.1.2.2.3.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2.2.3">𝑖</ci></apply><ci id="S2.SS4.2.p1.11.m11.1.1.2.3.cmml" xref="S2.SS4.2.p1.11.m11.1.1.2.3">𝑐</ci></apply><apply id="S2.SS4.2.p1.11.m11.1.1.3.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.11.m11.1.1.3.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.11.m11.1.1.3.2.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3.2">ℬ</ci><apply id="S2.SS4.2.p1.11.m11.1.1.3.3.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3.3"><times id="S2.SS4.2.p1.11.m11.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3.3.1"></times><cn id="S2.SS4.2.p1.11.m11.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.11.m11.1.1.3.3.2">2</cn><ci id="S2.SS4.2.p1.11.m11.1.1.3.3.3.cmml" xref="S2.SS4.2.p1.11.m11.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.11.m11.1c">B_{i}^{c}\in\mathcal{B}_{2n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.11.m11.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, it is not hard to see that <math alttext="\phi^{-1}(B_{i}^{c})" class="ltx_Math" display="inline" id="S2.SS4.2.p1.12.m12.1"><semantics id="S2.SS4.2.p1.12.m12.1a"><mrow id="S2.SS4.2.p1.12.m12.1.1" xref="S2.SS4.2.p1.12.m12.1.1.cmml"><msup id="S2.SS4.2.p1.12.m12.1.1.3" xref="S2.SS4.2.p1.12.m12.1.1.3.cmml"><mi id="S2.SS4.2.p1.12.m12.1.1.3.2" xref="S2.SS4.2.p1.12.m12.1.1.3.2.cmml">ϕ</mi><mrow id="S2.SS4.2.p1.12.m12.1.1.3.3" xref="S2.SS4.2.p1.12.m12.1.1.3.3.cmml"><mo id="S2.SS4.2.p1.12.m12.1.1.3.3a" xref="S2.SS4.2.p1.12.m12.1.1.3.3.cmml">−</mo><mn id="S2.SS4.2.p1.12.m12.1.1.3.3.2" xref="S2.SS4.2.p1.12.m12.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS4.2.p1.12.m12.1.1.2" xref="S2.SS4.2.p1.12.m12.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.12.m12.1.1.1.1" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.12.m12.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.cmml">(</mo><msubsup id="S2.SS4.2.p1.12.m12.1.1.1.1.1" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.cmml"><mi id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.2" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.2.cmml">B</mi><mi id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.3" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.3.cmml">i</mi><mi id="S2.SS4.2.p1.12.m12.1.1.1.1.1.3" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.3.cmml">c</mi></msubsup><mo id="S2.SS4.2.p1.12.m12.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.12.m12.1b"><apply id="S2.SS4.2.p1.12.m12.1.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1"><times id="S2.SS4.2.p1.12.m12.1.1.2.cmml" xref="S2.SS4.2.p1.12.m12.1.1.2"></times><apply id="S2.SS4.2.p1.12.m12.1.1.3.cmml" xref="S2.SS4.2.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.12.m12.1.1.3.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1.3">superscript</csymbol><ci id="S2.SS4.2.p1.12.m12.1.1.3.2.cmml" xref="S2.SS4.2.p1.12.m12.1.1.3.2">italic-ϕ</ci><apply id="S2.SS4.2.p1.12.m12.1.1.3.3.cmml" xref="S2.SS4.2.p1.12.m12.1.1.3.3"><minus id="S2.SS4.2.p1.12.m12.1.1.3.3.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1.3.3"></minus><cn id="S2.SS4.2.p1.12.m12.1.1.3.3.2.cmml" type="integer" xref="S2.SS4.2.p1.12.m12.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS4.2.p1.12.m12.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p1.12.m12.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1">superscript</csymbol><apply id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.1.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.2.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.2">𝐵</ci><ci id="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.3.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S2.SS4.2.p1.12.m12.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.12.m12.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.12.m12.1c">\phi^{-1}(B_{i}^{c})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.12.m12.1d">italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT )</annotation></semantics></math> also corresponds to either a union of rows or a union of columns. This completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS4.p4"> <p class="ltx_p" id="S2.SS4.p4.1">An advantage of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem13" title="Lemma 13 (Tight transference from graph complexity to circuit complexity). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">13</span></a> over existing results connecting graph complexity and circuit complexity is that it offers a tighter connection between these two models by focusing on a convenient complexity measure (intersection complexity instead of circuit complexity).<span class="ltx_note ltx_role_footnote" id="footnote9"><sup class="ltx_note_mark">9</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">9</sup><span class="ltx_tag ltx_tag_note">9</span> In the Magnification Lemma of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite>, it is already implicitly shown that <math alttext="D_{\cap}(f_{G}\mid{\mathcal{B}}_{2n})\geq D_{\cap}(G\mid{\mathcal{G}}_{N,N})" class="ltx_Math" display="inline" id="footnote9.m1.4"><semantics id="footnote9.m1.4b"><mrow id="footnote9.m1.4.4" xref="footnote9.m1.4.4.cmml"><mrow id="footnote9.m1.3.3.1" xref="footnote9.m1.3.3.1.cmml"><msub id="footnote9.m1.3.3.1.3" xref="footnote9.m1.3.3.1.3.cmml"><mi id="footnote9.m1.3.3.1.3.2" xref="footnote9.m1.3.3.1.3.2.cmml">D</mi><mo id="footnote9.m1.3.3.1.3.3" xref="footnote9.m1.3.3.1.3.3.cmml">∩</mo></msub><mo id="footnote9.m1.3.3.1.2" xref="footnote9.m1.3.3.1.2.cmml">⁢</mo><mrow id="footnote9.m1.3.3.1.1.1" xref="footnote9.m1.3.3.1.1.1.1.cmml"><mo id="footnote9.m1.3.3.1.1.1.2" stretchy="false" xref="footnote9.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="footnote9.m1.3.3.1.1.1.1" xref="footnote9.m1.3.3.1.1.1.1.cmml"><msub id="footnote9.m1.3.3.1.1.1.1.2" xref="footnote9.m1.3.3.1.1.1.1.2.cmml"><mi id="footnote9.m1.3.3.1.1.1.1.2.2" xref="footnote9.m1.3.3.1.1.1.1.2.2.cmml">f</mi><mi id="footnote9.m1.3.3.1.1.1.1.2.3" xref="footnote9.m1.3.3.1.1.1.1.2.3.cmml">G</mi></msub><mo id="footnote9.m1.3.3.1.1.1.1.1" xref="footnote9.m1.3.3.1.1.1.1.1.cmml">∣</mo><msub id="footnote9.m1.3.3.1.1.1.1.3" xref="footnote9.m1.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m1.3.3.1.1.1.1.3.2" xref="footnote9.m1.3.3.1.1.1.1.3.2.cmml">ℬ</mi><mrow id="footnote9.m1.3.3.1.1.1.1.3.3" xref="footnote9.m1.3.3.1.1.1.1.3.3.cmml"><mn id="footnote9.m1.3.3.1.1.1.1.3.3.2" xref="footnote9.m1.3.3.1.1.1.1.3.3.2.cmml">2</mn><mo id="footnote9.m1.3.3.1.1.1.1.3.3.1" xref="footnote9.m1.3.3.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="footnote9.m1.3.3.1.1.1.1.3.3.3" xref="footnote9.m1.3.3.1.1.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><mo id="footnote9.m1.3.3.1.1.1.3" stretchy="false" xref="footnote9.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="footnote9.m1.4.4.3" xref="footnote9.m1.4.4.3.cmml">≥</mo><mrow id="footnote9.m1.4.4.2" xref="footnote9.m1.4.4.2.cmml"><msub id="footnote9.m1.4.4.2.3" xref="footnote9.m1.4.4.2.3.cmml"><mi id="footnote9.m1.4.4.2.3.2" xref="footnote9.m1.4.4.2.3.2.cmml">D</mi><mo id="footnote9.m1.4.4.2.3.3" xref="footnote9.m1.4.4.2.3.3.cmml">∩</mo></msub><mo id="footnote9.m1.4.4.2.2" xref="footnote9.m1.4.4.2.2.cmml">⁢</mo><mrow id="footnote9.m1.4.4.2.1.1" xref="footnote9.m1.4.4.2.1.1.1.cmml"><mo id="footnote9.m1.4.4.2.1.1.2" stretchy="false" xref="footnote9.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="footnote9.m1.4.4.2.1.1.1" xref="footnote9.m1.4.4.2.1.1.1.cmml"><mi id="footnote9.m1.4.4.2.1.1.1.2" xref="footnote9.m1.4.4.2.1.1.1.2.cmml">G</mi><mo id="footnote9.m1.4.4.2.1.1.1.1" xref="footnote9.m1.4.4.2.1.1.1.1.cmml">∣</mo><msub id="footnote9.m1.4.4.2.1.1.1.3" xref="footnote9.m1.4.4.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" 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id="footnote9.m1.3.3.1.3.2.cmml" xref="footnote9.m1.3.3.1.3.2">𝐷</ci><intersect id="footnote9.m1.3.3.1.3.3.cmml" xref="footnote9.m1.3.3.1.3.3"></intersect></apply><apply id="footnote9.m1.3.3.1.1.1.1.cmml" xref="footnote9.m1.3.3.1.1.1"><csymbol cd="latexml" id="footnote9.m1.3.3.1.1.1.1.1.cmml" xref="footnote9.m1.3.3.1.1.1.1.1">conditional</csymbol><apply id="footnote9.m1.3.3.1.1.1.1.2.cmml" xref="footnote9.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="footnote9.m1.3.3.1.1.1.1.2.1.cmml" xref="footnote9.m1.3.3.1.1.1.1.2">subscript</csymbol><ci id="footnote9.m1.3.3.1.1.1.1.2.2.cmml" xref="footnote9.m1.3.3.1.1.1.1.2.2">𝑓</ci><ci id="footnote9.m1.3.3.1.1.1.1.2.3.cmml" xref="footnote9.m1.3.3.1.1.1.1.2.3">𝐺</ci></apply><apply id="footnote9.m1.3.3.1.1.1.1.3.cmml" xref="footnote9.m1.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m1.3.3.1.1.1.1.3.1.cmml" xref="footnote9.m1.3.3.1.1.1.1.3">subscript</csymbol><ci id="footnote9.m1.3.3.1.1.1.1.3.2.cmml" xref="footnote9.m1.3.3.1.1.1.1.3.2">ℬ</ci><apply id="footnote9.m1.3.3.1.1.1.1.3.3.cmml" xref="footnote9.m1.3.3.1.1.1.1.3.3"><times id="footnote9.m1.3.3.1.1.1.1.3.3.1.cmml" xref="footnote9.m1.3.3.1.1.1.1.3.3.1"></times><cn id="footnote9.m1.3.3.1.1.1.1.3.3.2.cmml" type="integer" xref="footnote9.m1.3.3.1.1.1.1.3.3.2">2</cn><ci id="footnote9.m1.3.3.1.1.1.1.3.3.3.cmml" xref="footnote9.m1.3.3.1.1.1.1.3.3.3">𝑛</ci></apply></apply></apply></apply><apply id="footnote9.m1.4.4.2.cmml" xref="footnote9.m1.4.4.2"><times id="footnote9.m1.4.4.2.2.cmml" xref="footnote9.m1.4.4.2.2"></times><apply id="footnote9.m1.4.4.2.3.cmml" xref="footnote9.m1.4.4.2.3"><csymbol cd="ambiguous" id="footnote9.m1.4.4.2.3.1.cmml" xref="footnote9.m1.4.4.2.3">subscript</csymbol><ci id="footnote9.m1.4.4.2.3.2.cmml" xref="footnote9.m1.4.4.2.3.2">𝐷</ci><intersect id="footnote9.m1.4.4.2.3.3.cmml" xref="footnote9.m1.4.4.2.3.3"></intersect></apply><apply id="footnote9.m1.4.4.2.1.1.1.cmml" xref="footnote9.m1.4.4.2.1.1"><csymbol cd="latexml" id="footnote9.m1.4.4.2.1.1.1.1.cmml" xref="footnote9.m1.4.4.2.1.1.1.1">conditional</csymbol><ci id="footnote9.m1.4.4.2.1.1.1.2.cmml" xref="footnote9.m1.4.4.2.1.1.1.2">𝐺</ci><apply id="footnote9.m1.4.4.2.1.1.1.3.cmml" xref="footnote9.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m1.4.4.2.1.1.1.3.1.cmml" xref="footnote9.m1.4.4.2.1.1.1.3">subscript</csymbol><ci id="footnote9.m1.4.4.2.1.1.1.3.2.cmml" xref="footnote9.m1.4.4.2.1.1.1.3.2">𝒢</ci><list id="footnote9.m1.2.2.2.3.cmml" xref="footnote9.m1.2.2.2.4"><ci id="footnote9.m1.1.1.1.1.cmml" xref="footnote9.m1.1.1.1.1">𝑁</ci><ci id="footnote9.m1.2.2.2.2.cmml" xref="footnote9.m1.2.2.2.2">𝑁</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m1.4d">D_{\cap}(f_{G}\mid{\mathcal{B}}_{2n})\geq D_{\cap}(G\mid{\mathcal{G}}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="footnote9.m1.4e">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) ≥ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>. However, the literature in graph complexity focuses on the relationship between <math alttext="D(f_{G}\mid{\mathcal{B}}_{2n})" class="ltx_Math" display="inline" id="footnote9.m2.1"><semantics id="footnote9.m2.1b"><mrow id="footnote9.m2.1.1" xref="footnote9.m2.1.1.cmml"><mi id="footnote9.m2.1.1.3" xref="footnote9.m2.1.1.3.cmml">D</mi><mo id="footnote9.m2.1.1.2" xref="footnote9.m2.1.1.2.cmml">⁢</mo><mrow id="footnote9.m2.1.1.1.1" xref="footnote9.m2.1.1.1.1.1.cmml"><mo id="footnote9.m2.1.1.1.1.2" stretchy="false" xref="footnote9.m2.1.1.1.1.1.cmml">(</mo><mrow id="footnote9.m2.1.1.1.1.1" xref="footnote9.m2.1.1.1.1.1.cmml"><msub id="footnote9.m2.1.1.1.1.1.2" xref="footnote9.m2.1.1.1.1.1.2.cmml"><mi id="footnote9.m2.1.1.1.1.1.2.2" xref="footnote9.m2.1.1.1.1.1.2.2.cmml">f</mi><mi id="footnote9.m2.1.1.1.1.1.2.3" xref="footnote9.m2.1.1.1.1.1.2.3.cmml">G</mi></msub><mo id="footnote9.m2.1.1.1.1.1.1" xref="footnote9.m2.1.1.1.1.1.1.cmml">∣</mo><msub id="footnote9.m2.1.1.1.1.1.3" xref="footnote9.m2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m2.1.1.1.1.1.3.2" xref="footnote9.m2.1.1.1.1.1.3.2.cmml">ℬ</mi><mrow id="footnote9.m2.1.1.1.1.1.3.3" xref="footnote9.m2.1.1.1.1.1.3.3.cmml"><mn id="footnote9.m2.1.1.1.1.1.3.3.2" xref="footnote9.m2.1.1.1.1.1.3.3.2.cmml">2</mn><mo id="footnote9.m2.1.1.1.1.1.3.3.1" xref="footnote9.m2.1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="footnote9.m2.1.1.1.1.1.3.3.3" xref="footnote9.m2.1.1.1.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><mo id="footnote9.m2.1.1.1.1.3" stretchy="false" xref="footnote9.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m2.1c"><apply id="footnote9.m2.1.1.cmml" xref="footnote9.m2.1.1"><times id="footnote9.m2.1.1.2.cmml" xref="footnote9.m2.1.1.2"></times><ci id="footnote9.m2.1.1.3.cmml" xref="footnote9.m2.1.1.3">𝐷</ci><apply id="footnote9.m2.1.1.1.1.1.cmml" xref="footnote9.m2.1.1.1.1"><csymbol cd="latexml" id="footnote9.m2.1.1.1.1.1.1.cmml" xref="footnote9.m2.1.1.1.1.1.1">conditional</csymbol><apply id="footnote9.m2.1.1.1.1.1.2.cmml" xref="footnote9.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="footnote9.m2.1.1.1.1.1.2.1.cmml" xref="footnote9.m2.1.1.1.1.1.2">subscript</csymbol><ci id="footnote9.m2.1.1.1.1.1.2.2.cmml" xref="footnote9.m2.1.1.1.1.1.2.2">𝑓</ci><ci id="footnote9.m2.1.1.1.1.1.2.3.cmml" xref="footnote9.m2.1.1.1.1.1.2.3">𝐺</ci></apply><apply id="footnote9.m2.1.1.1.1.1.3.cmml" xref="footnote9.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m2.1.1.1.1.1.3.1.cmml" xref="footnote9.m2.1.1.1.1.1.3">subscript</csymbol><ci id="footnote9.m2.1.1.1.1.1.3.2.cmml" xref="footnote9.m2.1.1.1.1.1.3.2">ℬ</ci><apply id="footnote9.m2.1.1.1.1.1.3.3.cmml" xref="footnote9.m2.1.1.1.1.1.3.3"><times id="footnote9.m2.1.1.1.1.1.3.3.1.cmml" xref="footnote9.m2.1.1.1.1.1.3.3.1"></times><cn id="footnote9.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="footnote9.m2.1.1.1.1.1.3.3.2">2</cn><ci id="footnote9.m2.1.1.1.1.1.3.3.3.cmml" xref="footnote9.m2.1.1.1.1.1.3.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m2.1d">D(f_{G}\mid{\mathcal{B}}_{2n})</annotation><annotation encoding="application/x-llamapun" id="footnote9.m2.1e">italic_D ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="D(G\mid{\mathcal{G}}_{N,N})" class="ltx_Math" display="inline" id="footnote9.m3.3"><semantics id="footnote9.m3.3b"><mrow id="footnote9.m3.3.3" xref="footnote9.m3.3.3.cmml"><mi id="footnote9.m3.3.3.3" xref="footnote9.m3.3.3.3.cmml">D</mi><mo id="footnote9.m3.3.3.2" xref="footnote9.m3.3.3.2.cmml">⁢</mo><mrow id="footnote9.m3.3.3.1.1" xref="footnote9.m3.3.3.1.1.1.cmml"><mo id="footnote9.m3.3.3.1.1.2" stretchy="false" xref="footnote9.m3.3.3.1.1.1.cmml">(</mo><mrow id="footnote9.m3.3.3.1.1.1" xref="footnote9.m3.3.3.1.1.1.cmml"><mi id="footnote9.m3.3.3.1.1.1.2" xref="footnote9.m3.3.3.1.1.1.2.cmml">G</mi><mo id="footnote9.m3.3.3.1.1.1.1" xref="footnote9.m3.3.3.1.1.1.1.cmml">∣</mo><msub id="footnote9.m3.3.3.1.1.1.3" xref="footnote9.m3.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m3.3.3.1.1.1.3.2" xref="footnote9.m3.3.3.1.1.1.3.2.cmml">𝒢</mi><mrow id="footnote9.m3.2.2.2.4" xref="footnote9.m3.2.2.2.3.cmml"><mi id="footnote9.m3.1.1.1.1" xref="footnote9.m3.1.1.1.1.cmml">N</mi><mo id="footnote9.m3.2.2.2.4.1" xref="footnote9.m3.2.2.2.3.cmml">,</mo><mi id="footnote9.m3.2.2.2.2" xref="footnote9.m3.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="footnote9.m3.3.3.1.1.3" stretchy="false" xref="footnote9.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m3.3c"><apply id="footnote9.m3.3.3.cmml" xref="footnote9.m3.3.3"><times id="footnote9.m3.3.3.2.cmml" xref="footnote9.m3.3.3.2"></times><ci id="footnote9.m3.3.3.3.cmml" xref="footnote9.m3.3.3.3">𝐷</ci><apply id="footnote9.m3.3.3.1.1.1.cmml" xref="footnote9.m3.3.3.1.1"><csymbol cd="latexml" id="footnote9.m3.3.3.1.1.1.1.cmml" xref="footnote9.m3.3.3.1.1.1.1">conditional</csymbol><ci id="footnote9.m3.3.3.1.1.1.2.cmml" xref="footnote9.m3.3.3.1.1.1.2">𝐺</ci><apply id="footnote9.m3.3.3.1.1.1.3.cmml" xref="footnote9.m3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m3.3.3.1.1.1.3.1.cmml" xref="footnote9.m3.3.3.1.1.1.3">subscript</csymbol><ci id="footnote9.m3.3.3.1.1.1.3.2.cmml" xref="footnote9.m3.3.3.1.1.1.3.2">𝒢</ci><list id="footnote9.m3.2.2.2.3.cmml" xref="footnote9.m3.2.2.2.4"><ci id="footnote9.m3.1.1.1.1.cmml" xref="footnote9.m3.1.1.1.1">𝑁</ci><ci id="footnote9.m3.2.2.2.2.cmml" xref="footnote9.m3.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m3.3d">D(G\mid{\mathcal{G}}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="footnote9.m3.3e">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>, where there is a constant factor loss. In particular, the best transference bound known is <math alttext="D(f_{G}\mid{\mathcal{B}}_{2n})\geq D(G\mid{\mathcal{G}}_{N,N})-(4+o(1))N" class="ltx_Math" display="inline" id="footnote9.m4.6"><semantics id="footnote9.m4.6b"><mrow id="footnote9.m4.6.6" xref="footnote9.m4.6.6.cmml"><mrow id="footnote9.m4.4.4.1" xref="footnote9.m4.4.4.1.cmml"><mi id="footnote9.m4.4.4.1.3" xref="footnote9.m4.4.4.1.3.cmml">D</mi><mo id="footnote9.m4.4.4.1.2" xref="footnote9.m4.4.4.1.2.cmml">⁢</mo><mrow id="footnote9.m4.4.4.1.1.1" xref="footnote9.m4.4.4.1.1.1.1.cmml"><mo id="footnote9.m4.4.4.1.1.1.2" stretchy="false" xref="footnote9.m4.4.4.1.1.1.1.cmml">(</mo><mrow id="footnote9.m4.4.4.1.1.1.1" xref="footnote9.m4.4.4.1.1.1.1.cmml"><msub id="footnote9.m4.4.4.1.1.1.1.2" xref="footnote9.m4.4.4.1.1.1.1.2.cmml"><mi id="footnote9.m4.4.4.1.1.1.1.2.2" xref="footnote9.m4.4.4.1.1.1.1.2.2.cmml">f</mi><mi id="footnote9.m4.4.4.1.1.1.1.2.3" xref="footnote9.m4.4.4.1.1.1.1.2.3.cmml">G</mi></msub><mo id="footnote9.m4.4.4.1.1.1.1.1" xref="footnote9.m4.4.4.1.1.1.1.1.cmml">∣</mo><msub id="footnote9.m4.4.4.1.1.1.1.3" xref="footnote9.m4.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m4.4.4.1.1.1.1.3.2" xref="footnote9.m4.4.4.1.1.1.1.3.2.cmml">ℬ</mi><mrow id="footnote9.m4.4.4.1.1.1.1.3.3" xref="footnote9.m4.4.4.1.1.1.1.3.3.cmml"><mn id="footnote9.m4.4.4.1.1.1.1.3.3.2" xref="footnote9.m4.4.4.1.1.1.1.3.3.2.cmml">2</mn><mo id="footnote9.m4.4.4.1.1.1.1.3.3.1" xref="footnote9.m4.4.4.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="footnote9.m4.4.4.1.1.1.1.3.3.3" xref="footnote9.m4.4.4.1.1.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><mo id="footnote9.m4.4.4.1.1.1.3" stretchy="false" xref="footnote9.m4.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="footnote9.m4.6.6.4" xref="footnote9.m4.6.6.4.cmml">≥</mo><mrow id="footnote9.m4.6.6.3" xref="footnote9.m4.6.6.3.cmml"><mrow id="footnote9.m4.5.5.2.1" xref="footnote9.m4.5.5.2.1.cmml"><mi id="footnote9.m4.5.5.2.1.3" xref="footnote9.m4.5.5.2.1.3.cmml">D</mi><mo id="footnote9.m4.5.5.2.1.2" xref="footnote9.m4.5.5.2.1.2.cmml">⁢</mo><mrow id="footnote9.m4.5.5.2.1.1.1" xref="footnote9.m4.5.5.2.1.1.1.1.cmml"><mo id="footnote9.m4.5.5.2.1.1.1.2" stretchy="false" xref="footnote9.m4.5.5.2.1.1.1.1.cmml">(</mo><mrow id="footnote9.m4.5.5.2.1.1.1.1" xref="footnote9.m4.5.5.2.1.1.1.1.cmml"><mi id="footnote9.m4.5.5.2.1.1.1.1.2" xref="footnote9.m4.5.5.2.1.1.1.1.2.cmml">G</mi><mo id="footnote9.m4.5.5.2.1.1.1.1.1" xref="footnote9.m4.5.5.2.1.1.1.1.1.cmml">∣</mo><msub id="footnote9.m4.5.5.2.1.1.1.1.3" xref="footnote9.m4.5.5.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m4.5.5.2.1.1.1.1.3.2" xref="footnote9.m4.5.5.2.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="footnote9.m4.2.2.2.4" xref="footnote9.m4.2.2.2.3.cmml"><mi id="footnote9.m4.1.1.1.1" xref="footnote9.m4.1.1.1.1.cmml">N</mi><mo id="footnote9.m4.2.2.2.4.1" xref="footnote9.m4.2.2.2.3.cmml">,</mo><mi id="footnote9.m4.2.2.2.2" xref="footnote9.m4.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="footnote9.m4.5.5.2.1.1.1.3" stretchy="false" xref="footnote9.m4.5.5.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="footnote9.m4.6.6.3.3" xref="footnote9.m4.6.6.3.3.cmml">−</mo><mrow id="footnote9.m4.6.6.3.2" xref="footnote9.m4.6.6.3.2.cmml"><mrow id="footnote9.m4.6.6.3.2.1.1" xref="footnote9.m4.6.6.3.2.1.1.1.cmml"><mo id="footnote9.m4.6.6.3.2.1.1.2" stretchy="false" xref="footnote9.m4.6.6.3.2.1.1.1.cmml">(</mo><mrow id="footnote9.m4.6.6.3.2.1.1.1" xref="footnote9.m4.6.6.3.2.1.1.1.cmml"><mn id="footnote9.m4.6.6.3.2.1.1.1.2" xref="footnote9.m4.6.6.3.2.1.1.1.2.cmml">4</mn><mo id="footnote9.m4.6.6.3.2.1.1.1.1" xref="footnote9.m4.6.6.3.2.1.1.1.1.cmml">+</mo><mrow id="footnote9.m4.6.6.3.2.1.1.1.3" xref="footnote9.m4.6.6.3.2.1.1.1.3.cmml"><mi id="footnote9.m4.6.6.3.2.1.1.1.3.2" xref="footnote9.m4.6.6.3.2.1.1.1.3.2.cmml">o</mi><mo id="footnote9.m4.6.6.3.2.1.1.1.3.1" xref="footnote9.m4.6.6.3.2.1.1.1.3.1.cmml">⁢</mo><mrow 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id="footnote9.m4.4.4.1.3.cmml" xref="footnote9.m4.4.4.1.3">𝐷</ci><apply id="footnote9.m4.4.4.1.1.1.1.cmml" xref="footnote9.m4.4.4.1.1.1"><csymbol cd="latexml" id="footnote9.m4.4.4.1.1.1.1.1.cmml" xref="footnote9.m4.4.4.1.1.1.1.1">conditional</csymbol><apply id="footnote9.m4.4.4.1.1.1.1.2.cmml" xref="footnote9.m4.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="footnote9.m4.4.4.1.1.1.1.2.1.cmml" xref="footnote9.m4.4.4.1.1.1.1.2">subscript</csymbol><ci id="footnote9.m4.4.4.1.1.1.1.2.2.cmml" xref="footnote9.m4.4.4.1.1.1.1.2.2">𝑓</ci><ci id="footnote9.m4.4.4.1.1.1.1.2.3.cmml" xref="footnote9.m4.4.4.1.1.1.1.2.3">𝐺</ci></apply><apply id="footnote9.m4.4.4.1.1.1.1.3.cmml" xref="footnote9.m4.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m4.4.4.1.1.1.1.3.1.cmml" xref="footnote9.m4.4.4.1.1.1.1.3">subscript</csymbol><ci id="footnote9.m4.4.4.1.1.1.1.3.2.cmml" xref="footnote9.m4.4.4.1.1.1.1.3.2">ℬ</ci><apply id="footnote9.m4.4.4.1.1.1.1.3.3.cmml" xref="footnote9.m4.4.4.1.1.1.1.3.3"><times id="footnote9.m4.4.4.1.1.1.1.3.3.1.cmml" xref="footnote9.m4.4.4.1.1.1.1.3.3.1"></times><cn id="footnote9.m4.4.4.1.1.1.1.3.3.2.cmml" type="integer" xref="footnote9.m4.4.4.1.1.1.1.3.3.2">2</cn><ci id="footnote9.m4.4.4.1.1.1.1.3.3.3.cmml" xref="footnote9.m4.4.4.1.1.1.1.3.3.3">𝑛</ci></apply></apply></apply></apply><apply id="footnote9.m4.6.6.3.cmml" xref="footnote9.m4.6.6.3"><minus id="footnote9.m4.6.6.3.3.cmml" xref="footnote9.m4.6.6.3.3"></minus><apply id="footnote9.m4.5.5.2.1.cmml" xref="footnote9.m4.5.5.2.1"><times id="footnote9.m4.5.5.2.1.2.cmml" xref="footnote9.m4.5.5.2.1.2"></times><ci id="footnote9.m4.5.5.2.1.3.cmml" xref="footnote9.m4.5.5.2.1.3">𝐷</ci><apply id="footnote9.m4.5.5.2.1.1.1.1.cmml" xref="footnote9.m4.5.5.2.1.1.1"><csymbol cd="latexml" id="footnote9.m4.5.5.2.1.1.1.1.1.cmml" xref="footnote9.m4.5.5.2.1.1.1.1.1">conditional</csymbol><ci id="footnote9.m4.5.5.2.1.1.1.1.2.cmml" xref="footnote9.m4.5.5.2.1.1.1.1.2">𝐺</ci><apply id="footnote9.m4.5.5.2.1.1.1.1.3.cmml" xref="footnote9.m4.5.5.2.1.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m4.5.5.2.1.1.1.1.3.1.cmml" xref="footnote9.m4.5.5.2.1.1.1.1.3">subscript</csymbol><ci id="footnote9.m4.5.5.2.1.1.1.1.3.2.cmml" xref="footnote9.m4.5.5.2.1.1.1.1.3.2">𝒢</ci><list id="footnote9.m4.2.2.2.3.cmml" xref="footnote9.m4.2.2.2.4"><ci id="footnote9.m4.1.1.1.1.cmml" xref="footnote9.m4.1.1.1.1">𝑁</ci><ci id="footnote9.m4.2.2.2.2.cmml" xref="footnote9.m4.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="footnote9.m4.6.6.3.2.cmml" xref="footnote9.m4.6.6.3.2"><times id="footnote9.m4.6.6.3.2.2.cmml" xref="footnote9.m4.6.6.3.2.2"></times><apply id="footnote9.m4.6.6.3.2.1.1.1.cmml" xref="footnote9.m4.6.6.3.2.1.1"><plus id="footnote9.m4.6.6.3.2.1.1.1.1.cmml" xref="footnote9.m4.6.6.3.2.1.1.1.1"></plus><cn id="footnote9.m4.6.6.3.2.1.1.1.2.cmml" type="integer" xref="footnote9.m4.6.6.3.2.1.1.1.2">4</cn><apply id="footnote9.m4.6.6.3.2.1.1.1.3.cmml" xref="footnote9.m4.6.6.3.2.1.1.1.3"><times id="footnote9.m4.6.6.3.2.1.1.1.3.1.cmml" xref="footnote9.m4.6.6.3.2.1.1.1.3.1"></times><ci id="footnote9.m4.6.6.3.2.1.1.1.3.2.cmml" xref="footnote9.m4.6.6.3.2.1.1.1.3.2">𝑜</ci><cn id="footnote9.m4.3.3.cmml" type="integer" xref="footnote9.m4.3.3">1</cn></apply></apply><ci id="footnote9.m4.6.6.3.2.3.cmml" xref="footnote9.m4.6.6.3.2.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m4.6d">D(f_{G}\mid{\mathcal{B}}_{2n})\geq D(G\mid{\mathcal{G}}_{N,N})-(4+o(1))N</annotation><annotation encoding="application/x-llamapun" id="footnote9.m4.6e">italic_D ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) ≥ italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) - ( 4 + italic_o ( 1 ) ) italic_N</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib8" title="">8</a>]</cite>, citing <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib2" title="">2</a>]</cite>). This means that only a <math alttext="\Omega(N)" class="ltx_Math" display="inline" id="footnote9.m5.1"><semantics id="footnote9.m5.1b"><mrow id="footnote9.m5.1.2" xref="footnote9.m5.1.2.cmml"><mi id="footnote9.m5.1.2.2" mathvariant="normal" xref="footnote9.m5.1.2.2.cmml">Ω</mi><mo id="footnote9.m5.1.2.1" xref="footnote9.m5.1.2.1.cmml">⁢</mo><mrow id="footnote9.m5.1.2.3.2" xref="footnote9.m5.1.2.cmml"><mo id="footnote9.m5.1.2.3.2.1" stretchy="false" xref="footnote9.m5.1.2.cmml">(</mo><mi id="footnote9.m5.1.1" xref="footnote9.m5.1.1.cmml">N</mi><mo id="footnote9.m5.1.2.3.2.2" stretchy="false" xref="footnote9.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m5.1c"><apply id="footnote9.m5.1.2.cmml" xref="footnote9.m5.1.2"><times id="footnote9.m5.1.2.1.cmml" xref="footnote9.m5.1.2.1"></times><ci id="footnote9.m5.1.2.2.cmml" xref="footnote9.m5.1.2.2">Ω</ci><ci id="footnote9.m5.1.1.cmml" xref="footnote9.m5.1.1">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m5.1d">\Omega(N)</annotation><annotation encoding="application/x-llamapun" id="footnote9.m5.1e">roman_Ω ( italic_N )</annotation></semantics></math> lower bound on <math alttext="D(G\mid{\mathcal{G}}_{N,N})" class="ltx_Math" display="inline" id="footnote9.m6.3"><semantics id="footnote9.m6.3b"><mrow id="footnote9.m6.3.3" xref="footnote9.m6.3.3.cmml"><mi id="footnote9.m6.3.3.3" xref="footnote9.m6.3.3.3.cmml">D</mi><mo id="footnote9.m6.3.3.2" xref="footnote9.m6.3.3.2.cmml">⁢</mo><mrow id="footnote9.m6.3.3.1.1" xref="footnote9.m6.3.3.1.1.1.cmml"><mo id="footnote9.m6.3.3.1.1.2" stretchy="false" xref="footnote9.m6.3.3.1.1.1.cmml">(</mo><mrow id="footnote9.m6.3.3.1.1.1" xref="footnote9.m6.3.3.1.1.1.cmml"><mi id="footnote9.m6.3.3.1.1.1.2" xref="footnote9.m6.3.3.1.1.1.2.cmml">G</mi><mo id="footnote9.m6.3.3.1.1.1.1" xref="footnote9.m6.3.3.1.1.1.1.cmml">∣</mo><msub id="footnote9.m6.3.3.1.1.1.3" xref="footnote9.m6.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m6.3.3.1.1.1.3.2" xref="footnote9.m6.3.3.1.1.1.3.2.cmml">𝒢</mi><mrow id="footnote9.m6.2.2.2.4" xref="footnote9.m6.2.2.2.3.cmml"><mi id="footnote9.m6.1.1.1.1" xref="footnote9.m6.1.1.1.1.cmml">N</mi><mo id="footnote9.m6.2.2.2.4.1" xref="footnote9.m6.2.2.2.3.cmml">,</mo><mi id="footnote9.m6.2.2.2.2" xref="footnote9.m6.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="footnote9.m6.3.3.1.1.3" stretchy="false" xref="footnote9.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m6.3c"><apply id="footnote9.m6.3.3.cmml" xref="footnote9.m6.3.3"><times id="footnote9.m6.3.3.2.cmml" xref="footnote9.m6.3.3.2"></times><ci id="footnote9.m6.3.3.3.cmml" xref="footnote9.m6.3.3.3">𝐷</ci><apply id="footnote9.m6.3.3.1.1.1.cmml" xref="footnote9.m6.3.3.1.1"><csymbol cd="latexml" id="footnote9.m6.3.3.1.1.1.1.cmml" xref="footnote9.m6.3.3.1.1.1.1">conditional</csymbol><ci id="footnote9.m6.3.3.1.1.1.2.cmml" xref="footnote9.m6.3.3.1.1.1.2">𝐺</ci><apply id="footnote9.m6.3.3.1.1.1.3.cmml" xref="footnote9.m6.3.3.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m6.3.3.1.1.1.3.1.cmml" xref="footnote9.m6.3.3.1.1.1.3">subscript</csymbol><ci id="footnote9.m6.3.3.1.1.1.3.2.cmml" xref="footnote9.m6.3.3.1.1.1.3.2">𝒢</ci><list id="footnote9.m6.2.2.2.3.cmml" xref="footnote9.m6.2.2.2.4"><ci id="footnote9.m6.1.1.1.1.cmml" xref="footnote9.m6.1.1.1.1">𝑁</ci><ci id="footnote9.m6.2.2.2.2.cmml" xref="footnote9.m6.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m6.3d">D(G\mid{\mathcal{G}}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="footnote9.m6.3e">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> would imply a meaningful bound on <math alttext="D(f_{G}\mid{\mathcal{B}}_{2n})" class="ltx_Math" display="inline" id="footnote9.m7.1"><semantics id="footnote9.m7.1b"><mrow id="footnote9.m7.1.1" xref="footnote9.m7.1.1.cmml"><mi id="footnote9.m7.1.1.3" xref="footnote9.m7.1.1.3.cmml">D</mi><mo id="footnote9.m7.1.1.2" xref="footnote9.m7.1.1.2.cmml">⁢</mo><mrow id="footnote9.m7.1.1.1.1" xref="footnote9.m7.1.1.1.1.1.cmml"><mo id="footnote9.m7.1.1.1.1.2" stretchy="false" xref="footnote9.m7.1.1.1.1.1.cmml">(</mo><mrow id="footnote9.m7.1.1.1.1.1" xref="footnote9.m7.1.1.1.1.1.cmml"><msub id="footnote9.m7.1.1.1.1.1.2" xref="footnote9.m7.1.1.1.1.1.2.cmml"><mi id="footnote9.m7.1.1.1.1.1.2.2" xref="footnote9.m7.1.1.1.1.1.2.2.cmml">f</mi><mi id="footnote9.m7.1.1.1.1.1.2.3" xref="footnote9.m7.1.1.1.1.1.2.3.cmml">G</mi></msub><mo id="footnote9.m7.1.1.1.1.1.1" xref="footnote9.m7.1.1.1.1.1.1.cmml">∣</mo><msub id="footnote9.m7.1.1.1.1.1.3" xref="footnote9.m7.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote9.m7.1.1.1.1.1.3.2" xref="footnote9.m7.1.1.1.1.1.3.2.cmml">ℬ</mi><mrow id="footnote9.m7.1.1.1.1.1.3.3" xref="footnote9.m7.1.1.1.1.1.3.3.cmml"><mn id="footnote9.m7.1.1.1.1.1.3.3.2" xref="footnote9.m7.1.1.1.1.1.3.3.2.cmml">2</mn><mo id="footnote9.m7.1.1.1.1.1.3.3.1" xref="footnote9.m7.1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="footnote9.m7.1.1.1.1.1.3.3.3" xref="footnote9.m7.1.1.1.1.1.3.3.3.cmml">n</mi></mrow></msub></mrow><mo id="footnote9.m7.1.1.1.1.3" stretchy="false" xref="footnote9.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m7.1c"><apply id="footnote9.m7.1.1.cmml" xref="footnote9.m7.1.1"><times id="footnote9.m7.1.1.2.cmml" xref="footnote9.m7.1.1.2"></times><ci id="footnote9.m7.1.1.3.cmml" xref="footnote9.m7.1.1.3">𝐷</ci><apply id="footnote9.m7.1.1.1.1.1.cmml" xref="footnote9.m7.1.1.1.1"><csymbol cd="latexml" id="footnote9.m7.1.1.1.1.1.1.cmml" xref="footnote9.m7.1.1.1.1.1.1">conditional</csymbol><apply id="footnote9.m7.1.1.1.1.1.2.cmml" xref="footnote9.m7.1.1.1.1.1.2"><csymbol cd="ambiguous" id="footnote9.m7.1.1.1.1.1.2.1.cmml" xref="footnote9.m7.1.1.1.1.1.2">subscript</csymbol><ci id="footnote9.m7.1.1.1.1.1.2.2.cmml" xref="footnote9.m7.1.1.1.1.1.2.2">𝑓</ci><ci id="footnote9.m7.1.1.1.1.1.2.3.cmml" xref="footnote9.m7.1.1.1.1.1.2.3">𝐺</ci></apply><apply id="footnote9.m7.1.1.1.1.1.3.cmml" xref="footnote9.m7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="footnote9.m7.1.1.1.1.1.3.1.cmml" xref="footnote9.m7.1.1.1.1.1.3">subscript</csymbol><ci id="footnote9.m7.1.1.1.1.1.3.2.cmml" xref="footnote9.m7.1.1.1.1.1.3.2">ℬ</ci><apply id="footnote9.m7.1.1.1.1.1.3.3.cmml" xref="footnote9.m7.1.1.1.1.1.3.3"><times id="footnote9.m7.1.1.1.1.1.3.3.1.cmml" xref="footnote9.m7.1.1.1.1.1.3.3.1"></times><cn id="footnote9.m7.1.1.1.1.1.3.3.2.cmml" type="integer" xref="footnote9.m7.1.1.1.1.1.3.3.2">2</cn><ci id="footnote9.m7.1.1.1.1.1.3.3.3.cmml" xref="footnote9.m7.1.1.1.1.1.3.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m7.1d">D(f_{G}\mid{\mathcal{B}}_{2n})</annotation><annotation encoding="application/x-llamapun" id="footnote9.m7.1e">italic_D ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, whereas our setting allows us to transfer a <math alttext="(1+\varepsilon)\log N" class="ltx_Math" display="inline" id="footnote9.m8.1"><semantics id="footnote9.m8.1b"><mrow id="footnote9.m8.1.1" xref="footnote9.m8.1.1.cmml"><mrow id="footnote9.m8.1.1.1.1" xref="footnote9.m8.1.1.1.1.1.cmml"><mo id="footnote9.m8.1.1.1.1.2" stretchy="false" xref="footnote9.m8.1.1.1.1.1.cmml">(</mo><mrow id="footnote9.m8.1.1.1.1.1" xref="footnote9.m8.1.1.1.1.1.cmml"><mn id="footnote9.m8.1.1.1.1.1.2" xref="footnote9.m8.1.1.1.1.1.2.cmml">1</mn><mo id="footnote9.m8.1.1.1.1.1.1" xref="footnote9.m8.1.1.1.1.1.1.cmml">+</mo><mi id="footnote9.m8.1.1.1.1.1.3" xref="footnote9.m8.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="footnote9.m8.1.1.1.1.3" stretchy="false" xref="footnote9.m8.1.1.1.1.1.cmml">)</mo></mrow><mo id="footnote9.m8.1.1.2" lspace="0.167em" xref="footnote9.m8.1.1.2.cmml">⁢</mo><mrow id="footnote9.m8.1.1.3" xref="footnote9.m8.1.1.3.cmml"><mi id="footnote9.m8.1.1.3.1" xref="footnote9.m8.1.1.3.1.cmml">log</mi><mo id="footnote9.m8.1.1.3b" lspace="0.167em" xref="footnote9.m8.1.1.3.cmml">⁡</mo><mi id="footnote9.m8.1.1.3.2" xref="footnote9.m8.1.1.3.2.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m8.1c"><apply id="footnote9.m8.1.1.cmml" xref="footnote9.m8.1.1"><times id="footnote9.m8.1.1.2.cmml" xref="footnote9.m8.1.1.2"></times><apply id="footnote9.m8.1.1.1.1.1.cmml" xref="footnote9.m8.1.1.1.1"><plus id="footnote9.m8.1.1.1.1.1.1.cmml" xref="footnote9.m8.1.1.1.1.1.1"></plus><cn id="footnote9.m8.1.1.1.1.1.2.cmml" type="integer" xref="footnote9.m8.1.1.1.1.1.2">1</cn><ci id="footnote9.m8.1.1.1.1.1.3.cmml" xref="footnote9.m8.1.1.1.1.1.3">𝜀</ci></apply><apply id="footnote9.m8.1.1.3.cmml" xref="footnote9.m8.1.1.3"><log id="footnote9.m8.1.1.3.1.cmml" xref="footnote9.m8.1.1.3.1"></log><ci id="footnote9.m8.1.1.3.2.cmml" xref="footnote9.m8.1.1.3.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m8.1d">(1+\varepsilon)\log N</annotation><annotation encoding="application/x-llamapun" id="footnote9.m8.1e">( 1 + italic_ε ) roman_log italic_N</annotation></semantics></math> graph complexity lower bound into a <math alttext="(1+\varepsilon)n" class="ltx_Math" display="inline" id="footnote9.m9.1"><semantics id="footnote9.m9.1b"><mrow id="footnote9.m9.1.1" xref="footnote9.m9.1.1.cmml"><mrow id="footnote9.m9.1.1.1.1" xref="footnote9.m9.1.1.1.1.1.cmml"><mo id="footnote9.m9.1.1.1.1.2" stretchy="false" xref="footnote9.m9.1.1.1.1.1.cmml">(</mo><mrow id="footnote9.m9.1.1.1.1.1" xref="footnote9.m9.1.1.1.1.1.cmml"><mn id="footnote9.m9.1.1.1.1.1.2" xref="footnote9.m9.1.1.1.1.1.2.cmml">1</mn><mo id="footnote9.m9.1.1.1.1.1.1" xref="footnote9.m9.1.1.1.1.1.1.cmml">+</mo><mi id="footnote9.m9.1.1.1.1.1.3" xref="footnote9.m9.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="footnote9.m9.1.1.1.1.3" stretchy="false" xref="footnote9.m9.1.1.1.1.1.cmml">)</mo></mrow><mo id="footnote9.m9.1.1.2" xref="footnote9.m9.1.1.2.cmml">⁢</mo><mi id="footnote9.m9.1.1.3" xref="footnote9.m9.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m9.1c"><apply id="footnote9.m9.1.1.cmml" xref="footnote9.m9.1.1"><times id="footnote9.m9.1.1.2.cmml" xref="footnote9.m9.1.1.2"></times><apply id="footnote9.m9.1.1.1.1.1.cmml" xref="footnote9.m9.1.1.1.1"><plus id="footnote9.m9.1.1.1.1.1.1.cmml" xref="footnote9.m9.1.1.1.1.1.1"></plus><cn id="footnote9.m9.1.1.1.1.1.2.cmml" type="integer" xref="footnote9.m9.1.1.1.1.1.2">1</cn><ci id="footnote9.m9.1.1.1.1.1.3.cmml" xref="footnote9.m9.1.1.1.1.1.3">𝜀</ci></apply><ci id="footnote9.m9.1.1.3.cmml" xref="footnote9.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m9.1d">(1+\varepsilon)n</annotation><annotation encoding="application/x-llamapun" id="footnote9.m9.1e">( 1 + italic_ε ) italic_n</annotation></semantics></math> circuit lower bound. </span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_remark" id="Thmtheorem14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem14.1.1.1">Remark 14</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem14.2.2"> </span>(Circuit lower bounds from graph complexity lower bounds)<span class="ltx_text ltx_font_bold" id="Thmtheorem14.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem14.p1"> <p class="ltx_p" id="Thmtheorem14.p1.27"><span class="ltx_text ltx_font_italic" id="Thmtheorem14.p1.27.27">Let <math alttext="C\geq 1" class="ltx_Math" display="inline" id="Thmtheorem14.p1.1.1.m1.1"><semantics id="Thmtheorem14.p1.1.1.m1.1a"><mrow id="Thmtheorem14.p1.1.1.m1.1.1" xref="Thmtheorem14.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem14.p1.1.1.m1.1.1.2" xref="Thmtheorem14.p1.1.1.m1.1.1.2.cmml">C</mi><mo id="Thmtheorem14.p1.1.1.m1.1.1.1" xref="Thmtheorem14.p1.1.1.m1.1.1.1.cmml">≥</mo><mn id="Thmtheorem14.p1.1.1.m1.1.1.3" xref="Thmtheorem14.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.1.1.m1.1b"><apply id="Thmtheorem14.p1.1.1.m1.1.1.cmml" xref="Thmtheorem14.p1.1.1.m1.1.1"><geq id="Thmtheorem14.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem14.p1.1.1.m1.1.1.1"></geq><ci id="Thmtheorem14.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem14.p1.1.1.m1.1.1.2">𝐶</ci><cn id="Thmtheorem14.p1.1.1.m1.1.1.3.cmml" type="integer" xref="Thmtheorem14.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.1.1.m1.1c">C\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.1.1.m1.1d">italic_C ≥ 1</annotation></semantics></math> be a constant. We note that a lower bound of the form <math alttext="C\cdot\log N" class="ltx_Math" display="inline" id="Thmtheorem14.p1.2.2.m2.1"><semantics id="Thmtheorem14.p1.2.2.m2.1a"><mrow id="Thmtheorem14.p1.2.2.m2.1.1" xref="Thmtheorem14.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem14.p1.2.2.m2.1.1.2" xref="Thmtheorem14.p1.2.2.m2.1.1.2.cmml">C</mi><mo id="Thmtheorem14.p1.2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem14.p1.2.2.m2.1.1.1.cmml">⋅</mo><mrow id="Thmtheorem14.p1.2.2.m2.1.1.3" xref="Thmtheorem14.p1.2.2.m2.1.1.3.cmml"><mi id="Thmtheorem14.p1.2.2.m2.1.1.3.1" xref="Thmtheorem14.p1.2.2.m2.1.1.3.1.cmml">log</mi><mo id="Thmtheorem14.p1.2.2.m2.1.1.3a" lspace="0.167em" xref="Thmtheorem14.p1.2.2.m2.1.1.3.cmml">⁡</mo><mi id="Thmtheorem14.p1.2.2.m2.1.1.3.2" xref="Thmtheorem14.p1.2.2.m2.1.1.3.2.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.2.2.m2.1b"><apply id="Thmtheorem14.p1.2.2.m2.1.1.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1"><ci id="Thmtheorem14.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1.1">⋅</ci><ci id="Thmtheorem14.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1.2">𝐶</ci><apply id="Thmtheorem14.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1.3"><log id="Thmtheorem14.p1.2.2.m2.1.1.3.1.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1.3.1"></log><ci id="Thmtheorem14.p1.2.2.m2.1.1.3.2.cmml" xref="Thmtheorem14.p1.2.2.m2.1.1.3.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.2.2.m2.1c">C\cdot\log N</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.2.2.m2.1d">italic_C ⋅ roman_log italic_N</annotation></semantics></math> on <math alttext="D_{\cap}(H\mid\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="Thmtheorem14.p1.3.3.m3.3"><semantics id="Thmtheorem14.p1.3.3.m3.3a"><mrow id="Thmtheorem14.p1.3.3.m3.3.3" xref="Thmtheorem14.p1.3.3.m3.3.3.cmml"><msub id="Thmtheorem14.p1.3.3.m3.3.3.3" xref="Thmtheorem14.p1.3.3.m3.3.3.3.cmml"><mi id="Thmtheorem14.p1.3.3.m3.3.3.3.2" xref="Thmtheorem14.p1.3.3.m3.3.3.3.2.cmml">D</mi><mo id="Thmtheorem14.p1.3.3.m3.3.3.3.3" xref="Thmtheorem14.p1.3.3.m3.3.3.3.3.cmml">∩</mo></msub><mo id="Thmtheorem14.p1.3.3.m3.3.3.2" xref="Thmtheorem14.p1.3.3.m3.3.3.2.cmml">⁢</mo><mrow id="Thmtheorem14.p1.3.3.m3.3.3.1.1" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.cmml"><mo id="Thmtheorem14.p1.3.3.m3.3.3.1.1.2" stretchy="false" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.cmml">(</mo><mrow id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.cmml"><mi id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.2" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.2.cmml">H</mi><mo id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.1" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.1.cmml">∣</mo><msub id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.2" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.2.cmml">𝒢</mi><mrow id="Thmtheorem14.p1.3.3.m3.2.2.2.4" xref="Thmtheorem14.p1.3.3.m3.2.2.2.3.cmml"><mi id="Thmtheorem14.p1.3.3.m3.1.1.1.1" xref="Thmtheorem14.p1.3.3.m3.1.1.1.1.cmml">N</mi><mo id="Thmtheorem14.p1.3.3.m3.2.2.2.4.1" xref="Thmtheorem14.p1.3.3.m3.2.2.2.3.cmml">,</mo><mi id="Thmtheorem14.p1.3.3.m3.2.2.2.2" xref="Thmtheorem14.p1.3.3.m3.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="Thmtheorem14.p1.3.3.m3.3.3.1.1.3" stretchy="false" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.3.3.m3.3b"><apply id="Thmtheorem14.p1.3.3.m3.3.3.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3"><times id="Thmtheorem14.p1.3.3.m3.3.3.2.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.2"></times><apply id="Thmtheorem14.p1.3.3.m3.3.3.3.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.3"><csymbol cd="ambiguous" id="Thmtheorem14.p1.3.3.m3.3.3.3.1.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.3">subscript</csymbol><ci id="Thmtheorem14.p1.3.3.m3.3.3.3.2.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.3.2">𝐷</ci><intersect id="Thmtheorem14.p1.3.3.m3.3.3.3.3.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.3.3"></intersect></apply><apply id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1"><csymbol cd="latexml" id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.1">conditional</csymbol><ci id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.2.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.2">𝐻</ci><apply id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.1.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3">subscript</csymbol><ci id="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.2.cmml" xref="Thmtheorem14.p1.3.3.m3.3.3.1.1.1.3.2">𝒢</ci><list id="Thmtheorem14.p1.3.3.m3.2.2.2.3.cmml" xref="Thmtheorem14.p1.3.3.m3.2.2.2.4"><ci id="Thmtheorem14.p1.3.3.m3.1.1.1.1.cmml" xref="Thmtheorem14.p1.3.3.m3.1.1.1.1">𝑁</ci><ci id="Thmtheorem14.p1.3.3.m3.2.2.2.2.cmml" xref="Thmtheorem14.p1.3.3.m3.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.3.3.m3.3c">D_{\cap}(H\mid\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.3.3.m3.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_H ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> for an explicit graph <math alttext="H" class="ltx_Math" display="inline" id="Thmtheorem14.p1.4.4.m4.1"><semantics id="Thmtheorem14.p1.4.4.m4.1a"><mi id="Thmtheorem14.p1.4.4.m4.1.1" xref="Thmtheorem14.p1.4.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.4.4.m4.1b"><ci id="Thmtheorem14.p1.4.4.m4.1.1.cmml" xref="Thmtheorem14.p1.4.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.4.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.4.4.m4.1d">italic_H</annotation></semantics></math> can be translated into the same lower bound on the circuit complexity of a related explicit Boolean function. In more detail, let <math alttext="f_{H}\colon\{0,1\}^{2n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.5.5.m5.4"><semantics id="Thmtheorem14.p1.5.5.m5.4a"><mrow id="Thmtheorem14.p1.5.5.m5.4.5" xref="Thmtheorem14.p1.5.5.m5.4.5.cmml"><msub id="Thmtheorem14.p1.5.5.m5.4.5.2" xref="Thmtheorem14.p1.5.5.m5.4.5.2.cmml"><mi id="Thmtheorem14.p1.5.5.m5.4.5.2.2" xref="Thmtheorem14.p1.5.5.m5.4.5.2.2.cmml">f</mi><mi id="Thmtheorem14.p1.5.5.m5.4.5.2.3" xref="Thmtheorem14.p1.5.5.m5.4.5.2.3.cmml">H</mi></msub><mo id="Thmtheorem14.p1.5.5.m5.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem14.p1.5.5.m5.4.5.1.cmml">:</mo><mrow id="Thmtheorem14.p1.5.5.m5.4.5.3" xref="Thmtheorem14.p1.5.5.m5.4.5.3.cmml"><msup id="Thmtheorem14.p1.5.5.m5.4.5.3.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.cmml"><mrow id="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.1.cmml"><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.1.cmml">{</mo><mn id="Thmtheorem14.p1.5.5.m5.1.1" xref="Thmtheorem14.p1.5.5.m5.1.1.cmml">0</mn><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.2.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.1.cmml">,</mo><mn id="Thmtheorem14.p1.5.5.m5.2.2" xref="Thmtheorem14.p1.5.5.m5.2.2.cmml">1</mn><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.2.3" stretchy="false" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.cmml"><mn id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.2.cmml">2</mn><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.1" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.1.cmml">⁢</mo><mi id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.3" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.3.cmml">n</mi></mrow></msup><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.1" stretchy="false" xref="Thmtheorem14.p1.5.5.m5.4.5.3.1.cmml">→</mo><mrow id="Thmtheorem14.p1.5.5.m5.4.5.3.3.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.3.1.cmml"><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem14.p1.5.5.m5.4.5.3.3.1.cmml">{</mo><mn id="Thmtheorem14.p1.5.5.m5.3.3" xref="Thmtheorem14.p1.5.5.m5.3.3.cmml">0</mn><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.3.2.2" xref="Thmtheorem14.p1.5.5.m5.4.5.3.3.1.cmml">,</mo><mn id="Thmtheorem14.p1.5.5.m5.4.4" xref="Thmtheorem14.p1.5.5.m5.4.4.cmml">1</mn><mo id="Thmtheorem14.p1.5.5.m5.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem14.p1.5.5.m5.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.5.5.m5.4b"><apply id="Thmtheorem14.p1.5.5.m5.4.5.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5"><ci id="Thmtheorem14.p1.5.5.m5.4.5.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.1">:</ci><apply id="Thmtheorem14.p1.5.5.m5.4.5.2.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.5.5.m5.4.5.2.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.2">subscript</csymbol><ci id="Thmtheorem14.p1.5.5.m5.4.5.2.2.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.2.2">𝑓</ci><ci id="Thmtheorem14.p1.5.5.m5.4.5.2.3.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.2.3">𝐻</ci></apply><apply id="Thmtheorem14.p1.5.5.m5.4.5.3.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3"><ci id="Thmtheorem14.p1.5.5.m5.4.5.3.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.1">→</ci><apply id="Thmtheorem14.p1.5.5.m5.4.5.3.2.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.5.5.m5.4.5.3.2.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2">superscript</csymbol><set id="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.2.2"><cn id="Thmtheorem14.p1.5.5.m5.1.1.cmml" type="integer" xref="Thmtheorem14.p1.5.5.m5.1.1">0</cn><cn id="Thmtheorem14.p1.5.5.m5.2.2.cmml" type="integer" xref="Thmtheorem14.p1.5.5.m5.2.2">1</cn></set><apply id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3"><times id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.1"></times><cn id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.2.cmml" type="integer" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.2">2</cn><ci id="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.3.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.2.3.3">𝑛</ci></apply></apply><set id="Thmtheorem14.p1.5.5.m5.4.5.3.3.1.cmml" xref="Thmtheorem14.p1.5.5.m5.4.5.3.3.2"><cn id="Thmtheorem14.p1.5.5.m5.3.3.cmml" type="integer" xref="Thmtheorem14.p1.5.5.m5.3.3">0</cn><cn id="Thmtheorem14.p1.5.5.m5.4.4.cmml" type="integer" xref="Thmtheorem14.p1.5.5.m5.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.5.5.m5.4c">f_{H}\colon\{0,1\}^{2n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.5.5.m5.4d">italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> be the Boolean function corresponding to a bipartite graph <math alttext="H\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem14.p1.6.6.m6.2"><semantics id="Thmtheorem14.p1.6.6.m6.2a"><mrow id="Thmtheorem14.p1.6.6.m6.2.3" xref="Thmtheorem14.p1.6.6.m6.2.3.cmml"><mi id="Thmtheorem14.p1.6.6.m6.2.3.2" xref="Thmtheorem14.p1.6.6.m6.2.3.2.cmml">H</mi><mo id="Thmtheorem14.p1.6.6.m6.2.3.1" xref="Thmtheorem14.p1.6.6.m6.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem14.p1.6.6.m6.2.3.3" xref="Thmtheorem14.p1.6.6.m6.2.3.3.cmml"><mrow id="Thmtheorem14.p1.6.6.m6.2.3.3.2.2" xref="Thmtheorem14.p1.6.6.m6.2.3.3.2.1.cmml"><mo id="Thmtheorem14.p1.6.6.m6.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem14.p1.6.6.m6.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem14.p1.6.6.m6.1.1" xref="Thmtheorem14.p1.6.6.m6.1.1.cmml">N</mi><mo id="Thmtheorem14.p1.6.6.m6.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem14.p1.6.6.m6.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem14.p1.6.6.m6.2.3.3.1" rspace="0.222em" xref="Thmtheorem14.p1.6.6.m6.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem14.p1.6.6.m6.2.3.3.3.2" xref="Thmtheorem14.p1.6.6.m6.2.3.3.3.1.cmml"><mo id="Thmtheorem14.p1.6.6.m6.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem14.p1.6.6.m6.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem14.p1.6.6.m6.2.2" xref="Thmtheorem14.p1.6.6.m6.2.2.cmml">N</mi><mo id="Thmtheorem14.p1.6.6.m6.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem14.p1.6.6.m6.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.6.6.m6.2b"><apply id="Thmtheorem14.p1.6.6.m6.2.3.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3"><subset id="Thmtheorem14.p1.6.6.m6.2.3.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.1"></subset><ci id="Thmtheorem14.p1.6.6.m6.2.3.2.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.2">𝐻</ci><apply id="Thmtheorem14.p1.6.6.m6.2.3.3.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3"><times id="Thmtheorem14.p1.6.6.m6.2.3.3.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3.1"></times><apply id="Thmtheorem14.p1.6.6.m6.2.3.3.2.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem14.p1.6.6.m6.2.3.3.2.1.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem14.p1.6.6.m6.1.1.cmml" xref="Thmtheorem14.p1.6.6.m6.1.1">𝑁</ci></apply><apply id="Thmtheorem14.p1.6.6.m6.2.3.3.3.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem14.p1.6.6.m6.2.3.3.3.1.1.cmml" xref="Thmtheorem14.p1.6.6.m6.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem14.p1.6.6.m6.2.2.cmml" xref="Thmtheorem14.p1.6.6.m6.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.6.6.m6.2c">H\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.6.6.m6.2d">italic_H ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>. Now consider the function <math alttext="F\colon\{0,1\}^{1+2n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.7.7.m7.4"><semantics id="Thmtheorem14.p1.7.7.m7.4a"><mrow id="Thmtheorem14.p1.7.7.m7.4.5" xref="Thmtheorem14.p1.7.7.m7.4.5.cmml"><mi id="Thmtheorem14.p1.7.7.m7.4.5.2" xref="Thmtheorem14.p1.7.7.m7.4.5.2.cmml">F</mi><mo id="Thmtheorem14.p1.7.7.m7.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem14.p1.7.7.m7.4.5.1.cmml">:</mo><mrow id="Thmtheorem14.p1.7.7.m7.4.5.3" xref="Thmtheorem14.p1.7.7.m7.4.5.3.cmml"><msup id="Thmtheorem14.p1.7.7.m7.4.5.3.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.cmml"><mrow id="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.1.cmml"><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.1.cmml">{</mo><mn id="Thmtheorem14.p1.7.7.m7.1.1" xref="Thmtheorem14.p1.7.7.m7.1.1.cmml">0</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.2.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.1.cmml">,</mo><mn id="Thmtheorem14.p1.7.7.m7.2.2" xref="Thmtheorem14.p1.7.7.m7.2.2.cmml">1</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.2.3" stretchy="false" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.cmml"><mn id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.2.cmml">1</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.1" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.1.cmml">+</mo><mrow id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.cmml"><mn id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.2.cmml">2</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.1" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.1.cmml">⁢</mo><mi id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.3" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.3.cmml">n</mi></mrow></mrow></msup><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.1" stretchy="false" xref="Thmtheorem14.p1.7.7.m7.4.5.3.1.cmml">→</mo><mrow id="Thmtheorem14.p1.7.7.m7.4.5.3.3.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.3.1.cmml"><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem14.p1.7.7.m7.4.5.3.3.1.cmml">{</mo><mn id="Thmtheorem14.p1.7.7.m7.3.3" xref="Thmtheorem14.p1.7.7.m7.3.3.cmml">0</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.3.2.2" xref="Thmtheorem14.p1.7.7.m7.4.5.3.3.1.cmml">,</mo><mn id="Thmtheorem14.p1.7.7.m7.4.4" xref="Thmtheorem14.p1.7.7.m7.4.4.cmml">1</mn><mo id="Thmtheorem14.p1.7.7.m7.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem14.p1.7.7.m7.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.7.7.m7.4b"><apply id="Thmtheorem14.p1.7.7.m7.4.5.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5"><ci id="Thmtheorem14.p1.7.7.m7.4.5.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.1">:</ci><ci id="Thmtheorem14.p1.7.7.m7.4.5.2.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.2">𝐹</ci><apply id="Thmtheorem14.p1.7.7.m7.4.5.3.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3"><ci id="Thmtheorem14.p1.7.7.m7.4.5.3.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.1">→</ci><apply id="Thmtheorem14.p1.7.7.m7.4.5.3.2.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.7.7.m7.4.5.3.2.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2">superscript</csymbol><set id="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.2.2"><cn id="Thmtheorem14.p1.7.7.m7.1.1.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.1.1">0</cn><cn id="Thmtheorem14.p1.7.7.m7.2.2.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.2.2">1</cn></set><apply id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3"><plus id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.1"></plus><cn id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.2.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.2">1</cn><apply id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3"><times id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.1"></times><cn id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.2.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.2">2</cn><ci id="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.3.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.2.3.3.3">𝑛</ci></apply></apply></apply><set id="Thmtheorem14.p1.7.7.m7.4.5.3.3.1.cmml" xref="Thmtheorem14.p1.7.7.m7.4.5.3.3.2"><cn id="Thmtheorem14.p1.7.7.m7.3.3.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.3.3">0</cn><cn id="Thmtheorem14.p1.7.7.m7.4.4.cmml" type="integer" xref="Thmtheorem14.p1.7.7.m7.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.7.7.m7.4c">F\colon\{0,1\}^{1+2n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.7.7.m7.4d">italic_F : { 0 , 1 } start_POSTSUPERSCRIPT 1 + 2 italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> defined as follows. The value <math alttext="F(b,z)=f_{H}(z)" class="ltx_Math" display="inline" id="Thmtheorem14.p1.8.8.m8.3"><semantics id="Thmtheorem14.p1.8.8.m8.3a"><mrow id="Thmtheorem14.p1.8.8.m8.3.4" xref="Thmtheorem14.p1.8.8.m8.3.4.cmml"><mrow id="Thmtheorem14.p1.8.8.m8.3.4.2" xref="Thmtheorem14.p1.8.8.m8.3.4.2.cmml"><mi id="Thmtheorem14.p1.8.8.m8.3.4.2.2" xref="Thmtheorem14.p1.8.8.m8.3.4.2.2.cmml">F</mi><mo id="Thmtheorem14.p1.8.8.m8.3.4.2.1" xref="Thmtheorem14.p1.8.8.m8.3.4.2.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.8.8.m8.3.4.2.3.2" xref="Thmtheorem14.p1.8.8.m8.3.4.2.3.1.cmml"><mo id="Thmtheorem14.p1.8.8.m8.3.4.2.3.2.1" stretchy="false" xref="Thmtheorem14.p1.8.8.m8.3.4.2.3.1.cmml">(</mo><mi id="Thmtheorem14.p1.8.8.m8.1.1" xref="Thmtheorem14.p1.8.8.m8.1.1.cmml">b</mi><mo id="Thmtheorem14.p1.8.8.m8.3.4.2.3.2.2" xref="Thmtheorem14.p1.8.8.m8.3.4.2.3.1.cmml">,</mo><mi id="Thmtheorem14.p1.8.8.m8.2.2" xref="Thmtheorem14.p1.8.8.m8.2.2.cmml">z</mi><mo id="Thmtheorem14.p1.8.8.m8.3.4.2.3.2.3" stretchy="false" xref="Thmtheorem14.p1.8.8.m8.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem14.p1.8.8.m8.3.4.1" xref="Thmtheorem14.p1.8.8.m8.3.4.1.cmml">=</mo><mrow id="Thmtheorem14.p1.8.8.m8.3.4.3" xref="Thmtheorem14.p1.8.8.m8.3.4.3.cmml"><msub id="Thmtheorem14.p1.8.8.m8.3.4.3.2" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2.cmml"><mi id="Thmtheorem14.p1.8.8.m8.3.4.3.2.2" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2.2.cmml">f</mi><mi id="Thmtheorem14.p1.8.8.m8.3.4.3.2.3" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2.3.cmml">H</mi></msub><mo id="Thmtheorem14.p1.8.8.m8.3.4.3.1" xref="Thmtheorem14.p1.8.8.m8.3.4.3.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.8.8.m8.3.4.3.3.2" xref="Thmtheorem14.p1.8.8.m8.3.4.3.cmml"><mo id="Thmtheorem14.p1.8.8.m8.3.4.3.3.2.1" stretchy="false" xref="Thmtheorem14.p1.8.8.m8.3.4.3.cmml">(</mo><mi id="Thmtheorem14.p1.8.8.m8.3.3" xref="Thmtheorem14.p1.8.8.m8.3.3.cmml">z</mi><mo id="Thmtheorem14.p1.8.8.m8.3.4.3.3.2.2" stretchy="false" xref="Thmtheorem14.p1.8.8.m8.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.8.8.m8.3b"><apply id="Thmtheorem14.p1.8.8.m8.3.4.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4"><eq id="Thmtheorem14.p1.8.8.m8.3.4.1.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.1"></eq><apply id="Thmtheorem14.p1.8.8.m8.3.4.2.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.2"><times id="Thmtheorem14.p1.8.8.m8.3.4.2.1.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.2.1"></times><ci id="Thmtheorem14.p1.8.8.m8.3.4.2.2.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.2.2">𝐹</ci><interval closure="open" id="Thmtheorem14.p1.8.8.m8.3.4.2.3.1.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.2.3.2"><ci id="Thmtheorem14.p1.8.8.m8.1.1.cmml" xref="Thmtheorem14.p1.8.8.m8.1.1">𝑏</ci><ci id="Thmtheorem14.p1.8.8.m8.2.2.cmml" xref="Thmtheorem14.p1.8.8.m8.2.2">𝑧</ci></interval></apply><apply id="Thmtheorem14.p1.8.8.m8.3.4.3.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3"><times id="Thmtheorem14.p1.8.8.m8.3.4.3.1.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3.1"></times><apply id="Thmtheorem14.p1.8.8.m8.3.4.3.2.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.8.8.m8.3.4.3.2.1.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2">subscript</csymbol><ci id="Thmtheorem14.p1.8.8.m8.3.4.3.2.2.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2.2">𝑓</ci><ci id="Thmtheorem14.p1.8.8.m8.3.4.3.2.3.cmml" xref="Thmtheorem14.p1.8.8.m8.3.4.3.2.3">𝐻</ci></apply><ci id="Thmtheorem14.p1.8.8.m8.3.3.cmml" xref="Thmtheorem14.p1.8.8.m8.3.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.8.8.m8.3c">F(b,z)=f_{H}(z)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.8.8.m8.3d">italic_F ( italic_b , italic_z ) = italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> if the input bit <math alttext="b=1" class="ltx_Math" display="inline" id="Thmtheorem14.p1.9.9.m9.1"><semantics id="Thmtheorem14.p1.9.9.m9.1a"><mrow id="Thmtheorem14.p1.9.9.m9.1.1" xref="Thmtheorem14.p1.9.9.m9.1.1.cmml"><mi id="Thmtheorem14.p1.9.9.m9.1.1.2" xref="Thmtheorem14.p1.9.9.m9.1.1.2.cmml">b</mi><mo id="Thmtheorem14.p1.9.9.m9.1.1.1" xref="Thmtheorem14.p1.9.9.m9.1.1.1.cmml">=</mo><mn id="Thmtheorem14.p1.9.9.m9.1.1.3" xref="Thmtheorem14.p1.9.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.9.9.m9.1b"><apply id="Thmtheorem14.p1.9.9.m9.1.1.cmml" xref="Thmtheorem14.p1.9.9.m9.1.1"><eq id="Thmtheorem14.p1.9.9.m9.1.1.1.cmml" xref="Thmtheorem14.p1.9.9.m9.1.1.1"></eq><ci id="Thmtheorem14.p1.9.9.m9.1.1.2.cmml" xref="Thmtheorem14.p1.9.9.m9.1.1.2">𝑏</ci><cn id="Thmtheorem14.p1.9.9.m9.1.1.3.cmml" type="integer" xref="Thmtheorem14.p1.9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.9.9.m9.1c">b=1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.9.9.m9.1d">italic_b = 1</annotation></semantics></math>, and <math alttext="F(b,z)=\overline{f_{H}}(z)=1-f_{H}(z)" class="ltx_Math" display="inline" id="Thmtheorem14.p1.10.10.m10.4"><semantics id="Thmtheorem14.p1.10.10.m10.4a"><mrow id="Thmtheorem14.p1.10.10.m10.4.5" xref="Thmtheorem14.p1.10.10.m10.4.5.cmml"><mrow id="Thmtheorem14.p1.10.10.m10.4.5.2" xref="Thmtheorem14.p1.10.10.m10.4.5.2.cmml"><mi id="Thmtheorem14.p1.10.10.m10.4.5.2.2" xref="Thmtheorem14.p1.10.10.m10.4.5.2.2.cmml">F</mi><mo id="Thmtheorem14.p1.10.10.m10.4.5.2.1" xref="Thmtheorem14.p1.10.10.m10.4.5.2.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.2.3.2" xref="Thmtheorem14.p1.10.10.m10.4.5.2.3.1.cmml"><mo id="Thmtheorem14.p1.10.10.m10.4.5.2.3.2.1" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.2.3.1.cmml">(</mo><mi id="Thmtheorem14.p1.10.10.m10.1.1" xref="Thmtheorem14.p1.10.10.m10.1.1.cmml">b</mi><mo id="Thmtheorem14.p1.10.10.m10.4.5.2.3.2.2" xref="Thmtheorem14.p1.10.10.m10.4.5.2.3.1.cmml">,</mo><mi id="Thmtheorem14.p1.10.10.m10.2.2" xref="Thmtheorem14.p1.10.10.m10.2.2.cmml">z</mi><mo id="Thmtheorem14.p1.10.10.m10.4.5.2.3.2.3" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem14.p1.10.10.m10.4.5.3" xref="Thmtheorem14.p1.10.10.m10.4.5.3.cmml">=</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.4" xref="Thmtheorem14.p1.10.10.m10.4.5.4.cmml"><mover accent="true" id="Thmtheorem14.p1.10.10.m10.4.5.4.2" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.cmml"><msub id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.cmml"><mi id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.2" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.2.cmml">f</mi><mi id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.3" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.3.cmml">H</mi></msub><mo id="Thmtheorem14.p1.10.10.m10.4.5.4.2.1" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.1.cmml">¯</mo></mover><mo id="Thmtheorem14.p1.10.10.m10.4.5.4.1" xref="Thmtheorem14.p1.10.10.m10.4.5.4.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.4.3.2" xref="Thmtheorem14.p1.10.10.m10.4.5.4.cmml"><mo id="Thmtheorem14.p1.10.10.m10.4.5.4.3.2.1" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.4.cmml">(</mo><mi id="Thmtheorem14.p1.10.10.m10.3.3" xref="Thmtheorem14.p1.10.10.m10.3.3.cmml">z</mi><mo id="Thmtheorem14.p1.10.10.m10.4.5.4.3.2.2" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.4.cmml">)</mo></mrow></mrow><mo id="Thmtheorem14.p1.10.10.m10.4.5.5" xref="Thmtheorem14.p1.10.10.m10.4.5.5.cmml">=</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.6" xref="Thmtheorem14.p1.10.10.m10.4.5.6.cmml"><mn id="Thmtheorem14.p1.10.10.m10.4.5.6.2" xref="Thmtheorem14.p1.10.10.m10.4.5.6.2.cmml">1</mn><mo id="Thmtheorem14.p1.10.10.m10.4.5.6.1" xref="Thmtheorem14.p1.10.10.m10.4.5.6.1.cmml">−</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.6.3" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.cmml"><msub id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.cmml"><mi id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.2" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.2.cmml">f</mi><mi id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.3" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.3.cmml">H</mi></msub><mo id="Thmtheorem14.p1.10.10.m10.4.5.6.3.1" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.10.10.m10.4.5.6.3.3.2" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.cmml"><mo id="Thmtheorem14.p1.10.10.m10.4.5.6.3.3.2.1" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.cmml">(</mo><mi id="Thmtheorem14.p1.10.10.m10.4.4" xref="Thmtheorem14.p1.10.10.m10.4.4.cmml">z</mi><mo id="Thmtheorem14.p1.10.10.m10.4.5.6.3.3.2.2" stretchy="false" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.10.10.m10.4b"><apply id="Thmtheorem14.p1.10.10.m10.4.5.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5"><and id="Thmtheorem14.p1.10.10.m10.4.5a.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5"></and><apply id="Thmtheorem14.p1.10.10.m10.4.5b.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5"><eq id="Thmtheorem14.p1.10.10.m10.4.5.3.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.3"></eq><apply id="Thmtheorem14.p1.10.10.m10.4.5.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.2"><times id="Thmtheorem14.p1.10.10.m10.4.5.2.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.2.1"></times><ci id="Thmtheorem14.p1.10.10.m10.4.5.2.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.2.2">𝐹</ci><interval closure="open" id="Thmtheorem14.p1.10.10.m10.4.5.2.3.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.2.3.2"><ci id="Thmtheorem14.p1.10.10.m10.1.1.cmml" xref="Thmtheorem14.p1.10.10.m10.1.1">𝑏</ci><ci id="Thmtheorem14.p1.10.10.m10.2.2.cmml" xref="Thmtheorem14.p1.10.10.m10.2.2">𝑧</ci></interval></apply><apply id="Thmtheorem14.p1.10.10.m10.4.5.4.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4"><times id="Thmtheorem14.p1.10.10.m10.4.5.4.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.1"></times><apply id="Thmtheorem14.p1.10.10.m10.4.5.4.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2"><ci id="Thmtheorem14.p1.10.10.m10.4.5.4.2.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.1">¯</ci><apply id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2">subscript</csymbol><ci id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.2">𝑓</ci><ci id="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.3.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.4.2.2.3">𝐻</ci></apply></apply><ci id="Thmtheorem14.p1.10.10.m10.3.3.cmml" xref="Thmtheorem14.p1.10.10.m10.3.3">𝑧</ci></apply></apply><apply id="Thmtheorem14.p1.10.10.m10.4.5c.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5"><eq id="Thmtheorem14.p1.10.10.m10.4.5.5.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.5"></eq><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem14.p1.10.10.m10.4.5.4.cmml" id="Thmtheorem14.p1.10.10.m10.4.5d.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5"></share><apply id="Thmtheorem14.p1.10.10.m10.4.5.6.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6"><minus id="Thmtheorem14.p1.10.10.m10.4.5.6.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.1"></minus><cn id="Thmtheorem14.p1.10.10.m10.4.5.6.2.cmml" type="integer" xref="Thmtheorem14.p1.10.10.m10.4.5.6.2">1</cn><apply id="Thmtheorem14.p1.10.10.m10.4.5.6.3.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3"><times id="Thmtheorem14.p1.10.10.m10.4.5.6.3.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.1"></times><apply id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2"><csymbol cd="ambiguous" id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.1.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2">subscript</csymbol><ci id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.2.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.2">𝑓</ci><ci id="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.3.cmml" xref="Thmtheorem14.p1.10.10.m10.4.5.6.3.2.3">𝐻</ci></apply><ci id="Thmtheorem14.p1.10.10.m10.4.4.cmml" xref="Thmtheorem14.p1.10.10.m10.4.4">𝑧</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.10.10.m10.4c">F(b,z)=\overline{f_{H}}(z)=1-f_{H}(z)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.10.10.m10.4d">italic_F ( italic_b , italic_z ) = over¯ start_ARG italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_ARG ( italic_z ) = 1 - italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> if <math alttext="b=0" class="ltx_Math" display="inline" id="Thmtheorem14.p1.11.11.m11.1"><semantics id="Thmtheorem14.p1.11.11.m11.1a"><mrow id="Thmtheorem14.p1.11.11.m11.1.1" xref="Thmtheorem14.p1.11.11.m11.1.1.cmml"><mi id="Thmtheorem14.p1.11.11.m11.1.1.2" xref="Thmtheorem14.p1.11.11.m11.1.1.2.cmml">b</mi><mo id="Thmtheorem14.p1.11.11.m11.1.1.1" xref="Thmtheorem14.p1.11.11.m11.1.1.1.cmml">=</mo><mn id="Thmtheorem14.p1.11.11.m11.1.1.3" xref="Thmtheorem14.p1.11.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.11.11.m11.1b"><apply id="Thmtheorem14.p1.11.11.m11.1.1.cmml" xref="Thmtheorem14.p1.11.11.m11.1.1"><eq id="Thmtheorem14.p1.11.11.m11.1.1.1.cmml" xref="Thmtheorem14.p1.11.11.m11.1.1.1"></eq><ci id="Thmtheorem14.p1.11.11.m11.1.1.2.cmml" xref="Thmtheorem14.p1.11.11.m11.1.1.2">𝑏</ci><cn id="Thmtheorem14.p1.11.11.m11.1.1.3.cmml" type="integer" xref="Thmtheorem14.p1.11.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.11.11.m11.1c">b=0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.11.11.m11.1d">italic_b = 0</annotation></semantics></math>. Note that if <math alttext="H" class="ltx_Math" display="inline" id="Thmtheorem14.p1.12.12.m12.1"><semantics id="Thmtheorem14.p1.12.12.m12.1a"><mi id="Thmtheorem14.p1.12.12.m12.1.1" xref="Thmtheorem14.p1.12.12.m12.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.12.12.m12.1b"><ci id="Thmtheorem14.p1.12.12.m12.1.1.cmml" xref="Thmtheorem14.p1.12.12.m12.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.12.12.m12.1c">H</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.12.12.m12.1d">italic_H</annotation></semantics></math> can be computed in time <math alttext="\mathsf{poly}(N)" class="ltx_Math" display="inline" id="Thmtheorem14.p1.13.13.m13.1"><semantics id="Thmtheorem14.p1.13.13.m13.1a"><mrow id="Thmtheorem14.p1.13.13.m13.1.2" xref="Thmtheorem14.p1.13.13.m13.1.2.cmml"><mi id="Thmtheorem14.p1.13.13.m13.1.2.2" xref="Thmtheorem14.p1.13.13.m13.1.2.2.cmml">𝗉𝗈𝗅𝗒</mi><mo id="Thmtheorem14.p1.13.13.m13.1.2.1" xref="Thmtheorem14.p1.13.13.m13.1.2.1.cmml">⁢</mo><mrow id="Thmtheorem14.p1.13.13.m13.1.2.3.2" xref="Thmtheorem14.p1.13.13.m13.1.2.cmml"><mo id="Thmtheorem14.p1.13.13.m13.1.2.3.2.1" stretchy="false" xref="Thmtheorem14.p1.13.13.m13.1.2.cmml">(</mo><mi id="Thmtheorem14.p1.13.13.m13.1.1" xref="Thmtheorem14.p1.13.13.m13.1.1.cmml">N</mi><mo id="Thmtheorem14.p1.13.13.m13.1.2.3.2.2" stretchy="false" xref="Thmtheorem14.p1.13.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.13.13.m13.1b"><apply id="Thmtheorem14.p1.13.13.m13.1.2.cmml" xref="Thmtheorem14.p1.13.13.m13.1.2"><times id="Thmtheorem14.p1.13.13.m13.1.2.1.cmml" xref="Thmtheorem14.p1.13.13.m13.1.2.1"></times><ci id="Thmtheorem14.p1.13.13.m13.1.2.2.cmml" xref="Thmtheorem14.p1.13.13.m13.1.2.2">𝗉𝗈𝗅𝗒</ci><ci id="Thmtheorem14.p1.13.13.m13.1.1.cmml" xref="Thmtheorem14.p1.13.13.m13.1.1">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.13.13.m13.1c">\mathsf{poly}(N)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.13.13.m13.1d">sansserif_poly ( italic_N )</annotation></semantics></math> then the corresponding function <math alttext="F" class="ltx_Math" display="inline" id="Thmtheorem14.p1.14.14.m14.1"><semantics id="Thmtheorem14.p1.14.14.m14.1a"><mi id="Thmtheorem14.p1.14.14.m14.1.1" xref="Thmtheorem14.p1.14.14.m14.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.14.14.m14.1b"><ci id="Thmtheorem14.p1.14.14.m14.1.1.cmml" xref="Thmtheorem14.p1.14.14.m14.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.14.14.m14.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.14.14.m14.1d">italic_F</annotation></semantics></math> is in <math alttext="E=\mathsf{DTIME}[2^{O(m)}]" class="ltx_Math" display="inline" id="Thmtheorem14.p1.15.15.m15.2"><semantics id="Thmtheorem14.p1.15.15.m15.2a"><mrow id="Thmtheorem14.p1.15.15.m15.2.2" xref="Thmtheorem14.p1.15.15.m15.2.2.cmml"><mi id="Thmtheorem14.p1.15.15.m15.2.2.3" xref="Thmtheorem14.p1.15.15.m15.2.2.3.cmml">E</mi><mo id="Thmtheorem14.p1.15.15.m15.2.2.2" xref="Thmtheorem14.p1.15.15.m15.2.2.2.cmml">=</mo><mrow id="Thmtheorem14.p1.15.15.m15.2.2.1" xref="Thmtheorem14.p1.15.15.m15.2.2.1.cmml"><mi id="Thmtheorem14.p1.15.15.m15.2.2.1.3" xref="Thmtheorem14.p1.15.15.m15.2.2.1.3.cmml">𝖣𝖳𝖨𝖬𝖤</mi><mo id="Thmtheorem14.p1.15.15.m15.2.2.1.2" xref="Thmtheorem14.p1.15.15.m15.2.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.2.cmml"><mo id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.2" stretchy="false" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.2.1.cmml">[</mo><msup id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.cmml"><mn id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.2" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.2.cmml">2</mn><mrow id="Thmtheorem14.p1.15.15.m15.1.1.1" xref="Thmtheorem14.p1.15.15.m15.1.1.1.cmml"><mi id="Thmtheorem14.p1.15.15.m15.1.1.1.3" xref="Thmtheorem14.p1.15.15.m15.1.1.1.3.cmml">O</mi><mo id="Thmtheorem14.p1.15.15.m15.1.1.1.2" xref="Thmtheorem14.p1.15.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem14.p1.15.15.m15.1.1.1.4.2" xref="Thmtheorem14.p1.15.15.m15.1.1.1.cmml"><mo id="Thmtheorem14.p1.15.15.m15.1.1.1.4.2.1" stretchy="false" xref="Thmtheorem14.p1.15.15.m15.1.1.1.cmml">(</mo><mi id="Thmtheorem14.p1.15.15.m15.1.1.1.1" xref="Thmtheorem14.p1.15.15.m15.1.1.1.1.cmml">m</mi><mo id="Thmtheorem14.p1.15.15.m15.1.1.1.4.2.2" stretchy="false" xref="Thmtheorem14.p1.15.15.m15.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.15.15.m15.2b"><apply id="Thmtheorem14.p1.15.15.m15.2.2.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2"><eq id="Thmtheorem14.p1.15.15.m15.2.2.2.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.2"></eq><ci id="Thmtheorem14.p1.15.15.m15.2.2.3.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.3">𝐸</ci><apply id="Thmtheorem14.p1.15.15.m15.2.2.1.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1"><times id="Thmtheorem14.p1.15.15.m15.2.2.1.2.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.2"></times><ci id="Thmtheorem14.p1.15.15.m15.2.2.1.3.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.3">𝖣𝖳𝖨𝖬𝖤</ci><apply id="Thmtheorem14.p1.15.15.m15.2.2.1.1.2.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1"><csymbol cd="latexml" id="Thmtheorem14.p1.15.15.m15.2.2.1.1.2.1.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.2">delimited-[]</csymbol><apply id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.1.cmml" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1">superscript</csymbol><cn id="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.2.cmml" type="integer" xref="Thmtheorem14.p1.15.15.m15.2.2.1.1.1.1.2">2</cn><apply id="Thmtheorem14.p1.15.15.m15.1.1.1.cmml" xref="Thmtheorem14.p1.15.15.m15.1.1.1"><times id="Thmtheorem14.p1.15.15.m15.1.1.1.2.cmml" xref="Thmtheorem14.p1.15.15.m15.1.1.1.2"></times><ci id="Thmtheorem14.p1.15.15.m15.1.1.1.3.cmml" xref="Thmtheorem14.p1.15.15.m15.1.1.1.3">𝑂</ci><ci id="Thmtheorem14.p1.15.15.m15.1.1.1.1.cmml" xref="Thmtheorem14.p1.15.15.m15.1.1.1.1">𝑚</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.15.15.m15.2c">E=\mathsf{DTIME}[2^{O(m)}]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.15.15.m15.2d">italic_E = sansserif_DTIME [ 2 start_POSTSUPERSCRIPT italic_O ( italic_m ) end_POSTSUPERSCRIPT ]</annotation></semantics></math>, where <math alttext="m=2n+1" class="ltx_Math" display="inline" id="Thmtheorem14.p1.16.16.m16.1"><semantics id="Thmtheorem14.p1.16.16.m16.1a"><mrow id="Thmtheorem14.p1.16.16.m16.1.1" xref="Thmtheorem14.p1.16.16.m16.1.1.cmml"><mi id="Thmtheorem14.p1.16.16.m16.1.1.2" xref="Thmtheorem14.p1.16.16.m16.1.1.2.cmml">m</mi><mo id="Thmtheorem14.p1.16.16.m16.1.1.1" xref="Thmtheorem14.p1.16.16.m16.1.1.1.cmml">=</mo><mrow id="Thmtheorem14.p1.16.16.m16.1.1.3" xref="Thmtheorem14.p1.16.16.m16.1.1.3.cmml"><mrow id="Thmtheorem14.p1.16.16.m16.1.1.3.2" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.cmml"><mn id="Thmtheorem14.p1.16.16.m16.1.1.3.2.2" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.2.cmml">2</mn><mo id="Thmtheorem14.p1.16.16.m16.1.1.3.2.1" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.1.cmml">⁢</mo><mi id="Thmtheorem14.p1.16.16.m16.1.1.3.2.3" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.3.cmml">n</mi></mrow><mo id="Thmtheorem14.p1.16.16.m16.1.1.3.1" xref="Thmtheorem14.p1.16.16.m16.1.1.3.1.cmml">+</mo><mn id="Thmtheorem14.p1.16.16.m16.1.1.3.3" xref="Thmtheorem14.p1.16.16.m16.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.16.16.m16.1b"><apply id="Thmtheorem14.p1.16.16.m16.1.1.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1"><eq id="Thmtheorem14.p1.16.16.m16.1.1.1.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.1"></eq><ci id="Thmtheorem14.p1.16.16.m16.1.1.2.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.2">𝑚</ci><apply id="Thmtheorem14.p1.16.16.m16.1.1.3.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.3"><plus id="Thmtheorem14.p1.16.16.m16.1.1.3.1.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.3.1"></plus><apply id="Thmtheorem14.p1.16.16.m16.1.1.3.2.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2"><times id="Thmtheorem14.p1.16.16.m16.1.1.3.2.1.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.1"></times><cn id="Thmtheorem14.p1.16.16.m16.1.1.3.2.2.cmml" type="integer" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.2">2</cn><ci id="Thmtheorem14.p1.16.16.m16.1.1.3.2.3.cmml" xref="Thmtheorem14.p1.16.16.m16.1.1.3.2.3">𝑛</ci></apply><cn id="Thmtheorem14.p1.16.16.m16.1.1.3.3.cmml" type="integer" xref="Thmtheorem14.p1.16.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.16.16.m16.1c">m=2n+1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.16.16.m16.1d">italic_m = 2 italic_n + 1</annotation></semantics></math> is the input length of <math alttext="F" class="ltx_Math" display="inline" id="Thmtheorem14.p1.17.17.m17.1"><semantics id="Thmtheorem14.p1.17.17.m17.1a"><mi id="Thmtheorem14.p1.17.17.m17.1.1" xref="Thmtheorem14.p1.17.17.m17.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.17.17.m17.1b"><ci id="Thmtheorem14.p1.17.17.m17.1.1.cmml" xref="Thmtheorem14.p1.17.17.m17.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.17.17.m17.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.17.17.m17.1d">italic_F</annotation></semantics></math>. Moreover, if <math alttext="D_{\cap}(H\mid\mathcal{G}_{N,N})\geq C\cdot\log N" class="ltx_Math" display="inline" id="Thmtheorem14.p1.18.18.m18.3"><semantics id="Thmtheorem14.p1.18.18.m18.3a"><mrow id="Thmtheorem14.p1.18.18.m18.3.3" xref="Thmtheorem14.p1.18.18.m18.3.3.cmml"><mrow id="Thmtheorem14.p1.18.18.m18.3.3.1" xref="Thmtheorem14.p1.18.18.m18.3.3.1.cmml"><msub id="Thmtheorem14.p1.18.18.m18.3.3.1.3" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3.cmml"><mi id="Thmtheorem14.p1.18.18.m18.3.3.1.3.2" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3.2.cmml">D</mi><mo id="Thmtheorem14.p1.18.18.m18.3.3.1.3.3" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem14.p1.18.18.m18.3.3.1.2" xref="Thmtheorem14.p1.18.18.m18.3.3.1.2.cmml">⁢</mo><mrow id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.cmml"><mo id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.2" stretchy="false" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.cmml"><mi id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.2" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.2.cmml">H</mi><mo id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.1" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.1.cmml">∣</mo><msub id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.2" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="Thmtheorem14.p1.18.18.m18.2.2.2.4" xref="Thmtheorem14.p1.18.18.m18.2.2.2.3.cmml"><mi id="Thmtheorem14.p1.18.18.m18.1.1.1.1" xref="Thmtheorem14.p1.18.18.m18.1.1.1.1.cmml">N</mi><mo id="Thmtheorem14.p1.18.18.m18.2.2.2.4.1" xref="Thmtheorem14.p1.18.18.m18.2.2.2.3.cmml">,</mo><mi id="Thmtheorem14.p1.18.18.m18.2.2.2.2" xref="Thmtheorem14.p1.18.18.m18.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.3" stretchy="false" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem14.p1.18.18.m18.3.3.2" xref="Thmtheorem14.p1.18.18.m18.3.3.2.cmml">≥</mo><mrow id="Thmtheorem14.p1.18.18.m18.3.3.3" xref="Thmtheorem14.p1.18.18.m18.3.3.3.cmml"><mi id="Thmtheorem14.p1.18.18.m18.3.3.3.2" xref="Thmtheorem14.p1.18.18.m18.3.3.3.2.cmml">C</mi><mo id="Thmtheorem14.p1.18.18.m18.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem14.p1.18.18.m18.3.3.3.1.cmml">⋅</mo><mrow id="Thmtheorem14.p1.18.18.m18.3.3.3.3" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.cmml"><mi id="Thmtheorem14.p1.18.18.m18.3.3.3.3.1" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.1.cmml">log</mi><mo id="Thmtheorem14.p1.18.18.m18.3.3.3.3a" lspace="0.167em" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.cmml">⁡</mo><mi id="Thmtheorem14.p1.18.18.m18.3.3.3.3.2" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.2.cmml">N</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.18.18.m18.3b"><apply id="Thmtheorem14.p1.18.18.m18.3.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3"><geq id="Thmtheorem14.p1.18.18.m18.3.3.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.2"></geq><apply id="Thmtheorem14.p1.18.18.m18.3.3.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1"><times id="Thmtheorem14.p1.18.18.m18.3.3.1.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.2"></times><apply id="Thmtheorem14.p1.18.18.m18.3.3.1.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3"><csymbol cd="ambiguous" id="Thmtheorem14.p1.18.18.m18.3.3.1.3.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3">subscript</csymbol><ci id="Thmtheorem14.p1.18.18.m18.3.3.1.3.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3.2">𝐷</ci><intersect id="Thmtheorem14.p1.18.18.m18.3.3.1.3.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.3.3"></intersect></apply><apply id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1"><csymbol cd="latexml" id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.2">𝐻</ci><apply id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3">subscript</csymbol><ci id="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.1.1.1.1.3.2">𝒢</ci><list id="Thmtheorem14.p1.18.18.m18.2.2.2.3.cmml" xref="Thmtheorem14.p1.18.18.m18.2.2.2.4"><ci id="Thmtheorem14.p1.18.18.m18.1.1.1.1.cmml" xref="Thmtheorem14.p1.18.18.m18.1.1.1.1">𝑁</ci><ci id="Thmtheorem14.p1.18.18.m18.2.2.2.2.cmml" xref="Thmtheorem14.p1.18.18.m18.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="Thmtheorem14.p1.18.18.m18.3.3.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3"><ci id="Thmtheorem14.p1.18.18.m18.3.3.3.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3.1">⋅</ci><ci id="Thmtheorem14.p1.18.18.m18.3.3.3.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3.2">𝐶</ci><apply id="Thmtheorem14.p1.18.18.m18.3.3.3.3.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3"><log id="Thmtheorem14.p1.18.18.m18.3.3.3.3.1.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.1"></log><ci id="Thmtheorem14.p1.18.18.m18.3.3.3.3.2.cmml" xref="Thmtheorem14.p1.18.18.m18.3.3.3.3.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.18.18.m18.3c">D_{\cap}(H\mid\mathcal{G}_{N,N})\geq C\cdot\log N</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.18.18.m18.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_H ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≥ italic_C ⋅ roman_log italic_N</annotation></semantics></math> then any Boolean circuit computing <math alttext="F" class="ltx_Math" display="inline" id="Thmtheorem14.p1.19.19.m19.1"><semantics id="Thmtheorem14.p1.19.19.m19.1a"><mi id="Thmtheorem14.p1.19.19.m19.1.1" xref="Thmtheorem14.p1.19.19.m19.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.19.19.m19.1b"><ci id="Thmtheorem14.p1.19.19.m19.1.1.cmml" xref="Thmtheorem14.p1.19.19.m19.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.19.19.m19.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.19.19.m19.1d">italic_F</annotation></semantics></math> must contain at least <math alttext="C\cdot 2n" class="ltx_Math" display="inline" id="Thmtheorem14.p1.20.20.m20.1"><semantics id="Thmtheorem14.p1.20.20.m20.1a"><mrow id="Thmtheorem14.p1.20.20.m20.1.1" xref="Thmtheorem14.p1.20.20.m20.1.1.cmml"><mrow id="Thmtheorem14.p1.20.20.m20.1.1.2" xref="Thmtheorem14.p1.20.20.m20.1.1.2.cmml"><mi id="Thmtheorem14.p1.20.20.m20.1.1.2.2" xref="Thmtheorem14.p1.20.20.m20.1.1.2.2.cmml">C</mi><mo id="Thmtheorem14.p1.20.20.m20.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem14.p1.20.20.m20.1.1.2.1.cmml">⋅</mo><mn id="Thmtheorem14.p1.20.20.m20.1.1.2.3" xref="Thmtheorem14.p1.20.20.m20.1.1.2.3.cmml">2</mn></mrow><mo id="Thmtheorem14.p1.20.20.m20.1.1.1" xref="Thmtheorem14.p1.20.20.m20.1.1.1.cmml">⁢</mo><mi id="Thmtheorem14.p1.20.20.m20.1.1.3" xref="Thmtheorem14.p1.20.20.m20.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.20.20.m20.1b"><apply id="Thmtheorem14.p1.20.20.m20.1.1.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1"><times id="Thmtheorem14.p1.20.20.m20.1.1.1.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1.1"></times><apply id="Thmtheorem14.p1.20.20.m20.1.1.2.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1.2"><ci id="Thmtheorem14.p1.20.20.m20.1.1.2.1.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1.2.1">⋅</ci><ci id="Thmtheorem14.p1.20.20.m20.1.1.2.2.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1.2.2">𝐶</ci><cn id="Thmtheorem14.p1.20.20.m20.1.1.2.3.cmml" type="integer" xref="Thmtheorem14.p1.20.20.m20.1.1.2.3">2</cn></apply><ci id="Thmtheorem14.p1.20.20.m20.1.1.3.cmml" xref="Thmtheorem14.p1.20.20.m20.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.20.20.m20.1c">C\cdot 2n</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.20.20.m20.1d">italic_C ⋅ 2 italic_n</annotation></semantics></math> <math alttext="\mathsf{AND}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.21.21.m21.1"><semantics id="Thmtheorem14.p1.21.21.m21.1a"><mi id="Thmtheorem14.p1.21.21.m21.1.1" xref="Thmtheorem14.p1.21.21.m21.1.1.cmml">𝖠𝖭𝖣</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.21.21.m21.1b"><ci id="Thmtheorem14.p1.21.21.m21.1.1.cmml" xref="Thmtheorem14.p1.21.21.m21.1.1">𝖠𝖭𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.21.21.m21.1c">\mathsf{AND}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.21.21.m21.1d">sansserif_AND</annotation></semantics></math> and <math alttext="\mathsf{OR}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.22.22.m22.1"><semantics id="Thmtheorem14.p1.22.22.m22.1a"><mi id="Thmtheorem14.p1.22.22.m22.1.1" xref="Thmtheorem14.p1.22.22.m22.1.1.cmml">𝖮𝖱</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.22.22.m22.1b"><ci id="Thmtheorem14.p1.22.22.m22.1.1.cmml" xref="Thmtheorem14.p1.22.22.m22.1.1">𝖮𝖱</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.22.22.m22.1c">\mathsf{OR}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.22.22.m22.1d">sansserif_OR</annotation></semantics></math> gates in total <em class="ltx_emph ltx_font_upright" id="Thmtheorem14.p1.27.27.1">(</em>assuming a circuit model with access to input literals and without <math alttext="\mathsf{NOT}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.23.23.m23.1"><semantics id="Thmtheorem14.p1.23.23.m23.1a"><mi id="Thmtheorem14.p1.23.23.m23.1.1" xref="Thmtheorem14.p1.23.23.m23.1.1.cmml">𝖭𝖮𝖳</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.23.23.m23.1b"><ci id="Thmtheorem14.p1.23.23.m23.1.1.cmml" xref="Thmtheorem14.p1.23.23.m23.1.1">𝖭𝖮𝖳</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.23.23.m23.1c">\mathsf{NOT}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.23.23.m23.1d">sansserif_NOT</annotation></semantics></math> gates<em class="ltx_emph ltx_font_upright" id="Thmtheorem14.p1.27.27.2">)</em>. This follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem13" title="Lemma 13 (Tight transference from graph complexity to circuit complexity). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">13</span></a> and Boolean duality, i.e., that the <math alttext="\mathsf{AND}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.24.24.m24.1"><semantics id="Thmtheorem14.p1.24.24.m24.1a"><mi id="Thmtheorem14.p1.24.24.m24.1.1" xref="Thmtheorem14.p1.24.24.m24.1.1.cmml">𝖠𝖭𝖣</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.24.24.m24.1b"><ci id="Thmtheorem14.p1.24.24.m24.1.1.cmml" xref="Thmtheorem14.p1.24.24.m24.1.1">𝖠𝖭𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.24.24.m24.1c">\mathsf{AND}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.24.24.m24.1d">sansserif_AND</annotation></semantics></math> complexity of a Boolean function coincides with the <math alttext="\mathsf{OR}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.25.25.m25.1"><semantics id="Thmtheorem14.p1.25.25.m25.1a"><mi id="Thmtheorem14.p1.25.25.m25.1.1" xref="Thmtheorem14.p1.25.25.m25.1.1.cmml">𝖮𝖱</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.25.25.m25.1b"><ci id="Thmtheorem14.p1.25.25.m25.1.1.cmml" xref="Thmtheorem14.p1.25.25.m25.1.1">𝖮𝖱</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.25.25.m25.1c">\mathsf{OR}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.25.25.m25.1d">sansserif_OR</annotation></semantics></math> complexity of its negation. Formally, letting <math alttext="\mathcal{B}_{\ell}" class="ltx_Math" display="inline" id="Thmtheorem14.p1.26.26.m26.1"><semantics id="Thmtheorem14.p1.26.26.m26.1a"><msub id="Thmtheorem14.p1.26.26.m26.1.1" xref="Thmtheorem14.p1.26.26.m26.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem14.p1.26.26.m26.1.1.2" xref="Thmtheorem14.p1.26.26.m26.1.1.2.cmml">ℬ</mi><mi id="Thmtheorem14.p1.26.26.m26.1.1.3" mathvariant="normal" xref="Thmtheorem14.p1.26.26.m26.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.26.26.m26.1b"><apply id="Thmtheorem14.p1.26.26.m26.1.1.cmml" xref="Thmtheorem14.p1.26.26.m26.1.1"><csymbol cd="ambiguous" id="Thmtheorem14.p1.26.26.m26.1.1.1.cmml" xref="Thmtheorem14.p1.26.26.m26.1.1">subscript</csymbol><ci id="Thmtheorem14.p1.26.26.m26.1.1.2.cmml" xref="Thmtheorem14.p1.26.26.m26.1.1.2">ℬ</ci><ci id="Thmtheorem14.p1.26.26.m26.1.1.3.cmml" xref="Thmtheorem14.p1.26.26.m26.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.26.26.m26.1c">\mathcal{B}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.26.26.m26.1d">caligraphic_B start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> denote the standard set of generators in the Boolean circuit complexity of <math alttext="\ell" class="ltx_Math" display="inline" id="Thmtheorem14.p1.27.27.m27.1"><semantics id="Thmtheorem14.p1.27.27.m27.1a"><mi id="Thmtheorem14.p1.27.27.m27.1.1" mathvariant="normal" xref="Thmtheorem14.p1.27.27.m27.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem14.p1.27.27.m27.1b"><ci id="Thmtheorem14.p1.27.27.m27.1.1.cmml" xref="Thmtheorem14.p1.27.27.m27.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem14.p1.27.27.m27.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem14.p1.27.27.m27.1d">roman_ℓ</annotation></semantics></math>-bit Boolean functions, we have:</span></p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx3"> <tbody id="S2.Ex12"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D(F\mid\mathcal{B}_{m})" class="ltx_Math" display="inline" id="S2.Ex8.m1.1"><semantics id="S2.Ex8.m1.1a"><mrow id="S2.Ex8.m1.1.1" xref="S2.Ex8.m1.1.1.cmml"><mi id="S2.Ex8.m1.1.1.3" xref="S2.Ex8.m1.1.1.3.cmml">D</mi><mo id="S2.Ex8.m1.1.1.2" xref="S2.Ex8.m1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex8.m1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.cmml"><mo id="S2.Ex8.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex8.m1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.1.1.1.1.1.2" xref="S2.Ex8.m1.1.1.1.1.1.2.cmml">F</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.cmml">∣</mo><msub id="S2.Ex8.m1.1.1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex8.m1.1.1.1.1.1.3.2" xref="S2.Ex8.m1.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex8.m1.1.1.1.1.1.3.3" xref="S2.Ex8.m1.1.1.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex8.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex8.m1.1b"><apply id="S2.Ex8.m1.1.1.cmml" xref="S2.Ex8.m1.1.1"><times id="S2.Ex8.m1.1.1.2.cmml" xref="S2.Ex8.m1.1.1.2"></times><ci id="S2.Ex8.m1.1.1.3.cmml" xref="S2.Ex8.m1.1.1.3">𝐷</ci><apply id="S2.Ex8.m1.1.1.1.1.1.cmml" xref="S2.Ex8.m1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex8.m1.1.1.1.1.1.1.cmml" xref="S2.Ex8.m1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.Ex8.m1.1.1.1.1.1.2.cmml" xref="S2.Ex8.m1.1.1.1.1.1.2">𝐹</ci><apply id="S2.Ex8.m1.1.1.1.1.1.3.cmml" xref="S2.Ex8.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m1.1.1.1.1.1.3.1.cmml" xref="S2.Ex8.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m1.1.1.1.1.1.3.2.cmml" xref="S2.Ex8.m1.1.1.1.1.1.3.2">ℬ</ci><ci id="S2.Ex8.m1.1.1.1.1.1.3.3.cmml" xref="S2.Ex8.m1.1.1.1.1.1.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.1c">\displaystyle D(F\mid\mathcal{B}_{m})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.1d">italic_D ( italic_F ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S2.Ex8.m2.1"><semantics id="S2.Ex8.m2.1a"><mo id="S2.Ex8.m2.1.1" xref="S2.Ex8.m2.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex8.m2.1b"><geq id="S2.Ex8.m2.1.1.cmml" xref="S2.Ex8.m2.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m2.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m2.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle D_{\cap}(F\mid\mathcal{B}_{m})+D_{\cup}(F\mid\mathcal{B}_{m})" class="ltx_Math" display="inline" id="S2.Ex8.m3.2"><semantics id="S2.Ex8.m3.2a"><mrow id="S2.Ex8.m3.2.2" xref="S2.Ex8.m3.2.2.cmml"><mrow id="S2.Ex8.m3.1.1.1" xref="S2.Ex8.m3.1.1.1.cmml"><msub id="S2.Ex8.m3.1.1.1.3" xref="S2.Ex8.m3.1.1.1.3.cmml"><mi id="S2.Ex8.m3.1.1.1.3.2" xref="S2.Ex8.m3.1.1.1.3.2.cmml">D</mi><mo id="S2.Ex8.m3.1.1.1.3.3" xref="S2.Ex8.m3.1.1.1.3.3.cmml">∩</mo></msub><mo id="S2.Ex8.m3.1.1.1.2" xref="S2.Ex8.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex8.m3.1.1.1.1.1" xref="S2.Ex8.m3.1.1.1.1.1.1.cmml"><mo id="S2.Ex8.m3.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex8.m3.1.1.1.1.1.1" xref="S2.Ex8.m3.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m3.1.1.1.1.1.1.2" xref="S2.Ex8.m3.1.1.1.1.1.1.2.cmml">F</mi><mo id="S2.Ex8.m3.1.1.1.1.1.1.1" xref="S2.Ex8.m3.1.1.1.1.1.1.1.cmml">∣</mo><msub id="S2.Ex8.m3.1.1.1.1.1.1.3" xref="S2.Ex8.m3.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex8.m3.1.1.1.1.1.1.3.2" xref="S2.Ex8.m3.1.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex8.m3.1.1.1.1.1.1.3.3" xref="S2.Ex8.m3.1.1.1.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex8.m3.1.1.1.1.1.3" stretchy="false" xref="S2.Ex8.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex8.m3.2.2.3" xref="S2.Ex8.m3.2.2.3.cmml">+</mo><mrow id="S2.Ex8.m3.2.2.2" xref="S2.Ex8.m3.2.2.2.cmml"><msub id="S2.Ex8.m3.2.2.2.3" xref="S2.Ex8.m3.2.2.2.3.cmml"><mi id="S2.Ex8.m3.2.2.2.3.2" xref="S2.Ex8.m3.2.2.2.3.2.cmml">D</mi><mo id="S2.Ex8.m3.2.2.2.3.3" xref="S2.Ex8.m3.2.2.2.3.3.cmml">∪</mo></msub><mo id="S2.Ex8.m3.2.2.2.2" xref="S2.Ex8.m3.2.2.2.2.cmml">⁢</mo><mrow id="S2.Ex8.m3.2.2.2.1.1" xref="S2.Ex8.m3.2.2.2.1.1.1.cmml"><mo id="S2.Ex8.m3.2.2.2.1.1.2" stretchy="false" xref="S2.Ex8.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex8.m3.2.2.2.1.1.1" xref="S2.Ex8.m3.2.2.2.1.1.1.cmml"><mi id="S2.Ex8.m3.2.2.2.1.1.1.2" xref="S2.Ex8.m3.2.2.2.1.1.1.2.cmml">F</mi><mo id="S2.Ex8.m3.2.2.2.1.1.1.1" xref="S2.Ex8.m3.2.2.2.1.1.1.1.cmml">∣</mo><msub id="S2.Ex8.m3.2.2.2.1.1.1.3" xref="S2.Ex8.m3.2.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex8.m3.2.2.2.1.1.1.3.2" xref="S2.Ex8.m3.2.2.2.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex8.m3.2.2.2.1.1.1.3.3" xref="S2.Ex8.m3.2.2.2.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex8.m3.2.2.2.1.1.3" stretchy="false" xref="S2.Ex8.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex8.m3.2b"><apply id="S2.Ex8.m3.2.2.cmml" xref="S2.Ex8.m3.2.2"><plus id="S2.Ex8.m3.2.2.3.cmml" xref="S2.Ex8.m3.2.2.3"></plus><apply id="S2.Ex8.m3.1.1.1.cmml" xref="S2.Ex8.m3.1.1.1"><times id="S2.Ex8.m3.1.1.1.2.cmml" xref="S2.Ex8.m3.1.1.1.2"></times><apply id="S2.Ex8.m3.1.1.1.3.cmml" xref="S2.Ex8.m3.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m3.1.1.1.3.1.cmml" xref="S2.Ex8.m3.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m3.1.1.1.3.2.cmml" xref="S2.Ex8.m3.1.1.1.3.2">𝐷</ci><intersect id="S2.Ex8.m3.1.1.1.3.3.cmml" xref="S2.Ex8.m3.1.1.1.3.3"></intersect></apply><apply id="S2.Ex8.m3.1.1.1.1.1.1.cmml" xref="S2.Ex8.m3.1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex8.m3.1.1.1.1.1.1.1.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.Ex8.m3.1.1.1.1.1.1.2.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.2">𝐹</ci><apply id="S2.Ex8.m3.1.1.1.1.1.1.3.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m3.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m3.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.3.2">ℬ</ci><ci id="S2.Ex8.m3.1.1.1.1.1.1.3.3.cmml" xref="S2.Ex8.m3.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="S2.Ex8.m3.2.2.2.cmml" xref="S2.Ex8.m3.2.2.2"><times id="S2.Ex8.m3.2.2.2.2.cmml" xref="S2.Ex8.m3.2.2.2.2"></times><apply id="S2.Ex8.m3.2.2.2.3.cmml" xref="S2.Ex8.m3.2.2.2.3"><csymbol cd="ambiguous" id="S2.Ex8.m3.2.2.2.3.1.cmml" xref="S2.Ex8.m3.2.2.2.3">subscript</csymbol><ci id="S2.Ex8.m3.2.2.2.3.2.cmml" xref="S2.Ex8.m3.2.2.2.3.2">𝐷</ci><union id="S2.Ex8.m3.2.2.2.3.3.cmml" xref="S2.Ex8.m3.2.2.2.3.3"></union></apply><apply id="S2.Ex8.m3.2.2.2.1.1.1.cmml" xref="S2.Ex8.m3.2.2.2.1.1"><csymbol cd="latexml" id="S2.Ex8.m3.2.2.2.1.1.1.1.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.1">conditional</csymbol><ci id="S2.Ex8.m3.2.2.2.1.1.1.2.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.2">𝐹</ci><apply id="S2.Ex8.m3.2.2.2.1.1.1.3.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m3.2.2.2.1.1.1.3.1.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m3.2.2.2.1.1.1.3.2.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.3.2">ℬ</ci><ci id="S2.Ex8.m3.2.2.2.1.1.1.3.3.cmml" xref="S2.Ex8.m3.2.2.2.1.1.1.3.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m3.2c">\displaystyle D_{\cap}(F\mid\mathcal{B}_{m})+D_{\cup}(F\mid\mathcal{B}_{m})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m3.2d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_F ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) + italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_F ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S2.Ex9.m1.1"><semantics id="S2.Ex9.m1.1a"><mo id="S2.Ex9.m1.1.1" xref="S2.Ex9.m1.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex9.m1.1b"><geq id="S2.Ex9.m1.1.1.cmml" xref="S2.Ex9.m1.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m1.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle D_{\cap}(f_{H}\mid\mathcal{B}_{m})+D_{\cup}(\overline{f_{H}}\mid% \mathcal{B}_{m})-O(1)" class="ltx_Math" display="inline" id="S2.Ex9.m2.3"><semantics id="S2.Ex9.m2.3a"><mrow id="S2.Ex9.m2.3.3" xref="S2.Ex9.m2.3.3.cmml"><mrow id="S2.Ex9.m2.3.3.2" xref="S2.Ex9.m2.3.3.2.cmml"><mrow id="S2.Ex9.m2.2.2.1.1" xref="S2.Ex9.m2.2.2.1.1.cmml"><msub id="S2.Ex9.m2.2.2.1.1.3" xref="S2.Ex9.m2.2.2.1.1.3.cmml"><mi id="S2.Ex9.m2.2.2.1.1.3.2" xref="S2.Ex9.m2.2.2.1.1.3.2.cmml">D</mi><mo id="S2.Ex9.m2.2.2.1.1.3.3" xref="S2.Ex9.m2.2.2.1.1.3.3.cmml">∩</mo></msub><mo id="S2.Ex9.m2.2.2.1.1.2" xref="S2.Ex9.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9.m2.2.2.1.1.1.1" xref="S2.Ex9.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.Ex9.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.Ex9.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex9.m2.2.2.1.1.1.1.1" xref="S2.Ex9.m2.2.2.1.1.1.1.1.cmml"><msub id="S2.Ex9.m2.2.2.1.1.1.1.1.2" xref="S2.Ex9.m2.2.2.1.1.1.1.1.2.cmml"><mi id="S2.Ex9.m2.2.2.1.1.1.1.1.2.2" xref="S2.Ex9.m2.2.2.1.1.1.1.1.2.2.cmml">f</mi><mi id="S2.Ex9.m2.2.2.1.1.1.1.1.2.3" xref="S2.Ex9.m2.2.2.1.1.1.1.1.2.3.cmml">H</mi></msub><mo id="S2.Ex9.m2.2.2.1.1.1.1.1.1" xref="S2.Ex9.m2.2.2.1.1.1.1.1.1.cmml">∣</mo><msub id="S2.Ex9.m2.2.2.1.1.1.1.1.3" xref="S2.Ex9.m2.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m2.2.2.1.1.1.1.1.3.2" xref="S2.Ex9.m2.2.2.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex9.m2.2.2.1.1.1.1.1.3.3" xref="S2.Ex9.m2.2.2.1.1.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex9.m2.2.2.1.1.1.1.3" stretchy="false" xref="S2.Ex9.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex9.m2.3.3.2.3" xref="S2.Ex9.m2.3.3.2.3.cmml">+</mo><mrow id="S2.Ex9.m2.3.3.2.2" xref="S2.Ex9.m2.3.3.2.2.cmml"><msub id="S2.Ex9.m2.3.3.2.2.3" xref="S2.Ex9.m2.3.3.2.2.3.cmml"><mi id="S2.Ex9.m2.3.3.2.2.3.2" xref="S2.Ex9.m2.3.3.2.2.3.2.cmml">D</mi><mo id="S2.Ex9.m2.3.3.2.2.3.3" xref="S2.Ex9.m2.3.3.2.2.3.3.cmml">∪</mo></msub><mo id="S2.Ex9.m2.3.3.2.2.2" xref="S2.Ex9.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.Ex9.m2.3.3.2.2.1.1" xref="S2.Ex9.m2.3.3.2.2.1.1.1.cmml"><mo id="S2.Ex9.m2.3.3.2.2.1.1.2" stretchy="false" xref="S2.Ex9.m2.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex9.m2.3.3.2.2.1.1.1" xref="S2.Ex9.m2.3.3.2.2.1.1.1.cmml"><mover accent="true" id="S2.Ex9.m2.3.3.2.2.1.1.1.2" xref="S2.Ex9.m2.3.3.2.2.1.1.1.2.cmml"><msub id="S2.Ex9.m2.3.3.2.2.1.1.1.2.2" xref="S2.Ex9.m2.3.3.2.2.1.1.1.2.2.cmml"><mi id="S2.Ex9.m2.3.3.2.2.1.1.1.2.2.2" xref="S2.Ex9.m2.3.3.2.2.1.1.1.2.2.2.cmml">f</mi><mi id="S2.Ex9.m2.3.3.2.2.1.1.1.2.2.3" xref="S2.Ex9.m2.3.3.2.2.1.1.1.2.2.3.cmml">H</mi></msub><mo id="S2.Ex9.m2.3.3.2.2.1.1.1.2.1" xref="S2.Ex9.m2.3.3.2.2.1.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Ex9.m2.3.3.2.2.1.1.1.1" xref="S2.Ex9.m2.3.3.2.2.1.1.1.1.cmml">∣</mo><msub id="S2.Ex9.m2.3.3.2.2.1.1.1.3" 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id="S2.Ex9.m2.3.3.2.2.1.1.1.3.2.cmml" xref="S2.Ex9.m2.3.3.2.2.1.1.1.3.2">ℬ</ci><ci id="S2.Ex9.m2.3.3.2.2.1.1.1.3.3.cmml" xref="S2.Ex9.m2.3.3.2.2.1.1.1.3.3">𝑚</ci></apply></apply></apply></apply><apply id="S2.Ex9.m2.3.3.4.cmml" xref="S2.Ex9.m2.3.3.4"><times id="S2.Ex9.m2.3.3.4.1.cmml" xref="S2.Ex9.m2.3.3.4.1"></times><ci id="S2.Ex9.m2.3.3.4.2.cmml" xref="S2.Ex9.m2.3.3.4.2">𝑂</ci><cn id="S2.Ex9.m2.1.1.cmml" type="integer" xref="S2.Ex9.m2.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m2.3c">\displaystyle D_{\cap}(f_{H}\mid\mathcal{B}_{m})+D_{\cup}(\overline{f_{H}}\mid% \mathcal{B}_{m})-O(1)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m2.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) + italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( over¯ start_ARG italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_ARG ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) - italic_O ( 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex10.m1.1"><semantics id="S2.Ex10.m1.1a"><mo id="S2.Ex10.m1.1.1" xref="S2.Ex10.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex10.m1.1b"><eq id="S2.Ex10.m1.1.1.cmml" xref="S2.Ex10.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle D_{\cap}(f_{H}\mid\mathcal{B}_{m})+D_{\cap}(f_{H}\mid\mathcal{B}% _{m})-O(1)" class="ltx_Math" display="inline" id="S2.Ex10.m2.3"><semantics id="S2.Ex10.m2.3a"><mrow id="S2.Ex10.m2.3.3" xref="S2.Ex10.m2.3.3.cmml"><mrow id="S2.Ex10.m2.3.3.2" xref="S2.Ex10.m2.3.3.2.cmml"><mrow id="S2.Ex10.m2.2.2.1.1" xref="S2.Ex10.m2.2.2.1.1.cmml"><msub id="S2.Ex10.m2.2.2.1.1.3" xref="S2.Ex10.m2.2.2.1.1.3.cmml"><mi id="S2.Ex10.m2.2.2.1.1.3.2" xref="S2.Ex10.m2.2.2.1.1.3.2.cmml">D</mi><mo id="S2.Ex10.m2.2.2.1.1.3.3" xref="S2.Ex10.m2.2.2.1.1.3.3.cmml">∩</mo></msub><mo id="S2.Ex10.m2.2.2.1.1.2" xref="S2.Ex10.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Ex10.m2.2.2.1.1.1.1" xref="S2.Ex10.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.Ex10.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.Ex10.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex10.m2.2.2.1.1.1.1.1" xref="S2.Ex10.m2.2.2.1.1.1.1.1.cmml"><msub id="S2.Ex10.m2.2.2.1.1.1.1.1.2" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2.cmml"><mi id="S2.Ex10.m2.2.2.1.1.1.1.1.2.2" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2.2.cmml">f</mi><mi id="S2.Ex10.m2.2.2.1.1.1.1.1.2.3" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2.3.cmml">H</mi></msub><mo id="S2.Ex10.m2.2.2.1.1.1.1.1.1" xref="S2.Ex10.m2.2.2.1.1.1.1.1.1.cmml">∣</mo><msub id="S2.Ex10.m2.2.2.1.1.1.1.1.3" xref="S2.Ex10.m2.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex10.m2.2.2.1.1.1.1.1.3.2" xref="S2.Ex10.m2.2.2.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex10.m2.2.2.1.1.1.1.1.3.3" xref="S2.Ex10.m2.2.2.1.1.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex10.m2.2.2.1.1.1.1.3" stretchy="false" xref="S2.Ex10.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex10.m2.3.3.2.3" xref="S2.Ex10.m2.3.3.2.3.cmml">+</mo><mrow id="S2.Ex10.m2.3.3.2.2" xref="S2.Ex10.m2.3.3.2.2.cmml"><msub id="S2.Ex10.m2.3.3.2.2.3" xref="S2.Ex10.m2.3.3.2.2.3.cmml"><mi id="S2.Ex10.m2.3.3.2.2.3.2" xref="S2.Ex10.m2.3.3.2.2.3.2.cmml">D</mi><mo id="S2.Ex10.m2.3.3.2.2.3.3" xref="S2.Ex10.m2.3.3.2.2.3.3.cmml">∩</mo></msub><mo id="S2.Ex10.m2.3.3.2.2.2" xref="S2.Ex10.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.Ex10.m2.3.3.2.2.1.1" xref="S2.Ex10.m2.3.3.2.2.1.1.1.cmml"><mo id="S2.Ex10.m2.3.3.2.2.1.1.2" stretchy="false" xref="S2.Ex10.m2.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex10.m2.3.3.2.2.1.1.1" xref="S2.Ex10.m2.3.3.2.2.1.1.1.cmml"><msub id="S2.Ex10.m2.3.3.2.2.1.1.1.2" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2.cmml"><mi id="S2.Ex10.m2.3.3.2.2.1.1.1.2.2" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2.2.cmml">f</mi><mi id="S2.Ex10.m2.3.3.2.2.1.1.1.2.3" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2.3.cmml">H</mi></msub><mo id="S2.Ex10.m2.3.3.2.2.1.1.1.1" xref="S2.Ex10.m2.3.3.2.2.1.1.1.1.cmml">∣</mo><msub id="S2.Ex10.m2.3.3.2.2.1.1.1.3" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex10.m2.3.3.2.2.1.1.1.3.2" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.Ex10.m2.3.3.2.2.1.1.1.3.3" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3.3.cmml">m</mi></msub></mrow><mo id="S2.Ex10.m2.3.3.2.2.1.1.3" stretchy="false" xref="S2.Ex10.m2.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex10.m2.3.3.3" xref="S2.Ex10.m2.3.3.3.cmml">−</mo><mrow id="S2.Ex10.m2.3.3.4" xref="S2.Ex10.m2.3.3.4.cmml"><mi id="S2.Ex10.m2.3.3.4.2" xref="S2.Ex10.m2.3.3.4.2.cmml">O</mi><mo id="S2.Ex10.m2.3.3.4.1" xref="S2.Ex10.m2.3.3.4.1.cmml">⁢</mo><mrow id="S2.Ex10.m2.3.3.4.3.2" xref="S2.Ex10.m2.3.3.4.cmml"><mo id="S2.Ex10.m2.3.3.4.3.2.1" stretchy="false" xref="S2.Ex10.m2.3.3.4.cmml">(</mo><mn id="S2.Ex10.m2.1.1" xref="S2.Ex10.m2.1.1.cmml">1</mn><mo id="S2.Ex10.m2.3.3.4.3.2.2" stretchy="false" xref="S2.Ex10.m2.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex10.m2.3b"><apply id="S2.Ex10.m2.3.3.cmml" xref="S2.Ex10.m2.3.3"><minus id="S2.Ex10.m2.3.3.3.cmml" xref="S2.Ex10.m2.3.3.3"></minus><apply id="S2.Ex10.m2.3.3.2.cmml" xref="S2.Ex10.m2.3.3.2"><plus id="S2.Ex10.m2.3.3.2.3.cmml" xref="S2.Ex10.m2.3.3.2.3"></plus><apply id="S2.Ex10.m2.2.2.1.1.cmml" xref="S2.Ex10.m2.2.2.1.1"><times id="S2.Ex10.m2.2.2.1.1.2.cmml" xref="S2.Ex10.m2.2.2.1.1.2"></times><apply id="S2.Ex10.m2.2.2.1.1.3.cmml" xref="S2.Ex10.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.Ex10.m2.2.2.1.1.3.1.cmml" xref="S2.Ex10.m2.2.2.1.1.3">subscript</csymbol><ci id="S2.Ex10.m2.2.2.1.1.3.2.cmml" xref="S2.Ex10.m2.2.2.1.1.3.2">𝐷</ci><intersect id="S2.Ex10.m2.2.2.1.1.3.3.cmml" xref="S2.Ex10.m2.2.2.1.1.3.3"></intersect></apply><apply id="S2.Ex10.m2.2.2.1.1.1.1.1.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1"><csymbol cd="latexml" id="S2.Ex10.m2.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1.1.1">conditional</csymbol><apply id="S2.Ex10.m2.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex10.m2.2.2.1.1.1.1.1.2.1.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="S2.Ex10.m2.2.2.1.1.1.1.1.2.2.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2.2">𝑓</ci><ci id="S2.Ex10.m2.2.2.1.1.1.1.1.2.3.cmml" xref="S2.Ex10.m2.2.2.1.1.1.1.1.2.3">𝐻</ci></apply><apply 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xref="S2.Ex10.m2.3.3.2.2.1.1.1.1">conditional</csymbol><apply id="S2.Ex10.m2.3.3.2.2.1.1.1.2.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex10.m2.3.3.2.2.1.1.1.2.1.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2">subscript</csymbol><ci id="S2.Ex10.m2.3.3.2.2.1.1.1.2.2.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2.2">𝑓</ci><ci id="S2.Ex10.m2.3.3.2.2.1.1.1.2.3.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.2.3">𝐻</ci></apply><apply id="S2.Ex10.m2.3.3.2.2.1.1.1.3.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex10.m2.3.3.2.2.1.1.1.3.1.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3">subscript</csymbol><ci id="S2.Ex10.m2.3.3.2.2.1.1.1.3.2.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3.2">ℬ</ci><ci id="S2.Ex10.m2.3.3.2.2.1.1.1.3.3.cmml" xref="S2.Ex10.m2.3.3.2.2.1.1.1.3.3">𝑚</ci></apply></apply></apply></apply><apply id="S2.Ex10.m2.3.3.4.cmml" xref="S2.Ex10.m2.3.3.4"><times id="S2.Ex10.m2.3.3.4.1.cmml" xref="S2.Ex10.m2.3.3.4.1"></times><ci id="S2.Ex10.m2.3.3.4.2.cmml" xref="S2.Ex10.m2.3.3.4.2">𝑂</ci><cn id="S2.Ex10.m2.1.1.cmml" type="integer" xref="S2.Ex10.m2.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m2.3c">\displaystyle D_{\cap}(f_{H}\mid\mathcal{B}_{m})+D_{\cap}(f_{H}\mid\mathcal{B}% _{m})-O(1)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m2.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) + italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∣ caligraphic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) - italic_O ( 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S2.Ex11.m1.1"><semantics id="S2.Ex11.m1.1a"><mo id="S2.Ex11.m1.1.1" xref="S2.Ex11.m1.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex11.m1.1b"><geq id="S2.Ex11.m1.1.1.cmml" xref="S2.Ex11.m1.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex11.m1.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S2.Ex11.m1.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 2\cdot D_{\cap}(H\mid\mathcal{G}_{N,N})-O(1)" class="ltx_Math" display="inline" id="S2.Ex11.m2.4"><semantics id="S2.Ex11.m2.4a"><mrow id="S2.Ex11.m2.4.4" xref="S2.Ex11.m2.4.4.cmml"><mrow id="S2.Ex11.m2.4.4.1" xref="S2.Ex11.m2.4.4.1.cmml"><mrow id="S2.Ex11.m2.4.4.1.3" xref="S2.Ex11.m2.4.4.1.3.cmml"><mn id="S2.Ex11.m2.4.4.1.3.2" xref="S2.Ex11.m2.4.4.1.3.2.cmml">2</mn><mo id="S2.Ex11.m2.4.4.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex11.m2.4.4.1.3.1.cmml">⋅</mo><msub id="S2.Ex11.m2.4.4.1.3.3" xref="S2.Ex11.m2.4.4.1.3.3.cmml"><mi id="S2.Ex11.m2.4.4.1.3.3.2" xref="S2.Ex11.m2.4.4.1.3.3.2.cmml">D</mi><mo id="S2.Ex11.m2.4.4.1.3.3.3" xref="S2.Ex11.m2.4.4.1.3.3.3.cmml">∩</mo></msub></mrow><mo id="S2.Ex11.m2.4.4.1.2" xref="S2.Ex11.m2.4.4.1.2.cmml">⁢</mo><mrow id="S2.Ex11.m2.4.4.1.1.1" xref="S2.Ex11.m2.4.4.1.1.1.1.cmml"><mo id="S2.Ex11.m2.4.4.1.1.1.2" stretchy="false" xref="S2.Ex11.m2.4.4.1.1.1.1.cmml">(</mo><mrow id="S2.Ex11.m2.4.4.1.1.1.1" xref="S2.Ex11.m2.4.4.1.1.1.1.cmml"><mi id="S2.Ex11.m2.4.4.1.1.1.1.2" xref="S2.Ex11.m2.4.4.1.1.1.1.2.cmml">H</mi><mo id="S2.Ex11.m2.4.4.1.1.1.1.1" xref="S2.Ex11.m2.4.4.1.1.1.1.1.cmml">∣</mo><msub id="S2.Ex11.m2.4.4.1.1.1.1.3" xref="S2.Ex11.m2.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex11.m2.4.4.1.1.1.1.3.2" xref="S2.Ex11.m2.4.4.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S2.Ex11.m2.2.2.2.4" xref="S2.Ex11.m2.2.2.2.3.cmml"><mi id="S2.Ex11.m2.1.1.1.1" xref="S2.Ex11.m2.1.1.1.1.cmml">N</mi><mo id="S2.Ex11.m2.2.2.2.4.1" xref="S2.Ex11.m2.2.2.2.3.cmml">,</mo><mi id="S2.Ex11.m2.2.2.2.2" xref="S2.Ex11.m2.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S2.Ex11.m2.4.4.1.1.1.3" stretchy="false" xref="S2.Ex11.m2.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex11.m2.4.4.2" xref="S2.Ex11.m2.4.4.2.cmml">−</mo><mrow id="S2.Ex11.m2.4.4.3" xref="S2.Ex11.m2.4.4.3.cmml"><mi id="S2.Ex11.m2.4.4.3.2" xref="S2.Ex11.m2.4.4.3.2.cmml">O</mi><mo id="S2.Ex11.m2.4.4.3.1" xref="S2.Ex11.m2.4.4.3.1.cmml">⁢</mo><mrow id="S2.Ex11.m2.4.4.3.3.2" xref="S2.Ex11.m2.4.4.3.cmml"><mo id="S2.Ex11.m2.4.4.3.3.2.1" stretchy="false" xref="S2.Ex11.m2.4.4.3.cmml">(</mo><mn id="S2.Ex11.m2.3.3" xref="S2.Ex11.m2.3.3.cmml">1</mn><mo id="S2.Ex11.m2.4.4.3.3.2.2" stretchy="false" xref="S2.Ex11.m2.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex11.m2.4b"><apply id="S2.Ex11.m2.4.4.cmml" xref="S2.Ex11.m2.4.4"><minus id="S2.Ex11.m2.4.4.2.cmml" xref="S2.Ex11.m2.4.4.2"></minus><apply id="S2.Ex11.m2.4.4.1.cmml" xref="S2.Ex11.m2.4.4.1"><times id="S2.Ex11.m2.4.4.1.2.cmml" xref="S2.Ex11.m2.4.4.1.2"></times><apply id="S2.Ex11.m2.4.4.1.3.cmml" xref="S2.Ex11.m2.4.4.1.3"><ci id="S2.Ex11.m2.4.4.1.3.1.cmml" xref="S2.Ex11.m2.4.4.1.3.1">⋅</ci><cn id="S2.Ex11.m2.4.4.1.3.2.cmml" type="integer" xref="S2.Ex11.m2.4.4.1.3.2">2</cn><apply id="S2.Ex11.m2.4.4.1.3.3.cmml" xref="S2.Ex11.m2.4.4.1.3.3"><csymbol cd="ambiguous" id="S2.Ex11.m2.4.4.1.3.3.1.cmml" xref="S2.Ex11.m2.4.4.1.3.3">subscript</csymbol><ci id="S2.Ex11.m2.4.4.1.3.3.2.cmml" xref="S2.Ex11.m2.4.4.1.3.3.2">𝐷</ci><intersect id="S2.Ex11.m2.4.4.1.3.3.3.cmml" xref="S2.Ex11.m2.4.4.1.3.3.3"></intersect></apply></apply><apply id="S2.Ex11.m2.4.4.1.1.1.1.cmml" xref="S2.Ex11.m2.4.4.1.1.1"><csymbol cd="latexml" id="S2.Ex11.m2.4.4.1.1.1.1.1.cmml" xref="S2.Ex11.m2.4.4.1.1.1.1.1">conditional</csymbol><ci id="S2.Ex11.m2.4.4.1.1.1.1.2.cmml" xref="S2.Ex11.m2.4.4.1.1.1.1.2">𝐻</ci><apply id="S2.Ex11.m2.4.4.1.1.1.1.3.cmml" xref="S2.Ex11.m2.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex11.m2.4.4.1.1.1.1.3.1.cmml" xref="S2.Ex11.m2.4.4.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex11.m2.4.4.1.1.1.1.3.2.cmml" xref="S2.Ex11.m2.4.4.1.1.1.1.3.2">𝒢</ci><list id="S2.Ex11.m2.2.2.2.3.cmml" xref="S2.Ex11.m2.2.2.2.4"><ci id="S2.Ex11.m2.1.1.1.1.cmml" xref="S2.Ex11.m2.1.1.1.1">𝑁</ci><ci id="S2.Ex11.m2.2.2.2.2.cmml" xref="S2.Ex11.m2.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="S2.Ex11.m2.4.4.3.cmml" xref="S2.Ex11.m2.4.4.3"><times id="S2.Ex11.m2.4.4.3.1.cmml" xref="S2.Ex11.m2.4.4.3.1"></times><ci id="S2.Ex11.m2.4.4.3.2.cmml" xref="S2.Ex11.m2.4.4.3.2">𝑂</ci><cn id="S2.Ex11.m2.3.3.cmml" type="integer" xref="S2.Ex11.m2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex11.m2.4c">\displaystyle 2\cdot D_{\cap}(H\mid\mathcal{G}_{N,N})-O(1)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex11.m2.4d">2 ⋅ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_H ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) - italic_O ( 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\geq" class="ltx_Math" display="inline" id="S2.Ex12.m1.1"><semantics id="S2.Ex12.m1.1a"><mo id="S2.Ex12.m1.1.1" xref="S2.Ex12.m1.1.1.cmml">≥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex12.m1.1b"><geq id="S2.Ex12.m1.1.1.cmml" xref="S2.Ex12.m1.1.1"></geq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m1.1c">\displaystyle\geq</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m1.1d">≥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 2\cdot C\cdot\log N=C\cdot 2n=C\cdot m-O(1)." class="ltx_Math" display="inline" id="S2.Ex12.m2.2"><semantics id="S2.Ex12.m2.2a"><mrow id="S2.Ex12.m2.2.2.1" xref="S2.Ex12.m2.2.2.1.1.cmml"><mrow id="S2.Ex12.m2.2.2.1.1" xref="S2.Ex12.m2.2.2.1.1.cmml"><mrow id="S2.Ex12.m2.2.2.1.1.2" xref="S2.Ex12.m2.2.2.1.1.2.cmml"><mn id="S2.Ex12.m2.2.2.1.1.2.2" xref="S2.Ex12.m2.2.2.1.1.2.2.cmml">2</mn><mo id="S2.Ex12.m2.2.2.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex12.m2.2.2.1.1.2.1.cmml">⋅</mo><mi id="S2.Ex12.m2.2.2.1.1.2.3" xref="S2.Ex12.m2.2.2.1.1.2.3.cmml">C</mi><mo id="S2.Ex12.m2.2.2.1.1.2.1a" lspace="0.222em" rspace="0.222em" xref="S2.Ex12.m2.2.2.1.1.2.1.cmml">⋅</mo><mrow id="S2.Ex12.m2.2.2.1.1.2.4" xref="S2.Ex12.m2.2.2.1.1.2.4.cmml"><mi id="S2.Ex12.m2.2.2.1.1.2.4.1" xref="S2.Ex12.m2.2.2.1.1.2.4.1.cmml">log</mi><mo id="S2.Ex12.m2.2.2.1.1.2.4a" lspace="0.167em" xref="S2.Ex12.m2.2.2.1.1.2.4.cmml">⁡</mo><mi id="S2.Ex12.m2.2.2.1.1.2.4.2" xref="S2.Ex12.m2.2.2.1.1.2.4.2.cmml">N</mi></mrow></mrow><mo id="S2.Ex12.m2.2.2.1.1.3" xref="S2.Ex12.m2.2.2.1.1.3.cmml">=</mo><mrow id="S2.Ex12.m2.2.2.1.1.4" xref="S2.Ex12.m2.2.2.1.1.4.cmml"><mrow id="S2.Ex12.m2.2.2.1.1.4.2" xref="S2.Ex12.m2.2.2.1.1.4.2.cmml"><mi id="S2.Ex12.m2.2.2.1.1.4.2.2" xref="S2.Ex12.m2.2.2.1.1.4.2.2.cmml">C</mi><mo id="S2.Ex12.m2.2.2.1.1.4.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex12.m2.2.2.1.1.4.2.1.cmml">⋅</mo><mn id="S2.Ex12.m2.2.2.1.1.4.2.3" xref="S2.Ex12.m2.2.2.1.1.4.2.3.cmml">2</mn></mrow><mo id="S2.Ex12.m2.2.2.1.1.4.1" xref="S2.Ex12.m2.2.2.1.1.4.1.cmml">⁢</mo><mi id="S2.Ex12.m2.2.2.1.1.4.3" xref="S2.Ex12.m2.2.2.1.1.4.3.cmml">n</mi></mrow><mo id="S2.Ex12.m2.2.2.1.1.5" xref="S2.Ex12.m2.2.2.1.1.5.cmml">=</mo><mrow id="S2.Ex12.m2.2.2.1.1.6" xref="S2.Ex12.m2.2.2.1.1.6.cmml"><mrow id="S2.Ex12.m2.2.2.1.1.6.2" xref="S2.Ex12.m2.2.2.1.1.6.2.cmml"><mi id="S2.Ex12.m2.2.2.1.1.6.2.2" xref="S2.Ex12.m2.2.2.1.1.6.2.2.cmml">C</mi><mo id="S2.Ex12.m2.2.2.1.1.6.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex12.m2.2.2.1.1.6.2.1.cmml">⋅</mo><mi id="S2.Ex12.m2.2.2.1.1.6.2.3" xref="S2.Ex12.m2.2.2.1.1.6.2.3.cmml">m</mi></mrow><mo id="S2.Ex12.m2.2.2.1.1.6.1" xref="S2.Ex12.m2.2.2.1.1.6.1.cmml">−</mo><mrow id="S2.Ex12.m2.2.2.1.1.6.3" xref="S2.Ex12.m2.2.2.1.1.6.3.cmml"><mi id="S2.Ex12.m2.2.2.1.1.6.3.2" xref="S2.Ex12.m2.2.2.1.1.6.3.2.cmml">O</mi><mo id="S2.Ex12.m2.2.2.1.1.6.3.1" xref="S2.Ex12.m2.2.2.1.1.6.3.1.cmml">⁢</mo><mrow id="S2.Ex12.m2.2.2.1.1.6.3.3.2" xref="S2.Ex12.m2.2.2.1.1.6.3.cmml"><mo id="S2.Ex12.m2.2.2.1.1.6.3.3.2.1" stretchy="false" xref="S2.Ex12.m2.2.2.1.1.6.3.cmml">(</mo><mn id="S2.Ex12.m2.1.1" xref="S2.Ex12.m2.1.1.cmml">1</mn><mo id="S2.Ex12.m2.2.2.1.1.6.3.3.2.2" stretchy="false" xref="S2.Ex12.m2.2.2.1.1.6.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex12.m2.2.2.1.2" lspace="0em" xref="S2.Ex12.m2.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex12.m2.2b"><apply id="S2.Ex12.m2.2.2.1.1.cmml" xref="S2.Ex12.m2.2.2.1"><and id="S2.Ex12.m2.2.2.1.1a.cmml" xref="S2.Ex12.m2.2.2.1"></and><apply id="S2.Ex12.m2.2.2.1.1b.cmml" xref="S2.Ex12.m2.2.2.1"><eq id="S2.Ex12.m2.2.2.1.1.3.cmml" xref="S2.Ex12.m2.2.2.1.1.3"></eq><apply id="S2.Ex12.m2.2.2.1.1.2.cmml" xref="S2.Ex12.m2.2.2.1.1.2"><ci id="S2.Ex12.m2.2.2.1.1.2.1.cmml" xref="S2.Ex12.m2.2.2.1.1.2.1">⋅</ci><cn id="S2.Ex12.m2.2.2.1.1.2.2.cmml" type="integer" xref="S2.Ex12.m2.2.2.1.1.2.2">2</cn><ci id="S2.Ex12.m2.2.2.1.1.2.3.cmml" xref="S2.Ex12.m2.2.2.1.1.2.3">𝐶</ci><apply id="S2.Ex12.m2.2.2.1.1.2.4.cmml" xref="S2.Ex12.m2.2.2.1.1.2.4"><log id="S2.Ex12.m2.2.2.1.1.2.4.1.cmml" xref="S2.Ex12.m2.2.2.1.1.2.4.1"></log><ci id="S2.Ex12.m2.2.2.1.1.2.4.2.cmml" xref="S2.Ex12.m2.2.2.1.1.2.4.2">𝑁</ci></apply></apply><apply id="S2.Ex12.m2.2.2.1.1.4.cmml" xref="S2.Ex12.m2.2.2.1.1.4"><times id="S2.Ex12.m2.2.2.1.1.4.1.cmml" xref="S2.Ex12.m2.2.2.1.1.4.1"></times><apply id="S2.Ex12.m2.2.2.1.1.4.2.cmml" xref="S2.Ex12.m2.2.2.1.1.4.2"><ci id="S2.Ex12.m2.2.2.1.1.4.2.1.cmml" xref="S2.Ex12.m2.2.2.1.1.4.2.1">⋅</ci><ci id="S2.Ex12.m2.2.2.1.1.4.2.2.cmml" xref="S2.Ex12.m2.2.2.1.1.4.2.2">𝐶</ci><cn id="S2.Ex12.m2.2.2.1.1.4.2.3.cmml" type="integer" xref="S2.Ex12.m2.2.2.1.1.4.2.3">2</cn></apply><ci id="S2.Ex12.m2.2.2.1.1.4.3.cmml" xref="S2.Ex12.m2.2.2.1.1.4.3">𝑛</ci></apply></apply><apply id="S2.Ex12.m2.2.2.1.1c.cmml" xref="S2.Ex12.m2.2.2.1"><eq id="S2.Ex12.m2.2.2.1.1.5.cmml" xref="S2.Ex12.m2.2.2.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S2.Ex12.m2.2.2.1.1.4.cmml" id="S2.Ex12.m2.2.2.1.1d.cmml" xref="S2.Ex12.m2.2.2.1"></share><apply id="S2.Ex12.m2.2.2.1.1.6.cmml" xref="S2.Ex12.m2.2.2.1.1.6"><minus id="S2.Ex12.m2.2.2.1.1.6.1.cmml" xref="S2.Ex12.m2.2.2.1.1.6.1"></minus><apply id="S2.Ex12.m2.2.2.1.1.6.2.cmml" xref="S2.Ex12.m2.2.2.1.1.6.2"><ci id="S2.Ex12.m2.2.2.1.1.6.2.1.cmml" xref="S2.Ex12.m2.2.2.1.1.6.2.1">⋅</ci><ci id="S2.Ex12.m2.2.2.1.1.6.2.2.cmml" xref="S2.Ex12.m2.2.2.1.1.6.2.2">𝐶</ci><ci id="S2.Ex12.m2.2.2.1.1.6.2.3.cmml" xref="S2.Ex12.m2.2.2.1.1.6.2.3">𝑚</ci></apply><apply id="S2.Ex12.m2.2.2.1.1.6.3.cmml" xref="S2.Ex12.m2.2.2.1.1.6.3"><times id="S2.Ex12.m2.2.2.1.1.6.3.1.cmml" xref="S2.Ex12.m2.2.2.1.1.6.3.1"></times><ci id="S2.Ex12.m2.2.2.1.1.6.3.2.cmml" xref="S2.Ex12.m2.2.2.1.1.6.3.2">𝑂</ci><cn id="S2.Ex12.m2.1.1.cmml" type="integer" xref="S2.Ex12.m2.1.1">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m2.2c">\displaystyle 2\cdot C\cdot\log N=C\cdot 2n=C\cdot m-O(1).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m2.2d">2 ⋅ italic_C ⋅ roman_log italic_N = italic_C ⋅ 2 italic_n = italic_C ⋅ italic_m - italic_O ( 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="Thmtheorem15"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem15.1.1.1">Remark 15</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem15.2.2"> </span>(Graph complexity lower bounds from circuit complexity lower bounds)<span class="ltx_text ltx_font_bold" id="Thmtheorem15.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem15.p1"> <p class="ltx_p" id="Thmtheorem15.p1.6"><span class="ltx_text ltx_font_italic" id="Thmtheorem15.p1.6.6">It is not hard to show by <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem12" title="Lemma 12. ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">12</span></a> and a similar argument that a lower bound of the form <math alttext="\omega(2^{n}\cdot n)" class="ltx_Math" display="inline" id="Thmtheorem15.p1.1.1.m1.1"><semantics id="Thmtheorem15.p1.1.1.m1.1a"><mrow id="Thmtheorem15.p1.1.1.m1.1.1" xref="Thmtheorem15.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem15.p1.1.1.m1.1.1.3" xref="Thmtheorem15.p1.1.1.m1.1.1.3.cmml">ω</mi><mo id="Thmtheorem15.p1.1.1.m1.1.1.2" xref="Thmtheorem15.p1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem15.p1.1.1.m1.1.1.1.1" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="Thmtheorem15.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.cmml"><msup id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.cmml"><mn id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.2" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.2.cmml">2</mn><mi id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.3" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.3.cmml">n</mi></msup><mo id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.1.cmml">⋅</mo><mi id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.3" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="Thmtheorem15.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.1.1.m1.1b"><apply id="Thmtheorem15.p1.1.1.m1.1.1.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1"><times id="Thmtheorem15.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.2"></times><ci id="Thmtheorem15.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.3">𝜔</ci><apply id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1"><ci id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.1">⋅</ci><apply id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.1.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2">superscript</csymbol><cn id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.2.cmml" type="integer" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.2">2</cn><ci id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.3.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="Thmtheorem15.p1.1.1.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.1.1.m1.1c">\omega(2^{n}\cdot n)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.1.1.m1.1d">italic_ω ( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ⋅ italic_n )</annotation></semantics></math> on the circuit complexity of a function <math alttext="h\colon\{0,1\}^{2n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem15.p1.2.2.m2.4"><semantics id="Thmtheorem15.p1.2.2.m2.4a"><mrow id="Thmtheorem15.p1.2.2.m2.4.5" xref="Thmtheorem15.p1.2.2.m2.4.5.cmml"><mi id="Thmtheorem15.p1.2.2.m2.4.5.2" xref="Thmtheorem15.p1.2.2.m2.4.5.2.cmml">h</mi><mo id="Thmtheorem15.p1.2.2.m2.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem15.p1.2.2.m2.4.5.1.cmml">:</mo><mrow id="Thmtheorem15.p1.2.2.m2.4.5.3" xref="Thmtheorem15.p1.2.2.m2.4.5.3.cmml"><msup id="Thmtheorem15.p1.2.2.m2.4.5.3.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.cmml"><mrow id="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.1.cmml"><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.1.cmml">{</mo><mn id="Thmtheorem15.p1.2.2.m2.1.1" xref="Thmtheorem15.p1.2.2.m2.1.1.cmml">0</mn><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.2.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.1.cmml">,</mo><mn id="Thmtheorem15.p1.2.2.m2.2.2" xref="Thmtheorem15.p1.2.2.m2.2.2.cmml">1</mn><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.2.3" stretchy="false" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.cmml"><mn id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.2.cmml">2</mn><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.1" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.1.cmml">⁢</mo><mi id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.3" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.3.cmml">n</mi></mrow></msup><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.1" stretchy="false" xref="Thmtheorem15.p1.2.2.m2.4.5.3.1.cmml">→</mo><mrow id="Thmtheorem15.p1.2.2.m2.4.5.3.3.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.3.1.cmml"><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem15.p1.2.2.m2.4.5.3.3.1.cmml">{</mo><mn id="Thmtheorem15.p1.2.2.m2.3.3" xref="Thmtheorem15.p1.2.2.m2.3.3.cmml">0</mn><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.3.2.2" xref="Thmtheorem15.p1.2.2.m2.4.5.3.3.1.cmml">,</mo><mn id="Thmtheorem15.p1.2.2.m2.4.4" xref="Thmtheorem15.p1.2.2.m2.4.4.cmml">1</mn><mo id="Thmtheorem15.p1.2.2.m2.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem15.p1.2.2.m2.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.2.2.m2.4b"><apply id="Thmtheorem15.p1.2.2.m2.4.5.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5"><ci id="Thmtheorem15.p1.2.2.m2.4.5.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.1">:</ci><ci id="Thmtheorem15.p1.2.2.m2.4.5.2.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.2">ℎ</ci><apply id="Thmtheorem15.p1.2.2.m2.4.5.3.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3"><ci id="Thmtheorem15.p1.2.2.m2.4.5.3.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.1">→</ci><apply id="Thmtheorem15.p1.2.2.m2.4.5.3.2.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="Thmtheorem15.p1.2.2.m2.4.5.3.2.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2">superscript</csymbol><set id="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.2.2"><cn id="Thmtheorem15.p1.2.2.m2.1.1.cmml" type="integer" xref="Thmtheorem15.p1.2.2.m2.1.1">0</cn><cn id="Thmtheorem15.p1.2.2.m2.2.2.cmml" type="integer" xref="Thmtheorem15.p1.2.2.m2.2.2">1</cn></set><apply id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3"><times id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.1"></times><cn id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.2.cmml" type="integer" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.2">2</cn><ci id="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.3.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.2.3.3">𝑛</ci></apply></apply><set id="Thmtheorem15.p1.2.2.m2.4.5.3.3.1.cmml" xref="Thmtheorem15.p1.2.2.m2.4.5.3.3.2"><cn id="Thmtheorem15.p1.2.2.m2.3.3.cmml" type="integer" xref="Thmtheorem15.p1.2.2.m2.3.3">0</cn><cn id="Thmtheorem15.p1.2.2.m2.4.4.cmml" type="integer" xref="Thmtheorem15.p1.2.2.m2.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.2.2.m2.4c">h\colon\{0,1\}^{2n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.2.2.m2.4d">italic_h : { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> implies a <math alttext="\omega(N)" class="ltx_Math" display="inline" id="Thmtheorem15.p1.3.3.m3.1"><semantics id="Thmtheorem15.p1.3.3.m3.1a"><mrow id="Thmtheorem15.p1.3.3.m3.1.2" xref="Thmtheorem15.p1.3.3.m3.1.2.cmml"><mi id="Thmtheorem15.p1.3.3.m3.1.2.2" xref="Thmtheorem15.p1.3.3.m3.1.2.2.cmml">ω</mi><mo id="Thmtheorem15.p1.3.3.m3.1.2.1" xref="Thmtheorem15.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="Thmtheorem15.p1.3.3.m3.1.2.3.2" xref="Thmtheorem15.p1.3.3.m3.1.2.cmml"><mo id="Thmtheorem15.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="Thmtheorem15.p1.3.3.m3.1.2.cmml">(</mo><mi id="Thmtheorem15.p1.3.3.m3.1.1" xref="Thmtheorem15.p1.3.3.m3.1.1.cmml">N</mi><mo id="Thmtheorem15.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="Thmtheorem15.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.3.3.m3.1b"><apply id="Thmtheorem15.p1.3.3.m3.1.2.cmml" xref="Thmtheorem15.p1.3.3.m3.1.2"><times id="Thmtheorem15.p1.3.3.m3.1.2.1.cmml" xref="Thmtheorem15.p1.3.3.m3.1.2.1"></times><ci id="Thmtheorem15.p1.3.3.m3.1.2.2.cmml" xref="Thmtheorem15.p1.3.3.m3.1.2.2">𝜔</ci><ci id="Thmtheorem15.p1.3.3.m3.1.1.cmml" xref="Thmtheorem15.p1.3.3.m3.1.1">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.3.3.m3.1c">\omega(N)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.3.3.m3.1d">italic_ω ( italic_N )</annotation></semantics></math> lower bound in graph complexity, where <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="Thmtheorem15.p1.4.4.m4.1"><semantics id="Thmtheorem15.p1.4.4.m4.1a"><mrow id="Thmtheorem15.p1.4.4.m4.1.1" xref="Thmtheorem15.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem15.p1.4.4.m4.1.1.2" xref="Thmtheorem15.p1.4.4.m4.1.1.2.cmml">N</mi><mo id="Thmtheorem15.p1.4.4.m4.1.1.1" xref="Thmtheorem15.p1.4.4.m4.1.1.1.cmml">=</mo><msup id="Thmtheorem15.p1.4.4.m4.1.1.3" xref="Thmtheorem15.p1.4.4.m4.1.1.3.cmml"><mn id="Thmtheorem15.p1.4.4.m4.1.1.3.2" xref="Thmtheorem15.p1.4.4.m4.1.1.3.2.cmml">2</mn><mi id="Thmtheorem15.p1.4.4.m4.1.1.3.3" xref="Thmtheorem15.p1.4.4.m4.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.4.4.m4.1b"><apply id="Thmtheorem15.p1.4.4.m4.1.1.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1"><eq id="Thmtheorem15.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1.1"></eq><ci id="Thmtheorem15.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1.2">𝑁</ci><apply id="Thmtheorem15.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem15.p1.4.4.m4.1.1.3.1.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1.3">superscript</csymbol><cn id="Thmtheorem15.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="Thmtheorem15.p1.4.4.m4.1.1.3.2">2</cn><ci id="Thmtheorem15.p1.4.4.m4.1.1.3.3.cmml" xref="Thmtheorem15.p1.4.4.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.4.4.m4.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.4.4.m4.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> as usual. On the other hand, note that by a counting argument there exist graphs computed by a single <em class="ltx_emph ltx_font_upright" id="Thmtheorem15.p1.6.6.1">(</em>unbounded fan-in<em class="ltx_emph ltx_font_upright" id="Thmtheorem15.p1.6.6.2">)</em> union whose corresponding <math alttext="2n" class="ltx_Math" display="inline" id="Thmtheorem15.p1.5.5.m5.1"><semantics id="Thmtheorem15.p1.5.5.m5.1a"><mrow id="Thmtheorem15.p1.5.5.m5.1.1" xref="Thmtheorem15.p1.5.5.m5.1.1.cmml"><mn id="Thmtheorem15.p1.5.5.m5.1.1.2" xref="Thmtheorem15.p1.5.5.m5.1.1.2.cmml">2</mn><mo id="Thmtheorem15.p1.5.5.m5.1.1.1" xref="Thmtheorem15.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi id="Thmtheorem15.p1.5.5.m5.1.1.3" xref="Thmtheorem15.p1.5.5.m5.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.5.5.m5.1b"><apply id="Thmtheorem15.p1.5.5.m5.1.1.cmml" xref="Thmtheorem15.p1.5.5.m5.1.1"><times id="Thmtheorem15.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem15.p1.5.5.m5.1.1.1"></times><cn id="Thmtheorem15.p1.5.5.m5.1.1.2.cmml" type="integer" xref="Thmtheorem15.p1.5.5.m5.1.1.2">2</cn><ci id="Thmtheorem15.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem15.p1.5.5.m5.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.5.5.m5.1c">2n</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.5.5.m5.1d">2 italic_n</annotation></semantics></math>-bit Boolean function has circuit complexity <math alttext="\Omega(2^{n}/n)" class="ltx_Math" display="inline" id="Thmtheorem15.p1.6.6.m6.1"><semantics id="Thmtheorem15.p1.6.6.m6.1a"><mrow id="Thmtheorem15.p1.6.6.m6.1.1" xref="Thmtheorem15.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem15.p1.6.6.m6.1.1.3" mathvariant="normal" xref="Thmtheorem15.p1.6.6.m6.1.1.3.cmml">Ω</mi><mo id="Thmtheorem15.p1.6.6.m6.1.1.2" xref="Thmtheorem15.p1.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem15.p1.6.6.m6.1.1.1.1" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.cmml"><mo id="Thmtheorem15.p1.6.6.m6.1.1.1.1.2" stretchy="false" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.cmml"><msup id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.cmml"><mn id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.2" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.2.cmml">2</mn><mi id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.3" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.3.cmml">n</mi></msup><mo id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.1" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.1.cmml">/</mo><mi id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.3" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="Thmtheorem15.p1.6.6.m6.1.1.1.1.3" stretchy="false" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem15.p1.6.6.m6.1b"><apply id="Thmtheorem15.p1.6.6.m6.1.1.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1"><times id="Thmtheorem15.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.2"></times><ci id="Thmtheorem15.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.3">Ω</ci><apply id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1"><divide id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.1"></divide><apply id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.1.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2">superscript</csymbol><cn id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.2.cmml" type="integer" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.2">2</cn><ci id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.3.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.3.cmml" xref="Thmtheorem15.p1.6.6.m6.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem15.p1.6.6.m6.1c">\Omega(2^{n}/n)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem15.p1.6.6.m6.1d">roman_Ω ( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT / italic_n )</annotation></semantics></math>. In particular, it follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem9" title="Lemma 9 (Immediate from [21]). ‣ 2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">9</span></a> that a Boolean function can have exponential intersection complexity, while the corresponding graph has zero intersection complexity.</span></p> </div> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S2.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.5 </span>Cyclic Discrete Complexity</h3> <div class="ltx_para" id="S2.SS5.p1"> <p class="ltx_p" id="S2.SS5.p1.23">We introduce a variant of the complexity measure <math alttext="D(\cdot\mid\cdot)" class="ltx_math_unparsed" display="inline" id="S2.SS5.p1.1.m1.1"><semantics id="S2.SS5.p1.1.m1.1a"><mrow id="S2.SS5.p1.1.m1.1b"><mi id="S2.SS5.p1.1.m1.1.1">D</mi><mrow id="S2.SS5.p1.1.m1.1.2"><mo id="S2.SS5.p1.1.m1.1.2.1" stretchy="false">(</mo><mo id="S2.SS5.p1.1.m1.1.2.2" lspace="0em" rspace="0em">⋅</mo><mo id="S2.SS5.p1.1.m1.1.2.3" lspace="0em" rspace="0em">∣</mo><mo id="S2.SS5.p1.1.m1.1.2.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.SS5.p1.1.m1.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS5.p1.1.m1.1c">D(\cdot\mid\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.1.m1.1d">italic_D ( ⋅ ∣ ⋅ )</annotation></semantics></math> that allows cyclic constructions. Formally, we use <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS5.p1.2.m2.1"><semantics id="S2.SS5.p1.2.m2.1a"><mrow id="S2.SS5.p1.2.m2.1.1" xref="S2.SS5.p1.2.m2.1.1.cmml"><mi id="S2.SS5.p1.2.m2.1.1.3" xref="S2.SS5.p1.2.m2.1.1.3.cmml">D</mi><mo id="S2.SS5.p1.2.m2.1.1.2" xref="S2.SS5.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.SS5.p1.2.m2.1.1.1.1" xref="S2.SS5.p1.2.m2.1.1.1.1.1.cmml"><mo id="S2.SS5.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S2.SS5.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.p1.2.m2.1.1.1.1.1" xref="S2.SS5.p1.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS5.p1.2.m2.1.1.1.1.1.2" xref="S2.SS5.p1.2.m2.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS5.p1.2.m2.1.1.1.1.1.1" xref="S2.SS5.p1.2.m2.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.2.m2.1.1.1.1.1.3" xref="S2.SS5.p1.2.m2.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS5.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S2.SS5.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.2.m2.1b"><apply id="S2.SS5.p1.2.m2.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1"><times id="S2.SS5.p1.2.m2.1.1.2.cmml" xref="S2.SS5.p1.2.m2.1.1.2"></times><ci id="S2.SS5.p1.2.m2.1.1.3.cmml" xref="S2.SS5.p1.2.m2.1.1.3">𝐷</ci><apply id="S2.SS5.p1.2.m2.1.1.1.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1.1.1"><csymbol cd="latexml" id="S2.SS5.p1.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS5.p1.2.m2.1.1.1.1.1.2.cmml" xref="S2.SS5.p1.2.m2.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS5.p1.2.m2.1.1.1.1.1.3.cmml" xref="S2.SS5.p1.2.m2.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.2.m2.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.2.m2.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> to denote the <em class="ltx_emph ltx_font_italic" id="S2.SS5.p1.23.1">cyclic discrete complexity</em> of <math alttext="A" class="ltx_Math" display="inline" id="S2.SS5.p1.3.m3.1"><semantics id="S2.SS5.p1.3.m3.1a"><mi id="S2.SS5.p1.3.m3.1.1" xref="S2.SS5.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.3.m3.1b"><ci id="S2.SS5.p1.3.m3.1.1.cmml" xref="S2.SS5.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.3.m3.1d">italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.p1.4.m4.1"><semantics id="S2.SS5.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.4.m4.1.1" xref="S2.SS5.p1.4.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.4.m4.1b"><ci id="S2.SS5.p1.4.m4.1.1.cmml" xref="S2.SS5.p1.4.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.4.m4.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.4.m4.1d">caligraphic_B</annotation></semantics></math>, defined as follows. We consider a <em class="ltx_emph ltx_font_italic" id="S2.SS5.p1.23.2">syntactic sequence</em> <math alttext="I_{1},\ldots,I_{t}" class="ltx_Math" display="inline" id="S2.SS5.p1.5.m5.3"><semantics id="S2.SS5.p1.5.m5.3a"><mrow id="S2.SS5.p1.5.m5.3.3.2" xref="S2.SS5.p1.5.m5.3.3.3.cmml"><msub id="S2.SS5.p1.5.m5.2.2.1.1" xref="S2.SS5.p1.5.m5.2.2.1.1.cmml"><mi id="S2.SS5.p1.5.m5.2.2.1.1.2" xref="S2.SS5.p1.5.m5.2.2.1.1.2.cmml">I</mi><mn id="S2.SS5.p1.5.m5.2.2.1.1.3" xref="S2.SS5.p1.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS5.p1.5.m5.3.3.2.3" xref="S2.SS5.p1.5.m5.3.3.3.cmml">,</mo><mi id="S2.SS5.p1.5.m5.1.1" mathvariant="normal" xref="S2.SS5.p1.5.m5.1.1.cmml">…</mi><mo id="S2.SS5.p1.5.m5.3.3.2.4" xref="S2.SS5.p1.5.m5.3.3.3.cmml">,</mo><msub id="S2.SS5.p1.5.m5.3.3.2.2" xref="S2.SS5.p1.5.m5.3.3.2.2.cmml"><mi id="S2.SS5.p1.5.m5.3.3.2.2.2" xref="S2.SS5.p1.5.m5.3.3.2.2.2.cmml">I</mi><mi id="S2.SS5.p1.5.m5.3.3.2.2.3" xref="S2.SS5.p1.5.m5.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.5.m5.3b"><list id="S2.SS5.p1.5.m5.3.3.3.cmml" xref="S2.SS5.p1.5.m5.3.3.2"><apply id="S2.SS5.p1.5.m5.2.2.1.1.cmml" xref="S2.SS5.p1.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.5.m5.2.2.1.1.1.cmml" xref="S2.SS5.p1.5.m5.2.2.1.1">subscript</csymbol><ci id="S2.SS5.p1.5.m5.2.2.1.1.2.cmml" xref="S2.SS5.p1.5.m5.2.2.1.1.2">𝐼</ci><cn id="S2.SS5.p1.5.m5.2.2.1.1.3.cmml" type="integer" xref="S2.SS5.p1.5.m5.2.2.1.1.3">1</cn></apply><ci id="S2.SS5.p1.5.m5.1.1.cmml" xref="S2.SS5.p1.5.m5.1.1">…</ci><apply id="S2.SS5.p1.5.m5.3.3.2.2.cmml" xref="S2.SS5.p1.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS5.p1.5.m5.3.3.2.2.1.cmml" xref="S2.SS5.p1.5.m5.3.3.2.2">subscript</csymbol><ci id="S2.SS5.p1.5.m5.3.3.2.2.2.cmml" xref="S2.SS5.p1.5.m5.3.3.2.2.2">𝐼</ci><ci id="S2.SS5.p1.5.m5.3.3.2.2.3.cmml" xref="S2.SS5.p1.5.m5.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.5.m5.3c">I_{1},\ldots,I_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.5.m5.3d">italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, together with a fixed operation of the form <math alttext="I_{i}=K_{i_{1}}\star_{i}K_{i_{2}}" class="ltx_Math" display="inline" id="S2.SS5.p1.6.m6.1"><semantics id="S2.SS5.p1.6.m6.1a"><mrow id="S2.SS5.p1.6.m6.1.1" xref="S2.SS5.p1.6.m6.1.1.cmml"><msub id="S2.SS5.p1.6.m6.1.1.2" xref="S2.SS5.p1.6.m6.1.1.2.cmml"><mi id="S2.SS5.p1.6.m6.1.1.2.2" xref="S2.SS5.p1.6.m6.1.1.2.2.cmml">I</mi><mi id="S2.SS5.p1.6.m6.1.1.2.3" xref="S2.SS5.p1.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS5.p1.6.m6.1.1.1" xref="S2.SS5.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS5.p1.6.m6.1.1.3" xref="S2.SS5.p1.6.m6.1.1.3.cmml"><msub id="S2.SS5.p1.6.m6.1.1.3.2" xref="S2.SS5.p1.6.m6.1.1.3.2.cmml"><mi id="S2.SS5.p1.6.m6.1.1.3.2.2" xref="S2.SS5.p1.6.m6.1.1.3.2.2.cmml">K</mi><msub id="S2.SS5.p1.6.m6.1.1.3.2.3" xref="S2.SS5.p1.6.m6.1.1.3.2.3.cmml"><mi id="S2.SS5.p1.6.m6.1.1.3.2.3.2" xref="S2.SS5.p1.6.m6.1.1.3.2.3.2.cmml">i</mi><mn id="S2.SS5.p1.6.m6.1.1.3.2.3.3" xref="S2.SS5.p1.6.m6.1.1.3.2.3.3.cmml">1</mn></msub></msub><msub id="S2.SS5.p1.6.m6.1.1.3.1" xref="S2.SS5.p1.6.m6.1.1.3.1.cmml"><mo id="S2.SS5.p1.6.m6.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS5.p1.6.m6.1.1.3.1.2.cmml">⋆</mo><mi id="S2.SS5.p1.6.m6.1.1.3.1.3" xref="S2.SS5.p1.6.m6.1.1.3.1.3.cmml">i</mi></msub><msub id="S2.SS5.p1.6.m6.1.1.3.3" xref="S2.SS5.p1.6.m6.1.1.3.3.cmml"><mi id="S2.SS5.p1.6.m6.1.1.3.3.2" xref="S2.SS5.p1.6.m6.1.1.3.3.2.cmml">K</mi><msub id="S2.SS5.p1.6.m6.1.1.3.3.3" xref="S2.SS5.p1.6.m6.1.1.3.3.3.cmml"><mi id="S2.SS5.p1.6.m6.1.1.3.3.3.2" xref="S2.SS5.p1.6.m6.1.1.3.3.3.2.cmml">i</mi><mn id="S2.SS5.p1.6.m6.1.1.3.3.3.3" xref="S2.SS5.p1.6.m6.1.1.3.3.3.3.cmml">2</mn></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.6.m6.1b"><apply id="S2.SS5.p1.6.m6.1.1.cmml" xref="S2.SS5.p1.6.m6.1.1"><eq id="S2.SS5.p1.6.m6.1.1.1.cmml" xref="S2.SS5.p1.6.m6.1.1.1"></eq><apply id="S2.SS5.p1.6.m6.1.1.2.cmml" xref="S2.SS5.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.2.1.cmml" xref="S2.SS5.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.2.2.cmml" xref="S2.SS5.p1.6.m6.1.1.2.2">𝐼</ci><ci id="S2.SS5.p1.6.m6.1.1.2.3.cmml" xref="S2.SS5.p1.6.m6.1.1.2.3">𝑖</ci></apply><apply id="S2.SS5.p1.6.m6.1.1.3.cmml" xref="S2.SS5.p1.6.m6.1.1.3"><apply id="S2.SS5.p1.6.m6.1.1.3.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.3.1.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.1">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.3.1.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.1.2">⋆</ci><ci id="S2.SS5.p1.6.m6.1.1.3.1.3.cmml" xref="S2.SS5.p1.6.m6.1.1.3.1.3">𝑖</ci></apply><apply id="S2.SS5.p1.6.m6.1.1.3.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.3.2.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.3.2.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2.2">𝐾</ci><apply id="S2.SS5.p1.6.m6.1.1.3.2.3.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.3.2.3.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2.3">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.3.2.3.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.2.3.2">𝑖</ci><cn id="S2.SS5.p1.6.m6.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS5.p1.6.m6.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.6.m6.1.1.3.3.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.3.3.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.3.3.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3.2">𝐾</ci><apply id="S2.SS5.p1.6.m6.1.1.3.3.3.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.1.1.3.3.3.1.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3.3">subscript</csymbol><ci id="S2.SS5.p1.6.m6.1.1.3.3.3.2.cmml" xref="S2.SS5.p1.6.m6.1.1.3.3.3.2">𝑖</ci><cn id="S2.SS5.p1.6.m6.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS5.p1.6.m6.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.6.m6.1c">I_{i}=K_{i_{1}}\star_{i}K_{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.6.m6.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="K_{i_{1}},K_{i_{2}}\in\{I_{1},\ldots,I_{t}\}\cup\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.p1.7.m7.5"><semantics id="S2.SS5.p1.7.m7.5a"><mrow id="S2.SS5.p1.7.m7.5.5" xref="S2.SS5.p1.7.m7.5.5.cmml"><mrow id="S2.SS5.p1.7.m7.3.3.2.2" xref="S2.SS5.p1.7.m7.3.3.2.3.cmml"><msub id="S2.SS5.p1.7.m7.2.2.1.1.1" xref="S2.SS5.p1.7.m7.2.2.1.1.1.cmml"><mi id="S2.SS5.p1.7.m7.2.2.1.1.1.2" xref="S2.SS5.p1.7.m7.2.2.1.1.1.2.cmml">K</mi><msub id="S2.SS5.p1.7.m7.2.2.1.1.1.3" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3.cmml"><mi id="S2.SS5.p1.7.m7.2.2.1.1.1.3.2" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3.2.cmml">i</mi><mn id="S2.SS5.p1.7.m7.2.2.1.1.1.3.3" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3.3.cmml">1</mn></msub></msub><mo id="S2.SS5.p1.7.m7.3.3.2.2.3" xref="S2.SS5.p1.7.m7.3.3.2.3.cmml">,</mo><msub id="S2.SS5.p1.7.m7.3.3.2.2.2" xref="S2.SS5.p1.7.m7.3.3.2.2.2.cmml"><mi id="S2.SS5.p1.7.m7.3.3.2.2.2.2" xref="S2.SS5.p1.7.m7.3.3.2.2.2.2.cmml">K</mi><msub id="S2.SS5.p1.7.m7.3.3.2.2.2.3" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3.cmml"><mi id="S2.SS5.p1.7.m7.3.3.2.2.2.3.2" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3.2.cmml">i</mi><mn id="S2.SS5.p1.7.m7.3.3.2.2.2.3.3" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3.3.cmml">2</mn></msub></msub></mrow><mo id="S2.SS5.p1.7.m7.5.5.5" xref="S2.SS5.p1.7.m7.5.5.5.cmml">∈</mo><mrow id="S2.SS5.p1.7.m7.5.5.4" xref="S2.SS5.p1.7.m7.5.5.4.cmml"><mrow id="S2.SS5.p1.7.m7.5.5.4.2.2" xref="S2.SS5.p1.7.m7.5.5.4.2.3.cmml"><mo id="S2.SS5.p1.7.m7.5.5.4.2.2.3" stretchy="false" xref="S2.SS5.p1.7.m7.5.5.4.2.3.cmml">{</mo><msub id="S2.SS5.p1.7.m7.4.4.3.1.1.1" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1.cmml"><mi id="S2.SS5.p1.7.m7.4.4.3.1.1.1.2" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1.2.cmml">I</mi><mn id="S2.SS5.p1.7.m7.4.4.3.1.1.1.3" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS5.p1.7.m7.5.5.4.2.2.4" xref="S2.SS5.p1.7.m7.5.5.4.2.3.cmml">,</mo><mi id="S2.SS5.p1.7.m7.1.1" mathvariant="normal" xref="S2.SS5.p1.7.m7.1.1.cmml">…</mi><mo id="S2.SS5.p1.7.m7.5.5.4.2.2.5" xref="S2.SS5.p1.7.m7.5.5.4.2.3.cmml">,</mo><msub id="S2.SS5.p1.7.m7.5.5.4.2.2.2" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2.cmml"><mi id="S2.SS5.p1.7.m7.5.5.4.2.2.2.2" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2.2.cmml">I</mi><mi id="S2.SS5.p1.7.m7.5.5.4.2.2.2.3" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2.3.cmml">t</mi></msub><mo id="S2.SS5.p1.7.m7.5.5.4.2.2.6" stretchy="false" xref="S2.SS5.p1.7.m7.5.5.4.2.3.cmml">}</mo></mrow><mo id="S2.SS5.p1.7.m7.5.5.4.3" xref="S2.SS5.p1.7.m7.5.5.4.3.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.7.m7.5.5.4.4" xref="S2.SS5.p1.7.m7.5.5.4.4.cmml">ℬ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.7.m7.5b"><apply id="S2.SS5.p1.7.m7.5.5.cmml" xref="S2.SS5.p1.7.m7.5.5"><in id="S2.SS5.p1.7.m7.5.5.5.cmml" xref="S2.SS5.p1.7.m7.5.5.5"></in><list id="S2.SS5.p1.7.m7.3.3.2.3.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2"><apply id="S2.SS5.p1.7.m7.2.2.1.1.1.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.2.2.1.1.1.1.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S2.SS5.p1.7.m7.2.2.1.1.1.2.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1.2">𝐾</ci><apply id="S2.SS5.p1.7.m7.2.2.1.1.1.3.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.2.2.1.1.1.3.1.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3">subscript</csymbol><ci id="S2.SS5.p1.7.m7.2.2.1.1.1.3.2.cmml" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3.2">𝑖</ci><cn id="S2.SS5.p1.7.m7.2.2.1.1.1.3.3.cmml" type="integer" xref="S2.SS5.p1.7.m7.2.2.1.1.1.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.7.m7.3.3.2.2.2.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.3.3.2.2.2.1.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2">subscript</csymbol><ci id="S2.SS5.p1.7.m7.3.3.2.2.2.2.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2.2">𝐾</ci><apply id="S2.SS5.p1.7.m7.3.3.2.2.2.3.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.3.3.2.2.2.3.1.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3">subscript</csymbol><ci id="S2.SS5.p1.7.m7.3.3.2.2.2.3.2.cmml" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3.2">𝑖</ci><cn id="S2.SS5.p1.7.m7.3.3.2.2.2.3.3.cmml" type="integer" xref="S2.SS5.p1.7.m7.3.3.2.2.2.3.3">2</cn></apply></apply></list><apply id="S2.SS5.p1.7.m7.5.5.4.cmml" xref="S2.SS5.p1.7.m7.5.5.4"><union id="S2.SS5.p1.7.m7.5.5.4.3.cmml" xref="S2.SS5.p1.7.m7.5.5.4.3"></union><set id="S2.SS5.p1.7.m7.5.5.4.2.3.cmml" xref="S2.SS5.p1.7.m7.5.5.4.2.2"><apply id="S2.SS5.p1.7.m7.4.4.3.1.1.1.cmml" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.4.4.3.1.1.1.1.cmml" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1">subscript</csymbol><ci id="S2.SS5.p1.7.m7.4.4.3.1.1.1.2.cmml" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1.2">𝐼</ci><cn id="S2.SS5.p1.7.m7.4.4.3.1.1.1.3.cmml" type="integer" xref="S2.SS5.p1.7.m7.4.4.3.1.1.1.3">1</cn></apply><ci id="S2.SS5.p1.7.m7.1.1.cmml" xref="S2.SS5.p1.7.m7.1.1">…</ci><apply id="S2.SS5.p1.7.m7.5.5.4.2.2.2.cmml" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p1.7.m7.5.5.4.2.2.2.1.cmml" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2">subscript</csymbol><ci id="S2.SS5.p1.7.m7.5.5.4.2.2.2.2.cmml" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2.2">𝐼</ci><ci id="S2.SS5.p1.7.m7.5.5.4.2.2.2.3.cmml" xref="S2.SS5.p1.7.m7.5.5.4.2.2.2.3">𝑡</ci></apply></set><ci id="S2.SS5.p1.7.m7.5.5.4.4.cmml" xref="S2.SS5.p1.7.m7.5.5.4.4">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.7.m7.5c">K_{i_{1}},K_{i_{2}}\in\{I_{1},\ldots,I_{t}\}\cup\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.7.m7.5d">italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ { italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } ∪ caligraphic_B</annotation></semantics></math> and <math alttext="\star_{i}\in\{\cap,\cup\}" class="ltx_math_unparsed" display="inline" id="S2.SS5.p1.8.m8.1"><semantics id="S2.SS5.p1.8.m8.1a"><mrow id="S2.SS5.p1.8.m8.1b"><msub id="S2.SS5.p1.8.m8.1.2"><mo id="S2.SS5.p1.8.m8.1.2.2">⋆</mo><mi id="S2.SS5.p1.8.m8.1.2.3">i</mi></msub><mo id="S2.SS5.p1.8.m8.1.1" lspace="0em">∈</mo><mrow id="S2.SS5.p1.8.m8.1.3"><mo id="S2.SS5.p1.8.m8.1.3.1" stretchy="false">{</mo><mo id="S2.SS5.p1.8.m8.1.3.2" lspace="0em" rspace="0em">∩</mo><mo id="S2.SS5.p1.8.m8.1.3.3" rspace="0em">,</mo><mo id="S2.SS5.p1.8.m8.1.3.4" lspace="0em" rspace="0em">∪</mo><mo id="S2.SS5.p1.8.m8.1.3.5" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS5.p1.8.m8.1c">\star_{i}\in\{\cap,\cup\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.8.m8.1d">⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { ∩ , ∪ }</annotation></semantics></math>, for each <math alttext="i\in[t]" class="ltx_Math" display="inline" id="S2.SS5.p1.9.m9.1"><semantics id="S2.SS5.p1.9.m9.1a"><mrow id="S2.SS5.p1.9.m9.1.2" xref="S2.SS5.p1.9.m9.1.2.cmml"><mi id="S2.SS5.p1.9.m9.1.2.2" xref="S2.SS5.p1.9.m9.1.2.2.cmml">i</mi><mo id="S2.SS5.p1.9.m9.1.2.1" xref="S2.SS5.p1.9.m9.1.2.1.cmml">∈</mo><mrow id="S2.SS5.p1.9.m9.1.2.3.2" xref="S2.SS5.p1.9.m9.1.2.3.1.cmml"><mo id="S2.SS5.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S2.SS5.p1.9.m9.1.2.3.1.1.cmml">[</mo><mi id="S2.SS5.p1.9.m9.1.1" xref="S2.SS5.p1.9.m9.1.1.cmml">t</mi><mo id="S2.SS5.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S2.SS5.p1.9.m9.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.9.m9.1b"><apply id="S2.SS5.p1.9.m9.1.2.cmml" xref="S2.SS5.p1.9.m9.1.2"><in id="S2.SS5.p1.9.m9.1.2.1.cmml" xref="S2.SS5.p1.9.m9.1.2.1"></in><ci id="S2.SS5.p1.9.m9.1.2.2.cmml" xref="S2.SS5.p1.9.m9.1.2.2">𝑖</ci><apply id="S2.SS5.p1.9.m9.1.2.3.1.cmml" xref="S2.SS5.p1.9.m9.1.2.3.2"><csymbol cd="latexml" id="S2.SS5.p1.9.m9.1.2.3.1.1.cmml" xref="S2.SS5.p1.9.m9.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS5.p1.9.m9.1.1.cmml" xref="S2.SS5.p1.9.m9.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.9.m9.1c">i\in[t]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.9.m9.1d">italic_i ∈ [ italic_t ]</annotation></semantics></math>. (Notice that we do not require <math alttext="i_{1},i_{2}<i" class="ltx_Math" display="inline" id="S2.SS5.p1.10.m10.2"><semantics id="S2.SS5.p1.10.m10.2a"><mrow id="S2.SS5.p1.10.m10.2.2" xref="S2.SS5.p1.10.m10.2.2.cmml"><mrow id="S2.SS5.p1.10.m10.2.2.2.2" xref="S2.SS5.p1.10.m10.2.2.2.3.cmml"><msub id="S2.SS5.p1.10.m10.1.1.1.1.1" xref="S2.SS5.p1.10.m10.1.1.1.1.1.cmml"><mi id="S2.SS5.p1.10.m10.1.1.1.1.1.2" xref="S2.SS5.p1.10.m10.1.1.1.1.1.2.cmml">i</mi><mn id="S2.SS5.p1.10.m10.1.1.1.1.1.3" xref="S2.SS5.p1.10.m10.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS5.p1.10.m10.2.2.2.2.3" xref="S2.SS5.p1.10.m10.2.2.2.3.cmml">,</mo><msub id="S2.SS5.p1.10.m10.2.2.2.2.2" xref="S2.SS5.p1.10.m10.2.2.2.2.2.cmml"><mi id="S2.SS5.p1.10.m10.2.2.2.2.2.2" xref="S2.SS5.p1.10.m10.2.2.2.2.2.2.cmml">i</mi><mn id="S2.SS5.p1.10.m10.2.2.2.2.2.3" xref="S2.SS5.p1.10.m10.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S2.SS5.p1.10.m10.2.2.3" xref="S2.SS5.p1.10.m10.2.2.3.cmml"><</mo><mi id="S2.SS5.p1.10.m10.2.2.4" xref="S2.SS5.p1.10.m10.2.2.4.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.10.m10.2b"><apply id="S2.SS5.p1.10.m10.2.2.cmml" xref="S2.SS5.p1.10.m10.2.2"><lt id="S2.SS5.p1.10.m10.2.2.3.cmml" xref="S2.SS5.p1.10.m10.2.2.3"></lt><list id="S2.SS5.p1.10.m10.2.2.2.3.cmml" xref="S2.SS5.p1.10.m10.2.2.2.2"><apply id="S2.SS5.p1.10.m10.1.1.1.1.1.cmml" xref="S2.SS5.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.10.m10.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p1.10.m10.1.1.1.1.1.2.cmml" xref="S2.SS5.p1.10.m10.1.1.1.1.1.2">𝑖</ci><cn id="S2.SS5.p1.10.m10.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p1.10.m10.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS5.p1.10.m10.2.2.2.2.2.cmml" xref="S2.SS5.p1.10.m10.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p1.10.m10.2.2.2.2.2.1.cmml" xref="S2.SS5.p1.10.m10.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.p1.10.m10.2.2.2.2.2.2.cmml" xref="S2.SS5.p1.10.m10.2.2.2.2.2.2">𝑖</ci><cn id="S2.SS5.p1.10.m10.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS5.p1.10.m10.2.2.2.2.2.3">2</cn></apply></list><ci id="S2.SS5.p1.10.m10.2.2.4.cmml" xref="S2.SS5.p1.10.m10.2.2.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.10.m10.2c">i_{1},i_{2}<i</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.10.m10.2d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_i</annotation></semantics></math>.) The syntactic sequence is viewed as a formal description instead of an actual construction, and it is evaluated as follows. Initially, <math alttext="I_{i}^{0}\stackrel{{\scriptstyle\rm def}}{{=}}\emptyset" class="ltx_Math" display="inline" id="S2.SS5.p1.11.m11.1"><semantics id="S2.SS5.p1.11.m11.1a"><mrow id="S2.SS5.p1.11.m11.1.1" xref="S2.SS5.p1.11.m11.1.1.cmml"><msubsup id="S2.SS5.p1.11.m11.1.1.2" xref="S2.SS5.p1.11.m11.1.1.2.cmml"><mi id="S2.SS5.p1.11.m11.1.1.2.2.2" xref="S2.SS5.p1.11.m11.1.1.2.2.2.cmml">I</mi><mi id="S2.SS5.p1.11.m11.1.1.2.2.3" xref="S2.SS5.p1.11.m11.1.1.2.2.3.cmml">i</mi><mn id="S2.SS5.p1.11.m11.1.1.2.3" xref="S2.SS5.p1.11.m11.1.1.2.3.cmml">0</mn></msubsup><mover id="S2.SS5.p1.11.m11.1.1.1" xref="S2.SS5.p1.11.m11.1.1.1.cmml"><mo id="S2.SS5.p1.11.m11.1.1.1.2" xref="S2.SS5.p1.11.m11.1.1.1.2.cmml">=</mo><mi id="S2.SS5.p1.11.m11.1.1.1.3" xref="S2.SS5.p1.11.m11.1.1.1.3.cmml">def</mi></mover><mi id="S2.SS5.p1.11.m11.1.1.3" mathvariant="normal" xref="S2.SS5.p1.11.m11.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.11.m11.1b"><apply id="S2.SS5.p1.11.m11.1.1.cmml" xref="S2.SS5.p1.11.m11.1.1"><apply id="S2.SS5.p1.11.m11.1.1.1.cmml" xref="S2.SS5.p1.11.m11.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.11.m11.1.1.1.1.cmml" xref="S2.SS5.p1.11.m11.1.1.1">superscript</csymbol><eq id="S2.SS5.p1.11.m11.1.1.1.2.cmml" xref="S2.SS5.p1.11.m11.1.1.1.2"></eq><ci id="S2.SS5.p1.11.m11.1.1.1.3.cmml" xref="S2.SS5.p1.11.m11.1.1.1.3">def</ci></apply><apply id="S2.SS5.p1.11.m11.1.1.2.cmml" xref="S2.SS5.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.11.m11.1.1.2.1.cmml" xref="S2.SS5.p1.11.m11.1.1.2">superscript</csymbol><apply id="S2.SS5.p1.11.m11.1.1.2.2.cmml" xref="S2.SS5.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.11.m11.1.1.2.2.1.cmml" xref="S2.SS5.p1.11.m11.1.1.2">subscript</csymbol><ci id="S2.SS5.p1.11.m11.1.1.2.2.2.cmml" xref="S2.SS5.p1.11.m11.1.1.2.2.2">𝐼</ci><ci id="S2.SS5.p1.11.m11.1.1.2.2.3.cmml" xref="S2.SS5.p1.11.m11.1.1.2.2.3">𝑖</ci></apply><cn id="S2.SS5.p1.11.m11.1.1.2.3.cmml" type="integer" xref="S2.SS5.p1.11.m11.1.1.2.3">0</cn></apply><emptyset id="S2.SS5.p1.11.m11.1.1.3.cmml" xref="S2.SS5.p1.11.m11.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.11.m11.1c">I_{i}^{0}\stackrel{{\scriptstyle\rm def}}{{=}}\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.11.m11.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP ∅</annotation></semantics></math> for each <math alttext="i\in[t]" class="ltx_Math" display="inline" id="S2.SS5.p1.12.m12.1"><semantics id="S2.SS5.p1.12.m12.1a"><mrow id="S2.SS5.p1.12.m12.1.2" xref="S2.SS5.p1.12.m12.1.2.cmml"><mi id="S2.SS5.p1.12.m12.1.2.2" xref="S2.SS5.p1.12.m12.1.2.2.cmml">i</mi><mo id="S2.SS5.p1.12.m12.1.2.1" xref="S2.SS5.p1.12.m12.1.2.1.cmml">∈</mo><mrow id="S2.SS5.p1.12.m12.1.2.3.2" xref="S2.SS5.p1.12.m12.1.2.3.1.cmml"><mo id="S2.SS5.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S2.SS5.p1.12.m12.1.2.3.1.1.cmml">[</mo><mi id="S2.SS5.p1.12.m12.1.1" xref="S2.SS5.p1.12.m12.1.1.cmml">t</mi><mo id="S2.SS5.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S2.SS5.p1.12.m12.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.12.m12.1b"><apply id="S2.SS5.p1.12.m12.1.2.cmml" xref="S2.SS5.p1.12.m12.1.2"><in id="S2.SS5.p1.12.m12.1.2.1.cmml" xref="S2.SS5.p1.12.m12.1.2.1"></in><ci id="S2.SS5.p1.12.m12.1.2.2.cmml" xref="S2.SS5.p1.12.m12.1.2.2">𝑖</ci><apply id="S2.SS5.p1.12.m12.1.2.3.1.cmml" xref="S2.SS5.p1.12.m12.1.2.3.2"><csymbol cd="latexml" id="S2.SS5.p1.12.m12.1.2.3.1.1.cmml" xref="S2.SS5.p1.12.m12.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS5.p1.12.m12.1.1.cmml" xref="S2.SS5.p1.12.m12.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.12.m12.1c">i\in[t]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.12.m12.1d">italic_i ∈ [ italic_t ]</annotation></semantics></math>. Then, for every <math alttext="j>0" class="ltx_Math" display="inline" id="S2.SS5.p1.13.m13.1"><semantics id="S2.SS5.p1.13.m13.1a"><mrow id="S2.SS5.p1.13.m13.1.1" xref="S2.SS5.p1.13.m13.1.1.cmml"><mi id="S2.SS5.p1.13.m13.1.1.2" xref="S2.SS5.p1.13.m13.1.1.2.cmml">j</mi><mo id="S2.SS5.p1.13.m13.1.1.1" xref="S2.SS5.p1.13.m13.1.1.1.cmml">></mo><mn id="S2.SS5.p1.13.m13.1.1.3" xref="S2.SS5.p1.13.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.13.m13.1b"><apply id="S2.SS5.p1.13.m13.1.1.cmml" xref="S2.SS5.p1.13.m13.1.1"><gt id="S2.SS5.p1.13.m13.1.1.1.cmml" xref="S2.SS5.p1.13.m13.1.1.1"></gt><ci id="S2.SS5.p1.13.m13.1.1.2.cmml" xref="S2.SS5.p1.13.m13.1.1.2">𝑗</ci><cn id="S2.SS5.p1.13.m13.1.1.3.cmml" type="integer" xref="S2.SS5.p1.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.13.m13.1c">j>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.13.m13.1d">italic_j > 0</annotation></semantics></math>, <math alttext="I^{j}_{i}\stackrel{{\scriptstyle\rm def}}{{=}}I^{j-1}\cup(K^{j-1}_{i_{1}}\star% _{i}K^{j-1}_{i_{2}})" class="ltx_Math" display="inline" id="S2.SS5.p1.14.m14.1"><semantics id="S2.SS5.p1.14.m14.1a"><mrow id="S2.SS5.p1.14.m14.1.1" xref="S2.SS5.p1.14.m14.1.1.cmml"><msubsup id="S2.SS5.p1.14.m14.1.1.3" xref="S2.SS5.p1.14.m14.1.1.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.3.2.2" xref="S2.SS5.p1.14.m14.1.1.3.2.2.cmml">I</mi><mi id="S2.SS5.p1.14.m14.1.1.3.3" xref="S2.SS5.p1.14.m14.1.1.3.3.cmml">i</mi><mi id="S2.SS5.p1.14.m14.1.1.3.2.3" xref="S2.SS5.p1.14.m14.1.1.3.2.3.cmml">j</mi></msubsup><mover id="S2.SS5.p1.14.m14.1.1.2" xref="S2.SS5.p1.14.m14.1.1.2.cmml"><mo id="S2.SS5.p1.14.m14.1.1.2.2" xref="S2.SS5.p1.14.m14.1.1.2.2.cmml">=</mo><mi id="S2.SS5.p1.14.m14.1.1.2.3" xref="S2.SS5.p1.14.m14.1.1.2.3.cmml">def</mi></mover><mrow id="S2.SS5.p1.14.m14.1.1.1" xref="S2.SS5.p1.14.m14.1.1.1.cmml"><msup id="S2.SS5.p1.14.m14.1.1.1.3" xref="S2.SS5.p1.14.m14.1.1.1.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.3.2" xref="S2.SS5.p1.14.m14.1.1.1.3.2.cmml">I</mi><mrow id="S2.SS5.p1.14.m14.1.1.1.3.3" xref="S2.SS5.p1.14.m14.1.1.1.3.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.3.3.2" xref="S2.SS5.p1.14.m14.1.1.1.3.3.2.cmml">j</mi><mo id="S2.SS5.p1.14.m14.1.1.1.3.3.1" xref="S2.SS5.p1.14.m14.1.1.1.3.3.1.cmml">−</mo><mn id="S2.SS5.p1.14.m14.1.1.1.3.3.3" xref="S2.SS5.p1.14.m14.1.1.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S2.SS5.p1.14.m14.1.1.1.2" xref="S2.SS5.p1.14.m14.1.1.1.2.cmml">∪</mo><mrow id="S2.SS5.p1.14.m14.1.1.1.1.1" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.cmml"><mo id="S2.SS5.p1.14.m14.1.1.1.1.1.2" stretchy="false" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.p1.14.m14.1.1.1.1.1.1" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.cmml"><msubsup id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.2" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.2.cmml">K</mi><msub 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xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.3.cmml">i</mi></msub><msubsup id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.2" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.2.cmml">K</mi><msub id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.2" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.2.cmml">i</mi><mn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.3" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.3.cmml">2</mn></msub><mrow id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.cmml"><mi id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.2" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.2.cmml">j</mi><mo id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.1" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.1.cmml">−</mo><mn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.3" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow><mo id="S2.SS5.p1.14.m14.1.1.1.1.1.3" stretchy="false" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.14.m14.1b"><apply id="S2.SS5.p1.14.m14.1.1.cmml" xref="S2.SS5.p1.14.m14.1.1"><apply id="S2.SS5.p1.14.m14.1.1.2.cmml" xref="S2.SS5.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.2.1.cmml" xref="S2.SS5.p1.14.m14.1.1.2">superscript</csymbol><eq id="S2.SS5.p1.14.m14.1.1.2.2.cmml" xref="S2.SS5.p1.14.m14.1.1.2.2"></eq><ci id="S2.SS5.p1.14.m14.1.1.2.3.cmml" xref="S2.SS5.p1.14.m14.1.1.2.3">def</ci></apply><apply id="S2.SS5.p1.14.m14.1.1.3.cmml" xref="S2.SS5.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.3">subscript</csymbol><apply id="S2.SS5.p1.14.m14.1.1.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.3.2.1.cmml" xref="S2.SS5.p1.14.m14.1.1.3">superscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.3.2.2.cmml" 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xref="S2.SS5.p1.14.m14.1.1.1.3.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1"><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.2">⋆</ci><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2">subscript</csymbol><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.2">𝐾</ci><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3"><minus id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.1"></minus><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.2">𝑗</ci><cn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.2.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3">subscript</csymbol><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.2">𝐾</ci><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3"><minus id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.1"></minus><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.2">𝑗</ci><cn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.1.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.2.cmml" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.2">𝑖</ci><cn id="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS5.p1.14.m14.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.14.m14.1c">I^{j}_{i}\stackrel{{\scriptstyle\rm def}}{{=}}I^{j-1}\cup(K^{j-1}_{i_{1}}\star% _{i}K^{j-1}_{i_{2}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.14.m14.1d">italic_I start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP italic_I start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT ∪ ( italic_K start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>, where the sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.p1.15.m15.1"><semantics id="S2.SS5.p1.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.15.m15.1.1" xref="S2.SS5.p1.15.m15.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.15.m15.1b"><ci id="S2.SS5.p1.15.m15.1.1.cmml" xref="S2.SS5.p1.15.m15.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.15.m15.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.15.m15.1d">caligraphic_B</annotation></semantics></math> remain fixed throughout the evaluation. We say that the syntactic sequence generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS5.p1.16.m16.1"><semantics id="S2.SS5.p1.16.m16.1a"><mi id="S2.SS5.p1.16.m16.1.1" xref="S2.SS5.p1.16.m16.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.16.m16.1b"><ci id="S2.SS5.p1.16.m16.1.1.cmml" xref="S2.SS5.p1.16.m16.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.16.m16.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.16.m16.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.p1.17.m17.1"><semantics id="S2.SS5.p1.17.m17.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.17.m17.1.1" xref="S2.SS5.p1.17.m17.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.17.m17.1b"><ci id="S2.SS5.p1.17.m17.1.1.cmml" xref="S2.SS5.p1.17.m17.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.17.m17.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.17.m17.1d">caligraphic_B</annotation></semantics></math> if there exists <math alttext="j\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS5.p1.18.m18.1"><semantics id="S2.SS5.p1.18.m18.1a"><mrow id="S2.SS5.p1.18.m18.1.1" xref="S2.SS5.p1.18.m18.1.1.cmml"><mi id="S2.SS5.p1.18.m18.1.1.2" xref="S2.SS5.p1.18.m18.1.1.2.cmml">j</mi><mo id="S2.SS5.p1.18.m18.1.1.1" xref="S2.SS5.p1.18.m18.1.1.1.cmml">∈</mo><mi id="S2.SS5.p1.18.m18.1.1.3" xref="S2.SS5.p1.18.m18.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.18.m18.1b"><apply id="S2.SS5.p1.18.m18.1.1.cmml" xref="S2.SS5.p1.18.m18.1.1"><in id="S2.SS5.p1.18.m18.1.1.1.cmml" xref="S2.SS5.p1.18.m18.1.1.1"></in><ci id="S2.SS5.p1.18.m18.1.1.2.cmml" xref="S2.SS5.p1.18.m18.1.1.2">𝑗</ci><ci id="S2.SS5.p1.18.m18.1.1.3.cmml" xref="S2.SS5.p1.18.m18.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.18.m18.1c">j\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.18.m18.1d">italic_j ∈ blackboard_N</annotation></semantics></math> such that <math alttext="I_{t}^{j^{\prime}}=A" class="ltx_Math" display="inline" id="S2.SS5.p1.19.m19.1"><semantics id="S2.SS5.p1.19.m19.1a"><mrow id="S2.SS5.p1.19.m19.1.1" xref="S2.SS5.p1.19.m19.1.1.cmml"><msubsup id="S2.SS5.p1.19.m19.1.1.2" xref="S2.SS5.p1.19.m19.1.1.2.cmml"><mi id="S2.SS5.p1.19.m19.1.1.2.2.2" xref="S2.SS5.p1.19.m19.1.1.2.2.2.cmml">I</mi><mi id="S2.SS5.p1.19.m19.1.1.2.2.3" xref="S2.SS5.p1.19.m19.1.1.2.2.3.cmml">t</mi><msup id="S2.SS5.p1.19.m19.1.1.2.3" xref="S2.SS5.p1.19.m19.1.1.2.3.cmml"><mi id="S2.SS5.p1.19.m19.1.1.2.3.2" xref="S2.SS5.p1.19.m19.1.1.2.3.2.cmml">j</mi><mo id="S2.SS5.p1.19.m19.1.1.2.3.3" xref="S2.SS5.p1.19.m19.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="S2.SS5.p1.19.m19.1.1.1" xref="S2.SS5.p1.19.m19.1.1.1.cmml">=</mo><mi id="S2.SS5.p1.19.m19.1.1.3" xref="S2.SS5.p1.19.m19.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.19.m19.1b"><apply id="S2.SS5.p1.19.m19.1.1.cmml" xref="S2.SS5.p1.19.m19.1.1"><eq id="S2.SS5.p1.19.m19.1.1.1.cmml" xref="S2.SS5.p1.19.m19.1.1.1"></eq><apply id="S2.SS5.p1.19.m19.1.1.2.cmml" xref="S2.SS5.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.19.m19.1.1.2.1.cmml" xref="S2.SS5.p1.19.m19.1.1.2">superscript</csymbol><apply id="S2.SS5.p1.19.m19.1.1.2.2.cmml" xref="S2.SS5.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.19.m19.1.1.2.2.1.cmml" xref="S2.SS5.p1.19.m19.1.1.2">subscript</csymbol><ci id="S2.SS5.p1.19.m19.1.1.2.2.2.cmml" xref="S2.SS5.p1.19.m19.1.1.2.2.2">𝐼</ci><ci id="S2.SS5.p1.19.m19.1.1.2.2.3.cmml" xref="S2.SS5.p1.19.m19.1.1.2.2.3">𝑡</ci></apply><apply id="S2.SS5.p1.19.m19.1.1.2.3.cmml" xref="S2.SS5.p1.19.m19.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.19.m19.1.1.2.3.1.cmml" xref="S2.SS5.p1.19.m19.1.1.2.3">superscript</csymbol><ci id="S2.SS5.p1.19.m19.1.1.2.3.2.cmml" xref="S2.SS5.p1.19.m19.1.1.2.3.2">𝑗</ci><ci id="S2.SS5.p1.19.m19.1.1.2.3.3.cmml" xref="S2.SS5.p1.19.m19.1.1.2.3.3">′</ci></apply></apply><ci id="S2.SS5.p1.19.m19.1.1.3.cmml" xref="S2.SS5.p1.19.m19.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.19.m19.1c">I_{t}^{j^{\prime}}=A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.19.m19.1d">italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT = italic_A</annotation></semantics></math> for every <math alttext="j^{\prime}\geq j" class="ltx_Math" display="inline" id="S2.SS5.p1.20.m20.1"><semantics id="S2.SS5.p1.20.m20.1a"><mrow id="S2.SS5.p1.20.m20.1.1" xref="S2.SS5.p1.20.m20.1.1.cmml"><msup id="S2.SS5.p1.20.m20.1.1.2" xref="S2.SS5.p1.20.m20.1.1.2.cmml"><mi id="S2.SS5.p1.20.m20.1.1.2.2" xref="S2.SS5.p1.20.m20.1.1.2.2.cmml">j</mi><mo id="S2.SS5.p1.20.m20.1.1.2.3" xref="S2.SS5.p1.20.m20.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS5.p1.20.m20.1.1.1" xref="S2.SS5.p1.20.m20.1.1.1.cmml">≥</mo><mi id="S2.SS5.p1.20.m20.1.1.3" xref="S2.SS5.p1.20.m20.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.20.m20.1b"><apply id="S2.SS5.p1.20.m20.1.1.cmml" xref="S2.SS5.p1.20.m20.1.1"><geq id="S2.SS5.p1.20.m20.1.1.1.cmml" xref="S2.SS5.p1.20.m20.1.1.1"></geq><apply id="S2.SS5.p1.20.m20.1.1.2.cmml" xref="S2.SS5.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p1.20.m20.1.1.2.1.cmml" xref="S2.SS5.p1.20.m20.1.1.2">superscript</csymbol><ci id="S2.SS5.p1.20.m20.1.1.2.2.cmml" xref="S2.SS5.p1.20.m20.1.1.2.2">𝑗</ci><ci id="S2.SS5.p1.20.m20.1.1.2.3.cmml" xref="S2.SS5.p1.20.m20.1.1.2.3">′</ci></apply><ci id="S2.SS5.p1.20.m20.1.1.3.cmml" xref="S2.SS5.p1.20.m20.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.20.m20.1c">j^{\prime}\geq j</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.20.m20.1d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_j</annotation></semantics></math>. Finally, we let <math alttext="D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S2.SS5.p1.21.m21.1"><semantics id="S2.SS5.p1.21.m21.1a"><mrow id="S2.SS5.p1.21.m21.1.1" xref="S2.SS5.p1.21.m21.1.1.cmml"><mi id="S2.SS5.p1.21.m21.1.1.3" xref="S2.SS5.p1.21.m21.1.1.3.cmml">D</mi><mo id="S2.SS5.p1.21.m21.1.1.2" xref="S2.SS5.p1.21.m21.1.1.2.cmml">⁢</mo><mrow id="S2.SS5.p1.21.m21.1.1.1.1" xref="S2.SS5.p1.21.m21.1.1.1.1.1.cmml"><mo id="S2.SS5.p1.21.m21.1.1.1.1.2" stretchy="false" xref="S2.SS5.p1.21.m21.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.p1.21.m21.1.1.1.1.1" xref="S2.SS5.p1.21.m21.1.1.1.1.1.cmml"><mi id="S2.SS5.p1.21.m21.1.1.1.1.1.2" xref="S2.SS5.p1.21.m21.1.1.1.1.1.2.cmml">A</mi><mo id="S2.SS5.p1.21.m21.1.1.1.1.1.1" xref="S2.SS5.p1.21.m21.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.21.m21.1.1.1.1.1.3" xref="S2.SS5.p1.21.m21.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.SS5.p1.21.m21.1.1.1.1.3" stretchy="false" xref="S2.SS5.p1.21.m21.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.21.m21.1b"><apply id="S2.SS5.p1.21.m21.1.1.cmml" xref="S2.SS5.p1.21.m21.1.1"><times id="S2.SS5.p1.21.m21.1.1.2.cmml" xref="S2.SS5.p1.21.m21.1.1.2"></times><ci id="S2.SS5.p1.21.m21.1.1.3.cmml" xref="S2.SS5.p1.21.m21.1.1.3">𝐷</ci><apply id="S2.SS5.p1.21.m21.1.1.1.1.1.cmml" xref="S2.SS5.p1.21.m21.1.1.1.1"><csymbol cd="latexml" id="S2.SS5.p1.21.m21.1.1.1.1.1.1.cmml" xref="S2.SS5.p1.21.m21.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS5.p1.21.m21.1.1.1.1.1.2.cmml" xref="S2.SS5.p1.21.m21.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS5.p1.21.m21.1.1.1.1.1.3.cmml" xref="S2.SS5.p1.21.m21.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.21.m21.1c">D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.21.m21.1d">italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math> denote the minimum length <math alttext="t" class="ltx_Math" display="inline" id="S2.SS5.p1.22.m22.1"><semantics id="S2.SS5.p1.22.m22.1a"><mi id="S2.SS5.p1.22.m22.1.1" xref="S2.SS5.p1.22.m22.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.22.m22.1b"><ci id="S2.SS5.p1.22.m22.1.1.cmml" xref="S2.SS5.p1.22.m22.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.22.m22.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.22.m22.1d">italic_t</annotation></semantics></math> of such a sequence, if it exists. The complexity measure <math alttext="D_{\cap}" class="ltx_Math" display="inline" id="S2.SS5.p1.23.m23.1"><semantics id="S2.SS5.p1.23.m23.1a"><msub id="S2.SS5.p1.23.m23.1.1" xref="S2.SS5.p1.23.m23.1.1.cmml"><mi id="S2.SS5.p1.23.m23.1.1.2" xref="S2.SS5.p1.23.m23.1.1.2.cmml">D</mi><mo id="S2.SS5.p1.23.m23.1.1.3" xref="S2.SS5.p1.23.m23.1.1.3.cmml">∩</mo></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.23.m23.1b"><apply id="S2.SS5.p1.23.m23.1.1.cmml" xref="S2.SS5.p1.23.m23.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.23.m23.1.1.1.cmml" xref="S2.SS5.p1.23.m23.1.1">subscript</csymbol><ci id="S2.SS5.p1.23.m23.1.1.2.cmml" xref="S2.SS5.p1.23.m23.1.1.2">𝐷</ci><intersect id="S2.SS5.p1.23.m23.1.1.3.cmml" xref="S2.SS5.p1.23.m23.1.1.3"></intersect></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.23.m23.1c">D_{\cap}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.23.m23.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT</annotation></semantics></math> is defined analogously, and only takes into account the number of intersection operations in the definition of the syntactic sequence.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem16"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem16.1.1.1">Lemma 16</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem16.2.2"> </span>(Convergence of the evaluation procedure)<span class="ltx_text ltx_font_bold" id="Thmtheorem16.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem16.p1"> <p class="ltx_p" id="Thmtheorem16.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem16.p1.3.3">Suppose <math alttext="I_{1},\ldots,I_{t}" class="ltx_Math" display="inline" id="Thmtheorem16.p1.1.1.m1.3"><semantics id="Thmtheorem16.p1.1.1.m1.3a"><mrow id="Thmtheorem16.p1.1.1.m1.3.3.2" xref="Thmtheorem16.p1.1.1.m1.3.3.3.cmml"><msub id="Thmtheorem16.p1.1.1.m1.2.2.1.1" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1.cmml"><mi id="Thmtheorem16.p1.1.1.m1.2.2.1.1.2" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1.2.cmml">I</mi><mn id="Thmtheorem16.p1.1.1.m1.2.2.1.1.3" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmtheorem16.p1.1.1.m1.3.3.2.3" xref="Thmtheorem16.p1.1.1.m1.3.3.3.cmml">,</mo><mi id="Thmtheorem16.p1.1.1.m1.1.1" mathvariant="normal" xref="Thmtheorem16.p1.1.1.m1.1.1.cmml">…</mi><mo id="Thmtheorem16.p1.1.1.m1.3.3.2.4" xref="Thmtheorem16.p1.1.1.m1.3.3.3.cmml">,</mo><msub id="Thmtheorem16.p1.1.1.m1.3.3.2.2" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2.cmml"><mi id="Thmtheorem16.p1.1.1.m1.3.3.2.2.2" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2.2.cmml">I</mi><mi id="Thmtheorem16.p1.1.1.m1.3.3.2.2.3" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem16.p1.1.1.m1.3b"><list id="Thmtheorem16.p1.1.1.m1.3.3.3.cmml" xref="Thmtheorem16.p1.1.1.m1.3.3.2"><apply id="Thmtheorem16.p1.1.1.m1.2.2.1.1.cmml" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="Thmtheorem16.p1.1.1.m1.2.2.1.1.1.cmml" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1">subscript</csymbol><ci id="Thmtheorem16.p1.1.1.m1.2.2.1.1.2.cmml" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1.2">𝐼</ci><cn id="Thmtheorem16.p1.1.1.m1.2.2.1.1.3.cmml" type="integer" xref="Thmtheorem16.p1.1.1.m1.2.2.1.1.3">1</cn></apply><ci id="Thmtheorem16.p1.1.1.m1.1.1.cmml" xref="Thmtheorem16.p1.1.1.m1.1.1">…</ci><apply id="Thmtheorem16.p1.1.1.m1.3.3.2.2.cmml" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="Thmtheorem16.p1.1.1.m1.3.3.2.2.1.cmml" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2">subscript</csymbol><ci id="Thmtheorem16.p1.1.1.m1.3.3.2.2.2.cmml" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2.2">𝐼</ci><ci id="Thmtheorem16.p1.1.1.m1.3.3.2.2.3.cmml" xref="Thmtheorem16.p1.1.1.m1.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem16.p1.1.1.m1.3c">I_{1},\ldots,I_{t}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem16.p1.1.1.m1.3d">italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> together with the corresponding <math alttext="\star_{i}" class="ltx_Math" display="inline" id="Thmtheorem16.p1.2.2.m2.1"><semantics id="Thmtheorem16.p1.2.2.m2.1a"><msub id="Thmtheorem16.p1.2.2.m2.1.1" xref="Thmtheorem16.p1.2.2.m2.1.1.cmml"><mo id="Thmtheorem16.p1.2.2.m2.1.1.2" xref="Thmtheorem16.p1.2.2.m2.1.1.2.cmml">⋆</mo><mi id="Thmtheorem16.p1.2.2.m2.1.1.3" xref="Thmtheorem16.p1.2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem16.p1.2.2.m2.1b"><apply id="Thmtheorem16.p1.2.2.m2.1.1.cmml" xref="Thmtheorem16.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmtheorem16.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem16.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmtheorem16.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem16.p1.2.2.m2.1.1.2">⋆</ci><ci id="Thmtheorem16.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem16.p1.2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem16.p1.2.2.m2.1c">\star_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem16.p1.2.2.m2.1d">⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> operations define a syntactic sequence. Then, for every <math alttext="j\geq t" class="ltx_Math" display="inline" id="Thmtheorem16.p1.3.3.m3.1"><semantics id="Thmtheorem16.p1.3.3.m3.1a"><mrow id="Thmtheorem16.p1.3.3.m3.1.1" xref="Thmtheorem16.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem16.p1.3.3.m3.1.1.2" xref="Thmtheorem16.p1.3.3.m3.1.1.2.cmml">j</mi><mo id="Thmtheorem16.p1.3.3.m3.1.1.1" xref="Thmtheorem16.p1.3.3.m3.1.1.1.cmml">≥</mo><mi id="Thmtheorem16.p1.3.3.m3.1.1.3" xref="Thmtheorem16.p1.3.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem16.p1.3.3.m3.1b"><apply id="Thmtheorem16.p1.3.3.m3.1.1.cmml" xref="Thmtheorem16.p1.3.3.m3.1.1"><geq id="Thmtheorem16.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem16.p1.3.3.m3.1.1.1"></geq><ci id="Thmtheorem16.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem16.p1.3.3.m3.1.1.2">𝑗</ci><ci id="Thmtheorem16.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem16.p1.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem16.p1.3.3.m3.1c">j\geq t</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem16.p1.3.3.m3.1d">italic_j ≥ italic_t</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="I^{j+1}_{i}\;=\;I^{j}_{i}." class="ltx_Math" display="block" id="S2.Ex13.m1.1"><semantics id="S2.Ex13.m1.1a"><mrow id="S2.Ex13.m1.1.1.1" xref="S2.Ex13.m1.1.1.1.1.cmml"><mrow id="S2.Ex13.m1.1.1.1.1" xref="S2.Ex13.m1.1.1.1.1.cmml"><msubsup id="S2.Ex13.m1.1.1.1.1.2" xref="S2.Ex13.m1.1.1.1.1.2.cmml"><mi id="S2.Ex13.m1.1.1.1.1.2.2.2" xref="S2.Ex13.m1.1.1.1.1.2.2.2.cmml">I</mi><mi id="S2.Ex13.m1.1.1.1.1.2.3" xref="S2.Ex13.m1.1.1.1.1.2.3.cmml">i</mi><mrow id="S2.Ex13.m1.1.1.1.1.2.2.3" xref="S2.Ex13.m1.1.1.1.1.2.2.3.cmml"><mi id="S2.Ex13.m1.1.1.1.1.2.2.3.2" xref="S2.Ex13.m1.1.1.1.1.2.2.3.2.cmml">j</mi><mo id="S2.Ex13.m1.1.1.1.1.2.2.3.1" xref="S2.Ex13.m1.1.1.1.1.2.2.3.1.cmml">+</mo><mn id="S2.Ex13.m1.1.1.1.1.2.2.3.3" xref="S2.Ex13.m1.1.1.1.1.2.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S2.Ex13.m1.1.1.1.1.1" lspace="0.558em" rspace="0.558em" xref="S2.Ex13.m1.1.1.1.1.1.cmml">=</mo><msubsup id="S2.Ex13.m1.1.1.1.1.3" xref="S2.Ex13.m1.1.1.1.1.3.cmml"><mi id="S2.Ex13.m1.1.1.1.1.3.2.2" xref="S2.Ex13.m1.1.1.1.1.3.2.2.cmml">I</mi><mi id="S2.Ex13.m1.1.1.1.1.3.3" xref="S2.Ex13.m1.1.1.1.1.3.3.cmml">i</mi><mi id="S2.Ex13.m1.1.1.1.1.3.2.3" xref="S2.Ex13.m1.1.1.1.1.3.2.3.cmml">j</mi></msubsup></mrow><mo id="S2.Ex13.m1.1.1.1.2" lspace="0em" xref="S2.Ex13.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex13.m1.1b"><apply id="S2.Ex13.m1.1.1.1.1.cmml" xref="S2.Ex13.m1.1.1.1"><eq id="S2.Ex13.m1.1.1.1.1.1.cmml" xref="S2.Ex13.m1.1.1.1.1.1"></eq><apply id="S2.Ex13.m1.1.1.1.1.2.cmml" xref="S2.Ex13.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex13.m1.1.1.1.1.2.1.cmml" xref="S2.Ex13.m1.1.1.1.1.2">subscript</csymbol><apply id="S2.Ex13.m1.1.1.1.1.2.2.cmml" xref="S2.Ex13.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex13.m1.1.1.1.1.2.2.1.cmml" xref="S2.Ex13.m1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex13.m1.1.1.1.1.2.2.2.cmml" xref="S2.Ex13.m1.1.1.1.1.2.2.2">𝐼</ci><apply id="S2.Ex13.m1.1.1.1.1.2.2.3.cmml" xref="S2.Ex13.m1.1.1.1.1.2.2.3"><plus id="S2.Ex13.m1.1.1.1.1.2.2.3.1.cmml" xref="S2.Ex13.m1.1.1.1.1.2.2.3.1"></plus><ci id="S2.Ex13.m1.1.1.1.1.2.2.3.2.cmml" xref="S2.Ex13.m1.1.1.1.1.2.2.3.2">𝑗</ci><cn id="S2.Ex13.m1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S2.Ex13.m1.1.1.1.1.2.2.3.3">1</cn></apply></apply><ci id="S2.Ex13.m1.1.1.1.1.2.3.cmml" xref="S2.Ex13.m1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S2.Ex13.m1.1.1.1.1.3.cmml" xref="S2.Ex13.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex13.m1.1.1.1.1.3.1.cmml" xref="S2.Ex13.m1.1.1.1.1.3">subscript</csymbol><apply id="S2.Ex13.m1.1.1.1.1.3.2.cmml" xref="S2.Ex13.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex13.m1.1.1.1.1.3.2.1.cmml" xref="S2.Ex13.m1.1.1.1.1.3">superscript</csymbol><ci id="S2.Ex13.m1.1.1.1.1.3.2.2.cmml" xref="S2.Ex13.m1.1.1.1.1.3.2.2">𝐼</ci><ci id="S2.Ex13.m1.1.1.1.1.3.2.3.cmml" xref="S2.Ex13.m1.1.1.1.1.3.2.3">𝑗</ci></apply><ci id="S2.Ex13.m1.1.1.1.1.3.3.cmml" xref="S2.Ex13.m1.1.1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex13.m1.1c">I^{j+1}_{i}\;=\;I^{j}_{i}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex13.m1.1d">italic_I start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem16.p1.4"><span class="ltx_text ltx_font_italic" id="Thmtheorem16.p1.4.1">In other words, the evaluation converges after at most <math alttext="t" class="ltx_Math" display="inline" id="Thmtheorem16.p1.4.1.m1.1"><semantics id="Thmtheorem16.p1.4.1.m1.1a"><mi id="Thmtheorem16.p1.4.1.m1.1.1" xref="Thmtheorem16.p1.4.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem16.p1.4.1.m1.1b"><ci id="Thmtheorem16.p1.4.1.m1.1.1.cmml" xref="Thmtheorem16.p1.4.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem16.p1.4.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem16.p1.4.1.m1.1d">italic_t</annotation></semantics></math> steps.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS5.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS5.1.p1"> <p class="ltx_p" id="S2.SS5.1.p1.3">The evaluation is monotone, in the sense that an element <math alttext="v\in\Gamma" class="ltx_Math" display="inline" id="S2.SS5.1.p1.1.m1.1"><semantics id="S2.SS5.1.p1.1.m1.1a"><mrow id="S2.SS5.1.p1.1.m1.1.1" xref="S2.SS5.1.p1.1.m1.1.1.cmml"><mi id="S2.SS5.1.p1.1.m1.1.1.2" xref="S2.SS5.1.p1.1.m1.1.1.2.cmml">v</mi><mo id="S2.SS5.1.p1.1.m1.1.1.1" xref="S2.SS5.1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S2.SS5.1.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.SS5.1.p1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.1.p1.1.m1.1b"><apply id="S2.SS5.1.p1.1.m1.1.1.cmml" xref="S2.SS5.1.p1.1.m1.1.1"><in id="S2.SS5.1.p1.1.m1.1.1.1.cmml" xref="S2.SS5.1.p1.1.m1.1.1.1"></in><ci id="S2.SS5.1.p1.1.m1.1.1.2.cmml" xref="S2.SS5.1.p1.1.m1.1.1.2">𝑣</ci><ci id="S2.SS5.1.p1.1.m1.1.1.3.cmml" xref="S2.SS5.1.p1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.1.p1.1.m1.1c">v\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.1.p1.1.m1.1d">italic_v ∈ roman_Γ</annotation></semantics></math> added to a set during the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS5.1.p1.2.m2.1"><semantics id="S2.SS5.1.p1.2.m2.1a"><mi id="S2.SS5.1.p1.2.m2.1.1" xref="S2.SS5.1.p1.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.1.p1.2.m2.1b"><ci id="S2.SS5.1.p1.2.m2.1.1.cmml" xref="S2.SS5.1.p1.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.1.p1.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.1.p1.2.m2.1d">italic_j</annotation></semantics></math>-th step of the evaluation cannot be removed in subsequent updates. From the point of view of this fixed element, if it is not added to a new set during an update, it won’t be added to new sets in subsequent updates. Consequently, each set in the sequence converges after at most <math alttext="t" class="ltx_Math" display="inline" id="S2.SS5.1.p1.3.m3.1"><semantics id="S2.SS5.1.p1.3.m3.1a"><mi id="S2.SS5.1.p1.3.m3.1.1" xref="S2.SS5.1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.1.p1.3.m3.1b"><ci id="S2.SS5.1.p1.3.m3.1.1.cmml" xref="S2.SS5.1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.1.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.1.p1.3.m3.1d">italic_t</annotation></semantics></math> iterations. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmtheorem17"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem17.1.1.1">Corollary 17</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem17.2.2"> </span>(Cyclic discrete complexity versus discrete complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem17.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem17.p1"> <p class="ltx_p" id="Thmtheorem17.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem17.p1.2.2">For every set <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem17.p1.1.1.m1.1"><semantics id="Thmtheorem17.p1.1.1.m1.1a"><mrow id="Thmtheorem17.p1.1.1.m1.1.1" xref="Thmtheorem17.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem17.p1.1.1.m1.1.1.2" xref="Thmtheorem17.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem17.p1.1.1.m1.1.1.1" xref="Thmtheorem17.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem17.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem17.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem17.p1.1.1.m1.1b"><apply id="Thmtheorem17.p1.1.1.m1.1.1.cmml" xref="Thmtheorem17.p1.1.1.m1.1.1"><subset id="Thmtheorem17.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem17.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem17.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem17.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem17.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem17.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem17.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem17.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> and family <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="Thmtheorem17.p1.2.2.m2.1"><semantics id="Thmtheorem17.p1.2.2.m2.1a"><mrow id="Thmtheorem17.p1.2.2.m2.1.2" xref="Thmtheorem17.p1.2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem17.p1.2.2.m2.1.2.2" xref="Thmtheorem17.p1.2.2.m2.1.2.2.cmml">ℬ</mi><mo id="Thmtheorem17.p1.2.2.m2.1.2.1" xref="Thmtheorem17.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem17.p1.2.2.m2.1.2.3" xref="Thmtheorem17.p1.2.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem17.p1.2.2.m2.1.2.3.2" xref="Thmtheorem17.p1.2.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem17.p1.2.2.m2.1.2.3.1" xref="Thmtheorem17.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem17.p1.2.2.m2.1.2.3.3.2" xref="Thmtheorem17.p1.2.2.m2.1.2.3.cmml"><mo id="Thmtheorem17.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem17.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmtheorem17.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem17.p1.2.2.m2.1.1.cmml">Γ</mi><mo id="Thmtheorem17.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem17.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem17.p1.2.2.m2.1b"><apply id="Thmtheorem17.p1.2.2.m2.1.2.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2"><subset id="Thmtheorem17.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2.1"></subset><ci id="Thmtheorem17.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2.2">ℬ</ci><apply id="Thmtheorem17.p1.2.2.m2.1.2.3.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2.3"><times id="Thmtheorem17.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2.3.1"></times><ci id="Thmtheorem17.p1.2.2.m2.1.2.3.2.cmml" xref="Thmtheorem17.p1.2.2.m2.1.2.3.2">𝒫</ci><ci id="Thmtheorem17.p1.2.2.m2.1.1.cmml" xref="Thmtheorem17.p1.2.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem17.p1.2.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem17.p1.2.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math> of generators,</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D_{\cap}(A\mid\mathcal{B})\;\leq\;D_{\cap}(A\mid\mathcal{B})\;\leq\;D_{\cap}(A% \mid\mathcal{B})^{2}." class="ltx_Math" display="block" id="S2.Ex14.m1.1"><semantics id="S2.Ex14.m1.1a"><mrow id="S2.Ex14.m1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.cmml"><mrow id="S2.Ex14.m1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.cmml"><mrow id="S2.Ex14.m1.1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.1.cmml"><msub id="S2.Ex14.m1.1.1.1.1.1.3" xref="S2.Ex14.m1.1.1.1.1.1.3.cmml"><mi id="S2.Ex14.m1.1.1.1.1.1.3.2" xref="S2.Ex14.m1.1.1.1.1.1.3.2.cmml">D</mi><mo id="S2.Ex14.m1.1.1.1.1.1.3.3" xref="S2.Ex14.m1.1.1.1.1.1.3.3.cmml">∩</mo></msub><mo id="S2.Ex14.m1.1.1.1.1.1.2" xref="S2.Ex14.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex14.m1.1.1.1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex14.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex14.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex14.m1.1.1.1.1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex14.m1.1.1.1.1.1.1.1.1.2" xref="S2.Ex14.m1.1.1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.Ex14.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" 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</tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S2.SS5.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS5.2.p1"> <p class="ltx_p" id="S2.SS5.2.p1.7">For the first inequality, observe that from every construction of <math alttext="A" class="ltx_Math" display="inline" id="S2.SS5.2.p1.1.m1.1"><semantics id="S2.SS5.2.p1.1.m1.1a"><mi id="S2.SS5.2.p1.1.m1.1.1" xref="S2.SS5.2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.1.m1.1b"><ci id="S2.SS5.2.p1.1.m1.1.1.cmml" xref="S2.SS5.2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.1.m1.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.2.p1.2.m2.1"><semantics id="S2.SS5.2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.2.p1.2.m2.1.1" xref="S2.SS5.2.p1.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.2.m2.1b"><ci id="S2.SS5.2.p1.2.m2.1.1.cmml" xref="S2.SS5.2.p1.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.2.m2.1d">caligraphic_B</annotation></semantics></math> we can define an acyclic syntactic sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS5.2.p1.3.m3.1"><semantics id="S2.SS5.2.p1.3.m3.1a"><mi id="S2.SS5.2.p1.3.m3.1.1" xref="S2.SS5.2.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.3.m3.1b"><ci id="S2.SS5.2.p1.3.m3.1.1.cmml" xref="S2.SS5.2.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.3.m3.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.2.p1.4.m4.1"><semantics id="S2.SS5.2.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.2.p1.4.m4.1.1" xref="S2.SS5.2.p1.4.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.4.m4.1b"><ci id="S2.SS5.2.p1.4.m4.1.1.cmml" xref="S2.SS5.2.p1.4.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.4.m4.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.4.m4.1d">caligraphic_B</annotation></semantics></math>. For the second inequality, simply unfold the evaluation of the syntactic sequence into a sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S2.SS5.2.p1.5.m5.1"><semantics id="S2.SS5.2.p1.5.m5.1a"><mi id="S2.SS5.2.p1.5.m5.1.1" xref="S2.SS5.2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.5.m5.1b"><ci id="S2.SS5.2.p1.5.m5.1.1.cmml" xref="S2.SS5.2.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.5.m5.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS5.2.p1.6.m6.1"><semantics id="S2.SS5.2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.2.p1.6.m6.1.1" xref="S2.SS5.2.p1.6.m6.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.2.p1.6.m6.1b"><ci id="S2.SS5.2.p1.6.m6.1.1.cmml" xref="S2.SS5.2.p1.6.m6.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.6.m6.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.6.m6.1d">caligraphic_B</annotation></semantics></math>. Since the additional union operations coming from the update step <math alttext="I^{j}_{i}=I^{j-1}\cup(K^{j-1}_{i_{1}}\star_{i}K^{j-1}_{i_{2}})" class="ltx_Math" display="inline" id="S2.SS5.2.p1.7.m7.1"><semantics id="S2.SS5.2.p1.7.m7.1a"><mrow id="S2.SS5.2.p1.7.m7.1.1" xref="S2.SS5.2.p1.7.m7.1.1.cmml"><msubsup id="S2.SS5.2.p1.7.m7.1.1.3" xref="S2.SS5.2.p1.7.m7.1.1.3.cmml"><mi id="S2.SS5.2.p1.7.m7.1.1.3.2.2" xref="S2.SS5.2.p1.7.m7.1.1.3.2.2.cmml">I</mi><mi id="S2.SS5.2.p1.7.m7.1.1.3.3" xref="S2.SS5.2.p1.7.m7.1.1.3.3.cmml">i</mi><mi id="S2.SS5.2.p1.7.m7.1.1.3.2.3" xref="S2.SS5.2.p1.7.m7.1.1.3.2.3.cmml">j</mi></msubsup><mo id="S2.SS5.2.p1.7.m7.1.1.2" xref="S2.SS5.2.p1.7.m7.1.1.2.cmml">=</mo><mrow id="S2.SS5.2.p1.7.m7.1.1.1" xref="S2.SS5.2.p1.7.m7.1.1.1.cmml"><msup id="S2.SS5.2.p1.7.m7.1.1.1.3" xref="S2.SS5.2.p1.7.m7.1.1.1.3.cmml"><mi id="S2.SS5.2.p1.7.m7.1.1.1.3.2" xref="S2.SS5.2.p1.7.m7.1.1.1.3.2.cmml">I</mi><mrow id="S2.SS5.2.p1.7.m7.1.1.1.3.3" xref="S2.SS5.2.p1.7.m7.1.1.1.3.3.cmml"><mi id="S2.SS5.2.p1.7.m7.1.1.1.3.3.2" xref="S2.SS5.2.p1.7.m7.1.1.1.3.3.2.cmml">j</mi><mo id="S2.SS5.2.p1.7.m7.1.1.1.3.3.1" xref="S2.SS5.2.p1.7.m7.1.1.1.3.3.1.cmml">−</mo><mn id="S2.SS5.2.p1.7.m7.1.1.1.3.3.3" xref="S2.SS5.2.p1.7.m7.1.1.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S2.SS5.2.p1.7.m7.1.1.1.2" xref="S2.SS5.2.p1.7.m7.1.1.1.2.cmml">∪</mo><mrow id="S2.SS5.2.p1.7.m7.1.1.1.1.1" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS5.2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.cmml"><msubsup id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.cmml"><mi id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.2.2" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.2.2.cmml">K</mi><msub id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.3" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.3.cmml"><mi id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.2.3.2" 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xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.2">𝐾</ci><apply id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3"><minus id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.1.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.1"></minus><ci id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.2.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.2">𝑗</ci><cn id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.1.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.2.cmml" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.2">𝑖</ci><cn id="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS5.2.p1.7.m7.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.2.p1.7.m7.1c">I^{j}_{i}=I^{j-1}\cup(K^{j-1}_{i_{1}}\star_{i}K^{j-1}_{i_{2}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.2.p1.7.m7.1d">italic_I start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT ∪ ( italic_K start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> do not increase intersection complexity, the claimed upper bound follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem16" title="Lemma 16 (Convergence of the evaluation procedure). ‣ 2.5 Cyclic Discrete Complexity ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">16</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS5.p2"> <p class="ltx_p" id="S2.SS5.p2.1">We will employ cyclic discrete complexity in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.4</span></a> to exactly characterize the power of the fusion method as a framework to lower bound discrete complexity. We finish this section with a concrete example that is relevant in the context of the fusion method (cf. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS3" title="3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.3</span></a>). <br class="ltx_break"/></p> </div> <div class="ltx_para ltx_noindent" id="S2.SS5.p3"> <p class="ltx_p" id="S2.SS5.p3.16"><span class="ltx_text ltx_font_bold" id="S2.SS5.p3.1.1">Example: The Fusion Problem <math alttext="\Pi_{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.SS5.p3.1.1.m1.1"><semantics id="S2.SS5.p3.1.1.m1.1a"><msub id="S2.SS5.p3.1.1.m1.1.1" xref="S2.SS5.p3.1.1.m1.1.1.cmml"><mi id="S2.SS5.p3.1.1.m1.1.1.2" mathvariant="normal" xref="S2.SS5.p3.1.1.m1.1.1.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p3.1.1.m1.1.1.3" xref="S2.SS5.p3.1.1.m1.1.1.3.cmml">ℛ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.1.1.m1.1b"><apply id="S2.SS5.p3.1.1.m1.1.1.cmml" xref="S2.SS5.p3.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.1.1.m1.1.1.1.cmml" xref="S2.SS5.p3.1.1.m1.1.1">subscript</csymbol><ci id="S2.SS5.p3.1.1.m1.1.1.2.cmml" xref="S2.SS5.p3.1.1.m1.1.1.2">Π</ci><ci id="S2.SS5.p3.1.1.m1.1.1.3.cmml" xref="S2.SS5.p3.1.1.m1.1.1.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.1.1.m1.1c">\Pi_{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.1.1.m1.1d">roman_Π start_POSTSUBSCRIPT caligraphic_R end_POSTSUBSCRIPT</annotation></semantics></math>.</span> Let <math alttext="[m]=\{1,\ldots,m\}" class="ltx_Math" display="inline" id="S2.SS5.p3.2.m1.4"><semantics id="S2.SS5.p3.2.m1.4a"><mrow id="S2.SS5.p3.2.m1.4.5" xref="S2.SS5.p3.2.m1.4.5.cmml"><mrow id="S2.SS5.p3.2.m1.4.5.2.2" xref="S2.SS5.p3.2.m1.4.5.2.1.cmml"><mo id="S2.SS5.p3.2.m1.4.5.2.2.1" stretchy="false" xref="S2.SS5.p3.2.m1.4.5.2.1.1.cmml">[</mo><mi id="S2.SS5.p3.2.m1.1.1" xref="S2.SS5.p3.2.m1.1.1.cmml">m</mi><mo id="S2.SS5.p3.2.m1.4.5.2.2.2" stretchy="false" xref="S2.SS5.p3.2.m1.4.5.2.1.1.cmml">]</mo></mrow><mo id="S2.SS5.p3.2.m1.4.5.1" xref="S2.SS5.p3.2.m1.4.5.1.cmml">=</mo><mrow id="S2.SS5.p3.2.m1.4.5.3.2" xref="S2.SS5.p3.2.m1.4.5.3.1.cmml"><mo id="S2.SS5.p3.2.m1.4.5.3.2.1" stretchy="false" xref="S2.SS5.p3.2.m1.4.5.3.1.cmml">{</mo><mn id="S2.SS5.p3.2.m1.2.2" xref="S2.SS5.p3.2.m1.2.2.cmml">1</mn><mo id="S2.SS5.p3.2.m1.4.5.3.2.2" xref="S2.SS5.p3.2.m1.4.5.3.1.cmml">,</mo><mi id="S2.SS5.p3.2.m1.3.3" mathvariant="normal" xref="S2.SS5.p3.2.m1.3.3.cmml">…</mi><mo id="S2.SS5.p3.2.m1.4.5.3.2.3" xref="S2.SS5.p3.2.m1.4.5.3.1.cmml">,</mo><mi id="S2.SS5.p3.2.m1.4.4" xref="S2.SS5.p3.2.m1.4.4.cmml">m</mi><mo id="S2.SS5.p3.2.m1.4.5.3.2.4" stretchy="false" xref="S2.SS5.p3.2.m1.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.2.m1.4b"><apply id="S2.SS5.p3.2.m1.4.5.cmml" xref="S2.SS5.p3.2.m1.4.5"><eq id="S2.SS5.p3.2.m1.4.5.1.cmml" xref="S2.SS5.p3.2.m1.4.5.1"></eq><apply id="S2.SS5.p3.2.m1.4.5.2.1.cmml" xref="S2.SS5.p3.2.m1.4.5.2.2"><csymbol cd="latexml" id="S2.SS5.p3.2.m1.4.5.2.1.1.cmml" xref="S2.SS5.p3.2.m1.4.5.2.2.1">delimited-[]</csymbol><ci id="S2.SS5.p3.2.m1.1.1.cmml" xref="S2.SS5.p3.2.m1.1.1">𝑚</ci></apply><set id="S2.SS5.p3.2.m1.4.5.3.1.cmml" xref="S2.SS5.p3.2.m1.4.5.3.2"><cn id="S2.SS5.p3.2.m1.2.2.cmml" type="integer" xref="S2.SS5.p3.2.m1.2.2">1</cn><ci id="S2.SS5.p3.2.m1.3.3.cmml" xref="S2.SS5.p3.2.m1.3.3">…</ci><ci id="S2.SS5.p3.2.m1.4.4.cmml" xref="S2.SS5.p3.2.m1.4.4">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.2.m1.4c">[m]=\{1,\ldots,m\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.2.m1.4d">[ italic_m ] = { 1 , … , italic_m }</annotation></semantics></math>, <math alttext="Y\subseteq[m]" class="ltx_Math" display="inline" id="S2.SS5.p3.3.m2.1"><semantics id="S2.SS5.p3.3.m2.1a"><mrow 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id="S2.SS5.p3.4.m3.1.1.cmml" xref="S2.SS5.p3.4.m3.1.1">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.4.m3.1c">[m]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.4.m3.1d">[ italic_m ]</annotation></semantics></math>, and <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS5.p3.5.m4.1"><semantics id="S2.SS5.p3.5.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p3.5.m4.1.1" xref="S2.SS5.p3.5.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.5.m4.1b"><ci id="S2.SS5.p3.5.m4.1.1.cmml" xref="S2.SS5.p3.5.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.5.m4.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.5.m4.1d">caligraphic_R</annotation></semantics></math> be a <em class="ltx_emph ltx_font_italic" id="S2.SS5.p3.16.2">fixed</em> set of rules encoded by a set of triples of the form <math alttext="(a,b,c)" class="ltx_Math" display="inline" id="S2.SS5.p3.6.m5.3"><semantics id="S2.SS5.p3.6.m5.3a"><mrow id="S2.SS5.p3.6.m5.3.4.2" xref="S2.SS5.p3.6.m5.3.4.1.cmml"><mo id="S2.SS5.p3.6.m5.3.4.2.1" stretchy="false" xref="S2.SS5.p3.6.m5.3.4.1.cmml">(</mo><mi id="S2.SS5.p3.6.m5.1.1" xref="S2.SS5.p3.6.m5.1.1.cmml">a</mi><mo id="S2.SS5.p3.6.m5.3.4.2.2" xref="S2.SS5.p3.6.m5.3.4.1.cmml">,</mo><mi id="S2.SS5.p3.6.m5.2.2" xref="S2.SS5.p3.6.m5.2.2.cmml">b</mi><mo id="S2.SS5.p3.6.m5.3.4.2.3" xref="S2.SS5.p3.6.m5.3.4.1.cmml">,</mo><mi id="S2.SS5.p3.6.m5.3.3" xref="S2.SS5.p3.6.m5.3.3.cmml">c</mi><mo id="S2.SS5.p3.6.m5.3.4.2.4" stretchy="false" xref="S2.SS5.p3.6.m5.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.6.m5.3b"><vector id="S2.SS5.p3.6.m5.3.4.1.cmml" xref="S2.SS5.p3.6.m5.3.4.2"><ci id="S2.SS5.p3.6.m5.1.1.cmml" xref="S2.SS5.p3.6.m5.1.1">𝑎</ci><ci id="S2.SS5.p3.6.m5.2.2.cmml" xref="S2.SS5.p3.6.m5.2.2">𝑏</ci><ci id="S2.SS5.p3.6.m5.3.3.cmml" xref="S2.SS5.p3.6.m5.3.3">𝑐</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.6.m5.3c">(a,b,c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.6.m5.3d">( italic_a , italic_b , italic_c )</annotation></semantics></math>, where <math alttext="a,b,c\in[m]" class="ltx_Math" display="inline" id="S2.SS5.p3.7.m6.4"><semantics id="S2.SS5.p3.7.m6.4a"><mrow id="S2.SS5.p3.7.m6.4.5" xref="S2.SS5.p3.7.m6.4.5.cmml"><mrow id="S2.SS5.p3.7.m6.4.5.2.2" xref="S2.SS5.p3.7.m6.4.5.2.1.cmml"><mi id="S2.SS5.p3.7.m6.2.2" xref="S2.SS5.p3.7.m6.2.2.cmml">a</mi><mo id="S2.SS5.p3.7.m6.4.5.2.2.1" xref="S2.SS5.p3.7.m6.4.5.2.1.cmml">,</mo><mi id="S2.SS5.p3.7.m6.3.3" xref="S2.SS5.p3.7.m6.3.3.cmml">b</mi><mo id="S2.SS5.p3.7.m6.4.5.2.2.2" xref="S2.SS5.p3.7.m6.4.5.2.1.cmml">,</mo><mi id="S2.SS5.p3.7.m6.4.4" xref="S2.SS5.p3.7.m6.4.4.cmml">c</mi></mrow><mo id="S2.SS5.p3.7.m6.4.5.1" xref="S2.SS5.p3.7.m6.4.5.1.cmml">∈</mo><mrow id="S2.SS5.p3.7.m6.4.5.3.2" xref="S2.SS5.p3.7.m6.4.5.3.1.cmml"><mo id="S2.SS5.p3.7.m6.4.5.3.2.1" stretchy="false" xref="S2.SS5.p3.7.m6.4.5.3.1.1.cmml">[</mo><mi id="S2.SS5.p3.7.m6.1.1" xref="S2.SS5.p3.7.m6.1.1.cmml">m</mi><mo id="S2.SS5.p3.7.m6.4.5.3.2.2" stretchy="false" xref="S2.SS5.p3.7.m6.4.5.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.7.m6.4b"><apply id="S2.SS5.p3.7.m6.4.5.cmml" xref="S2.SS5.p3.7.m6.4.5"><in id="S2.SS5.p3.7.m6.4.5.1.cmml" xref="S2.SS5.p3.7.m6.4.5.1"></in><list id="S2.SS5.p3.7.m6.4.5.2.1.cmml" xref="S2.SS5.p3.7.m6.4.5.2.2"><ci id="S2.SS5.p3.7.m6.2.2.cmml" xref="S2.SS5.p3.7.m6.2.2">𝑎</ci><ci id="S2.SS5.p3.7.m6.3.3.cmml" xref="S2.SS5.p3.7.m6.3.3">𝑏</ci><ci id="S2.SS5.p3.7.m6.4.4.cmml" xref="S2.SS5.p3.7.m6.4.4">𝑐</ci></list><apply id="S2.SS5.p3.7.m6.4.5.3.1.cmml" xref="S2.SS5.p3.7.m6.4.5.3.2"><csymbol cd="latexml" id="S2.SS5.p3.7.m6.4.5.3.1.1.cmml" xref="S2.SS5.p3.7.m6.4.5.3.2.1">delimited-[]</csymbol><ci id="S2.SS5.p3.7.m6.1.1.cmml" xref="S2.SS5.p3.7.m6.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.7.m6.4c">a,b,c\in[m]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.7.m6.4d">italic_a , italic_b , italic_c ∈ [ italic_m ]</annotation></semantics></math> are arbitrary. The meaning of a rule <math alttext="(a,b,c)" class="ltx_Math" display="inline" id="S2.SS5.p3.8.m7.3"><semantics id="S2.SS5.p3.8.m7.3a"><mrow id="S2.SS5.p3.8.m7.3.4.2" xref="S2.SS5.p3.8.m7.3.4.1.cmml"><mo id="S2.SS5.p3.8.m7.3.4.2.1" stretchy="false" xref="S2.SS5.p3.8.m7.3.4.1.cmml">(</mo><mi id="S2.SS5.p3.8.m7.1.1" xref="S2.SS5.p3.8.m7.1.1.cmml">a</mi><mo id="S2.SS5.p3.8.m7.3.4.2.2" xref="S2.SS5.p3.8.m7.3.4.1.cmml">,</mo><mi id="S2.SS5.p3.8.m7.2.2" xref="S2.SS5.p3.8.m7.2.2.cmml">b</mi><mo id="S2.SS5.p3.8.m7.3.4.2.3" xref="S2.SS5.p3.8.m7.3.4.1.cmml">,</mo><mi id="S2.SS5.p3.8.m7.3.3" xref="S2.SS5.p3.8.m7.3.3.cmml">c</mi><mo id="S2.SS5.p3.8.m7.3.4.2.4" stretchy="false" xref="S2.SS5.p3.8.m7.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.8.m7.3b"><vector id="S2.SS5.p3.8.m7.3.4.1.cmml" xref="S2.SS5.p3.8.m7.3.4.2"><ci id="S2.SS5.p3.8.m7.1.1.cmml" xref="S2.SS5.p3.8.m7.1.1">𝑎</ci><ci id="S2.SS5.p3.8.m7.2.2.cmml" xref="S2.SS5.p3.8.m7.2.2">𝑏</ci><ci id="S2.SS5.p3.8.m7.3.3.cmml" xref="S2.SS5.p3.8.m7.3.3">𝑐</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.8.m7.3c">(a,b,c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.8.m7.3d">( italic_a , italic_b , italic_c )</annotation></semantics></math> is that the element <math alttext="c" class="ltx_Math" display="inline" id="S2.SS5.p3.9.m8.1"><semantics id="S2.SS5.p3.9.m8.1a"><mi id="S2.SS5.p3.9.m8.1.1" xref="S2.SS5.p3.9.m8.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.9.m8.1b"><ci id="S2.SS5.p3.9.m8.1.1.cmml" xref="S2.SS5.p3.9.m8.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.9.m8.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.9.m8.1d">italic_c</annotation></semantics></math> should be added to <math alttext="Y" class="ltx_Math" display="inline" id="S2.SS5.p3.10.m9.1"><semantics id="S2.SS5.p3.10.m9.1a"><mi id="S2.SS5.p3.10.m9.1.1" xref="S2.SS5.p3.10.m9.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.10.m9.1b"><ci id="S2.SS5.p3.10.m9.1.1.cmml" xref="S2.SS5.p3.10.m9.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.10.m9.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.10.m9.1d">italic_Y</annotation></semantics></math> in case this set already contains elements <math alttext="a" class="ltx_Math" display="inline" id="S2.SS5.p3.11.m10.1"><semantics id="S2.SS5.p3.11.m10.1a"><mi id="S2.SS5.p3.11.m10.1.1" xref="S2.SS5.p3.11.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.11.m10.1b"><ci id="S2.SS5.p3.11.m10.1.1.cmml" xref="S2.SS5.p3.11.m10.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.11.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.11.m10.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S2.SS5.p3.12.m11.1"><semantics id="S2.SS5.p3.12.m11.1a"><mi id="S2.SS5.p3.12.m11.1.1" xref="S2.SS5.p3.12.m11.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.12.m11.1b"><ci id="S2.SS5.p3.12.m11.1.1.cmml" xref="S2.SS5.p3.12.m11.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.12.m11.1c">b</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.12.m11.1d">italic_b</annotation></semantics></math>. We let <math alttext="\Pi_{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.SS5.p3.13.m12.1"><semantics id="S2.SS5.p3.13.m12.1a"><msub id="S2.SS5.p3.13.m12.1.1" xref="S2.SS5.p3.13.m12.1.1.cmml"><mi id="S2.SS5.p3.13.m12.1.1.2" mathvariant="normal" xref="S2.SS5.p3.13.m12.1.1.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p3.13.m12.1.1.3" xref="S2.SS5.p3.13.m12.1.1.3.cmml">ℛ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.13.m12.1b"><apply id="S2.SS5.p3.13.m12.1.1.cmml" xref="S2.SS5.p3.13.m12.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.13.m12.1.1.1.cmml" xref="S2.SS5.p3.13.m12.1.1">subscript</csymbol><ci id="S2.SS5.p3.13.m12.1.1.2.cmml" xref="S2.SS5.p3.13.m12.1.1.2">Π</ci><ci id="S2.SS5.p3.13.m12.1.1.3.cmml" xref="S2.SS5.p3.13.m12.1.1.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.13.m12.1c">\Pi_{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.13.m12.1d">roman_Π start_POSTSUBSCRIPT caligraphic_R end_POSTSUBSCRIPT</annotation></semantics></math> be the following computational problem: Given an arbitrary initial set <math alttext="Y\subseteq[m]" class="ltx_Math" display="inline" id="S2.SS5.p3.14.m13.1"><semantics id="S2.SS5.p3.14.m13.1a"><mrow id="S2.SS5.p3.14.m13.1.2" xref="S2.SS5.p3.14.m13.1.2.cmml"><mi id="S2.SS5.p3.14.m13.1.2.2" xref="S2.SS5.p3.14.m13.1.2.2.cmml">Y</mi><mo id="S2.SS5.p3.14.m13.1.2.1" xref="S2.SS5.p3.14.m13.1.2.1.cmml">⊆</mo><mrow id="S2.SS5.p3.14.m13.1.2.3.2" xref="S2.SS5.p3.14.m13.1.2.3.1.cmml"><mo id="S2.SS5.p3.14.m13.1.2.3.2.1" stretchy="false" xref="S2.SS5.p3.14.m13.1.2.3.1.1.cmml">[</mo><mi id="S2.SS5.p3.14.m13.1.1" xref="S2.SS5.p3.14.m13.1.1.cmml">m</mi><mo id="S2.SS5.p3.14.m13.1.2.3.2.2" stretchy="false" xref="S2.SS5.p3.14.m13.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.14.m13.1b"><apply id="S2.SS5.p3.14.m13.1.2.cmml" xref="S2.SS5.p3.14.m13.1.2"><subset id="S2.SS5.p3.14.m13.1.2.1.cmml" xref="S2.SS5.p3.14.m13.1.2.1"></subset><ci id="S2.SS5.p3.14.m13.1.2.2.cmml" xref="S2.SS5.p3.14.m13.1.2.2">𝑌</ci><apply id="S2.SS5.p3.14.m13.1.2.3.1.cmml" xref="S2.SS5.p3.14.m13.1.2.3.2"><csymbol cd="latexml" id="S2.SS5.p3.14.m13.1.2.3.1.1.cmml" xref="S2.SS5.p3.14.m13.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS5.p3.14.m13.1.1.cmml" xref="S2.SS5.p3.14.m13.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.14.m13.1c">Y\subseteq[m]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.14.m13.1d">italic_Y ⊆ [ italic_m ]</annotation></semantics></math> as an input instance, is the top element <math alttext="m" class="ltx_Math" display="inline" id="S2.SS5.p3.15.m14.1"><semantics id="S2.SS5.p3.15.m14.1a"><mi id="S2.SS5.p3.15.m14.1.1" xref="S2.SS5.p3.15.m14.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.15.m14.1b"><ci id="S2.SS5.p3.15.m14.1.1.cmml" xref="S2.SS5.p3.15.m14.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.15.m14.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.15.m14.1d">italic_m</annotation></semantics></math> eventually added to <math alttext="Y" class="ltx_Math" display="inline" id="S2.SS5.p3.16.m15.1"><semantics id="S2.SS5.p3.16.m15.1a"><mi id="S2.SS5.p3.16.m15.1.1" xref="S2.SS5.p3.16.m15.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.16.m15.1b"><ci id="S2.SS5.p3.16.m15.1.1.cmml" xref="S2.SS5.p3.16.m15.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.16.m15.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.16.m15.1d">italic_Y</annotation></semantics></math>? (Observe that this problem is closely related to the GEN Boolean function investigated in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib17" title="">17</a>]</cite> and related works.)</p> </div> <div class="ltx_para" id="S2.SS5.p4"> <p class="ltx_p" id="S2.SS5.p4.13">Note that, for every fixed set <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS5.p4.1.m1.1"><semantics id="S2.SS5.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.1.m1.1.1" xref="S2.SS5.p4.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.1.m1.1b"><ci id="S2.SS5.p4.1.m1.1.1.cmml" xref="S2.SS5.p4.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.1.m1.1d">caligraphic_R</annotation></semantics></math> of rules, <math alttext="\Pi_{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.SS5.p4.2.m2.1"><semantics id="S2.SS5.p4.2.m2.1a"><msub id="S2.SS5.p4.2.m2.1.1" xref="S2.SS5.p4.2.m2.1.1.cmml"><mi id="S2.SS5.p4.2.m2.1.1.2" mathvariant="normal" xref="S2.SS5.p4.2.m2.1.1.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.2.m2.1.1.3" xref="S2.SS5.p4.2.m2.1.1.3.cmml">ℛ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.2.m2.1b"><apply id="S2.SS5.p4.2.m2.1.1.cmml" xref="S2.SS5.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.2.m2.1.1.1.cmml" xref="S2.SS5.p4.2.m2.1.1">subscript</csymbol><ci id="S2.SS5.p4.2.m2.1.1.2.cmml" xref="S2.SS5.p4.2.m2.1.1.2">Π</ci><ci id="S2.SS5.p4.2.m2.1.1.3.cmml" xref="S2.SS5.p4.2.m2.1.1.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.2.m2.1c">\Pi_{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.2.m2.1d">roman_Π start_POSTSUBSCRIPT caligraphic_R end_POSTSUBSCRIPT</annotation></semantics></math> can be decided by a cyclic monotone Boolean circuit that contains exactly <math alttext="|\mathcal{R}|" class="ltx_Math" display="inline" id="S2.SS5.p4.3.m3.1"><semantics id="S2.SS5.p4.3.m3.1a"><mrow id="S2.SS5.p4.3.m3.1.2.2" xref="S2.SS5.p4.3.m3.1.2.1.cmml"><mo id="S2.SS5.p4.3.m3.1.2.2.1" stretchy="false" xref="S2.SS5.p4.3.m3.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.3.m3.1.1" xref="S2.SS5.p4.3.m3.1.1.cmml">ℛ</mi><mo id="S2.SS5.p4.3.m3.1.2.2.2" stretchy="false" xref="S2.SS5.p4.3.m3.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.3.m3.1b"><apply id="S2.SS5.p4.3.m3.1.2.1.cmml" xref="S2.SS5.p4.3.m3.1.2.2"><abs id="S2.SS5.p4.3.m3.1.2.1.1.cmml" xref="S2.SS5.p4.3.m3.1.2.2.1"></abs><ci id="S2.SS5.p4.3.m3.1.1.cmml" xref="S2.SS5.p4.3.m3.1.1">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.3.m3.1c">|\mathcal{R}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.3.m3.1d">| caligraphic_R |</annotation></semantics></math> fan-in two AND gates. Indeed, it is enough to consider a circuit over input variables <math alttext="y_{1},\ldots,y_{m}" class="ltx_Math" display="inline" id="S2.SS5.p4.4.m4.3"><semantics id="S2.SS5.p4.4.m4.3a"><mrow id="S2.SS5.p4.4.m4.3.3.2" xref="S2.SS5.p4.4.m4.3.3.3.cmml"><msub id="S2.SS5.p4.4.m4.2.2.1.1" xref="S2.SS5.p4.4.m4.2.2.1.1.cmml"><mi id="S2.SS5.p4.4.m4.2.2.1.1.2" xref="S2.SS5.p4.4.m4.2.2.1.1.2.cmml">y</mi><mn id="S2.SS5.p4.4.m4.2.2.1.1.3" xref="S2.SS5.p4.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS5.p4.4.m4.3.3.2.3" xref="S2.SS5.p4.4.m4.3.3.3.cmml">,</mo><mi id="S2.SS5.p4.4.m4.1.1" mathvariant="normal" xref="S2.SS5.p4.4.m4.1.1.cmml">…</mi><mo id="S2.SS5.p4.4.m4.3.3.2.4" xref="S2.SS5.p4.4.m4.3.3.3.cmml">,</mo><msub id="S2.SS5.p4.4.m4.3.3.2.2" xref="S2.SS5.p4.4.m4.3.3.2.2.cmml"><mi id="S2.SS5.p4.4.m4.3.3.2.2.2" xref="S2.SS5.p4.4.m4.3.3.2.2.2.cmml">y</mi><mi id="S2.SS5.p4.4.m4.3.3.2.2.3" xref="S2.SS5.p4.4.m4.3.3.2.2.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.4.m4.3b"><list id="S2.SS5.p4.4.m4.3.3.3.cmml" xref="S2.SS5.p4.4.m4.3.3.2"><apply id="S2.SS5.p4.4.m4.2.2.1.1.cmml" xref="S2.SS5.p4.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.4.m4.2.2.1.1.1.cmml" xref="S2.SS5.p4.4.m4.2.2.1.1">subscript</csymbol><ci id="S2.SS5.p4.4.m4.2.2.1.1.2.cmml" xref="S2.SS5.p4.4.m4.2.2.1.1.2">𝑦</ci><cn id="S2.SS5.p4.4.m4.2.2.1.1.3.cmml" type="integer" xref="S2.SS5.p4.4.m4.2.2.1.1.3">1</cn></apply><ci id="S2.SS5.p4.4.m4.1.1.cmml" xref="S2.SS5.p4.4.m4.1.1">…</ci><apply id="S2.SS5.p4.4.m4.3.3.2.2.cmml" xref="S2.SS5.p4.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS5.p4.4.m4.3.3.2.2.1.cmml" xref="S2.SS5.p4.4.m4.3.3.2.2">subscript</csymbol><ci id="S2.SS5.p4.4.m4.3.3.2.2.2.cmml" xref="S2.SS5.p4.4.m4.3.3.2.2.2">𝑦</ci><ci id="S2.SS5.p4.4.m4.3.3.2.2.3.cmml" xref="S2.SS5.p4.4.m4.3.3.2.2.3">𝑚</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.4.m4.3c">y_{1},\ldots,y_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.4.m4.3d">italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> that contains three additional layers of gates, described as follows. The first layer contains fan-in two OR gates <math alttext="f_{1},\ldots,f_{m}" class="ltx_Math" display="inline" id="S2.SS5.p4.5.m5.3"><semantics id="S2.SS5.p4.5.m5.3a"><mrow id="S2.SS5.p4.5.m5.3.3.2" xref="S2.SS5.p4.5.m5.3.3.3.cmml"><msub id="S2.SS5.p4.5.m5.2.2.1.1" xref="S2.SS5.p4.5.m5.2.2.1.1.cmml"><mi id="S2.SS5.p4.5.m5.2.2.1.1.2" xref="S2.SS5.p4.5.m5.2.2.1.1.2.cmml">f</mi><mn id="S2.SS5.p4.5.m5.2.2.1.1.3" xref="S2.SS5.p4.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS5.p4.5.m5.3.3.2.3" xref="S2.SS5.p4.5.m5.3.3.3.cmml">,</mo><mi id="S2.SS5.p4.5.m5.1.1" mathvariant="normal" xref="S2.SS5.p4.5.m5.1.1.cmml">…</mi><mo id="S2.SS5.p4.5.m5.3.3.2.4" xref="S2.SS5.p4.5.m5.3.3.3.cmml">,</mo><msub id="S2.SS5.p4.5.m5.3.3.2.2" xref="S2.SS5.p4.5.m5.3.3.2.2.cmml"><mi id="S2.SS5.p4.5.m5.3.3.2.2.2" xref="S2.SS5.p4.5.m5.3.3.2.2.2.cmml">f</mi><mi id="S2.SS5.p4.5.m5.3.3.2.2.3" xref="S2.SS5.p4.5.m5.3.3.2.2.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.5.m5.3b"><list id="S2.SS5.p4.5.m5.3.3.3.cmml" xref="S2.SS5.p4.5.m5.3.3.2"><apply id="S2.SS5.p4.5.m5.2.2.1.1.cmml" xref="S2.SS5.p4.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.5.m5.2.2.1.1.1.cmml" xref="S2.SS5.p4.5.m5.2.2.1.1">subscript</csymbol><ci id="S2.SS5.p4.5.m5.2.2.1.1.2.cmml" xref="S2.SS5.p4.5.m5.2.2.1.1.2">𝑓</ci><cn id="S2.SS5.p4.5.m5.2.2.1.1.3.cmml" type="integer" xref="S2.SS5.p4.5.m5.2.2.1.1.3">1</cn></apply><ci id="S2.SS5.p4.5.m5.1.1.cmml" xref="S2.SS5.p4.5.m5.1.1">…</ci><apply id="S2.SS5.p4.5.m5.3.3.2.2.cmml" xref="S2.SS5.p4.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS5.p4.5.m5.3.3.2.2.1.cmml" xref="S2.SS5.p4.5.m5.3.3.2.2">subscript</csymbol><ci id="S2.SS5.p4.5.m5.3.3.2.2.2.cmml" xref="S2.SS5.p4.5.m5.3.3.2.2.2">𝑓</ci><ci id="S2.SS5.p4.5.m5.3.3.2.2.3.cmml" xref="S2.SS5.p4.5.m5.3.3.2.2.3">𝑚</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.5.m5.3c">f_{1},\ldots,f_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.5.m5.3d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, where each <math alttext="f_{i}" class="ltx_Math" display="inline" id="S2.SS5.p4.6.m6.1"><semantics id="S2.SS5.p4.6.m6.1a"><msub id="S2.SS5.p4.6.m6.1.1" xref="S2.SS5.p4.6.m6.1.1.cmml"><mi id="S2.SS5.p4.6.m6.1.1.2" xref="S2.SS5.p4.6.m6.1.1.2.cmml">f</mi><mi id="S2.SS5.p4.6.m6.1.1.3" xref="S2.SS5.p4.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.6.m6.1b"><apply id="S2.SS5.p4.6.m6.1.1.cmml" xref="S2.SS5.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.6.m6.1.1.1.cmml" xref="S2.SS5.p4.6.m6.1.1">subscript</csymbol><ci id="S2.SS5.p4.6.m6.1.1.2.cmml" xref="S2.SS5.p4.6.m6.1.1.2">𝑓</ci><ci id="S2.SS5.p4.6.m6.1.1.3.cmml" xref="S2.SS5.p4.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.6.m6.1c">f_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.6.m6.1d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is fed by the input variable <math alttext="y_{i}" class="ltx_Math" display="inline" id="S2.SS5.p4.7.m7.1"><semantics id="S2.SS5.p4.7.m7.1a"><msub id="S2.SS5.p4.7.m7.1.1" xref="S2.SS5.p4.7.m7.1.1.cmml"><mi id="S2.SS5.p4.7.m7.1.1.2" xref="S2.SS5.p4.7.m7.1.1.2.cmml">y</mi><mi id="S2.SS5.p4.7.m7.1.1.3" xref="S2.SS5.p4.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.7.m7.1b"><apply id="S2.SS5.p4.7.m7.1.1.cmml" xref="S2.SS5.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.7.m7.1.1.1.cmml" xref="S2.SS5.p4.7.m7.1.1">subscript</csymbol><ci id="S2.SS5.p4.7.m7.1.1.2.cmml" xref="S2.SS5.p4.7.m7.1.1.2">𝑦</ci><ci id="S2.SS5.p4.7.m7.1.1.3.cmml" xref="S2.SS5.p4.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.7.m7.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.7.m7.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and by a corresponding gate <math alttext="h_{i}" class="ltx_Math" display="inline" id="S2.SS5.p4.8.m8.1"><semantics id="S2.SS5.p4.8.m8.1a"><msub id="S2.SS5.p4.8.m8.1.1" xref="S2.SS5.p4.8.m8.1.1.cmml"><mi id="S2.SS5.p4.8.m8.1.1.2" xref="S2.SS5.p4.8.m8.1.1.2.cmml">h</mi><mi id="S2.SS5.p4.8.m8.1.1.3" xref="S2.SS5.p4.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.8.m8.1b"><apply id="S2.SS5.p4.8.m8.1.1.cmml" xref="S2.SS5.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.8.m8.1.1.1.cmml" xref="S2.SS5.p4.8.m8.1.1">subscript</csymbol><ci id="S2.SS5.p4.8.m8.1.1.2.cmml" xref="S2.SS5.p4.8.m8.1.1.2">ℎ</ci><ci id="S2.SS5.p4.8.m8.1.1.3.cmml" xref="S2.SS5.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.8.m8.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.8.m8.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in the third layer. Each rule <math alttext="(a,b,c)\in\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS5.p4.9.m9.3"><semantics id="S2.SS5.p4.9.m9.3a"><mrow id="S2.SS5.p4.9.m9.3.4" xref="S2.SS5.p4.9.m9.3.4.cmml"><mrow id="S2.SS5.p4.9.m9.3.4.2.2" xref="S2.SS5.p4.9.m9.3.4.2.1.cmml"><mo id="S2.SS5.p4.9.m9.3.4.2.2.1" stretchy="false" xref="S2.SS5.p4.9.m9.3.4.2.1.cmml">(</mo><mi id="S2.SS5.p4.9.m9.1.1" xref="S2.SS5.p4.9.m9.1.1.cmml">a</mi><mo id="S2.SS5.p4.9.m9.3.4.2.2.2" xref="S2.SS5.p4.9.m9.3.4.2.1.cmml">,</mo><mi id="S2.SS5.p4.9.m9.2.2" xref="S2.SS5.p4.9.m9.2.2.cmml">b</mi><mo id="S2.SS5.p4.9.m9.3.4.2.2.3" xref="S2.SS5.p4.9.m9.3.4.2.1.cmml">,</mo><mi id="S2.SS5.p4.9.m9.3.3" xref="S2.SS5.p4.9.m9.3.3.cmml">c</mi><mo id="S2.SS5.p4.9.m9.3.4.2.2.4" stretchy="false" xref="S2.SS5.p4.9.m9.3.4.2.1.cmml">)</mo></mrow><mo id="S2.SS5.p4.9.m9.3.4.1" xref="S2.SS5.p4.9.m9.3.4.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.9.m9.3.4.3" xref="S2.SS5.p4.9.m9.3.4.3.cmml">ℛ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.9.m9.3b"><apply id="S2.SS5.p4.9.m9.3.4.cmml" xref="S2.SS5.p4.9.m9.3.4"><in id="S2.SS5.p4.9.m9.3.4.1.cmml" xref="S2.SS5.p4.9.m9.3.4.1"></in><vector id="S2.SS5.p4.9.m9.3.4.2.1.cmml" xref="S2.SS5.p4.9.m9.3.4.2.2"><ci id="S2.SS5.p4.9.m9.1.1.cmml" xref="S2.SS5.p4.9.m9.1.1">𝑎</ci><ci id="S2.SS5.p4.9.m9.2.2.cmml" xref="S2.SS5.p4.9.m9.2.2">𝑏</ci><ci id="S2.SS5.p4.9.m9.3.3.cmml" xref="S2.SS5.p4.9.m9.3.3">𝑐</ci></vector><ci id="S2.SS5.p4.9.m9.3.4.3.cmml" xref="S2.SS5.p4.9.m9.3.4.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.9.m9.3c">(a,b,c)\in\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.9.m9.3d">( italic_a , italic_b , italic_c ) ∈ caligraphic_R</annotation></semantics></math> gives rise to a fan-in two AND gate <math alttext="g_{a,b,c}" class="ltx_Math" display="inline" id="S2.SS5.p4.10.m10.3"><semantics id="S2.SS5.p4.10.m10.3a"><msub id="S2.SS5.p4.10.m10.3.4" xref="S2.SS5.p4.10.m10.3.4.cmml"><mi id="S2.SS5.p4.10.m10.3.4.2" xref="S2.SS5.p4.10.m10.3.4.2.cmml">g</mi><mrow id="S2.SS5.p4.10.m10.3.3.3.5" xref="S2.SS5.p4.10.m10.3.3.3.4.cmml"><mi id="S2.SS5.p4.10.m10.1.1.1.1" xref="S2.SS5.p4.10.m10.1.1.1.1.cmml">a</mi><mo id="S2.SS5.p4.10.m10.3.3.3.5.1" xref="S2.SS5.p4.10.m10.3.3.3.4.cmml">,</mo><mi id="S2.SS5.p4.10.m10.2.2.2.2" xref="S2.SS5.p4.10.m10.2.2.2.2.cmml">b</mi><mo id="S2.SS5.p4.10.m10.3.3.3.5.2" xref="S2.SS5.p4.10.m10.3.3.3.4.cmml">,</mo><mi id="S2.SS5.p4.10.m10.3.3.3.3" xref="S2.SS5.p4.10.m10.3.3.3.3.cmml">c</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.10.m10.3b"><apply id="S2.SS5.p4.10.m10.3.4.cmml" xref="S2.SS5.p4.10.m10.3.4"><csymbol cd="ambiguous" id="S2.SS5.p4.10.m10.3.4.1.cmml" xref="S2.SS5.p4.10.m10.3.4">subscript</csymbol><ci id="S2.SS5.p4.10.m10.3.4.2.cmml" xref="S2.SS5.p4.10.m10.3.4.2">𝑔</ci><list id="S2.SS5.p4.10.m10.3.3.3.4.cmml" xref="S2.SS5.p4.10.m10.3.3.3.5"><ci id="S2.SS5.p4.10.m10.1.1.1.1.cmml" xref="S2.SS5.p4.10.m10.1.1.1.1">𝑎</ci><ci id="S2.SS5.p4.10.m10.2.2.2.2.cmml" xref="S2.SS5.p4.10.m10.2.2.2.2">𝑏</ci><ci id="S2.SS5.p4.10.m10.3.3.3.3.cmml" xref="S2.SS5.p4.10.m10.3.3.3.3">𝑐</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.10.m10.3c">g_{a,b,c}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.10.m10.3d">italic_g start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT</annotation></semantics></math> in the second layer of the circuit, where <math alttext="g_{a,b,c}=f_{a}\wedge f_{b}" class="ltx_Math" display="inline" id="S2.SS5.p4.11.m11.3"><semantics id="S2.SS5.p4.11.m11.3a"><mrow id="S2.SS5.p4.11.m11.3.4" xref="S2.SS5.p4.11.m11.3.4.cmml"><msub id="S2.SS5.p4.11.m11.3.4.2" xref="S2.SS5.p4.11.m11.3.4.2.cmml"><mi id="S2.SS5.p4.11.m11.3.4.2.2" xref="S2.SS5.p4.11.m11.3.4.2.2.cmml">g</mi><mrow id="S2.SS5.p4.11.m11.3.3.3.5" xref="S2.SS5.p4.11.m11.3.3.3.4.cmml"><mi id="S2.SS5.p4.11.m11.1.1.1.1" xref="S2.SS5.p4.11.m11.1.1.1.1.cmml">a</mi><mo id="S2.SS5.p4.11.m11.3.3.3.5.1" xref="S2.SS5.p4.11.m11.3.3.3.4.cmml">,</mo><mi id="S2.SS5.p4.11.m11.2.2.2.2" xref="S2.SS5.p4.11.m11.2.2.2.2.cmml">b</mi><mo id="S2.SS5.p4.11.m11.3.3.3.5.2" xref="S2.SS5.p4.11.m11.3.3.3.4.cmml">,</mo><mi id="S2.SS5.p4.11.m11.3.3.3.3" xref="S2.SS5.p4.11.m11.3.3.3.3.cmml">c</mi></mrow></msub><mo id="S2.SS5.p4.11.m11.3.4.1" xref="S2.SS5.p4.11.m11.3.4.1.cmml">=</mo><mrow id="S2.SS5.p4.11.m11.3.4.3" xref="S2.SS5.p4.11.m11.3.4.3.cmml"><msub id="S2.SS5.p4.11.m11.3.4.3.2" xref="S2.SS5.p4.11.m11.3.4.3.2.cmml"><mi id="S2.SS5.p4.11.m11.3.4.3.2.2" xref="S2.SS5.p4.11.m11.3.4.3.2.2.cmml">f</mi><mi id="S2.SS5.p4.11.m11.3.4.3.2.3" xref="S2.SS5.p4.11.m11.3.4.3.2.3.cmml">a</mi></msub><mo id="S2.SS5.p4.11.m11.3.4.3.1" xref="S2.SS5.p4.11.m11.3.4.3.1.cmml">∧</mo><msub id="S2.SS5.p4.11.m11.3.4.3.3" xref="S2.SS5.p4.11.m11.3.4.3.3.cmml"><mi id="S2.SS5.p4.11.m11.3.4.3.3.2" xref="S2.SS5.p4.11.m11.3.4.3.3.2.cmml">f</mi><mi id="S2.SS5.p4.11.m11.3.4.3.3.3" xref="S2.SS5.p4.11.m11.3.4.3.3.3.cmml">b</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.11.m11.3b"><apply id="S2.SS5.p4.11.m11.3.4.cmml" xref="S2.SS5.p4.11.m11.3.4"><eq id="S2.SS5.p4.11.m11.3.4.1.cmml" xref="S2.SS5.p4.11.m11.3.4.1"></eq><apply id="S2.SS5.p4.11.m11.3.4.2.cmml" xref="S2.SS5.p4.11.m11.3.4.2"><csymbol cd="ambiguous" id="S2.SS5.p4.11.m11.3.4.2.1.cmml" xref="S2.SS5.p4.11.m11.3.4.2">subscript</csymbol><ci id="S2.SS5.p4.11.m11.3.4.2.2.cmml" xref="S2.SS5.p4.11.m11.3.4.2.2">𝑔</ci><list id="S2.SS5.p4.11.m11.3.3.3.4.cmml" xref="S2.SS5.p4.11.m11.3.3.3.5"><ci id="S2.SS5.p4.11.m11.1.1.1.1.cmml" xref="S2.SS5.p4.11.m11.1.1.1.1">𝑎</ci><ci id="S2.SS5.p4.11.m11.2.2.2.2.cmml" xref="S2.SS5.p4.11.m11.2.2.2.2">𝑏</ci><ci id="S2.SS5.p4.11.m11.3.3.3.3.cmml" xref="S2.SS5.p4.11.m11.3.3.3.3">𝑐</ci></list></apply><apply id="S2.SS5.p4.11.m11.3.4.3.cmml" xref="S2.SS5.p4.11.m11.3.4.3"><and id="S2.SS5.p4.11.m11.3.4.3.1.cmml" xref="S2.SS5.p4.11.m11.3.4.3.1"></and><apply id="S2.SS5.p4.11.m11.3.4.3.2.cmml" xref="S2.SS5.p4.11.m11.3.4.3.2"><csymbol cd="ambiguous" id="S2.SS5.p4.11.m11.3.4.3.2.1.cmml" xref="S2.SS5.p4.11.m11.3.4.3.2">subscript</csymbol><ci id="S2.SS5.p4.11.m11.3.4.3.2.2.cmml" xref="S2.SS5.p4.11.m11.3.4.3.2.2">𝑓</ci><ci id="S2.SS5.p4.11.m11.3.4.3.2.3.cmml" xref="S2.SS5.p4.11.m11.3.4.3.2.3">𝑎</ci></apply><apply id="S2.SS5.p4.11.m11.3.4.3.3.cmml" xref="S2.SS5.p4.11.m11.3.4.3.3"><csymbol cd="ambiguous" id="S2.SS5.p4.11.m11.3.4.3.3.1.cmml" xref="S2.SS5.p4.11.m11.3.4.3.3">subscript</csymbol><ci id="S2.SS5.p4.11.m11.3.4.3.3.2.cmml" xref="S2.SS5.p4.11.m11.3.4.3.3.2">𝑓</ci><ci id="S2.SS5.p4.11.m11.3.4.3.3.3.cmml" xref="S2.SS5.p4.11.m11.3.4.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.11.m11.3c">g_{a,b,c}=f_{a}\wedge f_{b}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.11.m11.3d">italic_g start_POSTSUBSCRIPT italic_a , italic_b , italic_c end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∧ italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Finally, in the third layer we have for each <math alttext="i\in[m]" class="ltx_Math" display="inline" id="S2.SS5.p4.12.m12.1"><semantics id="S2.SS5.p4.12.m12.1a"><mrow id="S2.SS5.p4.12.m12.1.2" xref="S2.SS5.p4.12.m12.1.2.cmml"><mi id="S2.SS5.p4.12.m12.1.2.2" xref="S2.SS5.p4.12.m12.1.2.2.cmml">i</mi><mo id="S2.SS5.p4.12.m12.1.2.1" xref="S2.SS5.p4.12.m12.1.2.1.cmml">∈</mo><mrow id="S2.SS5.p4.12.m12.1.2.3.2" xref="S2.SS5.p4.12.m12.1.2.3.1.cmml"><mo id="S2.SS5.p4.12.m12.1.2.3.2.1" stretchy="false" xref="S2.SS5.p4.12.m12.1.2.3.1.1.cmml">[</mo><mi id="S2.SS5.p4.12.m12.1.1" xref="S2.SS5.p4.12.m12.1.1.cmml">m</mi><mo id="S2.SS5.p4.12.m12.1.2.3.2.2" stretchy="false" xref="S2.SS5.p4.12.m12.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.12.m12.1b"><apply id="S2.SS5.p4.12.m12.1.2.cmml" xref="S2.SS5.p4.12.m12.1.2"><in id="S2.SS5.p4.12.m12.1.2.1.cmml" xref="S2.SS5.p4.12.m12.1.2.1"></in><ci id="S2.SS5.p4.12.m12.1.2.2.cmml" xref="S2.SS5.p4.12.m12.1.2.2">𝑖</ci><apply id="S2.SS5.p4.12.m12.1.2.3.1.cmml" xref="S2.SS5.p4.12.m12.1.2.3.2"><csymbol cd="latexml" id="S2.SS5.p4.12.m12.1.2.3.1.1.cmml" xref="S2.SS5.p4.12.m12.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS5.p4.12.m12.1.1.cmml" xref="S2.SS5.p4.12.m12.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.12.m12.1c">i\in[m]</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.12.m12.1d">italic_i ∈ [ italic_m ]</annotation></semantics></math> a corresponding OR gate <math alttext="h_{i}" class="ltx_Math" display="inline" id="S2.SS5.p4.13.m13.1"><semantics id="S2.SS5.p4.13.m13.1a"><msub id="S2.SS5.p4.13.m13.1.1" xref="S2.SS5.p4.13.m13.1.1.cmml"><mi id="S2.SS5.p4.13.m13.1.1.2" xref="S2.SS5.p4.13.m13.1.1.2.cmml">h</mi><mi id="S2.SS5.p4.13.m13.1.1.3" xref="S2.SS5.p4.13.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.13.m13.1b"><apply id="S2.SS5.p4.13.m13.1.1.cmml" xref="S2.SS5.p4.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.13.m13.1.1.1.cmml" xref="S2.SS5.p4.13.m13.1.1">subscript</csymbol><ci id="S2.SS5.p4.13.m13.1.1.2.cmml" xref="S2.SS5.p4.13.m13.1.1.2">ℎ</ci><ci id="S2.SS5.p4.13.m13.1.1.3.cmml" xref="S2.SS5.p4.13.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.13.m13.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.13.m13.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="h_{i}=\bigvee_{u,v\in[m],(u,v,i)\in\mathcal{R}}g_{u,v,i}." class="ltx_Math" display="block" id="S2.Ex15.m1.12"><semantics id="S2.Ex15.m1.12a"><mrow id="S2.Ex15.m1.12.12.1" xref="S2.Ex15.m1.12.12.1.1.cmml"><mrow id="S2.Ex15.m1.12.12.1.1" xref="S2.Ex15.m1.12.12.1.1.cmml"><msub id="S2.Ex15.m1.12.12.1.1.2" xref="S2.Ex15.m1.12.12.1.1.2.cmml"><mi id="S2.Ex15.m1.12.12.1.1.2.2" xref="S2.Ex15.m1.12.12.1.1.2.2.cmml">h</mi><mi id="S2.Ex15.m1.12.12.1.1.2.3" xref="S2.Ex15.m1.12.12.1.1.2.3.cmml">i</mi></msub><mo id="S2.Ex15.m1.12.12.1.1.1" rspace="0.111em" xref="S2.Ex15.m1.12.12.1.1.1.cmml">=</mo><mrow id="S2.Ex15.m1.12.12.1.1.3" xref="S2.Ex15.m1.12.12.1.1.3.cmml"><munder id="S2.Ex15.m1.12.12.1.1.3.1" xref="S2.Ex15.m1.12.12.1.1.3.1.cmml"><mo id="S2.Ex15.m1.12.12.1.1.3.1.2" movablelimits="false" xref="S2.Ex15.m1.12.12.1.1.3.1.2.cmml">⋁</mo><mrow id="S2.Ex15.m1.8.8.8.8" xref="S2.Ex15.m1.8.8.8.9.cmml"><mrow id="S2.Ex15.m1.7.7.7.7.1" xref="S2.Ex15.m1.7.7.7.7.1.cmml"><mrow id="S2.Ex15.m1.7.7.7.7.1.2.2" xref="S2.Ex15.m1.7.7.7.7.1.2.1.cmml"><mi id="S2.Ex15.m1.2.2.2.2" xref="S2.Ex15.m1.2.2.2.2.cmml">u</mi><mo id="S2.Ex15.m1.7.7.7.7.1.2.2.1" xref="S2.Ex15.m1.7.7.7.7.1.2.1.cmml">,</mo><mi id="S2.Ex15.m1.3.3.3.3" xref="S2.Ex15.m1.3.3.3.3.cmml">v</mi></mrow><mo id="S2.Ex15.m1.7.7.7.7.1.1" xref="S2.Ex15.m1.7.7.7.7.1.1.cmml">∈</mo><mrow id="S2.Ex15.m1.7.7.7.7.1.3.2" xref="S2.Ex15.m1.7.7.7.7.1.3.1.cmml"><mo id="S2.Ex15.m1.7.7.7.7.1.3.2.1" stretchy="false" xref="S2.Ex15.m1.7.7.7.7.1.3.1.1.cmml">[</mo><mi id="S2.Ex15.m1.1.1.1.1" xref="S2.Ex15.m1.1.1.1.1.cmml">m</mi><mo id="S2.Ex15.m1.7.7.7.7.1.3.2.2" stretchy="false" xref="S2.Ex15.m1.7.7.7.7.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S2.Ex15.m1.8.8.8.8.3" xref="S2.Ex15.m1.8.8.8.9a.cmml">,</mo><mrow id="S2.Ex15.m1.8.8.8.8.2" xref="S2.Ex15.m1.8.8.8.8.2.cmml"><mrow id="S2.Ex15.m1.8.8.8.8.2.2.2" xref="S2.Ex15.m1.8.8.8.8.2.2.1.cmml"><mo id="S2.Ex15.m1.8.8.8.8.2.2.2.1" stretchy="false" xref="S2.Ex15.m1.8.8.8.8.2.2.1.cmml">(</mo><mi id="S2.Ex15.m1.4.4.4.4" xref="S2.Ex15.m1.4.4.4.4.cmml">u</mi><mo id="S2.Ex15.m1.8.8.8.8.2.2.2.2" xref="S2.Ex15.m1.8.8.8.8.2.2.1.cmml">,</mo><mi id="S2.Ex15.m1.5.5.5.5" xref="S2.Ex15.m1.5.5.5.5.cmml">v</mi><mo id="S2.Ex15.m1.8.8.8.8.2.2.2.3" xref="S2.Ex15.m1.8.8.8.8.2.2.1.cmml">,</mo><mi id="S2.Ex15.m1.6.6.6.6" xref="S2.Ex15.m1.6.6.6.6.cmml">i</mi><mo id="S2.Ex15.m1.8.8.8.8.2.2.2.4" stretchy="false" xref="S2.Ex15.m1.8.8.8.8.2.2.1.cmml">)</mo></mrow><mo id="S2.Ex15.m1.8.8.8.8.2.1" xref="S2.Ex15.m1.8.8.8.8.2.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex15.m1.8.8.8.8.2.3" xref="S2.Ex15.m1.8.8.8.8.2.3.cmml">ℛ</mi></mrow></mrow></munder><msub id="S2.Ex15.m1.12.12.1.1.3.2" xref="S2.Ex15.m1.12.12.1.1.3.2.cmml"><mi id="S2.Ex15.m1.12.12.1.1.3.2.2" xref="S2.Ex15.m1.12.12.1.1.3.2.2.cmml">g</mi><mrow id="S2.Ex15.m1.11.11.3.5" xref="S2.Ex15.m1.11.11.3.4.cmml"><mi id="S2.Ex15.m1.9.9.1.1" xref="S2.Ex15.m1.9.9.1.1.cmml">u</mi><mo id="S2.Ex15.m1.11.11.3.5.1" xref="S2.Ex15.m1.11.11.3.4.cmml">,</mo><mi id="S2.Ex15.m1.10.10.2.2" xref="S2.Ex15.m1.10.10.2.2.cmml">v</mi><mo id="S2.Ex15.m1.11.11.3.5.2" 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xref="S2.Ex15.m1.11.11.3.3">𝑖</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex15.m1.12c">h_{i}=\bigvee_{u,v\in[m],(u,v,i)\in\mathcal{R}}g_{u,v,i}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex15.m1.12d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ⋁ start_POSTSUBSCRIPT italic_u , italic_v ∈ [ italic_m ] , ( italic_u , italic_v , italic_i ) ∈ caligraphic_R end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_u , italic_v , italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.p4.16">(We stress that <em class="ltx_emph ltx_font_italic" id="S2.SS5.p4.16.1">unbounded fan-in</em> gates are used only to simplify the description of the circuit.) It is easy to see that the gate <math alttext="f_{m}" class="ltx_Math" display="inline" id="S2.SS5.p4.14.m1.1"><semantics id="S2.SS5.p4.14.m1.1a"><msub id="S2.SS5.p4.14.m1.1.1" xref="S2.SS5.p4.14.m1.1.1.cmml"><mi id="S2.SS5.p4.14.m1.1.1.2" xref="S2.SS5.p4.14.m1.1.1.2.cmml">f</mi><mi id="S2.SS5.p4.14.m1.1.1.3" xref="S2.SS5.p4.14.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.14.m1.1b"><apply id="S2.SS5.p4.14.m1.1.1.cmml" xref="S2.SS5.p4.14.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.14.m1.1.1.1.cmml" xref="S2.SS5.p4.14.m1.1.1">subscript</csymbol><ci id="S2.SS5.p4.14.m1.1.1.2.cmml" xref="S2.SS5.p4.14.m1.1.1.2">𝑓</ci><ci id="S2.SS5.p4.14.m1.1.1.3.cmml" xref="S2.SS5.p4.14.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.14.m1.1c">f_{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.14.m1.1d">italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> computes <math alttext="\Pi_{\mathcal{R}}" class="ltx_Math" display="inline" id="S2.SS5.p4.15.m2.1"><semantics id="S2.SS5.p4.15.m2.1a"><msub id="S2.SS5.p4.15.m2.1.1" xref="S2.SS5.p4.15.m2.1.1.cmml"><mi id="S2.SS5.p4.15.m2.1.1.2" mathvariant="normal" xref="S2.SS5.p4.15.m2.1.1.2.cmml">Π</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.15.m2.1.1.3" xref="S2.SS5.p4.15.m2.1.1.3.cmml">ℛ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.15.m2.1b"><apply id="S2.SS5.p4.15.m2.1.1.cmml" xref="S2.SS5.p4.15.m2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p4.15.m2.1.1.1.cmml" xref="S2.SS5.p4.15.m2.1.1">subscript</csymbol><ci id="S2.SS5.p4.15.m2.1.1.2.cmml" xref="S2.SS5.p4.15.m2.1.1.2">Π</ci><ci id="S2.SS5.p4.15.m2.1.1.3.cmml" xref="S2.SS5.p4.15.m2.1.1.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.15.m2.1c">\Pi_{\mathcal{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.15.m2.1d">roman_Π start_POSTSUBSCRIPT caligraphic_R end_POSTSUBSCRIPT</annotation></semantics></math> after at most <math alttext="O(|\mathcal{R}|)" class="ltx_Math" display="inline" id="S2.SS5.p4.16.m3.2"><semantics id="S2.SS5.p4.16.m3.2a"><mrow id="S2.SS5.p4.16.m3.2.2" xref="S2.SS5.p4.16.m3.2.2.cmml"><mi id="S2.SS5.p4.16.m3.2.2.3" xref="S2.SS5.p4.16.m3.2.2.3.cmml">O</mi><mo id="S2.SS5.p4.16.m3.2.2.2" xref="S2.SS5.p4.16.m3.2.2.2.cmml">⁢</mo><mrow id="S2.SS5.p4.16.m3.2.2.1.1" xref="S2.SS5.p4.16.m3.2.2.cmml"><mo id="S2.SS5.p4.16.m3.2.2.1.1.2" stretchy="false" xref="S2.SS5.p4.16.m3.2.2.cmml">(</mo><mrow id="S2.SS5.p4.16.m3.2.2.1.1.1.2" xref="S2.SS5.p4.16.m3.2.2.1.1.1.1.cmml"><mo id="S2.SS5.p4.16.m3.2.2.1.1.1.2.1" stretchy="false" xref="S2.SS5.p4.16.m3.2.2.1.1.1.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p4.16.m3.1.1" xref="S2.SS5.p4.16.m3.1.1.cmml">ℛ</mi><mo id="S2.SS5.p4.16.m3.2.2.1.1.1.2.2" stretchy="false" xref="S2.SS5.p4.16.m3.2.2.1.1.1.1.1.cmml">|</mo></mrow><mo id="S2.SS5.p4.16.m3.2.2.1.1.3" stretchy="false" xref="S2.SS5.p4.16.m3.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p4.16.m3.2b"><apply id="S2.SS5.p4.16.m3.2.2.cmml" xref="S2.SS5.p4.16.m3.2.2"><times id="S2.SS5.p4.16.m3.2.2.2.cmml" xref="S2.SS5.p4.16.m3.2.2.2"></times><ci id="S2.SS5.p4.16.m3.2.2.3.cmml" xref="S2.SS5.p4.16.m3.2.2.3">𝑂</ci><apply id="S2.SS5.p4.16.m3.2.2.1.1.1.1.cmml" xref="S2.SS5.p4.16.m3.2.2.1.1.1.2"><abs id="S2.SS5.p4.16.m3.2.2.1.1.1.1.1.cmml" xref="S2.SS5.p4.16.m3.2.2.1.1.1.2.1"></abs><ci id="S2.SS5.p4.16.m3.1.1.cmml" xref="S2.SS5.p4.16.m3.1.1">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p4.16.m3.2c">O(|\mathcal{R}|)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p4.16.m3.2d">italic_O ( | caligraphic_R | )</annotation></semantics></math> iterations of the evaluation procedure.</p> </div> </section> </section> <section class="ltx_section ltx_indent_first" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Characterizations of Discrete Complexity via Set-Theoretic Fusion</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The technique presented in this section can be seen as a set-theoretic formulation of some results from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib16" title="">16</a>]</cite> and <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>. The tighter characterization that appears in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS4" title="3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.4</span></a> is an adaptation of a result from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite>.</p> </div> <section class="ltx_subsection ltx_indent_first" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Definitions and notation</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.5">For convenience, let <math alttext="U\stackrel{{\scriptstyle\rm def}}{{=}}A^{c}=\Gamma\setminus A" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">U</mi><mover id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml"><mo id="S3.SS1.p1.1.m1.1.1.3.2" xref="S3.SS1.p1.1.m1.1.1.3.2.cmml">=</mo><mi id="S3.SS1.p1.1.m1.1.1.3.3" xref="S3.SS1.p1.1.m1.1.1.3.3.cmml">def</mi></mover><msup id="S3.SS1.p1.1.m1.1.1.4" xref="S3.SS1.p1.1.m1.1.1.4.cmml"><mi id="S3.SS1.p1.1.m1.1.1.4.2" xref="S3.SS1.p1.1.m1.1.1.4.2.cmml">A</mi><mi id="S3.SS1.p1.1.m1.1.1.4.3" xref="S3.SS1.p1.1.m1.1.1.4.3.cmml">c</mi></msup><mo id="S3.SS1.p1.1.m1.1.1.5" xref="S3.SS1.p1.1.m1.1.1.5.cmml">=</mo><mrow id="S3.SS1.p1.1.m1.1.1.6" xref="S3.SS1.p1.1.m1.1.1.6.cmml"><mi id="S3.SS1.p1.1.m1.1.1.6.2" mathvariant="normal" xref="S3.SS1.p1.1.m1.1.1.6.2.cmml">Γ</mi><mo id="S3.SS1.p1.1.m1.1.1.6.1" xref="S3.SS1.p1.1.m1.1.1.6.1.cmml">∖</mo><mi id="S3.SS1.p1.1.m1.1.1.6.3" xref="S3.SS1.p1.1.m1.1.1.6.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><and id="S3.SS1.p1.1.m1.1.1a.cmml" xref="S3.SS1.p1.1.m1.1.1"></and><apply id="S3.SS1.p1.1.m1.1.1b.cmml" xref="S3.SS1.p1.1.m1.1.1"><apply id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.p1.1.m1.1.1.3">superscript</csymbol><eq id="S3.SS1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS1.p1.1.m1.1.1.3.2"></eq><ci id="S3.SS1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3.3">def</ci></apply><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑈</ci><apply id="S3.SS1.p1.1.m1.1.1.4.cmml" xref="S3.SS1.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.1.1.4.1.cmml" xref="S3.SS1.p1.1.m1.1.1.4">superscript</csymbol><ci id="S3.SS1.p1.1.m1.1.1.4.2.cmml" xref="S3.SS1.p1.1.m1.1.1.4.2">𝐴</ci><ci id="S3.SS1.p1.1.m1.1.1.4.3.cmml" xref="S3.SS1.p1.1.m1.1.1.4.3">𝑐</ci></apply></apply><apply id="S3.SS1.p1.1.m1.1.1c.cmml" xref="S3.SS1.p1.1.m1.1.1"><eq id="S3.SS1.p1.1.m1.1.1.5.cmml" xref="S3.SS1.p1.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS1.p1.1.m1.1.1.4.cmml" id="S3.SS1.p1.1.m1.1.1d.cmml" xref="S3.SS1.p1.1.m1.1.1"></share><apply id="S3.SS1.p1.1.m1.1.1.6.cmml" xref="S3.SS1.p1.1.m1.1.1.6"><setdiff id="S3.SS1.p1.1.m1.1.1.6.1.cmml" xref="S3.SS1.p1.1.m1.1.1.6.1"></setdiff><ci id="S3.SS1.p1.1.m1.1.1.6.2.cmml" xref="S3.SS1.p1.1.m1.1.1.6.2">Γ</ci><ci id="S3.SS1.p1.1.m1.1.1.6.3.cmml" xref="S3.SS1.p1.1.m1.1.1.6.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">U\stackrel{{\scriptstyle\rm def}}{{=}}A^{c}=\Gamma\setminus A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_U start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = roman_Γ ∖ italic_A</annotation></semantics></math>, where <math alttext="\Gamma" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" mathvariant="normal" xref="S3.SS1.p1.2.m2.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">roman_Γ</annotation></semantics></math> is the ambient space. We assume from now on that <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_A</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S3.SS1.p1.5.1">non-trivial</em>, i.e., both <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mi id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_A</annotation></semantics></math> and <math alttext="A^{c}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><msup id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml"><mi id="S3.SS1.p1.5.m5.1.1.2" xref="S3.SS1.p1.5.m5.1.1.2.cmml">A</mi><mi id="S3.SS1.p1.5.m5.1.1.3" xref="S3.SS1.p1.5.m5.1.1.3.cmml">c</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><apply id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.SS1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.p1.5.m5.1.1.2">𝐴</ci><ci id="S3.SS1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.p1.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">A^{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math> are non-empty.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem18"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem18.1.1.1">Definition 18</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem18.2.2"> </span>(Semi-filter)<span class="ltx_text ltx_font_bold" id="Thmtheorem18.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem18.p1"> <p class="ltx_p" id="Thmtheorem18.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem18.p1.2.2">We say that a non-empty family <math alttext="\mathcal{F}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="Thmtheorem18.p1.1.1.m1.1"><semantics id="Thmtheorem18.p1.1.1.m1.1a"><mrow id="Thmtheorem18.p1.1.1.m1.1.2" xref="Thmtheorem18.p1.1.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem18.p1.1.1.m1.1.2.2" xref="Thmtheorem18.p1.1.1.m1.1.2.2.cmml">ℱ</mi><mo id="Thmtheorem18.p1.1.1.m1.1.2.1" xref="Thmtheorem18.p1.1.1.m1.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem18.p1.1.1.m1.1.2.3" xref="Thmtheorem18.p1.1.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem18.p1.1.1.m1.1.2.3.2" xref="Thmtheorem18.p1.1.1.m1.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem18.p1.1.1.m1.1.2.3.1" xref="Thmtheorem18.p1.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem18.p1.1.1.m1.1.2.3.3.2" xref="Thmtheorem18.p1.1.1.m1.1.2.3.cmml"><mo id="Thmtheorem18.p1.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem18.p1.1.1.m1.1.2.3.cmml">(</mo><mi id="Thmtheorem18.p1.1.1.m1.1.1" xref="Thmtheorem18.p1.1.1.m1.1.1.cmml">U</mi><mo id="Thmtheorem18.p1.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem18.p1.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem18.p1.1.1.m1.1b"><apply id="Thmtheorem18.p1.1.1.m1.1.2.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2"><subset id="Thmtheorem18.p1.1.1.m1.1.2.1.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2.1"></subset><ci id="Thmtheorem18.p1.1.1.m1.1.2.2.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2.2">ℱ</ci><apply id="Thmtheorem18.p1.1.1.m1.1.2.3.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2.3"><times id="Thmtheorem18.p1.1.1.m1.1.2.3.1.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2.3.1"></times><ci id="Thmtheorem18.p1.1.1.m1.1.2.3.2.cmml" xref="Thmtheorem18.p1.1.1.m1.1.2.3.2">𝒫</ci><ci id="Thmtheorem18.p1.1.1.m1.1.1.cmml" xref="Thmtheorem18.p1.1.1.m1.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem18.p1.1.1.m1.1c">\mathcal{F}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem18.p1.1.1.m1.1d">caligraphic_F ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> of sets is a <em class="ltx_emph ltx_font_upright" id="Thmtheorem18.p1.2.2.1">semi-filter</em> over <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem18.p1.2.2.m2.1"><semantics id="Thmtheorem18.p1.2.2.m2.1a"><mi id="Thmtheorem18.p1.2.2.m2.1.1" xref="Thmtheorem18.p1.2.2.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem18.p1.2.2.m2.1b"><ci id="Thmtheorem18.p1.2.2.m2.1.1.cmml" xref="Thmtheorem18.p1.2.2.m2.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem18.p1.2.2.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem18.p1.2.2.m2.1d">italic_U</annotation></semantics></math> if the following hold:</span></p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.3"><em class="ltx_emph" id="S3.I1.i1.p1.3.1">(upward closure)</em><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.3.2"> If </span><math alttext="U_{1}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><msub id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2.2" xref="S3.I1.i1.p1.1.m1.1.1.2.2.cmml">U</mi><mn id="S3.I1.i1.p1.1.m1.1.1.2.3" xref="S3.I1.i1.p1.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><in id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"></in><apply id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.2">𝑈</ci><cn id="S3.I1.i1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.2.3">1</cn></apply><ci id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">U_{1}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.3.3"> and </span><math alttext="U_{1}\subseteq U_{2}\subseteq U" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mrow id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml"><msub id="S3.I1.i1.p1.2.m2.1.1.2" xref="S3.I1.i1.p1.2.m2.1.1.2.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.2.2" xref="S3.I1.i1.p1.2.m2.1.1.2.2.cmml">U</mi><mn id="S3.I1.i1.p1.2.m2.1.1.2.3" xref="S3.I1.i1.p1.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.I1.i1.p1.2.m2.1.1.3" xref="S3.I1.i1.p1.2.m2.1.1.3.cmml">⊆</mo><msub id="S3.I1.i1.p1.2.m2.1.1.4" xref="S3.I1.i1.p1.2.m2.1.1.4.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.4.2" xref="S3.I1.i1.p1.2.m2.1.1.4.2.cmml">U</mi><mn id="S3.I1.i1.p1.2.m2.1.1.4.3" xref="S3.I1.i1.p1.2.m2.1.1.4.3.cmml">2</mn></msub><mo id="S3.I1.i1.p1.2.m2.1.1.5" xref="S3.I1.i1.p1.2.m2.1.1.5.cmml">⊆</mo><mi id="S3.I1.i1.p1.2.m2.1.1.6" xref="S3.I1.i1.p1.2.m2.1.1.6.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><apply id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><and id="S3.I1.i1.p1.2.m2.1.1a.cmml" xref="S3.I1.i1.p1.2.m2.1.1"></and><apply id="S3.I1.i1.p1.2.m2.1.1b.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><subset id="S3.I1.i1.p1.2.m2.1.1.3.cmml" xref="S3.I1.i1.p1.2.m2.1.1.3"></subset><apply id="S3.I1.i1.p1.2.m2.1.1.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.2.m2.1.1.2.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2.2">𝑈</ci><cn id="S3.I1.i1.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S3.I1.i1.p1.2.m2.1.1.2.3">1</cn></apply><apply id="S3.I1.i1.p1.2.m2.1.1.4.cmml" xref="S3.I1.i1.p1.2.m2.1.1.4"><csymbol cd="ambiguous" id="S3.I1.i1.p1.2.m2.1.1.4.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1.4">subscript</csymbol><ci id="S3.I1.i1.p1.2.m2.1.1.4.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.4.2">𝑈</ci><cn id="S3.I1.i1.p1.2.m2.1.1.4.3.cmml" type="integer" xref="S3.I1.i1.p1.2.m2.1.1.4.3">2</cn></apply></apply><apply id="S3.I1.i1.p1.2.m2.1.1c.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><subset id="S3.I1.i1.p1.2.m2.1.1.5.cmml" xref="S3.I1.i1.p1.2.m2.1.1.5"></subset><share href="https://arxiv.org/html/2503.14117v1#S3.I1.i1.p1.2.m2.1.1.4.cmml" id="S3.I1.i1.p1.2.m2.1.1d.cmml" xref="S3.I1.i1.p1.2.m2.1.1"></share><ci id="S3.I1.i1.p1.2.m2.1.1.6.cmml" xref="S3.I1.i1.p1.2.m2.1.1.6">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">U_{1}\subseteq U_{2}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊆ italic_U</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.3.4">, then </span><math alttext="U_{2}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><mrow id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><msub id="S3.I1.i1.p1.3.m3.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.2.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.2.2" xref="S3.I1.i1.p1.3.m3.1.1.2.2.cmml">U</mi><mn id="S3.I1.i1.p1.3.m3.1.1.2.3" xref="S3.I1.i1.p1.3.m3.1.1.2.3.cmml">2</mn></msub><mo id="S3.I1.i1.p1.3.m3.1.1.1" xref="S3.I1.i1.p1.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><in id="S3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1"></in><apply id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2.2">𝑈</ci><cn id="S3.I1.i1.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S3.I1.i1.p1.3.m3.1.1.2.3">2</cn></apply><ci id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">U_{2}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.3.5">.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1"><em class="ltx_emph" id="S3.I1.i2.p1.1.1">(non-trivial)</em><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.1.2"> </span><math alttext="\emptyset\notin\mathcal{F}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" mathvariant="normal" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">∅</mi><mo id="S3.I1.i2.p1.1.m1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml">∉</mo><mi class="ltx_font_mathcaligraphic" id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><notin id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1"></notin><emptyset id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2"></emptyset><ci id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">\emptyset\notin\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">∅ ∉ caligraphic_F</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.1.3">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem19"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem19.2.1.1">Definition 19</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem19.3.2"> </span>(Semi-filter above <math alttext="w" class="ltx_Math" display="inline" id="Thmtheorem19.1.m1.1"><semantics id="Thmtheorem19.1.m1.1b"><mi id="Thmtheorem19.1.m1.1.1" xref="Thmtheorem19.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem19.1.m1.1c"><ci id="Thmtheorem19.1.m1.1.1.cmml" xref="Thmtheorem19.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.1.m1.1d">w</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.1.m1.1e">italic_w</annotation></semantics></math>)<span class="ltx_text ltx_font_bold" id="Thmtheorem19.4.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem19.p1"> <p class="ltx_p" id="Thmtheorem19.p1.7"><span class="ltx_text ltx_font_italic" id="Thmtheorem19.p1.7.7">We say that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem19.p1.1.1.m1.1"><semantics id="Thmtheorem19.p1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem19.p1.1.1.m1.1.1" xref="Thmtheorem19.p1.1.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.1.1.m1.1b"><ci id="Thmtheorem19.p1.1.1.m1.1.1.cmml" xref="Thmtheorem19.p1.1.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.1.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.1.1.m1.1d">caligraphic_F</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="Thmtheorem19.p1.7.7.3">above</em> an element <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="Thmtheorem19.p1.2.2.m2.1"><semantics id="Thmtheorem19.p1.2.2.m2.1a"><mrow id="Thmtheorem19.p1.2.2.m2.1.1" xref="Thmtheorem19.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem19.p1.2.2.m2.1.1.2" xref="Thmtheorem19.p1.2.2.m2.1.1.2.cmml">w</mi><mo id="Thmtheorem19.p1.2.2.m2.1.1.1" xref="Thmtheorem19.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="Thmtheorem19.p1.2.2.m2.1.1.3" mathvariant="normal" xref="Thmtheorem19.p1.2.2.m2.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.2.2.m2.1b"><apply id="Thmtheorem19.p1.2.2.m2.1.1.cmml" xref="Thmtheorem19.p1.2.2.m2.1.1"><in id="Thmtheorem19.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem19.p1.2.2.m2.1.1.1"></in><ci id="Thmtheorem19.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem19.p1.2.2.m2.1.1.2">𝑤</ci><ci id="Thmtheorem19.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem19.p1.2.2.m2.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.2.2.m2.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.2.2.m2.1d">italic_w ∈ roman_Γ</annotation></semantics></math> <em class="ltx_emph ltx_font_upright" id="Thmtheorem19.p1.4.4.2">(with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem19.p1.3.3.1.m1.1"><semantics id="Thmtheorem19.p1.3.3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem19.p1.3.3.1.m1.1.1" xref="Thmtheorem19.p1.3.3.1.m1.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.3.3.1.m1.1b"><ci id="Thmtheorem19.p1.3.3.1.m1.1.1.cmml" xref="Thmtheorem19.p1.3.3.1.m1.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.3.3.1.m1.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.3.3.1.m1.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U=A^{c}" class="ltx_Math" display="inline" id="Thmtheorem19.p1.4.4.2.m2.1"><semantics id="Thmtheorem19.p1.4.4.2.m2.1a"><mrow id="Thmtheorem19.p1.4.4.2.m2.1.1" xref="Thmtheorem19.p1.4.4.2.m2.1.1.cmml"><mi id="Thmtheorem19.p1.4.4.2.m2.1.1.2" xref="Thmtheorem19.p1.4.4.2.m2.1.1.2.cmml">U</mi><mo id="Thmtheorem19.p1.4.4.2.m2.1.1.1" xref="Thmtheorem19.p1.4.4.2.m2.1.1.1.cmml">=</mo><msup id="Thmtheorem19.p1.4.4.2.m2.1.1.3" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3.cmml"><mi id="Thmtheorem19.p1.4.4.2.m2.1.1.3.2" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3.2.cmml">A</mi><mi id="Thmtheorem19.p1.4.4.2.m2.1.1.3.3" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3.3.cmml">c</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.4.4.2.m2.1b"><apply id="Thmtheorem19.p1.4.4.2.m2.1.1.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1"><eq id="Thmtheorem19.p1.4.4.2.m2.1.1.1.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.1"></eq><ci id="Thmtheorem19.p1.4.4.2.m2.1.1.2.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.2">𝑈</ci><apply id="Thmtheorem19.p1.4.4.2.m2.1.1.3.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem19.p1.4.4.2.m2.1.1.3.1.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3">superscript</csymbol><ci id="Thmtheorem19.p1.4.4.2.m2.1.1.3.2.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3.2">𝐴</ci><ci id="Thmtheorem19.p1.4.4.2.m2.1.1.3.3.cmml" xref="Thmtheorem19.p1.4.4.2.m2.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.4.4.2.m2.1c">U=A^{c}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.4.4.2.m2.1d">italic_U = italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math>)</em> if the following condition holds. For every <math alttext="B\in\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem19.p1.5.5.m3.1"><semantics id="Thmtheorem19.p1.5.5.m3.1a"><mrow id="Thmtheorem19.p1.5.5.m3.1.1" xref="Thmtheorem19.p1.5.5.m3.1.1.cmml"><mi id="Thmtheorem19.p1.5.5.m3.1.1.2" xref="Thmtheorem19.p1.5.5.m3.1.1.2.cmml">B</mi><mo id="Thmtheorem19.p1.5.5.m3.1.1.1" xref="Thmtheorem19.p1.5.5.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem19.p1.5.5.m3.1.1.3" xref="Thmtheorem19.p1.5.5.m3.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.5.5.m3.1b"><apply id="Thmtheorem19.p1.5.5.m3.1.1.cmml" xref="Thmtheorem19.p1.5.5.m3.1.1"><in id="Thmtheorem19.p1.5.5.m3.1.1.1.cmml" xref="Thmtheorem19.p1.5.5.m3.1.1.1"></in><ci id="Thmtheorem19.p1.5.5.m3.1.1.2.cmml" xref="Thmtheorem19.p1.5.5.m3.1.1.2">𝐵</ci><ci id="Thmtheorem19.p1.5.5.m3.1.1.3.cmml" xref="Thmtheorem19.p1.5.5.m3.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.5.5.m3.1c">B\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.5.5.m3.1d">italic_B ∈ caligraphic_B</annotation></semantics></math>, if <math alttext="w\in B" class="ltx_Math" display="inline" id="Thmtheorem19.p1.6.6.m4.1"><semantics id="Thmtheorem19.p1.6.6.m4.1a"><mrow id="Thmtheorem19.p1.6.6.m4.1.1" xref="Thmtheorem19.p1.6.6.m4.1.1.cmml"><mi id="Thmtheorem19.p1.6.6.m4.1.1.2" xref="Thmtheorem19.p1.6.6.m4.1.1.2.cmml">w</mi><mo id="Thmtheorem19.p1.6.6.m4.1.1.1" xref="Thmtheorem19.p1.6.6.m4.1.1.1.cmml">∈</mo><mi id="Thmtheorem19.p1.6.6.m4.1.1.3" xref="Thmtheorem19.p1.6.6.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.6.6.m4.1b"><apply id="Thmtheorem19.p1.6.6.m4.1.1.cmml" xref="Thmtheorem19.p1.6.6.m4.1.1"><in id="Thmtheorem19.p1.6.6.m4.1.1.1.cmml" xref="Thmtheorem19.p1.6.6.m4.1.1.1"></in><ci id="Thmtheorem19.p1.6.6.m4.1.1.2.cmml" xref="Thmtheorem19.p1.6.6.m4.1.1.2">𝑤</ci><ci id="Thmtheorem19.p1.6.6.m4.1.1.3.cmml" xref="Thmtheorem19.p1.6.6.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.6.6.m4.1c">w\in B</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.6.6.m4.1d">italic_w ∈ italic_B</annotation></semantics></math> then <math alttext="B_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem19.p1.7.7.m5.1"><semantics id="Thmtheorem19.p1.7.7.m5.1a"><mrow id="Thmtheorem19.p1.7.7.m5.1.1" xref="Thmtheorem19.p1.7.7.m5.1.1.cmml"><msub id="Thmtheorem19.p1.7.7.m5.1.1.2" xref="Thmtheorem19.p1.7.7.m5.1.1.2.cmml"><mi id="Thmtheorem19.p1.7.7.m5.1.1.2.2" xref="Thmtheorem19.p1.7.7.m5.1.1.2.2.cmml">B</mi><mi id="Thmtheorem19.p1.7.7.m5.1.1.2.3" xref="Thmtheorem19.p1.7.7.m5.1.1.2.3.cmml">U</mi></msub><mo id="Thmtheorem19.p1.7.7.m5.1.1.1" xref="Thmtheorem19.p1.7.7.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem19.p1.7.7.m5.1.1.3" xref="Thmtheorem19.p1.7.7.m5.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem19.p1.7.7.m5.1b"><apply id="Thmtheorem19.p1.7.7.m5.1.1.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1"><in id="Thmtheorem19.p1.7.7.m5.1.1.1.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.1"></in><apply id="Thmtheorem19.p1.7.7.m5.1.1.2.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem19.p1.7.7.m5.1.1.2.1.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.2">subscript</csymbol><ci id="Thmtheorem19.p1.7.7.m5.1.1.2.2.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.2.2">𝐵</ci><ci id="Thmtheorem19.p1.7.7.m5.1.1.2.3.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.2.3">𝑈</ci></apply><ci id="Thmtheorem19.p1.7.7.m5.1.1.3.cmml" xref="Thmtheorem19.p1.7.7.m5.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem19.p1.7.7.m5.1c">B_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem19.p1.7.7.m5.1d">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.F2" title="In 3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">2</span></a> illustrates <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem19" title="Definition 19 (Semi-filter above 𝑤). ‣ 3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">19</span></a> in the particularly simple and attractive 2-dimensional framework of graph complexity considered in this work.</p> </div> <figure class="ltx_figure" id="S3.F2"> <table class="ltx_equation ltx_eqn_table" id="S3.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\setcounter{MaxMatrixCols}{22}\begin{matrix}\cdot&\leavevmode\hbox to11.67pt{% \vbox to11.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-5.55522pt% \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{% rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }% \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}% \pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}% }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{% rgb}{0.7,0.7,0.7}\pgfsys@color@gray@fill{0.7}\pgfsys@invoke{ }% \pgfsys@fill@opacity{0.8}\pgfsys@invoke{ }\pgfsys@invoke{ }{0.0}{1.0}{0.0pt}{0% .0pt}{ 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}\pgfsys@endscope}{{{{}}\pgfsys@beginscope% \pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-2.22221pt}% \pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}% {0}\pgfsys@invoke{ }\hbox{\pgfsys@fill@opacity{1}\pgfsys@stroke@opacity{1}{$% \bullet$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}% \pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}% \lxSVG@closescope\endpgfpicture}}&\cdot\par\end{matrix}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex16.m1.1d">start_ARG start_ROW start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL end_ROW start_ROW start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL italic_w end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL end_ROW start_ROW start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL end_ROW start_ROW start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL end_ROW start_ROW start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL end_ROW start_ROW start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL start_CELL ⋅ end_CELL start_CELL ∙ end_CELL start_CELL ∙ end_CELL start_CELL ⋅ end_CELL end_ROW end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>In this example, <math alttext="\Gamma=[6]\times[22]" class="ltx_Math" display="inline" id="S3.F2.16.m1.2"><semantics id="S3.F2.16.m1.2b"><mrow id="S3.F2.16.m1.2.3" xref="S3.F2.16.m1.2.3.cmml"><mi id="S3.F2.16.m1.2.3.2" mathvariant="normal" xref="S3.F2.16.m1.2.3.2.cmml">Γ</mi><mo id="S3.F2.16.m1.2.3.1" xref="S3.F2.16.m1.2.3.1.cmml">=</mo><mrow id="S3.F2.16.m1.2.3.3" xref="S3.F2.16.m1.2.3.3.cmml"><mrow id="S3.F2.16.m1.2.3.3.2.2" xref="S3.F2.16.m1.2.3.3.2.1.cmml"><mo id="S3.F2.16.m1.2.3.3.2.2.1" stretchy="false" xref="S3.F2.16.m1.2.3.3.2.1.1.cmml">[</mo><mn id="S3.F2.16.m1.1.1" xref="S3.F2.16.m1.1.1.cmml">6</mn><mo id="S3.F2.16.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S3.F2.16.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.F2.16.m1.2.3.3.1" rspace="0.222em" xref="S3.F2.16.m1.2.3.3.1.cmml">×</mo><mrow id="S3.F2.16.m1.2.3.3.3.2" xref="S3.F2.16.m1.2.3.3.3.1.cmml"><mo id="S3.F2.16.m1.2.3.3.3.2.1" stretchy="false" xref="S3.F2.16.m1.2.3.3.3.1.1.cmml">[</mo><mn id="S3.F2.16.m1.2.2" xref="S3.F2.16.m1.2.2.cmml">22</mn><mo id="S3.F2.16.m1.2.3.3.3.2.2" stretchy="false" xref="S3.F2.16.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.16.m1.2c"><apply id="S3.F2.16.m1.2.3.cmml" xref="S3.F2.16.m1.2.3"><eq id="S3.F2.16.m1.2.3.1.cmml" xref="S3.F2.16.m1.2.3.1"></eq><ci id="S3.F2.16.m1.2.3.2.cmml" xref="S3.F2.16.m1.2.3.2">Γ</ci><apply id="S3.F2.16.m1.2.3.3.cmml" xref="S3.F2.16.m1.2.3.3"><times id="S3.F2.16.m1.2.3.3.1.cmml" xref="S3.F2.16.m1.2.3.3.1"></times><apply id="S3.F2.16.m1.2.3.3.2.1.cmml" xref="S3.F2.16.m1.2.3.3.2.2"><csymbol cd="latexml" id="S3.F2.16.m1.2.3.3.2.1.1.cmml" xref="S3.F2.16.m1.2.3.3.2.2.1">delimited-[]</csymbol><cn id="S3.F2.16.m1.1.1.cmml" type="integer" xref="S3.F2.16.m1.1.1">6</cn></apply><apply id="S3.F2.16.m1.2.3.3.3.1.cmml" xref="S3.F2.16.m1.2.3.3.3.2"><csymbol cd="latexml" id="S3.F2.16.m1.2.3.3.3.1.1.cmml" xref="S3.F2.16.m1.2.3.3.3.2.1">delimited-[]</csymbol><cn id="S3.F2.16.m1.2.2.cmml" type="integer" xref="S3.F2.16.m1.2.2">22</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.16.m1.2d">\Gamma=[6]\times[22]</annotation><annotation encoding="application/x-llamapun" id="S3.F2.16.m1.2e">roman_Γ = [ 6 ] × [ 22 ]</annotation></semantics></math>, <math alttext="\mathcal{B}=\mathcal{G}_{6,22}" class="ltx_Math" display="inline" id="S3.F2.17.m2.2"><semantics id="S3.F2.17.m2.2b"><mrow id="S3.F2.17.m2.2.3" xref="S3.F2.17.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F2.17.m2.2.3.2" xref="S3.F2.17.m2.2.3.2.cmml">ℬ</mi><mo id="S3.F2.17.m2.2.3.1" xref="S3.F2.17.m2.2.3.1.cmml">=</mo><msub id="S3.F2.17.m2.2.3.3" xref="S3.F2.17.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F2.17.m2.2.3.3.2" xref="S3.F2.17.m2.2.3.3.2.cmml">𝒢</mi><mrow id="S3.F2.17.m2.2.2.2.4" xref="S3.F2.17.m2.2.2.2.3.cmml"><mn id="S3.F2.17.m2.1.1.1.1" xref="S3.F2.17.m2.1.1.1.1.cmml">6</mn><mo id="S3.F2.17.m2.2.2.2.4.1" xref="S3.F2.17.m2.2.2.2.3.cmml">,</mo><mn id="S3.F2.17.m2.2.2.2.2" xref="S3.F2.17.m2.2.2.2.2.cmml">22</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.17.m2.2c"><apply id="S3.F2.17.m2.2.3.cmml" xref="S3.F2.17.m2.2.3"><eq id="S3.F2.17.m2.2.3.1.cmml" xref="S3.F2.17.m2.2.3.1"></eq><ci id="S3.F2.17.m2.2.3.2.cmml" xref="S3.F2.17.m2.2.3.2">ℬ</ci><apply id="S3.F2.17.m2.2.3.3.cmml" xref="S3.F2.17.m2.2.3.3"><csymbol cd="ambiguous" id="S3.F2.17.m2.2.3.3.1.cmml" xref="S3.F2.17.m2.2.3.3">subscript</csymbol><ci id="S3.F2.17.m2.2.3.3.2.cmml" xref="S3.F2.17.m2.2.3.3.2">𝒢</ci><list id="S3.F2.17.m2.2.2.2.3.cmml" xref="S3.F2.17.m2.2.2.2.4"><cn id="S3.F2.17.m2.1.1.1.1.cmml" type="integer" xref="S3.F2.17.m2.1.1.1.1">6</cn><cn id="S3.F2.17.m2.2.2.2.2.cmml" type="integer" xref="S3.F2.17.m2.2.2.2.2">22</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.17.m2.2d">\mathcal{B}=\mathcal{G}_{6,22}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.17.m2.2e">caligraphic_B = caligraphic_G start_POSTSUBSCRIPT 6 , 22 end_POSTSUBSCRIPT</annotation></semantics></math> (as in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS2" title="2.2.2 Bipartite graph complexity ‣ 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.2.2</span></a>), and the <math alttext="\{\cdot,\bullet,w\}" class="ltx_Math" display="inline" id="S3.F2.18.m3.3"><semantics id="S3.F2.18.m3.3b"><mrow id="S3.F2.18.m3.3.4.2" xref="S3.F2.18.m3.3.4.1.cmml"><mo id="S3.F2.18.m3.3.4.2.1" stretchy="false" xref="S3.F2.18.m3.3.4.1.cmml">{</mo><mo id="S3.F2.18.m3.1.1" lspace="0em" rspace="0em" xref="S3.F2.18.m3.1.1.cmml">⋅</mo><mo id="S3.F2.18.m3.3.4.2.2" rspace="0em" xref="S3.F2.18.m3.3.4.1.cmml">,</mo><mo id="S3.F2.18.m3.2.2" lspace="0em" rspace="0em" xref="S3.F2.18.m3.2.2.cmml">∙</mo><mo id="S3.F2.18.m3.3.4.2.3" xref="S3.F2.18.m3.3.4.1.cmml">,</mo><mi id="S3.F2.18.m3.3.3" xref="S3.F2.18.m3.3.3.cmml">w</mi><mo id="S3.F2.18.m3.3.4.2.4" stretchy="false" xref="S3.F2.18.m3.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.18.m3.3c"><set id="S3.F2.18.m3.3.4.1.cmml" xref="S3.F2.18.m3.3.4.2"><ci id="S3.F2.18.m3.1.1.cmml" xref="S3.F2.18.m3.1.1">⋅</ci><ci id="S3.F2.18.m3.2.2.cmml" xref="S3.F2.18.m3.2.2">∙</ci><ci id="S3.F2.18.m3.3.3.cmml" xref="S3.F2.18.m3.3.3">𝑤</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.18.m3.3d">\{\cdot,\bullet,w\}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.18.m3.3e">{ ⋅ , ∙ , italic_w }</annotation></semantics></math>-valued matrix above encodes <math alttext="U=G^{c}" class="ltx_Math" display="inline" id="S3.F2.19.m4.1"><semantics id="S3.F2.19.m4.1b"><mrow id="S3.F2.19.m4.1.1" xref="S3.F2.19.m4.1.1.cmml"><mi id="S3.F2.19.m4.1.1.2" xref="S3.F2.19.m4.1.1.2.cmml">U</mi><mo id="S3.F2.19.m4.1.1.1" xref="S3.F2.19.m4.1.1.1.cmml">=</mo><msup id="S3.F2.19.m4.1.1.3" xref="S3.F2.19.m4.1.1.3.cmml"><mi id="S3.F2.19.m4.1.1.3.2" xref="S3.F2.19.m4.1.1.3.2.cmml">G</mi><mi id="S3.F2.19.m4.1.1.3.3" xref="S3.F2.19.m4.1.1.3.3.cmml">c</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.19.m4.1c"><apply id="S3.F2.19.m4.1.1.cmml" xref="S3.F2.19.m4.1.1"><eq id="S3.F2.19.m4.1.1.1.cmml" xref="S3.F2.19.m4.1.1.1"></eq><ci id="S3.F2.19.m4.1.1.2.cmml" xref="S3.F2.19.m4.1.1.2">𝑈</ci><apply id="S3.F2.19.m4.1.1.3.cmml" xref="S3.F2.19.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F2.19.m4.1.1.3.1.cmml" xref="S3.F2.19.m4.1.1.3">superscript</csymbol><ci id="S3.F2.19.m4.1.1.3.2.cmml" xref="S3.F2.19.m4.1.1.3.2">𝐺</ci><ci id="S3.F2.19.m4.1.1.3.3.cmml" xref="S3.F2.19.m4.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.19.m4.1d">U=G^{c}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.19.m4.1e">italic_U = italic_G start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math> (rectangles with <math alttext="\bullet" class="ltx_Math" display="inline" id="S3.F2.20.m5.1"><semantics id="S3.F2.20.m5.1b"><mo id="S3.F2.20.m5.1.1" xref="S3.F2.20.m5.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S3.F2.20.m5.1c"><ci id="S3.F2.20.m5.1.1.cmml" xref="S3.F2.20.m5.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.20.m5.1d">\bullet</annotation><annotation encoding="application/x-llamapun" id="S3.F2.20.m5.1e">∙</annotation></semantics></math>), where <math alttext="G\subseteq\Gamma" class="ltx_Math" display="inline" id="S3.F2.21.m6.1"><semantics id="S3.F2.21.m6.1b"><mrow id="S3.F2.21.m6.1.1" xref="S3.F2.21.m6.1.1.cmml"><mi id="S3.F2.21.m6.1.1.2" xref="S3.F2.21.m6.1.1.2.cmml">G</mi><mo id="S3.F2.21.m6.1.1.1" xref="S3.F2.21.m6.1.1.1.cmml">⊆</mo><mi id="S3.F2.21.m6.1.1.3" mathvariant="normal" xref="S3.F2.21.m6.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.21.m6.1c"><apply id="S3.F2.21.m6.1.1.cmml" xref="S3.F2.21.m6.1.1"><subset id="S3.F2.21.m6.1.1.1.cmml" xref="S3.F2.21.m6.1.1.1"></subset><ci id="S3.F2.21.m6.1.1.2.cmml" xref="S3.F2.21.m6.1.1.2">𝐺</ci><ci id="S3.F2.21.m6.1.1.3.cmml" xref="S3.F2.21.m6.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.21.m6.1d">G\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.F2.21.m6.1e">italic_G ⊆ roman_Γ</annotation></semantics></math> (locations with <math alttext="\cdot" class="ltx_Math" display="inline" id="S3.F2.22.m7.1"><semantics id="S3.F2.22.m7.1b"><mo id="S3.F2.22.m7.1.1" xref="S3.F2.22.m7.1.1.cmml">⋅</mo><annotation-xml encoding="MathML-Content" id="S3.F2.22.m7.1c"><ci id="S3.F2.22.m7.1.1.cmml" xref="S3.F2.22.m7.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.22.m7.1d">\cdot</annotation><annotation encoding="application/x-llamapun" id="S3.F2.22.m7.1e">⋅</annotation></semantics></math> and <math alttext="w" class="ltx_Math" display="inline" id="S3.F2.23.m8.1"><semantics id="S3.F2.23.m8.1b"><mi id="S3.F2.23.m8.1.1" xref="S3.F2.23.m8.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.F2.23.m8.1c"><ci id="S3.F2.23.m8.1.1.cmml" xref="S3.F2.23.m8.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.23.m8.1d">w</annotation><annotation encoding="application/x-llamapun" id="S3.F2.23.m8.1e">italic_w</annotation></semantics></math>) can be interpreted as a bipartite graph. If a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.F2.24.m9.1"><semantics id="S3.F2.24.m9.1b"><mi class="ltx_font_mathcaligraphic" id="S3.F2.24.m9.1.1" xref="S3.F2.24.m9.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.F2.24.m9.1c"><ci id="S3.F2.24.m9.1.1.cmml" xref="S3.F2.24.m9.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.24.m9.1d">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.24.m9.1e">caligraphic_F</annotation></semantics></math> over <math alttext="U" class="ltx_Math" display="inline" id="S3.F2.25.m10.1"><semantics id="S3.F2.25.m10.1b"><mi id="S3.F2.25.m10.1.1" xref="S3.F2.25.m10.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.F2.25.m10.1c"><ci id="S3.F2.25.m10.1.1.cmml" xref="S3.F2.25.m10.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.25.m10.1d">U</annotation><annotation encoding="application/x-llamapun" id="S3.F2.25.m10.1e">italic_U</annotation></semantics></math> is above <math alttext="w\in G" class="ltx_Math" display="inline" id="S3.F2.26.m11.1"><semantics id="S3.F2.26.m11.1b"><mrow id="S3.F2.26.m11.1.1" xref="S3.F2.26.m11.1.1.cmml"><mi id="S3.F2.26.m11.1.1.2" xref="S3.F2.26.m11.1.1.2.cmml">w</mi><mo id="S3.F2.26.m11.1.1.1" xref="S3.F2.26.m11.1.1.1.cmml">∈</mo><mi id="S3.F2.26.m11.1.1.3" xref="S3.F2.26.m11.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.26.m11.1c"><apply id="S3.F2.26.m11.1.1.cmml" xref="S3.F2.26.m11.1.1"><in id="S3.F2.26.m11.1.1.1.cmml" xref="S3.F2.26.m11.1.1.1"></in><ci id="S3.F2.26.m11.1.1.2.cmml" xref="S3.F2.26.m11.1.1.2">𝑤</ci><ci id="S3.F2.26.m11.1.1.3.cmml" xref="S3.F2.26.m11.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.26.m11.1d">w\in G</annotation><annotation encoding="application/x-llamapun" id="S3.F2.26.m11.1e">italic_w ∈ italic_G</annotation></semantics></math> (corresponding to coordinates <math alttext="(2,15)" class="ltx_Math" display="inline" id="S3.F2.27.m12.2"><semantics id="S3.F2.27.m12.2b"><mrow id="S3.F2.27.m12.2.3.2" xref="S3.F2.27.m12.2.3.1.cmml"><mo id="S3.F2.27.m12.2.3.2.1" stretchy="false" xref="S3.F2.27.m12.2.3.1.cmml">(</mo><mn id="S3.F2.27.m12.1.1" xref="S3.F2.27.m12.1.1.cmml">2</mn><mo id="S3.F2.27.m12.2.3.2.2" xref="S3.F2.27.m12.2.3.1.cmml">,</mo><mn id="S3.F2.27.m12.2.2" xref="S3.F2.27.m12.2.2.cmml">15</mn><mo id="S3.F2.27.m12.2.3.2.3" stretchy="false" xref="S3.F2.27.m12.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.27.m12.2c"><interval closure="open" id="S3.F2.27.m12.2.3.1.cmml" xref="S3.F2.27.m12.2.3.2"><cn id="S3.F2.27.m12.1.1.cmml" type="integer" xref="S3.F2.27.m12.1.1">2</cn><cn id="S3.F2.27.m12.2.2.cmml" type="integer" xref="S3.F2.27.m12.2.2">15</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.27.m12.2d">(2,15)</annotation><annotation encoding="application/x-llamapun" id="S3.F2.27.m12.2e">( 2 , 15 )</annotation></semantics></math>), then it must contain the distinguished subsets of <math alttext="U" class="ltx_Math" display="inline" id="S3.F2.28.m13.1"><semantics id="S3.F2.28.m13.1b"><mi id="S3.F2.28.m13.1.1" xref="S3.F2.28.m13.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.F2.28.m13.1c"><ci id="S3.F2.28.m13.1.1.cmml" xref="S3.F2.28.m13.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.28.m13.1d">U</annotation><annotation encoding="application/x-llamapun" id="S3.F2.28.m13.1e">italic_U</annotation></semantics></math> represented in blue <math alttext="(R_{2}\cap U)" class="ltx_Math" display="inline" id="S3.F2.29.m14.1"><semantics id="S3.F2.29.m14.1b"><mrow id="S3.F2.29.m14.1.1.1" xref="S3.F2.29.m14.1.1.1.1.cmml"><mo id="S3.F2.29.m14.1.1.1.2" stretchy="false" xref="S3.F2.29.m14.1.1.1.1.cmml">(</mo><mrow id="S3.F2.29.m14.1.1.1.1" xref="S3.F2.29.m14.1.1.1.1.cmml"><msub id="S3.F2.29.m14.1.1.1.1.2" xref="S3.F2.29.m14.1.1.1.1.2.cmml"><mi id="S3.F2.29.m14.1.1.1.1.2.2" xref="S3.F2.29.m14.1.1.1.1.2.2.cmml">R</mi><mn id="S3.F2.29.m14.1.1.1.1.2.3" xref="S3.F2.29.m14.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S3.F2.29.m14.1.1.1.1.1" xref="S3.F2.29.m14.1.1.1.1.1.cmml">∩</mo><mi id="S3.F2.29.m14.1.1.1.1.3" xref="S3.F2.29.m14.1.1.1.1.3.cmml">U</mi></mrow><mo id="S3.F2.29.m14.1.1.1.3" stretchy="false" xref="S3.F2.29.m14.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.29.m14.1c"><apply id="S3.F2.29.m14.1.1.1.1.cmml" xref="S3.F2.29.m14.1.1.1"><intersect id="S3.F2.29.m14.1.1.1.1.1.cmml" xref="S3.F2.29.m14.1.1.1.1.1"></intersect><apply id="S3.F2.29.m14.1.1.1.1.2.cmml" xref="S3.F2.29.m14.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.F2.29.m14.1.1.1.1.2.1.cmml" xref="S3.F2.29.m14.1.1.1.1.2">subscript</csymbol><ci id="S3.F2.29.m14.1.1.1.1.2.2.cmml" xref="S3.F2.29.m14.1.1.1.1.2.2">𝑅</ci><cn id="S3.F2.29.m14.1.1.1.1.2.3.cmml" type="integer" xref="S3.F2.29.m14.1.1.1.1.2.3">2</cn></apply><ci id="S3.F2.29.m14.1.1.1.1.3.cmml" xref="S3.F2.29.m14.1.1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.29.m14.1d">(R_{2}\cap U)</annotation><annotation encoding="application/x-llamapun" id="S3.F2.29.m14.1e">( italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∩ italic_U )</annotation></semantics></math> and in orange <math alttext="(C_{15}\cap U)" class="ltx_Math" display="inline" id="S3.F2.30.m15.1"><semantics id="S3.F2.30.m15.1b"><mrow id="S3.F2.30.m15.1.1.1" xref="S3.F2.30.m15.1.1.1.1.cmml"><mo id="S3.F2.30.m15.1.1.1.2" stretchy="false" xref="S3.F2.30.m15.1.1.1.1.cmml">(</mo><mrow id="S3.F2.30.m15.1.1.1.1" xref="S3.F2.30.m15.1.1.1.1.cmml"><msub id="S3.F2.30.m15.1.1.1.1.2" xref="S3.F2.30.m15.1.1.1.1.2.cmml"><mi id="S3.F2.30.m15.1.1.1.1.2.2" xref="S3.F2.30.m15.1.1.1.1.2.2.cmml">C</mi><mn id="S3.F2.30.m15.1.1.1.1.2.3" xref="S3.F2.30.m15.1.1.1.1.2.3.cmml">15</mn></msub><mo id="S3.F2.30.m15.1.1.1.1.1" xref="S3.F2.30.m15.1.1.1.1.1.cmml">∩</mo><mi id="S3.F2.30.m15.1.1.1.1.3" xref="S3.F2.30.m15.1.1.1.1.3.cmml">U</mi></mrow><mo id="S3.F2.30.m15.1.1.1.3" stretchy="false" xref="S3.F2.30.m15.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.30.m15.1c"><apply id="S3.F2.30.m15.1.1.1.1.cmml" xref="S3.F2.30.m15.1.1.1"><intersect id="S3.F2.30.m15.1.1.1.1.1.cmml" xref="S3.F2.30.m15.1.1.1.1.1"></intersect><apply id="S3.F2.30.m15.1.1.1.1.2.cmml" xref="S3.F2.30.m15.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.F2.30.m15.1.1.1.1.2.1.cmml" xref="S3.F2.30.m15.1.1.1.1.2">subscript</csymbol><ci id="S3.F2.30.m15.1.1.1.1.2.2.cmml" xref="S3.F2.30.m15.1.1.1.1.2.2">𝐶</ci><cn id="S3.F2.30.m15.1.1.1.1.2.3.cmml" type="integer" xref="S3.F2.30.m15.1.1.1.1.2.3">15</cn></apply><ci id="S3.F2.30.m15.1.1.1.1.3.cmml" xref="S3.F2.30.m15.1.1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.30.m15.1d">(C_{15}\cap U)</annotation><annotation encoding="application/x-llamapun" id="S3.F2.30.m15.1e">( italic_C start_POSTSUBSCRIPT 15 end_POSTSUBSCRIPT ∩ italic_U )</annotation></semantics></math>, respectively.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.3">Intuitively, semi-filters will be used to produce counter-examples to the correctness of a candidate construction of a set <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mi id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><ci id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">caligraphic_B</annotation></semantics></math> that is more efficient than <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mrow id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><msub id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml"><mi id="S3.SS1.p3.3.m3.1.1.3.2" xref="S3.SS1.p3.3.m3.1.1.3.2.cmml">D</mi><mo id="S3.SS1.p3.3.m3.1.1.3.3" xref="S3.SS1.p3.3.m3.1.1.3.3.cmml">∩</mo></msub><mo id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.p3.3.m3.1.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS1.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS1.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.p3.3.m3.1.1.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.1.1.1.2" xref="S3.SS1.p3.3.m3.1.1.1.1.1.2.cmml">A</mi><mo id="S3.SS1.p3.3.m3.1.1.1.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.3.m3.1.1.1.1.1.3" xref="S3.SS1.p3.3.m3.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS1.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S3.SS1.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><times id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2"></times><apply id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.3.1.cmml" xref="S3.SS1.p3.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.3.2.cmml" xref="S3.SS1.p3.3.m3.1.1.3.2">𝐷</ci><intersect id="S3.SS1.p3.3.m3.1.1.3.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3.3"></intersect></apply><apply id="S3.SS1.p3.3.m3.1.1.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1.1"><csymbol cd="latexml" id="S3.SS1.p3.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1.1.1.1">conditional</csymbol><ci id="S3.SS1.p3.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.1.1.1.2">𝐴</ci><ci id="S3.SS1.p3.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math>. This will become clear in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS2" title="3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem20"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem20.1.1.1">Definition 20</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem20.2.2"> </span>(Preservation of pairs of subsets)<span class="ltx_text ltx_font_bold" id="Thmtheorem20.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem20.p1"> <p class="ltx_p" id="Thmtheorem20.p1.10"><span class="ltx_text ltx_font_italic" id="Thmtheorem20.p1.10.10">Let <math alttext="\Lambda=\{(E_{1},H_{1}),\ldots,(E_{\ell},H_{\ell})\}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.1.1.m1.3"><semantics id="Thmtheorem20.p1.1.1.m1.3a"><mrow id="Thmtheorem20.p1.1.1.m1.3.3" xref="Thmtheorem20.p1.1.1.m1.3.3.cmml"><mi id="Thmtheorem20.p1.1.1.m1.3.3.4" mathvariant="normal" xref="Thmtheorem20.p1.1.1.m1.3.3.4.cmml">Λ</mi><mo id="Thmtheorem20.p1.1.1.m1.3.3.3" xref="Thmtheorem20.p1.1.1.m1.3.3.3.cmml">=</mo><mrow id="Thmtheorem20.p1.1.1.m1.3.3.2.2" xref="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml"><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.3" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml">{</mo><mrow id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.3.cmml"><mo id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.3" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.3.cmml">(</mo><msub id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.cmml"><mi id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.2" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.2.cmml">E</mi><mn id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.3" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.4" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.3.cmml">,</mo><msub id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.cmml"><mi id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.2" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.2.cmml">H</mi><mn id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.3" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.3.cmml">1</mn></msub><mo id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.5" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.3.cmml">)</mo></mrow><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.4" xref="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml">,</mo><mi id="Thmtheorem20.p1.1.1.m1.1.1" mathvariant="normal" xref="Thmtheorem20.p1.1.1.m1.1.1.cmml">…</mi><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.5" xref="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml">,</mo><mrow id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.3.cmml"><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.3" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.3.cmml">(</mo><msub id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.cmml"><mi id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.2" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.2.cmml">E</mi><mi id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.3" mathvariant="normal" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.3.cmml">ℓ</mi></msub><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.4" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.3.cmml">,</mo><msub id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.cmml"><mi id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.2" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.2.cmml">H</mi><mi id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.3" mathvariant="normal" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.3.cmml">ℓ</mi></msub><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.5" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.3.cmml">)</mo></mrow><mo id="Thmtheorem20.p1.1.1.m1.3.3.2.2.6" stretchy="false" xref="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.1.1.m1.3b"><apply id="Thmtheorem20.p1.1.1.m1.3.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3"><eq id="Thmtheorem20.p1.1.1.m1.3.3.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.3"></eq><ci id="Thmtheorem20.p1.1.1.m1.3.3.4.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.4">Λ</ci><set id="Thmtheorem20.p1.1.1.m1.3.3.2.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2"><interval closure="open" id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.3.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2"><apply id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.2.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.2">𝐸</ci><cn id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.1.1.3">1</cn></apply><apply id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.1.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2">subscript</csymbol><ci id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.2.cmml" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.2">𝐻</ci><cn id="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.3.cmml" type="integer" xref="Thmtheorem20.p1.1.1.m1.2.2.1.1.1.2.2.3">1</cn></apply></interval><ci id="Thmtheorem20.p1.1.1.m1.1.1.cmml" xref="Thmtheorem20.p1.1.1.m1.1.1">…</ci><interval closure="open" id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2"><apply id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.1.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1">subscript</csymbol><ci id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.2.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.2">𝐸</ci><ci id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.1.1.3">ℓ</ci></apply><apply id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.1.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2">subscript</csymbol><ci id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.2.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.2">𝐻</ci><ci id="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.3.cmml" xref="Thmtheorem20.p1.1.1.m1.3.3.2.2.2.2.2.3">ℓ</ci></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.1.1.m1.3c">\Lambda=\{(E_{1},H_{1}),\ldots,(E_{\ell},H_{\ell})\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.1.1.m1.3d">roman_Λ = { ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , ( italic_E start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) }</annotation></semantics></math> be a family of pairs of subsets of <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem20.p1.2.2.m2.1"><semantics id="Thmtheorem20.p1.2.2.m2.1a"><mi id="Thmtheorem20.p1.2.2.m2.1.1" xref="Thmtheorem20.p1.2.2.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.2.2.m2.1b"><ci id="Thmtheorem20.p1.2.2.m2.1.1.cmml" xref="Thmtheorem20.p1.2.2.m2.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.2.2.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.2.2.m2.1d">italic_U</annotation></semantics></math>. We say that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.3.3.m3.1"><semantics id="Thmtheorem20.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem20.p1.3.3.m3.1.1" xref="Thmtheorem20.p1.3.3.m3.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.3.3.m3.1b"><ci id="Thmtheorem20.p1.3.3.m3.1.1.cmml" xref="Thmtheorem20.p1.3.3.m3.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.3.3.m3.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.3.3.m3.1d">caligraphic_F</annotation></semantics></math> <em class="ltx_emph ltx_font_upright" id="Thmtheorem20.p1.10.10.1">preserves</em> a pair <math alttext="(E_{i},H_{i})" class="ltx_Math" display="inline" id="Thmtheorem20.p1.4.4.m4.2"><semantics id="Thmtheorem20.p1.4.4.m4.2a"><mrow id="Thmtheorem20.p1.4.4.m4.2.2.2" xref="Thmtheorem20.p1.4.4.m4.2.2.3.cmml"><mo id="Thmtheorem20.p1.4.4.m4.2.2.2.3" stretchy="false" xref="Thmtheorem20.p1.4.4.m4.2.2.3.cmml">(</mo><msub id="Thmtheorem20.p1.4.4.m4.1.1.1.1" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1.cmml"><mi id="Thmtheorem20.p1.4.4.m4.1.1.1.1.2" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1.2.cmml">E</mi><mi id="Thmtheorem20.p1.4.4.m4.1.1.1.1.3" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmtheorem20.p1.4.4.m4.2.2.2.4" xref="Thmtheorem20.p1.4.4.m4.2.2.3.cmml">,</mo><msub id="Thmtheorem20.p1.4.4.m4.2.2.2.2" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2.cmml"><mi id="Thmtheorem20.p1.4.4.m4.2.2.2.2.2" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2.2.cmml">H</mi><mi id="Thmtheorem20.p1.4.4.m4.2.2.2.2.3" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2.3.cmml">i</mi></msub><mo id="Thmtheorem20.p1.4.4.m4.2.2.2.5" stretchy="false" xref="Thmtheorem20.p1.4.4.m4.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.4.4.m4.2b"><interval closure="open" id="Thmtheorem20.p1.4.4.m4.2.2.3.cmml" xref="Thmtheorem20.p1.4.4.m4.2.2.2"><apply id="Thmtheorem20.p1.4.4.m4.1.1.1.1.cmml" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem20.p1.4.4.m4.1.1.1.1.1.cmml" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1">subscript</csymbol><ci id="Thmtheorem20.p1.4.4.m4.1.1.1.1.2.cmml" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1.2">𝐸</ci><ci id="Thmtheorem20.p1.4.4.m4.1.1.1.1.3.cmml" xref="Thmtheorem20.p1.4.4.m4.1.1.1.1.3">𝑖</ci></apply><apply id="Thmtheorem20.p1.4.4.m4.2.2.2.2.cmml" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.4.4.m4.2.2.2.2.1.cmml" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2">subscript</csymbol><ci id="Thmtheorem20.p1.4.4.m4.2.2.2.2.2.cmml" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2.2">𝐻</ci><ci id="Thmtheorem20.p1.4.4.m4.2.2.2.2.3.cmml" xref="Thmtheorem20.p1.4.4.m4.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.4.4.m4.2c">(E_{i},H_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.4.4.m4.2d">( italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> if <math alttext="E_{i}\in\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.5.5.m5.1"><semantics id="Thmtheorem20.p1.5.5.m5.1a"><mrow id="Thmtheorem20.p1.5.5.m5.1.1" xref="Thmtheorem20.p1.5.5.m5.1.1.cmml"><msub id="Thmtheorem20.p1.5.5.m5.1.1.2" xref="Thmtheorem20.p1.5.5.m5.1.1.2.cmml"><mi id="Thmtheorem20.p1.5.5.m5.1.1.2.2" xref="Thmtheorem20.p1.5.5.m5.1.1.2.2.cmml">E</mi><mi id="Thmtheorem20.p1.5.5.m5.1.1.2.3" xref="Thmtheorem20.p1.5.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="Thmtheorem20.p1.5.5.m5.1.1.1" xref="Thmtheorem20.p1.5.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem20.p1.5.5.m5.1.1.3" xref="Thmtheorem20.p1.5.5.m5.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.5.5.m5.1b"><apply id="Thmtheorem20.p1.5.5.m5.1.1.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1"><in id="Thmtheorem20.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.1"></in><apply id="Thmtheorem20.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.5.5.m5.1.1.2.1.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="Thmtheorem20.p1.5.5.m5.1.1.2.2.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.2.2">𝐸</ci><ci id="Thmtheorem20.p1.5.5.m5.1.1.2.3.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.2.3">𝑖</ci></apply><ci id="Thmtheorem20.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem20.p1.5.5.m5.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.5.5.m5.1c">E_{i}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.5.5.m5.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math> and <math alttext="H_{i}\in\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.6.6.m6.1"><semantics id="Thmtheorem20.p1.6.6.m6.1a"><mrow id="Thmtheorem20.p1.6.6.m6.1.1" xref="Thmtheorem20.p1.6.6.m6.1.1.cmml"><msub id="Thmtheorem20.p1.6.6.m6.1.1.2" xref="Thmtheorem20.p1.6.6.m6.1.1.2.cmml"><mi id="Thmtheorem20.p1.6.6.m6.1.1.2.2" xref="Thmtheorem20.p1.6.6.m6.1.1.2.2.cmml">H</mi><mi id="Thmtheorem20.p1.6.6.m6.1.1.2.3" xref="Thmtheorem20.p1.6.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="Thmtheorem20.p1.6.6.m6.1.1.1" xref="Thmtheorem20.p1.6.6.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem20.p1.6.6.m6.1.1.3" xref="Thmtheorem20.p1.6.6.m6.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.6.6.m6.1b"><apply id="Thmtheorem20.p1.6.6.m6.1.1.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1"><in id="Thmtheorem20.p1.6.6.m6.1.1.1.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.1"></in><apply id="Thmtheorem20.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.6.6.m6.1.1.2.1.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="Thmtheorem20.p1.6.6.m6.1.1.2.2.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.2.2">𝐻</ci><ci id="Thmtheorem20.p1.6.6.m6.1.1.2.3.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.2.3">𝑖</ci></apply><ci id="Thmtheorem20.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem20.p1.6.6.m6.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.6.6.m6.1c">H_{i}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.6.6.m6.1d">italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math> imply <math alttext="E_{i}\cap H_{i}\in\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.7.7.m7.1"><semantics id="Thmtheorem20.p1.7.7.m7.1a"><mrow id="Thmtheorem20.p1.7.7.m7.1.1" xref="Thmtheorem20.p1.7.7.m7.1.1.cmml"><mrow id="Thmtheorem20.p1.7.7.m7.1.1.2" xref="Thmtheorem20.p1.7.7.m7.1.1.2.cmml"><msub id="Thmtheorem20.p1.7.7.m7.1.1.2.2" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2.cmml"><mi id="Thmtheorem20.p1.7.7.m7.1.1.2.2.2" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2.2.cmml">E</mi><mi id="Thmtheorem20.p1.7.7.m7.1.1.2.2.3" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2.3.cmml">i</mi></msub><mo id="Thmtheorem20.p1.7.7.m7.1.1.2.1" xref="Thmtheorem20.p1.7.7.m7.1.1.2.1.cmml">∩</mo><msub id="Thmtheorem20.p1.7.7.m7.1.1.2.3" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3.cmml"><mi id="Thmtheorem20.p1.7.7.m7.1.1.2.3.2" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3.2.cmml">H</mi><mi id="Thmtheorem20.p1.7.7.m7.1.1.2.3.3" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="Thmtheorem20.p1.7.7.m7.1.1.1" xref="Thmtheorem20.p1.7.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem20.p1.7.7.m7.1.1.3" xref="Thmtheorem20.p1.7.7.m7.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.7.7.m7.1b"><apply id="Thmtheorem20.p1.7.7.m7.1.1.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1"><in id="Thmtheorem20.p1.7.7.m7.1.1.1.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.1"></in><apply id="Thmtheorem20.p1.7.7.m7.1.1.2.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2"><intersect id="Thmtheorem20.p1.7.7.m7.1.1.2.1.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.1"></intersect><apply id="Thmtheorem20.p1.7.7.m7.1.1.2.2.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="Thmtheorem20.p1.7.7.m7.1.1.2.2.1.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2">subscript</csymbol><ci id="Thmtheorem20.p1.7.7.m7.1.1.2.2.2.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2.2">𝐸</ci><ci id="Thmtheorem20.p1.7.7.m7.1.1.2.2.3.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.2.3">𝑖</ci></apply><apply id="Thmtheorem20.p1.7.7.m7.1.1.2.3.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="Thmtheorem20.p1.7.7.m7.1.1.2.3.1.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3">subscript</csymbol><ci id="Thmtheorem20.p1.7.7.m7.1.1.2.3.2.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3.2">𝐻</ci><ci id="Thmtheorem20.p1.7.7.m7.1.1.2.3.3.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.2.3.3">𝑖</ci></apply></apply><ci id="Thmtheorem20.p1.7.7.m7.1.1.3.cmml" xref="Thmtheorem20.p1.7.7.m7.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.7.7.m7.1c">E_{i}\cap H_{i}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.7.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>. We say that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem20.p1.8.8.m8.1"><semantics id="Thmtheorem20.p1.8.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem20.p1.8.8.m8.1.1" xref="Thmtheorem20.p1.8.8.m8.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.8.8.m8.1b"><ci id="Thmtheorem20.p1.8.8.m8.1.1.cmml" xref="Thmtheorem20.p1.8.8.m8.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.8.8.m8.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.8.8.m8.1d">caligraphic_F</annotation></semantics></math> <em class="ltx_emph ltx_font_upright" id="Thmtheorem20.p1.10.10.2">preserves</em> <math alttext="\Lambda" class="ltx_Math" display="inline" id="Thmtheorem20.p1.9.9.m9.1"><semantics id="Thmtheorem20.p1.9.9.m9.1a"><mi id="Thmtheorem20.p1.9.9.m9.1.1" mathvariant="normal" xref="Thmtheorem20.p1.9.9.m9.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.9.9.m9.1b"><ci id="Thmtheorem20.p1.9.9.m9.1.1.cmml" xref="Thmtheorem20.p1.9.9.m9.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.9.9.m9.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.9.9.m9.1d">roman_Λ</annotation></semantics></math> if it preserves every pair in <math alttext="\Lambda" class="ltx_Math" display="inline" id="Thmtheorem20.p1.10.10.m10.1"><semantics id="Thmtheorem20.p1.10.10.m10.1a"><mi id="Thmtheorem20.p1.10.10.m10.1.1" mathvariant="normal" xref="Thmtheorem20.p1.10.10.m10.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem20.p1.10.10.m10.1b"><ci id="Thmtheorem20.p1.10.10.m10.1.1.cmml" xref="Thmtheorem20.p1.10.10.m10.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem20.p1.10.10.m10.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem20.p1.10.10.m10.1d">roman_Λ</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.2">We now introduce a measure of the <em class="ltx_emph ltx_font_italic" id="S3.SS1.p4.2.1">cover complexity</em> of <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml"><mi id="S3.SS1.p4.1.m1.1.1.2" xref="S3.SS1.p4.1.m1.1.1.2.cmml">A</mi><mo id="S3.SS1.p4.1.m1.1.1.1" xref="S3.SS1.p4.1.m1.1.1.1.cmml">⊆</mo><mi id="S3.SS1.p4.1.m1.1.1.3" mathvariant="normal" xref="S3.SS1.p4.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1"><subset id="S3.SS1.p4.1.m1.1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1.1"></subset><ci id="S3.SS1.p4.1.m1.1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2">𝐴</ci><ci id="S3.SS1.p4.1.m1.1.1.3.cmml" xref="S3.SS1.p4.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> with respect to a family <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mrow id="S3.SS1.p4.2.m2.1.2" xref="S3.SS1.p4.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.2.m2.1.2.2" xref="S3.SS1.p4.2.m2.1.2.2.cmml">ℬ</mi><mo id="S3.SS1.p4.2.m2.1.2.1" xref="S3.SS1.p4.2.m2.1.2.1.cmml">⊆</mo><mrow id="S3.SS1.p4.2.m2.1.2.3" xref="S3.SS1.p4.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.2.m2.1.2.3.2" xref="S3.SS1.p4.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="S3.SS1.p4.2.m2.1.2.3.1" xref="S3.SS1.p4.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.2.m2.1.2.3.3.2" xref="S3.SS1.p4.2.m2.1.2.3.cmml"><mo id="S3.SS1.p4.2.m2.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.p4.2.m2.1.2.3.cmml">(</mo><mi id="S3.SS1.p4.2.m2.1.1" mathvariant="normal" xref="S3.SS1.p4.2.m2.1.1.cmml">Γ</mi><mo id="S3.SS1.p4.2.m2.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.p4.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><apply id="S3.SS1.p4.2.m2.1.2.cmml" xref="S3.SS1.p4.2.m2.1.2"><subset id="S3.SS1.p4.2.m2.1.2.1.cmml" xref="S3.SS1.p4.2.m2.1.2.1"></subset><ci id="S3.SS1.p4.2.m2.1.2.2.cmml" xref="S3.SS1.p4.2.m2.1.2.2">ℬ</ci><apply id="S3.SS1.p4.2.m2.1.2.3.cmml" xref="S3.SS1.p4.2.m2.1.2.3"><times id="S3.SS1.p4.2.m2.1.2.3.1.cmml" xref="S3.SS1.p4.2.m2.1.2.3.1"></times><ci id="S3.SS1.p4.2.m2.1.2.3.2.cmml" xref="S3.SS1.p4.2.m2.1.2.3.2">𝒫</ci><ci id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem21"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem21.1.1.1">Definition 21</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem21.2.2"> </span>(Cover complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem21.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem21.p1"> <p class="ltx_p" id="Thmtheorem21.p1.9"><span class="ltx_text ltx_font_italic" id="Thmtheorem21.p1.9.9">We let <math alttext="\rho(A,\mathcal{B})\in\mathbb{N}\cup\{\infty\}" class="ltx_Math" display="inline" id="Thmtheorem21.p1.1.1.m1.3"><semantics id="Thmtheorem21.p1.1.1.m1.3a"><mrow id="Thmtheorem21.p1.1.1.m1.3.4" xref="Thmtheorem21.p1.1.1.m1.3.4.cmml"><mrow id="Thmtheorem21.p1.1.1.m1.3.4.2" xref="Thmtheorem21.p1.1.1.m1.3.4.2.cmml"><mi id="Thmtheorem21.p1.1.1.m1.3.4.2.2" xref="Thmtheorem21.p1.1.1.m1.3.4.2.2.cmml">ρ</mi><mo id="Thmtheorem21.p1.1.1.m1.3.4.2.1" xref="Thmtheorem21.p1.1.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="Thmtheorem21.p1.1.1.m1.3.4.2.3.2" xref="Thmtheorem21.p1.1.1.m1.3.4.2.3.1.cmml"><mo id="Thmtheorem21.p1.1.1.m1.3.4.2.3.2.1" stretchy="false" xref="Thmtheorem21.p1.1.1.m1.3.4.2.3.1.cmml">(</mo><mi id="Thmtheorem21.p1.1.1.m1.1.1" xref="Thmtheorem21.p1.1.1.m1.1.1.cmml">A</mi><mo id="Thmtheorem21.p1.1.1.m1.3.4.2.3.2.2" xref="Thmtheorem21.p1.1.1.m1.3.4.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem21.p1.1.1.m1.2.2" xref="Thmtheorem21.p1.1.1.m1.2.2.cmml">ℬ</mi><mo id="Thmtheorem21.p1.1.1.m1.3.4.2.3.2.3" stretchy="false" xref="Thmtheorem21.p1.1.1.m1.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem21.p1.1.1.m1.3.4.1" xref="Thmtheorem21.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="Thmtheorem21.p1.1.1.m1.3.4.3" xref="Thmtheorem21.p1.1.1.m1.3.4.3.cmml"><mi id="Thmtheorem21.p1.1.1.m1.3.4.3.2" xref="Thmtheorem21.p1.1.1.m1.3.4.3.2.cmml">ℕ</mi><mo id="Thmtheorem21.p1.1.1.m1.3.4.3.1" xref="Thmtheorem21.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="Thmtheorem21.p1.1.1.m1.3.4.3.3.2" xref="Thmtheorem21.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="Thmtheorem21.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="Thmtheorem21.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="Thmtheorem21.p1.1.1.m1.3.3" mathvariant="normal" xref="Thmtheorem21.p1.1.1.m1.3.3.cmml">∞</mi><mo id="Thmtheorem21.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="Thmtheorem21.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.1.1.m1.3b"><apply id="Thmtheorem21.p1.1.1.m1.3.4.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4"><in id="Thmtheorem21.p1.1.1.m1.3.4.1.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.1"></in><apply id="Thmtheorem21.p1.1.1.m1.3.4.2.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.2"><times id="Thmtheorem21.p1.1.1.m1.3.4.2.1.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.2.1"></times><ci id="Thmtheorem21.p1.1.1.m1.3.4.2.2.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.2.2">𝜌</ci><interval closure="open" id="Thmtheorem21.p1.1.1.m1.3.4.2.3.1.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.2.3.2"><ci id="Thmtheorem21.p1.1.1.m1.1.1.cmml" xref="Thmtheorem21.p1.1.1.m1.1.1">𝐴</ci><ci id="Thmtheorem21.p1.1.1.m1.2.2.cmml" xref="Thmtheorem21.p1.1.1.m1.2.2">ℬ</ci></interval></apply><apply id="Thmtheorem21.p1.1.1.m1.3.4.3.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.3"><union id="Thmtheorem21.p1.1.1.m1.3.4.3.1.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.3.1"></union><ci id="Thmtheorem21.p1.1.1.m1.3.4.3.2.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.3.2">ℕ</ci><set id="Thmtheorem21.p1.1.1.m1.3.4.3.3.1.cmml" xref="Thmtheorem21.p1.1.1.m1.3.4.3.3.2"><infinity id="Thmtheorem21.p1.1.1.m1.3.3.cmml" xref="Thmtheorem21.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.1.1.m1.3c">\rho(A,\mathcal{B})\in\mathbb{N}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.1.1.m1.3d">italic_ρ ( italic_A , caligraphic_B ) ∈ blackboard_N ∪ { ∞ }</annotation></semantics></math> be the minimum size of a collection <math alttext="\Lambda" class="ltx_Math" display="inline" id="Thmtheorem21.p1.2.2.m2.1"><semantics id="Thmtheorem21.p1.2.2.m2.1a"><mi id="Thmtheorem21.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem21.p1.2.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.2.2.m2.1b"><ci id="Thmtheorem21.p1.2.2.m2.1.1.cmml" xref="Thmtheorem21.p1.2.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.2.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.2.2.m2.1d">roman_Λ</annotation></semantics></math> of pairs of subsets of <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem21.p1.3.3.m3.1"><semantics id="Thmtheorem21.p1.3.3.m3.1a"><mi id="Thmtheorem21.p1.3.3.m3.1.1" xref="Thmtheorem21.p1.3.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.3.3.m3.1b"><ci id="Thmtheorem21.p1.3.3.m3.1.1.cmml" xref="Thmtheorem21.p1.3.3.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.3.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.3.3.m3.1d">italic_U</annotation></semantics></math> such that there is no semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem21.p1.4.4.m4.1"><semantics id="Thmtheorem21.p1.4.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem21.p1.4.4.m4.1.1" xref="Thmtheorem21.p1.4.4.m4.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.4.4.m4.1b"><ci id="Thmtheorem21.p1.4.4.m4.1.1.cmml" xref="Thmtheorem21.p1.4.4.m4.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.4.4.m4.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.4.4.m4.1d">caligraphic_F</annotation></semantics></math> over <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem21.p1.5.5.m5.1"><semantics id="Thmtheorem21.p1.5.5.m5.1a"><mi id="Thmtheorem21.p1.5.5.m5.1.1" xref="Thmtheorem21.p1.5.5.m5.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.5.5.m5.1b"><ci id="Thmtheorem21.p1.5.5.m5.1.1.cmml" xref="Thmtheorem21.p1.5.5.m5.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.5.5.m5.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.5.5.m5.1d">italic_U</annotation></semantics></math> that preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="Thmtheorem21.p1.6.6.m6.1"><semantics id="Thmtheorem21.p1.6.6.m6.1a"><mi id="Thmtheorem21.p1.6.6.m6.1.1" mathvariant="normal" xref="Thmtheorem21.p1.6.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.6.6.m6.1b"><ci id="Thmtheorem21.p1.6.6.m6.1.1.cmml" xref="Thmtheorem21.p1.6.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.6.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.6.6.m6.1d">roman_Λ</annotation></semantics></math> and is above an element <math alttext="a\in A" class="ltx_Math" display="inline" id="Thmtheorem21.p1.7.7.m7.1"><semantics id="Thmtheorem21.p1.7.7.m7.1a"><mrow id="Thmtheorem21.p1.7.7.m7.1.1" xref="Thmtheorem21.p1.7.7.m7.1.1.cmml"><mi id="Thmtheorem21.p1.7.7.m7.1.1.2" xref="Thmtheorem21.p1.7.7.m7.1.1.2.cmml">a</mi><mo id="Thmtheorem21.p1.7.7.m7.1.1.1" xref="Thmtheorem21.p1.7.7.m7.1.1.1.cmml">∈</mo><mi id="Thmtheorem21.p1.7.7.m7.1.1.3" xref="Thmtheorem21.p1.7.7.m7.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.7.7.m7.1b"><apply id="Thmtheorem21.p1.7.7.m7.1.1.cmml" xref="Thmtheorem21.p1.7.7.m7.1.1"><in id="Thmtheorem21.p1.7.7.m7.1.1.1.cmml" xref="Thmtheorem21.p1.7.7.m7.1.1.1"></in><ci id="Thmtheorem21.p1.7.7.m7.1.1.2.cmml" xref="Thmtheorem21.p1.7.7.m7.1.1.2">𝑎</ci><ci id="Thmtheorem21.p1.7.7.m7.1.1.3.cmml" xref="Thmtheorem21.p1.7.7.m7.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.7.7.m7.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.7.7.m7.1d">italic_a ∈ italic_A</annotation></semantics></math> <em class="ltx_emph ltx_font_upright" id="Thmtheorem21.p1.9.9.1">(</em>with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem21.p1.8.8.m8.1"><semantics id="Thmtheorem21.p1.8.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem21.p1.8.8.m8.1.1" xref="Thmtheorem21.p1.8.8.m8.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.8.8.m8.1b"><ci id="Thmtheorem21.p1.8.8.m8.1.1.cmml" xref="Thmtheorem21.p1.8.8.m8.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.8.8.m8.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.8.8.m8.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem21.p1.9.9.m9.1"><semantics id="Thmtheorem21.p1.9.9.m9.1a"><mi id="Thmtheorem21.p1.9.9.m9.1.1" xref="Thmtheorem21.p1.9.9.m9.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem21.p1.9.9.m9.1b"><ci id="Thmtheorem21.p1.9.9.m9.1.1.cmml" xref="Thmtheorem21.p1.9.9.m9.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem21.p1.9.9.m9.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem21.p1.9.9.m9.1d">italic_U</annotation></semantics></math><em class="ltx_emph ltx_font_upright" id="Thmtheorem21.p1.9.9.2">)</em>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.1">The definition of cover complexity considered here is with respect to semi-filters (essentially, monotone functions which are not equal to the constant function which outputs 1). In the context of circuit complexity, notions of cover complexity with respect to other types of Boolean functions (such as ultrafilters and linear functions) have been considered, yielding characterizations of different circuit models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib19" title="">19</a>]</cite>. If we ask that in every pair at least one of the sets is the intersection of a generator with <math alttext="U" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><mi id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><ci id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">italic_U</annotation></semantics></math>, we obtain characterizations of branching models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib20" title="">20</a>]</cite> (such as branching programs). In <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS3" title="4.3 Nondeterministic graph complexity ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>, we will consider the 2-dimensional cover problem with ultrafilters.</p> </div> <section class="ltx_paragraph ltx_indentfirst" id="S3.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Cover Graph of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.1.m1.1"><semantics id="S3.SS1.SSS0.Px1.1.m1.1b"><mi id="S3.SS1.SSS0.Px1.1.m1.1.1" xref="S3.SS1.SSS0.Px1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.1.m1.1c"><ci id="S3.SS1.SSS0.Px1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.1.m1.1d">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.1.m1.1e">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.2.m2.1"><semantics id="S3.SS1.SSS0.Px1.2.m2.1b"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.2.m2.1.1" xref="S3.SS1.SSS0.Px1.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.2.m2.1c"><ci id="S3.SS1.SSS0.Px1.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px1.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.2.m2.1d">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.2.m2.1e">caligraphic_B</annotation></semantics></math>.</h5> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p1.1">In order to get more intuition about the notion of cover complexity, consider an undirected bipartite graph <math alttext="\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.1.m1.5"><semantics id="S3.SS1.SSS0.Px1.p1.1.m1.5a"><mrow id="S3.SS1.SSS0.Px1.p1.1.m1.5.5" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.cmml"><msub id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.4" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.4.cmml"><mi id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.4.2" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.4.2.cmml">Φ</mi><mrow id="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.4" xref="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.cmml">A</mi><mo id="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.4.1" xref="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.2" xref="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.2.cmml">ℬ</mi></mrow></msub><mo id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.3.cmml">=</mo><mrow 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xref="S3.SS1.SSS0.Px1.p1.1.m1.2.2.2.2">ℬ</ci></list></apply><vector id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2"><apply id="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.4.4.1.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply><apply id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.5.5.2.2.2.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply><ci id="S3.SS1.SSS0.Px1.p1.1.m1.3.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.3.3">ℰ</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.1.m1.5c">\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.1.m1.5d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT = ( italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT , caligraphic_E )</annotation></semantics></math>, where</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx4"> <tbody id="S3.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.Ex17.m1.1"><semantics id="S3.Ex17.m1.1a"><msub id="S3.Ex17.m1.1.1" xref="S3.Ex17.m1.1.1.cmml"><mi id="S3.Ex17.m1.1.1.2" xref="S3.Ex17.m1.1.1.2.cmml">V</mi><mi id="S3.Ex17.m1.1.1.3" xref="S3.Ex17.m1.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Ex17.m1.1b"><apply id="S3.Ex17.m1.1.1.cmml" xref="S3.Ex17.m1.1.1"><csymbol cd="ambiguous" id="S3.Ex17.m1.1.1.1.cmml" xref="S3.Ex17.m1.1.1">subscript</csymbol><ci id="S3.Ex17.m1.1.1.2.cmml" xref="S3.Ex17.m1.1.1.2">𝑉</ci><ci id="S3.Ex17.m1.1.1.3.cmml" xref="S3.Ex17.m1.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m1.1c">\displaystyle V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m1.1d">italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}" class="ltx_Math" display="inline" id="S3.Ex17.m2.1"><semantics id="S3.Ex17.m2.1a"><mover id="S3.Ex17.m2.1.1" xref="S3.Ex17.m2.1.1.cmml"><mo id="S3.Ex17.m2.1.1.2" xref="S3.Ex17.m2.1.1.2.cmml">=</mo><mi id="S3.Ex17.m2.1.1.3" xref="S3.Ex17.m2.1.1.3.cmml">def</mi></mover><annotation-xml encoding="MathML-Content" id="S3.Ex17.m2.1b"><apply id="S3.Ex17.m2.1.1.cmml" xref="S3.Ex17.m2.1.1"><csymbol cd="ambiguous" id="S3.Ex17.m2.1.1.1.cmml" xref="S3.Ex17.m2.1.1">superscript</csymbol><eq id="S3.Ex17.m2.1.1.2.cmml" xref="S3.Ex17.m2.1.1.2"></eq><ci id="S3.Ex17.m2.1.1.3.cmml" xref="S3.Ex17.m2.1.1.3">def</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m2.1c">\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m2.1d">start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\{(E,H)\mid E,H\subseteq U\}," class="ltx_Math" display="inline" id="S3.Ex17.m3.5"><semantics id="S3.Ex17.m3.5a"><mrow id="S3.Ex17.m3.5.5.1"><mrow id="S3.Ex17.m3.5.5.1.1.2" xref="S3.Ex17.m3.5.5.1.1.3.cmml"><mo id="S3.Ex17.m3.5.5.1.1.2.3" stretchy="false" xref="S3.Ex17.m3.5.5.1.1.3.1.cmml">{</mo><mrow id="S3.Ex17.m3.5.5.1.1.1.1.2" xref="S3.Ex17.m3.5.5.1.1.1.1.1.cmml"><mo id="S3.Ex17.m3.5.5.1.1.1.1.2.1" stretchy="false" xref="S3.Ex17.m3.5.5.1.1.1.1.1.cmml">(</mo><mi id="S3.Ex17.m3.1.1" xref="S3.Ex17.m3.1.1.cmml">E</mi><mo id="S3.Ex17.m3.5.5.1.1.1.1.2.2" xref="S3.Ex17.m3.5.5.1.1.1.1.1.cmml">,</mo><mi id="S3.Ex17.m3.2.2" xref="S3.Ex17.m3.2.2.cmml">H</mi><mo id="S3.Ex17.m3.5.5.1.1.1.1.2.3" stretchy="false" xref="S3.Ex17.m3.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo fence="true" id="S3.Ex17.m3.5.5.1.1.2.4" lspace="0em" rspace="0em" xref="S3.Ex17.m3.5.5.1.1.3.1.cmml">∣</mo><mrow id="S3.Ex17.m3.5.5.1.1.2.2" xref="S3.Ex17.m3.5.5.1.1.2.2.cmml"><mrow id="S3.Ex17.m3.5.5.1.1.2.2.2.2" xref="S3.Ex17.m3.5.5.1.1.2.2.2.1.cmml"><mi id="S3.Ex17.m3.3.3" xref="S3.Ex17.m3.3.3.cmml">E</mi><mo id="S3.Ex17.m3.5.5.1.1.2.2.2.2.1" xref="S3.Ex17.m3.5.5.1.1.2.2.2.1.cmml">,</mo><mi id="S3.Ex17.m3.4.4" xref="S3.Ex17.m3.4.4.cmml">H</mi></mrow><mo id="S3.Ex17.m3.5.5.1.1.2.2.1" xref="S3.Ex17.m3.5.5.1.1.2.2.1.cmml">⊆</mo><mi id="S3.Ex17.m3.5.5.1.1.2.2.3" xref="S3.Ex17.m3.5.5.1.1.2.2.3.cmml">U</mi></mrow><mo id="S3.Ex17.m3.5.5.1.1.2.5" stretchy="false" xref="S3.Ex17.m3.5.5.1.1.3.1.cmml">}</mo></mrow><mo id="S3.Ex17.m3.5.5.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex17.m3.5b"><apply id="S3.Ex17.m3.5.5.1.1.3.cmml" xref="S3.Ex17.m3.5.5.1.1.2"><csymbol cd="latexml" id="S3.Ex17.m3.5.5.1.1.3.1.cmml" xref="S3.Ex17.m3.5.5.1.1.2.3">conditional-set</csymbol><interval closure="open" id="S3.Ex17.m3.5.5.1.1.1.1.1.cmml" xref="S3.Ex17.m3.5.5.1.1.1.1.2"><ci id="S3.Ex17.m3.1.1.cmml" xref="S3.Ex17.m3.1.1">𝐸</ci><ci id="S3.Ex17.m3.2.2.cmml" xref="S3.Ex17.m3.2.2">𝐻</ci></interval><apply id="S3.Ex17.m3.5.5.1.1.2.2.cmml" xref="S3.Ex17.m3.5.5.1.1.2.2"><subset id="S3.Ex17.m3.5.5.1.1.2.2.1.cmml" xref="S3.Ex17.m3.5.5.1.1.2.2.1"></subset><list id="S3.Ex17.m3.5.5.1.1.2.2.2.1.cmml" xref="S3.Ex17.m3.5.5.1.1.2.2.2.2"><ci id="S3.Ex17.m3.3.3.cmml" xref="S3.Ex17.m3.3.3">𝐸</ci><ci id="S3.Ex17.m3.4.4.cmml" xref="S3.Ex17.m3.4.4">𝐻</ci></list><ci id="S3.Ex17.m3.5.5.1.1.2.2.3.cmml" xref="S3.Ex17.m3.5.5.1.1.2.2.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m3.5c">\displaystyle\{(E,H)\mid E,H\subseteq U\},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m3.5d">{ ( italic_E , italic_H ) ∣ italic_E , italic_H ⊆ italic_U } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.Ex18.m1.1"><semantics id="S3.Ex18.m1.1a"><msub id="S3.Ex18.m1.1.1" xref="S3.Ex18.m1.1.1.cmml"><mi id="S3.Ex18.m1.1.1.2" xref="S3.Ex18.m1.1.1.2.cmml">V</mi><mi id="S3.Ex18.m1.1.1.3" xref="S3.Ex18.m1.1.1.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Ex18.m1.1b"><apply id="S3.Ex18.m1.1.1.cmml" xref="S3.Ex18.m1.1.1"><csymbol cd="ambiguous" id="S3.Ex18.m1.1.1.1.cmml" xref="S3.Ex18.m1.1.1">subscript</csymbol><ci id="S3.Ex18.m1.1.1.2.cmml" xref="S3.Ex18.m1.1.1.2">𝑉</ci><ci id="S3.Ex18.m1.1.1.3.cmml" xref="S3.Ex18.m1.1.1.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m1.1c">\displaystyle V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m1.1d">italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}" class="ltx_Math" display="inline" id="S3.Ex18.m2.1"><semantics id="S3.Ex18.m2.1a"><mover id="S3.Ex18.m2.1.1" xref="S3.Ex18.m2.1.1.cmml"><mo id="S3.Ex18.m2.1.1.2" xref="S3.Ex18.m2.1.1.2.cmml">=</mo><mi id="S3.Ex18.m2.1.1.3" xref="S3.Ex18.m2.1.1.3.cmml">def</mi></mover><annotation-xml encoding="MathML-Content" id="S3.Ex18.m2.1b"><apply id="S3.Ex18.m2.1.1.cmml" xref="S3.Ex18.m2.1.1"><csymbol cd="ambiguous" id="S3.Ex18.m2.1.1.1.cmml" xref="S3.Ex18.m2.1.1">superscript</csymbol><eq id="S3.Ex18.m2.1.1.2.cmml" xref="S3.Ex18.m2.1.1.2"></eq><ci id="S3.Ex18.m2.1.1.3.cmml" xref="S3.Ex18.m2.1.1.3">def</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m2.1c">\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m2.1d">start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\{\mathcal{F}\subseteq\mathcal{P}(U)\mid\mathcal{F}~{}\text{is a % semi-filter and}~{}\mathcal{F}~{}\text{is above some}~{}a\in A\}," class="ltx_Math" display="inline" id="S3.Ex18.m3.2"><semantics id="S3.Ex18.m3.2a"><mrow id="S3.Ex18.m3.2.2.1"><mrow id="S3.Ex18.m3.2.2.1.1.2" xref="S3.Ex18.m3.2.2.1.1.3.cmml"><mo id="S3.Ex18.m3.2.2.1.1.2.3" stretchy="false" xref="S3.Ex18.m3.2.2.1.1.3.1.cmml">{</mo><mrow id="S3.Ex18.m3.2.2.1.1.1.1" xref="S3.Ex18.m3.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex18.m3.2.2.1.1.1.1.2" xref="S3.Ex18.m3.2.2.1.1.1.1.2.cmml">ℱ</mi><mo id="S3.Ex18.m3.2.2.1.1.1.1.1" xref="S3.Ex18.m3.2.2.1.1.1.1.1.cmml">⊆</mo><mrow id="S3.Ex18.m3.2.2.1.1.1.1.3" xref="S3.Ex18.m3.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex18.m3.2.2.1.1.1.1.3.2" xref="S3.Ex18.m3.2.2.1.1.1.1.3.2.cmml">𝒫</mi><mo id="S3.Ex18.m3.2.2.1.1.1.1.3.1" xref="S3.Ex18.m3.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex18.m3.2.2.1.1.1.1.3.3.2" xref="S3.Ex18.m3.2.2.1.1.1.1.3.cmml"><mo id="S3.Ex18.m3.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S3.Ex18.m3.2.2.1.1.1.1.3.cmml">(</mo><mi id="S3.Ex18.m3.1.1" xref="S3.Ex18.m3.1.1.cmml">U</mi><mo id="S3.Ex18.m3.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S3.Ex18.m3.2.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S3.Ex18.m3.2.2.1.1.2.4" lspace="0em" rspace="0em" xref="S3.Ex18.m3.2.2.1.1.3.1.cmml">∣</mo><mrow id="S3.Ex18.m3.2.2.1.1.2.2" xref="S3.Ex18.m3.2.2.1.1.2.2.cmml"><mrow id="S3.Ex18.m3.2.2.1.1.2.2.2" xref="S3.Ex18.m3.2.2.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex18.m3.2.2.1.1.2.2.2.2" xref="S3.Ex18.m3.2.2.1.1.2.2.2.2.cmml">ℱ</mi><mo id="S3.Ex18.m3.2.2.1.1.2.2.2.1" lspace="0.330em" xref="S3.Ex18.m3.2.2.1.1.2.2.2.1.cmml">⁢</mo><mtext id="S3.Ex18.m3.2.2.1.1.2.2.2.3" xref="S3.Ex18.m3.2.2.1.1.2.2.2.3a.cmml">is a semi-filter and</mtext><mo id="S3.Ex18.m3.2.2.1.1.2.2.2.1a" lspace="0.330em" xref="S3.Ex18.m3.2.2.1.1.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex18.m3.2.2.1.1.2.2.2.4" xref="S3.Ex18.m3.2.2.1.1.2.2.2.4.cmml">ℱ</mi><mo id="S3.Ex18.m3.2.2.1.1.2.2.2.1b" lspace="0.330em" xref="S3.Ex18.m3.2.2.1.1.2.2.2.1.cmml">⁢</mo><mtext id="S3.Ex18.m3.2.2.1.1.2.2.2.5" xref="S3.Ex18.m3.2.2.1.1.2.2.2.5a.cmml">is above some</mtext><mo id="S3.Ex18.m3.2.2.1.1.2.2.2.1c" lspace="0.330em" xref="S3.Ex18.m3.2.2.1.1.2.2.2.1.cmml">⁢</mo><mi id="S3.Ex18.m3.2.2.1.1.2.2.2.6" xref="S3.Ex18.m3.2.2.1.1.2.2.2.6.cmml">a</mi></mrow><mo id="S3.Ex18.m3.2.2.1.1.2.2.1" xref="S3.Ex18.m3.2.2.1.1.2.2.1.cmml">∈</mo><mi id="S3.Ex18.m3.2.2.1.1.2.2.3" xref="S3.Ex18.m3.2.2.1.1.2.2.3.cmml">A</mi></mrow><mo id="S3.Ex18.m3.2.2.1.1.2.5" stretchy="false" xref="S3.Ex18.m3.2.2.1.1.3.1.cmml">}</mo></mrow><mo id="S3.Ex18.m3.2.2.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex18.m3.2b"><apply id="S3.Ex18.m3.2.2.1.1.3.cmml" xref="S3.Ex18.m3.2.2.1.1.2"><csymbol cd="latexml" id="S3.Ex18.m3.2.2.1.1.3.1.cmml" xref="S3.Ex18.m3.2.2.1.1.2.3">conditional-set</csymbol><apply id="S3.Ex18.m3.2.2.1.1.1.1.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1"><subset id="S3.Ex18.m3.2.2.1.1.1.1.1.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1.1"></subset><ci id="S3.Ex18.m3.2.2.1.1.1.1.2.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1.2">ℱ</ci><apply id="S3.Ex18.m3.2.2.1.1.1.1.3.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1.3"><times id="S3.Ex18.m3.2.2.1.1.1.1.3.1.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1.3.1"></times><ci id="S3.Ex18.m3.2.2.1.1.1.1.3.2.cmml" xref="S3.Ex18.m3.2.2.1.1.1.1.3.2">𝒫</ci><ci id="S3.Ex18.m3.1.1.cmml" xref="S3.Ex18.m3.1.1">𝑈</ci></apply></apply><apply id="S3.Ex18.m3.2.2.1.1.2.2.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2"><in id="S3.Ex18.m3.2.2.1.1.2.2.1.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.1"></in><apply id="S3.Ex18.m3.2.2.1.1.2.2.2.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2"><times id="S3.Ex18.m3.2.2.1.1.2.2.2.1.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.1"></times><ci id="S3.Ex18.m3.2.2.1.1.2.2.2.2.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.2">ℱ</ci><ci id="S3.Ex18.m3.2.2.1.1.2.2.2.3a.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.3"><mtext id="S3.Ex18.m3.2.2.1.1.2.2.2.3.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.3">is a semi-filter and</mtext></ci><ci id="S3.Ex18.m3.2.2.1.1.2.2.2.4.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.4">ℱ</ci><ci id="S3.Ex18.m3.2.2.1.1.2.2.2.5a.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.5"><mtext id="S3.Ex18.m3.2.2.1.1.2.2.2.5.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.5">is above some</mtext></ci><ci id="S3.Ex18.m3.2.2.1.1.2.2.2.6.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.2.6">𝑎</ci></apply><ci id="S3.Ex18.m3.2.2.1.1.2.2.3.cmml" xref="S3.Ex18.m3.2.2.1.1.2.2.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m3.2c">\displaystyle\{\mathcal{F}\subseteq\mathcal{P}(U)\mid\mathcal{F}~{}\text{is a % semi-filter and}~{}\mathcal{F}~{}\text{is above some}~{}a\in A\},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m3.2d">{ caligraphic_F ⊆ caligraphic_P ( italic_U ) ∣ caligraphic_F is a semi-filter and caligraphic_F is above some italic_a ∈ italic_A } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p1.12">and there is an edge <math alttext="e\in\mathcal{E}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.2.m1.1"><semantics id="S3.SS1.SSS0.Px1.p1.2.m1.1a"><mrow id="S3.SS1.SSS0.Px1.p1.2.m1.1.1" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.2" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.3.cmml">ℰ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.2.m1.1b"><apply id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1"><in id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.1"></in><ci id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS0.Px1.p1.2.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.2.m1.1.1.3">ℰ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.2.m1.1c">e\in\mathcal{E}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.2.m1.1d">italic_e ∈ caligraphic_E</annotation></semantics></math> connecting <math alttext="(E,H)\in V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.3.m2.2"><semantics id="S3.SS1.SSS0.Px1.p1.3.m2.2a"><mrow id="S3.SS1.SSS0.Px1.p1.3.m2.2.3" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.cmml"><mrow id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.2" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.1.cmml"><mo id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.2.1" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.1.cmml">(</mo><mi id="S3.SS1.SSS0.Px1.p1.3.m2.1.1" xref="S3.SS1.SSS0.Px1.p1.3.m2.1.1.cmml">E</mi><mo id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.2.2" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.1.cmml">,</mo><mi id="S3.SS1.SSS0.Px1.p1.3.m2.2.2" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.2.cmml">H</mi><mo id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.2.3" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.1" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.1.cmml">∈</mo><msub id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.2" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.3" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.3.m2.2b"><apply id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3"><in id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.1"></in><interval closure="open" id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.2.2"><ci id="S3.SS1.SSS0.Px1.p1.3.m2.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.1.1">𝐸</ci><ci id="S3.SS1.SSS0.Px1.p1.3.m2.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.2">𝐻</ci></interval><apply id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.3.cmml" xref="S3.SS1.SSS0.Px1.p1.3.m2.2.3.3.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.3.m2.2c">(E,H)\in V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.3.m2.2d">( italic_E , italic_H ) ∈ italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{F}\in V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.4.m3.1"><semantics id="S3.SS1.SSS0.Px1.p1.4.m3.1a"><mrow id="S3.SS1.SSS0.Px1.p1.4.m3.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.2" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.2.cmml">ℱ</mi><mo id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.1" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.1.cmml">∈</mo><msub id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.2" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.3" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.4.m3.1b"><apply id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1"><in id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.1"></in><ci id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.2">ℱ</ci><apply id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.3.cmml" xref="S3.SS1.SSS0.Px1.p1.4.m3.1.1.3.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.4.m3.1c">\mathcal{F}\in V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.4.m3.1d">caligraphic_F ∈ italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.5.m4.1"><semantics id="S3.SS1.SSS0.Px1.p1.5.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.5.m4.1.1" xref="S3.SS1.SSS0.Px1.p1.5.m4.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.5.m4.1b"><ci id="S3.SS1.SSS0.Px1.p1.5.m4.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.5.m4.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.5.m4.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.5.m4.1d">caligraphic_F</annotation></semantics></math> does not preserve <math alttext="(E,H)" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.6.m5.2"><semantics id="S3.SS1.SSS0.Px1.p1.6.m5.2a"><mrow id="S3.SS1.SSS0.Px1.p1.6.m5.2.3.2" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.3.1.cmml"><mo id="S3.SS1.SSS0.Px1.p1.6.m5.2.3.2.1" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.3.1.cmml">(</mo><mi id="S3.SS1.SSS0.Px1.p1.6.m5.1.1" xref="S3.SS1.SSS0.Px1.p1.6.m5.1.1.cmml">E</mi><mo id="S3.SS1.SSS0.Px1.p1.6.m5.2.3.2.2" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.3.1.cmml">,</mo><mi id="S3.SS1.SSS0.Px1.p1.6.m5.2.2" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.2.cmml">H</mi><mo id="S3.SS1.SSS0.Px1.p1.6.m5.2.3.2.3" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.6.m5.2b"><interval closure="open" id="S3.SS1.SSS0.Px1.p1.6.m5.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.3.2"><ci id="S3.SS1.SSS0.Px1.p1.6.m5.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m5.1.1">𝐸</ci><ci id="S3.SS1.SSS0.Px1.p1.6.m5.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.6.m5.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.6.m5.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.6.m5.2d">( italic_E , italic_H )</annotation></semantics></math>. Then <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.7.m6.2"><semantics id="S3.SS1.SSS0.Px1.p1.7.m6.2a"><mrow id="S3.SS1.SSS0.Px1.p1.7.m6.2.3" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.2" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.2.cmml">ρ</mi><mo id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.1" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.2" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.1.cmml"><mo id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.1.cmml">(</mo><mi id="S3.SS1.SSS0.Px1.p1.7.m6.1.1" xref="S3.SS1.SSS0.Px1.p1.7.m6.1.1.cmml">A</mi><mo id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.2.2" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.7.m6.2.2" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.2.cmml">ℬ</mi><mo id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.2.3" stretchy="false" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.7.m6.2b"><apply id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3"><times id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.1"></times><ci id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.2">𝜌</ci><interval closure="open" id="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.3.3.2"><ci id="S3.SS1.SSS0.Px1.p1.7.m6.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.1.1">𝐴</ci><ci id="S3.SS1.SSS0.Px1.p1.7.m6.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.7.m6.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.7.m6.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.7.m6.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> is the minimum number of vertices in <math alttext="V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.8.m7.1"><semantics id="S3.SS1.SSS0.Px1.p1.8.m7.1a"><msub id="S3.SS1.SSS0.Px1.p1.8.m7.1.1" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.2" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.3" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.8.m7.1b"><apply id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.8.m7.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.8.m7.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.8.m7.1c">V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.8.m7.1d">italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math> whose adjacent edges cover all the vertices in <math alttext="V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.9.m8.1"><semantics id="S3.SS1.SSS0.Px1.p1.9.m8.1a"><msub id="S3.SS1.SSS0.Px1.p1.9.m8.1.1" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.2" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.3" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.9.m8.1b"><apply id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p1.9.m8.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.9.m8.1.1.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.9.m8.1c">V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.9.m8.1d">italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math>. For convenience, we say that <math alttext="\Phi_{A,\mathcal{B}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.10.m9.2"><semantics id="S3.SS1.SSS0.Px1.p1.10.m9.2a"><msub id="S3.SS1.SSS0.Px1.p1.10.m9.2.3" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.10.m9.2.3.2" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.3.2.cmml">Φ</mi><mrow id="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.4" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.3.cmml"><mi id="S3.SS1.SSS0.Px1.p1.10.m9.1.1.1.1" xref="S3.SS1.SSS0.Px1.p1.10.m9.1.1.1.1.cmml">A</mi><mo id="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.4.1" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.2" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.2.cmml">ℬ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.10.m9.2b"><apply id="S3.SS1.SSS0.Px1.p1.10.m9.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p1.10.m9.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.3">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p1.10.m9.2.3.2.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.3.2">Φ</ci><list id="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.4"><ci id="S3.SS1.SSS0.Px1.p1.10.m9.1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.1.1.1.1">𝐴</ci><ci id="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px1.p1.10.m9.2.2.2.2">ℬ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.10.m9.2c">\Phi_{A,\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.10.m9.2d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.12.1">cover graph</em> of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.11.m10.1"><semantics id="S3.SS1.SSS0.Px1.p1.11.m10.1a"><mi id="S3.SS1.SSS0.Px1.p1.11.m10.1.1" xref="S3.SS1.SSS0.Px1.p1.11.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.11.m10.1b"><ci id="S3.SS1.SSS0.Px1.p1.11.m10.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.11.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.11.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.11.m10.1d">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p1.12.m11.1"><semantics id="S3.SS1.SSS0.Px1.p1.12.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p1.12.m11.1.1" xref="S3.SS1.SSS0.Px1.p1.12.m11.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.12.m11.1b"><ci id="S3.SS1.SSS0.Px1.p1.12.m11.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.12.m11.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.12.m11.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.12.m11.1d">caligraphic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p2.3">Note that a set of vertices in <math alttext="V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p2.1.m1.1"><semantics id="S3.SS1.SSS0.Px1.p2.1.m1.1a"><msub id="S3.SS1.SSS0.Px1.p2.1.m1.1.1" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p2.1.m1.1b"><apply id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.1.m1.1c">V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.1.m1.1d">italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math> whose adjacent edges cover all of the vertices in <math alttext="V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p2.2.m2.1"><semantics id="S3.SS1.SSS0.Px1.p2.2.m2.1a"><msub id="S3.SS1.SSS0.Px1.p2.2.m2.1.1" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.2" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.2.cmml">V</mi><mi id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.3" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p2.2.m2.1b"><apply id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.2">𝑉</ci><ci id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.2.m2.1c">V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.2.m2.1d">italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math> is also known as a <em class="ltx_emph ltx_font_italic" id="S3.SS1.SSS0.Px1.p2.3.1">dominating set</em> in graph theory. Moreover, identifying vertices with their neighbourhoods, the value of <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p2.3.m3.2"><semantics id="S3.SS1.SSS0.Px1.p2.3.m3.2a"><mrow id="S3.SS1.SSS0.Px1.p2.3.m3.2.3" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.cmml"><mi id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.2" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.2.cmml">ρ</mi><mo id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.1" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.2" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.1.cmml"><mo id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.1.cmml">(</mo><mi id="S3.SS1.SSS0.Px1.p2.3.m3.1.1" xref="S3.SS1.SSS0.Px1.p2.3.m3.1.1.cmml">A</mi><mo id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.2.2" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.SSS0.Px1.p2.3.m3.2.2" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.2.cmml">ℬ</mi><mo id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p2.3.m3.2b"><apply id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3"><times id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.1"></times><ci id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.2.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.2">𝜌</ci><interval closure="open" id="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.1.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.3.3.2"><ci id="S3.SS1.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.1.1">𝐴</ci><ci id="S3.SS1.SSS0.Px1.p2.3.m3.2.2.cmml" xref="S3.SS1.SSS0.Px1.p2.3.m3.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.3.m3.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.3.m3.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> is equivalent to the optimal value of a set cover problem.</p> </div> </section> </section> <section class="ltx_subsection ltx_indent_first" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Discrete complexity lower bounds using the fusion method</h3> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem22"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem22.1.1.1">Theorem 22</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem22.2.2"> </span>(Fusion lower bound)<span class="ltx_text ltx_font_bold" id="Thmtheorem22.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem22.p1"> <p class="ltx_p" id="Thmtheorem22.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem22.p1.2.2">Let <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem22.p1.1.1.m1.1"><semantics id="Thmtheorem22.p1.1.1.m1.1a"><mrow id="Thmtheorem22.p1.1.1.m1.1.1" xref="Thmtheorem22.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem22.p1.1.1.m1.1.1.2" xref="Thmtheorem22.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem22.p1.1.1.m1.1.1.1" xref="Thmtheorem22.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem22.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem22.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem22.p1.1.1.m1.1b"><apply id="Thmtheorem22.p1.1.1.m1.1.1.cmml" xref="Thmtheorem22.p1.1.1.m1.1.1"><subset id="Thmtheorem22.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem22.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem22.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem22.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem22.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem22.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem22.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem22.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> be non-trivial, and <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="Thmtheorem22.p1.2.2.m2.1"><semantics id="Thmtheorem22.p1.2.2.m2.1a"><mrow id="Thmtheorem22.p1.2.2.m2.1.2" xref="Thmtheorem22.p1.2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem22.p1.2.2.m2.1.2.2" xref="Thmtheorem22.p1.2.2.m2.1.2.2.cmml">ℬ</mi><mo id="Thmtheorem22.p1.2.2.m2.1.2.1" xref="Thmtheorem22.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem22.p1.2.2.m2.1.2.3" xref="Thmtheorem22.p1.2.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem22.p1.2.2.m2.1.2.3.2" xref="Thmtheorem22.p1.2.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem22.p1.2.2.m2.1.2.3.1" xref="Thmtheorem22.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem22.p1.2.2.m2.1.2.3.3.2" xref="Thmtheorem22.p1.2.2.m2.1.2.3.cmml"><mo id="Thmtheorem22.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem22.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmtheorem22.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem22.p1.2.2.m2.1.1.cmml">Γ</mi><mo id="Thmtheorem22.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem22.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem22.p1.2.2.m2.1b"><apply id="Thmtheorem22.p1.2.2.m2.1.2.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2"><subset id="Thmtheorem22.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2.1"></subset><ci id="Thmtheorem22.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2.2">ℬ</ci><apply id="Thmtheorem22.p1.2.2.m2.1.2.3.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2.3"><times id="Thmtheorem22.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2.3.1"></times><ci id="Thmtheorem22.p1.2.2.m2.1.2.3.2.cmml" xref="Thmtheorem22.p1.2.2.m2.1.2.3.2">𝒫</ci><ci id="Thmtheorem22.p1.2.2.m2.1.1.cmml" xref="Thmtheorem22.p1.2.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem22.p1.2.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem22.p1.2.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math> be a non-empty family of generators. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(A,\mathcal{B})\;\leq\;D_{\cap}(A\mid\mathcal{B})." class="ltx_Math" display="block" id="S3.Ex19.m1.3"><semantics id="S3.Ex19.m1.3a"><mrow id="S3.Ex19.m1.3.3.1" xref="S3.Ex19.m1.3.3.1.1.cmml"><mrow id="S3.Ex19.m1.3.3.1.1" xref="S3.Ex19.m1.3.3.1.1.cmml"><mrow id="S3.Ex19.m1.3.3.1.1.3" xref="S3.Ex19.m1.3.3.1.1.3.cmml"><mi id="S3.Ex19.m1.3.3.1.1.3.2" xref="S3.Ex19.m1.3.3.1.1.3.2.cmml">ρ</mi><mo id="S3.Ex19.m1.3.3.1.1.3.1" xref="S3.Ex19.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex19.m1.3.3.1.1.3.3.2" xref="S3.Ex19.m1.3.3.1.1.3.3.1.cmml"><mo id="S3.Ex19.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.Ex19.m1.3.3.1.1.3.3.1.cmml">(</mo><mi id="S3.Ex19.m1.1.1" xref="S3.Ex19.m1.1.1.cmml">A</mi><mo id="S3.Ex19.m1.3.3.1.1.3.3.2.2" xref="S3.Ex19.m1.3.3.1.1.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex19.m1.2.2" xref="S3.Ex19.m1.2.2.cmml">ℬ</mi><mo id="S3.Ex19.m1.3.3.1.1.3.3.2.3" rspace="0.280em" stretchy="false" xref="S3.Ex19.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex19.m1.3.3.1.1.2" rspace="0.558em" xref="S3.Ex19.m1.3.3.1.1.2.cmml">≤</mo><mrow id="S3.Ex19.m1.3.3.1.1.1" xref="S3.Ex19.m1.3.3.1.1.1.cmml"><msub id="S3.Ex19.m1.3.3.1.1.1.3" xref="S3.Ex19.m1.3.3.1.1.1.3.cmml"><mi id="S3.Ex19.m1.3.3.1.1.1.3.2" xref="S3.Ex19.m1.3.3.1.1.1.3.2.cmml">D</mi><mo id="S3.Ex19.m1.3.3.1.1.1.3.3" xref="S3.Ex19.m1.3.3.1.1.1.3.3.cmml">∩</mo></msub><mo id="S3.Ex19.m1.3.3.1.1.1.2" xref="S3.Ex19.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex19.m1.3.3.1.1.1.1.1" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S3.Ex19.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex19.m1.3.3.1.1.1.1.1.1" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex19.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex19.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex19.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.Ex19.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex19.m1.3.3.1.2" lspace="0em" xref="S3.Ex19.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex19.m1.3b"><apply id="S3.Ex19.m1.3.3.1.1.cmml" xref="S3.Ex19.m1.3.3.1"><leq id="S3.Ex19.m1.3.3.1.1.2.cmml" xref="S3.Ex19.m1.3.3.1.1.2"></leq><apply id="S3.Ex19.m1.3.3.1.1.3.cmml" xref="S3.Ex19.m1.3.3.1.1.3"><times id="S3.Ex19.m1.3.3.1.1.3.1.cmml" xref="S3.Ex19.m1.3.3.1.1.3.1"></times><ci id="S3.Ex19.m1.3.3.1.1.3.2.cmml" xref="S3.Ex19.m1.3.3.1.1.3.2">𝜌</ci><interval closure="open" id="S3.Ex19.m1.3.3.1.1.3.3.1.cmml" xref="S3.Ex19.m1.3.3.1.1.3.3.2"><ci id="S3.Ex19.m1.1.1.cmml" xref="S3.Ex19.m1.1.1">𝐴</ci><ci id="S3.Ex19.m1.2.2.cmml" xref="S3.Ex19.m1.2.2">ℬ</ci></interval></apply><apply id="S3.Ex19.m1.3.3.1.1.1.cmml" xref="S3.Ex19.m1.3.3.1.1.1"><times id="S3.Ex19.m1.3.3.1.1.1.2.cmml" xref="S3.Ex19.m1.3.3.1.1.1.2"></times><apply id="S3.Ex19.m1.3.3.1.1.1.3.cmml" xref="S3.Ex19.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex19.m1.3.3.1.1.1.3.1.cmml" xref="S3.Ex19.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S3.Ex19.m1.3.3.1.1.1.3.2.cmml" xref="S3.Ex19.m1.3.3.1.1.1.3.2">𝐷</ci><intersect id="S3.Ex19.m1.3.3.1.1.1.3.3.cmml" xref="S3.Ex19.m1.3.3.1.1.1.3.3"></intersect></apply><apply id="S3.Ex19.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex19.m1.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S3.Ex19.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.1">conditional</csymbol><ci id="S3.Ex19.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex19.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S3.Ex19.m1.3.3.1.1.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m1.3c">\rho(A,\mathcal{B})\;\leq\;D_{\cap}(A\mid\mathcal{B}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m1.3d">italic_ρ ( italic_A , caligraphic_B ) ≤ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem22.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem22.p1.3.1">In other words, the cover complexity of a non-trivial set lower bounds its intersection complexity.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.8">Before proving the result, it is instructive to consider an example. Assume <math alttext="\Gamma=[N]\times[N]" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.2"><semantics id="S3.SS2.p1.1.m1.2a"><mrow id="S3.SS2.p1.1.m1.2.3" xref="S3.SS2.p1.1.m1.2.3.cmml"><mi id="S3.SS2.p1.1.m1.2.3.2" mathvariant="normal" xref="S3.SS2.p1.1.m1.2.3.2.cmml">Γ</mi><mo id="S3.SS2.p1.1.m1.2.3.1" xref="S3.SS2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S3.SS2.p1.1.m1.2.3.3" xref="S3.SS2.p1.1.m1.2.3.3.cmml"><mrow id="S3.SS2.p1.1.m1.2.3.3.2.2" xref="S3.SS2.p1.1.m1.2.3.3.2.1.cmml"><mo id="S3.SS2.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S3.SS2.p1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">N</mi><mo id="S3.SS2.p1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S3.SS2.p1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.SS2.p1.1.m1.2.3.3.1" rspace="0.222em" xref="S3.SS2.p1.1.m1.2.3.3.1.cmml">×</mo><mrow id="S3.SS2.p1.1.m1.2.3.3.3.2" xref="S3.SS2.p1.1.m1.2.3.3.3.1.cmml"><mo id="S3.SS2.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S3.SS2.p1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="S3.SS2.p1.1.m1.2.2" xref="S3.SS2.p1.1.m1.2.2.cmml">N</mi><mo id="S3.SS2.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S3.SS2.p1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.2b"><apply id="S3.SS2.p1.1.m1.2.3.cmml" xref="S3.SS2.p1.1.m1.2.3"><eq id="S3.SS2.p1.1.m1.2.3.1.cmml" xref="S3.SS2.p1.1.m1.2.3.1"></eq><ci id="S3.SS2.p1.1.m1.2.3.2.cmml" xref="S3.SS2.p1.1.m1.2.3.2">Γ</ci><apply id="S3.SS2.p1.1.m1.2.3.3.cmml" xref="S3.SS2.p1.1.m1.2.3.3"><times id="S3.SS2.p1.1.m1.2.3.3.1.cmml" xref="S3.SS2.p1.1.m1.2.3.3.1"></times><apply id="S3.SS2.p1.1.m1.2.3.3.2.1.cmml" xref="S3.SS2.p1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="S3.SS2.p1.1.m1.2.3.3.2.1.1.cmml" xref="S3.SS2.p1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝑁</ci></apply><apply id="S3.SS2.p1.1.m1.2.3.3.3.1.cmml" xref="S3.SS2.p1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="S3.SS2.p1.1.m1.2.3.3.3.1.1.cmml" xref="S3.SS2.p1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.p1.1.m1.2.2.cmml" xref="S3.SS2.p1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.2c">\Gamma=[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.2d">roman_Γ = [ italic_N ] × [ italic_N ]</annotation></semantics></math> and <math alttext="\mathcal{B}=\mathcal{R}_{N}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mrow id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.2.m2.1.1.2" xref="S3.SS2.p1.2.m2.1.1.2.cmml">ℬ</mi><mo id="S3.SS2.p1.2.m2.1.1.1" xref="S3.SS2.p1.2.m2.1.1.1.cmml">=</mo><msub id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.2.m2.1.1.3.2" xref="S3.SS2.p1.2.m2.1.1.3.2.cmml">ℛ</mi><mi id="S3.SS2.p1.2.m2.1.1.3.3" xref="S3.SS2.p1.2.m2.1.1.3.3.cmml">N</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><apply id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1"><eq id="S3.SS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1.1"></eq><ci id="S3.SS2.p1.2.m2.1.1.2.cmml" xref="S3.SS2.p1.2.m2.1.1.2">ℬ</ci><apply id="S3.SS2.p1.2.m2.1.1.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.3.1.cmml" xref="S3.SS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.2.m2.1.1.3.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.2">ℛ</ci><ci id="S3.SS2.p1.2.m2.1.1.3.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\mathcal{B}=\mathcal{R}_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">caligraphic_B = caligraphic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> are instantiated as in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS4" title="2.2.4 Combinatorial rectangles from communication complexity ‣ 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.2.4</span></a>, where we noted that <math alttext="D_{\cap}(G\mid\mathcal{R}_{N})" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mrow id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><msub id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml"><mi id="S3.SS2.p1.3.m3.1.1.3.2" xref="S3.SS2.p1.3.m3.1.1.3.2.cmml">D</mi><mo id="S3.SS2.p1.3.m3.1.1.3.3" xref="S3.SS2.p1.3.m3.1.1.3.3.cmml">∩</mo></msub><mo id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p1.3.m3.1.1.1.1" xref="S3.SS2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow 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xref="S3.SS2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.3.m3.1.1.3.2.cmml" xref="S3.SS2.p1.3.m3.1.1.3.2">𝐷</ci><intersect id="S3.SS2.p1.3.m3.1.1.3.3.cmml" xref="S3.SS2.p1.3.m3.1.1.3.3"></intersect></apply><apply id="S3.SS2.p1.3.m3.1.1.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1"><csymbol cd="latexml" id="S3.SS2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.1">conditional</csymbol><ci id="S3.SS2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.2">𝐺</ci><apply id="S3.SS2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.3.2">ℛ</ci><ci id="S3.SS2.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p1.3.m3.1.1.1.1.1.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">D_{\cap}(G\mid\mathcal{R}_{N})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT )</annotation></semantics></math> is always zero. Indeed, <math alttext="\rho(G,\mathcal{R}_{N})=0" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.2"><semantics id="S3.SS2.p1.4.m4.2a"><mrow id="S3.SS2.p1.4.m4.2.2" xref="S3.SS2.p1.4.m4.2.2.cmml"><mrow id="S3.SS2.p1.4.m4.2.2.1" xref="S3.SS2.p1.4.m4.2.2.1.cmml"><mi id="S3.SS2.p1.4.m4.2.2.1.3" xref="S3.SS2.p1.4.m4.2.2.1.3.cmml">ρ</mi><mo id="S3.SS2.p1.4.m4.2.2.1.2" xref="S3.SS2.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS2.p1.4.m4.2.2.1.1.1" xref="S3.SS2.p1.4.m4.2.2.1.1.2.cmml"><mo id="S3.SS2.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S3.SS2.p1.4.m4.2.2.1.1.2.cmml">(</mo><mi id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml">G</mi><mo id="S3.SS2.p1.4.m4.2.2.1.1.1.3" xref="S3.SS2.p1.4.m4.2.2.1.1.2.cmml">,</mo><msub id="S3.SS2.p1.4.m4.2.2.1.1.1.1" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.4.m4.2.2.1.1.1.1.2" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1.2.cmml">ℛ</mi><mi id="S3.SS2.p1.4.m4.2.2.1.1.1.1.3" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1.3.cmml">N</mi></msub><mo id="S3.SS2.p1.4.m4.2.2.1.1.1.4" stretchy="false" xref="S3.SS2.p1.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p1.4.m4.2.2.2" xref="S3.SS2.p1.4.m4.2.2.2.cmml">=</mo><mn id="S3.SS2.p1.4.m4.2.2.3" xref="S3.SS2.p1.4.m4.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.2b"><apply id="S3.SS2.p1.4.m4.2.2.cmml" xref="S3.SS2.p1.4.m4.2.2"><eq id="S3.SS2.p1.4.m4.2.2.2.cmml" xref="S3.SS2.p1.4.m4.2.2.2"></eq><apply id="S3.SS2.p1.4.m4.2.2.1.cmml" xref="S3.SS2.p1.4.m4.2.2.1"><times id="S3.SS2.p1.4.m4.2.2.1.2.cmml" xref="S3.SS2.p1.4.m4.2.2.1.2"></times><ci id="S3.SS2.p1.4.m4.2.2.1.3.cmml" xref="S3.SS2.p1.4.m4.2.2.1.3">𝜌</ci><interval closure="open" id="S3.SS2.p1.4.m4.2.2.1.1.2.cmml" xref="S3.SS2.p1.4.m4.2.2.1.1.1"><ci id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">𝐺</ci><apply id="S3.SS2.p1.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1.2">ℛ</ci><ci id="S3.SS2.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S3.SS2.p1.4.m4.2.2.1.1.1.1.3">𝑁</ci></apply></interval></apply><cn id="S3.SS2.p1.4.m4.2.2.3.cmml" type="integer" xref="S3.SS2.p1.4.m4.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.2c">\rho(G,\mathcal{R}_{N})=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.2d">italic_ρ ( italic_G , caligraphic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> for every non-trivial <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.2"><semantics id="S3.SS2.p1.5.m5.2a"><mrow id="S3.SS2.p1.5.m5.2.3" xref="S3.SS2.p1.5.m5.2.3.cmml"><mi id="S3.SS2.p1.5.m5.2.3.2" xref="S3.SS2.p1.5.m5.2.3.2.cmml">G</mi><mo id="S3.SS2.p1.5.m5.2.3.1" xref="S3.SS2.p1.5.m5.2.3.1.cmml">⊆</mo><mrow id="S3.SS2.p1.5.m5.2.3.3" xref="S3.SS2.p1.5.m5.2.3.3.cmml"><mrow id="S3.SS2.p1.5.m5.2.3.3.2.2" xref="S3.SS2.p1.5.m5.2.3.3.2.1.cmml"><mo id="S3.SS2.p1.5.m5.2.3.3.2.2.1" stretchy="false" xref="S3.SS2.p1.5.m5.2.3.3.2.1.1.cmml">[</mo><mi id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml">N</mi><mo id="S3.SS2.p1.5.m5.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S3.SS2.p1.5.m5.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.SS2.p1.5.m5.2.3.3.1" rspace="0.222em" xref="S3.SS2.p1.5.m5.2.3.3.1.cmml">×</mo><mrow id="S3.SS2.p1.5.m5.2.3.3.3.2" xref="S3.SS2.p1.5.m5.2.3.3.3.1.cmml"><mo id="S3.SS2.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S3.SS2.p1.5.m5.2.3.3.3.1.1.cmml">[</mo><mi id="S3.SS2.p1.5.m5.2.2" xref="S3.SS2.p1.5.m5.2.2.cmml">N</mi><mo id="S3.SS2.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S3.SS2.p1.5.m5.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.2b"><apply id="S3.SS2.p1.5.m5.2.3.cmml" xref="S3.SS2.p1.5.m5.2.3"><subset id="S3.SS2.p1.5.m5.2.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.1"></subset><ci id="S3.SS2.p1.5.m5.2.3.2.cmml" xref="S3.SS2.p1.5.m5.2.3.2">𝐺</ci><apply id="S3.SS2.p1.5.m5.2.3.3.cmml" xref="S3.SS2.p1.5.m5.2.3.3"><times id="S3.SS2.p1.5.m5.2.3.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.1"></times><apply id="S3.SS2.p1.5.m5.2.3.3.2.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.2.2"><csymbol cd="latexml" id="S3.SS2.p1.5.m5.2.3.3.2.1.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1">𝑁</ci></apply><apply id="S3.SS2.p1.5.m5.2.3.3.3.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.3.2"><csymbol cd="latexml" id="S3.SS2.p1.5.m5.2.3.3.3.1.1.cmml" xref="S3.SS2.p1.5.m5.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.p1.5.m5.2.2.cmml" xref="S3.SS2.p1.5.m5.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>, since in the corresponding cover graph <math alttext="\Phi_{G,\mathcal{R}_{N}}" class="ltx_Math" display="inline" id="S3.SS2.p1.6.m6.2"><semantics id="S3.SS2.p1.6.m6.2a"><msub id="S3.SS2.p1.6.m6.2.3" xref="S3.SS2.p1.6.m6.2.3.cmml"><mi id="S3.SS2.p1.6.m6.2.3.2" mathvariant="normal" xref="S3.SS2.p1.6.m6.2.3.2.cmml">Φ</mi><mrow id="S3.SS2.p1.6.m6.2.2.2.2" xref="S3.SS2.p1.6.m6.2.2.2.3.cmml"><mi id="S3.SS2.p1.6.m6.1.1.1.1" xref="S3.SS2.p1.6.m6.1.1.1.1.cmml">G</mi><mo id="S3.SS2.p1.6.m6.2.2.2.2.2" xref="S3.SS2.p1.6.m6.2.2.2.3.cmml">,</mo><msub id="S3.SS2.p1.6.m6.2.2.2.2.1" xref="S3.SS2.p1.6.m6.2.2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.6.m6.2.2.2.2.1.2" xref="S3.SS2.p1.6.m6.2.2.2.2.1.2.cmml">ℛ</mi><mi id="S3.SS2.p1.6.m6.2.2.2.2.1.3" xref="S3.SS2.p1.6.m6.2.2.2.2.1.3.cmml">N</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m6.2b"><apply id="S3.SS2.p1.6.m6.2.3.cmml" xref="S3.SS2.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.2.3.1.cmml" xref="S3.SS2.p1.6.m6.2.3">subscript</csymbol><ci id="S3.SS2.p1.6.m6.2.3.2.cmml" xref="S3.SS2.p1.6.m6.2.3.2">Φ</ci><list id="S3.SS2.p1.6.m6.2.2.2.3.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2"><ci id="S3.SS2.p1.6.m6.1.1.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1.1.1">𝐺</ci><apply id="S3.SS2.p1.6.m6.2.2.2.2.1.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.2.2.2.2.1.1.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S3.SS2.p1.6.m6.2.2.2.2.1.2.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.1.2">ℛ</ci><ci id="S3.SS2.p1.6.m6.2.2.2.2.1.3.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.1.3">𝑁</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m6.2c">\Phi_{G,\mathcal{R}_{N}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.6.m6.2d">roman_Φ start_POSTSUBSCRIPT italic_G , caligraphic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> the vertex set <math alttext="V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS2.p1.7.m7.1"><semantics id="S3.SS2.p1.7.m7.1a"><msub id="S3.SS2.p1.7.m7.1.1" xref="S3.SS2.p1.7.m7.1.1.cmml"><mi id="S3.SS2.p1.7.m7.1.1.2" xref="S3.SS2.p1.7.m7.1.1.2.cmml">V</mi><mi id="S3.SS2.p1.7.m7.1.1.3" xref="S3.SS2.p1.7.m7.1.1.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.7.m7.1b"><apply id="S3.SS2.p1.7.m7.1.1.cmml" xref="S3.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.7.m7.1.1.1.cmml" xref="S3.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS2.p1.7.m7.1.1.2.cmml" xref="S3.SS2.p1.7.m7.1.1.2">𝑉</ci><ci id="S3.SS2.p1.7.m7.1.1.3.cmml" xref="S3.SS2.p1.7.m7.1.1.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.7.m7.1c">V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.7.m7.1d">italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math> is empty (observe that if a semi-filter is above some <math alttext="a\in G" class="ltx_Math" display="inline" id="S3.SS2.p1.8.m8.1"><semantics id="S3.SS2.p1.8.m8.1a"><mrow id="S3.SS2.p1.8.m8.1.1" xref="S3.SS2.p1.8.m8.1.1.cmml"><mi id="S3.SS2.p1.8.m8.1.1.2" xref="S3.SS2.p1.8.m8.1.1.2.cmml">a</mi><mo id="S3.SS2.p1.8.m8.1.1.1" xref="S3.SS2.p1.8.m8.1.1.1.cmml">∈</mo><mi id="S3.SS2.p1.8.m8.1.1.3" xref="S3.SS2.p1.8.m8.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.8.m8.1b"><apply id="S3.SS2.p1.8.m8.1.1.cmml" xref="S3.SS2.p1.8.m8.1.1"><in id="S3.SS2.p1.8.m8.1.1.1.cmml" xref="S3.SS2.p1.8.m8.1.1.1"></in><ci id="S3.SS2.p1.8.m8.1.1.2.cmml" xref="S3.SS2.p1.8.m8.1.1.2">𝑎</ci><ci id="S3.SS2.p1.8.m8.1.1.3.cmml" xref="S3.SS2.p1.8.m8.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.8.m8.1c">a\in G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.8.m8.1d">italic_a ∈ italic_G</annotation></semantics></math>, then it must contain the empty set, which is contradictory).</p> </div> <div class="ltx_proof" id="S3.SS2.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.1.p1"> <p class="ltx_p" id="S3.SS2.1.p1.3">Let <math alttext="|\mathcal{B}|=m" class="ltx_Math" display="inline" id="S3.SS2.1.p1.1.m1.1"><semantics id="S3.SS2.1.p1.1.m1.1a"><mrow id="S3.SS2.1.p1.1.m1.1.2" xref="S3.SS2.1.p1.1.m1.1.2.cmml"><mrow id="S3.SS2.1.p1.1.m1.1.2.2.2" xref="S3.SS2.1.p1.1.m1.1.2.2.1.cmml"><mo id="S3.SS2.1.p1.1.m1.1.2.2.2.1" stretchy="false" xref="S3.SS2.1.p1.1.m1.1.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.1.m1.1.1" xref="S3.SS2.1.p1.1.m1.1.1.cmml">ℬ</mi><mo id="S3.SS2.1.p1.1.m1.1.2.2.2.2" stretchy="false" xref="S3.SS2.1.p1.1.m1.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.SS2.1.p1.1.m1.1.2.1" xref="S3.SS2.1.p1.1.m1.1.2.1.cmml">=</mo><mi id="S3.SS2.1.p1.1.m1.1.2.3" xref="S3.SS2.1.p1.1.m1.1.2.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.1.m1.1b"><apply id="S3.SS2.1.p1.1.m1.1.2.cmml" xref="S3.SS2.1.p1.1.m1.1.2"><eq id="S3.SS2.1.p1.1.m1.1.2.1.cmml" xref="S3.SS2.1.p1.1.m1.1.2.1"></eq><apply id="S3.SS2.1.p1.1.m1.1.2.2.1.cmml" xref="S3.SS2.1.p1.1.m1.1.2.2.2"><abs id="S3.SS2.1.p1.1.m1.1.2.2.1.1.cmml" xref="S3.SS2.1.p1.1.m1.1.2.2.2.1"></abs><ci id="S3.SS2.1.p1.1.m1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.1.1">ℬ</ci></apply><ci id="S3.SS2.1.p1.1.m1.1.2.3.cmml" xref="S3.SS2.1.p1.1.m1.1.2.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.1.m1.1c">|\mathcal{B}|=m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.1.m1.1d">| caligraphic_B | = italic_m</annotation></semantics></math> and <math alttext="D_{\cap}(A\mid\mathcal{B})=k" class="ltx_Math" display="inline" id="S3.SS2.1.p1.2.m2.1"><semantics id="S3.SS2.1.p1.2.m2.1a"><mrow id="S3.SS2.1.p1.2.m2.1.1" xref="S3.SS2.1.p1.2.m2.1.1.cmml"><mrow id="S3.SS2.1.p1.2.m2.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.cmml"><msub id="S3.SS2.1.p1.2.m2.1.1.1.3" xref="S3.SS2.1.p1.2.m2.1.1.1.3.cmml"><mi id="S3.SS2.1.p1.2.m2.1.1.1.3.2" xref="S3.SS2.1.p1.2.m2.1.1.1.3.2.cmml">D</mi><mo id="S3.SS2.1.p1.2.m2.1.1.1.3.3" xref="S3.SS2.1.p1.2.m2.1.1.1.3.3.cmml">∩</mo></msub><mo id="S3.SS2.1.p1.2.m2.1.1.1.2" xref="S3.SS2.1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.2.m2.1.1.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS2.1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.2" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.3" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS2.1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.1.p1.2.m2.1.1.2" xref="S3.SS2.1.p1.2.m2.1.1.2.cmml">=</mo><mi id="S3.SS2.1.p1.2.m2.1.1.3" xref="S3.SS2.1.p1.2.m2.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.2.m2.1b"><apply id="S3.SS2.1.p1.2.m2.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1"><eq id="S3.SS2.1.p1.2.m2.1.1.2.cmml" xref="S3.SS2.1.p1.2.m2.1.1.2"></eq><apply id="S3.SS2.1.p1.2.m2.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1"><times id="S3.SS2.1.p1.2.m2.1.1.1.2.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.2"></times><apply id="S3.SS2.1.p1.2.m2.1.1.1.3.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.2.m2.1.1.1.3.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.3">subscript</csymbol><ci id="S3.SS2.1.p1.2.m2.1.1.1.3.2.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.3.2">𝐷</ci><intersect id="S3.SS2.1.p1.2.m2.1.1.1.3.3.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.3.3"></intersect></apply><apply id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.1">conditional</csymbol><ci id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1.1.1.1.3">ℬ</ci></apply></apply><ci id="S3.SS2.1.p1.2.m2.1.1.3.cmml" xref="S3.SS2.1.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.2.m2.1c">D_{\cap}(A\mid\mathcal{B})=k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_k</annotation></semantics></math>. Assume toward a contradiction that <math alttext="k<\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.3.m3.2"><semantics id="S3.SS2.1.p1.3.m3.2a"><mrow id="S3.SS2.1.p1.3.m3.2.3" xref="S3.SS2.1.p1.3.m3.2.3.cmml"><mi id="S3.SS2.1.p1.3.m3.2.3.2" xref="S3.SS2.1.p1.3.m3.2.3.2.cmml">k</mi><mo id="S3.SS2.1.p1.3.m3.2.3.1" xref="S3.SS2.1.p1.3.m3.2.3.1.cmml"><</mo><mrow id="S3.SS2.1.p1.3.m3.2.3.3" xref="S3.SS2.1.p1.3.m3.2.3.3.cmml"><mi id="S3.SS2.1.p1.3.m3.2.3.3.2" xref="S3.SS2.1.p1.3.m3.2.3.3.2.cmml">ρ</mi><mo id="S3.SS2.1.p1.3.m3.2.3.3.1" xref="S3.SS2.1.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.3.m3.2.3.3.3.2" xref="S3.SS2.1.p1.3.m3.2.3.3.3.1.cmml"><mo id="S3.SS2.1.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S3.SS2.1.p1.3.m3.1.1" xref="S3.SS2.1.p1.3.m3.1.1.cmml">A</mi><mo id="S3.SS2.1.p1.3.m3.2.3.3.3.2.2" xref="S3.SS2.1.p1.3.m3.2.3.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.3.m3.2.2" xref="S3.SS2.1.p1.3.m3.2.2.cmml">ℬ</mi><mo id="S3.SS2.1.p1.3.m3.2.3.3.3.2.3" stretchy="false" xref="S3.SS2.1.p1.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.3.m3.2b"><apply id="S3.SS2.1.p1.3.m3.2.3.cmml" xref="S3.SS2.1.p1.3.m3.2.3"><lt id="S3.SS2.1.p1.3.m3.2.3.1.cmml" xref="S3.SS2.1.p1.3.m3.2.3.1"></lt><ci id="S3.SS2.1.p1.3.m3.2.3.2.cmml" xref="S3.SS2.1.p1.3.m3.2.3.2">𝑘</ci><apply id="S3.SS2.1.p1.3.m3.2.3.3.cmml" xref="S3.SS2.1.p1.3.m3.2.3.3"><times id="S3.SS2.1.p1.3.m3.2.3.3.1.cmml" xref="S3.SS2.1.p1.3.m3.2.3.3.1"></times><ci id="S3.SS2.1.p1.3.m3.2.3.3.2.cmml" xref="S3.SS2.1.p1.3.m3.2.3.3.2">𝜌</ci><interval closure="open" id="S3.SS2.1.p1.3.m3.2.3.3.3.1.cmml" xref="S3.SS2.1.p1.3.m3.2.3.3.3.2"><ci id="S3.SS2.1.p1.3.m3.1.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1">𝐴</ci><ci id="S3.SS2.1.p1.3.m3.2.2.cmml" xref="S3.SS2.1.p1.3.m3.2.2">ℬ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.3.m3.2c">k<\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.3.m3.2d">italic_k < italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}=A" class="ltx_Math" display="block" id="S3.E4.m1.6"><semantics id="S3.E4.m1.6a"><mrow id="S3.E4.m1.6.6" xref="S3.E4.m1.6.6.cmml"><mrow id="S3.E4.m1.6.6.4.4" xref="S3.E4.m1.6.6.4.5.cmml"><msup id="S3.E4.m1.3.3.1.1.1" xref="S3.E4.m1.3.3.1.1.1.cmml"><mi id="S3.E4.m1.3.3.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.2.cmml">C</mi><mn id="S3.E4.m1.3.3.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.3.cmml">1</mn></msup><mo id="S3.E4.m1.6.6.4.4.5" xref="S3.E4.m1.6.6.4.5.cmml">,</mo><mi id="S3.E4.m1.1.1" mathvariant="normal" xref="S3.E4.m1.1.1.cmml">…</mi><mo id="S3.E4.m1.6.6.4.4.6" xref="S3.E4.m1.6.6.4.5.cmml">,</mo><msup id="S3.E4.m1.4.4.2.2.2" xref="S3.E4.m1.4.4.2.2.2.cmml"><mi id="S3.E4.m1.4.4.2.2.2.2" xref="S3.E4.m1.4.4.2.2.2.2.cmml">C</mi><mi id="S3.E4.m1.4.4.2.2.2.3" xref="S3.E4.m1.4.4.2.2.2.3.cmml">m</mi></msup><mo id="S3.E4.m1.6.6.4.4.7" xref="S3.E4.m1.6.6.4.5.cmml">,</mo><msup id="S3.E4.m1.5.5.3.3.3" xref="S3.E4.m1.5.5.3.3.3.cmml"><mi id="S3.E4.m1.5.5.3.3.3.2" xref="S3.E4.m1.5.5.3.3.3.2.cmml">C</mi><mrow id="S3.E4.m1.5.5.3.3.3.3" xref="S3.E4.m1.5.5.3.3.3.3.cmml"><mi id="S3.E4.m1.5.5.3.3.3.3.2" xref="S3.E4.m1.5.5.3.3.3.3.2.cmml">m</mi><mo id="S3.E4.m1.5.5.3.3.3.3.1" xref="S3.E4.m1.5.5.3.3.3.3.1.cmml">+</mo><mn id="S3.E4.m1.5.5.3.3.3.3.3" xref="S3.E4.m1.5.5.3.3.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.E4.m1.6.6.4.4.8" xref="S3.E4.m1.6.6.4.5.cmml">,</mo><mi id="S3.E4.m1.2.2" mathvariant="normal" xref="S3.E4.m1.2.2.cmml">…</mi><mo id="S3.E4.m1.6.6.4.4.9" xref="S3.E4.m1.6.6.4.5.cmml">,</mo><msup id="S3.E4.m1.6.6.4.4.4" xref="S3.E4.m1.6.6.4.4.4.cmml"><mi id="S3.E4.m1.6.6.4.4.4.2" xref="S3.E4.m1.6.6.4.4.4.2.cmml">C</mi><mrow id="S3.E4.m1.6.6.4.4.4.3" xref="S3.E4.m1.6.6.4.4.4.3.cmml"><mi id="S3.E4.m1.6.6.4.4.4.3.2" xref="S3.E4.m1.6.6.4.4.4.3.2.cmml">m</mi><mo id="S3.E4.m1.6.6.4.4.4.3.1" xref="S3.E4.m1.6.6.4.4.4.3.1.cmml">+</mo><mi id="S3.E4.m1.6.6.4.4.4.3.3" xref="S3.E4.m1.6.6.4.4.4.3.3.cmml">t</mi></mrow></msup></mrow><mo id="S3.E4.m1.6.6.5" xref="S3.E4.m1.6.6.5.cmml">=</mo><mi id="S3.E4.m1.6.6.6" xref="S3.E4.m1.6.6.6.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.E4.m1.6b"><apply id="S3.E4.m1.6.6.cmml" xref="S3.E4.m1.6.6"><eq id="S3.E4.m1.6.6.5.cmml" xref="S3.E4.m1.6.6.5"></eq><list id="S3.E4.m1.6.6.4.5.cmml" xref="S3.E4.m1.6.6.4.4"><apply id="S3.E4.m1.3.3.1.1.1.cmml" xref="S3.E4.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.E4.m1.3.3.1.1.1.1.cmml" xref="S3.E4.m1.3.3.1.1.1">superscript</csymbol><ci id="S3.E4.m1.3.3.1.1.1.2.cmml" xref="S3.E4.m1.3.3.1.1.1.2">𝐶</ci><cn id="S3.E4.m1.3.3.1.1.1.3.cmml" type="integer" xref="S3.E4.m1.3.3.1.1.1.3">1</cn></apply><ci id="S3.E4.m1.1.1.cmml" xref="S3.E4.m1.1.1">…</ci><apply id="S3.E4.m1.4.4.2.2.2.cmml" xref="S3.E4.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S3.E4.m1.4.4.2.2.2.1.cmml" xref="S3.E4.m1.4.4.2.2.2">superscript</csymbol><ci id="S3.E4.m1.4.4.2.2.2.2.cmml" xref="S3.E4.m1.4.4.2.2.2.2">𝐶</ci><ci id="S3.E4.m1.4.4.2.2.2.3.cmml" xref="S3.E4.m1.4.4.2.2.2.3">𝑚</ci></apply><apply id="S3.E4.m1.5.5.3.3.3.cmml" xref="S3.E4.m1.5.5.3.3.3"><csymbol cd="ambiguous" id="S3.E4.m1.5.5.3.3.3.1.cmml" xref="S3.E4.m1.5.5.3.3.3">superscript</csymbol><ci id="S3.E4.m1.5.5.3.3.3.2.cmml" xref="S3.E4.m1.5.5.3.3.3.2">𝐶</ci><apply id="S3.E4.m1.5.5.3.3.3.3.cmml" xref="S3.E4.m1.5.5.3.3.3.3"><plus id="S3.E4.m1.5.5.3.3.3.3.1.cmml" xref="S3.E4.m1.5.5.3.3.3.3.1"></plus><ci id="S3.E4.m1.5.5.3.3.3.3.2.cmml" xref="S3.E4.m1.5.5.3.3.3.3.2">𝑚</ci><cn id="S3.E4.m1.5.5.3.3.3.3.3.cmml" type="integer" xref="S3.E4.m1.5.5.3.3.3.3.3">1</cn></apply></apply><ci id="S3.E4.m1.2.2.cmml" xref="S3.E4.m1.2.2">…</ci><apply id="S3.E4.m1.6.6.4.4.4.cmml" xref="S3.E4.m1.6.6.4.4.4"><csymbol cd="ambiguous" id="S3.E4.m1.6.6.4.4.4.1.cmml" xref="S3.E4.m1.6.6.4.4.4">superscript</csymbol><ci id="S3.E4.m1.6.6.4.4.4.2.cmml" xref="S3.E4.m1.6.6.4.4.4.2">𝐶</ci><apply id="S3.E4.m1.6.6.4.4.4.3.cmml" xref="S3.E4.m1.6.6.4.4.4.3"><plus id="S3.E4.m1.6.6.4.4.4.3.1.cmml" xref="S3.E4.m1.6.6.4.4.4.3.1"></plus><ci id="S3.E4.m1.6.6.4.4.4.3.2.cmml" xref="S3.E4.m1.6.6.4.4.4.3.2">𝑚</ci><ci id="S3.E4.m1.6.6.4.4.4.3.3.cmml" xref="S3.E4.m1.6.6.4.4.4.3.3">𝑡</ci></apply></apply></list><ci id="S3.E4.m1.6.6.6.cmml" xref="S3.E4.m1.6.6.6">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.6c">C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}=A</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.6d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_C start_POSTSUPERSCRIPT italic_m + 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT = italic_A</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.1.p1.10">be an extended sequence of complexity <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.1.p1.4.m1.1"><semantics id="S3.SS2.1.p1.4.m1.1a"><mi id="S3.SS2.1.p1.4.m1.1.1" xref="S3.SS2.1.p1.4.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.4.m1.1b"><ci id="S3.SS2.1.p1.4.m1.1.1.cmml" xref="S3.SS2.1.p1.4.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.4.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.4.m1.1d">italic_t</annotation></semantics></math> that generates <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.1.p1.5.m2.1"><semantics id="S3.SS2.1.p1.5.m2.1a"><mi id="S3.SS2.1.p1.5.m2.1.1" xref="S3.SS2.1.p1.5.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.5.m2.1b"><ci id="S3.SS2.1.p1.5.m2.1.1.cmml" xref="S3.SS2.1.p1.5.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.5.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.5.m2.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.6.m3.1"><semantics id="S3.SS2.1.p1.6.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.6.m3.1.1" xref="S3.SS2.1.p1.6.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.6.m3.1b"><ci id="S3.SS2.1.p1.6.m3.1.1.cmml" xref="S3.SS2.1.p1.6.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.6.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.6.m3.1d">caligraphic_B</annotation></semantics></math>, and suppose it has intersection complexity <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.1.p1.7.m4.1"><semantics id="S3.SS2.1.p1.7.m4.1a"><mi id="S3.SS2.1.p1.7.m4.1.1" xref="S3.SS2.1.p1.7.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.7.m4.1b"><ci id="S3.SS2.1.p1.7.m4.1.1.cmml" xref="S3.SS2.1.p1.7.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.7.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.7.m4.1d">italic_k</annotation></semantics></math>. Let <math alttext="U\stackrel{{\scriptstyle\rm def}}{{=}}A^{c}=\Gamma\setminus A" class="ltx_Math" display="inline" id="S3.SS2.1.p1.8.m5.1"><semantics id="S3.SS2.1.p1.8.m5.1a"><mrow id="S3.SS2.1.p1.8.m5.1.1" xref="S3.SS2.1.p1.8.m5.1.1.cmml"><mi id="S3.SS2.1.p1.8.m5.1.1.2" xref="S3.SS2.1.p1.8.m5.1.1.2.cmml">U</mi><mover id="S3.SS2.1.p1.8.m5.1.1.3" xref="S3.SS2.1.p1.8.m5.1.1.3.cmml"><mo id="S3.SS2.1.p1.8.m5.1.1.3.2" xref="S3.SS2.1.p1.8.m5.1.1.3.2.cmml">=</mo><mi id="S3.SS2.1.p1.8.m5.1.1.3.3" xref="S3.SS2.1.p1.8.m5.1.1.3.3.cmml">def</mi></mover><msup id="S3.SS2.1.p1.8.m5.1.1.4" xref="S3.SS2.1.p1.8.m5.1.1.4.cmml"><mi id="S3.SS2.1.p1.8.m5.1.1.4.2" xref="S3.SS2.1.p1.8.m5.1.1.4.2.cmml">A</mi><mi id="S3.SS2.1.p1.8.m5.1.1.4.3" xref="S3.SS2.1.p1.8.m5.1.1.4.3.cmml">c</mi></msup><mo id="S3.SS2.1.p1.8.m5.1.1.5" xref="S3.SS2.1.p1.8.m5.1.1.5.cmml">=</mo><mrow id="S3.SS2.1.p1.8.m5.1.1.6" xref="S3.SS2.1.p1.8.m5.1.1.6.cmml"><mi id="S3.SS2.1.p1.8.m5.1.1.6.2" mathvariant="normal" xref="S3.SS2.1.p1.8.m5.1.1.6.2.cmml">Γ</mi><mo id="S3.SS2.1.p1.8.m5.1.1.6.1" xref="S3.SS2.1.p1.8.m5.1.1.6.1.cmml">∖</mo><mi id="S3.SS2.1.p1.8.m5.1.1.6.3" xref="S3.SS2.1.p1.8.m5.1.1.6.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.8.m5.1b"><apply id="S3.SS2.1.p1.8.m5.1.1.cmml" xref="S3.SS2.1.p1.8.m5.1.1"><and id="S3.SS2.1.p1.8.m5.1.1a.cmml" xref="S3.SS2.1.p1.8.m5.1.1"></and><apply id="S3.SS2.1.p1.8.m5.1.1b.cmml" xref="S3.SS2.1.p1.8.m5.1.1"><apply id="S3.SS2.1.p1.8.m5.1.1.3.cmml" xref="S3.SS2.1.p1.8.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m5.1.1.3.1.cmml" xref="S3.SS2.1.p1.8.m5.1.1.3">superscript</csymbol><eq id="S3.SS2.1.p1.8.m5.1.1.3.2.cmml" xref="S3.SS2.1.p1.8.m5.1.1.3.2"></eq><ci id="S3.SS2.1.p1.8.m5.1.1.3.3.cmml" xref="S3.SS2.1.p1.8.m5.1.1.3.3">def</ci></apply><ci id="S3.SS2.1.p1.8.m5.1.1.2.cmml" xref="S3.SS2.1.p1.8.m5.1.1.2">𝑈</ci><apply id="S3.SS2.1.p1.8.m5.1.1.4.cmml" xref="S3.SS2.1.p1.8.m5.1.1.4"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m5.1.1.4.1.cmml" xref="S3.SS2.1.p1.8.m5.1.1.4">superscript</csymbol><ci id="S3.SS2.1.p1.8.m5.1.1.4.2.cmml" xref="S3.SS2.1.p1.8.m5.1.1.4.2">𝐴</ci><ci id="S3.SS2.1.p1.8.m5.1.1.4.3.cmml" xref="S3.SS2.1.p1.8.m5.1.1.4.3">𝑐</ci></apply></apply><apply id="S3.SS2.1.p1.8.m5.1.1c.cmml" xref="S3.SS2.1.p1.8.m5.1.1"><eq id="S3.SS2.1.p1.8.m5.1.1.5.cmml" xref="S3.SS2.1.p1.8.m5.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.1.p1.8.m5.1.1.4.cmml" id="S3.SS2.1.p1.8.m5.1.1d.cmml" xref="S3.SS2.1.p1.8.m5.1.1"></share><apply id="S3.SS2.1.p1.8.m5.1.1.6.cmml" xref="S3.SS2.1.p1.8.m5.1.1.6"><setdiff id="S3.SS2.1.p1.8.m5.1.1.6.1.cmml" xref="S3.SS2.1.p1.8.m5.1.1.6.1"></setdiff><ci id="S3.SS2.1.p1.8.m5.1.1.6.2.cmml" xref="S3.SS2.1.p1.8.m5.1.1.6.2">Γ</ci><ci id="S3.SS2.1.p1.8.m5.1.1.6.3.cmml" xref="S3.SS2.1.p1.8.m5.1.1.6.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.8.m5.1c">U\stackrel{{\scriptstyle\rm def}}{{=}}A^{c}=\Gamma\setminus A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.8.m5.1d">italic_U start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = roman_Γ ∖ italic_A</annotation></semantics></math>. Recall that, by assumption, both <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.1.p1.9.m6.1"><semantics id="S3.SS2.1.p1.9.m6.1a"><mi id="S3.SS2.1.p1.9.m6.1.1" xref="S3.SS2.1.p1.9.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.9.m6.1b"><ci id="S3.SS2.1.p1.9.m6.1.1.cmml" xref="S3.SS2.1.p1.9.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.9.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.9.m6.1d">italic_A</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="S3.SS2.1.p1.10.m7.1"><semantics id="S3.SS2.1.p1.10.m7.1a"><mi id="S3.SS2.1.p1.10.m7.1.1" xref="S3.SS2.1.p1.10.m7.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.10.m7.1b"><ci id="S3.SS2.1.p1.10.m7.1.1.cmml" xref="S3.SS2.1.p1.10.m7.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.10.m7.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.10.m7.1d">italic_U</annotation></semantics></math> are non-empty. Consider the corresponding relativized sequence</p> <table class="ltx_equation ltx_eqn_table" id="S3.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{1}_{U},\ldots,C^{m}_{U},C^{m+1}_{U},\ldots,C^{m+t}_{U}=\emptyset." class="ltx_Math" display="block" id="S3.E5.m1.3"><semantics id="S3.E5.m1.3a"><mrow id="S3.E5.m1.3.3.1" xref="S3.E5.m1.3.3.1.1.cmml"><mrow id="S3.E5.m1.3.3.1.1" xref="S3.E5.m1.3.3.1.1.cmml"><mrow id="S3.E5.m1.3.3.1.1.4.4" xref="S3.E5.m1.3.3.1.1.4.5.cmml"><msubsup id="S3.E5.m1.3.3.1.1.1.1.1" xref="S3.E5.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.E5.m1.3.3.1.1.1.1.1.2.2" xref="S3.E5.m1.3.3.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.E5.m1.3.3.1.1.1.1.1.3" xref="S3.E5.m1.3.3.1.1.1.1.1.3.cmml">U</mi><mn id="S3.E5.m1.3.3.1.1.1.1.1.2.3" xref="S3.E5.m1.3.3.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S3.E5.m1.3.3.1.1.4.4.5" xref="S3.E5.m1.3.3.1.1.4.5.cmml">,</mo><mi id="S3.E5.m1.1.1" mathvariant="normal" xref="S3.E5.m1.1.1.cmml">…</mi><mo id="S3.E5.m1.3.3.1.1.4.4.6" xref="S3.E5.m1.3.3.1.1.4.5.cmml">,</mo><msubsup id="S3.E5.m1.3.3.1.1.2.2.2" xref="S3.E5.m1.3.3.1.1.2.2.2.cmml"><mi id="S3.E5.m1.3.3.1.1.2.2.2.2.2" xref="S3.E5.m1.3.3.1.1.2.2.2.2.2.cmml">C</mi><mi id="S3.E5.m1.3.3.1.1.2.2.2.3" xref="S3.E5.m1.3.3.1.1.2.2.2.3.cmml">U</mi><mi id="S3.E5.m1.3.3.1.1.2.2.2.2.3" xref="S3.E5.m1.3.3.1.1.2.2.2.2.3.cmml">m</mi></msubsup><mo id="S3.E5.m1.3.3.1.1.4.4.7" xref="S3.E5.m1.3.3.1.1.4.5.cmml">,</mo><msubsup id="S3.E5.m1.3.3.1.1.3.3.3" xref="S3.E5.m1.3.3.1.1.3.3.3.cmml"><mi id="S3.E5.m1.3.3.1.1.3.3.3.2.2" xref="S3.E5.m1.3.3.1.1.3.3.3.2.2.cmml">C</mi><mi id="S3.E5.m1.3.3.1.1.3.3.3.3" xref="S3.E5.m1.3.3.1.1.3.3.3.3.cmml">U</mi><mrow id="S3.E5.m1.3.3.1.1.3.3.3.2.3" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.cmml"><mi id="S3.E5.m1.3.3.1.1.3.3.3.2.3.2" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.2.cmml">m</mi><mo id="S3.E5.m1.3.3.1.1.3.3.3.2.3.1" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.1.cmml">+</mo><mn id="S3.E5.m1.3.3.1.1.3.3.3.2.3.3" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S3.E5.m1.3.3.1.1.4.4.8" xref="S3.E5.m1.3.3.1.1.4.5.cmml">,</mo><mi id="S3.E5.m1.2.2" mathvariant="normal" xref="S3.E5.m1.2.2.cmml">…</mi><mo id="S3.E5.m1.3.3.1.1.4.4.9" xref="S3.E5.m1.3.3.1.1.4.5.cmml">,</mo><msubsup id="S3.E5.m1.3.3.1.1.4.4.4" xref="S3.E5.m1.3.3.1.1.4.4.4.cmml"><mi id="S3.E5.m1.3.3.1.1.4.4.4.2.2" xref="S3.E5.m1.3.3.1.1.4.4.4.2.2.cmml">C</mi><mi id="S3.E5.m1.3.3.1.1.4.4.4.3" xref="S3.E5.m1.3.3.1.1.4.4.4.3.cmml">U</mi><mrow id="S3.E5.m1.3.3.1.1.4.4.4.2.3" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.cmml"><mi id="S3.E5.m1.3.3.1.1.4.4.4.2.3.2" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.2.cmml">m</mi><mo id="S3.E5.m1.3.3.1.1.4.4.4.2.3.1" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.1.cmml">+</mo><mi id="S3.E5.m1.3.3.1.1.4.4.4.2.3.3" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.3.cmml">t</mi></mrow></msubsup></mrow><mo id="S3.E5.m1.3.3.1.1.5" xref="S3.E5.m1.3.3.1.1.5.cmml">=</mo><mi id="S3.E5.m1.3.3.1.1.6" mathvariant="normal" xref="S3.E5.m1.3.3.1.1.6.cmml">∅</mi></mrow><mo id="S3.E5.m1.3.3.1.2" lspace="0em" xref="S3.E5.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E5.m1.3b"><apply id="S3.E5.m1.3.3.1.1.cmml" xref="S3.E5.m1.3.3.1"><eq id="S3.E5.m1.3.3.1.1.5.cmml" xref="S3.E5.m1.3.3.1.1.5"></eq><list id="S3.E5.m1.3.3.1.1.4.5.cmml" xref="S3.E5.m1.3.3.1.1.4.4"><apply id="S3.E5.m1.3.3.1.1.1.1.1.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1">subscript</csymbol><apply id="S3.E5.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1">superscript</csymbol><ci id="S3.E5.m1.3.3.1.1.1.1.1.2.2.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1.2.2">𝐶</ci><cn id="S3.E5.m1.3.3.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.E5.m1.3.3.1.1.1.1.1.2.3">1</cn></apply><ci id="S3.E5.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.E5.m1.3.3.1.1.1.1.1.3">𝑈</ci></apply><ci id="S3.E5.m1.1.1.cmml" xref="S3.E5.m1.1.1">…</ci><apply id="S3.E5.m1.3.3.1.1.2.2.2.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2">subscript</csymbol><apply id="S3.E5.m1.3.3.1.1.2.2.2.2.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.2.2.2.2.1.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2">superscript</csymbol><ci id="S3.E5.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2.2.2">𝐶</ci><ci id="S3.E5.m1.3.3.1.1.2.2.2.2.3.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2.2.3">𝑚</ci></apply><ci id="S3.E5.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.E5.m1.3.3.1.1.2.2.2.3">𝑈</ci></apply><apply id="S3.E5.m1.3.3.1.1.3.3.3.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.3.3.3.1.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3">subscript</csymbol><apply id="S3.E5.m1.3.3.1.1.3.3.3.2.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.3.3.3.2.1.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3">superscript</csymbol><ci id="S3.E5.m1.3.3.1.1.3.3.3.2.2.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3.2.2">𝐶</ci><apply id="S3.E5.m1.3.3.1.1.3.3.3.2.3.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3"><plus id="S3.E5.m1.3.3.1.1.3.3.3.2.3.1.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.1"></plus><ci id="S3.E5.m1.3.3.1.1.3.3.3.2.3.2.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.2">𝑚</ci><cn id="S3.E5.m1.3.3.1.1.3.3.3.2.3.3.cmml" type="integer" xref="S3.E5.m1.3.3.1.1.3.3.3.2.3.3">1</cn></apply></apply><ci id="S3.E5.m1.3.3.1.1.3.3.3.3.cmml" xref="S3.E5.m1.3.3.1.1.3.3.3.3">𝑈</ci></apply><ci id="S3.E5.m1.2.2.cmml" xref="S3.E5.m1.2.2">…</ci><apply id="S3.E5.m1.3.3.1.1.4.4.4.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.4.4.4.1.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4">subscript</csymbol><apply id="S3.E5.m1.3.3.1.1.4.4.4.2.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4"><csymbol cd="ambiguous" id="S3.E5.m1.3.3.1.1.4.4.4.2.1.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4">superscript</csymbol><ci id="S3.E5.m1.3.3.1.1.4.4.4.2.2.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.2.2">𝐶</ci><apply id="S3.E5.m1.3.3.1.1.4.4.4.2.3.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3"><plus id="S3.E5.m1.3.3.1.1.4.4.4.2.3.1.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.1"></plus><ci id="S3.E5.m1.3.3.1.1.4.4.4.2.3.2.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.2">𝑚</ci><ci id="S3.E5.m1.3.3.1.1.4.4.4.2.3.3.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.2.3.3">𝑡</ci></apply></apply><ci id="S3.E5.m1.3.3.1.1.4.4.4.3.cmml" xref="S3.E5.m1.3.3.1.1.4.4.4.3">𝑈</ci></apply></list><emptyset id="S3.E5.m1.3.3.1.1.6.cmml" xref="S3.E5.m1.3.3.1.1.6"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.3c">C^{1}_{U},\ldots,C^{m}_{U},C^{m+1}_{U},\ldots,C^{m+t}_{U}=\emptyset.</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.3d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_m + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = ∅ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.1.p1.26">This extended sequence generates the empty set from <math alttext="\mathcal{B}_{U}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.11.m1.1"><semantics id="S3.SS2.1.p1.11.m1.1a"><msub id="S3.SS2.1.p1.11.m1.1.1" xref="S3.SS2.1.p1.11.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.11.m1.1.1.2" xref="S3.SS2.1.p1.11.m1.1.1.2.cmml">ℬ</mi><mi id="S3.SS2.1.p1.11.m1.1.1.3" xref="S3.SS2.1.p1.11.m1.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.11.m1.1b"><apply id="S3.SS2.1.p1.11.m1.1.1.cmml" xref="S3.SS2.1.p1.11.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.11.m1.1.1.1.cmml" xref="S3.SS2.1.p1.11.m1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.11.m1.1.1.2.cmml" xref="S3.SS2.1.p1.11.m1.1.1.2">ℬ</ci><ci id="S3.SS2.1.p1.11.m1.1.1.3.cmml" xref="S3.SS2.1.p1.11.m1.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.11.m1.1c">\mathcal{B}_{U}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.11.m1.1d">caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> and has intersection complexity <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.1.p1.12.m2.1"><semantics id="S3.SS2.1.p1.12.m2.1a"><mi id="S3.SS2.1.p1.12.m2.1.1" xref="S3.SS2.1.p1.12.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.12.m2.1b"><ci id="S3.SS2.1.p1.12.m2.1.1.cmml" xref="S3.SS2.1.p1.12.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.12.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.12.m2.1d">italic_k</annotation></semantics></math>. Let <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS2.1.p1.13.m3.1"><semantics id="S3.SS2.1.p1.13.m3.1a"><mi id="S3.SS2.1.p1.13.m3.1.1" mathvariant="normal" xref="S3.SS2.1.p1.13.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.13.m3.1b"><ci id="S3.SS2.1.p1.13.m3.1.1.cmml" xref="S3.SS2.1.p1.13.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.13.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.13.m3.1d">roman_Λ</annotation></semantics></math> be the set of intersection operations in this sequence. Note that each pair <math alttext="(C^{i}_{U},C^{j}_{U})\in\Lambda" class="ltx_Math" display="inline" id="S3.SS2.1.p1.14.m4.2"><semantics id="S3.SS2.1.p1.14.m4.2a"><mrow id="S3.SS2.1.p1.14.m4.2.2" xref="S3.SS2.1.p1.14.m4.2.2.cmml"><mrow id="S3.SS2.1.p1.14.m4.2.2.2.2" xref="S3.SS2.1.p1.14.m4.2.2.2.3.cmml"><mo id="S3.SS2.1.p1.14.m4.2.2.2.2.3" stretchy="false" xref="S3.SS2.1.p1.14.m4.2.2.2.3.cmml">(</mo><msubsup id="S3.SS2.1.p1.14.m4.1.1.1.1.1" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.cmml"><mi id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.2" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.SS2.1.p1.14.m4.1.1.1.1.1.3" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.3.cmml">U</mi><mi id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.3" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.1.p1.14.m4.2.2.2.2.4" xref="S3.SS2.1.p1.14.m4.2.2.2.3.cmml">,</mo><msubsup id="S3.SS2.1.p1.14.m4.2.2.2.2.2" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.cmml"><mi id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.2" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.2.cmml">C</mi><mi id="S3.SS2.1.p1.14.m4.2.2.2.2.2.3" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.3.cmml">U</mi><mi id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.3" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.3.cmml">j</mi></msubsup><mo id="S3.SS2.1.p1.14.m4.2.2.2.2.5" stretchy="false" xref="S3.SS2.1.p1.14.m4.2.2.2.3.cmml">)</mo></mrow><mo id="S3.SS2.1.p1.14.m4.2.2.3" xref="S3.SS2.1.p1.14.m4.2.2.3.cmml">∈</mo><mi id="S3.SS2.1.p1.14.m4.2.2.4" mathvariant="normal" xref="S3.SS2.1.p1.14.m4.2.2.4.cmml">Λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.14.m4.2b"><apply id="S3.SS2.1.p1.14.m4.2.2.cmml" xref="S3.SS2.1.p1.14.m4.2.2"><in id="S3.SS2.1.p1.14.m4.2.2.3.cmml" xref="S3.SS2.1.p1.14.m4.2.2.3"></in><interval closure="open" id="S3.SS2.1.p1.14.m4.2.2.2.3.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2"><apply id="S3.SS2.1.p1.14.m4.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.14.m4.1.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1">subscript</csymbol><apply id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.1.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.2.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.3.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS2.1.p1.14.m4.1.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.14.m4.1.1.1.1.1.3">𝑈</ci></apply><apply id="S3.SS2.1.p1.14.m4.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.14.m4.2.2.2.2.2.1.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2">subscript</csymbol><apply id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.1.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2">superscript</csymbol><ci id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.2">𝐶</ci><ci id="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.3.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.2.3">𝑗</ci></apply><ci id="S3.SS2.1.p1.14.m4.2.2.2.2.2.3.cmml" xref="S3.SS2.1.p1.14.m4.2.2.2.2.2.3">𝑈</ci></apply></interval><ci id="S3.SS2.1.p1.14.m4.2.2.4.cmml" xref="S3.SS2.1.p1.14.m4.2.2.4">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.14.m4.2c">(C^{i}_{U},C^{j}_{U})\in\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.14.m4.2d">( italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ) ∈ roman_Λ</annotation></semantics></math> satisfies <math alttext="C^{i}_{U},C^{j}_{U}\subseteq U" class="ltx_Math" display="inline" id="S3.SS2.1.p1.15.m5.2"><semantics id="S3.SS2.1.p1.15.m5.2a"><mrow id="S3.SS2.1.p1.15.m5.2.2" xref="S3.SS2.1.p1.15.m5.2.2.cmml"><mrow id="S3.SS2.1.p1.15.m5.2.2.2.2" xref="S3.SS2.1.p1.15.m5.2.2.2.3.cmml"><msubsup id="S3.SS2.1.p1.15.m5.1.1.1.1.1" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.cmml"><mi id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.2" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.SS2.1.p1.15.m5.1.1.1.1.1.3" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.3.cmml">U</mi><mi id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.3" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.1.p1.15.m5.2.2.2.2.3" xref="S3.SS2.1.p1.15.m5.2.2.2.3.cmml">,</mo><msubsup id="S3.SS2.1.p1.15.m5.2.2.2.2.2" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.cmml"><mi id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.2" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.2.cmml">C</mi><mi id="S3.SS2.1.p1.15.m5.2.2.2.2.2.3" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.3.cmml">U</mi><mi id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.3" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.3.cmml">j</mi></msubsup></mrow><mo id="S3.SS2.1.p1.15.m5.2.2.3" xref="S3.SS2.1.p1.15.m5.2.2.3.cmml">⊆</mo><mi id="S3.SS2.1.p1.15.m5.2.2.4" xref="S3.SS2.1.p1.15.m5.2.2.4.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.15.m5.2b"><apply id="S3.SS2.1.p1.15.m5.2.2.cmml" xref="S3.SS2.1.p1.15.m5.2.2"><subset id="S3.SS2.1.p1.15.m5.2.2.3.cmml" xref="S3.SS2.1.p1.15.m5.2.2.3"></subset><list id="S3.SS2.1.p1.15.m5.2.2.2.3.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2"><apply id="S3.SS2.1.p1.15.m5.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.15.m5.1.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1">subscript</csymbol><apply id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.1.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.2.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.3.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS2.1.p1.15.m5.1.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.15.m5.1.1.1.1.1.3">𝑈</ci></apply><apply id="S3.SS2.1.p1.15.m5.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.15.m5.2.2.2.2.2.1.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2">subscript</csymbol><apply id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.1.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2">superscript</csymbol><ci id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.2.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.2">𝐶</ci><ci id="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.3.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.2.3">𝑗</ci></apply><ci id="S3.SS2.1.p1.15.m5.2.2.2.2.2.3.cmml" xref="S3.SS2.1.p1.15.m5.2.2.2.2.2.3">𝑈</ci></apply></list><ci id="S3.SS2.1.p1.15.m5.2.2.4.cmml" xref="S3.SS2.1.p1.15.m5.2.2.4">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.15.m5.2c">C^{i}_{U},C^{j}_{U}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.15.m5.2d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ⊆ italic_U</annotation></semantics></math>, and that <math alttext="|\Lambda|\leq k<\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.16.m6.3"><semantics id="S3.SS2.1.p1.16.m6.3a"><mrow id="S3.SS2.1.p1.16.m6.3.4" xref="S3.SS2.1.p1.16.m6.3.4.cmml"><mrow id="S3.SS2.1.p1.16.m6.3.4.2.2" xref="S3.SS2.1.p1.16.m6.3.4.2.1.cmml"><mo id="S3.SS2.1.p1.16.m6.3.4.2.2.1" stretchy="false" xref="S3.SS2.1.p1.16.m6.3.4.2.1.1.cmml">|</mo><mi id="S3.SS2.1.p1.16.m6.1.1" mathvariant="normal" xref="S3.SS2.1.p1.16.m6.1.1.cmml">Λ</mi><mo id="S3.SS2.1.p1.16.m6.3.4.2.2.2" stretchy="false" xref="S3.SS2.1.p1.16.m6.3.4.2.1.1.cmml">|</mo></mrow><mo id="S3.SS2.1.p1.16.m6.3.4.3" xref="S3.SS2.1.p1.16.m6.3.4.3.cmml">≤</mo><mi id="S3.SS2.1.p1.16.m6.3.4.4" xref="S3.SS2.1.p1.16.m6.3.4.4.cmml">k</mi><mo id="S3.SS2.1.p1.16.m6.3.4.5" xref="S3.SS2.1.p1.16.m6.3.4.5.cmml"><</mo><mrow id="S3.SS2.1.p1.16.m6.3.4.6" xref="S3.SS2.1.p1.16.m6.3.4.6.cmml"><mi id="S3.SS2.1.p1.16.m6.3.4.6.2" xref="S3.SS2.1.p1.16.m6.3.4.6.2.cmml">ρ</mi><mo id="S3.SS2.1.p1.16.m6.3.4.6.1" xref="S3.SS2.1.p1.16.m6.3.4.6.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.16.m6.3.4.6.3.2" xref="S3.SS2.1.p1.16.m6.3.4.6.3.1.cmml"><mo id="S3.SS2.1.p1.16.m6.3.4.6.3.2.1" stretchy="false" xref="S3.SS2.1.p1.16.m6.3.4.6.3.1.cmml">(</mo><mi id="S3.SS2.1.p1.16.m6.2.2" xref="S3.SS2.1.p1.16.m6.2.2.cmml">A</mi><mo id="S3.SS2.1.p1.16.m6.3.4.6.3.2.2" xref="S3.SS2.1.p1.16.m6.3.4.6.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.16.m6.3.3" xref="S3.SS2.1.p1.16.m6.3.3.cmml">ℬ</mi><mo id="S3.SS2.1.p1.16.m6.3.4.6.3.2.3" stretchy="false" xref="S3.SS2.1.p1.16.m6.3.4.6.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.16.m6.3b"><apply id="S3.SS2.1.p1.16.m6.3.4.cmml" xref="S3.SS2.1.p1.16.m6.3.4"><and id="S3.SS2.1.p1.16.m6.3.4a.cmml" xref="S3.SS2.1.p1.16.m6.3.4"></and><apply id="S3.SS2.1.p1.16.m6.3.4b.cmml" xref="S3.SS2.1.p1.16.m6.3.4"><leq id="S3.SS2.1.p1.16.m6.3.4.3.cmml" xref="S3.SS2.1.p1.16.m6.3.4.3"></leq><apply id="S3.SS2.1.p1.16.m6.3.4.2.1.cmml" xref="S3.SS2.1.p1.16.m6.3.4.2.2"><abs id="S3.SS2.1.p1.16.m6.3.4.2.1.1.cmml" xref="S3.SS2.1.p1.16.m6.3.4.2.2.1"></abs><ci id="S3.SS2.1.p1.16.m6.1.1.cmml" xref="S3.SS2.1.p1.16.m6.1.1">Λ</ci></apply><ci id="S3.SS2.1.p1.16.m6.3.4.4.cmml" xref="S3.SS2.1.p1.16.m6.3.4.4">𝑘</ci></apply><apply id="S3.SS2.1.p1.16.m6.3.4c.cmml" xref="S3.SS2.1.p1.16.m6.3.4"><lt id="S3.SS2.1.p1.16.m6.3.4.5.cmml" xref="S3.SS2.1.p1.16.m6.3.4.5"></lt><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.1.p1.16.m6.3.4.4.cmml" id="S3.SS2.1.p1.16.m6.3.4d.cmml" xref="S3.SS2.1.p1.16.m6.3.4"></share><apply id="S3.SS2.1.p1.16.m6.3.4.6.cmml" xref="S3.SS2.1.p1.16.m6.3.4.6"><times id="S3.SS2.1.p1.16.m6.3.4.6.1.cmml" xref="S3.SS2.1.p1.16.m6.3.4.6.1"></times><ci id="S3.SS2.1.p1.16.m6.3.4.6.2.cmml" xref="S3.SS2.1.p1.16.m6.3.4.6.2">𝜌</ci><interval closure="open" id="S3.SS2.1.p1.16.m6.3.4.6.3.1.cmml" xref="S3.SS2.1.p1.16.m6.3.4.6.3.2"><ci id="S3.SS2.1.p1.16.m6.2.2.cmml" xref="S3.SS2.1.p1.16.m6.2.2">𝐴</ci><ci id="S3.SS2.1.p1.16.m6.3.3.cmml" xref="S3.SS2.1.p1.16.m6.3.3">ℬ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.16.m6.3c">|\Lambda|\leq k<\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.16.m6.3d">| roman_Λ | ≤ italic_k < italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math>. Let <math alttext="\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.17.m7.5"><semantics id="S3.SS2.1.p1.17.m7.5a"><mrow id="S3.SS2.1.p1.17.m7.5.5" xref="S3.SS2.1.p1.17.m7.5.5.cmml"><msub id="S3.SS2.1.p1.17.m7.5.5.4" xref="S3.SS2.1.p1.17.m7.5.5.4.cmml"><mi id="S3.SS2.1.p1.17.m7.5.5.4.2" mathvariant="normal" xref="S3.SS2.1.p1.17.m7.5.5.4.2.cmml">Φ</mi><mrow id="S3.SS2.1.p1.17.m7.2.2.2.4" xref="S3.SS2.1.p1.17.m7.2.2.2.3.cmml"><mi id="S3.SS2.1.p1.17.m7.1.1.1.1" xref="S3.SS2.1.p1.17.m7.1.1.1.1.cmml">A</mi><mo id="S3.SS2.1.p1.17.m7.2.2.2.4.1" xref="S3.SS2.1.p1.17.m7.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.17.m7.2.2.2.2" xref="S3.SS2.1.p1.17.m7.2.2.2.2.cmml">ℬ</mi></mrow></msub><mo id="S3.SS2.1.p1.17.m7.5.5.3" xref="S3.SS2.1.p1.17.m7.5.5.3.cmml">=</mo><mrow id="S3.SS2.1.p1.17.m7.5.5.2.2" xref="S3.SS2.1.p1.17.m7.5.5.2.3.cmml"><mo id="S3.SS2.1.p1.17.m7.5.5.2.2.3" stretchy="false" xref="S3.SS2.1.p1.17.m7.5.5.2.3.cmml">(</mo><msub id="S3.SS2.1.p1.17.m7.4.4.1.1.1" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1.cmml"><mi id="S3.SS2.1.p1.17.m7.4.4.1.1.1.2" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1.2.cmml">V</mi><mi id="S3.SS2.1.p1.17.m7.4.4.1.1.1.3" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub><mo id="S3.SS2.1.p1.17.m7.5.5.2.2.4" xref="S3.SS2.1.p1.17.m7.5.5.2.3.cmml">,</mo><msub id="S3.SS2.1.p1.17.m7.5.5.2.2.2" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2.cmml"><mi id="S3.SS2.1.p1.17.m7.5.5.2.2.2.2" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2.2.cmml">V</mi><mi id="S3.SS2.1.p1.17.m7.5.5.2.2.2.3" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub><mo id="S3.SS2.1.p1.17.m7.5.5.2.2.5" xref="S3.SS2.1.p1.17.m7.5.5.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.17.m7.3.3" xref="S3.SS2.1.p1.17.m7.3.3.cmml">ℰ</mi><mo id="S3.SS2.1.p1.17.m7.5.5.2.2.6" stretchy="false" xref="S3.SS2.1.p1.17.m7.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.17.m7.5b"><apply id="S3.SS2.1.p1.17.m7.5.5.cmml" xref="S3.SS2.1.p1.17.m7.5.5"><eq id="S3.SS2.1.p1.17.m7.5.5.3.cmml" xref="S3.SS2.1.p1.17.m7.5.5.3"></eq><apply id="S3.SS2.1.p1.17.m7.5.5.4.cmml" xref="S3.SS2.1.p1.17.m7.5.5.4"><csymbol cd="ambiguous" id="S3.SS2.1.p1.17.m7.5.5.4.1.cmml" xref="S3.SS2.1.p1.17.m7.5.5.4">subscript</csymbol><ci id="S3.SS2.1.p1.17.m7.5.5.4.2.cmml" xref="S3.SS2.1.p1.17.m7.5.5.4.2">Φ</ci><list id="S3.SS2.1.p1.17.m7.2.2.2.3.cmml" xref="S3.SS2.1.p1.17.m7.2.2.2.4"><ci id="S3.SS2.1.p1.17.m7.1.1.1.1.cmml" xref="S3.SS2.1.p1.17.m7.1.1.1.1">𝐴</ci><ci id="S3.SS2.1.p1.17.m7.2.2.2.2.cmml" xref="S3.SS2.1.p1.17.m7.2.2.2.2">ℬ</ci></list></apply><vector id="S3.SS2.1.p1.17.m7.5.5.2.3.cmml" xref="S3.SS2.1.p1.17.m7.5.5.2.2"><apply id="S3.SS2.1.p1.17.m7.4.4.1.1.1.cmml" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.17.m7.4.4.1.1.1.1.cmml" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.17.m7.4.4.1.1.1.2.cmml" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1.2">𝑉</ci><ci id="S3.SS2.1.p1.17.m7.4.4.1.1.1.3.cmml" xref="S3.SS2.1.p1.17.m7.4.4.1.1.1.3">𝗉𝖺𝗂𝗋𝗌</ci></apply><apply id="S3.SS2.1.p1.17.m7.5.5.2.2.2.cmml" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.17.m7.5.5.2.2.2.1.cmml" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2">subscript</csymbol><ci id="S3.SS2.1.p1.17.m7.5.5.2.2.2.2.cmml" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2.2">𝑉</ci><ci id="S3.SS2.1.p1.17.m7.5.5.2.2.2.3.cmml" xref="S3.SS2.1.p1.17.m7.5.5.2.2.2.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply><ci id="S3.SS2.1.p1.17.m7.3.3.cmml" xref="S3.SS2.1.p1.17.m7.3.3">ℰ</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.17.m7.5c">\Phi_{A,\mathcal{B}}=(V_{\mathsf{pairs}},V_{\mathsf{filters}},\mathcal{E})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.17.m7.5d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT = ( italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT , caligraphic_E )</annotation></semantics></math> be the cover graph of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.1.p1.18.m8.1"><semantics id="S3.SS2.1.p1.18.m8.1a"><mi id="S3.SS2.1.p1.18.m8.1.1" xref="S3.SS2.1.p1.18.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.18.m8.1b"><ci id="S3.SS2.1.p1.18.m8.1.1.cmml" xref="S3.SS2.1.p1.18.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.18.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.18.m8.1d">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.19.m9.1"><semantics id="S3.SS2.1.p1.19.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.19.m9.1.1" xref="S3.SS2.1.p1.19.m9.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.19.m9.1b"><ci id="S3.SS2.1.p1.19.m9.1.1.cmml" xref="S3.SS2.1.p1.19.m9.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.19.m9.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.19.m9.1d">caligraphic_B</annotation></semantics></math>. Since <math alttext="\Lambda\subseteq V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.20.m10.1"><semantics id="S3.SS2.1.p1.20.m10.1a"><mrow id="S3.SS2.1.p1.20.m10.1.1" xref="S3.SS2.1.p1.20.m10.1.1.cmml"><mi id="S3.SS2.1.p1.20.m10.1.1.2" mathvariant="normal" xref="S3.SS2.1.p1.20.m10.1.1.2.cmml">Λ</mi><mo id="S3.SS2.1.p1.20.m10.1.1.1" xref="S3.SS2.1.p1.20.m10.1.1.1.cmml">⊆</mo><msub id="S3.SS2.1.p1.20.m10.1.1.3" xref="S3.SS2.1.p1.20.m10.1.1.3.cmml"><mi id="S3.SS2.1.p1.20.m10.1.1.3.2" xref="S3.SS2.1.p1.20.m10.1.1.3.2.cmml">V</mi><mi id="S3.SS2.1.p1.20.m10.1.1.3.3" xref="S3.SS2.1.p1.20.m10.1.1.3.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.20.m10.1b"><apply id="S3.SS2.1.p1.20.m10.1.1.cmml" xref="S3.SS2.1.p1.20.m10.1.1"><subset id="S3.SS2.1.p1.20.m10.1.1.1.cmml" xref="S3.SS2.1.p1.20.m10.1.1.1"></subset><ci id="S3.SS2.1.p1.20.m10.1.1.2.cmml" xref="S3.SS2.1.p1.20.m10.1.1.2">Λ</ci><apply id="S3.SS2.1.p1.20.m10.1.1.3.cmml" xref="S3.SS2.1.p1.20.m10.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.20.m10.1.1.3.1.cmml" xref="S3.SS2.1.p1.20.m10.1.1.3">subscript</csymbol><ci id="S3.SS2.1.p1.20.m10.1.1.3.2.cmml" xref="S3.SS2.1.p1.20.m10.1.1.3.2">𝑉</ci><ci id="S3.SS2.1.p1.20.m10.1.1.3.3.cmml" xref="S3.SS2.1.p1.20.m10.1.1.3.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.20.m10.1c">\Lambda\subseteq V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.20.m10.1d">roman_Λ ⊆ italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="|\Lambda|<\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.21.m11.3"><semantics id="S3.SS2.1.p1.21.m11.3a"><mrow id="S3.SS2.1.p1.21.m11.3.4" xref="S3.SS2.1.p1.21.m11.3.4.cmml"><mrow id="S3.SS2.1.p1.21.m11.3.4.2.2" xref="S3.SS2.1.p1.21.m11.3.4.2.1.cmml"><mo id="S3.SS2.1.p1.21.m11.3.4.2.2.1" stretchy="false" xref="S3.SS2.1.p1.21.m11.3.4.2.1.1.cmml">|</mo><mi id="S3.SS2.1.p1.21.m11.3.3" mathvariant="normal" xref="S3.SS2.1.p1.21.m11.3.3.cmml">Λ</mi><mo id="S3.SS2.1.p1.21.m11.3.4.2.2.2" stretchy="false" xref="S3.SS2.1.p1.21.m11.3.4.2.1.1.cmml">|</mo></mrow><mo id="S3.SS2.1.p1.21.m11.3.4.1" xref="S3.SS2.1.p1.21.m11.3.4.1.cmml"><</mo><mrow id="S3.SS2.1.p1.21.m11.3.4.3" xref="S3.SS2.1.p1.21.m11.3.4.3.cmml"><mi id="S3.SS2.1.p1.21.m11.3.4.3.2" xref="S3.SS2.1.p1.21.m11.3.4.3.2.cmml">ρ</mi><mo id="S3.SS2.1.p1.21.m11.3.4.3.1" xref="S3.SS2.1.p1.21.m11.3.4.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.21.m11.3.4.3.3.2" xref="S3.SS2.1.p1.21.m11.3.4.3.3.1.cmml"><mo id="S3.SS2.1.p1.21.m11.3.4.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.21.m11.3.4.3.3.1.cmml">(</mo><mi id="S3.SS2.1.p1.21.m11.1.1" xref="S3.SS2.1.p1.21.m11.1.1.cmml">A</mi><mo id="S3.SS2.1.p1.21.m11.3.4.3.3.2.2" xref="S3.SS2.1.p1.21.m11.3.4.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.21.m11.2.2" xref="S3.SS2.1.p1.21.m11.2.2.cmml">ℬ</mi><mo id="S3.SS2.1.p1.21.m11.3.4.3.3.2.3" stretchy="false" xref="S3.SS2.1.p1.21.m11.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.21.m11.3b"><apply id="S3.SS2.1.p1.21.m11.3.4.cmml" xref="S3.SS2.1.p1.21.m11.3.4"><lt id="S3.SS2.1.p1.21.m11.3.4.1.cmml" xref="S3.SS2.1.p1.21.m11.3.4.1"></lt><apply id="S3.SS2.1.p1.21.m11.3.4.2.1.cmml" xref="S3.SS2.1.p1.21.m11.3.4.2.2"><abs id="S3.SS2.1.p1.21.m11.3.4.2.1.1.cmml" xref="S3.SS2.1.p1.21.m11.3.4.2.2.1"></abs><ci id="S3.SS2.1.p1.21.m11.3.3.cmml" xref="S3.SS2.1.p1.21.m11.3.3">Λ</ci></apply><apply id="S3.SS2.1.p1.21.m11.3.4.3.cmml" xref="S3.SS2.1.p1.21.m11.3.4.3"><times id="S3.SS2.1.p1.21.m11.3.4.3.1.cmml" xref="S3.SS2.1.p1.21.m11.3.4.3.1"></times><ci id="S3.SS2.1.p1.21.m11.3.4.3.2.cmml" xref="S3.SS2.1.p1.21.m11.3.4.3.2">𝜌</ci><interval closure="open" id="S3.SS2.1.p1.21.m11.3.4.3.3.1.cmml" xref="S3.SS2.1.p1.21.m11.3.4.3.3.2"><ci id="S3.SS2.1.p1.21.m11.1.1.cmml" xref="S3.SS2.1.p1.21.m11.1.1">𝐴</ci><ci id="S3.SS2.1.p1.21.m11.2.2.cmml" xref="S3.SS2.1.p1.21.m11.2.2">ℬ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.21.m11.3c">|\Lambda|<\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.21.m11.3d">| roman_Λ | < italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math>, there exists <math alttext="\mathcal{F}\in V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.22.m12.1"><semantics id="S3.SS2.1.p1.22.m12.1a"><mrow id="S3.SS2.1.p1.22.m12.1.1" xref="S3.SS2.1.p1.22.m12.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.22.m12.1.1.2" xref="S3.SS2.1.p1.22.m12.1.1.2.cmml">ℱ</mi><mo id="S3.SS2.1.p1.22.m12.1.1.1" xref="S3.SS2.1.p1.22.m12.1.1.1.cmml">∈</mo><msub id="S3.SS2.1.p1.22.m12.1.1.3" xref="S3.SS2.1.p1.22.m12.1.1.3.cmml"><mi id="S3.SS2.1.p1.22.m12.1.1.3.2" xref="S3.SS2.1.p1.22.m12.1.1.3.2.cmml">V</mi><mi id="S3.SS2.1.p1.22.m12.1.1.3.3" xref="S3.SS2.1.p1.22.m12.1.1.3.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.22.m12.1b"><apply id="S3.SS2.1.p1.22.m12.1.1.cmml" xref="S3.SS2.1.p1.22.m12.1.1"><in id="S3.SS2.1.p1.22.m12.1.1.1.cmml" xref="S3.SS2.1.p1.22.m12.1.1.1"></in><ci id="S3.SS2.1.p1.22.m12.1.1.2.cmml" xref="S3.SS2.1.p1.22.m12.1.1.2">ℱ</ci><apply id="S3.SS2.1.p1.22.m12.1.1.3.cmml" xref="S3.SS2.1.p1.22.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.22.m12.1.1.3.1.cmml" xref="S3.SS2.1.p1.22.m12.1.1.3">subscript</csymbol><ci id="S3.SS2.1.p1.22.m12.1.1.3.2.cmml" xref="S3.SS2.1.p1.22.m12.1.1.3.2">𝑉</ci><ci id="S3.SS2.1.p1.22.m12.1.1.3.3.cmml" xref="S3.SS2.1.p1.22.m12.1.1.3.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.22.m12.1c">\mathcal{F}\in V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.22.m12.1d">caligraphic_F ∈ italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math> that is not covered by the pairs in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS2.1.p1.23.m13.1"><semantics id="S3.SS2.1.p1.23.m13.1a"><mi id="S3.SS2.1.p1.23.m13.1.1" mathvariant="normal" xref="S3.SS2.1.p1.23.m13.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.23.m13.1b"><ci id="S3.SS2.1.p1.23.m13.1.1.cmml" xref="S3.SS2.1.p1.23.m13.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.23.m13.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.23.m13.1d">roman_Λ</annotation></semantics></math>. Let <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.SS2.1.p1.24.m14.1"><semantics id="S3.SS2.1.p1.24.m14.1a"><mrow id="S3.SS2.1.p1.24.m14.1.1" xref="S3.SS2.1.p1.24.m14.1.1.cmml"><mi id="S3.SS2.1.p1.24.m14.1.1.2" xref="S3.SS2.1.p1.24.m14.1.1.2.cmml">a</mi><mo id="S3.SS2.1.p1.24.m14.1.1.1" xref="S3.SS2.1.p1.24.m14.1.1.1.cmml">∈</mo><mi id="S3.SS2.1.p1.24.m14.1.1.3" xref="S3.SS2.1.p1.24.m14.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.24.m14.1b"><apply id="S3.SS2.1.p1.24.m14.1.1.cmml" xref="S3.SS2.1.p1.24.m14.1.1"><in id="S3.SS2.1.p1.24.m14.1.1.1.cmml" xref="S3.SS2.1.p1.24.m14.1.1.1"></in><ci id="S3.SS2.1.p1.24.m14.1.1.2.cmml" xref="S3.SS2.1.p1.24.m14.1.1.2">𝑎</ci><ci id="S3.SS2.1.p1.24.m14.1.1.3.cmml" xref="S3.SS2.1.p1.24.m14.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.24.m14.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.24.m14.1d">italic_a ∈ italic_A</annotation></semantics></math> be an element such that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.25.m15.1"><semantics id="S3.SS2.1.p1.25.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.25.m15.1.1" xref="S3.SS2.1.p1.25.m15.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.25.m15.1b"><ci id="S3.SS2.1.p1.25.m15.1.1.cmml" xref="S3.SS2.1.p1.25.m15.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.25.m15.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.25.m15.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="a" class="ltx_Math" display="inline" id="S3.SS2.1.p1.26.m16.1"><semantics id="S3.SS2.1.p1.26.m16.1a"><mi id="S3.SS2.1.p1.26.m16.1.1" xref="S3.SS2.1.p1.26.m16.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.26.m16.1b"><ci id="S3.SS2.1.p1.26.m16.1.1.cmml" xref="S3.SS2.1.p1.26.m16.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.26.m16.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.26.m16.1d">italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.2.p2"> <p class="ltx_p" id="S3.SS2.2.p2.18">We trace the construction in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E4" title="Equation 4 ‣ Proof. ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4</span></a> from the point of view of the element <math alttext="a" class="ltx_Math" display="inline" id="S3.SS2.2.p2.1.m1.1"><semantics id="S3.SS2.2.p2.1.m1.1a"><mi id="S3.SS2.2.p2.1.m1.1.1" xref="S3.SS2.2.p2.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.1.m1.1b"><ci id="S3.SS2.2.p2.1.m1.1.1.cmml" xref="S3.SS2.2.p2.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.1.m1.1d">italic_a</annotation></semantics></math>. Let <math alttext="\alpha_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.2.m2.1"><semantics id="S3.SS2.2.p2.2.m2.1a"><mrow id="S3.SS2.2.p2.2.m2.1.1" xref="S3.SS2.2.p2.2.m2.1.1.cmml"><msub id="S3.SS2.2.p2.2.m2.1.1.2" xref="S3.SS2.2.p2.2.m2.1.1.2.cmml"><mi id="S3.SS2.2.p2.2.m2.1.1.2.2" xref="S3.SS2.2.p2.2.m2.1.1.2.2.cmml">α</mi><mi id="S3.SS2.2.p2.2.m2.1.1.2.3" xref="S3.SS2.2.p2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.2.p2.2.m2.1.1.1" xref="S3.SS2.2.p2.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.2.m2.1.1.3" xref="S3.SS2.2.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.2.m2.1b"><apply id="S3.SS2.2.p2.2.m2.1.1.cmml" xref="S3.SS2.2.p2.2.m2.1.1"><eq id="S3.SS2.2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.2.p2.2.m2.1.1.1"></eq><apply id="S3.SS2.2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.2.m2.1.1.2.1.cmml" xref="S3.SS2.2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.2.m2.1.1.2.2.cmml" xref="S3.SS2.2.p2.2.m2.1.1.2.2">𝛼</ci><ci id="S3.SS2.2.p2.2.m2.1.1.2.3.cmml" xref="S3.SS2.2.p2.2.m2.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.2.p2.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.2.m2.1c">\alpha_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> if and only if <math alttext="a\in C_{i}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.3.m3.1"><semantics id="S3.SS2.2.p2.3.m3.1a"><mrow id="S3.SS2.2.p2.3.m3.1.1" xref="S3.SS2.2.p2.3.m3.1.1.cmml"><mi id="S3.SS2.2.p2.3.m3.1.1.2" xref="S3.SS2.2.p2.3.m3.1.1.2.cmml">a</mi><mo id="S3.SS2.2.p2.3.m3.1.1.1" xref="S3.SS2.2.p2.3.m3.1.1.1.cmml">∈</mo><msub id="S3.SS2.2.p2.3.m3.1.1.3" xref="S3.SS2.2.p2.3.m3.1.1.3.cmml"><mi id="S3.SS2.2.p2.3.m3.1.1.3.2" xref="S3.SS2.2.p2.3.m3.1.1.3.2.cmml">C</mi><mi id="S3.SS2.2.p2.3.m3.1.1.3.3" xref="S3.SS2.2.p2.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.3.m3.1b"><apply id="S3.SS2.2.p2.3.m3.1.1.cmml" xref="S3.SS2.2.p2.3.m3.1.1"><in id="S3.SS2.2.p2.3.m3.1.1.1.cmml" xref="S3.SS2.2.p2.3.m3.1.1.1"></in><ci id="S3.SS2.2.p2.3.m3.1.1.2.cmml" xref="S3.SS2.2.p2.3.m3.1.1.2">𝑎</ci><apply id="S3.SS2.2.p2.3.m3.1.1.3.cmml" xref="S3.SS2.2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.3.m3.1.1.3.1.cmml" xref="S3.SS2.2.p2.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.2.p2.3.m3.1.1.3.2.cmml" xref="S3.SS2.2.p2.3.m3.1.1.3.2">𝐶</ci><ci id="S3.SS2.2.p2.3.m3.1.1.3.3.cmml" xref="S3.SS2.2.p2.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.3.m3.1c">a\in C_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.3.m3.1d">italic_a ∈ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Observe that <math alttext="\alpha_{m+t}=1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.4.m4.1"><semantics id="S3.SS2.2.p2.4.m4.1a"><mrow id="S3.SS2.2.p2.4.m4.1.1" xref="S3.SS2.2.p2.4.m4.1.1.cmml"><msub id="S3.SS2.2.p2.4.m4.1.1.2" xref="S3.SS2.2.p2.4.m4.1.1.2.cmml"><mi id="S3.SS2.2.p2.4.m4.1.1.2.2" xref="S3.SS2.2.p2.4.m4.1.1.2.2.cmml">α</mi><mrow id="S3.SS2.2.p2.4.m4.1.1.2.3" xref="S3.SS2.2.p2.4.m4.1.1.2.3.cmml"><mi id="S3.SS2.2.p2.4.m4.1.1.2.3.2" xref="S3.SS2.2.p2.4.m4.1.1.2.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.4.m4.1.1.2.3.1" xref="S3.SS2.2.p2.4.m4.1.1.2.3.1.cmml">+</mo><mi id="S3.SS2.2.p2.4.m4.1.1.2.3.3" xref="S3.SS2.2.p2.4.m4.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.2.p2.4.m4.1.1.1" xref="S3.SS2.2.p2.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.4.m4.1.1.3" xref="S3.SS2.2.p2.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.4.m4.1b"><apply id="S3.SS2.2.p2.4.m4.1.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1"><eq id="S3.SS2.2.p2.4.m4.1.1.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1"></eq><apply id="S3.SS2.2.p2.4.m4.1.1.2.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.4.m4.1.1.2.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.4.m4.1.1.2.2.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2.2">𝛼</ci><apply id="S3.SS2.2.p2.4.m4.1.1.2.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2.3"><plus id="S3.SS2.2.p2.4.m4.1.1.2.3.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2.3.1"></plus><ci id="S3.SS2.2.p2.4.m4.1.1.2.3.2.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2.3.2">𝑚</ci><ci id="S3.SS2.2.p2.4.m4.1.1.2.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS2.2.p2.4.m4.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.4.m4.1c">\alpha_{m+t}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.4.m4.1d">italic_α start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math>, since <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.SS2.2.p2.5.m5.1"><semantics id="S3.SS2.2.p2.5.m5.1a"><mrow id="S3.SS2.2.p2.5.m5.1.1" xref="S3.SS2.2.p2.5.m5.1.1.cmml"><mi id="S3.SS2.2.p2.5.m5.1.1.2" xref="S3.SS2.2.p2.5.m5.1.1.2.cmml">a</mi><mo id="S3.SS2.2.p2.5.m5.1.1.1" xref="S3.SS2.2.p2.5.m5.1.1.1.cmml">∈</mo><mi id="S3.SS2.2.p2.5.m5.1.1.3" xref="S3.SS2.2.p2.5.m5.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.5.m5.1b"><apply id="S3.SS2.2.p2.5.m5.1.1.cmml" xref="S3.SS2.2.p2.5.m5.1.1"><in id="S3.SS2.2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.2.p2.5.m5.1.1.1"></in><ci id="S3.SS2.2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.2.p2.5.m5.1.1.2">𝑎</ci><ci id="S3.SS2.2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.2.p2.5.m5.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.5.m5.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.5.m5.1d">italic_a ∈ italic_A</annotation></semantics></math>. In order to derive a contradiction, we define a second sequence <math alttext="\beta_{i}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.6.m6.1"><semantics id="S3.SS2.2.p2.6.m6.1a"><msub id="S3.SS2.2.p2.6.m6.1.1" xref="S3.SS2.2.p2.6.m6.1.1.cmml"><mi id="S3.SS2.2.p2.6.m6.1.1.2" xref="S3.SS2.2.p2.6.m6.1.1.2.cmml">β</mi><mi id="S3.SS2.2.p2.6.m6.1.1.3" xref="S3.SS2.2.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.6.m6.1b"><apply id="S3.SS2.2.p2.6.m6.1.1.cmml" xref="S3.SS2.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.6.m6.1.1.1.cmml" xref="S3.SS2.2.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS2.2.p2.6.m6.1.1.2.cmml" xref="S3.SS2.2.p2.6.m6.1.1.2">𝛽</ci><ci id="S3.SS2.2.p2.6.m6.1.1.3.cmml" xref="S3.SS2.2.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.6.m6.1c">\beta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.6.m6.1d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> that depends on the semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.7.m7.1"><semantics id="S3.SS2.2.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.7.m7.1.1" xref="S3.SS2.2.p2.7.m7.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.7.m7.1b"><ci id="S3.SS2.2.p2.7.m7.1.1.cmml" xref="S3.SS2.2.p2.7.m7.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.7.m7.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.7.m7.1d">caligraphic_F</annotation></semantics></math> and on the relativized construction appearing in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E5" title="Equation 5 ‣ Proof. ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">5</span></a>. We let <math alttext="\beta_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.8.m8.1"><semantics id="S3.SS2.2.p2.8.m8.1a"><mrow id="S3.SS2.2.p2.8.m8.1.1" xref="S3.SS2.2.p2.8.m8.1.1.cmml"><msub id="S3.SS2.2.p2.8.m8.1.1.2" xref="S3.SS2.2.p2.8.m8.1.1.2.cmml"><mi id="S3.SS2.2.p2.8.m8.1.1.2.2" xref="S3.SS2.2.p2.8.m8.1.1.2.2.cmml">β</mi><mi id="S3.SS2.2.p2.8.m8.1.1.2.3" xref="S3.SS2.2.p2.8.m8.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.2.p2.8.m8.1.1.1" xref="S3.SS2.2.p2.8.m8.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.8.m8.1.1.3" xref="S3.SS2.2.p2.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.8.m8.1b"><apply id="S3.SS2.2.p2.8.m8.1.1.cmml" xref="S3.SS2.2.p2.8.m8.1.1"><eq id="S3.SS2.2.p2.8.m8.1.1.1.cmml" xref="S3.SS2.2.p2.8.m8.1.1.1"></eq><apply id="S3.SS2.2.p2.8.m8.1.1.2.cmml" xref="S3.SS2.2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.8.m8.1.1.2.1.cmml" xref="S3.SS2.2.p2.8.m8.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.8.m8.1.1.2.2.cmml" xref="S3.SS2.2.p2.8.m8.1.1.2.2">𝛽</ci><ci id="S3.SS2.2.p2.8.m8.1.1.2.3.cmml" xref="S3.SS2.2.p2.8.m8.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.2.p2.8.m8.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.8.m8.1c">\beta_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.8.m8.1d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> if and only if <math alttext="C^{i}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.9.m9.1"><semantics id="S3.SS2.2.p2.9.m9.1a"><mrow id="S3.SS2.2.p2.9.m9.1.1" xref="S3.SS2.2.p2.9.m9.1.1.cmml"><msubsup id="S3.SS2.2.p2.9.m9.1.1.2" xref="S3.SS2.2.p2.9.m9.1.1.2.cmml"><mi id="S3.SS2.2.p2.9.m9.1.1.2.2.2" xref="S3.SS2.2.p2.9.m9.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.2.p2.9.m9.1.1.2.3" xref="S3.SS2.2.p2.9.m9.1.1.2.3.cmml">U</mi><mi id="S3.SS2.2.p2.9.m9.1.1.2.2.3" xref="S3.SS2.2.p2.9.m9.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.2.p2.9.m9.1.1.1" xref="S3.SS2.2.p2.9.m9.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.9.m9.1.1.3" xref="S3.SS2.2.p2.9.m9.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.9.m9.1b"><apply id="S3.SS2.2.p2.9.m9.1.1.cmml" xref="S3.SS2.2.p2.9.m9.1.1"><in id="S3.SS2.2.p2.9.m9.1.1.1.cmml" xref="S3.SS2.2.p2.9.m9.1.1.1"></in><apply id="S3.SS2.2.p2.9.m9.1.1.2.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.9.m9.1.1.2.1.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2">subscript</csymbol><apply id="S3.SS2.2.p2.9.m9.1.1.2.2.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.9.m9.1.1.2.2.1.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2">superscript</csymbol><ci id="S3.SS2.2.p2.9.m9.1.1.2.2.2.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2.2.2">𝐶</ci><ci id="S3.SS2.2.p2.9.m9.1.1.2.2.3.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.2.p2.9.m9.1.1.2.3.cmml" xref="S3.SS2.2.p2.9.m9.1.1.2.3">𝑈</ci></apply><ci id="S3.SS2.2.p2.9.m9.1.1.3.cmml" xref="S3.SS2.2.p2.9.m9.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.9.m9.1c">C^{i}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.9.m9.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math> (recall that <math alttext="\mathcal{F}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.10.m10.1"><semantics id="S3.SS2.2.p2.10.m10.1a"><mrow id="S3.SS2.2.p2.10.m10.1.2" xref="S3.SS2.2.p2.10.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.10.m10.1.2.2" xref="S3.SS2.2.p2.10.m10.1.2.2.cmml">ℱ</mi><mo id="S3.SS2.2.p2.10.m10.1.2.1" xref="S3.SS2.2.p2.10.m10.1.2.1.cmml">⊆</mo><mrow id="S3.SS2.2.p2.10.m10.1.2.3" xref="S3.SS2.2.p2.10.m10.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.10.m10.1.2.3.2" xref="S3.SS2.2.p2.10.m10.1.2.3.2.cmml">𝒫</mi><mo id="S3.SS2.2.p2.10.m10.1.2.3.1" xref="S3.SS2.2.p2.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.10.m10.1.2.3.3.2" xref="S3.SS2.2.p2.10.m10.1.2.3.cmml"><mo id="S3.SS2.2.p2.10.m10.1.2.3.3.2.1" stretchy="false" xref="S3.SS2.2.p2.10.m10.1.2.3.cmml">(</mo><mi id="S3.SS2.2.p2.10.m10.1.1" xref="S3.SS2.2.p2.10.m10.1.1.cmml">U</mi><mo id="S3.SS2.2.p2.10.m10.1.2.3.3.2.2" stretchy="false" xref="S3.SS2.2.p2.10.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.10.m10.1b"><apply id="S3.SS2.2.p2.10.m10.1.2.cmml" xref="S3.SS2.2.p2.10.m10.1.2"><subset id="S3.SS2.2.p2.10.m10.1.2.1.cmml" xref="S3.SS2.2.p2.10.m10.1.2.1"></subset><ci id="S3.SS2.2.p2.10.m10.1.2.2.cmml" xref="S3.SS2.2.p2.10.m10.1.2.2">ℱ</ci><apply id="S3.SS2.2.p2.10.m10.1.2.3.cmml" xref="S3.SS2.2.p2.10.m10.1.2.3"><times id="S3.SS2.2.p2.10.m10.1.2.3.1.cmml" xref="S3.SS2.2.p2.10.m10.1.2.3.1"></times><ci id="S3.SS2.2.p2.10.m10.1.2.3.2.cmml" xref="S3.SS2.2.p2.10.m10.1.2.3.2">𝒫</ci><ci id="S3.SS2.2.p2.10.m10.1.1.cmml" xref="S3.SS2.2.p2.10.m10.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.10.m10.1c">\mathcal{F}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.10.m10.1d">caligraphic_F ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> and <math alttext="C^{i}_{U}\subseteq U" class="ltx_Math" display="inline" id="S3.SS2.2.p2.11.m11.1"><semantics id="S3.SS2.2.p2.11.m11.1a"><mrow id="S3.SS2.2.p2.11.m11.1.1" xref="S3.SS2.2.p2.11.m11.1.1.cmml"><msubsup id="S3.SS2.2.p2.11.m11.1.1.2" xref="S3.SS2.2.p2.11.m11.1.1.2.cmml"><mi id="S3.SS2.2.p2.11.m11.1.1.2.2.2" xref="S3.SS2.2.p2.11.m11.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.2.p2.11.m11.1.1.2.3" xref="S3.SS2.2.p2.11.m11.1.1.2.3.cmml">U</mi><mi id="S3.SS2.2.p2.11.m11.1.1.2.2.3" xref="S3.SS2.2.p2.11.m11.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.2.p2.11.m11.1.1.1" xref="S3.SS2.2.p2.11.m11.1.1.1.cmml">⊆</mo><mi id="S3.SS2.2.p2.11.m11.1.1.3" xref="S3.SS2.2.p2.11.m11.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.11.m11.1b"><apply id="S3.SS2.2.p2.11.m11.1.1.cmml" xref="S3.SS2.2.p2.11.m11.1.1"><subset id="S3.SS2.2.p2.11.m11.1.1.1.cmml" xref="S3.SS2.2.p2.11.m11.1.1.1"></subset><apply id="S3.SS2.2.p2.11.m11.1.1.2.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.11.m11.1.1.2.1.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2">subscript</csymbol><apply id="S3.SS2.2.p2.11.m11.1.1.2.2.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.11.m11.1.1.2.2.1.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2">superscript</csymbol><ci id="S3.SS2.2.p2.11.m11.1.1.2.2.2.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2.2.2">𝐶</ci><ci id="S3.SS2.2.p2.11.m11.1.1.2.2.3.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.2.p2.11.m11.1.1.2.3.cmml" xref="S3.SS2.2.p2.11.m11.1.1.2.3">𝑈</ci></apply><ci id="S3.SS2.2.p2.11.m11.1.1.3.cmml" xref="S3.SS2.2.p2.11.m11.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.11.m11.1c">C^{i}_{U}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.11.m11.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ⊆ italic_U</annotation></semantics></math>). Since <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.12.m12.1"><semantics id="S3.SS2.2.p2.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.12.m12.1.1" xref="S3.SS2.2.p2.12.m12.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.12.m12.1b"><ci id="S3.SS2.2.p2.12.m12.1.1.cmml" xref="S3.SS2.2.p2.12.m12.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.12.m12.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.12.m12.1d">caligraphic_F</annotation></semantics></math> is a semi-filter and <math alttext="C^{m+t}_{U}=\emptyset" class="ltx_Math" display="inline" id="S3.SS2.2.p2.13.m13.1"><semantics id="S3.SS2.2.p2.13.m13.1a"><mrow id="S3.SS2.2.p2.13.m13.1.1" xref="S3.SS2.2.p2.13.m13.1.1.cmml"><msubsup id="S3.SS2.2.p2.13.m13.1.1.2" xref="S3.SS2.2.p2.13.m13.1.1.2.cmml"><mi id="S3.SS2.2.p2.13.m13.1.1.2.2.2" xref="S3.SS2.2.p2.13.m13.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.2.p2.13.m13.1.1.2.3" xref="S3.SS2.2.p2.13.m13.1.1.2.3.cmml">U</mi><mrow id="S3.SS2.2.p2.13.m13.1.1.2.2.3" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.cmml"><mi id="S3.SS2.2.p2.13.m13.1.1.2.2.3.2" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.13.m13.1.1.2.2.3.1" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.1.cmml">+</mo><mi id="S3.SS2.2.p2.13.m13.1.1.2.2.3.3" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.3.cmml">t</mi></mrow></msubsup><mo id="S3.SS2.2.p2.13.m13.1.1.1" xref="S3.SS2.2.p2.13.m13.1.1.1.cmml">=</mo><mi id="S3.SS2.2.p2.13.m13.1.1.3" mathvariant="normal" xref="S3.SS2.2.p2.13.m13.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.13.m13.1b"><apply id="S3.SS2.2.p2.13.m13.1.1.cmml" xref="S3.SS2.2.p2.13.m13.1.1"><eq id="S3.SS2.2.p2.13.m13.1.1.1.cmml" xref="S3.SS2.2.p2.13.m13.1.1.1"></eq><apply id="S3.SS2.2.p2.13.m13.1.1.2.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.13.m13.1.1.2.1.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2">subscript</csymbol><apply id="S3.SS2.2.p2.13.m13.1.1.2.2.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.13.m13.1.1.2.2.1.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2">superscript</csymbol><ci id="S3.SS2.2.p2.13.m13.1.1.2.2.2.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.2.2">𝐶</ci><apply id="S3.SS2.2.p2.13.m13.1.1.2.2.3.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3"><plus id="S3.SS2.2.p2.13.m13.1.1.2.2.3.1.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.1"></plus><ci id="S3.SS2.2.p2.13.m13.1.1.2.2.3.2.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.2">𝑚</ci><ci id="S3.SS2.2.p2.13.m13.1.1.2.2.3.3.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.2.3.3">𝑡</ci></apply></apply><ci id="S3.SS2.2.p2.13.m13.1.1.2.3.cmml" xref="S3.SS2.2.p2.13.m13.1.1.2.3">𝑈</ci></apply><emptyset id="S3.SS2.2.p2.13.m13.1.1.3.cmml" xref="S3.SS2.2.p2.13.m13.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.13.m13.1c">C^{m+t}_{U}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.13.m13.1d">italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = ∅</annotation></semantics></math>, we get <math alttext="\beta_{m+t}=0" class="ltx_Math" display="inline" id="S3.SS2.2.p2.14.m14.1"><semantics id="S3.SS2.2.p2.14.m14.1a"><mrow id="S3.SS2.2.p2.14.m14.1.1" xref="S3.SS2.2.p2.14.m14.1.1.cmml"><msub id="S3.SS2.2.p2.14.m14.1.1.2" xref="S3.SS2.2.p2.14.m14.1.1.2.cmml"><mi id="S3.SS2.2.p2.14.m14.1.1.2.2" xref="S3.SS2.2.p2.14.m14.1.1.2.2.cmml">β</mi><mrow id="S3.SS2.2.p2.14.m14.1.1.2.3" xref="S3.SS2.2.p2.14.m14.1.1.2.3.cmml"><mi id="S3.SS2.2.p2.14.m14.1.1.2.3.2" xref="S3.SS2.2.p2.14.m14.1.1.2.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.14.m14.1.1.2.3.1" xref="S3.SS2.2.p2.14.m14.1.1.2.3.1.cmml">+</mo><mi id="S3.SS2.2.p2.14.m14.1.1.2.3.3" xref="S3.SS2.2.p2.14.m14.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.2.p2.14.m14.1.1.1" xref="S3.SS2.2.p2.14.m14.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.14.m14.1.1.3" xref="S3.SS2.2.p2.14.m14.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.14.m14.1b"><apply id="S3.SS2.2.p2.14.m14.1.1.cmml" xref="S3.SS2.2.p2.14.m14.1.1"><eq id="S3.SS2.2.p2.14.m14.1.1.1.cmml" xref="S3.SS2.2.p2.14.m14.1.1.1"></eq><apply id="S3.SS2.2.p2.14.m14.1.1.2.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.14.m14.1.1.2.1.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.14.m14.1.1.2.2.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2.2">𝛽</ci><apply id="S3.SS2.2.p2.14.m14.1.1.2.3.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2.3"><plus id="S3.SS2.2.p2.14.m14.1.1.2.3.1.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2.3.1"></plus><ci id="S3.SS2.2.p2.14.m14.1.1.2.3.2.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2.3.2">𝑚</ci><ci id="S3.SS2.2.p2.14.m14.1.1.2.3.3.cmml" xref="S3.SS2.2.p2.14.m14.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS2.2.p2.14.m14.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.14.m14.1c">\beta_{m+t}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.14.m14.1d">italic_β start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>. We complete the argument by showing that for every <math alttext="i\in[m+t]" class="ltx_Math" display="inline" id="S3.SS2.2.p2.15.m15.1"><semantics id="S3.SS2.2.p2.15.m15.1a"><mrow id="S3.SS2.2.p2.15.m15.1.1" xref="S3.SS2.2.p2.15.m15.1.1.cmml"><mi id="S3.SS2.2.p2.15.m15.1.1.3" xref="S3.SS2.2.p2.15.m15.1.1.3.cmml">i</mi><mo id="S3.SS2.2.p2.15.m15.1.1.2" xref="S3.SS2.2.p2.15.m15.1.1.2.cmml">∈</mo><mrow id="S3.SS2.2.p2.15.m15.1.1.1.1" xref="S3.SS2.2.p2.15.m15.1.1.1.2.cmml"><mo id="S3.SS2.2.p2.15.m15.1.1.1.1.2" stretchy="false" xref="S3.SS2.2.p2.15.m15.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS2.2.p2.15.m15.1.1.1.1.1" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.cmml"><mi id="S3.SS2.2.p2.15.m15.1.1.1.1.1.2" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.2.cmml">m</mi><mo id="S3.SS2.2.p2.15.m15.1.1.1.1.1.1" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS2.2.p2.15.m15.1.1.1.1.1.3" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.SS2.2.p2.15.m15.1.1.1.1.3" stretchy="false" xref="S3.SS2.2.p2.15.m15.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.15.m15.1b"><apply id="S3.SS2.2.p2.15.m15.1.1.cmml" xref="S3.SS2.2.p2.15.m15.1.1"><in id="S3.SS2.2.p2.15.m15.1.1.2.cmml" xref="S3.SS2.2.p2.15.m15.1.1.2"></in><ci id="S3.SS2.2.p2.15.m15.1.1.3.cmml" xref="S3.SS2.2.p2.15.m15.1.1.3">𝑖</ci><apply id="S3.SS2.2.p2.15.m15.1.1.1.2.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1"><csymbol cd="latexml" id="S3.SS2.2.p2.15.m15.1.1.1.2.1.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS2.2.p2.15.m15.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1"><plus id="S3.SS2.2.p2.15.m15.1.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.1"></plus><ci id="S3.SS2.2.p2.15.m15.1.1.1.1.1.2.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.2">𝑚</ci><ci id="S3.SS2.2.p2.15.m15.1.1.1.1.1.3.cmml" xref="S3.SS2.2.p2.15.m15.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.15.m15.1c">i\in[m+t]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.15.m15.1d">italic_i ∈ [ italic_m + italic_t ]</annotation></semantics></math>, <math alttext="\alpha_{i}\leq\beta_{i}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.16.m16.1"><semantics id="S3.SS2.2.p2.16.m16.1a"><mrow id="S3.SS2.2.p2.16.m16.1.1" xref="S3.SS2.2.p2.16.m16.1.1.cmml"><msub id="S3.SS2.2.p2.16.m16.1.1.2" xref="S3.SS2.2.p2.16.m16.1.1.2.cmml"><mi id="S3.SS2.2.p2.16.m16.1.1.2.2" xref="S3.SS2.2.p2.16.m16.1.1.2.2.cmml">α</mi><mi id="S3.SS2.2.p2.16.m16.1.1.2.3" xref="S3.SS2.2.p2.16.m16.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.2.p2.16.m16.1.1.1" xref="S3.SS2.2.p2.16.m16.1.1.1.cmml">≤</mo><msub id="S3.SS2.2.p2.16.m16.1.1.3" xref="S3.SS2.2.p2.16.m16.1.1.3.cmml"><mi id="S3.SS2.2.p2.16.m16.1.1.3.2" xref="S3.SS2.2.p2.16.m16.1.1.3.2.cmml">β</mi><mi id="S3.SS2.2.p2.16.m16.1.1.3.3" xref="S3.SS2.2.p2.16.m16.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.16.m16.1b"><apply id="S3.SS2.2.p2.16.m16.1.1.cmml" xref="S3.SS2.2.p2.16.m16.1.1"><leq id="S3.SS2.2.p2.16.m16.1.1.1.cmml" xref="S3.SS2.2.p2.16.m16.1.1.1"></leq><apply id="S3.SS2.2.p2.16.m16.1.1.2.cmml" xref="S3.SS2.2.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.16.m16.1.1.2.1.cmml" xref="S3.SS2.2.p2.16.m16.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.16.m16.1.1.2.2.cmml" xref="S3.SS2.2.p2.16.m16.1.1.2.2">𝛼</ci><ci id="S3.SS2.2.p2.16.m16.1.1.2.3.cmml" xref="S3.SS2.2.p2.16.m16.1.1.2.3">𝑖</ci></apply><apply id="S3.SS2.2.p2.16.m16.1.1.3.cmml" xref="S3.SS2.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.16.m16.1.1.3.1.cmml" xref="S3.SS2.2.p2.16.m16.1.1.3">subscript</csymbol><ci id="S3.SS2.2.p2.16.m16.1.1.3.2.cmml" xref="S3.SS2.2.p2.16.m16.1.1.3.2">𝛽</ci><ci id="S3.SS2.2.p2.16.m16.1.1.3.3.cmml" xref="S3.SS2.2.p2.16.m16.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.16.m16.1c">\alpha_{i}\leq\beta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.16.m16.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which is in contradiction to <math alttext="\alpha_{m+t}=1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.17.m17.1"><semantics id="S3.SS2.2.p2.17.m17.1a"><mrow id="S3.SS2.2.p2.17.m17.1.1" xref="S3.SS2.2.p2.17.m17.1.1.cmml"><msub id="S3.SS2.2.p2.17.m17.1.1.2" xref="S3.SS2.2.p2.17.m17.1.1.2.cmml"><mi id="S3.SS2.2.p2.17.m17.1.1.2.2" xref="S3.SS2.2.p2.17.m17.1.1.2.2.cmml">α</mi><mrow id="S3.SS2.2.p2.17.m17.1.1.2.3" xref="S3.SS2.2.p2.17.m17.1.1.2.3.cmml"><mi id="S3.SS2.2.p2.17.m17.1.1.2.3.2" xref="S3.SS2.2.p2.17.m17.1.1.2.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.17.m17.1.1.2.3.1" xref="S3.SS2.2.p2.17.m17.1.1.2.3.1.cmml">+</mo><mi id="S3.SS2.2.p2.17.m17.1.1.2.3.3" xref="S3.SS2.2.p2.17.m17.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.2.p2.17.m17.1.1.1" xref="S3.SS2.2.p2.17.m17.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.17.m17.1.1.3" xref="S3.SS2.2.p2.17.m17.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.17.m17.1b"><apply id="S3.SS2.2.p2.17.m17.1.1.cmml" xref="S3.SS2.2.p2.17.m17.1.1"><eq id="S3.SS2.2.p2.17.m17.1.1.1.cmml" xref="S3.SS2.2.p2.17.m17.1.1.1"></eq><apply id="S3.SS2.2.p2.17.m17.1.1.2.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.17.m17.1.1.2.1.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.17.m17.1.1.2.2.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2.2">𝛼</ci><apply id="S3.SS2.2.p2.17.m17.1.1.2.3.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2.3"><plus id="S3.SS2.2.p2.17.m17.1.1.2.3.1.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2.3.1"></plus><ci id="S3.SS2.2.p2.17.m17.1.1.2.3.2.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2.3.2">𝑚</ci><ci id="S3.SS2.2.p2.17.m17.1.1.2.3.3.cmml" xref="S3.SS2.2.p2.17.m17.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS2.2.p2.17.m17.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.17.m17.1c">\alpha_{m+t}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.17.m17.1d">italic_α start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{m+t}=0" class="ltx_Math" display="inline" id="S3.SS2.2.p2.18.m18.1"><semantics id="S3.SS2.2.p2.18.m18.1a"><mrow id="S3.SS2.2.p2.18.m18.1.1" xref="S3.SS2.2.p2.18.m18.1.1.cmml"><msub id="S3.SS2.2.p2.18.m18.1.1.2" xref="S3.SS2.2.p2.18.m18.1.1.2.cmml"><mi id="S3.SS2.2.p2.18.m18.1.1.2.2" xref="S3.SS2.2.p2.18.m18.1.1.2.2.cmml">β</mi><mrow id="S3.SS2.2.p2.18.m18.1.1.2.3" xref="S3.SS2.2.p2.18.m18.1.1.2.3.cmml"><mi id="S3.SS2.2.p2.18.m18.1.1.2.3.2" xref="S3.SS2.2.p2.18.m18.1.1.2.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.18.m18.1.1.2.3.1" xref="S3.SS2.2.p2.18.m18.1.1.2.3.1.cmml">+</mo><mi id="S3.SS2.2.p2.18.m18.1.1.2.3.3" xref="S3.SS2.2.p2.18.m18.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS2.2.p2.18.m18.1.1.1" xref="S3.SS2.2.p2.18.m18.1.1.1.cmml">=</mo><mn id="S3.SS2.2.p2.18.m18.1.1.3" xref="S3.SS2.2.p2.18.m18.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.18.m18.1b"><apply id="S3.SS2.2.p2.18.m18.1.1.cmml" xref="S3.SS2.2.p2.18.m18.1.1"><eq id="S3.SS2.2.p2.18.m18.1.1.1.cmml" xref="S3.SS2.2.p2.18.m18.1.1.1"></eq><apply id="S3.SS2.2.p2.18.m18.1.1.2.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.18.m18.1.1.2.1.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.18.m18.1.1.2.2.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2.2">𝛽</ci><apply id="S3.SS2.2.p2.18.m18.1.1.2.3.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2.3"><plus id="S3.SS2.2.p2.18.m18.1.1.2.3.1.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2.3.1"></plus><ci id="S3.SS2.2.p2.18.m18.1.1.2.3.2.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2.3.2">𝑚</ci><ci id="S3.SS2.2.p2.18.m18.1.1.2.3.3.cmml" xref="S3.SS2.2.p2.18.m18.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS2.2.p2.18.m18.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.18.m18.1c">\beta_{m+t}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.18.m18.1d">italic_β start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem23"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem23.1.1.1">Claim 23</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem23.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem23.p1"> <p class="ltx_p" id="Thmtheorem23.p1.8"><span class="ltx_text ltx_font_italic" id="Thmtheorem23.p1.8.8">Suppose <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem23.p1.1.1.m1.1"><semantics id="Thmtheorem23.p1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem23.p1.1.1.m1.1.1" xref="Thmtheorem23.p1.1.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.1.1.m1.1b"><ci id="Thmtheorem23.p1.1.1.m1.1.1.cmml" xref="Thmtheorem23.p1.1.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.1.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.1.1.m1.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="a\in A" class="ltx_Math" display="inline" id="Thmtheorem23.p1.2.2.m2.1"><semantics id="Thmtheorem23.p1.2.2.m2.1a"><mrow id="Thmtheorem23.p1.2.2.m2.1.1" xref="Thmtheorem23.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem23.p1.2.2.m2.1.1.2" xref="Thmtheorem23.p1.2.2.m2.1.1.2.cmml">a</mi><mo id="Thmtheorem23.p1.2.2.m2.1.1.1" xref="Thmtheorem23.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="Thmtheorem23.p1.2.2.m2.1.1.3" xref="Thmtheorem23.p1.2.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.2.2.m2.1b"><apply id="Thmtheorem23.p1.2.2.m2.1.1.cmml" xref="Thmtheorem23.p1.2.2.m2.1.1"><in id="Thmtheorem23.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem23.p1.2.2.m2.1.1.1"></in><ci id="Thmtheorem23.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem23.p1.2.2.m2.1.1.2">𝑎</ci><ci id="Thmtheorem23.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem23.p1.2.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.2.2.m2.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.2.2.m2.1d">italic_a ∈ italic_A</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem23.p1.3.3.m3.1"><semantics id="Thmtheorem23.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem23.p1.3.3.m3.1.1" xref="Thmtheorem23.p1.3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.3.3.m3.1b"><ci id="Thmtheorem23.p1.3.3.m3.1.1.cmml" xref="Thmtheorem23.p1.3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.3.3.m3.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem23.p1.4.4.m4.1"><semantics id="Thmtheorem23.p1.4.4.m4.1a"><mi id="Thmtheorem23.p1.4.4.m4.1.1" xref="Thmtheorem23.p1.4.4.m4.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.4.4.m4.1b"><ci id="Thmtheorem23.p1.4.4.m4.1.1.cmml" xref="Thmtheorem23.p1.4.4.m4.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.4.4.m4.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.4.4.m4.1d">italic_U</annotation></semantics></math>, and that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem23.p1.5.5.m5.1"><semantics id="Thmtheorem23.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem23.p1.5.5.m5.1.1" xref="Thmtheorem23.p1.5.5.m5.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.5.5.m5.1b"><ci id="Thmtheorem23.p1.5.5.m5.1.1.cmml" xref="Thmtheorem23.p1.5.5.m5.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.5.5.m5.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.5.5.m5.1d">caligraphic_F</annotation></semantics></math> preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="Thmtheorem23.p1.6.6.m6.1"><semantics id="Thmtheorem23.p1.6.6.m6.1a"><mi id="Thmtheorem23.p1.6.6.m6.1.1" mathvariant="normal" xref="Thmtheorem23.p1.6.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.6.6.m6.1b"><ci id="Thmtheorem23.p1.6.6.m6.1.1.cmml" xref="Thmtheorem23.p1.6.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.6.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.6.6.m6.1d">roman_Λ</annotation></semantics></math>, the set of intersection operations in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E5" title="Equation 5 ‣ Proof. ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">5</span></a>. Then for every <math alttext="i\in[m+t]" class="ltx_Math" display="inline" id="Thmtheorem23.p1.7.7.m7.1"><semantics id="Thmtheorem23.p1.7.7.m7.1a"><mrow id="Thmtheorem23.p1.7.7.m7.1.1" xref="Thmtheorem23.p1.7.7.m7.1.1.cmml"><mi id="Thmtheorem23.p1.7.7.m7.1.1.3" xref="Thmtheorem23.p1.7.7.m7.1.1.3.cmml">i</mi><mo id="Thmtheorem23.p1.7.7.m7.1.1.2" xref="Thmtheorem23.p1.7.7.m7.1.1.2.cmml">∈</mo><mrow id="Thmtheorem23.p1.7.7.m7.1.1.1.1" xref="Thmtheorem23.p1.7.7.m7.1.1.1.2.cmml"><mo id="Thmtheorem23.p1.7.7.m7.1.1.1.1.2" stretchy="false" xref="Thmtheorem23.p1.7.7.m7.1.1.1.2.1.cmml">[</mo><mrow id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.cmml"><mi id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.2" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.2.cmml">m</mi><mo id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.1" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.1.cmml">+</mo><mi id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.3" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="Thmtheorem23.p1.7.7.m7.1.1.1.1.3" stretchy="false" xref="Thmtheorem23.p1.7.7.m7.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.7.7.m7.1b"><apply id="Thmtheorem23.p1.7.7.m7.1.1.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1"><in id="Thmtheorem23.p1.7.7.m7.1.1.2.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.2"></in><ci id="Thmtheorem23.p1.7.7.m7.1.1.3.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.3">𝑖</ci><apply id="Thmtheorem23.p1.7.7.m7.1.1.1.2.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem23.p1.7.7.m7.1.1.1.2.1.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.2">delimited-[]</csymbol><apply id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1"><plus id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.1"></plus><ci id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.2.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.2">𝑚</ci><ci id="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.3.cmml" xref="Thmtheorem23.p1.7.7.m7.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.7.7.m7.1c">i\in[m+t]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.7.7.m7.1d">italic_i ∈ [ italic_m + italic_t ]</annotation></semantics></math>, <math alttext="\alpha_{i}\leq\beta_{i}" class="ltx_Math" display="inline" id="Thmtheorem23.p1.8.8.m8.1"><semantics id="Thmtheorem23.p1.8.8.m8.1a"><mrow id="Thmtheorem23.p1.8.8.m8.1.1" xref="Thmtheorem23.p1.8.8.m8.1.1.cmml"><msub id="Thmtheorem23.p1.8.8.m8.1.1.2" xref="Thmtheorem23.p1.8.8.m8.1.1.2.cmml"><mi id="Thmtheorem23.p1.8.8.m8.1.1.2.2" xref="Thmtheorem23.p1.8.8.m8.1.1.2.2.cmml">α</mi><mi id="Thmtheorem23.p1.8.8.m8.1.1.2.3" xref="Thmtheorem23.p1.8.8.m8.1.1.2.3.cmml">i</mi></msub><mo id="Thmtheorem23.p1.8.8.m8.1.1.1" xref="Thmtheorem23.p1.8.8.m8.1.1.1.cmml">≤</mo><msub id="Thmtheorem23.p1.8.8.m8.1.1.3" xref="Thmtheorem23.p1.8.8.m8.1.1.3.cmml"><mi id="Thmtheorem23.p1.8.8.m8.1.1.3.2" xref="Thmtheorem23.p1.8.8.m8.1.1.3.2.cmml">β</mi><mi id="Thmtheorem23.p1.8.8.m8.1.1.3.3" xref="Thmtheorem23.p1.8.8.m8.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem23.p1.8.8.m8.1b"><apply id="Thmtheorem23.p1.8.8.m8.1.1.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1"><leq id="Thmtheorem23.p1.8.8.m8.1.1.1.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.1"></leq><apply id="Thmtheorem23.p1.8.8.m8.1.1.2.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem23.p1.8.8.m8.1.1.2.1.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.2">subscript</csymbol><ci id="Thmtheorem23.p1.8.8.m8.1.1.2.2.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.2.2">𝛼</ci><ci id="Thmtheorem23.p1.8.8.m8.1.1.2.3.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.2.3">𝑖</ci></apply><apply id="Thmtheorem23.p1.8.8.m8.1.1.3.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem23.p1.8.8.m8.1.1.3.1.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="Thmtheorem23.p1.8.8.m8.1.1.3.2.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.3.2">𝛽</ci><ci id="Thmtheorem23.p1.8.8.m8.1.1.3.3.cmml" xref="Thmtheorem23.p1.8.8.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem23.p1.8.8.m8.1c">\alpha_{i}\leq\beta_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem23.p1.8.8.m8.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.3.p3"> <p class="ltx_p" id="S3.SS2.3.p3.14">The proof is by induction on <math alttext="i" class="ltx_Math" display="inline" id="S3.SS2.3.p3.1.m1.1"><semantics id="S3.SS2.3.p3.1.m1.1a"><mi id="S3.SS2.3.p3.1.m1.1.1" xref="S3.SS2.3.p3.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.1.m1.1b"><ci id="S3.SS2.3.p3.1.m1.1.1.cmml" xref="S3.SS2.3.p3.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.1.m1.1d">italic_i</annotation></semantics></math>. For the base case, we consider <math alttext="i\leq m" class="ltx_Math" display="inline" id="S3.SS2.3.p3.2.m2.1"><semantics id="S3.SS2.3.p3.2.m2.1a"><mrow id="S3.SS2.3.p3.2.m2.1.1" xref="S3.SS2.3.p3.2.m2.1.1.cmml"><mi id="S3.SS2.3.p3.2.m2.1.1.2" xref="S3.SS2.3.p3.2.m2.1.1.2.cmml">i</mi><mo id="S3.SS2.3.p3.2.m2.1.1.1" xref="S3.SS2.3.p3.2.m2.1.1.1.cmml">≤</mo><mi id="S3.SS2.3.p3.2.m2.1.1.3" xref="S3.SS2.3.p3.2.m2.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.2.m2.1b"><apply id="S3.SS2.3.p3.2.m2.1.1.cmml" xref="S3.SS2.3.p3.2.m2.1.1"><leq id="S3.SS2.3.p3.2.m2.1.1.1.cmml" xref="S3.SS2.3.p3.2.m2.1.1.1"></leq><ci id="S3.SS2.3.p3.2.m2.1.1.2.cmml" xref="S3.SS2.3.p3.2.m2.1.1.2">𝑖</ci><ci id="S3.SS2.3.p3.2.m2.1.1.3.cmml" xref="S3.SS2.3.p3.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.2.m2.1c">i\leq m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.2.m2.1d">italic_i ≤ italic_m</annotation></semantics></math>. Since <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.3.m3.1"><semantics id="S3.SS2.3.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.3.p3.3.m3.1.1" xref="S3.SS2.3.p3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.3.m3.1b"><ci id="S3.SS2.3.p3.3.m3.1.1.cmml" xref="S3.SS2.3.p3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.3.m3.1d">caligraphic_B</annotation></semantics></math> is non-empty, <math alttext="m\geq 1" class="ltx_Math" display="inline" id="S3.SS2.3.p3.4.m4.1"><semantics id="S3.SS2.3.p3.4.m4.1a"><mrow id="S3.SS2.3.p3.4.m4.1.1" xref="S3.SS2.3.p3.4.m4.1.1.cmml"><mi id="S3.SS2.3.p3.4.m4.1.1.2" xref="S3.SS2.3.p3.4.m4.1.1.2.cmml">m</mi><mo id="S3.SS2.3.p3.4.m4.1.1.1" xref="S3.SS2.3.p3.4.m4.1.1.1.cmml">≥</mo><mn id="S3.SS2.3.p3.4.m4.1.1.3" xref="S3.SS2.3.p3.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.4.m4.1b"><apply id="S3.SS2.3.p3.4.m4.1.1.cmml" xref="S3.SS2.3.p3.4.m4.1.1"><geq id="S3.SS2.3.p3.4.m4.1.1.1.cmml" xref="S3.SS2.3.p3.4.m4.1.1.1"></geq><ci id="S3.SS2.3.p3.4.m4.1.1.2.cmml" xref="S3.SS2.3.p3.4.m4.1.1.2">𝑚</ci><cn id="S3.SS2.3.p3.4.m4.1.1.3.cmml" type="integer" xref="S3.SS2.3.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.4.m4.1c">m\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.4.m4.1d">italic_m ≥ 1</annotation></semantics></math>. Now if <math alttext="\alpha_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.3.p3.5.m5.1"><semantics id="S3.SS2.3.p3.5.m5.1a"><mrow id="S3.SS2.3.p3.5.m5.1.1" xref="S3.SS2.3.p3.5.m5.1.1.cmml"><msub id="S3.SS2.3.p3.5.m5.1.1.2" xref="S3.SS2.3.p3.5.m5.1.1.2.cmml"><mi id="S3.SS2.3.p3.5.m5.1.1.2.2" xref="S3.SS2.3.p3.5.m5.1.1.2.2.cmml">α</mi><mi id="S3.SS2.3.p3.5.m5.1.1.2.3" xref="S3.SS2.3.p3.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.3.p3.5.m5.1.1.1" xref="S3.SS2.3.p3.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS2.3.p3.5.m5.1.1.3" xref="S3.SS2.3.p3.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.5.m5.1b"><apply id="S3.SS2.3.p3.5.m5.1.1.cmml" xref="S3.SS2.3.p3.5.m5.1.1"><eq id="S3.SS2.3.p3.5.m5.1.1.1.cmml" xref="S3.SS2.3.p3.5.m5.1.1.1"></eq><apply id="S3.SS2.3.p3.5.m5.1.1.2.cmml" xref="S3.SS2.3.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.5.m5.1.1.2.1.cmml" xref="S3.SS2.3.p3.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS2.3.p3.5.m5.1.1.2.2.cmml" xref="S3.SS2.3.p3.5.m5.1.1.2.2">𝛼</ci><ci id="S3.SS2.3.p3.5.m5.1.1.2.3.cmml" xref="S3.SS2.3.p3.5.m5.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.3.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.SS2.3.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.5.m5.1c">\alpha_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.5.m5.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>, then <math alttext="a\in C^{i}=B" class="ltx_Math" display="inline" id="S3.SS2.3.p3.6.m6.1"><semantics id="S3.SS2.3.p3.6.m6.1a"><mrow id="S3.SS2.3.p3.6.m6.1.1" xref="S3.SS2.3.p3.6.m6.1.1.cmml"><mi id="S3.SS2.3.p3.6.m6.1.1.2" xref="S3.SS2.3.p3.6.m6.1.1.2.cmml">a</mi><mo id="S3.SS2.3.p3.6.m6.1.1.3" xref="S3.SS2.3.p3.6.m6.1.1.3.cmml">∈</mo><msup id="S3.SS2.3.p3.6.m6.1.1.4" xref="S3.SS2.3.p3.6.m6.1.1.4.cmml"><mi id="S3.SS2.3.p3.6.m6.1.1.4.2" xref="S3.SS2.3.p3.6.m6.1.1.4.2.cmml">C</mi><mi id="S3.SS2.3.p3.6.m6.1.1.4.3" xref="S3.SS2.3.p3.6.m6.1.1.4.3.cmml">i</mi></msup><mo id="S3.SS2.3.p3.6.m6.1.1.5" xref="S3.SS2.3.p3.6.m6.1.1.5.cmml">=</mo><mi id="S3.SS2.3.p3.6.m6.1.1.6" xref="S3.SS2.3.p3.6.m6.1.1.6.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.6.m6.1b"><apply id="S3.SS2.3.p3.6.m6.1.1.cmml" xref="S3.SS2.3.p3.6.m6.1.1"><and id="S3.SS2.3.p3.6.m6.1.1a.cmml" xref="S3.SS2.3.p3.6.m6.1.1"></and><apply id="S3.SS2.3.p3.6.m6.1.1b.cmml" xref="S3.SS2.3.p3.6.m6.1.1"><in id="S3.SS2.3.p3.6.m6.1.1.3.cmml" xref="S3.SS2.3.p3.6.m6.1.1.3"></in><ci id="S3.SS2.3.p3.6.m6.1.1.2.cmml" xref="S3.SS2.3.p3.6.m6.1.1.2">𝑎</ci><apply id="S3.SS2.3.p3.6.m6.1.1.4.cmml" xref="S3.SS2.3.p3.6.m6.1.1.4"><csymbol cd="ambiguous" id="S3.SS2.3.p3.6.m6.1.1.4.1.cmml" xref="S3.SS2.3.p3.6.m6.1.1.4">superscript</csymbol><ci id="S3.SS2.3.p3.6.m6.1.1.4.2.cmml" xref="S3.SS2.3.p3.6.m6.1.1.4.2">𝐶</ci><ci id="S3.SS2.3.p3.6.m6.1.1.4.3.cmml" xref="S3.SS2.3.p3.6.m6.1.1.4.3">𝑖</ci></apply></apply><apply id="S3.SS2.3.p3.6.m6.1.1c.cmml" xref="S3.SS2.3.p3.6.m6.1.1"><eq id="S3.SS2.3.p3.6.m6.1.1.5.cmml" xref="S3.SS2.3.p3.6.m6.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.3.p3.6.m6.1.1.4.cmml" id="S3.SS2.3.p3.6.m6.1.1d.cmml" xref="S3.SS2.3.p3.6.m6.1.1"></share><ci id="S3.SS2.3.p3.6.m6.1.1.6.cmml" xref="S3.SS2.3.p3.6.m6.1.1.6">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.6.m6.1c">a\in C^{i}=B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.6.m6.1d">italic_a ∈ italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_B</annotation></semantics></math> for some <math alttext="B\in\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.7.m7.1"><semantics id="S3.SS2.3.p3.7.m7.1a"><mrow id="S3.SS2.3.p3.7.m7.1.1" xref="S3.SS2.3.p3.7.m7.1.1.cmml"><mi id="S3.SS2.3.p3.7.m7.1.1.2" xref="S3.SS2.3.p3.7.m7.1.1.2.cmml">B</mi><mo id="S3.SS2.3.p3.7.m7.1.1.1" xref="S3.SS2.3.p3.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.3.p3.7.m7.1.1.3" xref="S3.SS2.3.p3.7.m7.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.7.m7.1b"><apply id="S3.SS2.3.p3.7.m7.1.1.cmml" xref="S3.SS2.3.p3.7.m7.1.1"><in id="S3.SS2.3.p3.7.m7.1.1.1.cmml" xref="S3.SS2.3.p3.7.m7.1.1.1"></in><ci id="S3.SS2.3.p3.7.m7.1.1.2.cmml" xref="S3.SS2.3.p3.7.m7.1.1.2">𝐵</ci><ci id="S3.SS2.3.p3.7.m7.1.1.3.cmml" xref="S3.SS2.3.p3.7.m7.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.7.m7.1c">B\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.7.m7.1d">italic_B ∈ caligraphic_B</annotation></semantics></math>. Since <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.8.m8.1"><semantics id="S3.SS2.3.p3.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.3.p3.8.m8.1.1" xref="S3.SS2.3.p3.8.m8.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.8.m8.1b"><ci id="S3.SS2.3.p3.8.m8.1.1.cmml" xref="S3.SS2.3.p3.8.m8.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.8.m8.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.8.m8.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="a" class="ltx_Math" display="inline" id="S3.SS2.3.p3.9.m9.1"><semantics id="S3.SS2.3.p3.9.m9.1a"><mi id="S3.SS2.3.p3.9.m9.1.1" xref="S3.SS2.3.p3.9.m9.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.9.m9.1b"><ci id="S3.SS2.3.p3.9.m9.1.1.cmml" xref="S3.SS2.3.p3.9.m9.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.9.m9.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.9.m9.1d">italic_a</annotation></semantics></math> (with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.10.m10.1"><semantics id="S3.SS2.3.p3.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.3.p3.10.m10.1.1" xref="S3.SS2.3.p3.10.m10.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.10.m10.1b"><ci id="S3.SS2.3.p3.10.m10.1.1.cmml" xref="S3.SS2.3.p3.10.m10.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.10.m10.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.10.m10.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="S3.SS2.3.p3.11.m11.1"><semantics id="S3.SS2.3.p3.11.m11.1a"><mi id="S3.SS2.3.p3.11.m11.1.1" xref="S3.SS2.3.p3.11.m11.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.11.m11.1b"><ci id="S3.SS2.3.p3.11.m11.1.1.cmml" xref="S3.SS2.3.p3.11.m11.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.11.m11.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.11.m11.1d">italic_U</annotation></semantics></math>) and <math alttext="a\in B" class="ltx_Math" display="inline" id="S3.SS2.3.p3.12.m12.1"><semantics id="S3.SS2.3.p3.12.m12.1a"><mrow id="S3.SS2.3.p3.12.m12.1.1" xref="S3.SS2.3.p3.12.m12.1.1.cmml"><mi id="S3.SS2.3.p3.12.m12.1.1.2" xref="S3.SS2.3.p3.12.m12.1.1.2.cmml">a</mi><mo id="S3.SS2.3.p3.12.m12.1.1.1" xref="S3.SS2.3.p3.12.m12.1.1.1.cmml">∈</mo><mi id="S3.SS2.3.p3.12.m12.1.1.3" xref="S3.SS2.3.p3.12.m12.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.12.m12.1b"><apply id="S3.SS2.3.p3.12.m12.1.1.cmml" xref="S3.SS2.3.p3.12.m12.1.1"><in id="S3.SS2.3.p3.12.m12.1.1.1.cmml" xref="S3.SS2.3.p3.12.m12.1.1.1"></in><ci id="S3.SS2.3.p3.12.m12.1.1.2.cmml" xref="S3.SS2.3.p3.12.m12.1.1.2">𝑎</ci><ci id="S3.SS2.3.p3.12.m12.1.1.3.cmml" xref="S3.SS2.3.p3.12.m12.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.12.m12.1c">a\in B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.12.m12.1d">italic_a ∈ italic_B</annotation></semantics></math>, <math alttext="C^{i}_{U}=B_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.13.m13.1"><semantics id="S3.SS2.3.p3.13.m13.1a"><mrow id="S3.SS2.3.p3.13.m13.1.1" xref="S3.SS2.3.p3.13.m13.1.1.cmml"><msubsup id="S3.SS2.3.p3.13.m13.1.1.2" xref="S3.SS2.3.p3.13.m13.1.1.2.cmml"><mi id="S3.SS2.3.p3.13.m13.1.1.2.2.2" xref="S3.SS2.3.p3.13.m13.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.3.p3.13.m13.1.1.2.3" xref="S3.SS2.3.p3.13.m13.1.1.2.3.cmml">U</mi><mi id="S3.SS2.3.p3.13.m13.1.1.2.2.3" xref="S3.SS2.3.p3.13.m13.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.3.p3.13.m13.1.1.3" xref="S3.SS2.3.p3.13.m13.1.1.3.cmml">=</mo><msub id="S3.SS2.3.p3.13.m13.1.1.4" xref="S3.SS2.3.p3.13.m13.1.1.4.cmml"><mi id="S3.SS2.3.p3.13.m13.1.1.4.2" xref="S3.SS2.3.p3.13.m13.1.1.4.2.cmml">B</mi><mi id="S3.SS2.3.p3.13.m13.1.1.4.3" xref="S3.SS2.3.p3.13.m13.1.1.4.3.cmml">U</mi></msub><mo id="S3.SS2.3.p3.13.m13.1.1.5" xref="S3.SS2.3.p3.13.m13.1.1.5.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.3.p3.13.m13.1.1.6" xref="S3.SS2.3.p3.13.m13.1.1.6.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.13.m13.1b"><apply id="S3.SS2.3.p3.13.m13.1.1.cmml" xref="S3.SS2.3.p3.13.m13.1.1"><and id="S3.SS2.3.p3.13.m13.1.1a.cmml" xref="S3.SS2.3.p3.13.m13.1.1"></and><apply id="S3.SS2.3.p3.13.m13.1.1b.cmml" xref="S3.SS2.3.p3.13.m13.1.1"><eq id="S3.SS2.3.p3.13.m13.1.1.3.cmml" xref="S3.SS2.3.p3.13.m13.1.1.3"></eq><apply id="S3.SS2.3.p3.13.m13.1.1.2.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.13.m13.1.1.2.1.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2">subscript</csymbol><apply id="S3.SS2.3.p3.13.m13.1.1.2.2.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.13.m13.1.1.2.2.1.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2">superscript</csymbol><ci id="S3.SS2.3.p3.13.m13.1.1.2.2.2.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2.2.2">𝐶</ci><ci id="S3.SS2.3.p3.13.m13.1.1.2.2.3.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.3.p3.13.m13.1.1.2.3.cmml" xref="S3.SS2.3.p3.13.m13.1.1.2.3">𝑈</ci></apply><apply id="S3.SS2.3.p3.13.m13.1.1.4.cmml" xref="S3.SS2.3.p3.13.m13.1.1.4"><csymbol cd="ambiguous" id="S3.SS2.3.p3.13.m13.1.1.4.1.cmml" xref="S3.SS2.3.p3.13.m13.1.1.4">subscript</csymbol><ci id="S3.SS2.3.p3.13.m13.1.1.4.2.cmml" xref="S3.SS2.3.p3.13.m13.1.1.4.2">𝐵</ci><ci id="S3.SS2.3.p3.13.m13.1.1.4.3.cmml" xref="S3.SS2.3.p3.13.m13.1.1.4.3">𝑈</ci></apply></apply><apply id="S3.SS2.3.p3.13.m13.1.1c.cmml" xref="S3.SS2.3.p3.13.m13.1.1"><in id="S3.SS2.3.p3.13.m13.1.1.5.cmml" xref="S3.SS2.3.p3.13.m13.1.1.5"></in><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.3.p3.13.m13.1.1.4.cmml" id="S3.SS2.3.p3.13.m13.1.1d.cmml" xref="S3.SS2.3.p3.13.m13.1.1"></share><ci id="S3.SS2.3.p3.13.m13.1.1.6.cmml" xref="S3.SS2.3.p3.13.m13.1.1.6">ℱ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.13.m13.1c">C^{i}_{U}=B_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.13.m13.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>, and consequently <math alttext="\beta_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.3.p3.14.m14.1"><semantics id="S3.SS2.3.p3.14.m14.1a"><mrow id="S3.SS2.3.p3.14.m14.1.1" xref="S3.SS2.3.p3.14.m14.1.1.cmml"><msub id="S3.SS2.3.p3.14.m14.1.1.2" xref="S3.SS2.3.p3.14.m14.1.1.2.cmml"><mi id="S3.SS2.3.p3.14.m14.1.1.2.2" xref="S3.SS2.3.p3.14.m14.1.1.2.2.cmml">β</mi><mi id="S3.SS2.3.p3.14.m14.1.1.2.3" xref="S3.SS2.3.p3.14.m14.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.3.p3.14.m14.1.1.1" xref="S3.SS2.3.p3.14.m14.1.1.1.cmml">=</mo><mn id="S3.SS2.3.p3.14.m14.1.1.3" xref="S3.SS2.3.p3.14.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.14.m14.1b"><apply id="S3.SS2.3.p3.14.m14.1.1.cmml" xref="S3.SS2.3.p3.14.m14.1.1"><eq id="S3.SS2.3.p3.14.m14.1.1.1.cmml" xref="S3.SS2.3.p3.14.m14.1.1.1"></eq><apply id="S3.SS2.3.p3.14.m14.1.1.2.cmml" xref="S3.SS2.3.p3.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.14.m14.1.1.2.1.cmml" xref="S3.SS2.3.p3.14.m14.1.1.2">subscript</csymbol><ci id="S3.SS2.3.p3.14.m14.1.1.2.2.cmml" xref="S3.SS2.3.p3.14.m14.1.1.2.2">𝛽</ci><ci id="S3.SS2.3.p3.14.m14.1.1.2.3.cmml" xref="S3.SS2.3.p3.14.m14.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.3.p3.14.m14.1.1.3.cmml" type="integer" xref="S3.SS2.3.p3.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.14.m14.1c">\beta_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.14.m14.1d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>. This completes the base case.</p> </div> <div class="ltx_para" id="S3.SS2.4.p4"> <p class="ltx_p" id="S3.SS2.4.p4.17">The induction step follows from the induction hypothesis and the upward closure of <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.1.m1.1"><semantics id="S3.SS2.4.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.4.p4.1.m1.1.1" xref="S3.SS2.4.p4.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.1.m1.1b"><ci id="S3.SS2.4.p4.1.m1.1.1.cmml" xref="S3.SS2.4.p4.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.1.m1.1d">caligraphic_F</annotation></semantics></math> in the case of a union operation, and from the induction hypothesis and the fact that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.2.m2.1"><semantics id="S3.SS2.4.p4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.4.p4.2.m2.1.1" xref="S3.SS2.4.p4.2.m2.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.2.m2.1b"><ci id="S3.SS2.4.p4.2.m2.1.1.cmml" xref="S3.SS2.4.p4.2.m2.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.2.m2.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.2.m2.1d">caligraphic_F</annotation></semantics></math> preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS2.4.p4.3.m3.1"><semantics id="S3.SS2.4.p4.3.m3.1a"><mi id="S3.SS2.4.p4.3.m3.1.1" mathvariant="normal" xref="S3.SS2.4.p4.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.3.m3.1b"><ci id="S3.SS2.4.p4.3.m3.1.1.cmml" xref="S3.SS2.4.p4.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.3.m3.1d">roman_Λ</annotation></semantics></math> in the case of an intersection operation. For instance, suppose that <math alttext="C^{i}=C^{i_{1}}\cap C^{i_{2}}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.4.m4.1"><semantics id="S3.SS2.4.p4.4.m4.1a"><mrow id="S3.SS2.4.p4.4.m4.1.1" xref="S3.SS2.4.p4.4.m4.1.1.cmml"><msup id="S3.SS2.4.p4.4.m4.1.1.2" xref="S3.SS2.4.p4.4.m4.1.1.2.cmml"><mi id="S3.SS2.4.p4.4.m4.1.1.2.2" xref="S3.SS2.4.p4.4.m4.1.1.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.4.m4.1.1.2.3" xref="S3.SS2.4.p4.4.m4.1.1.2.3.cmml">i</mi></msup><mo id="S3.SS2.4.p4.4.m4.1.1.1" xref="S3.SS2.4.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS2.4.p4.4.m4.1.1.3" xref="S3.SS2.4.p4.4.m4.1.1.3.cmml"><msup id="S3.SS2.4.p4.4.m4.1.1.3.2" xref="S3.SS2.4.p4.4.m4.1.1.3.2.cmml"><mi id="S3.SS2.4.p4.4.m4.1.1.3.2.2" xref="S3.SS2.4.p4.4.m4.1.1.3.2.2.cmml">C</mi><msub id="S3.SS2.4.p4.4.m4.1.1.3.2.3" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3.cmml"><mi id="S3.SS2.4.p4.4.m4.1.1.3.2.3.2" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.4.m4.1.1.3.2.3.3" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3.3.cmml">1</mn></msub></msup><mo id="S3.SS2.4.p4.4.m4.1.1.3.1" xref="S3.SS2.4.p4.4.m4.1.1.3.1.cmml">∩</mo><msup id="S3.SS2.4.p4.4.m4.1.1.3.3" xref="S3.SS2.4.p4.4.m4.1.1.3.3.cmml"><mi id="S3.SS2.4.p4.4.m4.1.1.3.3.2" xref="S3.SS2.4.p4.4.m4.1.1.3.3.2.cmml">C</mi><msub id="S3.SS2.4.p4.4.m4.1.1.3.3.3" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3.cmml"><mi id="S3.SS2.4.p4.4.m4.1.1.3.3.3.2" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.4.m4.1.1.3.3.3.3" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3.3.cmml">2</mn></msub></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.4.m4.1b"><apply id="S3.SS2.4.p4.4.m4.1.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1"><eq id="S3.SS2.4.p4.4.m4.1.1.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.1"></eq><apply id="S3.SS2.4.p4.4.m4.1.1.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m4.1.1.2.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.2">superscript</csymbol><ci id="S3.SS2.4.p4.4.m4.1.1.2.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.2.2">𝐶</ci><ci id="S3.SS2.4.p4.4.m4.1.1.2.3.cmml" xref="S3.SS2.4.p4.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.SS2.4.p4.4.m4.1.1.3.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3"><intersect id="S3.SS2.4.p4.4.m4.1.1.3.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.1"></intersect><apply id="S3.SS2.4.p4.4.m4.1.1.3.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m4.1.1.3.2.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.SS2.4.p4.4.m4.1.1.3.2.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2.2">𝐶</ci><apply id="S3.SS2.4.p4.4.m4.1.1.3.2.3.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m4.1.1.3.2.3.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.4.m4.1.1.3.2.3.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.4.m4.1.1.3.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.4.m4.1.1.3.2.3.3">1</cn></apply></apply><apply id="S3.SS2.4.p4.4.m4.1.1.3.3.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m4.1.1.3.3.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3">superscript</csymbol><ci id="S3.SS2.4.p4.4.m4.1.1.3.3.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3.2">𝐶</ci><apply id="S3.SS2.4.p4.4.m4.1.1.3.3.3.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m4.1.1.3.3.3.1.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3">subscript</csymbol><ci id="S3.SS2.4.p4.4.m4.1.1.3.3.3.2.cmml" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3.2">𝑖</ci><cn id="S3.SS2.4.p4.4.m4.1.1.3.3.3.3.cmml" type="integer" xref="S3.SS2.4.p4.4.m4.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.4.m4.1c">C^{i}=C^{i_{1}}\cap C^{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.4.m4.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∩ italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="C^{i}_{U}=C^{i_{1}}_{U}\cap C^{i_{2}}_{U}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.5.m5.1"><semantics id="S3.SS2.4.p4.5.m5.1a"><mrow id="S3.SS2.4.p4.5.m5.1.1" xref="S3.SS2.4.p4.5.m5.1.1.cmml"><msubsup id="S3.SS2.4.p4.5.m5.1.1.2" xref="S3.SS2.4.p4.5.m5.1.1.2.cmml"><mi id="S3.SS2.4.p4.5.m5.1.1.2.2.2" xref="S3.SS2.4.p4.5.m5.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.5.m5.1.1.2.3" xref="S3.SS2.4.p4.5.m5.1.1.2.3.cmml">U</mi><mi id="S3.SS2.4.p4.5.m5.1.1.2.2.3" xref="S3.SS2.4.p4.5.m5.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.4.p4.5.m5.1.1.1" xref="S3.SS2.4.p4.5.m5.1.1.1.cmml">=</mo><mrow id="S3.SS2.4.p4.5.m5.1.1.3" xref="S3.SS2.4.p4.5.m5.1.1.3.cmml"><msubsup id="S3.SS2.4.p4.5.m5.1.1.3.2" xref="S3.SS2.4.p4.5.m5.1.1.3.2.cmml"><mi id="S3.SS2.4.p4.5.m5.1.1.3.2.2.2" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.5.m5.1.1.3.2.3" xref="S3.SS2.4.p4.5.m5.1.1.3.2.3.cmml">U</mi><msub id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.cmml"><mi id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.2" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.3" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.3.cmml">1</mn></msub></msubsup><mo id="S3.SS2.4.p4.5.m5.1.1.3.1" xref="S3.SS2.4.p4.5.m5.1.1.3.1.cmml">∩</mo><msubsup id="S3.SS2.4.p4.5.m5.1.1.3.3" xref="S3.SS2.4.p4.5.m5.1.1.3.3.cmml"><mi id="S3.SS2.4.p4.5.m5.1.1.3.3.2.2" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.5.m5.1.1.3.3.3" xref="S3.SS2.4.p4.5.m5.1.1.3.3.3.cmml">U</mi><msub id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.cmml"><mi id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.2" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.3" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.3.cmml">2</mn></msub></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.5.m5.1b"><apply id="S3.SS2.4.p4.5.m5.1.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1"><eq id="S3.SS2.4.p4.5.m5.1.1.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.1"></eq><apply id="S3.SS2.4.p4.5.m5.1.1.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.2.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2">subscript</csymbol><apply id="S3.SS2.4.p4.5.m5.1.1.2.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.2.2.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS2.4.p4.5.m5.1.1.2.2.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2.2.2">𝐶</ci><ci id="S3.SS2.4.p4.5.m5.1.1.2.2.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.4.p4.5.m5.1.1.2.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.2.3">𝑈</ci></apply><apply id="S3.SS2.4.p4.5.m5.1.1.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3"><intersect id="S3.SS2.4.p4.5.m5.1.1.3.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.1"></intersect><apply id="S3.SS2.4.p4.5.m5.1.1.3.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.2.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2">subscript</csymbol><apply id="S3.SS2.4.p4.5.m5.1.1.3.2.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.2.2.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2">superscript</csymbol><ci id="S3.SS2.4.p4.5.m5.1.1.3.2.2.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.2">𝐶</ci><apply id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.5.m5.1.1.3.2.2.3.3">1</cn></apply></apply><ci id="S3.SS2.4.p4.5.m5.1.1.3.2.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.2.3">𝑈</ci></apply><apply id="S3.SS2.4.p4.5.m5.1.1.3.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.3.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3">subscript</csymbol><apply id="S3.SS2.4.p4.5.m5.1.1.3.3.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.3.2.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3">superscript</csymbol><ci id="S3.SS2.4.p4.5.m5.1.1.3.3.2.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.2">𝐶</ci><apply id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.1.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.2.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.5.m5.1.1.3.3.2.3.3">2</cn></apply></apply><ci id="S3.SS2.4.p4.5.m5.1.1.3.3.3.cmml" xref="S3.SS2.4.p4.5.m5.1.1.3.3.3">𝑈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.5.m5.1c">C^{i}_{U}=C^{i_{1}}_{U}\cap C^{i_{2}}_{U}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.5.m5.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∩ italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>, respectively, where <math alttext="i_{1},i_{2}<i" class="ltx_Math" display="inline" id="S3.SS2.4.p4.6.m6.2"><semantics id="S3.SS2.4.p4.6.m6.2a"><mrow id="S3.SS2.4.p4.6.m6.2.2" xref="S3.SS2.4.p4.6.m6.2.2.cmml"><mrow id="S3.SS2.4.p4.6.m6.2.2.2.2" xref="S3.SS2.4.p4.6.m6.2.2.2.3.cmml"><msub id="S3.SS2.4.p4.6.m6.1.1.1.1.1" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1.cmml"><mi id="S3.SS2.4.p4.6.m6.1.1.1.1.1.2" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1.2.cmml">i</mi><mn id="S3.SS2.4.p4.6.m6.1.1.1.1.1.3" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.4.p4.6.m6.2.2.2.2.3" xref="S3.SS2.4.p4.6.m6.2.2.2.3.cmml">,</mo><msub id="S3.SS2.4.p4.6.m6.2.2.2.2.2" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2.cmml"><mi id="S3.SS2.4.p4.6.m6.2.2.2.2.2.2" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2.2.cmml">i</mi><mn id="S3.SS2.4.p4.6.m6.2.2.2.2.2.3" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S3.SS2.4.p4.6.m6.2.2.3" xref="S3.SS2.4.p4.6.m6.2.2.3.cmml"><</mo><mi id="S3.SS2.4.p4.6.m6.2.2.4" xref="S3.SS2.4.p4.6.m6.2.2.4.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.6.m6.2b"><apply id="S3.SS2.4.p4.6.m6.2.2.cmml" xref="S3.SS2.4.p4.6.m6.2.2"><lt id="S3.SS2.4.p4.6.m6.2.2.3.cmml" xref="S3.SS2.4.p4.6.m6.2.2.3"></lt><list id="S3.SS2.4.p4.6.m6.2.2.2.3.cmml" xref="S3.SS2.4.p4.6.m6.2.2.2.2"><apply id="S3.SS2.4.p4.6.m6.1.1.1.1.1.cmml" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.4.p4.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.4.p4.6.m6.1.1.1.1.1.2.cmml" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1.2">𝑖</ci><cn id="S3.SS2.4.p4.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.4.p4.6.m6.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS2.4.p4.6.m6.2.2.2.2.2.cmml" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.6.m6.2.2.2.2.2.1.cmml" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S3.SS2.4.p4.6.m6.2.2.2.2.2.2.cmml" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2.2">𝑖</ci><cn id="S3.SS2.4.p4.6.m6.2.2.2.2.2.3.cmml" type="integer" xref="S3.SS2.4.p4.6.m6.2.2.2.2.2.3">2</cn></apply></list><ci id="S3.SS2.4.p4.6.m6.2.2.4.cmml" xref="S3.SS2.4.p4.6.m6.2.2.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.6.m6.2c">i_{1},i_{2}<i</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.6.m6.2d">italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_i</annotation></semantics></math>. Assume that <math alttext="\alpha_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.4.p4.7.m7.1"><semantics id="S3.SS2.4.p4.7.m7.1a"><mrow id="S3.SS2.4.p4.7.m7.1.1" xref="S3.SS2.4.p4.7.m7.1.1.cmml"><msub id="S3.SS2.4.p4.7.m7.1.1.2" xref="S3.SS2.4.p4.7.m7.1.1.2.cmml"><mi id="S3.SS2.4.p4.7.m7.1.1.2.2" xref="S3.SS2.4.p4.7.m7.1.1.2.2.cmml">α</mi><mi id="S3.SS2.4.p4.7.m7.1.1.2.3" xref="S3.SS2.4.p4.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.4.p4.7.m7.1.1.1" xref="S3.SS2.4.p4.7.m7.1.1.1.cmml">=</mo><mn id="S3.SS2.4.p4.7.m7.1.1.3" xref="S3.SS2.4.p4.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.7.m7.1b"><apply id="S3.SS2.4.p4.7.m7.1.1.cmml" xref="S3.SS2.4.p4.7.m7.1.1"><eq id="S3.SS2.4.p4.7.m7.1.1.1.cmml" xref="S3.SS2.4.p4.7.m7.1.1.1"></eq><apply id="S3.SS2.4.p4.7.m7.1.1.2.cmml" xref="S3.SS2.4.p4.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.7.m7.1.1.2.1.cmml" xref="S3.SS2.4.p4.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS2.4.p4.7.m7.1.1.2.2.cmml" xref="S3.SS2.4.p4.7.m7.1.1.2.2">𝛼</ci><ci id="S3.SS2.4.p4.7.m7.1.1.2.3.cmml" xref="S3.SS2.4.p4.7.m7.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.4.p4.7.m7.1.1.3.cmml" type="integer" xref="S3.SS2.4.p4.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.7.m7.1c">\alpha_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.7.m7.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Then <math alttext="a\in C^{i}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.8.m8.1"><semantics id="S3.SS2.4.p4.8.m8.1a"><mrow id="S3.SS2.4.p4.8.m8.1.1" xref="S3.SS2.4.p4.8.m8.1.1.cmml"><mi id="S3.SS2.4.p4.8.m8.1.1.2" xref="S3.SS2.4.p4.8.m8.1.1.2.cmml">a</mi><mo id="S3.SS2.4.p4.8.m8.1.1.1" xref="S3.SS2.4.p4.8.m8.1.1.1.cmml">∈</mo><msup id="S3.SS2.4.p4.8.m8.1.1.3" xref="S3.SS2.4.p4.8.m8.1.1.3.cmml"><mi id="S3.SS2.4.p4.8.m8.1.1.3.2" xref="S3.SS2.4.p4.8.m8.1.1.3.2.cmml">C</mi><mi id="S3.SS2.4.p4.8.m8.1.1.3.3" xref="S3.SS2.4.p4.8.m8.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.8.m8.1b"><apply id="S3.SS2.4.p4.8.m8.1.1.cmml" xref="S3.SS2.4.p4.8.m8.1.1"><in id="S3.SS2.4.p4.8.m8.1.1.1.cmml" xref="S3.SS2.4.p4.8.m8.1.1.1"></in><ci id="S3.SS2.4.p4.8.m8.1.1.2.cmml" xref="S3.SS2.4.p4.8.m8.1.1.2">𝑎</ci><apply id="S3.SS2.4.p4.8.m8.1.1.3.cmml" xref="S3.SS2.4.p4.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.8.m8.1.1.3.1.cmml" xref="S3.SS2.4.p4.8.m8.1.1.3">superscript</csymbol><ci id="S3.SS2.4.p4.8.m8.1.1.3.2.cmml" xref="S3.SS2.4.p4.8.m8.1.1.3.2">𝐶</ci><ci id="S3.SS2.4.p4.8.m8.1.1.3.3.cmml" xref="S3.SS2.4.p4.8.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.8.m8.1c">a\in C^{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.8.m8.1d">italic_a ∈ italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, and consequently <math alttext="a\in C^{i_{1}}\cap C^{i_{2}}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.9.m9.1"><semantics id="S3.SS2.4.p4.9.m9.1a"><mrow id="S3.SS2.4.p4.9.m9.1.1" xref="S3.SS2.4.p4.9.m9.1.1.cmml"><mi id="S3.SS2.4.p4.9.m9.1.1.2" xref="S3.SS2.4.p4.9.m9.1.1.2.cmml">a</mi><mo id="S3.SS2.4.p4.9.m9.1.1.1" xref="S3.SS2.4.p4.9.m9.1.1.1.cmml">∈</mo><mrow id="S3.SS2.4.p4.9.m9.1.1.3" xref="S3.SS2.4.p4.9.m9.1.1.3.cmml"><msup id="S3.SS2.4.p4.9.m9.1.1.3.2" xref="S3.SS2.4.p4.9.m9.1.1.3.2.cmml"><mi id="S3.SS2.4.p4.9.m9.1.1.3.2.2" xref="S3.SS2.4.p4.9.m9.1.1.3.2.2.cmml">C</mi><msub id="S3.SS2.4.p4.9.m9.1.1.3.2.3" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3.cmml"><mi id="S3.SS2.4.p4.9.m9.1.1.3.2.3.2" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.9.m9.1.1.3.2.3.3" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3.3.cmml">1</mn></msub></msup><mo id="S3.SS2.4.p4.9.m9.1.1.3.1" xref="S3.SS2.4.p4.9.m9.1.1.3.1.cmml">∩</mo><msup id="S3.SS2.4.p4.9.m9.1.1.3.3" xref="S3.SS2.4.p4.9.m9.1.1.3.3.cmml"><mi id="S3.SS2.4.p4.9.m9.1.1.3.3.2" xref="S3.SS2.4.p4.9.m9.1.1.3.3.2.cmml">C</mi><msub id="S3.SS2.4.p4.9.m9.1.1.3.3.3" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3.cmml"><mi id="S3.SS2.4.p4.9.m9.1.1.3.3.3.2" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.9.m9.1.1.3.3.3.3" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3.3.cmml">2</mn></msub></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.9.m9.1b"><apply id="S3.SS2.4.p4.9.m9.1.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1"><in id="S3.SS2.4.p4.9.m9.1.1.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.1"></in><ci id="S3.SS2.4.p4.9.m9.1.1.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.2">𝑎</ci><apply id="S3.SS2.4.p4.9.m9.1.1.3.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3"><intersect id="S3.SS2.4.p4.9.m9.1.1.3.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.1"></intersect><apply id="S3.SS2.4.p4.9.m9.1.1.3.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.9.m9.1.1.3.2.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2">superscript</csymbol><ci id="S3.SS2.4.p4.9.m9.1.1.3.2.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2.2">𝐶</ci><apply id="S3.SS2.4.p4.9.m9.1.1.3.2.3.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.9.m9.1.1.3.2.3.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.9.m9.1.1.3.2.3.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.9.m9.1.1.3.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.9.m9.1.1.3.2.3.3">1</cn></apply></apply><apply id="S3.SS2.4.p4.9.m9.1.1.3.3.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.9.m9.1.1.3.3.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3">superscript</csymbol><ci id="S3.SS2.4.p4.9.m9.1.1.3.3.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3.2">𝐶</ci><apply id="S3.SS2.4.p4.9.m9.1.1.3.3.3.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.9.m9.1.1.3.3.3.1.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3">subscript</csymbol><ci id="S3.SS2.4.p4.9.m9.1.1.3.3.3.2.cmml" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3.2">𝑖</ci><cn id="S3.SS2.4.p4.9.m9.1.1.3.3.3.3.cmml" type="integer" xref="S3.SS2.4.p4.9.m9.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.9.m9.1c">a\in C^{i_{1}}\cap C^{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.9.m9.1d">italic_a ∈ italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∩ italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Using the induction hypothesis, <math alttext="1=\alpha_{i_{1}}=\alpha_{i_{2}}=\beta_{i_{1}}=\beta_{i_{2}}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.10.m10.1"><semantics id="S3.SS2.4.p4.10.m10.1a"><mrow id="S3.SS2.4.p4.10.m10.1.1" xref="S3.SS2.4.p4.10.m10.1.1.cmml"><mn id="S3.SS2.4.p4.10.m10.1.1.2" xref="S3.SS2.4.p4.10.m10.1.1.2.cmml">1</mn><mo id="S3.SS2.4.p4.10.m10.1.1.3" xref="S3.SS2.4.p4.10.m10.1.1.3.cmml">=</mo><msub id="S3.SS2.4.p4.10.m10.1.1.4" xref="S3.SS2.4.p4.10.m10.1.1.4.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.4.2" xref="S3.SS2.4.p4.10.m10.1.1.4.2.cmml">α</mi><msub id="S3.SS2.4.p4.10.m10.1.1.4.3" xref="S3.SS2.4.p4.10.m10.1.1.4.3.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.4.3.2" xref="S3.SS2.4.p4.10.m10.1.1.4.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.10.m10.1.1.4.3.3" xref="S3.SS2.4.p4.10.m10.1.1.4.3.3.cmml">1</mn></msub></msub><mo id="S3.SS2.4.p4.10.m10.1.1.5" xref="S3.SS2.4.p4.10.m10.1.1.5.cmml">=</mo><msub id="S3.SS2.4.p4.10.m10.1.1.6" xref="S3.SS2.4.p4.10.m10.1.1.6.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.6.2" xref="S3.SS2.4.p4.10.m10.1.1.6.2.cmml">α</mi><msub id="S3.SS2.4.p4.10.m10.1.1.6.3" xref="S3.SS2.4.p4.10.m10.1.1.6.3.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.6.3.2" xref="S3.SS2.4.p4.10.m10.1.1.6.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.10.m10.1.1.6.3.3" xref="S3.SS2.4.p4.10.m10.1.1.6.3.3.cmml">2</mn></msub></msub><mo id="S3.SS2.4.p4.10.m10.1.1.7" xref="S3.SS2.4.p4.10.m10.1.1.7.cmml">=</mo><msub id="S3.SS2.4.p4.10.m10.1.1.8" xref="S3.SS2.4.p4.10.m10.1.1.8.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.8.2" xref="S3.SS2.4.p4.10.m10.1.1.8.2.cmml">β</mi><msub id="S3.SS2.4.p4.10.m10.1.1.8.3" xref="S3.SS2.4.p4.10.m10.1.1.8.3.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.8.3.2" xref="S3.SS2.4.p4.10.m10.1.1.8.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.10.m10.1.1.8.3.3" xref="S3.SS2.4.p4.10.m10.1.1.8.3.3.cmml">1</mn></msub></msub><mo id="S3.SS2.4.p4.10.m10.1.1.9" xref="S3.SS2.4.p4.10.m10.1.1.9.cmml">=</mo><msub id="S3.SS2.4.p4.10.m10.1.1.10" xref="S3.SS2.4.p4.10.m10.1.1.10.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.10.2" xref="S3.SS2.4.p4.10.m10.1.1.10.2.cmml">β</mi><msub id="S3.SS2.4.p4.10.m10.1.1.10.3" xref="S3.SS2.4.p4.10.m10.1.1.10.3.cmml"><mi id="S3.SS2.4.p4.10.m10.1.1.10.3.2" xref="S3.SS2.4.p4.10.m10.1.1.10.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.10.m10.1.1.10.3.3" xref="S3.SS2.4.p4.10.m10.1.1.10.3.3.cmml">2</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.10.m10.1b"><apply id="S3.SS2.4.p4.10.m10.1.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1"><and id="S3.SS2.4.p4.10.m10.1.1a.cmml" xref="S3.SS2.4.p4.10.m10.1.1"></and><apply id="S3.SS2.4.p4.10.m10.1.1b.cmml" xref="S3.SS2.4.p4.10.m10.1.1"><eq id="S3.SS2.4.p4.10.m10.1.1.3.cmml" xref="S3.SS2.4.p4.10.m10.1.1.3"></eq><cn id="S3.SS2.4.p4.10.m10.1.1.2.cmml" type="integer" xref="S3.SS2.4.p4.10.m10.1.1.2">1</cn><apply id="S3.SS2.4.p4.10.m10.1.1.4.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.4.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.4.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4.2">𝛼</ci><apply id="S3.SS2.4.p4.10.m10.1.1.4.3.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.4.3.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4.3">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.4.3.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.4.3.2">𝑖</ci><cn id="S3.SS2.4.p4.10.m10.1.1.4.3.3.cmml" type="integer" xref="S3.SS2.4.p4.10.m10.1.1.4.3.3">1</cn></apply></apply></apply><apply id="S3.SS2.4.p4.10.m10.1.1c.cmml" xref="S3.SS2.4.p4.10.m10.1.1"><eq id="S3.SS2.4.p4.10.m10.1.1.5.cmml" xref="S3.SS2.4.p4.10.m10.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.4.p4.10.m10.1.1.4.cmml" id="S3.SS2.4.p4.10.m10.1.1d.cmml" xref="S3.SS2.4.p4.10.m10.1.1"></share><apply id="S3.SS2.4.p4.10.m10.1.1.6.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.6.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.6.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6.2">𝛼</ci><apply id="S3.SS2.4.p4.10.m10.1.1.6.3.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.6.3.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6.3">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.6.3.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.6.3.2">𝑖</ci><cn id="S3.SS2.4.p4.10.m10.1.1.6.3.3.cmml" type="integer" xref="S3.SS2.4.p4.10.m10.1.1.6.3.3">2</cn></apply></apply></apply><apply id="S3.SS2.4.p4.10.m10.1.1e.cmml" xref="S3.SS2.4.p4.10.m10.1.1"><eq id="S3.SS2.4.p4.10.m10.1.1.7.cmml" xref="S3.SS2.4.p4.10.m10.1.1.7"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.4.p4.10.m10.1.1.6.cmml" id="S3.SS2.4.p4.10.m10.1.1f.cmml" xref="S3.SS2.4.p4.10.m10.1.1"></share><apply id="S3.SS2.4.p4.10.m10.1.1.8.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.8.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.8.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8.2">𝛽</ci><apply id="S3.SS2.4.p4.10.m10.1.1.8.3.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.8.3.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8.3">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.8.3.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.8.3.2">𝑖</ci><cn id="S3.SS2.4.p4.10.m10.1.1.8.3.3.cmml" type="integer" xref="S3.SS2.4.p4.10.m10.1.1.8.3.3">1</cn></apply></apply></apply><apply id="S3.SS2.4.p4.10.m10.1.1g.cmml" xref="S3.SS2.4.p4.10.m10.1.1"><eq id="S3.SS2.4.p4.10.m10.1.1.9.cmml" xref="S3.SS2.4.p4.10.m10.1.1.9"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.4.p4.10.m10.1.1.8.cmml" id="S3.SS2.4.p4.10.m10.1.1h.cmml" xref="S3.SS2.4.p4.10.m10.1.1"></share><apply id="S3.SS2.4.p4.10.m10.1.1.10.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.10.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.10.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10.2">𝛽</ci><apply id="S3.SS2.4.p4.10.m10.1.1.10.3.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.10.m10.1.1.10.3.1.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10.3">subscript</csymbol><ci id="S3.SS2.4.p4.10.m10.1.1.10.3.2.cmml" xref="S3.SS2.4.p4.10.m10.1.1.10.3.2">𝑖</ci><cn id="S3.SS2.4.p4.10.m10.1.1.10.3.3.cmml" type="integer" xref="S3.SS2.4.p4.10.m10.1.1.10.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.10.m10.1c">1=\alpha_{i_{1}}=\alpha_{i_{2}}=\beta_{i_{1}}=\beta_{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.10.m10.1d">1 = italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, <math alttext="C^{i_{1}}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.11.m11.1"><semantics id="S3.SS2.4.p4.11.m11.1a"><mrow id="S3.SS2.4.p4.11.m11.1.1" xref="S3.SS2.4.p4.11.m11.1.1.cmml"><msubsup id="S3.SS2.4.p4.11.m11.1.1.2" xref="S3.SS2.4.p4.11.m11.1.1.2.cmml"><mi id="S3.SS2.4.p4.11.m11.1.1.2.2.2" xref="S3.SS2.4.p4.11.m11.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.11.m11.1.1.2.3" xref="S3.SS2.4.p4.11.m11.1.1.2.3.cmml">U</mi><msub id="S3.SS2.4.p4.11.m11.1.1.2.2.3" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3.cmml"><mi id="S3.SS2.4.p4.11.m11.1.1.2.2.3.2" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.11.m11.1.1.2.2.3.3" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3.3.cmml">1</mn></msub></msubsup><mo id="S3.SS2.4.p4.11.m11.1.1.1" xref="S3.SS2.4.p4.11.m11.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.4.p4.11.m11.1.1.3" xref="S3.SS2.4.p4.11.m11.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.11.m11.1b"><apply id="S3.SS2.4.p4.11.m11.1.1.cmml" xref="S3.SS2.4.p4.11.m11.1.1"><in id="S3.SS2.4.p4.11.m11.1.1.1.cmml" xref="S3.SS2.4.p4.11.m11.1.1.1"></in><apply id="S3.SS2.4.p4.11.m11.1.1.2.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.11.m11.1.1.2.1.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2">subscript</csymbol><apply id="S3.SS2.4.p4.11.m11.1.1.2.2.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.11.m11.1.1.2.2.1.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2">superscript</csymbol><ci id="S3.SS2.4.p4.11.m11.1.1.2.2.2.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2.2.2">𝐶</ci><apply id="S3.SS2.4.p4.11.m11.1.1.2.2.3.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.11.m11.1.1.2.2.3.1.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.11.m11.1.1.2.2.3.2.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.11.m11.1.1.2.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.11.m11.1.1.2.2.3.3">1</cn></apply></apply><ci id="S3.SS2.4.p4.11.m11.1.1.2.3.cmml" xref="S3.SS2.4.p4.11.m11.1.1.2.3">𝑈</ci></apply><ci id="S3.SS2.4.p4.11.m11.1.1.3.cmml" xref="S3.SS2.4.p4.11.m11.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.11.m11.1c">C^{i_{1}}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.11.m11.1d">italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math> and <math alttext="C^{i_{2}}_{U}\in F" class="ltx_Math" display="inline" id="S3.SS2.4.p4.12.m12.1"><semantics id="S3.SS2.4.p4.12.m12.1a"><mrow id="S3.SS2.4.p4.12.m12.1.1" xref="S3.SS2.4.p4.12.m12.1.1.cmml"><msubsup id="S3.SS2.4.p4.12.m12.1.1.2" xref="S3.SS2.4.p4.12.m12.1.1.2.cmml"><mi id="S3.SS2.4.p4.12.m12.1.1.2.2.2" xref="S3.SS2.4.p4.12.m12.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.12.m12.1.1.2.3" xref="S3.SS2.4.p4.12.m12.1.1.2.3.cmml">U</mi><msub id="S3.SS2.4.p4.12.m12.1.1.2.2.3" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3.cmml"><mi id="S3.SS2.4.p4.12.m12.1.1.2.2.3.2" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.12.m12.1.1.2.2.3.3" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3.3.cmml">2</mn></msub></msubsup><mo id="S3.SS2.4.p4.12.m12.1.1.1" xref="S3.SS2.4.p4.12.m12.1.1.1.cmml">∈</mo><mi id="S3.SS2.4.p4.12.m12.1.1.3" xref="S3.SS2.4.p4.12.m12.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.12.m12.1b"><apply id="S3.SS2.4.p4.12.m12.1.1.cmml" xref="S3.SS2.4.p4.12.m12.1.1"><in id="S3.SS2.4.p4.12.m12.1.1.1.cmml" xref="S3.SS2.4.p4.12.m12.1.1.1"></in><apply id="S3.SS2.4.p4.12.m12.1.1.2.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.12.m12.1.1.2.1.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2">subscript</csymbol><apply id="S3.SS2.4.p4.12.m12.1.1.2.2.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.12.m12.1.1.2.2.1.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2">superscript</csymbol><ci id="S3.SS2.4.p4.12.m12.1.1.2.2.2.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2.2.2">𝐶</ci><apply id="S3.SS2.4.p4.12.m12.1.1.2.2.3.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.12.m12.1.1.2.2.3.1.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.12.m12.1.1.2.2.3.2.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.12.m12.1.1.2.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.12.m12.1.1.2.2.3.3">2</cn></apply></apply><ci id="S3.SS2.4.p4.12.m12.1.1.2.3.cmml" xref="S3.SS2.4.p4.12.m12.1.1.2.3">𝑈</ci></apply><ci id="S3.SS2.4.p4.12.m12.1.1.3.cmml" xref="S3.SS2.4.p4.12.m12.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.12.m12.1c">C^{i_{2}}_{U}\in F</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.12.m12.1d">italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ italic_F</annotation></semantics></math>. Now using that <math alttext="(C^{i_{1}}_{U},C^{i_{2}}_{U})\in\Lambda" class="ltx_Math" display="inline" id="S3.SS2.4.p4.13.m13.2"><semantics id="S3.SS2.4.p4.13.m13.2a"><mrow id="S3.SS2.4.p4.13.m13.2.2" xref="S3.SS2.4.p4.13.m13.2.2.cmml"><mrow id="S3.SS2.4.p4.13.m13.2.2.2.2" xref="S3.SS2.4.p4.13.m13.2.2.2.3.cmml"><mo id="S3.SS2.4.p4.13.m13.2.2.2.2.3" stretchy="false" xref="S3.SS2.4.p4.13.m13.2.2.2.3.cmml">(</mo><msubsup id="S3.SS2.4.p4.13.m13.1.1.1.1.1" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.cmml"><mi id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.2" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.13.m13.1.1.1.1.1.3" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.3.cmml">U</mi><msub id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.cmml"><mi id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.2" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.3" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.3.cmml">1</mn></msub></msubsup><mo id="S3.SS2.4.p4.13.m13.2.2.2.2.4" xref="S3.SS2.4.p4.13.m13.2.2.2.3.cmml">,</mo><msubsup id="S3.SS2.4.p4.13.m13.2.2.2.2.2" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.cmml"><mi id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.2" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.13.m13.2.2.2.2.2.3" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.3.cmml">U</mi><msub id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.cmml"><mi id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.2" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.3" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.3.cmml">2</mn></msub></msubsup><mo id="S3.SS2.4.p4.13.m13.2.2.2.2.5" stretchy="false" xref="S3.SS2.4.p4.13.m13.2.2.2.3.cmml">)</mo></mrow><mo id="S3.SS2.4.p4.13.m13.2.2.3" xref="S3.SS2.4.p4.13.m13.2.2.3.cmml">∈</mo><mi id="S3.SS2.4.p4.13.m13.2.2.4" mathvariant="normal" xref="S3.SS2.4.p4.13.m13.2.2.4.cmml">Λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.13.m13.2b"><apply id="S3.SS2.4.p4.13.m13.2.2.cmml" xref="S3.SS2.4.p4.13.m13.2.2"><in id="S3.SS2.4.p4.13.m13.2.2.3.cmml" xref="S3.SS2.4.p4.13.m13.2.2.3"></in><interval closure="open" id="S3.SS2.4.p4.13.m13.2.2.2.3.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2"><apply id="S3.SS2.4.p4.13.m13.1.1.1.1.1.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.1.1.1.1.1.1.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1">subscript</csymbol><apply id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.1.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.2.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.2">𝐶</ci><apply id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.1.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.2.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S3.SS2.4.p4.13.m13.1.1.1.1.1.3.cmml" xref="S3.SS2.4.p4.13.m13.1.1.1.1.1.3">𝑈</ci></apply><apply id="S3.SS2.4.p4.13.m13.2.2.2.2.2.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.2.2.2.2.2.1.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2">subscript</csymbol><apply id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.1.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2">superscript</csymbol><ci id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.2.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.2">𝐶</ci><apply id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.1.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.2.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.2.3.3">2</cn></apply></apply><ci id="S3.SS2.4.p4.13.m13.2.2.2.2.2.3.cmml" xref="S3.SS2.4.p4.13.m13.2.2.2.2.2.3">𝑈</ci></apply></interval><ci id="S3.SS2.4.p4.13.m13.2.2.4.cmml" xref="S3.SS2.4.p4.13.m13.2.2.4">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.13.m13.2c">(C^{i_{1}}_{U},C^{i_{2}}_{U})\in\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.13.m13.2d">( italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ) ∈ roman_Λ</annotation></semantics></math> and that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.4.p4.14.m14.1"><semantics id="S3.SS2.4.p4.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.4.p4.14.m14.1.1" xref="S3.SS2.4.p4.14.m14.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.14.m14.1b"><ci id="S3.SS2.4.p4.14.m14.1.1.cmml" xref="S3.SS2.4.p4.14.m14.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.14.m14.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.14.m14.1d">caligraphic_F</annotation></semantics></math> preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS2.4.p4.15.m15.1"><semantics id="S3.SS2.4.p4.15.m15.1a"><mi id="S3.SS2.4.p4.15.m15.1.1" mathvariant="normal" xref="S3.SS2.4.p4.15.m15.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.15.m15.1b"><ci id="S3.SS2.4.p4.15.m15.1.1.cmml" xref="S3.SS2.4.p4.15.m15.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.15.m15.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.15.m15.1d">roman_Λ</annotation></semantics></math>, it follows that <math alttext="C^{i}_{U}=C^{i_{1}}_{U}\cap C^{i_{2}}_{U}\in F" class="ltx_Math" display="inline" id="S3.SS2.4.p4.16.m16.1"><semantics id="S3.SS2.4.p4.16.m16.1a"><mrow id="S3.SS2.4.p4.16.m16.1.1" xref="S3.SS2.4.p4.16.m16.1.1.cmml"><msubsup id="S3.SS2.4.p4.16.m16.1.1.2" xref="S3.SS2.4.p4.16.m16.1.1.2.cmml"><mi id="S3.SS2.4.p4.16.m16.1.1.2.2.2" xref="S3.SS2.4.p4.16.m16.1.1.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.16.m16.1.1.2.3" xref="S3.SS2.4.p4.16.m16.1.1.2.3.cmml">U</mi><mi id="S3.SS2.4.p4.16.m16.1.1.2.2.3" xref="S3.SS2.4.p4.16.m16.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS2.4.p4.16.m16.1.1.3" xref="S3.SS2.4.p4.16.m16.1.1.3.cmml">=</mo><mrow id="S3.SS2.4.p4.16.m16.1.1.4" xref="S3.SS2.4.p4.16.m16.1.1.4.cmml"><msubsup id="S3.SS2.4.p4.16.m16.1.1.4.2" xref="S3.SS2.4.p4.16.m16.1.1.4.2.cmml"><mi id="S3.SS2.4.p4.16.m16.1.1.4.2.2.2" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.16.m16.1.1.4.2.3" xref="S3.SS2.4.p4.16.m16.1.1.4.2.3.cmml">U</mi><msub id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.cmml"><mi id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.2" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.3" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.3.cmml">1</mn></msub></msubsup><mo id="S3.SS2.4.p4.16.m16.1.1.4.1" xref="S3.SS2.4.p4.16.m16.1.1.4.1.cmml">∩</mo><msubsup id="S3.SS2.4.p4.16.m16.1.1.4.3" xref="S3.SS2.4.p4.16.m16.1.1.4.3.cmml"><mi id="S3.SS2.4.p4.16.m16.1.1.4.3.2.2" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.2.cmml">C</mi><mi id="S3.SS2.4.p4.16.m16.1.1.4.3.3" xref="S3.SS2.4.p4.16.m16.1.1.4.3.3.cmml">U</mi><msub id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.cmml"><mi id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.2" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.2.cmml">i</mi><mn id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.3" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.3.cmml">2</mn></msub></msubsup></mrow><mo id="S3.SS2.4.p4.16.m16.1.1.5" xref="S3.SS2.4.p4.16.m16.1.1.5.cmml">∈</mo><mi id="S3.SS2.4.p4.16.m16.1.1.6" xref="S3.SS2.4.p4.16.m16.1.1.6.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.16.m16.1b"><apply id="S3.SS2.4.p4.16.m16.1.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1"><and id="S3.SS2.4.p4.16.m16.1.1a.cmml" xref="S3.SS2.4.p4.16.m16.1.1"></and><apply id="S3.SS2.4.p4.16.m16.1.1b.cmml" xref="S3.SS2.4.p4.16.m16.1.1"><eq id="S3.SS2.4.p4.16.m16.1.1.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.3"></eq><apply id="S3.SS2.4.p4.16.m16.1.1.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.2.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2">subscript</csymbol><apply id="S3.SS2.4.p4.16.m16.1.1.2.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.2.2.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2">superscript</csymbol><ci id="S3.SS2.4.p4.16.m16.1.1.2.2.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2.2.2">𝐶</ci><ci id="S3.SS2.4.p4.16.m16.1.1.2.2.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS2.4.p4.16.m16.1.1.2.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.2.3">𝑈</ci></apply><apply id="S3.SS2.4.p4.16.m16.1.1.4.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4"><intersect id="S3.SS2.4.p4.16.m16.1.1.4.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.1"></intersect><apply id="S3.SS2.4.p4.16.m16.1.1.4.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.2.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2">subscript</csymbol><apply id="S3.SS2.4.p4.16.m16.1.1.4.2.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.2.2.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2">superscript</csymbol><ci id="S3.SS2.4.p4.16.m16.1.1.4.2.2.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.2">𝐶</ci><apply id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.16.m16.1.1.4.2.2.3.3">1</cn></apply></apply><ci id="S3.SS2.4.p4.16.m16.1.1.4.2.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.2.3">𝑈</ci></apply><apply id="S3.SS2.4.p4.16.m16.1.1.4.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.3.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3">subscript</csymbol><apply id="S3.SS2.4.p4.16.m16.1.1.4.3.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.3.2.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3">superscript</csymbol><ci id="S3.SS2.4.p4.16.m16.1.1.4.3.2.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.2">𝐶</ci><apply id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3"><csymbol cd="ambiguous" id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.1.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3">subscript</csymbol><ci id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.2.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.2">𝑖</ci><cn id="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.3.cmml" type="integer" xref="S3.SS2.4.p4.16.m16.1.1.4.3.2.3.3">2</cn></apply></apply><ci id="S3.SS2.4.p4.16.m16.1.1.4.3.3.cmml" xref="S3.SS2.4.p4.16.m16.1.1.4.3.3">𝑈</ci></apply></apply></apply><apply id="S3.SS2.4.p4.16.m16.1.1c.cmml" xref="S3.SS2.4.p4.16.m16.1.1"><in id="S3.SS2.4.p4.16.m16.1.1.5.cmml" xref="S3.SS2.4.p4.16.m16.1.1.5"></in><share href="https://arxiv.org/html/2503.14117v1#S3.SS2.4.p4.16.m16.1.1.4.cmml" id="S3.SS2.4.p4.16.m16.1.1d.cmml" xref="S3.SS2.4.p4.16.m16.1.1"></share><ci id="S3.SS2.4.p4.16.m16.1.1.6.cmml" xref="S3.SS2.4.p4.16.m16.1.1.6">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.16.m16.1c">C^{i}_{U}=C^{i_{1}}_{U}\cap C^{i_{2}}_{U}\in F</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.16.m16.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∩ italic_C start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ italic_F</annotation></semantics></math>. In other words, <math alttext="\beta_{i}=1" class="ltx_Math" display="inline" id="S3.SS2.4.p4.17.m17.1"><semantics id="S3.SS2.4.p4.17.m17.1a"><mrow id="S3.SS2.4.p4.17.m17.1.1" xref="S3.SS2.4.p4.17.m17.1.1.cmml"><msub id="S3.SS2.4.p4.17.m17.1.1.2" xref="S3.SS2.4.p4.17.m17.1.1.2.cmml"><mi id="S3.SS2.4.p4.17.m17.1.1.2.2" xref="S3.SS2.4.p4.17.m17.1.1.2.2.cmml">β</mi><mi id="S3.SS2.4.p4.17.m17.1.1.2.3" xref="S3.SS2.4.p4.17.m17.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.4.p4.17.m17.1.1.1" xref="S3.SS2.4.p4.17.m17.1.1.1.cmml">=</mo><mn id="S3.SS2.4.p4.17.m17.1.1.3" xref="S3.SS2.4.p4.17.m17.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.17.m17.1b"><apply id="S3.SS2.4.p4.17.m17.1.1.cmml" xref="S3.SS2.4.p4.17.m17.1.1"><eq id="S3.SS2.4.p4.17.m17.1.1.1.cmml" xref="S3.SS2.4.p4.17.m17.1.1.1"></eq><apply id="S3.SS2.4.p4.17.m17.1.1.2.cmml" xref="S3.SS2.4.p4.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.17.m17.1.1.2.1.cmml" xref="S3.SS2.4.p4.17.m17.1.1.2">subscript</csymbol><ci id="S3.SS2.4.p4.17.m17.1.1.2.2.cmml" xref="S3.SS2.4.p4.17.m17.1.1.2.2">𝛽</ci><ci id="S3.SS2.4.p4.17.m17.1.1.2.3.cmml" xref="S3.SS2.4.p4.17.m17.1.1.2.3">𝑖</ci></apply><cn id="S3.SS2.4.p4.17.m17.1.1.3.cmml" type="integer" xref="S3.SS2.4.p4.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.17.m17.1c">\beta_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.17.m17.1d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>. The other case is similar. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS2.5.p5"> <p class="ltx_p" id="S3.SS2.5.p5.1">This establishes the claim and completes the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>. ∎</p> </div> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Set-theoretic fusion as a complete framework for lower bounds</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">In this section, we establish a converse to Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem24"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem24.1.1.1">Theorem 24</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem24.2.2"> </span>(Fusion upper bound)<span class="ltx_text ltx_font_bold" id="Thmtheorem24.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem24.p1"> <p class="ltx_p" id="Thmtheorem24.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem24.p1.2.2">Let <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem24.p1.1.1.m1.1"><semantics id="Thmtheorem24.p1.1.1.m1.1a"><mrow id="Thmtheorem24.p1.1.1.m1.1.1" xref="Thmtheorem24.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem24.p1.1.1.m1.1.1.2" xref="Thmtheorem24.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem24.p1.1.1.m1.1.1.1" xref="Thmtheorem24.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem24.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem24.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem24.p1.1.1.m1.1b"><apply id="Thmtheorem24.p1.1.1.m1.1.1.cmml" xref="Thmtheorem24.p1.1.1.m1.1.1"><subset id="Thmtheorem24.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem24.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem24.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem24.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem24.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem24.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem24.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem24.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> be non-trivial, and <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="Thmtheorem24.p1.2.2.m2.1"><semantics id="Thmtheorem24.p1.2.2.m2.1a"><mrow id="Thmtheorem24.p1.2.2.m2.1.2" xref="Thmtheorem24.p1.2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem24.p1.2.2.m2.1.2.2" xref="Thmtheorem24.p1.2.2.m2.1.2.2.cmml">ℬ</mi><mo id="Thmtheorem24.p1.2.2.m2.1.2.1" xref="Thmtheorem24.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem24.p1.2.2.m2.1.2.3" xref="Thmtheorem24.p1.2.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem24.p1.2.2.m2.1.2.3.2" xref="Thmtheorem24.p1.2.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem24.p1.2.2.m2.1.2.3.1" xref="Thmtheorem24.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem24.p1.2.2.m2.1.2.3.3.2" xref="Thmtheorem24.p1.2.2.m2.1.2.3.cmml"><mo id="Thmtheorem24.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem24.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmtheorem24.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem24.p1.2.2.m2.1.1.cmml">Γ</mi><mo id="Thmtheorem24.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem24.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem24.p1.2.2.m2.1b"><apply id="Thmtheorem24.p1.2.2.m2.1.2.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2"><subset id="Thmtheorem24.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2.1"></subset><ci id="Thmtheorem24.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2.2">ℬ</ci><apply id="Thmtheorem24.p1.2.2.m2.1.2.3.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2.3"><times id="Thmtheorem24.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2.3.1"></times><ci id="Thmtheorem24.p1.2.2.m2.1.2.3.2.cmml" xref="Thmtheorem24.p1.2.2.m2.1.2.3.2">𝒫</ci><ci id="Thmtheorem24.p1.2.2.m2.1.1.cmml" xref="Thmtheorem24.p1.2.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem24.p1.2.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem24.p1.2.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math> be a non-empty family of generators. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D_{\cap}(A\mid\mathcal{B})\;\leq\;\rho(A,\mathcal{B})^{2}." class="ltx_Math" display="block" id="S3.Ex20.m1.3"><semantics id="S3.Ex20.m1.3a"><mrow id="S3.Ex20.m1.3.3.1" xref="S3.Ex20.m1.3.3.1.1.cmml"><mrow id="S3.Ex20.m1.3.3.1.1" xref="S3.Ex20.m1.3.3.1.1.cmml"><mrow id="S3.Ex20.m1.3.3.1.1.1" xref="S3.Ex20.m1.3.3.1.1.1.cmml"><msub id="S3.Ex20.m1.3.3.1.1.1.3" xref="S3.Ex20.m1.3.3.1.1.1.3.cmml"><mi id="S3.Ex20.m1.3.3.1.1.1.3.2" xref="S3.Ex20.m1.3.3.1.1.1.3.2.cmml">D</mi><mo id="S3.Ex20.m1.3.3.1.1.1.3.3" xref="S3.Ex20.m1.3.3.1.1.1.3.3.cmml">∩</mo></msub><mo id="S3.Ex20.m1.3.3.1.1.1.2" xref="S3.Ex20.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex20.m1.3.3.1.1.1.1.1" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S3.Ex20.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex20.m1.3.3.1.1.1.1.1.1" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex20.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex20.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex20.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.Ex20.m1.3.3.1.1.1.1.1.3" rspace="0.280em" stretchy="false" xref="S3.Ex20.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex20.m1.3.3.1.1.2" rspace="0.558em" xref="S3.Ex20.m1.3.3.1.1.2.cmml">≤</mo><mrow id="S3.Ex20.m1.3.3.1.1.3" xref="S3.Ex20.m1.3.3.1.1.3.cmml"><mi id="S3.Ex20.m1.3.3.1.1.3.2" xref="S3.Ex20.m1.3.3.1.1.3.2.cmml">ρ</mi><mo id="S3.Ex20.m1.3.3.1.1.3.1" xref="S3.Ex20.m1.3.3.1.1.3.1.cmml">⁢</mo><msup id="S3.Ex20.m1.3.3.1.1.3.3" xref="S3.Ex20.m1.3.3.1.1.3.3.cmml"><mrow id="S3.Ex20.m1.3.3.1.1.3.3.2.2" 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xref="S3.Ex20.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Ex20.m1.3.3.1.1.3.3.1.cmml" xref="S3.Ex20.m1.3.3.1.1.3.3">superscript</csymbol><interval closure="open" id="S3.Ex20.m1.3.3.1.1.3.3.2.1.cmml" xref="S3.Ex20.m1.3.3.1.1.3.3.2.2"><ci id="S3.Ex20.m1.1.1.cmml" xref="S3.Ex20.m1.1.1">𝐴</ci><ci id="S3.Ex20.m1.2.2.cmml" xref="S3.Ex20.m1.2.2">ℬ</ci></interval><cn id="S3.Ex20.m1.3.3.1.1.3.3.3.cmml" type="integer" xref="S3.Ex20.m1.3.3.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex20.m1.3c">D_{\cap}(A\mid\mathcal{B})\;\leq\;\rho(A,\mathcal{B})^{2}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex20.m1.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_ρ ( italic_A , caligraphic_B ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="Thmtheorem25"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem25.1.1.1">Remark 25</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem25.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem25.p1"> <p class="ltx_p" id="Thmtheorem25.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem25.p1.5.5">It is important in the statements of Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a> that the characterization of <math alttext="D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="Thmtheorem25.p1.1.1.m1.1"><semantics id="Thmtheorem25.p1.1.1.m1.1a"><mrow id="Thmtheorem25.p1.1.1.m1.1.1" xref="Thmtheorem25.p1.1.1.m1.1.1.cmml"><msub id="Thmtheorem25.p1.1.1.m1.1.1.3" xref="Thmtheorem25.p1.1.1.m1.1.1.3.cmml"><mi id="Thmtheorem25.p1.1.1.m1.1.1.3.2" xref="Thmtheorem25.p1.1.1.m1.1.1.3.2.cmml">D</mi><mo id="Thmtheorem25.p1.1.1.m1.1.1.3.3" xref="Thmtheorem25.p1.1.1.m1.1.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem25.p1.1.1.m1.1.1.2" xref="Thmtheorem25.p1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem25.p1.1.1.m1.1.1.1.1" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="Thmtheorem25.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.cmml"><mi id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.2" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.1" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.3" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem25.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem25.p1.1.1.m1.1b"><apply id="Thmtheorem25.p1.1.1.m1.1.1.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1"><times id="Thmtheorem25.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.2"></times><apply id="Thmtheorem25.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem25.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="Thmtheorem25.p1.1.1.m1.1.1.3.2.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.3.2">𝐷</ci><intersect id="Thmtheorem25.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.3.3"></intersect></apply><apply id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="Thmtheorem25.p1.1.1.m1.1.1.1.1.1.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem25.p1.1.1.m1.1c">D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem25.p1.1.1.m1.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math> in terms of <math alttext="\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="Thmtheorem25.p1.2.2.m2.2"><semantics id="Thmtheorem25.p1.2.2.m2.2a"><mrow id="Thmtheorem25.p1.2.2.m2.2.3" xref="Thmtheorem25.p1.2.2.m2.2.3.cmml"><mi id="Thmtheorem25.p1.2.2.m2.2.3.2" xref="Thmtheorem25.p1.2.2.m2.2.3.2.cmml">ρ</mi><mo id="Thmtheorem25.p1.2.2.m2.2.3.1" xref="Thmtheorem25.p1.2.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem25.p1.2.2.m2.2.3.3.2" xref="Thmtheorem25.p1.2.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem25.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem25.p1.2.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem25.p1.2.2.m2.1.1" xref="Thmtheorem25.p1.2.2.m2.1.1.cmml">A</mi><mo id="Thmtheorem25.p1.2.2.m2.2.3.3.2.2" xref="Thmtheorem25.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem25.p1.2.2.m2.2.2" xref="Thmtheorem25.p1.2.2.m2.2.2.cmml">ℬ</mi><mo id="Thmtheorem25.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem25.p1.2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem25.p1.2.2.m2.2b"><apply id="Thmtheorem25.p1.2.2.m2.2.3.cmml" xref="Thmtheorem25.p1.2.2.m2.2.3"><times id="Thmtheorem25.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem25.p1.2.2.m2.2.3.1"></times><ci id="Thmtheorem25.p1.2.2.m2.2.3.2.cmml" xref="Thmtheorem25.p1.2.2.m2.2.3.2">𝜌</ci><interval closure="open" id="Thmtheorem25.p1.2.2.m2.2.3.3.1.cmml" xref="Thmtheorem25.p1.2.2.m2.2.3.3.2"><ci id="Thmtheorem25.p1.2.2.m2.1.1.cmml" xref="Thmtheorem25.p1.2.2.m2.1.1">𝐴</ci><ci id="Thmtheorem25.p1.2.2.m2.2.2.cmml" xref="Thmtheorem25.p1.2.2.m2.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem25.p1.2.2.m2.2c">\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem25.p1.2.2.m2.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> does not suffer a quantitative loss that depends on <math alttext="|\mathcal{B}|" class="ltx_Math" display="inline" id="Thmtheorem25.p1.3.3.m3.1"><semantics id="Thmtheorem25.p1.3.3.m3.1a"><mrow id="Thmtheorem25.p1.3.3.m3.1.2.2" xref="Thmtheorem25.p1.3.3.m3.1.2.1.cmml"><mo id="Thmtheorem25.p1.3.3.m3.1.2.2.1" stretchy="false" xref="Thmtheorem25.p1.3.3.m3.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem25.p1.3.3.m3.1.1" xref="Thmtheorem25.p1.3.3.m3.1.1.cmml">ℬ</mi><mo id="Thmtheorem25.p1.3.3.m3.1.2.2.2" stretchy="false" xref="Thmtheorem25.p1.3.3.m3.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem25.p1.3.3.m3.1b"><apply id="Thmtheorem25.p1.3.3.m3.1.2.1.cmml" xref="Thmtheorem25.p1.3.3.m3.1.2.2"><abs id="Thmtheorem25.p1.3.3.m3.1.2.1.1.cmml" xref="Thmtheorem25.p1.3.3.m3.1.2.2.1"></abs><ci id="Thmtheorem25.p1.3.3.m3.1.1.cmml" xref="Thmtheorem25.p1.3.3.m3.1.1">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem25.p1.3.3.m3.1c">|\mathcal{B}|</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem25.p1.3.3.m3.1d">| caligraphic_B |</annotation></semantics></math>. This allows us to apply the results in discrete spaces for which the number of generators in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem25.p1.4.4.m4.1"><semantics id="Thmtheorem25.p1.4.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem25.p1.4.4.m4.1.1" xref="Thmtheorem25.p1.4.4.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem25.p1.4.4.m4.1b"><ci id="Thmtheorem25.p1.4.4.m4.1.1.cmml" xref="Thmtheorem25.p1.4.4.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem25.p1.4.4.m4.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem25.p1.4.4.m4.1d">caligraphic_B</annotation></semantics></math> is large compared to the size of the ambient space <math alttext="\Gamma" class="ltx_Math" display="inline" id="Thmtheorem25.p1.5.5.m5.1"><semantics id="Thmtheorem25.p1.5.5.m5.1a"><mi id="Thmtheorem25.p1.5.5.m5.1.1" mathvariant="normal" xref="Thmtheorem25.p1.5.5.m5.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem25.p1.5.5.m5.1b"><ci id="Thmtheorem25.p1.5.5.m5.1.1.cmml" xref="Thmtheorem25.p1.5.5.m5.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem25.p1.5.5.m5.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem25.p1.5.5.m5.1d">roman_Γ</annotation></semantics></math>, such as in graph complexity.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.13"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.1.p1"> <p class="ltx_p" id="S3.SS3.1.p1.2">Let <math alttext="U=A^{c}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.1.m1.1"><semantics id="S3.SS3.1.p1.1.m1.1a"><mrow id="S3.SS3.1.p1.1.m1.1.1" xref="S3.SS3.1.p1.1.m1.1.1.cmml"><mi id="S3.SS3.1.p1.1.m1.1.1.2" xref="S3.SS3.1.p1.1.m1.1.1.2.cmml">U</mi><mo id="S3.SS3.1.p1.1.m1.1.1.1" xref="S3.SS3.1.p1.1.m1.1.1.1.cmml">=</mo><msup id="S3.SS3.1.p1.1.m1.1.1.3" xref="S3.SS3.1.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.1.p1.1.m1.1.1.3.2" xref="S3.SS3.1.p1.1.m1.1.1.3.2.cmml">A</mi><mi id="S3.SS3.1.p1.1.m1.1.1.3.3" xref="S3.SS3.1.p1.1.m1.1.1.3.3.cmml">c</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.1.m1.1b"><apply id="S3.SS3.1.p1.1.m1.1.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1"><eq id="S3.SS3.1.p1.1.m1.1.1.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1.1"></eq><ci id="S3.SS3.1.p1.1.m1.1.1.2.cmml" xref="S3.SS3.1.p1.1.m1.1.1.2">𝑈</ci><apply id="S3.SS3.1.p1.1.m1.1.1.3.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS3.1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3.2">𝐴</ci><ci id="S3.SS3.1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.1.m1.1c">U=A^{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.1.m1.1d">italic_U = italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="\rho(A,\mathcal{B})=t" class="ltx_Math" display="inline" id="S3.SS3.1.p1.2.m2.2"><semantics id="S3.SS3.1.p1.2.m2.2a"><mrow id="S3.SS3.1.p1.2.m2.2.3" xref="S3.SS3.1.p1.2.m2.2.3.cmml"><mrow id="S3.SS3.1.p1.2.m2.2.3.2" xref="S3.SS3.1.p1.2.m2.2.3.2.cmml"><mi id="S3.SS3.1.p1.2.m2.2.3.2.2" xref="S3.SS3.1.p1.2.m2.2.3.2.2.cmml">ρ</mi><mo id="S3.SS3.1.p1.2.m2.2.3.2.1" xref="S3.SS3.1.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.1.p1.2.m2.2.3.2.3.2" xref="S3.SS3.1.p1.2.m2.2.3.2.3.1.cmml"><mo id="S3.SS3.1.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.1.p1.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S3.SS3.1.p1.2.m2.1.1" xref="S3.SS3.1.p1.2.m2.1.1.cmml">A</mi><mo id="S3.SS3.1.p1.2.m2.2.3.2.3.2.2" xref="S3.SS3.1.p1.2.m2.2.3.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.1.p1.2.m2.2.2" xref="S3.SS3.1.p1.2.m2.2.2.cmml">ℬ</mi><mo id="S3.SS3.1.p1.2.m2.2.3.2.3.2.3" stretchy="false" xref="S3.SS3.1.p1.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.1.p1.2.m2.2.3.1" xref="S3.SS3.1.p1.2.m2.2.3.1.cmml">=</mo><mi id="S3.SS3.1.p1.2.m2.2.3.3" xref="S3.SS3.1.p1.2.m2.2.3.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.2.m2.2b"><apply id="S3.SS3.1.p1.2.m2.2.3.cmml" xref="S3.SS3.1.p1.2.m2.2.3"><eq id="S3.SS3.1.p1.2.m2.2.3.1.cmml" xref="S3.SS3.1.p1.2.m2.2.3.1"></eq><apply id="S3.SS3.1.p1.2.m2.2.3.2.cmml" xref="S3.SS3.1.p1.2.m2.2.3.2"><times id="S3.SS3.1.p1.2.m2.2.3.2.1.cmml" xref="S3.SS3.1.p1.2.m2.2.3.2.1"></times><ci id="S3.SS3.1.p1.2.m2.2.3.2.2.cmml" xref="S3.SS3.1.p1.2.m2.2.3.2.2">𝜌</ci><interval closure="open" id="S3.SS3.1.p1.2.m2.2.3.2.3.1.cmml" xref="S3.SS3.1.p1.2.m2.2.3.2.3.2"><ci id="S3.SS3.1.p1.2.m2.1.1.cmml" xref="S3.SS3.1.p1.2.m2.1.1">𝐴</ci><ci id="S3.SS3.1.p1.2.m2.2.2.cmml" xref="S3.SS3.1.p1.2.m2.2.2">ℬ</ci></interval></apply><ci id="S3.SS3.1.p1.2.m2.2.3.3.cmml" xref="S3.SS3.1.p1.2.m2.2.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.2.m2.2c">\rho(A,\mathcal{B})=t</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.2.m2.2d">italic_ρ ( italic_A , caligraphic_B ) = italic_t</annotation></semantics></math>, and assume that this is witnessed by a family</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Lambda=\{(H_{1},E_{1}),\ldots,(H_{t},E_{t})\}" class="ltx_Math" display="block" id="S3.Ex21.m1.3"><semantics id="S3.Ex21.m1.3a"><mrow id="S3.Ex21.m1.3.3" xref="S3.Ex21.m1.3.3.cmml"><mi id="S3.Ex21.m1.3.3.4" mathvariant="normal" xref="S3.Ex21.m1.3.3.4.cmml">Λ</mi><mo id="S3.Ex21.m1.3.3.3" xref="S3.Ex21.m1.3.3.3.cmml">=</mo><mrow id="S3.Ex21.m1.3.3.2.2" xref="S3.Ex21.m1.3.3.2.3.cmml"><mo id="S3.Ex21.m1.3.3.2.2.3" stretchy="false" xref="S3.Ex21.m1.3.3.2.3.cmml">{</mo><mrow id="S3.Ex21.m1.2.2.1.1.1.2" xref="S3.Ex21.m1.2.2.1.1.1.3.cmml"><mo id="S3.Ex21.m1.2.2.1.1.1.2.3" stretchy="false" xref="S3.Ex21.m1.2.2.1.1.1.3.cmml">(</mo><msub id="S3.Ex21.m1.2.2.1.1.1.1.1" xref="S3.Ex21.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.Ex21.m1.2.2.1.1.1.1.1.2" xref="S3.Ex21.m1.2.2.1.1.1.1.1.2.cmml">H</mi><mn id="S3.Ex21.m1.2.2.1.1.1.1.1.3" xref="S3.Ex21.m1.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex21.m1.2.2.1.1.1.2.4" xref="S3.Ex21.m1.2.2.1.1.1.3.cmml">,</mo><msub id="S3.Ex21.m1.2.2.1.1.1.2.2" xref="S3.Ex21.m1.2.2.1.1.1.2.2.cmml"><mi id="S3.Ex21.m1.2.2.1.1.1.2.2.2" xref="S3.Ex21.m1.2.2.1.1.1.2.2.2.cmml">E</mi><mn id="S3.Ex21.m1.2.2.1.1.1.2.2.3" xref="S3.Ex21.m1.2.2.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S3.Ex21.m1.2.2.1.1.1.2.5" stretchy="false" xref="S3.Ex21.m1.2.2.1.1.1.3.cmml">)</mo></mrow><mo id="S3.Ex21.m1.3.3.2.2.4" xref="S3.Ex21.m1.3.3.2.3.cmml">,</mo><mi id="S3.Ex21.m1.1.1" mathvariant="normal" xref="S3.Ex21.m1.1.1.cmml">…</mi><mo id="S3.Ex21.m1.3.3.2.2.5" xref="S3.Ex21.m1.3.3.2.3.cmml">,</mo><mrow id="S3.Ex21.m1.3.3.2.2.2.2" xref="S3.Ex21.m1.3.3.2.2.2.3.cmml"><mo id="S3.Ex21.m1.3.3.2.2.2.2.3" stretchy="false" xref="S3.Ex21.m1.3.3.2.2.2.3.cmml">(</mo><msub id="S3.Ex21.m1.3.3.2.2.2.1.1" xref="S3.Ex21.m1.3.3.2.2.2.1.1.cmml"><mi id="S3.Ex21.m1.3.3.2.2.2.1.1.2" xref="S3.Ex21.m1.3.3.2.2.2.1.1.2.cmml">H</mi><mi id="S3.Ex21.m1.3.3.2.2.2.1.1.3" xref="S3.Ex21.m1.3.3.2.2.2.1.1.3.cmml">t</mi></msub><mo id="S3.Ex21.m1.3.3.2.2.2.2.4" xref="S3.Ex21.m1.3.3.2.2.2.3.cmml">,</mo><msub id="S3.Ex21.m1.3.3.2.2.2.2.2" xref="S3.Ex21.m1.3.3.2.2.2.2.2.cmml"><mi id="S3.Ex21.m1.3.3.2.2.2.2.2.2" xref="S3.Ex21.m1.3.3.2.2.2.2.2.2.cmml">E</mi><mi id="S3.Ex21.m1.3.3.2.2.2.2.2.3" xref="S3.Ex21.m1.3.3.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S3.Ex21.m1.3.3.2.2.2.2.5" stretchy="false" xref="S3.Ex21.m1.3.3.2.2.2.3.cmml">)</mo></mrow><mo id="S3.Ex21.m1.3.3.2.2.6" stretchy="false" xref="S3.Ex21.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex21.m1.3b"><apply id="S3.Ex21.m1.3.3.cmml" xref="S3.Ex21.m1.3.3"><eq id="S3.Ex21.m1.3.3.3.cmml" xref="S3.Ex21.m1.3.3.3"></eq><ci id="S3.Ex21.m1.3.3.4.cmml" xref="S3.Ex21.m1.3.3.4">Λ</ci><set id="S3.Ex21.m1.3.3.2.3.cmml" xref="S3.Ex21.m1.3.3.2.2"><interval closure="open" id="S3.Ex21.m1.2.2.1.1.1.3.cmml" xref="S3.Ex21.m1.2.2.1.1.1.2"><apply id="S3.Ex21.m1.2.2.1.1.1.1.1.cmml" xref="S3.Ex21.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex21.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex21.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex21.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.Ex21.m1.2.2.1.1.1.1.1.2">𝐻</ci><cn id="S3.Ex21.m1.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex21.m1.2.2.1.1.1.1.1.3">1</cn></apply><apply id="S3.Ex21.m1.2.2.1.1.1.2.2.cmml" xref="S3.Ex21.m1.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex21.m1.2.2.1.1.1.2.2.1.cmml" xref="S3.Ex21.m1.2.2.1.1.1.2.2">subscript</csymbol><ci id="S3.Ex21.m1.2.2.1.1.1.2.2.2.cmml" xref="S3.Ex21.m1.2.2.1.1.1.2.2.2">𝐸</ci><cn id="S3.Ex21.m1.2.2.1.1.1.2.2.3.cmml" type="integer" xref="S3.Ex21.m1.2.2.1.1.1.2.2.3">1</cn></apply></interval><ci id="S3.Ex21.m1.1.1.cmml" xref="S3.Ex21.m1.1.1">…</ci><interval closure="open" id="S3.Ex21.m1.3.3.2.2.2.3.cmml" xref="S3.Ex21.m1.3.3.2.2.2.2"><apply id="S3.Ex21.m1.3.3.2.2.2.1.1.cmml" xref="S3.Ex21.m1.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex21.m1.3.3.2.2.2.1.1.1.cmml" xref="S3.Ex21.m1.3.3.2.2.2.1.1">subscript</csymbol><ci id="S3.Ex21.m1.3.3.2.2.2.1.1.2.cmml" xref="S3.Ex21.m1.3.3.2.2.2.1.1.2">𝐻</ci><ci id="S3.Ex21.m1.3.3.2.2.2.1.1.3.cmml" xref="S3.Ex21.m1.3.3.2.2.2.1.1.3">𝑡</ci></apply><apply id="S3.Ex21.m1.3.3.2.2.2.2.2.cmml" xref="S3.Ex21.m1.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex21.m1.3.3.2.2.2.2.2.1.cmml" xref="S3.Ex21.m1.3.3.2.2.2.2.2">subscript</csymbol><ci id="S3.Ex21.m1.3.3.2.2.2.2.2.2.cmml" xref="S3.Ex21.m1.3.3.2.2.2.2.2.2">𝐸</ci><ci id="S3.Ex21.m1.3.3.2.2.2.2.2.3.cmml" xref="S3.Ex21.m1.3.3.2.2.2.2.2.3">𝑡</ci></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex21.m1.3c">\Lambda=\{(H_{1},E_{1}),\ldots,(H_{t},E_{t})\}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex21.m1.3d">roman_Λ = { ( italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , ( italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.1.p1.4">of <math alttext="t" class="ltx_Math" display="inline" id="S3.SS3.1.p1.3.m1.1"><semantics id="S3.SS3.1.p1.3.m1.1a"><mi id="S3.SS3.1.p1.3.m1.1.1" xref="S3.SS3.1.p1.3.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.3.m1.1b"><ci id="S3.SS3.1.p1.3.m1.1.1.cmml" xref="S3.SS3.1.p1.3.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.3.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.3.m1.1d">italic_t</annotation></semantics></math> pairs of subsets of <math alttext="U" class="ltx_Math" display="inline" id="S3.SS3.1.p1.4.m2.1"><semantics id="S3.SS3.1.p1.4.m2.1a"><mi id="S3.SS3.1.p1.4.m2.1.1" xref="S3.SS3.1.p1.4.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.4.m2.1b"><ci id="S3.SS3.1.p1.4.m2.1.1.cmml" xref="S3.SS3.1.p1.4.m2.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.4.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.4.m2.1d">italic_U</annotation></semantics></math>. We let</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathfrak{F}_{\Lambda}=\{\mathcal{F}\subseteq\mathcal{P}(U)\mid\mathcal{F}~{}% \text{is a semi-filter that preserves}~{}\Lambda\}." class="ltx_Math" display="block" id="S3.Ex22.m1.2"><semantics id="S3.Ex22.m1.2a"><mrow id="S3.Ex22.m1.2.2.1" xref="S3.Ex22.m1.2.2.1.1.cmml"><mrow id="S3.Ex22.m1.2.2.1.1" xref="S3.Ex22.m1.2.2.1.1.cmml"><msub id="S3.Ex22.m1.2.2.1.1.4" xref="S3.Ex22.m1.2.2.1.1.4.cmml"><mi id="S3.Ex22.m1.2.2.1.1.4.2" xref="S3.Ex22.m1.2.2.1.1.4.2.cmml">𝔉</mi><mi id="S3.Ex22.m1.2.2.1.1.4.3" mathvariant="normal" xref="S3.Ex22.m1.2.2.1.1.4.3.cmml">Λ</mi></msub><mo id="S3.Ex22.m1.2.2.1.1.3" xref="S3.Ex22.m1.2.2.1.1.3.cmml">=</mo><mrow id="S3.Ex22.m1.2.2.1.1.2.2" xref="S3.Ex22.m1.2.2.1.1.2.3.cmml"><mo id="S3.Ex22.m1.2.2.1.1.2.2.3" stretchy="false" xref="S3.Ex22.m1.2.2.1.1.2.3.1.cmml">{</mo><mrow id="S3.Ex22.m1.2.2.1.1.1.1.1" xref="S3.Ex22.m1.2.2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex22.m1.2.2.1.1.1.1.1.2" xref="S3.Ex22.m1.2.2.1.1.1.1.1.2.cmml">ℱ</mi><mo id="S3.Ex22.m1.2.2.1.1.1.1.1.1" xref="S3.Ex22.m1.2.2.1.1.1.1.1.1.cmml">⊆</mo><mrow id="S3.Ex22.m1.2.2.1.1.1.1.1.3" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex22.m1.2.2.1.1.1.1.1.3.2" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.2.cmml">𝒫</mi><mo id="S3.Ex22.m1.2.2.1.1.1.1.1.3.1" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex22.m1.2.2.1.1.1.1.1.3.3.2" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.cmml"><mo id="S3.Ex22.m1.2.2.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.cmml">(</mo><mi id="S3.Ex22.m1.1.1" xref="S3.Ex22.m1.1.1.cmml">U</mi><mo id="S3.Ex22.m1.2.2.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S3.Ex22.m1.2.2.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S3.Ex22.m1.2.2.1.1.2.2.4" lspace="0em" rspace="0em" xref="S3.Ex22.m1.2.2.1.1.2.3.1.cmml">∣</mo><mrow id="S3.Ex22.m1.2.2.1.1.2.2.2" xref="S3.Ex22.m1.2.2.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex22.m1.2.2.1.1.2.2.2.2" xref="S3.Ex22.m1.2.2.1.1.2.2.2.2.cmml">ℱ</mi><mo id="S3.Ex22.m1.2.2.1.1.2.2.2.1" lspace="0.330em" xref="S3.Ex22.m1.2.2.1.1.2.2.2.1.cmml">⁢</mo><mtext id="S3.Ex22.m1.2.2.1.1.2.2.2.3" xref="S3.Ex22.m1.2.2.1.1.2.2.2.3a.cmml">is a semi-filter that preserves</mtext><mo id="S3.Ex22.m1.2.2.1.1.2.2.2.1a" lspace="0.330em" xref="S3.Ex22.m1.2.2.1.1.2.2.2.1.cmml">⁢</mo><mi id="S3.Ex22.m1.2.2.1.1.2.2.2.4" mathvariant="normal" xref="S3.Ex22.m1.2.2.1.1.2.2.2.4.cmml">Λ</mi></mrow><mo id="S3.Ex22.m1.2.2.1.1.2.2.5" stretchy="false" xref="S3.Ex22.m1.2.2.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S3.Ex22.m1.2.2.1.2" lspace="0em" xref="S3.Ex22.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex22.m1.2b"><apply id="S3.Ex22.m1.2.2.1.1.cmml" 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encoding="application/x-llamapun" id="S3.Ex22.m1.2d">fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT = { caligraphic_F ⊆ caligraphic_P ( italic_U ) ∣ caligraphic_F is a semi-filter that preserves roman_Λ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.1.p1.9">Recall the definition of the cover graph <math alttext="\Phi_{A,\mathcal{B}}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.5.m1.2"><semantics id="S3.SS3.1.p1.5.m1.2a"><msub id="S3.SS3.1.p1.5.m1.2.3" xref="S3.SS3.1.p1.5.m1.2.3.cmml"><mi id="S3.SS3.1.p1.5.m1.2.3.2" mathvariant="normal" xref="S3.SS3.1.p1.5.m1.2.3.2.cmml">Φ</mi><mrow id="S3.SS3.1.p1.5.m1.2.2.2.4" xref="S3.SS3.1.p1.5.m1.2.2.2.3.cmml"><mi id="S3.SS3.1.p1.5.m1.1.1.1.1" xref="S3.SS3.1.p1.5.m1.1.1.1.1.cmml">A</mi><mo id="S3.SS3.1.p1.5.m1.2.2.2.4.1" xref="S3.SS3.1.p1.5.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.1.p1.5.m1.2.2.2.2" xref="S3.SS3.1.p1.5.m1.2.2.2.2.cmml">ℬ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.5.m1.2b"><apply id="S3.SS3.1.p1.5.m1.2.3.cmml" xref="S3.SS3.1.p1.5.m1.2.3"><csymbol cd="ambiguous" id="S3.SS3.1.p1.5.m1.2.3.1.cmml" xref="S3.SS3.1.p1.5.m1.2.3">subscript</csymbol><ci id="S3.SS3.1.p1.5.m1.2.3.2.cmml" xref="S3.SS3.1.p1.5.m1.2.3.2">Φ</ci><list id="S3.SS3.1.p1.5.m1.2.2.2.3.cmml" xref="S3.SS3.1.p1.5.m1.2.2.2.4"><ci id="S3.SS3.1.p1.5.m1.1.1.1.1.cmml" xref="S3.SS3.1.p1.5.m1.1.1.1.1">𝐴</ci><ci id="S3.SS3.1.p1.5.m1.2.2.2.2.cmml" xref="S3.SS3.1.p1.5.m1.2.2.2.2">ℬ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.5.m1.2c">\Phi_{A,\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.5.m1.2d">roman_Φ start_POSTSUBSCRIPT italic_A , caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.1.p1.6.m2.1"><semantics id="S3.SS3.1.p1.6.m2.1a"><mi id="S3.SS3.1.p1.6.m2.1.1" xref="S3.SS3.1.p1.6.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.6.m2.1b"><ci id="S3.SS3.1.p1.6.m2.1.1.cmml" xref="S3.SS3.1.p1.6.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.6.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.6.m2.1d">italic_A</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.7.m3.1"><semantics id="S3.SS3.1.p1.7.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.1.p1.7.m3.1.1" xref="S3.SS3.1.p1.7.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.7.m3.1b"><ci id="S3.SS3.1.p1.7.m3.1.1.cmml" xref="S3.SS3.1.p1.7.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.7.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.7.m3.1d">caligraphic_B</annotation></semantics></math> (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1" title="3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.1</span></a>). Observe that, while <math alttext="\Lambda\subseteq V_{\mathsf{pairs}}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.8.m4.1"><semantics id="S3.SS3.1.p1.8.m4.1a"><mrow id="S3.SS3.1.p1.8.m4.1.1" xref="S3.SS3.1.p1.8.m4.1.1.cmml"><mi id="S3.SS3.1.p1.8.m4.1.1.2" mathvariant="normal" xref="S3.SS3.1.p1.8.m4.1.1.2.cmml">Λ</mi><mo id="S3.SS3.1.p1.8.m4.1.1.1" xref="S3.SS3.1.p1.8.m4.1.1.1.cmml">⊆</mo><msub id="S3.SS3.1.p1.8.m4.1.1.3" xref="S3.SS3.1.p1.8.m4.1.1.3.cmml"><mi id="S3.SS3.1.p1.8.m4.1.1.3.2" xref="S3.SS3.1.p1.8.m4.1.1.3.2.cmml">V</mi><mi id="S3.SS3.1.p1.8.m4.1.1.3.3" xref="S3.SS3.1.p1.8.m4.1.1.3.3.cmml">𝗉𝖺𝗂𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.8.m4.1b"><apply id="S3.SS3.1.p1.8.m4.1.1.cmml" xref="S3.SS3.1.p1.8.m4.1.1"><subset id="S3.SS3.1.p1.8.m4.1.1.1.cmml" xref="S3.SS3.1.p1.8.m4.1.1.1"></subset><ci id="S3.SS3.1.p1.8.m4.1.1.2.cmml" xref="S3.SS3.1.p1.8.m4.1.1.2">Λ</ci><apply id="S3.SS3.1.p1.8.m4.1.1.3.cmml" xref="S3.SS3.1.p1.8.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.1.p1.8.m4.1.1.3.1.cmml" xref="S3.SS3.1.p1.8.m4.1.1.3">subscript</csymbol><ci id="S3.SS3.1.p1.8.m4.1.1.3.2.cmml" xref="S3.SS3.1.p1.8.m4.1.1.3.2">𝑉</ci><ci id="S3.SS3.1.p1.8.m4.1.1.3.3.cmml" xref="S3.SS3.1.p1.8.m4.1.1.3.3">𝗉𝖺𝗂𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.8.m4.1c">\Lambda\subseteq V_{\mathsf{pairs}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.8.m4.1d">roman_Λ ⊆ italic_V start_POSTSUBSCRIPT sansserif_pairs end_POSTSUBSCRIPT</annotation></semantics></math>, it is not necessarily the case that <math alttext="\mathfrak{F}_{\Lambda}\subseteq V_{\mathsf{filters}}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.9.m5.1"><semantics id="S3.SS3.1.p1.9.m5.1a"><mrow id="S3.SS3.1.p1.9.m5.1.1" xref="S3.SS3.1.p1.9.m5.1.1.cmml"><msub id="S3.SS3.1.p1.9.m5.1.1.2" xref="S3.SS3.1.p1.9.m5.1.1.2.cmml"><mi id="S3.SS3.1.p1.9.m5.1.1.2.2" xref="S3.SS3.1.p1.9.m5.1.1.2.2.cmml">𝔉</mi><mi id="S3.SS3.1.p1.9.m5.1.1.2.3" mathvariant="normal" xref="S3.SS3.1.p1.9.m5.1.1.2.3.cmml">Λ</mi></msub><mo id="S3.SS3.1.p1.9.m5.1.1.1" xref="S3.SS3.1.p1.9.m5.1.1.1.cmml">⊆</mo><msub id="S3.SS3.1.p1.9.m5.1.1.3" xref="S3.SS3.1.p1.9.m5.1.1.3.cmml"><mi id="S3.SS3.1.p1.9.m5.1.1.3.2" xref="S3.SS3.1.p1.9.m5.1.1.3.2.cmml">V</mi><mi id="S3.SS3.1.p1.9.m5.1.1.3.3" xref="S3.SS3.1.p1.9.m5.1.1.3.3.cmml">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.9.m5.1b"><apply id="S3.SS3.1.p1.9.m5.1.1.cmml" xref="S3.SS3.1.p1.9.m5.1.1"><subset id="S3.SS3.1.p1.9.m5.1.1.1.cmml" xref="S3.SS3.1.p1.9.m5.1.1.1"></subset><apply id="S3.SS3.1.p1.9.m5.1.1.2.cmml" xref="S3.SS3.1.p1.9.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.1.p1.9.m5.1.1.2.1.cmml" xref="S3.SS3.1.p1.9.m5.1.1.2">subscript</csymbol><ci id="S3.SS3.1.p1.9.m5.1.1.2.2.cmml" xref="S3.SS3.1.p1.9.m5.1.1.2.2">𝔉</ci><ci id="S3.SS3.1.p1.9.m5.1.1.2.3.cmml" xref="S3.SS3.1.p1.9.m5.1.1.2.3">Λ</ci></apply><apply id="S3.SS3.1.p1.9.m5.1.1.3.cmml" xref="S3.SS3.1.p1.9.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.1.p1.9.m5.1.1.3.1.cmml" xref="S3.SS3.1.p1.9.m5.1.1.3">subscript</csymbol><ci id="S3.SS3.1.p1.9.m5.1.1.3.2.cmml" xref="S3.SS3.1.p1.9.m5.1.1.3.2">𝑉</ci><ci id="S3.SS3.1.p1.9.m5.1.1.3.3.cmml" xref="S3.SS3.1.p1.9.m5.1.1.3.3">𝖿𝗂𝗅𝗍𝖾𝗋𝗌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.9.m5.1c">\mathfrak{F}_{\Lambda}\subseteq V_{\mathsf{filters}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.9.m5.1d">fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ⊆ italic_V start_POSTSUBSCRIPT sansserif_filters end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem26"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem26.1.1.1">Claim 26</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem26.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem26.p1"> <p class="ltx_p" id="Thmtheorem26.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem26.p1.1.1">For every <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="Thmtheorem26.p1.1.1.m1.1"><semantics id="Thmtheorem26.p1.1.1.m1.1a"><mrow id="Thmtheorem26.p1.1.1.m1.1.1" xref="Thmtheorem26.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem26.p1.1.1.m1.1.1.2" xref="Thmtheorem26.p1.1.1.m1.1.1.2.cmml">w</mi><mo id="Thmtheorem26.p1.1.1.m1.1.1.1" xref="Thmtheorem26.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem26.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem26.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem26.p1.1.1.m1.1b"><apply id="Thmtheorem26.p1.1.1.m1.1.1.cmml" xref="Thmtheorem26.p1.1.1.m1.1.1"><in id="Thmtheorem26.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem26.p1.1.1.m1.1.1.1"></in><ci id="Thmtheorem26.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem26.p1.1.1.m1.1.1.2">𝑤</ci><ci id="Thmtheorem26.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem26.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem26.p1.1.1.m1.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem26.p1.1.1.m1.1d">italic_w ∈ roman_Γ</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w\in A\quad\text{if and only if}\quad\nexists\mathcal{F}\in\mathfrak{F}_{% \Lambda}~{}\text{that is above}~{}w~{}(\text{w.r.t.}~{}\mathcal{B}~{}\text{and% }~{}U)." class="ltx_Math" display="block" id="S3.Ex23.m1.3"><semantics id="S3.Ex23.m1.3a"><mrow id="S3.Ex23.m1.3.3.1"><mrow id="S3.Ex23.m1.3.3.1.1.2" xref="S3.Ex23.m1.3.3.1.1.3.cmml"><mrow id="S3.Ex23.m1.3.3.1.1.1.1" xref="S3.Ex23.m1.3.3.1.1.1.1.cmml"><mi id="S3.Ex23.m1.3.3.1.1.1.1.2" xref="S3.Ex23.m1.3.3.1.1.1.1.2.cmml">w</mi><mo id="S3.Ex23.m1.3.3.1.1.1.1.1" xref="S3.Ex23.m1.3.3.1.1.1.1.1.cmml">∈</mo><mrow id="S3.Ex23.m1.3.3.1.1.1.1.3.2" xref="S3.Ex23.m1.3.3.1.1.1.1.3.1.cmml"><mi id="S3.Ex23.m1.1.1" xref="S3.Ex23.m1.1.1.cmml">A</mi><mspace id="S3.Ex23.m1.3.3.1.1.1.1.3.2.1" width="1em" xref="S3.Ex23.m1.3.3.1.1.1.1.3.1.cmml"></mspace><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.2.2" xref="S3.Ex23.m1.2.2a.cmml">if and only if</mtext></mrow></mrow><mspace id="S3.Ex23.m1.3.3.1.1.2.3" width="1em" xref="S3.Ex23.m1.3.3.1.1.3a.cmml"></mspace><mrow id="S3.Ex23.m1.3.3.1.1.2.2" xref="S3.Ex23.m1.3.3.1.1.2.2.cmml"><mrow id="S3.Ex23.m1.3.3.1.1.2.2.3" xref="S3.Ex23.m1.3.3.1.1.2.2.3.cmml"><mi id="S3.Ex23.m1.3.3.1.1.2.2.3.2" mathvariant="normal" xref="S3.Ex23.m1.3.3.1.1.2.2.3.2.cmml">∄</mi><mo id="S3.Ex23.m1.3.3.1.1.2.2.3.1" xref="S3.Ex23.m1.3.3.1.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex23.m1.3.3.1.1.2.2.3.3" xref="S3.Ex23.m1.3.3.1.1.2.2.3.3.cmml">ℱ</mi></mrow><mo id="S3.Ex23.m1.3.3.1.1.2.2.2" xref="S3.Ex23.m1.3.3.1.1.2.2.2.cmml">∈</mo><mrow id="S3.Ex23.m1.3.3.1.1.2.2.1" xref="S3.Ex23.m1.3.3.1.1.2.2.1.cmml"><msub id="S3.Ex23.m1.3.3.1.1.2.2.1.3" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3.cmml"><mi id="S3.Ex23.m1.3.3.1.1.2.2.1.3.2" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3.2.cmml">𝔉</mi><mi id="S3.Ex23.m1.3.3.1.1.2.2.1.3.3" mathvariant="normal" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3.3.cmml">Λ</mi></msub><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.2" xref="S3.Ex23.m1.3.3.1.1.2.2.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.4" xref="S3.Ex23.m1.3.3.1.1.2.2.1.4a.cmml">that is above</mtext><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.2a" lspace="0.330em" xref="S3.Ex23.m1.3.3.1.1.2.2.1.2.cmml">⁢</mo><mi id="S3.Ex23.m1.3.3.1.1.2.2.1.5" xref="S3.Ex23.m1.3.3.1.1.2.2.1.5.cmml">w</mi><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.2b" lspace="0.330em" xref="S3.Ex23.m1.3.3.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.cmml"><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.2" stretchy="false" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2a.cmml">w.r.t.</mtext><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1" lspace="0.330em" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.3" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.3.cmml">ℬ</mi><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1a" lspace="0.330em" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4a.cmml">and</mtext><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1b" lspace="0.330em" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.5" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.5.cmml">U</mi></mrow><mo id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.3" stretchy="false" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.Ex23.m1.3.3.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex23.m1.3b"><apply id="S3.Ex23.m1.3.3.1.1.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.Ex23.m1.3.3.1.1.3a.cmml" xref="S3.Ex23.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S3.Ex23.m1.3.3.1.1.1.1.cmml" xref="S3.Ex23.m1.3.3.1.1.1.1"><in id="S3.Ex23.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex23.m1.3.3.1.1.1.1.1"></in><ci id="S3.Ex23.m1.3.3.1.1.1.1.2.cmml" xref="S3.Ex23.m1.3.3.1.1.1.1.2">𝑤</ci><list id="S3.Ex23.m1.3.3.1.1.1.1.3.1.cmml" xref="S3.Ex23.m1.3.3.1.1.1.1.3.2"><ci id="S3.Ex23.m1.1.1.cmml" xref="S3.Ex23.m1.1.1">𝐴</ci><ci id="S3.Ex23.m1.2.2a.cmml" xref="S3.Ex23.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.2.2.cmml" xref="S3.Ex23.m1.2.2">if and only if</mtext></ci></list></apply><apply id="S3.Ex23.m1.3.3.1.1.2.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2"><in id="S3.Ex23.m1.3.3.1.1.2.2.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.2"></in><apply id="S3.Ex23.m1.3.3.1.1.2.2.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.3"><times id="S3.Ex23.m1.3.3.1.1.2.2.3.1.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.3.1"></times><csymbol cd="latexml" id="S3.Ex23.m1.3.3.1.1.2.2.3.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.3.2">not-exists</csymbol><ci id="S3.Ex23.m1.3.3.1.1.2.2.3.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.3.3">ℱ</ci></apply><apply id="S3.Ex23.m1.3.3.1.1.2.2.1.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1"><times id="S3.Ex23.m1.3.3.1.1.2.2.1.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.2"></times><apply id="S3.Ex23.m1.3.3.1.1.2.2.1.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3"><csymbol cd="ambiguous" id="S3.Ex23.m1.3.3.1.1.2.2.1.3.1.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3">subscript</csymbol><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.3.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3.2">𝔉</ci><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.3.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.3.3">Λ</ci></apply><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.4a.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.4"><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.4.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.4">that is above</mtext></ci><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.5.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.5">𝑤</ci><apply id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1"><times id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.1"></times><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2a.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2"><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.2">w.r.t.</mtext></ci><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.3.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.3">ℬ</ci><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4a.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4"><mtext class="ltx_mathvariant_italic" id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.4">and</mtext></ci><ci id="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.5.cmml" xref="S3.Ex23.m1.3.3.1.1.2.2.1.1.1.1.5">𝑈</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex23.m1.3c">w\in A\quad\text{if and only if}\quad\nexists\mathcal{F}\in\mathfrak{F}_{% \Lambda}~{}\text{that is above}~{}w~{}(\text{w.r.t.}~{}\mathcal{B}~{}\text{and% }~{}U).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex23.m1.3d">italic_w ∈ italic_A if and only if ∄ caligraphic_F ∈ fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT that is above italic_w ( w.r.t. caligraphic_B and italic_U ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS3.2.p2"> <p class="ltx_p" id="S3.SS3.2.p2.10">In order to see this, notice that if <math alttext="w\in A" class="ltx_Math" display="inline" id="S3.SS3.2.p2.1.m1.1"><semantics id="S3.SS3.2.p2.1.m1.1a"><mrow id="S3.SS3.2.p2.1.m1.1.1" xref="S3.SS3.2.p2.1.m1.1.1.cmml"><mi id="S3.SS3.2.p2.1.m1.1.1.2" xref="S3.SS3.2.p2.1.m1.1.1.2.cmml">w</mi><mo id="S3.SS3.2.p2.1.m1.1.1.1" xref="S3.SS3.2.p2.1.m1.1.1.1.cmml">∈</mo><mi id="S3.SS3.2.p2.1.m1.1.1.3" xref="S3.SS3.2.p2.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.1.m1.1b"><apply id="S3.SS3.2.p2.1.m1.1.1.cmml" xref="S3.SS3.2.p2.1.m1.1.1"><in id="S3.SS3.2.p2.1.m1.1.1.1.cmml" xref="S3.SS3.2.p2.1.m1.1.1.1"></in><ci id="S3.SS3.2.p2.1.m1.1.1.2.cmml" xref="S3.SS3.2.p2.1.m1.1.1.2">𝑤</ci><ci id="S3.SS3.2.p2.1.m1.1.1.3.cmml" xref="S3.SS3.2.p2.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.1.m1.1c">w\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.1.m1.1d">italic_w ∈ italic_A</annotation></semantics></math> then indeed there is no such <math alttext="\mathcal{F}\in\mathfrak{F}_{\Lambda}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.2.m2.1"><semantics id="S3.SS3.2.p2.2.m2.1a"><mrow id="S3.SS3.2.p2.2.m2.1.1" xref="S3.SS3.2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.2.p2.2.m2.1.1.2" xref="S3.SS3.2.p2.2.m2.1.1.2.cmml">ℱ</mi><mo id="S3.SS3.2.p2.2.m2.1.1.1" xref="S3.SS3.2.p2.2.m2.1.1.1.cmml">∈</mo><msub id="S3.SS3.2.p2.2.m2.1.1.3" xref="S3.SS3.2.p2.2.m2.1.1.3.cmml"><mi id="S3.SS3.2.p2.2.m2.1.1.3.2" xref="S3.SS3.2.p2.2.m2.1.1.3.2.cmml">𝔉</mi><mi id="S3.SS3.2.p2.2.m2.1.1.3.3" mathvariant="normal" xref="S3.SS3.2.p2.2.m2.1.1.3.3.cmml">Λ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.2.m2.1b"><apply id="S3.SS3.2.p2.2.m2.1.1.cmml" xref="S3.SS3.2.p2.2.m2.1.1"><in id="S3.SS3.2.p2.2.m2.1.1.1.cmml" xref="S3.SS3.2.p2.2.m2.1.1.1"></in><ci id="S3.SS3.2.p2.2.m2.1.1.2.cmml" xref="S3.SS3.2.p2.2.m2.1.1.2">ℱ</ci><apply id="S3.SS3.2.p2.2.m2.1.1.3.cmml" xref="S3.SS3.2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.2.p2.2.m2.1.1.3.1.cmml" xref="S3.SS3.2.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.2.p2.2.m2.1.1.3.2.cmml" xref="S3.SS3.2.p2.2.m2.1.1.3.2">𝔉</ci><ci id="S3.SS3.2.p2.2.m2.1.1.3.3.cmml" xref="S3.SS3.2.p2.2.m2.1.1.3.3">Λ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.2.m2.1c">\mathcal{F}\in\mathfrak{F}_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.2.m2.1d">caligraphic_F ∈ fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math>, using the definitions of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.2.p2.3.m3.1"><semantics id="S3.SS3.2.p2.3.m3.1a"><mi id="S3.SS3.2.p2.3.m3.1.1" xref="S3.SS3.2.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.3.m3.1b"><ci id="S3.SS3.2.p2.3.m3.1.1.cmml" xref="S3.SS3.2.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.3.m3.1d">italic_ρ</annotation></semantics></math> and <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS3.2.p2.4.m4.1"><semantics id="S3.SS3.2.p2.4.m4.1a"><mi id="S3.SS3.2.p2.4.m4.1.1" mathvariant="normal" xref="S3.SS3.2.p2.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.4.m4.1b"><ci id="S3.SS3.2.p2.4.m4.1.1.cmml" xref="S3.SS3.2.p2.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.4.m4.1d">roman_Λ</annotation></semantics></math>. On the other hand, for <math alttext="w\notin A" class="ltx_Math" display="inline" id="S3.SS3.2.p2.5.m5.1"><semantics id="S3.SS3.2.p2.5.m5.1a"><mrow id="S3.SS3.2.p2.5.m5.1.1" xref="S3.SS3.2.p2.5.m5.1.1.cmml"><mi id="S3.SS3.2.p2.5.m5.1.1.2" xref="S3.SS3.2.p2.5.m5.1.1.2.cmml">w</mi><mo id="S3.SS3.2.p2.5.m5.1.1.1" xref="S3.SS3.2.p2.5.m5.1.1.1.cmml">∉</mo><mi id="S3.SS3.2.p2.5.m5.1.1.3" xref="S3.SS3.2.p2.5.m5.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.5.m5.1b"><apply id="S3.SS3.2.p2.5.m5.1.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1"><notin id="S3.SS3.2.p2.5.m5.1.1.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1.1"></notin><ci id="S3.SS3.2.p2.5.m5.1.1.2.cmml" xref="S3.SS3.2.p2.5.m5.1.1.2">𝑤</ci><ci id="S3.SS3.2.p2.5.m5.1.1.3.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.5.m5.1c">w\notin A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.5.m5.1d">italic_w ∉ italic_A</annotation></semantics></math>, it is easy to check that <math alttext="\mathcal{F}_{w}\stackrel{{\scriptstyle\rm def}}{{=}}\{U^{\prime}\subseteq U% \mid w\in U^{\prime}\}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.6.m6.2"><semantics id="S3.SS3.2.p2.6.m6.2a"><mrow id="S3.SS3.2.p2.6.m6.2.2" xref="S3.SS3.2.p2.6.m6.2.2.cmml"><msub id="S3.SS3.2.p2.6.m6.2.2.4" xref="S3.SS3.2.p2.6.m6.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.2.p2.6.m6.2.2.4.2" xref="S3.SS3.2.p2.6.m6.2.2.4.2.cmml">ℱ</mi><mi id="S3.SS3.2.p2.6.m6.2.2.4.3" xref="S3.SS3.2.p2.6.m6.2.2.4.3.cmml">w</mi></msub><mover id="S3.SS3.2.p2.6.m6.2.2.3" xref="S3.SS3.2.p2.6.m6.2.2.3.cmml"><mo id="S3.SS3.2.p2.6.m6.2.2.3.2" xref="S3.SS3.2.p2.6.m6.2.2.3.2.cmml">=</mo><mi id="S3.SS3.2.p2.6.m6.2.2.3.3" xref="S3.SS3.2.p2.6.m6.2.2.3.3.cmml">def</mi></mover><mrow id="S3.SS3.2.p2.6.m6.2.2.2.2" xref="S3.SS3.2.p2.6.m6.2.2.2.3.cmml"><mo id="S3.SS3.2.p2.6.m6.2.2.2.2.3" stretchy="false" xref="S3.SS3.2.p2.6.m6.2.2.2.3.1.cmml">{</mo><mrow id="S3.SS3.2.p2.6.m6.1.1.1.1.1" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.cmml"><msup id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.cmml"><mi id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.2" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.2.cmml">U</mi><mo id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.3" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS3.2.p2.6.m6.1.1.1.1.1.1" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.1.cmml">⊆</mo><mi id="S3.SS3.2.p2.6.m6.1.1.1.1.1.3" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.3.cmml">U</mi></mrow><mo fence="true" id="S3.SS3.2.p2.6.m6.2.2.2.2.4" lspace="0em" rspace="0em" xref="S3.SS3.2.p2.6.m6.2.2.2.3.1.cmml">∣</mo><mrow id="S3.SS3.2.p2.6.m6.2.2.2.2.2" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.cmml"><mi id="S3.SS3.2.p2.6.m6.2.2.2.2.2.2" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.2.cmml">w</mi><mo id="S3.SS3.2.p2.6.m6.2.2.2.2.2.1" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.1.cmml">∈</mo><msup id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.cmml"><mi id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.2" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.2.cmml">U</mi><mo id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.3" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS3.2.p2.6.m6.2.2.2.2.5" stretchy="false" xref="S3.SS3.2.p2.6.m6.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.6.m6.2b"><apply id="S3.SS3.2.p2.6.m6.2.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2"><apply id="S3.SS3.2.p2.6.m6.2.2.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.3"><csymbol cd="ambiguous" id="S3.SS3.2.p2.6.m6.2.2.3.1.cmml" xref="S3.SS3.2.p2.6.m6.2.2.3">superscript</csymbol><eq id="S3.SS3.2.p2.6.m6.2.2.3.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2.3.2"></eq><ci id="S3.SS3.2.p2.6.m6.2.2.3.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.3.3">def</ci></apply><apply id="S3.SS3.2.p2.6.m6.2.2.4.cmml" xref="S3.SS3.2.p2.6.m6.2.2.4"><csymbol cd="ambiguous" id="S3.SS3.2.p2.6.m6.2.2.4.1.cmml" xref="S3.SS3.2.p2.6.m6.2.2.4">subscript</csymbol><ci id="S3.SS3.2.p2.6.m6.2.2.4.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2.4.2">ℱ</ci><ci id="S3.SS3.2.p2.6.m6.2.2.4.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.4.3">𝑤</ci></apply><apply id="S3.SS3.2.p2.6.m6.2.2.2.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2"><csymbol cd="latexml" id="S3.SS3.2.p2.6.m6.2.2.2.3.1.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.3">conditional-set</csymbol><apply id="S3.SS3.2.p2.6.m6.1.1.1.1.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1"><subset id="S3.SS3.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.1"></subset><apply id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.2.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.2">𝑈</ci><ci id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.3">′</ci></apply><ci id="S3.SS3.2.p2.6.m6.1.1.1.1.1.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.3">𝑈</ci></apply><apply id="S3.SS3.2.p2.6.m6.2.2.2.2.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2"><in id="S3.SS3.2.p2.6.m6.2.2.2.2.2.1.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.1"></in><ci id="S3.SS3.2.p2.6.m6.2.2.2.2.2.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.2">𝑤</ci><apply id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.1.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3">superscript</csymbol><ci id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.2.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.2">𝑈</ci><ci id="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.3.cmml" xref="S3.SS3.2.p2.6.m6.2.2.2.2.2.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.6.m6.2c">\mathcal{F}_{w}\stackrel{{\scriptstyle\rm def}}{{=}}\{U^{\prime}\subseteq U% \mid w\in U^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.6.m6.2d">caligraphic_F start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_U ∣ italic_w ∈ italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math> is a semi-filter that preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS3.2.p2.7.m7.1"><semantics id="S3.SS3.2.p2.7.m7.1a"><mi id="S3.SS3.2.p2.7.m7.1.1" mathvariant="normal" xref="S3.SS3.2.p2.7.m7.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.7.m7.1b"><ci id="S3.SS3.2.p2.7.m7.1.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.7.m7.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.7.m7.1d">roman_Λ</annotation></semantics></math> and that is above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.2.p2.8.m8.1"><semantics id="S3.SS3.2.p2.8.m8.1a"><mi id="S3.SS3.2.p2.8.m8.1.1" xref="S3.SS3.2.p2.8.m8.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.8.m8.1b"><ci id="S3.SS3.2.p2.8.m8.1.1.cmml" xref="S3.SS3.2.p2.8.m8.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.8.m8.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.8.m8.1d">italic_w</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.9.m9.1"><semantics id="S3.SS3.2.p2.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.2.p2.9.m9.1.1" xref="S3.SS3.2.p2.9.m9.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.9.m9.1b"><ci id="S3.SS3.2.p2.9.m9.1.1.cmml" xref="S3.SS3.2.p2.9.m9.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.9.m9.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.9.m9.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="S3.SS3.2.p2.10.m10.1"><semantics id="S3.SS3.2.p2.10.m10.1a"><mi id="S3.SS3.2.p2.10.m10.1.1" xref="S3.SS3.2.p2.10.m10.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.10.m10.1b"><ci id="S3.SS3.2.p2.10.m10.1.1.cmml" xref="S3.SS3.2.p2.10.m10.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.10.m10.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.10.m10.1d">italic_U</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS3.3.p3"> <p class="ltx_p" id="S3.SS3.3.p3.10">This claim provides a criterion to determine if an element is in <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.3.p3.1.m1.1"><semantics id="S3.SS3.3.p3.1.m1.1a"><mi id="S3.SS3.3.p3.1.m1.1.1" xref="S3.SS3.3.p3.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.1.m1.1b"><ci id="S3.SS3.3.p3.1.m1.1.1.cmml" xref="S3.SS3.3.p3.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.1.m1.1d">italic_A</annotation></semantics></math>. This will be used in a construction of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.3.p3.2.m2.1"><semantics id="S3.SS3.3.p3.2.m2.1a"><mi id="S3.SS3.3.p3.2.m2.1.1" xref="S3.SS3.3.p3.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.2.m2.1b"><ci id="S3.SS3.3.p3.2.m2.1.1.cmml" xref="S3.SS3.3.p3.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.2.m2.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.3.p3.3.m3.1"><semantics id="S3.SS3.3.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.3.m3.1.1" xref="S3.SS3.3.p3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.3.m3.1b"><ci id="S3.SS3.3.p3.3.m3.1.1.cmml" xref="S3.SS3.3.p3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.3.m3.1d">caligraphic_B</annotation></semantics></math> showing that <math alttext="D_{\cap}(A\mid\mathcal{B})=O(\rho(A,\mathcal{B})^{2})" class="ltx_Math" display="inline" id="S3.SS3.3.p3.4.m4.4"><semantics id="S3.SS3.3.p3.4.m4.4a"><mrow id="S3.SS3.3.p3.4.m4.4.4" xref="S3.SS3.3.p3.4.m4.4.4.cmml"><mrow id="S3.SS3.3.p3.4.m4.3.3.1" xref="S3.SS3.3.p3.4.m4.3.3.1.cmml"><msub id="S3.SS3.3.p3.4.m4.3.3.1.3" xref="S3.SS3.3.p3.4.m4.3.3.1.3.cmml"><mi id="S3.SS3.3.p3.4.m4.3.3.1.3.2" xref="S3.SS3.3.p3.4.m4.3.3.1.3.2.cmml">D</mi><mo id="S3.SS3.3.p3.4.m4.3.3.1.3.3" xref="S3.SS3.3.p3.4.m4.3.3.1.3.3.cmml">∩</mo></msub><mo id="S3.SS3.3.p3.4.m4.3.3.1.2" xref="S3.SS3.3.p3.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS3.3.p3.4.m4.3.3.1.1.1" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.cmml"><mo id="S3.SS3.3.p3.4.m4.3.3.1.1.1.2" stretchy="false" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.cmml"><mi id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.2" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.2.cmml">A</mi><mo id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.1" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.3" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS3.3.p3.4.m4.3.3.1.1.1.3" stretchy="false" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.3.p3.4.m4.4.4.3" xref="S3.SS3.3.p3.4.m4.4.4.3.cmml">=</mo><mrow id="S3.SS3.3.p3.4.m4.4.4.2" xref="S3.SS3.3.p3.4.m4.4.4.2.cmml"><mi id="S3.SS3.3.p3.4.m4.4.4.2.3" xref="S3.SS3.3.p3.4.m4.4.4.2.3.cmml">O</mi><mo id="S3.SS3.3.p3.4.m4.4.4.2.2" xref="S3.SS3.3.p3.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="S3.SS3.3.p3.4.m4.4.4.2.1.1" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.cmml"><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.2" stretchy="false" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.cmml">(</mo><mrow id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.cmml"><mi id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.2" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.2.cmml">ρ</mi><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.1" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.1.cmml">⁢</mo><msup id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.cmml"><mrow id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.2" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.1.cmml"><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.2.1" stretchy="false" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.1.cmml">(</mo><mi id="S3.SS3.3.p3.4.m4.1.1" xref="S3.SS3.3.p3.4.m4.1.1.cmml">A</mi><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.2.2" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.4.m4.2.2" xref="S3.SS3.3.p3.4.m4.2.2.cmml">ℬ</mi><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.2.3" stretchy="false" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.1.cmml">)</mo></mrow><mn id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.3" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S3.SS3.3.p3.4.m4.4.4.2.1.1.3" stretchy="false" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.4.m4.4b"><apply id="S3.SS3.3.p3.4.m4.4.4.cmml" xref="S3.SS3.3.p3.4.m4.4.4"><eq id="S3.SS3.3.p3.4.m4.4.4.3.cmml" xref="S3.SS3.3.p3.4.m4.4.4.3"></eq><apply id="S3.SS3.3.p3.4.m4.3.3.1.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1"><times id="S3.SS3.3.p3.4.m4.3.3.1.2.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.2"></times><apply id="S3.SS3.3.p3.4.m4.3.3.1.3.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p3.4.m4.3.3.1.3.1.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.3">subscript</csymbol><ci id="S3.SS3.3.p3.4.m4.3.3.1.3.2.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.3.2">𝐷</ci><intersect id="S3.SS3.3.p3.4.m4.3.3.1.3.3.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.3.3"></intersect></apply><apply id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1"><csymbol cd="latexml" id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.1.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.1">conditional</csymbol><ci id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.2.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.2">𝐴</ci><ci id="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.3.cmml" xref="S3.SS3.3.p3.4.m4.3.3.1.1.1.1.3">ℬ</ci></apply></apply><apply id="S3.SS3.3.p3.4.m4.4.4.2.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2"><times id="S3.SS3.3.p3.4.m4.4.4.2.2.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.2"></times><ci id="S3.SS3.3.p3.4.m4.4.4.2.3.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.3">𝑂</ci><apply id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1"><times id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.1.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.1"></times><ci id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.2.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.2">𝜌</ci><apply id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.1.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3">superscript</csymbol><interval closure="open" id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.1.cmml" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.2.2"><ci id="S3.SS3.3.p3.4.m4.1.1.cmml" xref="S3.SS3.3.p3.4.m4.1.1">𝐴</ci><ci id="S3.SS3.3.p3.4.m4.2.2.cmml" xref="S3.SS3.3.p3.4.m4.2.2">ℬ</ci></interval><cn id="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS3.3.p3.4.m4.4.4.2.1.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.4.m4.4c">D_{\cap}(A\mid\mathcal{B})=O(\rho(A,\mathcal{B})^{2})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.4.m4.4d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_O ( italic_ρ ( italic_A , caligraphic_B ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>. The intuition is that, for a given <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="S3.SS3.3.p3.5.m5.1"><semantics id="S3.SS3.3.p3.5.m5.1a"><mrow id="S3.SS3.3.p3.5.m5.1.1" xref="S3.SS3.3.p3.5.m5.1.1.cmml"><mi id="S3.SS3.3.p3.5.m5.1.1.2" xref="S3.SS3.3.p3.5.m5.1.1.2.cmml">w</mi><mo id="S3.SS3.3.p3.5.m5.1.1.1" xref="S3.SS3.3.p3.5.m5.1.1.1.cmml">∈</mo><mi id="S3.SS3.3.p3.5.m5.1.1.3" mathvariant="normal" xref="S3.SS3.3.p3.5.m5.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.5.m5.1b"><apply id="S3.SS3.3.p3.5.m5.1.1.cmml" xref="S3.SS3.3.p3.5.m5.1.1"><in id="S3.SS3.3.p3.5.m5.1.1.1.cmml" xref="S3.SS3.3.p3.5.m5.1.1.1"></in><ci id="S3.SS3.3.p3.5.m5.1.1.2.cmml" xref="S3.SS3.3.p3.5.m5.1.1.2">𝑤</ci><ci id="S3.SS3.3.p3.5.m5.1.1.3.cmml" xref="S3.SS3.3.p3.5.m5.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.5.m5.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.5.m5.1d">italic_w ∈ roman_Γ</annotation></semantics></math>, we must check if there is <math alttext="\mathcal{F}\in\mathfrak{F}_{\Lambda}" class="ltx_Math" display="inline" id="S3.SS3.3.p3.6.m6.1"><semantics id="S3.SS3.3.p3.6.m6.1a"><mrow id="S3.SS3.3.p3.6.m6.1.1" xref="S3.SS3.3.p3.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.6.m6.1.1.2" xref="S3.SS3.3.p3.6.m6.1.1.2.cmml">ℱ</mi><mo id="S3.SS3.3.p3.6.m6.1.1.1" xref="S3.SS3.3.p3.6.m6.1.1.1.cmml">∈</mo><msub id="S3.SS3.3.p3.6.m6.1.1.3" xref="S3.SS3.3.p3.6.m6.1.1.3.cmml"><mi id="S3.SS3.3.p3.6.m6.1.1.3.2" xref="S3.SS3.3.p3.6.m6.1.1.3.2.cmml">𝔉</mi><mi id="S3.SS3.3.p3.6.m6.1.1.3.3" mathvariant="normal" xref="S3.SS3.3.p3.6.m6.1.1.3.3.cmml">Λ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.6.m6.1b"><apply id="S3.SS3.3.p3.6.m6.1.1.cmml" xref="S3.SS3.3.p3.6.m6.1.1"><in id="S3.SS3.3.p3.6.m6.1.1.1.cmml" xref="S3.SS3.3.p3.6.m6.1.1.1"></in><ci id="S3.SS3.3.p3.6.m6.1.1.2.cmml" xref="S3.SS3.3.p3.6.m6.1.1.2">ℱ</ci><apply id="S3.SS3.3.p3.6.m6.1.1.3.cmml" xref="S3.SS3.3.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p3.6.m6.1.1.3.1.cmml" xref="S3.SS3.3.p3.6.m6.1.1.3">subscript</csymbol><ci id="S3.SS3.3.p3.6.m6.1.1.3.2.cmml" xref="S3.SS3.3.p3.6.m6.1.1.3.2">𝔉</ci><ci id="S3.SS3.3.p3.6.m6.1.1.3.3.cmml" xref="S3.SS3.3.p3.6.m6.1.1.3.3">Λ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.6.m6.1c">\mathcal{F}\in\mathfrak{F}_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.6.m6.1d">caligraphic_F ∈ fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> that is above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.3.p3.7.m7.1"><semantics id="S3.SS3.3.p3.7.m7.1a"><mi id="S3.SS3.3.p3.7.m7.1.1" xref="S3.SS3.3.p3.7.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.7.m7.1b"><ci id="S3.SS3.3.p3.7.m7.1.1.cmml" xref="S3.SS3.3.p3.7.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.7.m7.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.7.m7.1d">italic_w</annotation></semantics></math> with respect to <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.3.p3.8.m8.1"><semantics id="S3.SS3.3.p3.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.8.m8.1.1" xref="S3.SS3.3.p3.8.m8.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.8.m8.1b"><ci id="S3.SS3.3.p3.8.m8.1.1.cmml" xref="S3.SS3.3.p3.8.m8.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.8.m8.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.8.m8.1d">caligraphic_B</annotation></semantics></math> and <math alttext="U" class="ltx_Math" display="inline" id="S3.SS3.3.p3.9.m9.1"><semantics id="S3.SS3.3.p3.9.m9.1a"><mi id="S3.SS3.3.p3.9.m9.1.1" xref="S3.SS3.3.p3.9.m9.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.9.m9.1b"><ci id="S3.SS3.3.p3.9.m9.1.1.cmml" xref="S3.SS3.3.p3.9.m9.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.9.m9.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.9.m9.1d">italic_U</annotation></semantics></math>. In order to achieve this, we inspect the <em class="ltx_emph ltx_font_italic" id="S3.SS3.3.p3.10.1">minimal family</em> <math alttext="\mathcal{G}_{w}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="S3.SS3.3.p3.10.m10.1"><semantics id="S3.SS3.3.p3.10.m10.1a"><mrow id="S3.SS3.3.p3.10.m10.1.2" xref="S3.SS3.3.p3.10.m10.1.2.cmml"><msub id="S3.SS3.3.p3.10.m10.1.2.2" xref="S3.SS3.3.p3.10.m10.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.10.m10.1.2.2.2" xref="S3.SS3.3.p3.10.m10.1.2.2.2.cmml">𝒢</mi><mi id="S3.SS3.3.p3.10.m10.1.2.2.3" xref="S3.SS3.3.p3.10.m10.1.2.2.3.cmml">w</mi></msub><mo id="S3.SS3.3.p3.10.m10.1.2.1" xref="S3.SS3.3.p3.10.m10.1.2.1.cmml">⊆</mo><mrow id="S3.SS3.3.p3.10.m10.1.2.3" xref="S3.SS3.3.p3.10.m10.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p3.10.m10.1.2.3.2" xref="S3.SS3.3.p3.10.m10.1.2.3.2.cmml">𝒫</mi><mo id="S3.SS3.3.p3.10.m10.1.2.3.1" xref="S3.SS3.3.p3.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS3.3.p3.10.m10.1.2.3.3.2" xref="S3.SS3.3.p3.10.m10.1.2.3.cmml"><mo id="S3.SS3.3.p3.10.m10.1.2.3.3.2.1" stretchy="false" xref="S3.SS3.3.p3.10.m10.1.2.3.cmml">(</mo><mi id="S3.SS3.3.p3.10.m10.1.1" xref="S3.SS3.3.p3.10.m10.1.1.cmml">U</mi><mo id="S3.SS3.3.p3.10.m10.1.2.3.3.2.2" stretchy="false" xref="S3.SS3.3.p3.10.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p3.10.m10.1b"><apply id="S3.SS3.3.p3.10.m10.1.2.cmml" xref="S3.SS3.3.p3.10.m10.1.2"><subset id="S3.SS3.3.p3.10.m10.1.2.1.cmml" xref="S3.SS3.3.p3.10.m10.1.2.1"></subset><apply id="S3.SS3.3.p3.10.m10.1.2.2.cmml" xref="S3.SS3.3.p3.10.m10.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.3.p3.10.m10.1.2.2.1.cmml" xref="S3.SS3.3.p3.10.m10.1.2.2">subscript</csymbol><ci id="S3.SS3.3.p3.10.m10.1.2.2.2.cmml" xref="S3.SS3.3.p3.10.m10.1.2.2.2">𝒢</ci><ci id="S3.SS3.3.p3.10.m10.1.2.2.3.cmml" xref="S3.SS3.3.p3.10.m10.1.2.2.3">𝑤</ci></apply><apply id="S3.SS3.3.p3.10.m10.1.2.3.cmml" xref="S3.SS3.3.p3.10.m10.1.2.3"><times id="S3.SS3.3.p3.10.m10.1.2.3.1.cmml" xref="S3.SS3.3.p3.10.m10.1.2.3.1"></times><ci id="S3.SS3.3.p3.10.m10.1.2.3.2.cmml" xref="S3.SS3.3.p3.10.m10.1.2.3.2">𝒫</ci><ci id="S3.SS3.3.p3.10.m10.1.1.cmml" xref="S3.SS3.3.p3.10.m10.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p3.10.m10.1c">\mathcal{G}_{w}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p3.10.m10.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> of sets that must be contained in any such (candidate) semi-filter.</p> </div> <div class="ltx_para" id="S3.SS3.4.p4"> <p class="ltx_p" id="S3.SS3.4.p4.5">For every <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="S3.SS3.4.p4.1.m1.1"><semantics id="S3.SS3.4.p4.1.m1.1a"><mrow id="S3.SS3.4.p4.1.m1.1.1" xref="S3.SS3.4.p4.1.m1.1.1.cmml"><mi id="S3.SS3.4.p4.1.m1.1.1.2" xref="S3.SS3.4.p4.1.m1.1.1.2.cmml">w</mi><mo id="S3.SS3.4.p4.1.m1.1.1.1" xref="S3.SS3.4.p4.1.m1.1.1.1.cmml">∈</mo><mi id="S3.SS3.4.p4.1.m1.1.1.3" mathvariant="normal" xref="S3.SS3.4.p4.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p4.1.m1.1b"><apply id="S3.SS3.4.p4.1.m1.1.1.cmml" xref="S3.SS3.4.p4.1.m1.1.1"><in id="S3.SS3.4.p4.1.m1.1.1.1.cmml" xref="S3.SS3.4.p4.1.m1.1.1.1"></in><ci id="S3.SS3.4.p4.1.m1.1.1.2.cmml" xref="S3.SS3.4.p4.1.m1.1.1.2">𝑤</ci><ci id="S3.SS3.4.p4.1.m1.1.1.3.cmml" xref="S3.SS3.4.p4.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p4.1.m1.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p4.1.m1.1d">italic_w ∈ roman_Γ</annotation></semantics></math>, we require <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.4.p4.2.m2.1"><semantics id="S3.SS3.4.p4.2.m2.1a"><msub id="S3.SS3.4.p4.2.m2.1.1" xref="S3.SS3.4.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.4.p4.2.m2.1.1.2" xref="S3.SS3.4.p4.2.m2.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.4.p4.2.m2.1.1.3" xref="S3.SS3.4.p4.2.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p4.2.m2.1b"><apply id="S3.SS3.4.p4.2.m2.1.1.cmml" xref="S3.SS3.4.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p4.2.m2.1.1.1.cmml" xref="S3.SS3.4.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.4.p4.2.m2.1.1.2.cmml" xref="S3.SS3.4.p4.2.m2.1.1.2">𝒢</ci><ci id="S3.SS3.4.p4.2.m2.1.1.3.cmml" xref="S3.SS3.4.p4.2.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p4.2.m2.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p4.2.m2.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> to be above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.4.p4.3.m3.1"><semantics id="S3.SS3.4.p4.3.m3.1a"><mi id="S3.SS3.4.p4.3.m3.1.1" xref="S3.SS3.4.p4.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p4.3.m3.1b"><ci id="S3.SS3.4.p4.3.m3.1.1.cmml" xref="S3.SS3.4.p4.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p4.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p4.3.m3.1d">italic_w</annotation></semantics></math>, upward-closed, and to preserve <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS3.4.p4.4.m4.1"><semantics id="S3.SS3.4.p4.4.m4.1a"><mi id="S3.SS3.4.p4.4.m4.1.1" mathvariant="normal" xref="S3.SS3.4.p4.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p4.4.m4.1b"><ci id="S3.SS3.4.p4.4.m4.1.1.cmml" xref="S3.SS3.4.p4.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p4.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p4.4.m4.1d">roman_Λ</annotation></semantics></math>. The rules for constructing <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.4.p4.5.m5.1"><semantics id="S3.SS3.4.p4.5.m5.1a"><msub id="S3.SS3.4.p4.5.m5.1.1" xref="S3.SS3.4.p4.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.4.p4.5.m5.1.1.2" xref="S3.SS3.4.p4.5.m5.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.4.p4.5.m5.1.1.3" xref="S3.SS3.4.p4.5.m5.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p4.5.m5.1b"><apply id="S3.SS3.4.p4.5.m5.1.1.cmml" xref="S3.SS3.4.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p4.5.m5.1.1.1.cmml" xref="S3.SS3.4.p4.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.4.p4.5.m5.1.1.2.cmml" xref="S3.SS3.4.p4.5.m5.1.1.2">𝒢</ci><ci id="S3.SS3.4.p4.5.m5.1.1.3.cmml" xref="S3.SS3.4.p4.5.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p4.5.m5.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p4.5.m5.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> are simple:</p> </div> <div class="ltx_para" id="S3.SS3.5.p5"> <ul class="ltx_itemize" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.6"><em class="ltx_emph ltx_font_italic" id="S3.I2.i1.p1.6.1">Base case</em>. If <math alttext="w\in B" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mrow id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.2" xref="S3.I2.i1.p1.1.m1.1.1.2.cmml">w</mi><mo id="S3.I2.i1.p1.1.m1.1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S3.I2.i1.p1.1.m1.1.1.3" xref="S3.I2.i1.p1.1.m1.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><apply id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1"><in id="S3.I2.i1.p1.1.m1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1"></in><ci id="S3.I2.i1.p1.1.m1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2">𝑤</ci><ci id="S3.I2.i1.p1.1.m1.1.1.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">w\in B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">italic_w ∈ italic_B</annotation></semantics></math> for <math alttext="B\in\mathcal{B}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mrow id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.2.cmml">B</mi><mo id="S3.I2.i1.p1.2.m2.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.2.m2.1.1.3" xref="S3.I2.i1.p1.2.m2.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1"><in id="S3.I2.i1.p1.2.m2.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1"></in><ci id="S3.I2.i1.p1.2.m2.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2">𝐵</ci><ci id="S3.I2.i1.p1.2.m2.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">B\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">italic_B ∈ caligraphic_B</annotation></semantics></math>, then add <math alttext="B_{U}=B\cap U" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><mrow id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><msub id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.2.2" xref="S3.I2.i1.p1.3.m3.1.1.2.2.cmml">B</mi><mi id="S3.I2.i1.p1.3.m3.1.1.2.3" xref="S3.I2.i1.p1.3.m3.1.1.2.3.cmml">U</mi></msub><mo id="S3.I2.i1.p1.3.m3.1.1.1" xref="S3.I2.i1.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.3.2" xref="S3.I2.i1.p1.3.m3.1.1.3.2.cmml">B</mi><mo id="S3.I2.i1.p1.3.m3.1.1.3.1" xref="S3.I2.i1.p1.3.m3.1.1.3.1.cmml">∩</mo><mi id="S3.I2.i1.p1.3.m3.1.1.3.3" xref="S3.I2.i1.p1.3.m3.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><eq id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.1"></eq><apply id="S3.I2.i1.p1.3.m3.1.1.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I2.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.2">𝐵</ci><ci id="S3.I2.i1.p1.3.m3.1.1.2.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.3">𝑈</ci></apply><apply id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3"><intersect id="S3.I2.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3.1"></intersect><ci id="S3.I2.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3.2">𝐵</ci><ci id="S3.I2.i1.p1.3.m3.1.1.3.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">B_{U}=B\cap U</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_B ∩ italic_U</annotation></semantics></math> to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.1"><semantics id="S3.I2.i1.p1.4.m4.1a"><msub id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.4.m4.1.1.2" xref="S3.I2.i1.p1.4.m4.1.1.2.cmml">𝒢</mi><mi id="S3.I2.i1.p1.4.m4.1.1.3" xref="S3.I2.i1.p1.4.m4.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.1b"><apply id="S3.I2.i1.p1.4.m4.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.4.m4.1.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1">subscript</csymbol><ci id="S3.I2.i1.p1.4.m4.1.1.2.cmml" xref="S3.I2.i1.p1.4.m4.1.1.2">𝒢</ci><ci id="S3.I2.i1.p1.4.m4.1.1.3.cmml" xref="S3.I2.i1.p1.4.m4.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, together with every set <math alttext="U^{\prime}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.5.m5.1"><semantics id="S3.I2.i1.p1.5.m5.1a"><msup id="S3.I2.i1.p1.5.m5.1.1" xref="S3.I2.i1.p1.5.m5.1.1.cmml"><mi id="S3.I2.i1.p1.5.m5.1.1.2" xref="S3.I2.i1.p1.5.m5.1.1.2.cmml">U</mi><mo id="S3.I2.i1.p1.5.m5.1.1.3" xref="S3.I2.i1.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.5.m5.1b"><apply id="S3.I2.i1.p1.5.m5.1.1.cmml" xref="S3.I2.i1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.5.m5.1.1.1.cmml" xref="S3.I2.i1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.5.m5.1.1.2.cmml" xref="S3.I2.i1.p1.5.m5.1.1.2">𝑈</ci><ci id="S3.I2.i1.p1.5.m5.1.1.3.cmml" xref="S3.I2.i1.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.5.m5.1c">U^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.5.m5.1d">italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="B_{U}\subseteq U^{\prime}\subseteq U" class="ltx_Math" display="inline" id="S3.I2.i1.p1.6.m6.1"><semantics id="S3.I2.i1.p1.6.m6.1a"><mrow id="S3.I2.i1.p1.6.m6.1.1" xref="S3.I2.i1.p1.6.m6.1.1.cmml"><msub id="S3.I2.i1.p1.6.m6.1.1.2" xref="S3.I2.i1.p1.6.m6.1.1.2.cmml"><mi id="S3.I2.i1.p1.6.m6.1.1.2.2" xref="S3.I2.i1.p1.6.m6.1.1.2.2.cmml">B</mi><mi id="S3.I2.i1.p1.6.m6.1.1.2.3" xref="S3.I2.i1.p1.6.m6.1.1.2.3.cmml">U</mi></msub><mo id="S3.I2.i1.p1.6.m6.1.1.3" xref="S3.I2.i1.p1.6.m6.1.1.3.cmml">⊆</mo><msup id="S3.I2.i1.p1.6.m6.1.1.4" xref="S3.I2.i1.p1.6.m6.1.1.4.cmml"><mi id="S3.I2.i1.p1.6.m6.1.1.4.2" xref="S3.I2.i1.p1.6.m6.1.1.4.2.cmml">U</mi><mo id="S3.I2.i1.p1.6.m6.1.1.4.3" xref="S3.I2.i1.p1.6.m6.1.1.4.3.cmml">′</mo></msup><mo id="S3.I2.i1.p1.6.m6.1.1.5" xref="S3.I2.i1.p1.6.m6.1.1.5.cmml">⊆</mo><mi id="S3.I2.i1.p1.6.m6.1.1.6" xref="S3.I2.i1.p1.6.m6.1.1.6.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.6.m6.1b"><apply id="S3.I2.i1.p1.6.m6.1.1.cmml" xref="S3.I2.i1.p1.6.m6.1.1"><and id="S3.I2.i1.p1.6.m6.1.1a.cmml" xref="S3.I2.i1.p1.6.m6.1.1"></and><apply id="S3.I2.i1.p1.6.m6.1.1b.cmml" xref="S3.I2.i1.p1.6.m6.1.1"><subset id="S3.I2.i1.p1.6.m6.1.1.3.cmml" xref="S3.I2.i1.p1.6.m6.1.1.3"></subset><apply id="S3.I2.i1.p1.6.m6.1.1.2.cmml" xref="S3.I2.i1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.6.m6.1.1.2.1.cmml" xref="S3.I2.i1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S3.I2.i1.p1.6.m6.1.1.2.2.cmml" xref="S3.I2.i1.p1.6.m6.1.1.2.2">𝐵</ci><ci id="S3.I2.i1.p1.6.m6.1.1.2.3.cmml" xref="S3.I2.i1.p1.6.m6.1.1.2.3">𝑈</ci></apply><apply id="S3.I2.i1.p1.6.m6.1.1.4.cmml" xref="S3.I2.i1.p1.6.m6.1.1.4"><csymbol cd="ambiguous" id="S3.I2.i1.p1.6.m6.1.1.4.1.cmml" xref="S3.I2.i1.p1.6.m6.1.1.4">superscript</csymbol><ci id="S3.I2.i1.p1.6.m6.1.1.4.2.cmml" xref="S3.I2.i1.p1.6.m6.1.1.4.2">𝑈</ci><ci id="S3.I2.i1.p1.6.m6.1.1.4.3.cmml" xref="S3.I2.i1.p1.6.m6.1.1.4.3">′</ci></apply></apply><apply id="S3.I2.i1.p1.6.m6.1.1c.cmml" xref="S3.I2.i1.p1.6.m6.1.1"><subset id="S3.I2.i1.p1.6.m6.1.1.5.cmml" xref="S3.I2.i1.p1.6.m6.1.1.5"></subset><share href="https://arxiv.org/html/2503.14117v1#S3.I2.i1.p1.6.m6.1.1.4.cmml" id="S3.I2.i1.p1.6.m6.1.1d.cmml" xref="S3.I2.i1.p1.6.m6.1.1"></share><ci id="S3.I2.i1.p1.6.m6.1.1.6.cmml" xref="S3.I2.i1.p1.6.m6.1.1.6">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.6.m6.1c">B_{U}\subseteq U^{\prime}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.6.m6.1d">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ⊆ italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_U</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.7"><em class="ltx_emph ltx_font_italic" id="S3.I2.i2.p1.7.1">Propagation step</em>. If both <math alttext="E_{i}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><msub id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.2.cmml">E</mi><mi id="S3.I2.i2.p1.1.m1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2">𝐸</ci><ci id="S3.I2.i2.p1.1.m1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">E_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{i}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><msub id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">H</mi><mi id="S3.I2.i2.p1.2.m2.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2">𝐻</ci><ci id="S3.I2.i2.p1.2.m2.1.1.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">H_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are in <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><msub id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml">𝒢</mi><mi id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">𝒢</ci><ci id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, add <math alttext="E_{i}\cap H_{i}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.1"><semantics id="S3.I2.i2.p1.4.m4.1a"><mrow id="S3.I2.i2.p1.4.m4.1.1" xref="S3.I2.i2.p1.4.m4.1.1.cmml"><msub id="S3.I2.i2.p1.4.m4.1.1.2" xref="S3.I2.i2.p1.4.m4.1.1.2.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.2.2" xref="S3.I2.i2.p1.4.m4.1.1.2.2.cmml">E</mi><mi id="S3.I2.i2.p1.4.m4.1.1.2.3" xref="S3.I2.i2.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.I2.i2.p1.4.m4.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.cmml">∩</mo><msub id="S3.I2.i2.p1.4.m4.1.1.3" xref="S3.I2.i2.p1.4.m4.1.1.3.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.3.2" xref="S3.I2.i2.p1.4.m4.1.1.3.2.cmml">H</mi><mi id="S3.I2.i2.p1.4.m4.1.1.3.3" xref="S3.I2.i2.p1.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.1b"><apply id="S3.I2.i2.p1.4.m4.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1"><intersect id="S3.I2.i2.p1.4.m4.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1"></intersect><apply id="S3.I2.i2.p1.4.m4.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.1.1.2.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.I2.i2.p1.4.m4.1.1.2.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2.2">𝐸</ci><ci id="S3.I2.i2.p1.4.m4.1.1.2.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.I2.i2.p1.4.m4.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.1.1.3.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.I2.i2.p1.4.m4.1.1.3.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3.2">𝐻</ci><ci id="S3.I2.i2.p1.4.m4.1.1.3.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.1c">E_{i}\cap H_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.5.m5.1"><semantics id="S3.I2.i2.p1.5.m5.1a"><msub id="S3.I2.i2.p1.5.m5.1.1" xref="S3.I2.i2.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i2.p1.5.m5.1.1.2" xref="S3.I2.i2.p1.5.m5.1.1.2.cmml">𝒢</mi><mi id="S3.I2.i2.p1.5.m5.1.1.3" xref="S3.I2.i2.p1.5.m5.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.5.m5.1b"><apply id="S3.I2.i2.p1.5.m5.1.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.5.m5.1.1.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.5.m5.1.1.2.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2">𝒢</ci><ci id="S3.I2.i2.p1.5.m5.1.1.3.cmml" xref="S3.I2.i2.p1.5.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.5.m5.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.5.m5.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, together with every set <math alttext="U^{\prime}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.6.m6.1"><semantics id="S3.I2.i2.p1.6.m6.1a"><msup id="S3.I2.i2.p1.6.m6.1.1" xref="S3.I2.i2.p1.6.m6.1.1.cmml"><mi id="S3.I2.i2.p1.6.m6.1.1.2" xref="S3.I2.i2.p1.6.m6.1.1.2.cmml">U</mi><mo id="S3.I2.i2.p1.6.m6.1.1.3" xref="S3.I2.i2.p1.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.6.m6.1b"><apply id="S3.I2.i2.p1.6.m6.1.1.cmml" xref="S3.I2.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.6.m6.1.1.1.cmml" xref="S3.I2.i2.p1.6.m6.1.1">superscript</csymbol><ci id="S3.I2.i2.p1.6.m6.1.1.2.cmml" xref="S3.I2.i2.p1.6.m6.1.1.2">𝑈</ci><ci id="S3.I2.i2.p1.6.m6.1.1.3.cmml" xref="S3.I2.i2.p1.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.6.m6.1c">U^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.6.m6.1d">italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="E_{i}\cap H_{i}\subseteq U^{\prime}\subseteq U" class="ltx_Math" display="inline" id="S3.I2.i2.p1.7.m7.1"><semantics id="S3.I2.i2.p1.7.m7.1a"><mrow id="S3.I2.i2.p1.7.m7.1.1" xref="S3.I2.i2.p1.7.m7.1.1.cmml"><mrow id="S3.I2.i2.p1.7.m7.1.1.2" xref="S3.I2.i2.p1.7.m7.1.1.2.cmml"><msub id="S3.I2.i2.p1.7.m7.1.1.2.2" xref="S3.I2.i2.p1.7.m7.1.1.2.2.cmml"><mi id="S3.I2.i2.p1.7.m7.1.1.2.2.2" xref="S3.I2.i2.p1.7.m7.1.1.2.2.2.cmml">E</mi><mi id="S3.I2.i2.p1.7.m7.1.1.2.2.3" xref="S3.I2.i2.p1.7.m7.1.1.2.2.3.cmml">i</mi></msub><mo id="S3.I2.i2.p1.7.m7.1.1.2.1" xref="S3.I2.i2.p1.7.m7.1.1.2.1.cmml">∩</mo><msub id="S3.I2.i2.p1.7.m7.1.1.2.3" xref="S3.I2.i2.p1.7.m7.1.1.2.3.cmml"><mi id="S3.I2.i2.p1.7.m7.1.1.2.3.2" xref="S3.I2.i2.p1.7.m7.1.1.2.3.2.cmml">H</mi><mi id="S3.I2.i2.p1.7.m7.1.1.2.3.3" xref="S3.I2.i2.p1.7.m7.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S3.I2.i2.p1.7.m7.1.1.3" xref="S3.I2.i2.p1.7.m7.1.1.3.cmml">⊆</mo><msup id="S3.I2.i2.p1.7.m7.1.1.4" xref="S3.I2.i2.p1.7.m7.1.1.4.cmml"><mi id="S3.I2.i2.p1.7.m7.1.1.4.2" xref="S3.I2.i2.p1.7.m7.1.1.4.2.cmml">U</mi><mo id="S3.I2.i2.p1.7.m7.1.1.4.3" xref="S3.I2.i2.p1.7.m7.1.1.4.3.cmml">′</mo></msup><mo id="S3.I2.i2.p1.7.m7.1.1.5" xref="S3.I2.i2.p1.7.m7.1.1.5.cmml">⊆</mo><mi id="S3.I2.i2.p1.7.m7.1.1.6" xref="S3.I2.i2.p1.7.m7.1.1.6.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.7.m7.1b"><apply id="S3.I2.i2.p1.7.m7.1.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1"><and id="S3.I2.i2.p1.7.m7.1.1a.cmml" xref="S3.I2.i2.p1.7.m7.1.1"></and><apply id="S3.I2.i2.p1.7.m7.1.1b.cmml" xref="S3.I2.i2.p1.7.m7.1.1"><subset id="S3.I2.i2.p1.7.m7.1.1.3.cmml" xref="S3.I2.i2.p1.7.m7.1.1.3"></subset><apply id="S3.I2.i2.p1.7.m7.1.1.2.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2"><intersect id="S3.I2.i2.p1.7.m7.1.1.2.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.1"></intersect><apply id="S3.I2.i2.p1.7.m7.1.1.2.2.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.7.m7.1.1.2.2.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="S3.I2.i2.p1.7.m7.1.1.2.2.2.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.2.2">𝐸</ci><ci id="S3.I2.i2.p1.7.m7.1.1.2.2.3.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.2.3">𝑖</ci></apply><apply id="S3.I2.i2.p1.7.m7.1.1.2.3.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.7.m7.1.1.2.3.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.3">subscript</csymbol><ci id="S3.I2.i2.p1.7.m7.1.1.2.3.2.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.3.2">𝐻</ci><ci id="S3.I2.i2.p1.7.m7.1.1.2.3.3.cmml" xref="S3.I2.i2.p1.7.m7.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S3.I2.i2.p1.7.m7.1.1.4.cmml" xref="S3.I2.i2.p1.7.m7.1.1.4"><csymbol cd="ambiguous" id="S3.I2.i2.p1.7.m7.1.1.4.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1.4">superscript</csymbol><ci id="S3.I2.i2.p1.7.m7.1.1.4.2.cmml" xref="S3.I2.i2.p1.7.m7.1.1.4.2">𝑈</ci><ci id="S3.I2.i2.p1.7.m7.1.1.4.3.cmml" xref="S3.I2.i2.p1.7.m7.1.1.4.3">′</ci></apply></apply><apply id="S3.I2.i2.p1.7.m7.1.1c.cmml" xref="S3.I2.i2.p1.7.m7.1.1"><subset id="S3.I2.i2.p1.7.m7.1.1.5.cmml" xref="S3.I2.i2.p1.7.m7.1.1.5"></subset><share href="https://arxiv.org/html/2503.14117v1#S3.I2.i2.p1.7.m7.1.1.4.cmml" id="S3.I2.i2.p1.7.m7.1.1d.cmml" xref="S3.I2.i2.p1.7.m7.1.1"></share><ci id="S3.I2.i2.p1.7.m7.1.1.6.cmml" xref="S3.I2.i2.p1.7.m7.1.1.6">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.7.m7.1c">E_{i}\cap H_{i}\subseteq U^{\prime}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_U</annotation></semantics></math>.</p> </div> </li> </ul> </div> <div class="ltx_para ltx_noindent" id="S3.SS3.6.p6"> <p class="ltx_p" id="S3.SS3.6.p6.1">We apply the base case once, and repeatedly invoke the propagation step until no new sets are added to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.6.p6.1.m1.1"><semantics id="S3.SS3.6.p6.1.m1.1a"><msub id="S3.SS3.6.p6.1.m1.1.1" xref="S3.SS3.6.p6.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.6.p6.1.m1.1.1.2" xref="S3.SS3.6.p6.1.m1.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.6.p6.1.m1.1.1.3" xref="S3.SS3.6.p6.1.m1.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p6.1.m1.1b"><apply id="S3.SS3.6.p6.1.m1.1.1.cmml" xref="S3.SS3.6.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.6.p6.1.m1.1.1.1.cmml" xref="S3.SS3.6.p6.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.6.p6.1.m1.1.1.2.cmml" xref="S3.SS3.6.p6.1.m1.1.1.2">𝒢</ci><ci id="S3.SS3.6.p6.1.m1.1.1.3.cmml" xref="S3.SS3.6.p6.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p6.1.m1.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p6.1.m1.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>. Clearly, this process terminates within a finite number of steps.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem27"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem27.1.1.1">Claim 27</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem27.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem27.p1"> <p class="ltx_p" id="Thmtheorem27.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem27.p1.3.3">For every <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="Thmtheorem27.p1.1.1.m1.1"><semantics id="Thmtheorem27.p1.1.1.m1.1a"><mrow id="Thmtheorem27.p1.1.1.m1.1.1" xref="Thmtheorem27.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem27.p1.1.1.m1.1.1.2" xref="Thmtheorem27.p1.1.1.m1.1.1.2.cmml">w</mi><mo id="Thmtheorem27.p1.1.1.m1.1.1.1" xref="Thmtheorem27.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem27.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem27.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem27.p1.1.1.m1.1b"><apply id="Thmtheorem27.p1.1.1.m1.1.1.cmml" xref="Thmtheorem27.p1.1.1.m1.1.1"><in id="Thmtheorem27.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem27.p1.1.1.m1.1.1.1"></in><ci id="Thmtheorem27.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem27.p1.1.1.m1.1.1.2">𝑤</ci><ci id="Thmtheorem27.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem27.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem27.p1.1.1.m1.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem27.p1.1.1.m1.1d">italic_w ∈ roman_Γ</annotation></semantics></math>, the empty set is added to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="Thmtheorem27.p1.2.2.m2.1"><semantics id="Thmtheorem27.p1.2.2.m2.1a"><msub id="Thmtheorem27.p1.2.2.m2.1.1" xref="Thmtheorem27.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem27.p1.2.2.m2.1.1.2" xref="Thmtheorem27.p1.2.2.m2.1.1.2.cmml">𝒢</mi><mi id="Thmtheorem27.p1.2.2.m2.1.1.3" xref="Thmtheorem27.p1.2.2.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem27.p1.2.2.m2.1b"><apply id="Thmtheorem27.p1.2.2.m2.1.1.cmml" xref="Thmtheorem27.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmtheorem27.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem27.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmtheorem27.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem27.p1.2.2.m2.1.1.2">𝒢</ci><ci id="Thmtheorem27.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem27.p1.2.2.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem27.p1.2.2.m2.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem27.p1.2.2.m2.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="w\in A" class="ltx_Math" display="inline" id="Thmtheorem27.p1.3.3.m3.1"><semantics id="Thmtheorem27.p1.3.3.m3.1a"><mrow id="Thmtheorem27.p1.3.3.m3.1.1" xref="Thmtheorem27.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem27.p1.3.3.m3.1.1.2" xref="Thmtheorem27.p1.3.3.m3.1.1.2.cmml">w</mi><mo id="Thmtheorem27.p1.3.3.m3.1.1.1" xref="Thmtheorem27.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="Thmtheorem27.p1.3.3.m3.1.1.3" xref="Thmtheorem27.p1.3.3.m3.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem27.p1.3.3.m3.1b"><apply id="Thmtheorem27.p1.3.3.m3.1.1.cmml" xref="Thmtheorem27.p1.3.3.m3.1.1"><in id="Thmtheorem27.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem27.p1.3.3.m3.1.1.1"></in><ci id="Thmtheorem27.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem27.p1.3.3.m3.1.1.2">𝑤</ci><ci id="Thmtheorem27.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem27.p1.3.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem27.p1.3.3.m3.1c">w\in A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem27.p1.3.3.m3.1d">italic_w ∈ italic_A</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.7.p7"> <p class="ltx_p" id="S3.SS3.7.p7.18">We argue that <math alttext="w\notin A" class="ltx_Math" display="inline" id="S3.SS3.7.p7.1.m1.1"><semantics id="S3.SS3.7.p7.1.m1.1a"><mrow id="S3.SS3.7.p7.1.m1.1.1" xref="S3.SS3.7.p7.1.m1.1.1.cmml"><mi id="S3.SS3.7.p7.1.m1.1.1.2" xref="S3.SS3.7.p7.1.m1.1.1.2.cmml">w</mi><mo id="S3.SS3.7.p7.1.m1.1.1.1" xref="S3.SS3.7.p7.1.m1.1.1.1.cmml">∉</mo><mi id="S3.SS3.7.p7.1.m1.1.1.3" xref="S3.SS3.7.p7.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.1.m1.1b"><apply id="S3.SS3.7.p7.1.m1.1.1.cmml" xref="S3.SS3.7.p7.1.m1.1.1"><notin id="S3.SS3.7.p7.1.m1.1.1.1.cmml" xref="S3.SS3.7.p7.1.m1.1.1.1"></notin><ci id="S3.SS3.7.p7.1.m1.1.1.2.cmml" xref="S3.SS3.7.p7.1.m1.1.1.2">𝑤</ci><ci id="S3.SS3.7.p7.1.m1.1.1.3.cmml" xref="S3.SS3.7.p7.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.1.m1.1c">w\notin A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.1.m1.1d">italic_w ∉ italic_A</annotation></semantics></math> if and only if <math alttext="\emptyset\notin\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.2.m2.1"><semantics id="S3.SS3.7.p7.2.m2.1a"><mrow id="S3.SS3.7.p7.2.m2.1.1" xref="S3.SS3.7.p7.2.m2.1.1.cmml"><mi id="S3.SS3.7.p7.2.m2.1.1.2" mathvariant="normal" xref="S3.SS3.7.p7.2.m2.1.1.2.cmml">∅</mi><mo id="S3.SS3.7.p7.2.m2.1.1.1" xref="S3.SS3.7.p7.2.m2.1.1.1.cmml">∉</mo><msub id="S3.SS3.7.p7.2.m2.1.1.3" xref="S3.SS3.7.p7.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.2.m2.1.1.3.2" xref="S3.SS3.7.p7.2.m2.1.1.3.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.2.m2.1.1.3.3" xref="S3.SS3.7.p7.2.m2.1.1.3.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.2.m2.1b"><apply id="S3.SS3.7.p7.2.m2.1.1.cmml" xref="S3.SS3.7.p7.2.m2.1.1"><notin id="S3.SS3.7.p7.2.m2.1.1.1.cmml" xref="S3.SS3.7.p7.2.m2.1.1.1"></notin><emptyset id="S3.SS3.7.p7.2.m2.1.1.2.cmml" xref="S3.SS3.7.p7.2.m2.1.1.2"></emptyset><apply id="S3.SS3.7.p7.2.m2.1.1.3.cmml" xref="S3.SS3.7.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.7.p7.2.m2.1.1.3.1.cmml" xref="S3.SS3.7.p7.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.7.p7.2.m2.1.1.3.2.cmml" xref="S3.SS3.7.p7.2.m2.1.1.3.2">𝒢</ci><ci id="S3.SS3.7.p7.2.m2.1.1.3.3.cmml" xref="S3.SS3.7.p7.2.m2.1.1.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.2.m2.1c">\emptyset\notin\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.2.m2.1d">∅ ∉ caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>. Clearly, if <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.3.m3.1"><semantics id="S3.SS3.7.p7.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.3.m3.1.1" xref="S3.SS3.7.p7.3.m3.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.3.m3.1b"><ci id="S3.SS3.7.p7.3.m3.1.1.cmml" xref="S3.SS3.7.p7.3.m3.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.3.m3.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.3.m3.1d">caligraphic_F</annotation></semantics></math> is a semi-filter that is above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.7.p7.4.m4.1"><semantics id="S3.SS3.7.p7.4.m4.1a"><mi id="S3.SS3.7.p7.4.m4.1.1" xref="S3.SS3.7.p7.4.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.4.m4.1b"><ci id="S3.SS3.7.p7.4.m4.1.1.cmml" xref="S3.SS3.7.p7.4.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.4.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.4.m4.1d">italic_w</annotation></semantics></math> and preserves <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS3.7.p7.5.m5.1"><semantics id="S3.SS3.7.p7.5.m5.1a"><mi id="S3.SS3.7.p7.5.m5.1.1" mathvariant="normal" xref="S3.SS3.7.p7.5.m5.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.5.m5.1b"><ci id="S3.SS3.7.p7.5.m5.1.1.cmml" xref="S3.SS3.7.p7.5.m5.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.5.m5.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.5.m5.1d">roman_Λ</annotation></semantics></math>, we must have <math alttext="\mathcal{G}_{w}\subseteq\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.6.m6.1"><semantics id="S3.SS3.7.p7.6.m6.1a"><mrow id="S3.SS3.7.p7.6.m6.1.1" xref="S3.SS3.7.p7.6.m6.1.1.cmml"><msub id="S3.SS3.7.p7.6.m6.1.1.2" xref="S3.SS3.7.p7.6.m6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.6.m6.1.1.2.2" xref="S3.SS3.7.p7.6.m6.1.1.2.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.6.m6.1.1.2.3" xref="S3.SS3.7.p7.6.m6.1.1.2.3.cmml">w</mi></msub><mo id="S3.SS3.7.p7.6.m6.1.1.1" xref="S3.SS3.7.p7.6.m6.1.1.1.cmml">⊆</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.6.m6.1.1.3" xref="S3.SS3.7.p7.6.m6.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.6.m6.1b"><apply id="S3.SS3.7.p7.6.m6.1.1.cmml" xref="S3.SS3.7.p7.6.m6.1.1"><subset id="S3.SS3.7.p7.6.m6.1.1.1.cmml" xref="S3.SS3.7.p7.6.m6.1.1.1"></subset><apply id="S3.SS3.7.p7.6.m6.1.1.2.cmml" xref="S3.SS3.7.p7.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.7.p7.6.m6.1.1.2.1.cmml" xref="S3.SS3.7.p7.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS3.7.p7.6.m6.1.1.2.2.cmml" xref="S3.SS3.7.p7.6.m6.1.1.2.2">𝒢</ci><ci id="S3.SS3.7.p7.6.m6.1.1.2.3.cmml" xref="S3.SS3.7.p7.6.m6.1.1.2.3">𝑤</ci></apply><ci id="S3.SS3.7.p7.6.m6.1.1.3.cmml" xref="S3.SS3.7.p7.6.m6.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.6.m6.1c">\mathcal{G}_{w}\subseteq\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.6.m6.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ⊆ caligraphic_F</annotation></semantics></math>. For <math alttext="w\notin A" class="ltx_Math" display="inline" id="S3.SS3.7.p7.7.m7.1"><semantics id="S3.SS3.7.p7.7.m7.1a"><mrow id="S3.SS3.7.p7.7.m7.1.1" xref="S3.SS3.7.p7.7.m7.1.1.cmml"><mi id="S3.SS3.7.p7.7.m7.1.1.2" xref="S3.SS3.7.p7.7.m7.1.1.2.cmml">w</mi><mo id="S3.SS3.7.p7.7.m7.1.1.1" xref="S3.SS3.7.p7.7.m7.1.1.1.cmml">∉</mo><mi id="S3.SS3.7.p7.7.m7.1.1.3" xref="S3.SS3.7.p7.7.m7.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.7.m7.1b"><apply id="S3.SS3.7.p7.7.m7.1.1.cmml" xref="S3.SS3.7.p7.7.m7.1.1"><notin id="S3.SS3.7.p7.7.m7.1.1.1.cmml" xref="S3.SS3.7.p7.7.m7.1.1.1"></notin><ci id="S3.SS3.7.p7.7.m7.1.1.2.cmml" xref="S3.SS3.7.p7.7.m7.1.1.2">𝑤</ci><ci id="S3.SS3.7.p7.7.m7.1.1.3.cmml" xref="S3.SS3.7.p7.7.m7.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.7.m7.1c">w\notin A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.7.m7.1d">italic_w ∉ italic_A</annotation></semantics></math>, the process described above cannot possibly add <math alttext="\emptyset" class="ltx_Math" display="inline" id="S3.SS3.7.p7.8.m8.1"><semantics id="S3.SS3.7.p7.8.m8.1a"><mi id="S3.SS3.7.p7.8.m8.1.1" mathvariant="normal" xref="S3.SS3.7.p7.8.m8.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.8.m8.1b"><emptyset id="S3.SS3.7.p7.8.m8.1.1.cmml" xref="S3.SS3.7.p7.8.m8.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.8.m8.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.8.m8.1d">∅</annotation></semantics></math> to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.9.m9.1"><semantics id="S3.SS3.7.p7.9.m9.1a"><msub id="S3.SS3.7.p7.9.m9.1.1" xref="S3.SS3.7.p7.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.9.m9.1.1.2" xref="S3.SS3.7.p7.9.m9.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.9.m9.1.1.3" xref="S3.SS3.7.p7.9.m9.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.9.m9.1b"><apply id="S3.SS3.7.p7.9.m9.1.1.cmml" xref="S3.SS3.7.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p7.9.m9.1.1.1.cmml" xref="S3.SS3.7.p7.9.m9.1.1">subscript</csymbol><ci id="S3.SS3.7.p7.9.m9.1.1.2.cmml" xref="S3.SS3.7.p7.9.m9.1.1.2">𝒢</ci><ci id="S3.SS3.7.p7.9.m9.1.1.3.cmml" xref="S3.SS3.7.p7.9.m9.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.9.m9.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.9.m9.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, since by Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem26" title="Claim 26. ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">26</span></a> there is a semi-filter <math alttext="\mathcal{F}\in\mathfrak{F}_{\Lambda}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.10.m10.1"><semantics id="S3.SS3.7.p7.10.m10.1a"><mrow id="S3.SS3.7.p7.10.m10.1.1" xref="S3.SS3.7.p7.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.10.m10.1.1.2" xref="S3.SS3.7.p7.10.m10.1.1.2.cmml">ℱ</mi><mo id="S3.SS3.7.p7.10.m10.1.1.1" xref="S3.SS3.7.p7.10.m10.1.1.1.cmml">∈</mo><msub id="S3.SS3.7.p7.10.m10.1.1.3" xref="S3.SS3.7.p7.10.m10.1.1.3.cmml"><mi id="S3.SS3.7.p7.10.m10.1.1.3.2" xref="S3.SS3.7.p7.10.m10.1.1.3.2.cmml">𝔉</mi><mi id="S3.SS3.7.p7.10.m10.1.1.3.3" mathvariant="normal" xref="S3.SS3.7.p7.10.m10.1.1.3.3.cmml">Λ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.10.m10.1b"><apply id="S3.SS3.7.p7.10.m10.1.1.cmml" xref="S3.SS3.7.p7.10.m10.1.1"><in id="S3.SS3.7.p7.10.m10.1.1.1.cmml" xref="S3.SS3.7.p7.10.m10.1.1.1"></in><ci id="S3.SS3.7.p7.10.m10.1.1.2.cmml" xref="S3.SS3.7.p7.10.m10.1.1.2">ℱ</ci><apply id="S3.SS3.7.p7.10.m10.1.1.3.cmml" xref="S3.SS3.7.p7.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.7.p7.10.m10.1.1.3.1.cmml" xref="S3.SS3.7.p7.10.m10.1.1.3">subscript</csymbol><ci id="S3.SS3.7.p7.10.m10.1.1.3.2.cmml" xref="S3.SS3.7.p7.10.m10.1.1.3.2">𝔉</ci><ci id="S3.SS3.7.p7.10.m10.1.1.3.3.cmml" xref="S3.SS3.7.p7.10.m10.1.1.3.3">Λ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.10.m10.1c">\mathcal{F}\in\mathfrak{F}_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.10.m10.1d">caligraphic_F ∈ fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> that is above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.7.p7.11.m11.1"><semantics id="S3.SS3.7.p7.11.m11.1a"><mi id="S3.SS3.7.p7.11.m11.1.1" xref="S3.SS3.7.p7.11.m11.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.11.m11.1b"><ci id="S3.SS3.7.p7.11.m11.1.1.cmml" xref="S3.SS3.7.p7.11.m11.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.11.m11.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.11.m11.1d">italic_w</annotation></semantics></math>, and <math alttext="\mathcal{G}_{w}\subseteq\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.12.m12.1"><semantics id="S3.SS3.7.p7.12.m12.1a"><mrow id="S3.SS3.7.p7.12.m12.1.1" xref="S3.SS3.7.p7.12.m12.1.1.cmml"><msub id="S3.SS3.7.p7.12.m12.1.1.2" xref="S3.SS3.7.p7.12.m12.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.12.m12.1.1.2.2" xref="S3.SS3.7.p7.12.m12.1.1.2.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.12.m12.1.1.2.3" xref="S3.SS3.7.p7.12.m12.1.1.2.3.cmml">w</mi></msub><mo id="S3.SS3.7.p7.12.m12.1.1.1" xref="S3.SS3.7.p7.12.m12.1.1.1.cmml">⊆</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.12.m12.1.1.3" xref="S3.SS3.7.p7.12.m12.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.12.m12.1b"><apply id="S3.SS3.7.p7.12.m12.1.1.cmml" xref="S3.SS3.7.p7.12.m12.1.1"><subset id="S3.SS3.7.p7.12.m12.1.1.1.cmml" xref="S3.SS3.7.p7.12.m12.1.1.1"></subset><apply id="S3.SS3.7.p7.12.m12.1.1.2.cmml" xref="S3.SS3.7.p7.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.7.p7.12.m12.1.1.2.1.cmml" xref="S3.SS3.7.p7.12.m12.1.1.2">subscript</csymbol><ci id="S3.SS3.7.p7.12.m12.1.1.2.2.cmml" xref="S3.SS3.7.p7.12.m12.1.1.2.2">𝒢</ci><ci id="S3.SS3.7.p7.12.m12.1.1.2.3.cmml" xref="S3.SS3.7.p7.12.m12.1.1.2.3">𝑤</ci></apply><ci id="S3.SS3.7.p7.12.m12.1.1.3.cmml" xref="S3.SS3.7.p7.12.m12.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.12.m12.1c">\mathcal{G}_{w}\subseteq\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.12.m12.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ⊆ caligraphic_F</annotation></semantics></math>. On the other hand, if this process terminates without adding <math alttext="\emptyset" class="ltx_Math" display="inline" id="S3.SS3.7.p7.13.m13.1"><semantics id="S3.SS3.7.p7.13.m13.1a"><mi id="S3.SS3.7.p7.13.m13.1.1" mathvariant="normal" xref="S3.SS3.7.p7.13.m13.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.13.m13.1b"><emptyset id="S3.SS3.7.p7.13.m13.1.1.cmml" xref="S3.SS3.7.p7.13.m13.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.13.m13.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.13.m13.1d">∅</annotation></semantics></math> to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.14.m14.1"><semantics id="S3.SS3.7.p7.14.m14.1a"><msub id="S3.SS3.7.p7.14.m14.1.1" xref="S3.SS3.7.p7.14.m14.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.14.m14.1.1.2" xref="S3.SS3.7.p7.14.m14.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.14.m14.1.1.3" xref="S3.SS3.7.p7.14.m14.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.14.m14.1b"><apply id="S3.SS3.7.p7.14.m14.1.1.cmml" xref="S3.SS3.7.p7.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p7.14.m14.1.1.1.cmml" xref="S3.SS3.7.p7.14.m14.1.1">subscript</csymbol><ci id="S3.SS3.7.p7.14.m14.1.1.2.cmml" xref="S3.SS3.7.p7.14.m14.1.1.2">𝒢</ci><ci id="S3.SS3.7.p7.14.m14.1.1.3.cmml" xref="S3.SS3.7.p7.14.m14.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.14.m14.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.14.m14.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, it is easy to see that <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.15.m15.1"><semantics id="S3.SS3.7.p7.15.m15.1a"><msub id="S3.SS3.7.p7.15.m15.1.1" xref="S3.SS3.7.p7.15.m15.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p7.15.m15.1.1.2" xref="S3.SS3.7.p7.15.m15.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.7.p7.15.m15.1.1.3" xref="S3.SS3.7.p7.15.m15.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.15.m15.1b"><apply id="S3.SS3.7.p7.15.m15.1.1.cmml" xref="S3.SS3.7.p7.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p7.15.m15.1.1.1.cmml" xref="S3.SS3.7.p7.15.m15.1.1">subscript</csymbol><ci id="S3.SS3.7.p7.15.m15.1.1.2.cmml" xref="S3.SS3.7.p7.15.m15.1.1.2">𝒢</ci><ci id="S3.SS3.7.p7.15.m15.1.1.3.cmml" xref="S3.SS3.7.p7.15.m15.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.15.m15.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.15.m15.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> is a semi-filter in <math alttext="\mathfrak{F}_{\Lambda}" class="ltx_Math" display="inline" id="S3.SS3.7.p7.16.m16.1"><semantics id="S3.SS3.7.p7.16.m16.1a"><msub id="S3.SS3.7.p7.16.m16.1.1" xref="S3.SS3.7.p7.16.m16.1.1.cmml"><mi id="S3.SS3.7.p7.16.m16.1.1.2" xref="S3.SS3.7.p7.16.m16.1.1.2.cmml">𝔉</mi><mi id="S3.SS3.7.p7.16.m16.1.1.3" mathvariant="normal" xref="S3.SS3.7.p7.16.m16.1.1.3.cmml">Λ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.16.m16.1b"><apply id="S3.SS3.7.p7.16.m16.1.1.cmml" xref="S3.SS3.7.p7.16.m16.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p7.16.m16.1.1.1.cmml" xref="S3.SS3.7.p7.16.m16.1.1">subscript</csymbol><ci id="S3.SS3.7.p7.16.m16.1.1.2.cmml" xref="S3.SS3.7.p7.16.m16.1.1.2">𝔉</ci><ci id="S3.SS3.7.p7.16.m16.1.1.3.cmml" xref="S3.SS3.7.p7.16.m16.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.16.m16.1c">\mathfrak{F}_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.16.m16.1d">fraktur_F start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> that is above <math alttext="w" class="ltx_Math" display="inline" id="S3.SS3.7.p7.17.m17.1"><semantics id="S3.SS3.7.p7.17.m17.1a"><mi id="S3.SS3.7.p7.17.m17.1.1" xref="S3.SS3.7.p7.17.m17.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.17.m17.1b"><ci id="S3.SS3.7.p7.17.m17.1.1.cmml" xref="S3.SS3.7.p7.17.m17.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.17.m17.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.17.m17.1d">italic_w</annotation></semantics></math>, which in turn implies that <math alttext="w\notin A" class="ltx_Math" display="inline" id="S3.SS3.7.p7.18.m18.1"><semantics id="S3.SS3.7.p7.18.m18.1a"><mrow id="S3.SS3.7.p7.18.m18.1.1" xref="S3.SS3.7.p7.18.m18.1.1.cmml"><mi id="S3.SS3.7.p7.18.m18.1.1.2" xref="S3.SS3.7.p7.18.m18.1.1.2.cmml">w</mi><mo id="S3.SS3.7.p7.18.m18.1.1.1" xref="S3.SS3.7.p7.18.m18.1.1.1.cmml">∉</mo><mi id="S3.SS3.7.p7.18.m18.1.1.3" xref="S3.SS3.7.p7.18.m18.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p7.18.m18.1b"><apply id="S3.SS3.7.p7.18.m18.1.1.cmml" xref="S3.SS3.7.p7.18.m18.1.1"><notin id="S3.SS3.7.p7.18.m18.1.1.1.cmml" xref="S3.SS3.7.p7.18.m18.1.1.1"></notin><ci id="S3.SS3.7.p7.18.m18.1.1.2.cmml" xref="S3.SS3.7.p7.18.m18.1.1.2">𝑤</ci><ci id="S3.SS3.7.p7.18.m18.1.1.3.cmml" xref="S3.SS3.7.p7.18.m18.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p7.18.m18.1c">w\notin A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p7.18.m18.1d">italic_w ∉ italic_A</annotation></semantics></math> via Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem26" title="Claim 26. ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">26</span></a>. This completes the proof of Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem27" title="Claim 27. ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">27</span></a>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS3.8.p8"> <p class="ltx_p" id="S3.SS3.8.p8.5">We now turn this discussion into the actual construction of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.8.p8.1.m1.1"><semantics id="S3.SS3.8.p8.1.m1.1a"><mi id="S3.SS3.8.p8.1.m1.1.1" xref="S3.SS3.8.p8.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.1.m1.1b"><ci id="S3.SS3.8.p8.1.m1.1.1.cmml" xref="S3.SS3.8.p8.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.1.m1.1d">italic_A</annotation></semantics></math> from the sets in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.8.p8.2.m2.1"><semantics id="S3.SS3.8.p8.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.8.p8.2.m2.1.1" xref="S3.SS3.8.p8.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.2.m2.1b"><ci id="S3.SS3.8.p8.2.m2.1.1.cmml" xref="S3.SS3.8.p8.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.2.m2.1d">caligraphic_B</annotation></semantics></math>. For convenience, we actually upper bound <math alttext="D_{\cap}(A\mid\mathcal{B}\cup\{\emptyset\})" class="ltx_Math" display="inline" id="S3.SS3.8.p8.3.m3.2"><semantics id="S3.SS3.8.p8.3.m3.2a"><mrow id="S3.SS3.8.p8.3.m3.2.2" xref="S3.SS3.8.p8.3.m3.2.2.cmml"><msub id="S3.SS3.8.p8.3.m3.2.2.3" xref="S3.SS3.8.p8.3.m3.2.2.3.cmml"><mi id="S3.SS3.8.p8.3.m3.2.2.3.2" xref="S3.SS3.8.p8.3.m3.2.2.3.2.cmml">D</mi><mo id="S3.SS3.8.p8.3.m3.2.2.3.3" xref="S3.SS3.8.p8.3.m3.2.2.3.3.cmml">∩</mo></msub><mo id="S3.SS3.8.p8.3.m3.2.2.2" xref="S3.SS3.8.p8.3.m3.2.2.2.cmml">⁢</mo><mrow id="S3.SS3.8.p8.3.m3.2.2.1.1" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.cmml"><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.2" stretchy="false" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS3.8.p8.3.m3.2.2.1.1.1" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.cmml"><mi id="S3.SS3.8.p8.3.m3.2.2.1.1.1.2" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.2.cmml">A</mi><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.1.1" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.1.cmml">∣</mo><mrow id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.2" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.2.cmml">ℬ</mi><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.1" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.1.cmml">∪</mo><mrow id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.2" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.1.cmml"><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.2.1" stretchy="false" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.1.cmml">{</mo><mi id="S3.SS3.8.p8.3.m3.1.1" mathvariant="normal" xref="S3.SS3.8.p8.3.m3.1.1.cmml">∅</mi><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.2.2" stretchy="false" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S3.SS3.8.p8.3.m3.2.2.1.1.3" stretchy="false" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.3.m3.2b"><apply id="S3.SS3.8.p8.3.m3.2.2.cmml" xref="S3.SS3.8.p8.3.m3.2.2"><times id="S3.SS3.8.p8.3.m3.2.2.2.cmml" xref="S3.SS3.8.p8.3.m3.2.2.2"></times><apply id="S3.SS3.8.p8.3.m3.2.2.3.cmml" xref="S3.SS3.8.p8.3.m3.2.2.3"><csymbol cd="ambiguous" id="S3.SS3.8.p8.3.m3.2.2.3.1.cmml" xref="S3.SS3.8.p8.3.m3.2.2.3">subscript</csymbol><ci id="S3.SS3.8.p8.3.m3.2.2.3.2.cmml" xref="S3.SS3.8.p8.3.m3.2.2.3.2">𝐷</ci><intersect id="S3.SS3.8.p8.3.m3.2.2.3.3.cmml" xref="S3.SS3.8.p8.3.m3.2.2.3.3"></intersect></apply><apply id="S3.SS3.8.p8.3.m3.2.2.1.1.1.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1"><csymbol cd="latexml" id="S3.SS3.8.p8.3.m3.2.2.1.1.1.1.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.1">conditional</csymbol><ci id="S3.SS3.8.p8.3.m3.2.2.1.1.1.2.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.2">𝐴</ci><apply id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3"><union id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.1.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.1"></union><ci id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.2.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.2">ℬ</ci><set id="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.1.cmml" xref="S3.SS3.8.p8.3.m3.2.2.1.1.1.3.3.2"><emptyset id="S3.SS3.8.p8.3.m3.1.1.cmml" xref="S3.SS3.8.p8.3.m3.1.1"></emptyset></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.3.m3.2c">D_{\cap}(A\mid\mathcal{B}\cup\{\emptyset\})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.3.m3.2d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ∪ { ∅ } )</annotation></semantics></math>, i.e., we freely use <math alttext="\emptyset" class="ltx_Math" display="inline" id="S3.SS3.8.p8.4.m4.1"><semantics id="S3.SS3.8.p8.4.m4.1a"><mi id="S3.SS3.8.p8.4.m4.1.1" mathvariant="normal" xref="S3.SS3.8.p8.4.m4.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.4.m4.1b"><emptyset id="S3.SS3.8.p8.4.m4.1.1.cmml" xref="S3.SS3.8.p8.4.m4.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.4.m4.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.4.m4.1d">∅</annotation></semantics></math> as a generator in the description of the sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.8.p8.5.m5.1"><semantics id="S3.SS3.8.p8.5.m5.1a"><mi id="S3.SS3.8.p8.5.m5.1.1" xref="S3.SS3.8.p8.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.5.m5.1b"><ci id="S3.SS3.8.p8.5.m5.1.1.cmml" xref="S3.SS3.8.p8.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.5.m5.1d">italic_A</annotation></semantics></math>. This is without loss of generality due to Fact <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem8" title="Fact 8. ‣ 2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">8</span></a>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\Omega\;\stackrel{{\scriptstyle\rm def}}{{=}}\;\mathcal{B}_{U}\cup\{E_{i}\}_{i% \in[t]}\cup\{H_{i}\}_{i\in[t]}\cup\{H_{i}\cap E_{i}\}_{i\in[t]}\cup\{\emptyset\}," class="ltx_Math" display="block" id="S3.Ex24.m1.5"><semantics id="S3.Ex24.m1.5a"><mrow id="S3.Ex24.m1.5.5.1" xref="S3.Ex24.m1.5.5.1.1.cmml"><mrow id="S3.Ex24.m1.5.5.1.1" xref="S3.Ex24.m1.5.5.1.1.cmml"><mi id="S3.Ex24.m1.5.5.1.1.5" mathvariant="normal" xref="S3.Ex24.m1.5.5.1.1.5.cmml">Ω</mi><mover id="S3.Ex24.m1.5.5.1.1.4" xref="S3.Ex24.m1.5.5.1.1.4.cmml"><mo id="S3.Ex24.m1.5.5.1.1.4.2" lspace="0.558em" xref="S3.Ex24.m1.5.5.1.1.4.2.cmml">=</mo><mi id="S3.Ex24.m1.5.5.1.1.4.3" xref="S3.Ex24.m1.5.5.1.1.4.3.cmml">def</mi></mover><mrow id="S3.Ex24.m1.5.5.1.1.3" xref="S3.Ex24.m1.5.5.1.1.3.cmml"><msub id="S3.Ex24.m1.5.5.1.1.3.5" xref="S3.Ex24.m1.5.5.1.1.3.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex24.m1.5.5.1.1.3.5.2" xref="S3.Ex24.m1.5.5.1.1.3.5.2.cmml">ℬ</mi><mi id="S3.Ex24.m1.5.5.1.1.3.5.3" xref="S3.Ex24.m1.5.5.1.1.3.5.3.cmml">U</mi></msub><mo id="S3.Ex24.m1.5.5.1.1.3.4" xref="S3.Ex24.m1.5.5.1.1.3.4.cmml">∪</mo><msub id="S3.Ex24.m1.5.5.1.1.1.1" xref="S3.Ex24.m1.5.5.1.1.1.1.cmml"><mrow id="S3.Ex24.m1.5.5.1.1.1.1.1.1" xref="S3.Ex24.m1.5.5.1.1.1.1.1.2.cmml"><mo id="S3.Ex24.m1.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex24.m1.5.5.1.1.1.1.1.2.cmml">{</mo><msub id="S3.Ex24.m1.5.5.1.1.1.1.1.1.1" xref="S3.Ex24.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex24.m1.5.5.1.1.1.1.1.1.1.2" xref="S3.Ex24.m1.5.5.1.1.1.1.1.1.1.2.cmml">E</mi><mi id="S3.Ex24.m1.5.5.1.1.1.1.1.1.1.3" xref="S3.Ex24.m1.5.5.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex24.m1.5.5.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex24.m1.5.5.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.Ex24.m1.1.1.1" xref="S3.Ex24.m1.1.1.1.cmml"><mi id="S3.Ex24.m1.1.1.1.3" xref="S3.Ex24.m1.1.1.1.3.cmml">i</mi><mo id="S3.Ex24.m1.1.1.1.2" xref="S3.Ex24.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex24.m1.1.1.1.4.2" xref="S3.Ex24.m1.1.1.1.4.1.cmml"><mo id="S3.Ex24.m1.1.1.1.4.2.1" stretchy="false" xref="S3.Ex24.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S3.Ex24.m1.1.1.1.1" xref="S3.Ex24.m1.1.1.1.1.cmml">t</mi><mo id="S3.Ex24.m1.1.1.1.4.2.2" stretchy="false" xref="S3.Ex24.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mo id="S3.Ex24.m1.5.5.1.1.3.4a" xref="S3.Ex24.m1.5.5.1.1.3.4.cmml">∪</mo><msub id="S3.Ex24.m1.5.5.1.1.2.2" xref="S3.Ex24.m1.5.5.1.1.2.2.cmml"><mrow id="S3.Ex24.m1.5.5.1.1.2.2.1.1" xref="S3.Ex24.m1.5.5.1.1.2.2.1.2.cmml"><mo id="S3.Ex24.m1.5.5.1.1.2.2.1.1.2" stretchy="false" xref="S3.Ex24.m1.5.5.1.1.2.2.1.2.cmml">{</mo><msub id="S3.Ex24.m1.5.5.1.1.2.2.1.1.1" xref="S3.Ex24.m1.5.5.1.1.2.2.1.1.1.cmml"><mi id="S3.Ex24.m1.5.5.1.1.2.2.1.1.1.2" xref="S3.Ex24.m1.5.5.1.1.2.2.1.1.1.2.cmml">H</mi><mi id="S3.Ex24.m1.5.5.1.1.2.2.1.1.1.3" xref="S3.Ex24.m1.5.5.1.1.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex24.m1.5.5.1.1.2.2.1.1.3" stretchy="false" xref="S3.Ex24.m1.5.5.1.1.2.2.1.2.cmml">}</mo></mrow><mrow id="S3.Ex24.m1.2.2.1" xref="S3.Ex24.m1.2.2.1.cmml"><mi id="S3.Ex24.m1.2.2.1.3" xref="S3.Ex24.m1.2.2.1.3.cmml">i</mi><mo id="S3.Ex24.m1.2.2.1.2" xref="S3.Ex24.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex24.m1.2.2.1.4.2" xref="S3.Ex24.m1.2.2.1.4.1.cmml"><mo id="S3.Ex24.m1.2.2.1.4.2.1" stretchy="false" xref="S3.Ex24.m1.2.2.1.4.1.1.cmml">[</mo><mi id="S3.Ex24.m1.2.2.1.1" xref="S3.Ex24.m1.2.2.1.1.cmml">t</mi><mo id="S3.Ex24.m1.2.2.1.4.2.2" stretchy="false" xref="S3.Ex24.m1.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mo id="S3.Ex24.m1.5.5.1.1.3.4b" xref="S3.Ex24.m1.5.5.1.1.3.4.cmml">∪</mo><msub id="S3.Ex24.m1.5.5.1.1.3.3" xref="S3.Ex24.m1.5.5.1.1.3.3.cmml"><mrow id="S3.Ex24.m1.5.5.1.1.3.3.1.1" xref="S3.Ex24.m1.5.5.1.1.3.3.1.2.cmml"><mo 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id="S3.Ex24.m1.5.5.1.1.3.6.1.cmml" xref="S3.Ex24.m1.5.5.1.1.3.6.2"><emptyset id="S3.Ex24.m1.4.4.cmml" xref="S3.Ex24.m1.4.4"></emptyset></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex24.m1.5c">\Omega\;\stackrel{{\scriptstyle\rm def}}{{=}}\;\mathcal{B}_{U}\cup\{E_{i}\}_{i% \in[t]}\cup\{H_{i}\}_{i\in[t]}\cup\{H_{i}\cap E_{i}\}_{i\in[t]}\cup\{\emptyset\},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex24.m1.5d">roman_Ω start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP caligraphic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∪ { italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_t ] end_POSTSUBSCRIPT ∪ { italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_t ] end_POSTSUBSCRIPT ∪ { italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_t ] end_POSTSUBSCRIPT ∪ { ∅ } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.8.p8.8">where we abuse notation and view <math alttext="\Omega" class="ltx_Math" display="inline" id="S3.SS3.8.p8.6.m1.1"><semantics id="S3.SS3.8.p8.6.m1.1a"><mi id="S3.SS3.8.p8.6.m1.1.1" mathvariant="normal" xref="S3.SS3.8.p8.6.m1.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.6.m1.1b"><ci id="S3.SS3.8.p8.6.m1.1.1.cmml" xref="S3.SS3.8.p8.6.m1.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.6.m1.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.6.m1.1d">roman_Ω</annotation></semantics></math> as a <em class="ltx_emph ltx_font_italic" id="S3.SS3.8.p8.8.1">multi-set</em>.<span class="ltx_note ltx_role_footnote" id="footnote10"><sup class="ltx_note_mark">10</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">10</sup><span class="ltx_tag ltx_tag_note">10</span>This is helpful in the argument. For instance, more than one set <math alttext="B\in\mathcal{B}" class="ltx_Math" display="inline" id="footnote10.m1.1"><semantics id="footnote10.m1.1b"><mrow id="footnote10.m1.1.1" xref="footnote10.m1.1.1.cmml"><mi id="footnote10.m1.1.1.2" xref="footnote10.m1.1.1.2.cmml">B</mi><mo id="footnote10.m1.1.1.1" xref="footnote10.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="footnote10.m1.1.1.3" xref="footnote10.m1.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m1.1c"><apply id="footnote10.m1.1.1.cmml" xref="footnote10.m1.1.1"><in id="footnote10.m1.1.1.1.cmml" xref="footnote10.m1.1.1.1"></in><ci id="footnote10.m1.1.1.2.cmml" xref="footnote10.m1.1.1.2">𝐵</ci><ci id="footnote10.m1.1.1.3.cmml" xref="footnote10.m1.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m1.1d">B\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="footnote10.m1.1e">italic_B ∈ caligraphic_B</annotation></semantics></math> might generate an empty set <math alttext="B_{U}=B\cap U\in\Omega" class="ltx_Math" display="inline" id="footnote10.m2.1"><semantics id="footnote10.m2.1b"><mrow id="footnote10.m2.1.1" xref="footnote10.m2.1.1.cmml"><msub id="footnote10.m2.1.1.2" xref="footnote10.m2.1.1.2.cmml"><mi id="footnote10.m2.1.1.2.2" xref="footnote10.m2.1.1.2.2.cmml">B</mi><mi id="footnote10.m2.1.1.2.3" xref="footnote10.m2.1.1.2.3.cmml">U</mi></msub><mo id="footnote10.m2.1.1.3" xref="footnote10.m2.1.1.3.cmml">=</mo><mrow id="footnote10.m2.1.1.4" xref="footnote10.m2.1.1.4.cmml"><mi id="footnote10.m2.1.1.4.2" xref="footnote10.m2.1.1.4.2.cmml">B</mi><mo id="footnote10.m2.1.1.4.1" xref="footnote10.m2.1.1.4.1.cmml">∩</mo><mi id="footnote10.m2.1.1.4.3" xref="footnote10.m2.1.1.4.3.cmml">U</mi></mrow><mo id="footnote10.m2.1.1.5" xref="footnote10.m2.1.1.5.cmml">∈</mo><mi id="footnote10.m2.1.1.6" mathvariant="normal" xref="footnote10.m2.1.1.6.cmml">Ω</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m2.1c"><apply id="footnote10.m2.1.1.cmml" xref="footnote10.m2.1.1"><and id="footnote10.m2.1.1a.cmml" xref="footnote10.m2.1.1"></and><apply id="footnote10.m2.1.1b.cmml" xref="footnote10.m2.1.1"><eq id="footnote10.m2.1.1.3.cmml" xref="footnote10.m2.1.1.3"></eq><apply id="footnote10.m2.1.1.2.cmml" xref="footnote10.m2.1.1.2"><csymbol cd="ambiguous" id="footnote10.m2.1.1.2.1.cmml" xref="footnote10.m2.1.1.2">subscript</csymbol><ci id="footnote10.m2.1.1.2.2.cmml" xref="footnote10.m2.1.1.2.2">𝐵</ci><ci id="footnote10.m2.1.1.2.3.cmml" xref="footnote10.m2.1.1.2.3">𝑈</ci></apply><apply id="footnote10.m2.1.1.4.cmml" xref="footnote10.m2.1.1.4"><intersect id="footnote10.m2.1.1.4.1.cmml" xref="footnote10.m2.1.1.4.1"></intersect><ci id="footnote10.m2.1.1.4.2.cmml" xref="footnote10.m2.1.1.4.2">𝐵</ci><ci id="footnote10.m2.1.1.4.3.cmml" xref="footnote10.m2.1.1.4.3">𝑈</ci></apply></apply><apply id="footnote10.m2.1.1c.cmml" xref="footnote10.m2.1.1"><in id="footnote10.m2.1.1.5.cmml" xref="footnote10.m2.1.1.5"></in><share href="https://arxiv.org/html/2503.14117v1#footnote10.m2.1.1.4.cmml" id="footnote10.m2.1.1d.cmml" xref="footnote10.m2.1.1"></share><ci id="footnote10.m2.1.1.6.cmml" xref="footnote10.m2.1.1.6">Ω</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m2.1d">B_{U}=B\cap U\in\Omega</annotation><annotation encoding="application/x-llamapun" id="footnote10.m2.1e">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_B ∩ italic_U ∈ roman_Ω</annotation></semantics></math>, but we will need to keep track of elements such that <math alttext="w\in B" class="ltx_Math" display="inline" id="footnote10.m3.1"><semantics id="footnote10.m3.1b"><mrow id="footnote10.m3.1.1" xref="footnote10.m3.1.1.cmml"><mi id="footnote10.m3.1.1.2" xref="footnote10.m3.1.1.2.cmml">w</mi><mo id="footnote10.m3.1.1.1" xref="footnote10.m3.1.1.1.cmml">∈</mo><mi id="footnote10.m3.1.1.3" xref="footnote10.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m3.1c"><apply id="footnote10.m3.1.1.cmml" xref="footnote10.m3.1.1"><in id="footnote10.m3.1.1.1.cmml" xref="footnote10.m3.1.1.1"></in><ci id="footnote10.m3.1.1.2.cmml" xref="footnote10.m3.1.1.2">𝑤</ci><ci id="footnote10.m3.1.1.3.cmml" xref="footnote10.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m3.1d">w\in B</annotation><annotation encoding="application/x-llamapun" id="footnote10.m3.1e">italic_w ∈ italic_B</annotation></semantics></math> and <math alttext="B_{U}=\emptyset" class="ltx_Math" display="inline" id="footnote10.m4.1"><semantics id="footnote10.m4.1b"><mrow id="footnote10.m4.1.1" xref="footnote10.m4.1.1.cmml"><msub id="footnote10.m4.1.1.2" xref="footnote10.m4.1.1.2.cmml"><mi id="footnote10.m4.1.1.2.2" xref="footnote10.m4.1.1.2.2.cmml">B</mi><mi id="footnote10.m4.1.1.2.3" xref="footnote10.m4.1.1.2.3.cmml">U</mi></msub><mo id="footnote10.m4.1.1.1" xref="footnote10.m4.1.1.1.cmml">=</mo><mi id="footnote10.m4.1.1.3" mathvariant="normal" xref="footnote10.m4.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m4.1c"><apply id="footnote10.m4.1.1.cmml" xref="footnote10.m4.1.1"><eq id="footnote10.m4.1.1.1.cmml" xref="footnote10.m4.1.1.1"></eq><apply id="footnote10.m4.1.1.2.cmml" xref="footnote10.m4.1.1.2"><csymbol cd="ambiguous" id="footnote10.m4.1.1.2.1.cmml" xref="footnote10.m4.1.1.2">subscript</csymbol><ci id="footnote10.m4.1.1.2.2.cmml" xref="footnote10.m4.1.1.2.2">𝐵</ci><ci id="footnote10.m4.1.1.2.3.cmml" xref="footnote10.m4.1.1.2.3">𝑈</ci></apply><emptyset id="footnote10.m4.1.1.3.cmml" xref="footnote10.m4.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m4.1d">B_{U}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="footnote10.m4.1e">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = ∅</annotation></semantics></math>.</span></span></span> For simplicity and in order to avoid extra terminology, we slightly abuse notation, and distinguish sets that are identical by the symbols representing them. This should be clear in each context, and the reader should keep in mind that we are simply translating the process that defines each <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.8.p8.7.m2.1"><semantics id="S3.SS3.8.p8.7.m2.1a"><msub id="S3.SS3.8.p8.7.m2.1.1" xref="S3.SS3.8.p8.7.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.8.p8.7.m2.1.1.2" xref="S3.SS3.8.p8.7.m2.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.8.p8.7.m2.1.1.3" xref="S3.SS3.8.p8.7.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.7.m2.1b"><apply id="S3.SS3.8.p8.7.m2.1.1.cmml" xref="S3.SS3.8.p8.7.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p8.7.m2.1.1.1.cmml" xref="S3.SS3.8.p8.7.m2.1.1">subscript</csymbol><ci id="S3.SS3.8.p8.7.m2.1.1.2.cmml" xref="S3.SS3.8.p8.7.m2.1.1.2">𝒢</ci><ci id="S3.SS3.8.p8.7.m2.1.1.3.cmml" xref="S3.SS3.8.p8.7.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.7.m2.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.7.m2.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> into a construction of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.8.p8.8.m3.1"><semantics id="S3.SS3.8.p8.8.m3.1a"><mi id="S3.SS3.8.p8.8.m3.1.1" xref="S3.SS3.8.p8.8.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p8.8.m3.1b"><ci id="S3.SS3.8.p8.8.m3.1.1.cmml" xref="S3.SS3.8.p8.8.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p8.8.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p8.8.m3.1d">italic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.9.p9"> <p class="ltx_p" id="S3.SS3.9.p9.14">Fix a set <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.9.p9.1.m1.1"><semantics id="S3.SS3.9.p9.1.m1.1a"><mi id="S3.SS3.9.p9.1.m1.1.1" xref="S3.SS3.9.p9.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.1.m1.1b"><ci id="S3.SS3.9.p9.1.m1.1.1.cmml" xref="S3.SS3.9.p9.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.1.m1.1d">italic_C</annotation></semantics></math> from the multi-set <math alttext="\Omega" class="ltx_Math" display="inline" id="S3.SS3.9.p9.2.m2.1"><semantics id="S3.SS3.9.p9.2.m2.1a"><mi id="S3.SS3.9.p9.2.m2.1.1" mathvariant="normal" xref="S3.SS3.9.p9.2.m2.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.2.m2.1b"><ci id="S3.SS3.9.p9.2.m2.1.1.cmml" xref="S3.SS3.9.p9.2.m2.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.2.m2.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.2.m2.1d">roman_Ω</annotation></semantics></math>. For an integer <math alttext="j\geq 1" class="ltx_Math" display="inline" id="S3.SS3.9.p9.3.m3.1"><semantics id="S3.SS3.9.p9.3.m3.1a"><mrow id="S3.SS3.9.p9.3.m3.1.1" xref="S3.SS3.9.p9.3.m3.1.1.cmml"><mi id="S3.SS3.9.p9.3.m3.1.1.2" xref="S3.SS3.9.p9.3.m3.1.1.2.cmml">j</mi><mo id="S3.SS3.9.p9.3.m3.1.1.1" xref="S3.SS3.9.p9.3.m3.1.1.1.cmml">≥</mo><mn id="S3.SS3.9.p9.3.m3.1.1.3" xref="S3.SS3.9.p9.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.3.m3.1b"><apply id="S3.SS3.9.p9.3.m3.1.1.cmml" xref="S3.SS3.9.p9.3.m3.1.1"><geq id="S3.SS3.9.p9.3.m3.1.1.1.cmml" xref="S3.SS3.9.p9.3.m3.1.1.1"></geq><ci id="S3.SS3.9.p9.3.m3.1.1.2.cmml" xref="S3.SS3.9.p9.3.m3.1.1.2">𝑗</ci><cn id="S3.SS3.9.p9.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.9.p9.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.3.m3.1c">j\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.3.m3.1d">italic_j ≥ 1</annotation></semantics></math>, we let <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.4.m4.1"><semantics id="S3.SS3.9.p9.4.m4.1a"><msubsup id="S3.SS3.9.p9.4.m4.1.1" xref="S3.SS3.9.p9.4.m4.1.1.cmml"><mi id="S3.SS3.9.p9.4.m4.1.1.2.2" xref="S3.SS3.9.p9.4.m4.1.1.2.2.cmml">S</mi><mi id="S3.SS3.9.p9.4.m4.1.1.3" xref="S3.SS3.9.p9.4.m4.1.1.3.cmml">C</mi><mi id="S3.SS3.9.p9.4.m4.1.1.2.3" xref="S3.SS3.9.p9.4.m4.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.4.m4.1b"><apply id="S3.SS3.9.p9.4.m4.1.1.cmml" xref="S3.SS3.9.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.4.m4.1.1.1.cmml" xref="S3.SS3.9.p9.4.m4.1.1">subscript</csymbol><apply id="S3.SS3.9.p9.4.m4.1.1.2.cmml" xref="S3.SS3.9.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.4.m4.1.1.2.1.cmml" xref="S3.SS3.9.p9.4.m4.1.1">superscript</csymbol><ci id="S3.SS3.9.p9.4.m4.1.1.2.2.cmml" xref="S3.SS3.9.p9.4.m4.1.1.2.2">𝑆</ci><ci id="S3.SS3.9.p9.4.m4.1.1.2.3.cmml" xref="S3.SS3.9.p9.4.m4.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.9.p9.4.m4.1.1.3.cmml" xref="S3.SS3.9.p9.4.m4.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.4.m4.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.4.m4.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> be the set of all <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="S3.SS3.9.p9.5.m5.1"><semantics id="S3.SS3.9.p9.5.m5.1a"><mrow id="S3.SS3.9.p9.5.m5.1.1" xref="S3.SS3.9.p9.5.m5.1.1.cmml"><mi id="S3.SS3.9.p9.5.m5.1.1.2" xref="S3.SS3.9.p9.5.m5.1.1.2.cmml">w</mi><mo id="S3.SS3.9.p9.5.m5.1.1.1" xref="S3.SS3.9.p9.5.m5.1.1.1.cmml">∈</mo><mi id="S3.SS3.9.p9.5.m5.1.1.3" mathvariant="normal" xref="S3.SS3.9.p9.5.m5.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.5.m5.1b"><apply id="S3.SS3.9.p9.5.m5.1.1.cmml" xref="S3.SS3.9.p9.5.m5.1.1"><in id="S3.SS3.9.p9.5.m5.1.1.1.cmml" xref="S3.SS3.9.p9.5.m5.1.1.1"></in><ci id="S3.SS3.9.p9.5.m5.1.1.2.cmml" xref="S3.SS3.9.p9.5.m5.1.1.2">𝑤</ci><ci id="S3.SS3.9.p9.5.m5.1.1.3.cmml" xref="S3.SS3.9.p9.5.m5.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.5.m5.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.5.m5.1d">italic_w ∈ roman_Γ</annotation></semantics></math> that have <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.9.p9.6.m6.1"><semantics id="S3.SS3.9.p9.6.m6.1a"><mi id="S3.SS3.9.p9.6.m6.1.1" xref="S3.SS3.9.p9.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.6.m6.1b"><ci id="S3.SS3.9.p9.6.m6.1.1.cmml" xref="S3.SS3.9.p9.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.6.m6.1d">italic_C</annotation></semantics></math> in <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.7.m7.1"><semantics id="S3.SS3.9.p9.7.m7.1a"><msub id="S3.SS3.9.p9.7.m7.1.1" xref="S3.SS3.9.p9.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.9.p9.7.m7.1.1.2" xref="S3.SS3.9.p9.7.m7.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.9.p9.7.m7.1.1.3" xref="S3.SS3.9.p9.7.m7.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.7.m7.1b"><apply id="S3.SS3.9.p9.7.m7.1.1.cmml" xref="S3.SS3.9.p9.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.7.m7.1.1.1.cmml" xref="S3.SS3.9.p9.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.9.p9.7.m7.1.1.2.cmml" xref="S3.SS3.9.p9.7.m7.1.1.2">𝒢</ci><ci id="S3.SS3.9.p9.7.m7.1.1.3.cmml" xref="S3.SS3.9.p9.7.m7.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.7.m7.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.7.m7.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> before the start of the <math alttext="j" class="ltx_Math" display="inline" id="S3.SS3.9.p9.8.m8.1"><semantics id="S3.SS3.9.p9.8.m8.1a"><mi id="S3.SS3.9.p9.8.m8.1.1" xref="S3.SS3.9.p9.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.8.m8.1b"><ci id="S3.SS3.9.p9.8.m8.1.1.cmml" xref="S3.SS3.9.p9.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.8.m8.1d">italic_j</annotation></semantics></math>-th iteration (propagation step) of the process described above. (Here we also view the sets <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.9.m9.1"><semantics id="S3.SS3.9.p9.9.m9.1a"><msubsup id="S3.SS3.9.p9.9.m9.1.1" xref="S3.SS3.9.p9.9.m9.1.1.cmml"><mi id="S3.SS3.9.p9.9.m9.1.1.2.2" xref="S3.SS3.9.p9.9.m9.1.1.2.2.cmml">S</mi><mi id="S3.SS3.9.p9.9.m9.1.1.3" xref="S3.SS3.9.p9.9.m9.1.1.3.cmml">C</mi><mi id="S3.SS3.9.p9.9.m9.1.1.2.3" xref="S3.SS3.9.p9.9.m9.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.9.m9.1b"><apply id="S3.SS3.9.p9.9.m9.1.1.cmml" xref="S3.SS3.9.p9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.9.m9.1.1.1.cmml" xref="S3.SS3.9.p9.9.m9.1.1">subscript</csymbol><apply id="S3.SS3.9.p9.9.m9.1.1.2.cmml" xref="S3.SS3.9.p9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.9.m9.1.1.2.1.cmml" xref="S3.SS3.9.p9.9.m9.1.1">superscript</csymbol><ci id="S3.SS3.9.p9.9.m9.1.1.2.2.cmml" xref="S3.SS3.9.p9.9.m9.1.1.2.2">𝑆</ci><ci id="S3.SS3.9.p9.9.m9.1.1.2.3.cmml" xref="S3.SS3.9.p9.9.m9.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.9.p9.9.m9.1.1.3.cmml" xref="S3.SS3.9.p9.9.m9.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.9.m9.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.9.m9.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> as different formal objects.) We construct each set <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.10.m10.1"><semantics id="S3.SS3.9.p9.10.m10.1a"><msubsup id="S3.SS3.9.p9.10.m10.1.1" xref="S3.SS3.9.p9.10.m10.1.1.cmml"><mi id="S3.SS3.9.p9.10.m10.1.1.2.2" xref="S3.SS3.9.p9.10.m10.1.1.2.2.cmml">S</mi><mi id="S3.SS3.9.p9.10.m10.1.1.3" xref="S3.SS3.9.p9.10.m10.1.1.3.cmml">C</mi><mi id="S3.SS3.9.p9.10.m10.1.1.2.3" xref="S3.SS3.9.p9.10.m10.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.10.m10.1b"><apply id="S3.SS3.9.p9.10.m10.1.1.cmml" xref="S3.SS3.9.p9.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.10.m10.1.1.1.cmml" xref="S3.SS3.9.p9.10.m10.1.1">subscript</csymbol><apply id="S3.SS3.9.p9.10.m10.1.1.2.cmml" xref="S3.SS3.9.p9.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p9.10.m10.1.1.2.1.cmml" xref="S3.SS3.9.p9.10.m10.1.1">superscript</csymbol><ci id="S3.SS3.9.p9.10.m10.1.1.2.2.cmml" xref="S3.SS3.9.p9.10.m10.1.1.2.2">𝑆</ci><ci id="S3.SS3.9.p9.10.m10.1.1.2.3.cmml" xref="S3.SS3.9.p9.10.m10.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.9.p9.10.m10.1.1.3.cmml" xref="S3.SS3.9.p9.10.m10.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.10.m10.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.10.m10.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="\mathcal{B}\cup\{\emptyset\}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.11.m11.1"><semantics id="S3.SS3.9.p9.11.m11.1a"><mrow id="S3.SS3.9.p9.11.m11.1.2" xref="S3.SS3.9.p9.11.m11.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.9.p9.11.m11.1.2.2" xref="S3.SS3.9.p9.11.m11.1.2.2.cmml">ℬ</mi><mo id="S3.SS3.9.p9.11.m11.1.2.1" xref="S3.SS3.9.p9.11.m11.1.2.1.cmml">∪</mo><mrow id="S3.SS3.9.p9.11.m11.1.2.3.2" xref="S3.SS3.9.p9.11.m11.1.2.3.1.cmml"><mo id="S3.SS3.9.p9.11.m11.1.2.3.2.1" stretchy="false" xref="S3.SS3.9.p9.11.m11.1.2.3.1.cmml">{</mo><mi id="S3.SS3.9.p9.11.m11.1.1" mathvariant="normal" xref="S3.SS3.9.p9.11.m11.1.1.cmml">∅</mi><mo id="S3.SS3.9.p9.11.m11.1.2.3.2.2" stretchy="false" xref="S3.SS3.9.p9.11.m11.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.11.m11.1b"><apply id="S3.SS3.9.p9.11.m11.1.2.cmml" xref="S3.SS3.9.p9.11.m11.1.2"><union id="S3.SS3.9.p9.11.m11.1.2.1.cmml" xref="S3.SS3.9.p9.11.m11.1.2.1"></union><ci id="S3.SS3.9.p9.11.m11.1.2.2.cmml" xref="S3.SS3.9.p9.11.m11.1.2.2">ℬ</ci><set id="S3.SS3.9.p9.11.m11.1.2.3.1.cmml" xref="S3.SS3.9.p9.11.m11.1.2.3.2"><emptyset id="S3.SS3.9.p9.11.m11.1.1.cmml" xref="S3.SS3.9.p9.11.m11.1.1"></emptyset></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.11.m11.1c">\mathcal{B}\cup\{\emptyset\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.11.m11.1d">caligraphic_B ∪ { ∅ }</annotation></semantics></math> by induction on <math alttext="j" class="ltx_Math" display="inline" id="S3.SS3.9.p9.12.m12.1"><semantics id="S3.SS3.9.p9.12.m12.1a"><mi id="S3.SS3.9.p9.12.m12.1.1" xref="S3.SS3.9.p9.12.m12.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.12.m12.1b"><ci id="S3.SS3.9.p9.12.m12.1.1.cmml" xref="S3.SS3.9.p9.12.m12.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.12.m12.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.12.m12.1d">italic_j</annotation></semantics></math>. By Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem27" title="Claim 27. ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">27</span></a>, for a large enough <math alttext="\ell\in\mathbb{N}" class="ltx_Math" display="inline" id="S3.SS3.9.p9.13.m13.1"><semantics id="S3.SS3.9.p9.13.m13.1a"><mrow id="S3.SS3.9.p9.13.m13.1.1" xref="S3.SS3.9.p9.13.m13.1.1.cmml"><mi id="S3.SS3.9.p9.13.m13.1.1.2" mathvariant="normal" xref="S3.SS3.9.p9.13.m13.1.1.2.cmml">ℓ</mi><mo id="S3.SS3.9.p9.13.m13.1.1.1" xref="S3.SS3.9.p9.13.m13.1.1.1.cmml">∈</mo><mi id="S3.SS3.9.p9.13.m13.1.1.3" xref="S3.SS3.9.p9.13.m13.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.13.m13.1b"><apply id="S3.SS3.9.p9.13.m13.1.1.cmml" xref="S3.SS3.9.p9.13.m13.1.1"><in id="S3.SS3.9.p9.13.m13.1.1.1.cmml" xref="S3.SS3.9.p9.13.m13.1.1.1"></in><ci id="S3.SS3.9.p9.13.m13.1.1.2.cmml" xref="S3.SS3.9.p9.13.m13.1.1.2">ℓ</ci><ci id="S3.SS3.9.p9.13.m13.1.1.3.cmml" xref="S3.SS3.9.p9.13.m13.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.13.m13.1c">\ell\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.13.m13.1d">roman_ℓ ∈ blackboard_N</annotation></semantics></math>, we get <math alttext="S^{\ell}_{\emptyset}=A" class="ltx_Math" display="inline" id="S3.SS3.9.p9.14.m14.1"><semantics id="S3.SS3.9.p9.14.m14.1a"><mrow id="S3.SS3.9.p9.14.m14.1.1" xref="S3.SS3.9.p9.14.m14.1.1.cmml"><msubsup id="S3.SS3.9.p9.14.m14.1.1.2" xref="S3.SS3.9.p9.14.m14.1.1.2.cmml"><mi id="S3.SS3.9.p9.14.m14.1.1.2.2.2" xref="S3.SS3.9.p9.14.m14.1.1.2.2.2.cmml">S</mi><mi id="S3.SS3.9.p9.14.m14.1.1.2.3" mathvariant="normal" xref="S3.SS3.9.p9.14.m14.1.1.2.3.cmml">∅</mi><mi id="S3.SS3.9.p9.14.m14.1.1.2.2.3" mathvariant="normal" xref="S3.SS3.9.p9.14.m14.1.1.2.2.3.cmml">ℓ</mi></msubsup><mo id="S3.SS3.9.p9.14.m14.1.1.1" xref="S3.SS3.9.p9.14.m14.1.1.1.cmml">=</mo><mi id="S3.SS3.9.p9.14.m14.1.1.3" xref="S3.SS3.9.p9.14.m14.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p9.14.m14.1b"><apply id="S3.SS3.9.p9.14.m14.1.1.cmml" xref="S3.SS3.9.p9.14.m14.1.1"><eq id="S3.SS3.9.p9.14.m14.1.1.1.cmml" xref="S3.SS3.9.p9.14.m14.1.1.1"></eq><apply id="S3.SS3.9.p9.14.m14.1.1.2.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.9.p9.14.m14.1.1.2.1.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2">subscript</csymbol><apply id="S3.SS3.9.p9.14.m14.1.1.2.2.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.9.p9.14.m14.1.1.2.2.1.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2">superscript</csymbol><ci id="S3.SS3.9.p9.14.m14.1.1.2.2.2.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2.2.2">𝑆</ci><ci id="S3.SS3.9.p9.14.m14.1.1.2.2.3.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2.2.3">ℓ</ci></apply><emptyset id="S3.SS3.9.p9.14.m14.1.1.2.3.cmml" xref="S3.SS3.9.p9.14.m14.1.1.2.3"></emptyset></apply><ci id="S3.SS3.9.p9.14.m14.1.1.3.cmml" xref="S3.SS3.9.p9.14.m14.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p9.14.m14.1c">S^{\ell}_{\emptyset}=A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p9.14.m14.1d">italic_S start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT = italic_A</annotation></semantics></math>, our final goal.</p> </div> <div class="ltx_para" id="S3.SS3.10.p10"> <p class="ltx_p" id="S3.SS3.10.p10.6">In the base case, i.e., for <math alttext="j=1" class="ltx_Math" display="inline" id="S3.SS3.10.p10.1.m1.1"><semantics id="S3.SS3.10.p10.1.m1.1a"><mrow id="S3.SS3.10.p10.1.m1.1.1" xref="S3.SS3.10.p10.1.m1.1.1.cmml"><mi id="S3.SS3.10.p10.1.m1.1.1.2" xref="S3.SS3.10.p10.1.m1.1.1.2.cmml">j</mi><mo id="S3.SS3.10.p10.1.m1.1.1.1" xref="S3.SS3.10.p10.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS3.10.p10.1.m1.1.1.3" xref="S3.SS3.10.p10.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.1.m1.1b"><apply id="S3.SS3.10.p10.1.m1.1.1.cmml" xref="S3.SS3.10.p10.1.m1.1.1"><eq id="S3.SS3.10.p10.1.m1.1.1.1.cmml" xref="S3.SS3.10.p10.1.m1.1.1.1"></eq><ci id="S3.SS3.10.p10.1.m1.1.1.2.cmml" xref="S3.SS3.10.p10.1.m1.1.1.2">𝑗</ci><cn id="S3.SS3.10.p10.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.10.p10.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.1.m1.1c">j=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.1.m1.1d">italic_j = 1</annotation></semantics></math>, we first set <math alttext="T^{1}_{B_{U}}=B" class="ltx_Math" display="inline" id="S3.SS3.10.p10.2.m2.1"><semantics id="S3.SS3.10.p10.2.m2.1a"><mrow id="S3.SS3.10.p10.2.m2.1.1" xref="S3.SS3.10.p10.2.m2.1.1.cmml"><msubsup id="S3.SS3.10.p10.2.m2.1.1.2" xref="S3.SS3.10.p10.2.m2.1.1.2.cmml"><mi id="S3.SS3.10.p10.2.m2.1.1.2.2.2" xref="S3.SS3.10.p10.2.m2.1.1.2.2.2.cmml">T</mi><msub id="S3.SS3.10.p10.2.m2.1.1.2.3" xref="S3.SS3.10.p10.2.m2.1.1.2.3.cmml"><mi id="S3.SS3.10.p10.2.m2.1.1.2.3.2" xref="S3.SS3.10.p10.2.m2.1.1.2.3.2.cmml">B</mi><mi id="S3.SS3.10.p10.2.m2.1.1.2.3.3" xref="S3.SS3.10.p10.2.m2.1.1.2.3.3.cmml">U</mi></msub><mn id="S3.SS3.10.p10.2.m2.1.1.2.2.3" xref="S3.SS3.10.p10.2.m2.1.1.2.2.3.cmml">1</mn></msubsup><mo id="S3.SS3.10.p10.2.m2.1.1.1" xref="S3.SS3.10.p10.2.m2.1.1.1.cmml">=</mo><mi id="S3.SS3.10.p10.2.m2.1.1.3" xref="S3.SS3.10.p10.2.m2.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.2.m2.1b"><apply id="S3.SS3.10.p10.2.m2.1.1.cmml" xref="S3.SS3.10.p10.2.m2.1.1"><eq id="S3.SS3.10.p10.2.m2.1.1.1.cmml" xref="S3.SS3.10.p10.2.m2.1.1.1"></eq><apply id="S3.SS3.10.p10.2.m2.1.1.2.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.10.p10.2.m2.1.1.2.1.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2">subscript</csymbol><apply id="S3.SS3.10.p10.2.m2.1.1.2.2.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.10.p10.2.m2.1.1.2.2.1.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2">superscript</csymbol><ci id="S3.SS3.10.p10.2.m2.1.1.2.2.2.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2.2.2">𝑇</ci><cn id="S3.SS3.10.p10.2.m2.1.1.2.2.3.cmml" type="integer" xref="S3.SS3.10.p10.2.m2.1.1.2.2.3">1</cn></apply><apply id="S3.SS3.10.p10.2.m2.1.1.2.3.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.10.p10.2.m2.1.1.2.3.1.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2.3">subscript</csymbol><ci id="S3.SS3.10.p10.2.m2.1.1.2.3.2.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2.3.2">𝐵</ci><ci id="S3.SS3.10.p10.2.m2.1.1.2.3.3.cmml" xref="S3.SS3.10.p10.2.m2.1.1.2.3.3">𝑈</ci></apply></apply><ci id="S3.SS3.10.p10.2.m2.1.1.3.cmml" xref="S3.SS3.10.p10.2.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.2.m2.1c">T^{1}_{B_{U}}=B</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.2.m2.1d">italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_B</annotation></semantics></math> for each <math alttext="B_{U}" class="ltx_Math" display="inline" id="S3.SS3.10.p10.3.m3.1"><semantics id="S3.SS3.10.p10.3.m3.1a"><msub id="S3.SS3.10.p10.3.m3.1.1" xref="S3.SS3.10.p10.3.m3.1.1.cmml"><mi id="S3.SS3.10.p10.3.m3.1.1.2" xref="S3.SS3.10.p10.3.m3.1.1.2.cmml">B</mi><mi id="S3.SS3.10.p10.3.m3.1.1.3" xref="S3.SS3.10.p10.3.m3.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.3.m3.1b"><apply id="S3.SS3.10.p10.3.m3.1.1.cmml" xref="S3.SS3.10.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.10.p10.3.m3.1.1.1.cmml" xref="S3.SS3.10.p10.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.10.p10.3.m3.1.1.2.cmml" xref="S3.SS3.10.p10.3.m3.1.1.2">𝐵</ci><ci id="S3.SS3.10.p10.3.m3.1.1.3.cmml" xref="S3.SS3.10.p10.3.m3.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.3.m3.1c">B_{U}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math> obtained from a set <math alttext="B\in\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.10.p10.4.m4.1"><semantics id="S3.SS3.10.p10.4.m4.1a"><mrow id="S3.SS3.10.p10.4.m4.1.1" xref="S3.SS3.10.p10.4.m4.1.1.cmml"><mi id="S3.SS3.10.p10.4.m4.1.1.2" xref="S3.SS3.10.p10.4.m4.1.1.2.cmml">B</mi><mo id="S3.SS3.10.p10.4.m4.1.1.1" xref="S3.SS3.10.p10.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.10.p10.4.m4.1.1.3" xref="S3.SS3.10.p10.4.m4.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.4.m4.1b"><apply id="S3.SS3.10.p10.4.m4.1.1.cmml" xref="S3.SS3.10.p10.4.m4.1.1"><in id="S3.SS3.10.p10.4.m4.1.1.1.cmml" xref="S3.SS3.10.p10.4.m4.1.1.1"></in><ci id="S3.SS3.10.p10.4.m4.1.1.2.cmml" xref="S3.SS3.10.p10.4.m4.1.1.2">𝐵</ci><ci id="S3.SS3.10.p10.4.m4.1.1.3.cmml" xref="S3.SS3.10.p10.4.m4.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.4.m4.1c">B\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.4.m4.1d">italic_B ∈ caligraphic_B</annotation></semantics></math>, and <math alttext="T^{1}_{I}=\emptyset" class="ltx_Math" display="inline" id="S3.SS3.10.p10.5.m5.1"><semantics id="S3.SS3.10.p10.5.m5.1a"><mrow id="S3.SS3.10.p10.5.m5.1.1" xref="S3.SS3.10.p10.5.m5.1.1.cmml"><msubsup id="S3.SS3.10.p10.5.m5.1.1.2" xref="S3.SS3.10.p10.5.m5.1.1.2.cmml"><mi id="S3.SS3.10.p10.5.m5.1.1.2.2.2" xref="S3.SS3.10.p10.5.m5.1.1.2.2.2.cmml">T</mi><mi id="S3.SS3.10.p10.5.m5.1.1.2.3" xref="S3.SS3.10.p10.5.m5.1.1.2.3.cmml">I</mi><mn id="S3.SS3.10.p10.5.m5.1.1.2.2.3" xref="S3.SS3.10.p10.5.m5.1.1.2.2.3.cmml">1</mn></msubsup><mo id="S3.SS3.10.p10.5.m5.1.1.1" xref="S3.SS3.10.p10.5.m5.1.1.1.cmml">=</mo><mi id="S3.SS3.10.p10.5.m5.1.1.3" mathvariant="normal" xref="S3.SS3.10.p10.5.m5.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.5.m5.1b"><apply id="S3.SS3.10.p10.5.m5.1.1.cmml" xref="S3.SS3.10.p10.5.m5.1.1"><eq id="S3.SS3.10.p10.5.m5.1.1.1.cmml" xref="S3.SS3.10.p10.5.m5.1.1.1"></eq><apply id="S3.SS3.10.p10.5.m5.1.1.2.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.10.p10.5.m5.1.1.2.1.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2">subscript</csymbol><apply id="S3.SS3.10.p10.5.m5.1.1.2.2.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.10.p10.5.m5.1.1.2.2.1.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS3.10.p10.5.m5.1.1.2.2.2.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2.2.2">𝑇</ci><cn id="S3.SS3.10.p10.5.m5.1.1.2.2.3.cmml" type="integer" xref="S3.SS3.10.p10.5.m5.1.1.2.2.3">1</cn></apply><ci id="S3.SS3.10.p10.5.m5.1.1.2.3.cmml" xref="S3.SS3.10.p10.5.m5.1.1.2.3">𝐼</ci></apply><emptyset id="S3.SS3.10.p10.5.m5.1.1.3.cmml" xref="S3.SS3.10.p10.5.m5.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.5.m5.1c">T^{1}_{I}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.5.m5.1d">italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = ∅</annotation></semantics></math> for every other set <math alttext="I\in\Omega" class="ltx_Math" display="inline" id="S3.SS3.10.p10.6.m6.1"><semantics id="S3.SS3.10.p10.6.m6.1a"><mrow id="S3.SS3.10.p10.6.m6.1.1" xref="S3.SS3.10.p10.6.m6.1.1.cmml"><mi id="S3.SS3.10.p10.6.m6.1.1.2" xref="S3.SS3.10.p10.6.m6.1.1.2.cmml">I</mi><mo id="S3.SS3.10.p10.6.m6.1.1.1" xref="S3.SS3.10.p10.6.m6.1.1.1.cmml">∈</mo><mi id="S3.SS3.10.p10.6.m6.1.1.3" mathvariant="normal" xref="S3.SS3.10.p10.6.m6.1.1.3.cmml">Ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.6.m6.1b"><apply id="S3.SS3.10.p10.6.m6.1.1.cmml" xref="S3.SS3.10.p10.6.m6.1.1"><in id="S3.SS3.10.p10.6.m6.1.1.1.cmml" xref="S3.SS3.10.p10.6.m6.1.1.1"></in><ci id="S3.SS3.10.p10.6.m6.1.1.2.cmml" xref="S3.SS3.10.p10.6.m6.1.1.2">𝐼</ci><ci id="S3.SS3.10.p10.6.m6.1.1.3.cmml" xref="S3.SS3.10.p10.6.m6.1.1.3">Ω</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.6.m6.1c">I\in\Omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.6.m6.1d">italic_I ∈ roman_Ω</annotation></semantics></math>. We then let</p> <table class="ltx_equation ltx_eqn_table" id="S3.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S^{1}_{C}\;=\;\bigcup_{C^{\prime}\in\Omega,C^{\prime}\subseteq C}T^{1}_{C^{% \prime}}," class="ltx_Math" display="block" id="S3.E6.m1.3"><semantics id="S3.E6.m1.3a"><mrow id="S3.E6.m1.3.3.1" xref="S3.E6.m1.3.3.1.1.cmml"><mrow id="S3.E6.m1.3.3.1.1" xref="S3.E6.m1.3.3.1.1.cmml"><msubsup id="S3.E6.m1.3.3.1.1.2" xref="S3.E6.m1.3.3.1.1.2.cmml"><mi id="S3.E6.m1.3.3.1.1.2.2.2" xref="S3.E6.m1.3.3.1.1.2.2.2.cmml">S</mi><mi id="S3.E6.m1.3.3.1.1.2.3" xref="S3.E6.m1.3.3.1.1.2.3.cmml">C</mi><mn id="S3.E6.m1.3.3.1.1.2.2.3" xref="S3.E6.m1.3.3.1.1.2.2.3.cmml">1</mn></msubsup><mo id="S3.E6.m1.3.3.1.1.1" lspace="0.558em" rspace="0.391em" xref="S3.E6.m1.3.3.1.1.1.cmml">=</mo><mrow id="S3.E6.m1.3.3.1.1.3" xref="S3.E6.m1.3.3.1.1.3.cmml"><munder id="S3.E6.m1.3.3.1.1.3.1" xref="S3.E6.m1.3.3.1.1.3.1.cmml"><mo id="S3.E6.m1.3.3.1.1.3.1.2" movablelimits="false" xref="S3.E6.m1.3.3.1.1.3.1.2.cmml">⋃</mo><mrow id="S3.E6.m1.2.2.2.2" xref="S3.E6.m1.2.2.2.3.cmml"><mrow id="S3.E6.m1.1.1.1.1.1" xref="S3.E6.m1.1.1.1.1.1.cmml"><msup id="S3.E6.m1.1.1.1.1.1.2" xref="S3.E6.m1.1.1.1.1.1.2.cmml"><mi id="S3.E6.m1.1.1.1.1.1.2.2" xref="S3.E6.m1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.E6.m1.1.1.1.1.1.2.3" xref="S3.E6.m1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S3.E6.m1.1.1.1.1.1.1" xref="S3.E6.m1.1.1.1.1.1.1.cmml">∈</mo><mi id="S3.E6.m1.1.1.1.1.1.3" mathvariant="normal" xref="S3.E6.m1.1.1.1.1.1.3.cmml">Ω</mi></mrow><mo id="S3.E6.m1.2.2.2.2.3" xref="S3.E6.m1.2.2.2.3a.cmml">,</mo><mrow id="S3.E6.m1.2.2.2.2.2" xref="S3.E6.m1.2.2.2.2.2.cmml"><msup id="S3.E6.m1.2.2.2.2.2.2" xref="S3.E6.m1.2.2.2.2.2.2.cmml"><mi id="S3.E6.m1.2.2.2.2.2.2.2" xref="S3.E6.m1.2.2.2.2.2.2.2.cmml">C</mi><mo id="S3.E6.m1.2.2.2.2.2.2.3" xref="S3.E6.m1.2.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S3.E6.m1.2.2.2.2.2.1" xref="S3.E6.m1.2.2.2.2.2.1.cmml">⊆</mo><mi id="S3.E6.m1.2.2.2.2.2.3" xref="S3.E6.m1.2.2.2.2.2.3.cmml">C</mi></mrow></mrow></munder><msubsup id="S3.E6.m1.3.3.1.1.3.2" xref="S3.E6.m1.3.3.1.1.3.2.cmml"><mi id="S3.E6.m1.3.3.1.1.3.2.2.2" xref="S3.E6.m1.3.3.1.1.3.2.2.2.cmml">T</mi><msup id="S3.E6.m1.3.3.1.1.3.2.3" xref="S3.E6.m1.3.3.1.1.3.2.3.cmml"><mi id="S3.E6.m1.3.3.1.1.3.2.3.2" xref="S3.E6.m1.3.3.1.1.3.2.3.2.cmml">C</mi><mo id="S3.E6.m1.3.3.1.1.3.2.3.3" xref="S3.E6.m1.3.3.1.1.3.2.3.3.cmml">′</mo></msup><mn id="S3.E6.m1.3.3.1.1.3.2.2.3" xref="S3.E6.m1.3.3.1.1.3.2.2.3.cmml">1</mn></msubsup></mrow></mrow><mo id="S3.E6.m1.3.3.1.2" xref="S3.E6.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E6.m1.3b"><apply id="S3.E6.m1.3.3.1.1.cmml" xref="S3.E6.m1.3.3.1"><eq id="S3.E6.m1.3.3.1.1.1.cmml" xref="S3.E6.m1.3.3.1.1.1"></eq><apply id="S3.E6.m1.3.3.1.1.2.cmml" xref="S3.E6.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.2.1.cmml" xref="S3.E6.m1.3.3.1.1.2">subscript</csymbol><apply id="S3.E6.m1.3.3.1.1.2.2.cmml" xref="S3.E6.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.2.2.1.cmml" xref="S3.E6.m1.3.3.1.1.2">superscript</csymbol><ci id="S3.E6.m1.3.3.1.1.2.2.2.cmml" xref="S3.E6.m1.3.3.1.1.2.2.2">𝑆</ci><cn id="S3.E6.m1.3.3.1.1.2.2.3.cmml" type="integer" xref="S3.E6.m1.3.3.1.1.2.2.3">1</cn></apply><ci id="S3.E6.m1.3.3.1.1.2.3.cmml" xref="S3.E6.m1.3.3.1.1.2.3">𝐶</ci></apply><apply id="S3.E6.m1.3.3.1.1.3.cmml" xref="S3.E6.m1.3.3.1.1.3"><apply id="S3.E6.m1.3.3.1.1.3.1.cmml" xref="S3.E6.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.3.1.1.cmml" xref="S3.E6.m1.3.3.1.1.3.1">subscript</csymbol><union id="S3.E6.m1.3.3.1.1.3.1.2.cmml" xref="S3.E6.m1.3.3.1.1.3.1.2"></union><apply id="S3.E6.m1.2.2.2.3.cmml" xref="S3.E6.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.E6.m1.2.2.2.3a.cmml" xref="S3.E6.m1.2.2.2.2.3">formulae-sequence</csymbol><apply id="S3.E6.m1.1.1.1.1.1.cmml" xref="S3.E6.m1.1.1.1.1.1"><in id="S3.E6.m1.1.1.1.1.1.1.cmml" xref="S3.E6.m1.1.1.1.1.1.1"></in><apply id="S3.E6.m1.1.1.1.1.1.2.cmml" xref="S3.E6.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E6.m1.1.1.1.1.1.2.1.cmml" xref="S3.E6.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.E6.m1.1.1.1.1.1.2.2.cmml" xref="S3.E6.m1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.E6.m1.1.1.1.1.1.2.3.cmml" xref="S3.E6.m1.1.1.1.1.1.2.3">′</ci></apply><ci id="S3.E6.m1.1.1.1.1.1.3.cmml" xref="S3.E6.m1.1.1.1.1.1.3">Ω</ci></apply><apply id="S3.E6.m1.2.2.2.2.2.cmml" xref="S3.E6.m1.2.2.2.2.2"><subset id="S3.E6.m1.2.2.2.2.2.1.cmml" xref="S3.E6.m1.2.2.2.2.2.1"></subset><apply id="S3.E6.m1.2.2.2.2.2.2.cmml" xref="S3.E6.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.E6.m1.2.2.2.2.2.2.1.cmml" xref="S3.E6.m1.2.2.2.2.2.2">superscript</csymbol><ci id="S3.E6.m1.2.2.2.2.2.2.2.cmml" xref="S3.E6.m1.2.2.2.2.2.2.2">𝐶</ci><ci id="S3.E6.m1.2.2.2.2.2.2.3.cmml" xref="S3.E6.m1.2.2.2.2.2.2.3">′</ci></apply><ci id="S3.E6.m1.2.2.2.2.2.3.cmml" xref="S3.E6.m1.2.2.2.2.2.3">𝐶</ci></apply></apply></apply><apply id="S3.E6.m1.3.3.1.1.3.2.cmml" xref="S3.E6.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.3.2.1.cmml" xref="S3.E6.m1.3.3.1.1.3.2">subscript</csymbol><apply id="S3.E6.m1.3.3.1.1.3.2.2.cmml" xref="S3.E6.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.E6.m1.3.3.1.1.3.2">superscript</csymbol><ci id="S3.E6.m1.3.3.1.1.3.2.2.2.cmml" xref="S3.E6.m1.3.3.1.1.3.2.2.2">𝑇</ci><cn id="S3.E6.m1.3.3.1.1.3.2.2.3.cmml" type="integer" xref="S3.E6.m1.3.3.1.1.3.2.2.3">1</cn></apply><apply id="S3.E6.m1.3.3.1.1.3.2.3.cmml" xref="S3.E6.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E6.m1.3.3.1.1.3.2.3.1.cmml" xref="S3.E6.m1.3.3.1.1.3.2.3">superscript</csymbol><ci id="S3.E6.m1.3.3.1.1.3.2.3.2.cmml" xref="S3.E6.m1.3.3.1.1.3.2.3.2">𝐶</ci><ci id="S3.E6.m1.3.3.1.1.3.2.3.3.cmml" xref="S3.E6.m1.3.3.1.1.3.2.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.3c">S^{1}_{C}\;=\;\bigcup_{C^{\prime}\in\Omega,C^{\prime}\subseteq C}T^{1}_{C^{% \prime}},</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.3d">italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = ⋃ start_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Ω , italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_C end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.10.p10.8">for each <math alttext="C\in\Omega" class="ltx_Math" display="inline" id="S3.SS3.10.p10.7.m1.1"><semantics id="S3.SS3.10.p10.7.m1.1a"><mrow id="S3.SS3.10.p10.7.m1.1.1" xref="S3.SS3.10.p10.7.m1.1.1.cmml"><mi id="S3.SS3.10.p10.7.m1.1.1.2" xref="S3.SS3.10.p10.7.m1.1.1.2.cmml">C</mi><mo id="S3.SS3.10.p10.7.m1.1.1.1" xref="S3.SS3.10.p10.7.m1.1.1.1.cmml">∈</mo><mi id="S3.SS3.10.p10.7.m1.1.1.3" mathvariant="normal" xref="S3.SS3.10.p10.7.m1.1.1.3.cmml">Ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.7.m1.1b"><apply id="S3.SS3.10.p10.7.m1.1.1.cmml" xref="S3.SS3.10.p10.7.m1.1.1"><in id="S3.SS3.10.p10.7.m1.1.1.1.cmml" xref="S3.SS3.10.p10.7.m1.1.1.1"></in><ci id="S3.SS3.10.p10.7.m1.1.1.2.cmml" xref="S3.SS3.10.p10.7.m1.1.1.2">𝐶</ci><ci id="S3.SS3.10.p10.7.m1.1.1.3.cmml" xref="S3.SS3.10.p10.7.m1.1.1.3">Ω</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.7.m1.1c">C\in\Omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.7.m1.1d">italic_C ∈ roman_Ω</annotation></semantics></math>. Observe that the base case satisfies the property in the definition of the sets <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.10.p10.8.m2.1"><semantics id="S3.SS3.10.p10.8.m2.1a"><msubsup id="S3.SS3.10.p10.8.m2.1.1" xref="S3.SS3.10.p10.8.m2.1.1.cmml"><mi id="S3.SS3.10.p10.8.m2.1.1.2.2" xref="S3.SS3.10.p10.8.m2.1.1.2.2.cmml">S</mi><mi id="S3.SS3.10.p10.8.m2.1.1.3" xref="S3.SS3.10.p10.8.m2.1.1.3.cmml">C</mi><mi id="S3.SS3.10.p10.8.m2.1.1.2.3" xref="S3.SS3.10.p10.8.m2.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.10.p10.8.m2.1b"><apply id="S3.SS3.10.p10.8.m2.1.1.cmml" xref="S3.SS3.10.p10.8.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.10.p10.8.m2.1.1.1.cmml" xref="S3.SS3.10.p10.8.m2.1.1">subscript</csymbol><apply id="S3.SS3.10.p10.8.m2.1.1.2.cmml" xref="S3.SS3.10.p10.8.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.10.p10.8.m2.1.1.2.1.cmml" xref="S3.SS3.10.p10.8.m2.1.1">superscript</csymbol><ci id="S3.SS3.10.p10.8.m2.1.1.2.2.cmml" xref="S3.SS3.10.p10.8.m2.1.1.2.2">𝑆</ci><ci id="S3.SS3.10.p10.8.m2.1.1.2.3.cmml" xref="S3.SS3.10.p10.8.m2.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.10.p10.8.m2.1.1.3.cmml" xref="S3.SS3.10.p10.8.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.10.p10.8.m2.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.10.p10.8.m2.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.11.p11"> <p class="ltx_p" id="S3.SS3.11.p11.3">Assume we have constructed <math alttext="S^{j-1}_{C}" class="ltx_Math" display="inline" id="S3.SS3.11.p11.1.m1.1"><semantics id="S3.SS3.11.p11.1.m1.1a"><msubsup id="S3.SS3.11.p11.1.m1.1.1" xref="S3.SS3.11.p11.1.m1.1.1.cmml"><mi id="S3.SS3.11.p11.1.m1.1.1.2.2" xref="S3.SS3.11.p11.1.m1.1.1.2.2.cmml">S</mi><mi id="S3.SS3.11.p11.1.m1.1.1.3" xref="S3.SS3.11.p11.1.m1.1.1.3.cmml">C</mi><mrow id="S3.SS3.11.p11.1.m1.1.1.2.3" xref="S3.SS3.11.p11.1.m1.1.1.2.3.cmml"><mi id="S3.SS3.11.p11.1.m1.1.1.2.3.2" xref="S3.SS3.11.p11.1.m1.1.1.2.3.2.cmml">j</mi><mo id="S3.SS3.11.p11.1.m1.1.1.2.3.1" xref="S3.SS3.11.p11.1.m1.1.1.2.3.1.cmml">−</mo><mn id="S3.SS3.11.p11.1.m1.1.1.2.3.3" xref="S3.SS3.11.p11.1.m1.1.1.2.3.3.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.1.m1.1b"><apply id="S3.SS3.11.p11.1.m1.1.1.cmml" xref="S3.SS3.11.p11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.1.m1.1.1.1.cmml" xref="S3.SS3.11.p11.1.m1.1.1">subscript</csymbol><apply id="S3.SS3.11.p11.1.m1.1.1.2.cmml" xref="S3.SS3.11.p11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.1.m1.1.1.2.1.cmml" xref="S3.SS3.11.p11.1.m1.1.1">superscript</csymbol><ci id="S3.SS3.11.p11.1.m1.1.1.2.2.cmml" xref="S3.SS3.11.p11.1.m1.1.1.2.2">𝑆</ci><apply id="S3.SS3.11.p11.1.m1.1.1.2.3.cmml" xref="S3.SS3.11.p11.1.m1.1.1.2.3"><minus id="S3.SS3.11.p11.1.m1.1.1.2.3.1.cmml" xref="S3.SS3.11.p11.1.m1.1.1.2.3.1"></minus><ci id="S3.SS3.11.p11.1.m1.1.1.2.3.2.cmml" xref="S3.SS3.11.p11.1.m1.1.1.2.3.2">𝑗</ci><cn id="S3.SS3.11.p11.1.m1.1.1.2.3.3.cmml" type="integer" xref="S3.SS3.11.p11.1.m1.1.1.2.3.3">1</cn></apply></apply><ci id="S3.SS3.11.p11.1.m1.1.1.3.cmml" xref="S3.SS3.11.p11.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.1.m1.1c">S^{j-1}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.1.m1.1d">italic_S start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>, for each <math alttext="C\in\Omega" class="ltx_Math" display="inline" id="S3.SS3.11.p11.2.m2.1"><semantics id="S3.SS3.11.p11.2.m2.1a"><mrow id="S3.SS3.11.p11.2.m2.1.1" xref="S3.SS3.11.p11.2.m2.1.1.cmml"><mi id="S3.SS3.11.p11.2.m2.1.1.2" xref="S3.SS3.11.p11.2.m2.1.1.2.cmml">C</mi><mo id="S3.SS3.11.p11.2.m2.1.1.1" xref="S3.SS3.11.p11.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS3.11.p11.2.m2.1.1.3" mathvariant="normal" xref="S3.SS3.11.p11.2.m2.1.1.3.cmml">Ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.2.m2.1b"><apply id="S3.SS3.11.p11.2.m2.1.1.cmml" xref="S3.SS3.11.p11.2.m2.1.1"><in id="S3.SS3.11.p11.2.m2.1.1.1.cmml" xref="S3.SS3.11.p11.2.m2.1.1.1"></in><ci id="S3.SS3.11.p11.2.m2.1.1.2.cmml" xref="S3.SS3.11.p11.2.m2.1.1.2">𝐶</ci><ci id="S3.SS3.11.p11.2.m2.1.1.3.cmml" xref="S3.SS3.11.p11.2.m2.1.1.3">Ω</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.2.m2.1c">C\in\Omega</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.2.m2.1d">italic_C ∈ roman_Ω</annotation></semantics></math>. We can construct each <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.11.p11.3.m3.1"><semantics id="S3.SS3.11.p11.3.m3.1a"><msubsup id="S3.SS3.11.p11.3.m3.1.1" xref="S3.SS3.11.p11.3.m3.1.1.cmml"><mi id="S3.SS3.11.p11.3.m3.1.1.2.2" xref="S3.SS3.11.p11.3.m3.1.1.2.2.cmml">S</mi><mi id="S3.SS3.11.p11.3.m3.1.1.3" xref="S3.SS3.11.p11.3.m3.1.1.3.cmml">C</mi><mi id="S3.SS3.11.p11.3.m3.1.1.2.3" xref="S3.SS3.11.p11.3.m3.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.3.m3.1b"><apply id="S3.SS3.11.p11.3.m3.1.1.cmml" xref="S3.SS3.11.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.3.m3.1.1.1.cmml" xref="S3.SS3.11.p11.3.m3.1.1">subscript</csymbol><apply id="S3.SS3.11.p11.3.m3.1.1.2.cmml" xref="S3.SS3.11.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.3.m3.1.1.2.1.cmml" xref="S3.SS3.11.p11.3.m3.1.1">superscript</csymbol><ci id="S3.SS3.11.p11.3.m3.1.1.2.2.cmml" xref="S3.SS3.11.p11.3.m3.1.1.2.2">𝑆</ci><ci id="S3.SS3.11.p11.3.m3.1.1.2.3.cmml" xref="S3.SS3.11.p11.3.m3.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.11.p11.3.m3.1.1.3.cmml" xref="S3.SS3.11.p11.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.3.m3.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.3.m3.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> from these sets as follows:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx5"> <tbody id="S3.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle T^{j}_{C}" class="ltx_Math" display="inline" id="S3.E7.m1.1"><semantics id="S3.E7.m1.1a"><msubsup id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml"><mi id="S3.E7.m1.1.1.2.2" xref="S3.E7.m1.1.1.2.2.cmml">T</mi><mi id="S3.E7.m1.1.1.3" xref="S3.E7.m1.1.1.3.cmml">C</mi><mi id="S3.E7.m1.1.1.2.3" xref="S3.E7.m1.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.E7.m1.1b"><apply id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.1.1"><csymbol cd="ambiguous" id="S3.E7.m1.1.1.1.cmml" xref="S3.E7.m1.1.1">subscript</csymbol><apply id="S3.E7.m1.1.1.2.cmml" xref="S3.E7.m1.1.1"><csymbol cd="ambiguous" id="S3.E7.m1.1.1.2.1.cmml" xref="S3.E7.m1.1.1">superscript</csymbol><ci id="S3.E7.m1.1.1.2.2.cmml" xref="S3.E7.m1.1.1.2.2">𝑇</ci><ci id="S3.E7.m1.1.1.2.3.cmml" xref="S3.E7.m1.1.1.2.3">𝑗</ci></apply><ci id="S3.E7.m1.1.1.3.cmml" xref="S3.E7.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.1c">\displaystyle T^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.1d">italic_T start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.E7.m2.1"><semantics id="S3.E7.m2.1a"><mo id="S3.E7.m2.1.1" xref="S3.E7.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.E7.m2.1b"><eq id="S3.E7.m2.1.1.cmml" xref="S3.E7.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle S^{j-1}_{C}\cup\bigcup_{\{i\in[t]\;\mid\;C=E_{i}\cap H_{i}\}}(S^% {j-1}_{E_{i}}\cap S^{j-1}_{H_{i}})," class="ltx_Math" display="inline" id="S3.E7.m3.4"><semantics id="S3.E7.m3.4a"><mrow id="S3.E7.m3.4.4.1" xref="S3.E7.m3.4.4.1.1.cmml"><mrow id="S3.E7.m3.4.4.1.1" xref="S3.E7.m3.4.4.1.1.cmml"><msubsup id="S3.E7.m3.4.4.1.1.3" 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end_POSTSUBSCRIPT ∪ ⋃ start_POSTSUBSCRIPT { italic_i ∈ [ italic_t ] ∣ italic_C = italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } end_POSTSUBSCRIPT ( italic_S start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ italic_S start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> <tbody id="S3.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.E8.m2.1"><semantics id="S3.E8.m2.1a"><mo id="S3.E8.m2.1.1" xref="S3.E8.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.E8.m2.1b"><eq id="S3.E8.m2.1.1.cmml" xref="S3.E8.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\bigcup_{C^{\prime}\in\Omega,C^{\prime}\subseteq C}T^{j}_{C^{% \prime}}." class="ltx_Math" display="inline" id="S3.E8.m3.3"><semantics id="S3.E8.m3.3a"><mrow 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xref="S3.E8.m3.3.3.1.1.2.2.3">𝑗</ci></apply><apply id="S3.E8.m3.3.3.1.1.2.3.cmml" xref="S3.E8.m3.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.E8.m3.3.3.1.1.2.3.1.cmml" xref="S3.E8.m3.3.3.1.1.2.3">superscript</csymbol><ci id="S3.E8.m3.3.3.1.1.2.3.2.cmml" xref="S3.E8.m3.3.3.1.1.2.3.2">𝐶</ci><ci id="S3.E8.m3.3.3.1.1.2.3.3.cmml" xref="S3.E8.m3.3.3.1.1.2.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m3.3c">\displaystyle\bigcup_{C^{\prime}\in\Omega,C^{\prime}\subseteq C}T^{j}_{C^{% \prime}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m3.3d">⋃ start_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Ω , italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_C end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.11.p11.8">Note that the definition of each <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.11.p11.4.m1.1"><semantics id="S3.SS3.11.p11.4.m1.1a"><msubsup id="S3.SS3.11.p11.4.m1.1.1" xref="S3.SS3.11.p11.4.m1.1.1.cmml"><mi id="S3.SS3.11.p11.4.m1.1.1.2.2" xref="S3.SS3.11.p11.4.m1.1.1.2.2.cmml">S</mi><mi id="S3.SS3.11.p11.4.m1.1.1.3" xref="S3.SS3.11.p11.4.m1.1.1.3.cmml">C</mi><mi id="S3.SS3.11.p11.4.m1.1.1.2.3" xref="S3.SS3.11.p11.4.m1.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.4.m1.1b"><apply id="S3.SS3.11.p11.4.m1.1.1.cmml" xref="S3.SS3.11.p11.4.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.4.m1.1.1.1.cmml" xref="S3.SS3.11.p11.4.m1.1.1">subscript</csymbol><apply id="S3.SS3.11.p11.4.m1.1.1.2.cmml" xref="S3.SS3.11.p11.4.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.4.m1.1.1.2.1.cmml" xref="S3.SS3.11.p11.4.m1.1.1">superscript</csymbol><ci id="S3.SS3.11.p11.4.m1.1.1.2.2.cmml" xref="S3.SS3.11.p11.4.m1.1.1.2.2">𝑆</ci><ci id="S3.SS3.11.p11.4.m1.1.1.2.3.cmml" xref="S3.SS3.11.p11.4.m1.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.11.p11.4.m1.1.1.3.cmml" xref="S3.SS3.11.p11.4.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.4.m1.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.4.m1.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> handles <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS3.11.p11.5.m2.1"><semantics id="S3.SS3.11.p11.5.m2.1a"><mi id="S3.SS3.11.p11.5.m2.1.1" mathvariant="normal" xref="S3.SS3.11.p11.5.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.5.m2.1b"><ci id="S3.SS3.11.p11.5.m2.1.1.cmml" xref="S3.SS3.11.p11.5.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.5.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.5.m2.1d">roman_Λ</annotation></semantics></math>-preservation and upward-closure, as in the propagation step. It is not difficult to show using the induction hypothesis that each set <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.11.p11.6.m3.1"><semantics id="S3.SS3.11.p11.6.m3.1a"><msubsup id="S3.SS3.11.p11.6.m3.1.1" xref="S3.SS3.11.p11.6.m3.1.1.cmml"><mi id="S3.SS3.11.p11.6.m3.1.1.2.2" xref="S3.SS3.11.p11.6.m3.1.1.2.2.cmml">S</mi><mi id="S3.SS3.11.p11.6.m3.1.1.3" xref="S3.SS3.11.p11.6.m3.1.1.3.cmml">C</mi><mi id="S3.SS3.11.p11.6.m3.1.1.2.3" xref="S3.SS3.11.p11.6.m3.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.6.m3.1b"><apply id="S3.SS3.11.p11.6.m3.1.1.cmml" xref="S3.SS3.11.p11.6.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.6.m3.1.1.1.cmml" xref="S3.SS3.11.p11.6.m3.1.1">subscript</csymbol><apply id="S3.SS3.11.p11.6.m3.1.1.2.cmml" xref="S3.SS3.11.p11.6.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.11.p11.6.m3.1.1.2.1.cmml" xref="S3.SS3.11.p11.6.m3.1.1">superscript</csymbol><ci id="S3.SS3.11.p11.6.m3.1.1.2.2.cmml" xref="S3.SS3.11.p11.6.m3.1.1.2.2">𝑆</ci><ci id="S3.SS3.11.p11.6.m3.1.1.2.3.cmml" xref="S3.SS3.11.p11.6.m3.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.11.p11.6.m3.1.1.3.cmml" xref="S3.SS3.11.p11.6.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.6.m3.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.6.m3.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> satisfies the required property (fix an element <math alttext="w\in\Gamma" class="ltx_Math" display="inline" id="S3.SS3.11.p11.7.m4.1"><semantics id="S3.SS3.11.p11.7.m4.1a"><mrow id="S3.SS3.11.p11.7.m4.1.1" xref="S3.SS3.11.p11.7.m4.1.1.cmml"><mi id="S3.SS3.11.p11.7.m4.1.1.2" xref="S3.SS3.11.p11.7.m4.1.1.2.cmml">w</mi><mo id="S3.SS3.11.p11.7.m4.1.1.1" xref="S3.SS3.11.p11.7.m4.1.1.1.cmml">∈</mo><mi id="S3.SS3.11.p11.7.m4.1.1.3" mathvariant="normal" xref="S3.SS3.11.p11.7.m4.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.7.m4.1b"><apply id="S3.SS3.11.p11.7.m4.1.1.cmml" xref="S3.SS3.11.p11.7.m4.1.1"><in id="S3.SS3.11.p11.7.m4.1.1.1.cmml" xref="S3.SS3.11.p11.7.m4.1.1.1"></in><ci id="S3.SS3.11.p11.7.m4.1.1.2.cmml" xref="S3.SS3.11.p11.7.m4.1.1.2">𝑤</ci><ci id="S3.SS3.11.p11.7.m4.1.1.3.cmml" xref="S3.SS3.11.p11.7.m4.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.7.m4.1c">w\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.7.m4.1d">italic_w ∈ roman_Γ</annotation></semantics></math>, and verify that it appears in the correct sets). This completes the construction of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.11.p11.8.m5.1"><semantics id="S3.SS3.11.p11.8.m5.1a"><mi id="S3.SS3.11.p11.8.m5.1.1" xref="S3.SS3.11.p11.8.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.11.p11.8.m5.1b"><ci id="S3.SS3.11.p11.8.m5.1.1.cmml" xref="S3.SS3.11.p11.8.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.11.p11.8.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.11.p11.8.m5.1d">italic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.12.p12"> <p class="ltx_p" id="S3.SS3.12.p12.19">In order to finish the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a>, we analyse the complexity of this construction. First, since each propagation step that introduces a new set to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.1.m1.1"><semantics id="S3.SS3.12.p12.1.m1.1a"><msub id="S3.SS3.12.p12.1.m1.1.1" xref="S3.SS3.12.p12.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.12.p12.1.m1.1.1.2" xref="S3.SS3.12.p12.1.m1.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.12.p12.1.m1.1.1.3" xref="S3.SS3.12.p12.1.m1.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.1.m1.1b"><apply id="S3.SS3.12.p12.1.m1.1.1.cmml" xref="S3.SS3.12.p12.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.1.m1.1.1.1.cmml" xref="S3.SS3.12.p12.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.12.p12.1.m1.1.1.2.cmml" xref="S3.SS3.12.p12.1.m1.1.1.2">𝒢</ci><ci id="S3.SS3.12.p12.1.m1.1.1.3.cmml" xref="S3.SS3.12.p12.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.1.m1.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.1.m1.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> adds at least one of the sets <math alttext="E_{i}\cap H_{i}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.2.m2.1"><semantics id="S3.SS3.12.p12.2.m2.1a"><mrow id="S3.SS3.12.p12.2.m2.1.1" xref="S3.SS3.12.p12.2.m2.1.1.cmml"><msub id="S3.SS3.12.p12.2.m2.1.1.2" xref="S3.SS3.12.p12.2.m2.1.1.2.cmml"><mi id="S3.SS3.12.p12.2.m2.1.1.2.2" xref="S3.SS3.12.p12.2.m2.1.1.2.2.cmml">E</mi><mi id="S3.SS3.12.p12.2.m2.1.1.2.3" xref="S3.SS3.12.p12.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.12.p12.2.m2.1.1.1" xref="S3.SS3.12.p12.2.m2.1.1.1.cmml">∩</mo><msub id="S3.SS3.12.p12.2.m2.1.1.3" xref="S3.SS3.12.p12.2.m2.1.1.3.cmml"><mi id="S3.SS3.12.p12.2.m2.1.1.3.2" xref="S3.SS3.12.p12.2.m2.1.1.3.2.cmml">H</mi><mi id="S3.SS3.12.p12.2.m2.1.1.3.3" xref="S3.SS3.12.p12.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" 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id="S3.SS3.12.p12.2.m2.1c">E_{i}\cap H_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.2.m2.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\mathcal{G}_{w}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.3.m3.1"><semantics id="S3.SS3.12.p12.3.m3.1a"><msub id="S3.SS3.12.p12.3.m3.1.1" xref="S3.SS3.12.p12.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.12.p12.3.m3.1.1.2" xref="S3.SS3.12.p12.3.m3.1.1.2.cmml">𝒢</mi><mi id="S3.SS3.12.p12.3.m3.1.1.3" xref="S3.SS3.12.p12.3.m3.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.3.m3.1b"><apply id="S3.SS3.12.p12.3.m3.1.1.cmml" xref="S3.SS3.12.p12.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.3.m3.1.1.1.cmml" xref="S3.SS3.12.p12.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.12.p12.3.m3.1.1.2.cmml" xref="S3.SS3.12.p12.3.m3.1.1.2">𝒢</ci><ci id="S3.SS3.12.p12.3.m3.1.1.3.cmml" xref="S3.SS3.12.p12.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.3.m3.1c">\mathcal{G}_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.3.m3.1d">caligraphic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, and there are at most <math alttext="t=|\Lambda|=\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS3.12.p12.4.m4.3"><semantics id="S3.SS3.12.p12.4.m4.3a"><mrow id="S3.SS3.12.p12.4.m4.3.4" xref="S3.SS3.12.p12.4.m4.3.4.cmml"><mi id="S3.SS3.12.p12.4.m4.3.4.2" xref="S3.SS3.12.p12.4.m4.3.4.2.cmml">t</mi><mo id="S3.SS3.12.p12.4.m4.3.4.3" xref="S3.SS3.12.p12.4.m4.3.4.3.cmml">=</mo><mrow id="S3.SS3.12.p12.4.m4.3.4.4.2" xref="S3.SS3.12.p12.4.m4.3.4.4.1.cmml"><mo id="S3.SS3.12.p12.4.m4.3.4.4.2.1" stretchy="false" xref="S3.SS3.12.p12.4.m4.3.4.4.1.1.cmml">|</mo><mi id="S3.SS3.12.p12.4.m4.1.1" mathvariant="normal" xref="S3.SS3.12.p12.4.m4.1.1.cmml">Λ</mi><mo id="S3.SS3.12.p12.4.m4.3.4.4.2.2" stretchy="false" xref="S3.SS3.12.p12.4.m4.3.4.4.1.1.cmml">|</mo></mrow><mo id="S3.SS3.12.p12.4.m4.3.4.5" xref="S3.SS3.12.p12.4.m4.3.4.5.cmml">=</mo><mrow id="S3.SS3.12.p12.4.m4.3.4.6" xref="S3.SS3.12.p12.4.m4.3.4.6.cmml"><mi id="S3.SS3.12.p12.4.m4.3.4.6.2" xref="S3.SS3.12.p12.4.m4.3.4.6.2.cmml">ρ</mi><mo id="S3.SS3.12.p12.4.m4.3.4.6.1" xref="S3.SS3.12.p12.4.m4.3.4.6.1.cmml">⁢</mo><mrow id="S3.SS3.12.p12.4.m4.3.4.6.3.2" xref="S3.SS3.12.p12.4.m4.3.4.6.3.1.cmml"><mo id="S3.SS3.12.p12.4.m4.3.4.6.3.2.1" stretchy="false" xref="S3.SS3.12.p12.4.m4.3.4.6.3.1.cmml">(</mo><mi id="S3.SS3.12.p12.4.m4.2.2" xref="S3.SS3.12.p12.4.m4.2.2.cmml">A</mi><mo id="S3.SS3.12.p12.4.m4.3.4.6.3.2.2" xref="S3.SS3.12.p12.4.m4.3.4.6.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.12.p12.4.m4.3.3" xref="S3.SS3.12.p12.4.m4.3.3.cmml">ℬ</mi><mo id="S3.SS3.12.p12.4.m4.3.4.6.3.2.3" stretchy="false" xref="S3.SS3.12.p12.4.m4.3.4.6.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.4.m4.3b"><apply id="S3.SS3.12.p12.4.m4.3.4.cmml" xref="S3.SS3.12.p12.4.m4.3.4"><and id="S3.SS3.12.p12.4.m4.3.4a.cmml" xref="S3.SS3.12.p12.4.m4.3.4"></and><apply id="S3.SS3.12.p12.4.m4.3.4b.cmml" xref="S3.SS3.12.p12.4.m4.3.4"><eq id="S3.SS3.12.p12.4.m4.3.4.3.cmml" xref="S3.SS3.12.p12.4.m4.3.4.3"></eq><ci id="S3.SS3.12.p12.4.m4.3.4.2.cmml" xref="S3.SS3.12.p12.4.m4.3.4.2">𝑡</ci><apply id="S3.SS3.12.p12.4.m4.3.4.4.1.cmml" xref="S3.SS3.12.p12.4.m4.3.4.4.2"><abs id="S3.SS3.12.p12.4.m4.3.4.4.1.1.cmml" xref="S3.SS3.12.p12.4.m4.3.4.4.2.1"></abs><ci id="S3.SS3.12.p12.4.m4.1.1.cmml" xref="S3.SS3.12.p12.4.m4.1.1">Λ</ci></apply></apply><apply id="S3.SS3.12.p12.4.m4.3.4c.cmml" xref="S3.SS3.12.p12.4.m4.3.4"><eq id="S3.SS3.12.p12.4.m4.3.4.5.cmml" xref="S3.SS3.12.p12.4.m4.3.4.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS3.12.p12.4.m4.3.4.4.cmml" id="S3.SS3.12.p12.4.m4.3.4d.cmml" xref="S3.SS3.12.p12.4.m4.3.4"></share><apply id="S3.SS3.12.p12.4.m4.3.4.6.cmml" xref="S3.SS3.12.p12.4.m4.3.4.6"><times id="S3.SS3.12.p12.4.m4.3.4.6.1.cmml" xref="S3.SS3.12.p12.4.m4.3.4.6.1"></times><ci id="S3.SS3.12.p12.4.m4.3.4.6.2.cmml" xref="S3.SS3.12.p12.4.m4.3.4.6.2">𝜌</ci><interval closure="open" id="S3.SS3.12.p12.4.m4.3.4.6.3.1.cmml" xref="S3.SS3.12.p12.4.m4.3.4.6.3.2"><ci id="S3.SS3.12.p12.4.m4.2.2.cmml" xref="S3.SS3.12.p12.4.m4.2.2">𝐴</ci><ci id="S3.SS3.12.p12.4.m4.3.3.cmml" xref="S3.SS3.12.p12.4.m4.3.3">ℬ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.4.m4.3c">t=|\Lambda|=\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.4.m4.3d">italic_t = | roman_Λ | = italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> such sets, it is sufficient in the construction above to take <math alttext="\ell=t+1" class="ltx_Math" display="inline" id="S3.SS3.12.p12.5.m5.1"><semantics id="S3.SS3.12.p12.5.m5.1a"><mrow id="S3.SS3.12.p12.5.m5.1.1" xref="S3.SS3.12.p12.5.m5.1.1.cmml"><mi id="S3.SS3.12.p12.5.m5.1.1.2" mathvariant="normal" xref="S3.SS3.12.p12.5.m5.1.1.2.cmml">ℓ</mi><mo id="S3.SS3.12.p12.5.m5.1.1.1" xref="S3.SS3.12.p12.5.m5.1.1.1.cmml">=</mo><mrow id="S3.SS3.12.p12.5.m5.1.1.3" xref="S3.SS3.12.p12.5.m5.1.1.3.cmml"><mi id="S3.SS3.12.p12.5.m5.1.1.3.2" xref="S3.SS3.12.p12.5.m5.1.1.3.2.cmml">t</mi><mo id="S3.SS3.12.p12.5.m5.1.1.3.1" xref="S3.SS3.12.p12.5.m5.1.1.3.1.cmml">+</mo><mn id="S3.SS3.12.p12.5.m5.1.1.3.3" xref="S3.SS3.12.p12.5.m5.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.5.m5.1b"><apply id="S3.SS3.12.p12.5.m5.1.1.cmml" xref="S3.SS3.12.p12.5.m5.1.1"><eq id="S3.SS3.12.p12.5.m5.1.1.1.cmml" xref="S3.SS3.12.p12.5.m5.1.1.1"></eq><ci id="S3.SS3.12.p12.5.m5.1.1.2.cmml" xref="S3.SS3.12.p12.5.m5.1.1.2">ℓ</ci><apply id="S3.SS3.12.p12.5.m5.1.1.3.cmml" xref="S3.SS3.12.p12.5.m5.1.1.3"><plus id="S3.SS3.12.p12.5.m5.1.1.3.1.cmml" xref="S3.SS3.12.p12.5.m5.1.1.3.1"></plus><ci id="S3.SS3.12.p12.5.m5.1.1.3.2.cmml" xref="S3.SS3.12.p12.5.m5.1.1.3.2">𝑡</ci><cn id="S3.SS3.12.p12.5.m5.1.1.3.3.cmml" type="integer" xref="S3.SS3.12.p12.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.5.m5.1c">\ell=t+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.5.m5.1d">roman_ℓ = italic_t + 1</annotation></semantics></math>. In particular, <math alttext="S^{t+1}_{\emptyset}=A" class="ltx_Math" display="inline" id="S3.SS3.12.p12.6.m6.1"><semantics id="S3.SS3.12.p12.6.m6.1a"><mrow id="S3.SS3.12.p12.6.m6.1.1" xref="S3.SS3.12.p12.6.m6.1.1.cmml"><msubsup id="S3.SS3.12.p12.6.m6.1.1.2" xref="S3.SS3.12.p12.6.m6.1.1.2.cmml"><mi id="S3.SS3.12.p12.6.m6.1.1.2.2.2" xref="S3.SS3.12.p12.6.m6.1.1.2.2.2.cmml">S</mi><mi id="S3.SS3.12.p12.6.m6.1.1.2.3" mathvariant="normal" xref="S3.SS3.12.p12.6.m6.1.1.2.3.cmml">∅</mi><mrow id="S3.SS3.12.p12.6.m6.1.1.2.2.3" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.cmml"><mi id="S3.SS3.12.p12.6.m6.1.1.2.2.3.2" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.2.cmml">t</mi><mo id="S3.SS3.12.p12.6.m6.1.1.2.2.3.1" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.1.cmml">+</mo><mn id="S3.SS3.12.p12.6.m6.1.1.2.2.3.3" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S3.SS3.12.p12.6.m6.1.1.1" xref="S3.SS3.12.p12.6.m6.1.1.1.cmml">=</mo><mi id="S3.SS3.12.p12.6.m6.1.1.3" xref="S3.SS3.12.p12.6.m6.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.6.m6.1b"><apply id="S3.SS3.12.p12.6.m6.1.1.cmml" xref="S3.SS3.12.p12.6.m6.1.1"><eq id="S3.SS3.12.p12.6.m6.1.1.1.cmml" xref="S3.SS3.12.p12.6.m6.1.1.1"></eq><apply id="S3.SS3.12.p12.6.m6.1.1.2.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.6.m6.1.1.2.1.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2">subscript</csymbol><apply id="S3.SS3.12.p12.6.m6.1.1.2.2.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.6.m6.1.1.2.2.1.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2">superscript</csymbol><ci id="S3.SS3.12.p12.6.m6.1.1.2.2.2.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2.2.2">𝑆</ci><apply id="S3.SS3.12.p12.6.m6.1.1.2.2.3.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3"><plus id="S3.SS3.12.p12.6.m6.1.1.2.2.3.1.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.1"></plus><ci id="S3.SS3.12.p12.6.m6.1.1.2.2.3.2.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.2">𝑡</ci><cn id="S3.SS3.12.p12.6.m6.1.1.2.2.3.3.cmml" type="integer" xref="S3.SS3.12.p12.6.m6.1.1.2.2.3.3">1</cn></apply></apply><emptyset id="S3.SS3.12.p12.6.m6.1.1.2.3.cmml" xref="S3.SS3.12.p12.6.m6.1.1.2.3"></emptyset></apply><ci id="S3.SS3.12.p12.6.m6.1.1.3.cmml" xref="S3.SS3.12.p12.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.6.m6.1c">S^{t+1}_{\emptyset}=A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.6.m6.1d">italic_S start_POSTSUPERSCRIPT italic_t + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT = italic_A</annotation></semantics></math>. Finally, each propagation step (which is associated to a fixed stage <math alttext="j\in[t]" class="ltx_Math" display="inline" id="S3.SS3.12.p12.7.m7.1"><semantics id="S3.SS3.12.p12.7.m7.1a"><mrow id="S3.SS3.12.p12.7.m7.1.2" xref="S3.SS3.12.p12.7.m7.1.2.cmml"><mi id="S3.SS3.12.p12.7.m7.1.2.2" xref="S3.SS3.12.p12.7.m7.1.2.2.cmml">j</mi><mo id="S3.SS3.12.p12.7.m7.1.2.1" xref="S3.SS3.12.p12.7.m7.1.2.1.cmml">∈</mo><mrow id="S3.SS3.12.p12.7.m7.1.2.3.2" xref="S3.SS3.12.p12.7.m7.1.2.3.1.cmml"><mo id="S3.SS3.12.p12.7.m7.1.2.3.2.1" stretchy="false" xref="S3.SS3.12.p12.7.m7.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.12.p12.7.m7.1.1" xref="S3.SS3.12.p12.7.m7.1.1.cmml">t</mi><mo id="S3.SS3.12.p12.7.m7.1.2.3.2.2" stretchy="false" xref="S3.SS3.12.p12.7.m7.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.7.m7.1b"><apply id="S3.SS3.12.p12.7.m7.1.2.cmml" xref="S3.SS3.12.p12.7.m7.1.2"><in id="S3.SS3.12.p12.7.m7.1.2.1.cmml" xref="S3.SS3.12.p12.7.m7.1.2.1"></in><ci id="S3.SS3.12.p12.7.m7.1.2.2.cmml" xref="S3.SS3.12.p12.7.m7.1.2.2">𝑗</ci><apply id="S3.SS3.12.p12.7.m7.1.2.3.1.cmml" xref="S3.SS3.12.p12.7.m7.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.12.p12.7.m7.1.2.3.1.1.cmml" xref="S3.SS3.12.p12.7.m7.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.12.p12.7.m7.1.1.cmml" xref="S3.SS3.12.p12.7.m7.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.7.m7.1c">j\in[t]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.7.m7.1d">italic_j ∈ [ italic_t ]</annotation></semantics></math> of the construction) only employs intersection operations for sets <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.12.p12.8.m8.1"><semantics id="S3.SS3.12.p12.8.m8.1a"><mi id="S3.SS3.12.p12.8.m8.1.1" xref="S3.SS3.12.p12.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.8.m8.1b"><ci id="S3.SS3.12.p12.8.m8.1.1.cmml" xref="S3.SS3.12.p12.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.8.m8.1d">italic_C</annotation></semantics></math> of the form <math alttext="E_{i}\cap H_{i}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.9.m9.1"><semantics id="S3.SS3.12.p12.9.m9.1a"><mrow id="S3.SS3.12.p12.9.m9.1.1" xref="S3.SS3.12.p12.9.m9.1.1.cmml"><msub id="S3.SS3.12.p12.9.m9.1.1.2" xref="S3.SS3.12.p12.9.m9.1.1.2.cmml"><mi id="S3.SS3.12.p12.9.m9.1.1.2.2" xref="S3.SS3.12.p12.9.m9.1.1.2.2.cmml">E</mi><mi id="S3.SS3.12.p12.9.m9.1.1.2.3" xref="S3.SS3.12.p12.9.m9.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.12.p12.9.m9.1.1.1" xref="S3.SS3.12.p12.9.m9.1.1.1.cmml">∩</mo><msub id="S3.SS3.12.p12.9.m9.1.1.3" xref="S3.SS3.12.p12.9.m9.1.1.3.cmml"><mi id="S3.SS3.12.p12.9.m9.1.1.3.2" xref="S3.SS3.12.p12.9.m9.1.1.3.2.cmml">H</mi><mi id="S3.SS3.12.p12.9.m9.1.1.3.3" xref="S3.SS3.12.p12.9.m9.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.9.m9.1b"><apply id="S3.SS3.12.p12.9.m9.1.1.cmml" xref="S3.SS3.12.p12.9.m9.1.1"><intersect id="S3.SS3.12.p12.9.m9.1.1.1.cmml" xref="S3.SS3.12.p12.9.m9.1.1.1"></intersect><apply id="S3.SS3.12.p12.9.m9.1.1.2.cmml" xref="S3.SS3.12.p12.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.9.m9.1.1.2.1.cmml" xref="S3.SS3.12.p12.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS3.12.p12.9.m9.1.1.2.2.cmml" xref="S3.SS3.12.p12.9.m9.1.1.2.2">𝐸</ci><ci id="S3.SS3.12.p12.9.m9.1.1.2.3.cmml" xref="S3.SS3.12.p12.9.m9.1.1.2.3">𝑖</ci></apply><apply id="S3.SS3.12.p12.9.m9.1.1.3.cmml" xref="S3.SS3.12.p12.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.12.p12.9.m9.1.1.3.1.cmml" xref="S3.SS3.12.p12.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS3.12.p12.9.m9.1.1.3.2.cmml" xref="S3.SS3.12.p12.9.m9.1.1.3.2">𝐻</ci><ci id="S3.SS3.12.p12.9.m9.1.1.3.3.cmml" xref="S3.SS3.12.p12.9.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.9.m9.1c">E_{i}\cap H_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.9.m9.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (in the corresponding definition of <math alttext="T^{i}_{C}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.10.m10.1"><semantics id="S3.SS3.12.p12.10.m10.1a"><msubsup id="S3.SS3.12.p12.10.m10.1.1" xref="S3.SS3.12.p12.10.m10.1.1.cmml"><mi id="S3.SS3.12.p12.10.m10.1.1.2.2" xref="S3.SS3.12.p12.10.m10.1.1.2.2.cmml">T</mi><mi id="S3.SS3.12.p12.10.m10.1.1.3" xref="S3.SS3.12.p12.10.m10.1.1.3.cmml">C</mi><mi id="S3.SS3.12.p12.10.m10.1.1.2.3" xref="S3.SS3.12.p12.10.m10.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.10.m10.1b"><apply id="S3.SS3.12.p12.10.m10.1.1.cmml" xref="S3.SS3.12.p12.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.10.m10.1.1.1.cmml" xref="S3.SS3.12.p12.10.m10.1.1">subscript</csymbol><apply id="S3.SS3.12.p12.10.m10.1.1.2.cmml" xref="S3.SS3.12.p12.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.10.m10.1.1.2.1.cmml" xref="S3.SS3.12.p12.10.m10.1.1">superscript</csymbol><ci id="S3.SS3.12.p12.10.m10.1.1.2.2.cmml" xref="S3.SS3.12.p12.10.m10.1.1.2.2">𝑇</ci><ci id="S3.SS3.12.p12.10.m10.1.1.2.3.cmml" xref="S3.SS3.12.p12.10.m10.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.12.p12.10.m10.1.1.3.cmml" xref="S3.SS3.12.p12.10.m10.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.10.m10.1c">T^{i}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.10.m10.1d">italic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>). Overall, among these sets, the <math alttext="j" class="ltx_Math" display="inline" id="S3.SS3.12.p12.11.m11.1"><semantics id="S3.SS3.12.p12.11.m11.1a"><mi id="S3.SS3.12.p12.11.m11.1.1" xref="S3.SS3.12.p12.11.m11.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.11.m11.1b"><ci id="S3.SS3.12.p12.11.m11.1.1.cmml" xref="S3.SS3.12.p12.11.m11.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.11.m11.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.11.m11.1d">italic_j</annotation></semantics></math>-th stage of the construction needs at most <math alttext="t" class="ltx_Math" display="inline" id="S3.SS3.12.p12.12.m12.1"><semantics id="S3.SS3.12.p12.12.m12.1a"><mi id="S3.SS3.12.p12.12.m12.1.1" xref="S3.SS3.12.p12.12.m12.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.12.m12.1b"><ci id="S3.SS3.12.p12.12.m12.1.1.cmml" xref="S3.SS3.12.p12.12.m12.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.12.m12.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.12.m12.1d">italic_t</annotation></semantics></math> intersections. To see this, note that sets <math alttext="S^{j}_{C}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.13.m13.1"><semantics id="S3.SS3.12.p12.13.m13.1a"><msubsup id="S3.SS3.12.p12.13.m13.1.1" xref="S3.SS3.12.p12.13.m13.1.1.cmml"><mi id="S3.SS3.12.p12.13.m13.1.1.2.2" xref="S3.SS3.12.p12.13.m13.1.1.2.2.cmml">S</mi><mi id="S3.SS3.12.p12.13.m13.1.1.3" xref="S3.SS3.12.p12.13.m13.1.1.3.cmml">C</mi><mi id="S3.SS3.12.p12.13.m13.1.1.2.3" xref="S3.SS3.12.p12.13.m13.1.1.2.3.cmml">j</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.13.m13.1b"><apply id="S3.SS3.12.p12.13.m13.1.1.cmml" xref="S3.SS3.12.p12.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.13.m13.1.1.1.cmml" xref="S3.SS3.12.p12.13.m13.1.1">subscript</csymbol><apply id="S3.SS3.12.p12.13.m13.1.1.2.cmml" xref="S3.SS3.12.p12.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.13.m13.1.1.2.1.cmml" xref="S3.SS3.12.p12.13.m13.1.1">superscript</csymbol><ci id="S3.SS3.12.p12.13.m13.1.1.2.2.cmml" xref="S3.SS3.12.p12.13.m13.1.1.2.2">𝑆</ci><ci id="S3.SS3.12.p12.13.m13.1.1.2.3.cmml" xref="S3.SS3.12.p12.13.m13.1.1.2.3">𝑗</ci></apply><ci id="S3.SS3.12.p12.13.m13.1.1.3.cmml" xref="S3.SS3.12.p12.13.m13.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.13.m13.1c">S^{j}_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.13.m13.1d">italic_S start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="C=E_{i}\cap H_{i}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.14.m14.1"><semantics id="S3.SS3.12.p12.14.m14.1a"><mrow id="S3.SS3.12.p12.14.m14.1.1" xref="S3.SS3.12.p12.14.m14.1.1.cmml"><mi id="S3.SS3.12.p12.14.m14.1.1.2" xref="S3.SS3.12.p12.14.m14.1.1.2.cmml">C</mi><mo id="S3.SS3.12.p12.14.m14.1.1.1" xref="S3.SS3.12.p12.14.m14.1.1.1.cmml">=</mo><mrow id="S3.SS3.12.p12.14.m14.1.1.3" xref="S3.SS3.12.p12.14.m14.1.1.3.cmml"><msub id="S3.SS3.12.p12.14.m14.1.1.3.2" xref="S3.SS3.12.p12.14.m14.1.1.3.2.cmml"><mi id="S3.SS3.12.p12.14.m14.1.1.3.2.2" xref="S3.SS3.12.p12.14.m14.1.1.3.2.2.cmml">E</mi><mi id="S3.SS3.12.p12.14.m14.1.1.3.2.3" xref="S3.SS3.12.p12.14.m14.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.SS3.12.p12.14.m14.1.1.3.1" xref="S3.SS3.12.p12.14.m14.1.1.3.1.cmml">∩</mo><msub id="S3.SS3.12.p12.14.m14.1.1.3.3" xref="S3.SS3.12.p12.14.m14.1.1.3.3.cmml"><mi id="S3.SS3.12.p12.14.m14.1.1.3.3.2" xref="S3.SS3.12.p12.14.m14.1.1.3.3.2.cmml">H</mi><mi id="S3.SS3.12.p12.14.m14.1.1.3.3.3" xref="S3.SS3.12.p12.14.m14.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.14.m14.1b"><apply id="S3.SS3.12.p12.14.m14.1.1.cmml" xref="S3.SS3.12.p12.14.m14.1.1"><eq id="S3.SS3.12.p12.14.m14.1.1.1.cmml" xref="S3.SS3.12.p12.14.m14.1.1.1"></eq><ci id="S3.SS3.12.p12.14.m14.1.1.2.cmml" xref="S3.SS3.12.p12.14.m14.1.1.2">𝐶</ci><apply id="S3.SS3.12.p12.14.m14.1.1.3.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3"><intersect id="S3.SS3.12.p12.14.m14.1.1.3.1.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.1"></intersect><apply id="S3.SS3.12.p12.14.m14.1.1.3.2.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.14.m14.1.1.3.2.1.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.2">subscript</csymbol><ci id="S3.SS3.12.p12.14.m14.1.1.3.2.2.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.2.2">𝐸</ci><ci id="S3.SS3.12.p12.14.m14.1.1.3.2.3.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.2.3">𝑖</ci></apply><apply id="S3.SS3.12.p12.14.m14.1.1.3.3.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.12.p12.14.m14.1.1.3.3.1.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.3">subscript</csymbol><ci id="S3.SS3.12.p12.14.m14.1.1.3.3.2.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.3.2">𝐻</ci><ci id="S3.SS3.12.p12.14.m14.1.1.3.3.3.cmml" xref="S3.SS3.12.p12.14.m14.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.14.m14.1c">C=E_{i}\cap H_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.14.m14.1d">italic_C = italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are only required to inspect the corresponding sets associated with pairs <math alttext="(E_{k},H_{k})" class="ltx_Math" display="inline" id="S3.SS3.12.p12.15.m15.2"><semantics id="S3.SS3.12.p12.15.m15.2a"><mrow id="S3.SS3.12.p12.15.m15.2.2.2" xref="S3.SS3.12.p12.15.m15.2.2.3.cmml"><mo id="S3.SS3.12.p12.15.m15.2.2.2.3" stretchy="false" xref="S3.SS3.12.p12.15.m15.2.2.3.cmml">(</mo><msub id="S3.SS3.12.p12.15.m15.1.1.1.1" xref="S3.SS3.12.p12.15.m15.1.1.1.1.cmml"><mi id="S3.SS3.12.p12.15.m15.1.1.1.1.2" xref="S3.SS3.12.p12.15.m15.1.1.1.1.2.cmml">E</mi><mi id="S3.SS3.12.p12.15.m15.1.1.1.1.3" xref="S3.SS3.12.p12.15.m15.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS3.12.p12.15.m15.2.2.2.4" xref="S3.SS3.12.p12.15.m15.2.2.3.cmml">,</mo><msub id="S3.SS3.12.p12.15.m15.2.2.2.2" xref="S3.SS3.12.p12.15.m15.2.2.2.2.cmml"><mi id="S3.SS3.12.p12.15.m15.2.2.2.2.2" xref="S3.SS3.12.p12.15.m15.2.2.2.2.2.cmml">H</mi><mi id="S3.SS3.12.p12.15.m15.2.2.2.2.3" xref="S3.SS3.12.p12.15.m15.2.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS3.12.p12.15.m15.2.2.2.5" stretchy="false" xref="S3.SS3.12.p12.15.m15.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.15.m15.2b"><interval closure="open" id="S3.SS3.12.p12.15.m15.2.2.3.cmml" xref="S3.SS3.12.p12.15.m15.2.2.2"><apply id="S3.SS3.12.p12.15.m15.1.1.1.1.cmml" xref="S3.SS3.12.p12.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.12.p12.15.m15.1.1.1.1.1.cmml" xref="S3.SS3.12.p12.15.m15.1.1.1.1">subscript</csymbol><ci id="S3.SS3.12.p12.15.m15.1.1.1.1.2.cmml" xref="S3.SS3.12.p12.15.m15.1.1.1.1.2">𝐸</ci><ci id="S3.SS3.12.p12.15.m15.1.1.1.1.3.cmml" xref="S3.SS3.12.p12.15.m15.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS3.12.p12.15.m15.2.2.2.2.cmml" xref="S3.SS3.12.p12.15.m15.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.15.m15.2.2.2.2.1.cmml" xref="S3.SS3.12.p12.15.m15.2.2.2.2">subscript</csymbol><ci id="S3.SS3.12.p12.15.m15.2.2.2.2.2.cmml" xref="S3.SS3.12.p12.15.m15.2.2.2.2.2">𝐻</ci><ci id="S3.SS3.12.p12.15.m15.2.2.2.2.3.cmml" xref="S3.SS3.12.p12.15.m15.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.15.m15.2c">(E_{k},H_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.15.m15.2d">( italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> with <math alttext="k\in[t]" class="ltx_Math" display="inline" id="S3.SS3.12.p12.16.m16.1"><semantics id="S3.SS3.12.p12.16.m16.1a"><mrow id="S3.SS3.12.p12.16.m16.1.2" xref="S3.SS3.12.p12.16.m16.1.2.cmml"><mi id="S3.SS3.12.p12.16.m16.1.2.2" xref="S3.SS3.12.p12.16.m16.1.2.2.cmml">k</mi><mo id="S3.SS3.12.p12.16.m16.1.2.1" xref="S3.SS3.12.p12.16.m16.1.2.1.cmml">∈</mo><mrow id="S3.SS3.12.p12.16.m16.1.2.3.2" xref="S3.SS3.12.p12.16.m16.1.2.3.1.cmml"><mo id="S3.SS3.12.p12.16.m16.1.2.3.2.1" stretchy="false" xref="S3.SS3.12.p12.16.m16.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.12.p12.16.m16.1.1" xref="S3.SS3.12.p12.16.m16.1.1.cmml">t</mi><mo id="S3.SS3.12.p12.16.m16.1.2.3.2.2" stretchy="false" xref="S3.SS3.12.p12.16.m16.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.16.m16.1b"><apply id="S3.SS3.12.p12.16.m16.1.2.cmml" xref="S3.SS3.12.p12.16.m16.1.2"><in id="S3.SS3.12.p12.16.m16.1.2.1.cmml" xref="S3.SS3.12.p12.16.m16.1.2.1"></in><ci id="S3.SS3.12.p12.16.m16.1.2.2.cmml" xref="S3.SS3.12.p12.16.m16.1.2.2">𝑘</ci><apply id="S3.SS3.12.p12.16.m16.1.2.3.1.cmml" xref="S3.SS3.12.p12.16.m16.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.12.p12.16.m16.1.2.3.1.1.cmml" xref="S3.SS3.12.p12.16.m16.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.12.p12.16.m16.1.1.cmml" xref="S3.SS3.12.p12.16.m16.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.16.m16.1c">k\in[t]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.16.m16.1d">italic_k ∈ [ italic_t ]</annotation></semantics></math> such that <math alttext="C=E_{k}\cap H_{k}" class="ltx_Math" display="inline" id="S3.SS3.12.p12.17.m17.1"><semantics id="S3.SS3.12.p12.17.m17.1a"><mrow id="S3.SS3.12.p12.17.m17.1.1" xref="S3.SS3.12.p12.17.m17.1.1.cmml"><mi id="S3.SS3.12.p12.17.m17.1.1.2" xref="S3.SS3.12.p12.17.m17.1.1.2.cmml">C</mi><mo id="S3.SS3.12.p12.17.m17.1.1.1" xref="S3.SS3.12.p12.17.m17.1.1.1.cmml">=</mo><mrow id="S3.SS3.12.p12.17.m17.1.1.3" xref="S3.SS3.12.p12.17.m17.1.1.3.cmml"><msub id="S3.SS3.12.p12.17.m17.1.1.3.2" xref="S3.SS3.12.p12.17.m17.1.1.3.2.cmml"><mi id="S3.SS3.12.p12.17.m17.1.1.3.2.2" xref="S3.SS3.12.p12.17.m17.1.1.3.2.2.cmml">E</mi><mi id="S3.SS3.12.p12.17.m17.1.1.3.2.3" xref="S3.SS3.12.p12.17.m17.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS3.12.p12.17.m17.1.1.3.1" xref="S3.SS3.12.p12.17.m17.1.1.3.1.cmml">∩</mo><msub id="S3.SS3.12.p12.17.m17.1.1.3.3" xref="S3.SS3.12.p12.17.m17.1.1.3.3.cmml"><mi id="S3.SS3.12.p12.17.m17.1.1.3.3.2" xref="S3.SS3.12.p12.17.m17.1.1.3.3.2.cmml">H</mi><mi id="S3.SS3.12.p12.17.m17.1.1.3.3.3" xref="S3.SS3.12.p12.17.m17.1.1.3.3.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.17.m17.1b"><apply id="S3.SS3.12.p12.17.m17.1.1.cmml" xref="S3.SS3.12.p12.17.m17.1.1"><eq id="S3.SS3.12.p12.17.m17.1.1.1.cmml" xref="S3.SS3.12.p12.17.m17.1.1.1"></eq><ci id="S3.SS3.12.p12.17.m17.1.1.2.cmml" xref="S3.SS3.12.p12.17.m17.1.1.2">𝐶</ci><apply id="S3.SS3.12.p12.17.m17.1.1.3.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3"><intersect id="S3.SS3.12.p12.17.m17.1.1.3.1.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.1"></intersect><apply id="S3.SS3.12.p12.17.m17.1.1.3.2.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.12.p12.17.m17.1.1.3.2.1.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.2">subscript</csymbol><ci id="S3.SS3.12.p12.17.m17.1.1.3.2.2.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.2.2">𝐸</ci><ci id="S3.SS3.12.p12.17.m17.1.1.3.2.3.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS3.12.p12.17.m17.1.1.3.3.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.12.p12.17.m17.1.1.3.3.1.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.3">subscript</csymbol><ci id="S3.SS3.12.p12.17.m17.1.1.3.3.2.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.3.2">𝐻</ci><ci id="S3.SS3.12.p12.17.m17.1.1.3.3.3.cmml" xref="S3.SS3.12.p12.17.m17.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.17.m17.1c">C=E_{k}\cap H_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.17.m17.1d">italic_C = italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, and such pairs are disjoint among the different sets <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.12.p12.18.m18.1"><semantics id="S3.SS3.12.p12.18.m18.1a"><mi id="S3.SS3.12.p12.18.m18.1.1" xref="S3.SS3.12.p12.18.m18.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.18.m18.1b"><ci id="S3.SS3.12.p12.18.m18.1.1.cmml" xref="S3.SS3.12.p12.18.m18.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.18.m18.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.18.m18.1d">italic_C</annotation></semantics></math> of this form. (There is no need to keep more than one such <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.12.p12.19.m19.1"><semantics id="S3.SS3.12.p12.19.m19.1a"><mi id="S3.SS3.12.p12.19.m19.1.1" xref="S3.SS3.12.p12.19.m19.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.12.p12.19.m19.1b"><ci id="S3.SS3.12.p12.19.m19.1.1.cmml" xref="S3.SS3.12.p12.19.m19.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.12.p12.19.m19.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.12.p12.19.m19.1d">italic_C</annotation></semantics></math> representing the same underlying set as a syntactical object in the construction.)</p> </div> <div class="ltx_para" id="S3.SS3.13.p13"> <p class="ltx_p" id="S3.SS3.13.p13.4">This immediately implies that <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.13.p13.1.m1.1"><semantics id="S3.SS3.13.p13.1.m1.1a"><mi id="S3.SS3.13.p13.1.m1.1.1" xref="S3.SS3.13.p13.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.13.p13.1.m1.1b"><ci id="S3.SS3.13.p13.1.m1.1.1.cmml" xref="S3.SS3.13.p13.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.13.p13.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.13.p13.1.m1.1d">italic_A</annotation></semantics></math> can be generated using at most <math alttext="t(t+1)" class="ltx_Math" display="inline" id="S3.SS3.13.p13.2.m2.1"><semantics id="S3.SS3.13.p13.2.m2.1a"><mrow id="S3.SS3.13.p13.2.m2.1.1" xref="S3.SS3.13.p13.2.m2.1.1.cmml"><mi id="S3.SS3.13.p13.2.m2.1.1.3" xref="S3.SS3.13.p13.2.m2.1.1.3.cmml">t</mi><mo id="S3.SS3.13.p13.2.m2.1.1.2" xref="S3.SS3.13.p13.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.13.p13.2.m2.1.1.1.1" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS3.13.p13.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.13.p13.2.m2.1.1.1.1.1" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.13.p13.2.m2.1.1.1.1.1.2" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S3.SS3.13.p13.2.m2.1.1.1.1.1.1" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.1.cmml">+</mo><mn id="S3.SS3.13.p13.2.m2.1.1.1.1.1.3" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS3.13.p13.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.13.p13.2.m2.1b"><apply id="S3.SS3.13.p13.2.m2.1.1.cmml" xref="S3.SS3.13.p13.2.m2.1.1"><times id="S3.SS3.13.p13.2.m2.1.1.2.cmml" xref="S3.SS3.13.p13.2.m2.1.1.2"></times><ci id="S3.SS3.13.p13.2.m2.1.1.3.cmml" xref="S3.SS3.13.p13.2.m2.1.1.3">𝑡</ci><apply id="S3.SS3.13.p13.2.m2.1.1.1.1.1.cmml" xref="S3.SS3.13.p13.2.m2.1.1.1.1"><plus id="S3.SS3.13.p13.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.1"></plus><ci id="S3.SS3.13.p13.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.2">𝑡</ci><cn id="S3.SS3.13.p13.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS3.13.p13.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.13.p13.2.m2.1c">t(t+1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.13.p13.2.m2.1d">italic_t ( italic_t + 1 )</annotation></semantics></math> intersections. However, since intersections are only added in steps <math alttext="j\in\left\{2,\dots,t+1\right\}" class="ltx_Math" display="inline" id="S3.SS3.13.p13.3.m3.3"><semantics id="S3.SS3.13.p13.3.m3.3a"><mrow id="S3.SS3.13.p13.3.m3.3.3" xref="S3.SS3.13.p13.3.m3.3.3.cmml"><mi id="S3.SS3.13.p13.3.m3.3.3.3" xref="S3.SS3.13.p13.3.m3.3.3.3.cmml">j</mi><mo id="S3.SS3.13.p13.3.m3.3.3.2" xref="S3.SS3.13.p13.3.m3.3.3.2.cmml">∈</mo><mrow id="S3.SS3.13.p13.3.m3.3.3.1.1" xref="S3.SS3.13.p13.3.m3.3.3.1.2.cmml"><mo id="S3.SS3.13.p13.3.m3.3.3.1.1.2" xref="S3.SS3.13.p13.3.m3.3.3.1.2.cmml">{</mo><mn id="S3.SS3.13.p13.3.m3.1.1" xref="S3.SS3.13.p13.3.m3.1.1.cmml">2</mn><mo id="S3.SS3.13.p13.3.m3.3.3.1.1.3" xref="S3.SS3.13.p13.3.m3.3.3.1.2.cmml">,</mo><mi id="S3.SS3.13.p13.3.m3.2.2" mathvariant="normal" xref="S3.SS3.13.p13.3.m3.2.2.cmml">…</mi><mo id="S3.SS3.13.p13.3.m3.3.3.1.1.4" xref="S3.SS3.13.p13.3.m3.3.3.1.2.cmml">,</mo><mrow id="S3.SS3.13.p13.3.m3.3.3.1.1.1" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.cmml"><mi id="S3.SS3.13.p13.3.m3.3.3.1.1.1.2" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.2.cmml">t</mi><mo id="S3.SS3.13.p13.3.m3.3.3.1.1.1.1" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.1.cmml">+</mo><mn id="S3.SS3.13.p13.3.m3.3.3.1.1.1.3" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS3.13.p13.3.m3.3.3.1.1.5" xref="S3.SS3.13.p13.3.m3.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.13.p13.3.m3.3b"><apply id="S3.SS3.13.p13.3.m3.3.3.cmml" xref="S3.SS3.13.p13.3.m3.3.3"><in id="S3.SS3.13.p13.3.m3.3.3.2.cmml" xref="S3.SS3.13.p13.3.m3.3.3.2"></in><ci id="S3.SS3.13.p13.3.m3.3.3.3.cmml" xref="S3.SS3.13.p13.3.m3.3.3.3">𝑗</ci><set id="S3.SS3.13.p13.3.m3.3.3.1.2.cmml" xref="S3.SS3.13.p13.3.m3.3.3.1.1"><cn id="S3.SS3.13.p13.3.m3.1.1.cmml" type="integer" xref="S3.SS3.13.p13.3.m3.1.1">2</cn><ci id="S3.SS3.13.p13.3.m3.2.2.cmml" xref="S3.SS3.13.p13.3.m3.2.2">…</ci><apply id="S3.SS3.13.p13.3.m3.3.3.1.1.1.cmml" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1"><plus id="S3.SS3.13.p13.3.m3.3.3.1.1.1.1.cmml" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.1"></plus><ci id="S3.SS3.13.p13.3.m3.3.3.1.1.1.2.cmml" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.2">𝑡</ci><cn id="S3.SS3.13.p13.3.m3.3.3.1.1.1.3.cmml" type="integer" xref="S3.SS3.13.p13.3.m3.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.13.p13.3.m3.3c">j\in\left\{2,\dots,t+1\right\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.13.p13.3.m3.3d">italic_j ∈ { 2 , … , italic_t + 1 }</annotation></semantics></math>, we obtain <math alttext="D_{\cap}(A\mid\mathcal{B})\leq\rho(A,\mathcal{B})^{2}" class="ltx_Math" display="inline" id="S3.SS3.13.p13.4.m4.3"><semantics id="S3.SS3.13.p13.4.m4.3a"><mrow id="S3.SS3.13.p13.4.m4.3.3" xref="S3.SS3.13.p13.4.m4.3.3.cmml"><mrow id="S3.SS3.13.p13.4.m4.3.3.1" xref="S3.SS3.13.p13.4.m4.3.3.1.cmml"><msub id="S3.SS3.13.p13.4.m4.3.3.1.3" xref="S3.SS3.13.p13.4.m4.3.3.1.3.cmml"><mi id="S3.SS3.13.p13.4.m4.3.3.1.3.2" xref="S3.SS3.13.p13.4.m4.3.3.1.3.2.cmml">D</mi><mo id="S3.SS3.13.p13.4.m4.3.3.1.3.3" xref="S3.SS3.13.p13.4.m4.3.3.1.3.3.cmml">∩</mo></msub><mo id="S3.SS3.13.p13.4.m4.3.3.1.2" xref="S3.SS3.13.p13.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS3.13.p13.4.m4.3.3.1.1.1" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.cmml"><mo id="S3.SS3.13.p13.4.m4.3.3.1.1.1.2" stretchy="false" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.cmml"><mi id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.2" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.2.cmml">A</mi><mo id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.1" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.3" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS3.13.p13.4.m4.3.3.1.1.1.3" stretchy="false" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.13.p13.4.m4.3.3.2" xref="S3.SS3.13.p13.4.m4.3.3.2.cmml">≤</mo><mrow id="S3.SS3.13.p13.4.m4.3.3.3" xref="S3.SS3.13.p13.4.m4.3.3.3.cmml"><mi id="S3.SS3.13.p13.4.m4.3.3.3.2" xref="S3.SS3.13.p13.4.m4.3.3.3.2.cmml">ρ</mi><mo id="S3.SS3.13.p13.4.m4.3.3.3.1" xref="S3.SS3.13.p13.4.m4.3.3.3.1.cmml">⁢</mo><msup id="S3.SS3.13.p13.4.m4.3.3.3.3" xref="S3.SS3.13.p13.4.m4.3.3.3.3.cmml"><mrow id="S3.SS3.13.p13.4.m4.3.3.3.3.2.2" xref="S3.SS3.13.p13.4.m4.3.3.3.3.2.1.cmml"><mo id="S3.SS3.13.p13.4.m4.3.3.3.3.2.2.1" stretchy="false" xref="S3.SS3.13.p13.4.m4.3.3.3.3.2.1.cmml">(</mo><mi id="S3.SS3.13.p13.4.m4.1.1" xref="S3.SS3.13.p13.4.m4.1.1.cmml">A</mi><mo id="S3.SS3.13.p13.4.m4.3.3.3.3.2.2.2" xref="S3.SS3.13.p13.4.m4.3.3.3.3.2.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.13.p13.4.m4.2.2" xref="S3.SS3.13.p13.4.m4.2.2.cmml">ℬ</mi><mo id="S3.SS3.13.p13.4.m4.3.3.3.3.2.2.3" stretchy="false" xref="S3.SS3.13.p13.4.m4.3.3.3.3.2.1.cmml">)</mo></mrow><mn id="S3.SS3.13.p13.4.m4.3.3.3.3.3" xref="S3.SS3.13.p13.4.m4.3.3.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.13.p13.4.m4.3b"><apply id="S3.SS3.13.p13.4.m4.3.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3"><leq id="S3.SS3.13.p13.4.m4.3.3.2.cmml" xref="S3.SS3.13.p13.4.m4.3.3.2"></leq><apply id="S3.SS3.13.p13.4.m4.3.3.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1"><times id="S3.SS3.13.p13.4.m4.3.3.1.2.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.2"></times><apply id="S3.SS3.13.p13.4.m4.3.3.1.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS3.13.p13.4.m4.3.3.1.3.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.3">subscript</csymbol><ci id="S3.SS3.13.p13.4.m4.3.3.1.3.2.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.3.2">𝐷</ci><intersect id="S3.SS3.13.p13.4.m4.3.3.1.3.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.3.3"></intersect></apply><apply id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1"><csymbol cd="latexml" id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.1">conditional</csymbol><ci id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.2.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.2">𝐴</ci><ci id="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3.1.1.1.1.3">ℬ</ci></apply></apply><apply id="S3.SS3.13.p13.4.m4.3.3.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3"><times id="S3.SS3.13.p13.4.m4.3.3.3.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3.1"></times><ci id="S3.SS3.13.p13.4.m4.3.3.3.2.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3.2">𝜌</ci><apply id="S3.SS3.13.p13.4.m4.3.3.3.3.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS3.13.p13.4.m4.3.3.3.3.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3.3">superscript</csymbol><interval closure="open" id="S3.SS3.13.p13.4.m4.3.3.3.3.2.1.cmml" xref="S3.SS3.13.p13.4.m4.3.3.3.3.2.2"><ci id="S3.SS3.13.p13.4.m4.1.1.cmml" xref="S3.SS3.13.p13.4.m4.1.1">𝐴</ci><ci id="S3.SS3.13.p13.4.m4.2.2.cmml" xref="S3.SS3.13.p13.4.m4.2.2">ℬ</ci></interval><cn id="S3.SS3.13.p13.4.m4.3.3.3.3.3.cmml" type="integer" xref="S3.SS3.13.p13.4.m4.3.3.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.13.p13.4.m4.3c">D_{\cap}(A\mid\mathcal{B})\leq\rho(A,\mathcal{B})^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.13.p13.4.m4.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_ρ ( italic_A , caligraphic_B ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, which completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">We take this opportunity to observe the following immediate consequence of Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a>. (A tighter relation between these measures is discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS3" title="2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.3</span></a>.)</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmtheorem28"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem28.1.1.1">Corollary 28</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem28.2.2"> </span>(Intersection complexity versus discrete complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem28.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem28.p1"> <p class="ltx_p" id="Thmtheorem28.p1.4"><span class="ltx_text ltx_font_italic" id="Thmtheorem28.p1.4.4">For every <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem28.p1.1.1.m1.1"><semantics id="Thmtheorem28.p1.1.1.m1.1a"><mrow id="Thmtheorem28.p1.1.1.m1.1.1" xref="Thmtheorem28.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem28.p1.1.1.m1.1.1.2" xref="Thmtheorem28.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem28.p1.1.1.m1.1.1.1" xref="Thmtheorem28.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem28.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem28.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem28.p1.1.1.m1.1b"><apply id="Thmtheorem28.p1.1.1.m1.1.1.cmml" xref="Thmtheorem28.p1.1.1.m1.1.1"><subset id="Thmtheorem28.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem28.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem28.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem28.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem28.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem28.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem28.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem28.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> and non-empty <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="Thmtheorem28.p1.2.2.m2.1"><semantics id="Thmtheorem28.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem28.p1.2.2.m2.1.1" xref="Thmtheorem28.p1.2.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem28.p1.2.2.m2.1b"><ci id="Thmtheorem28.p1.2.2.m2.1.1.cmml" xref="Thmtheorem28.p1.2.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem28.p1.2.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem28.p1.2.2.m2.1d">caligraphic_B</annotation></semantics></math>, if <math alttext="D_{\cap}(A\mid\mathcal{B})=t" class="ltx_Math" display="inline" id="Thmtheorem28.p1.3.3.m3.1"><semantics id="Thmtheorem28.p1.3.3.m3.1a"><mrow id="Thmtheorem28.p1.3.3.m3.1.1" xref="Thmtheorem28.p1.3.3.m3.1.1.cmml"><mrow id="Thmtheorem28.p1.3.3.m3.1.1.1" xref="Thmtheorem28.p1.3.3.m3.1.1.1.cmml"><msub id="Thmtheorem28.p1.3.3.m3.1.1.1.3" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3.cmml"><mi id="Thmtheorem28.p1.3.3.m3.1.1.1.3.2" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3.2.cmml">D</mi><mo id="Thmtheorem28.p1.3.3.m3.1.1.1.3.3" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3.3.cmml">∩</mo></msub><mo id="Thmtheorem28.p1.3.3.m3.1.1.1.2" xref="Thmtheorem28.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.2" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.1" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.3" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem28.p1.3.3.m3.1.1.2" xref="Thmtheorem28.p1.3.3.m3.1.1.2.cmml">=</mo><mi id="Thmtheorem28.p1.3.3.m3.1.1.3" xref="Thmtheorem28.p1.3.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem28.p1.3.3.m3.1b"><apply id="Thmtheorem28.p1.3.3.m3.1.1.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1"><eq id="Thmtheorem28.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.2"></eq><apply id="Thmtheorem28.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1"><times id="Thmtheorem28.p1.3.3.m3.1.1.1.2.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.2"></times><apply id="Thmtheorem28.p1.3.3.m3.1.1.1.3.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem28.p1.3.3.m3.1.1.1.3.1.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3">subscript</csymbol><ci id="Thmtheorem28.p1.3.3.m3.1.1.1.3.2.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3.2">𝐷</ci><intersect id="Thmtheorem28.p1.3.3.m3.1.1.1.3.3.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.3.3"></intersect></apply><apply id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.1.1.1.1.3">ℬ</ci></apply></apply><ci id="Thmtheorem28.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem28.p1.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem28.p1.3.3.m3.1c">D_{\cap}(A\mid\mathcal{B})=t</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem28.p1.3.3.m3.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) = italic_t</annotation></semantics></math> then <math alttext="D_{\cup}(A\mid\mathcal{B})\leq D(A\mid\mathcal{B})\leq O(t+|\mathcal{B}|)^{3}" class="ltx_Math" display="inline" id="Thmtheorem28.p1.4.4.m4.4"><semantics id="Thmtheorem28.p1.4.4.m4.4a"><mrow id="Thmtheorem28.p1.4.4.m4.4.4" xref="Thmtheorem28.p1.4.4.m4.4.4.cmml"><mrow id="Thmtheorem28.p1.4.4.m4.2.2.1" xref="Thmtheorem28.p1.4.4.m4.2.2.1.cmml"><msub id="Thmtheorem28.p1.4.4.m4.2.2.1.3" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3.cmml"><mi id="Thmtheorem28.p1.4.4.m4.2.2.1.3.2" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3.2.cmml">D</mi><mo id="Thmtheorem28.p1.4.4.m4.2.2.1.3.3" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3.3.cmml">∪</mo></msub><mo id="Thmtheorem28.p1.4.4.m4.2.2.1.2" xref="Thmtheorem28.p1.4.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.cmml"><mo id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.2" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.cmml"><mi id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.2" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.2.cmml">A</mi><mo id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.1" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.3" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.3" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem28.p1.4.4.m4.4.4.5" xref="Thmtheorem28.p1.4.4.m4.4.4.5.cmml">≤</mo><mrow id="Thmtheorem28.p1.4.4.m4.3.3.2" xref="Thmtheorem28.p1.4.4.m4.3.3.2.cmml"><mi id="Thmtheorem28.p1.4.4.m4.3.3.2.3" xref="Thmtheorem28.p1.4.4.m4.3.3.2.3.cmml">D</mi><mo id="Thmtheorem28.p1.4.4.m4.3.3.2.2" xref="Thmtheorem28.p1.4.4.m4.3.3.2.2.cmml">⁢</mo><mrow id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.cmml"><mo id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.2" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.cmml">(</mo><mrow id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.cmml"><mi id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.2" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.2.cmml">A</mi><mo id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.1" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.3" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.3.cmml">ℬ</mi></mrow><mo id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.3" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem28.p1.4.4.m4.4.4.6" xref="Thmtheorem28.p1.4.4.m4.4.4.6.cmml">≤</mo><mrow id="Thmtheorem28.p1.4.4.m4.4.4.3" xref="Thmtheorem28.p1.4.4.m4.4.4.3.cmml"><mi id="Thmtheorem28.p1.4.4.m4.4.4.3.3" xref="Thmtheorem28.p1.4.4.m4.4.4.3.3.cmml">O</mi><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.2" xref="Thmtheorem28.p1.4.4.m4.4.4.3.2.cmml">⁢</mo><msup id="Thmtheorem28.p1.4.4.m4.4.4.3.1" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.cmml"><mrow id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.cmml"><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.2" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.cmml"><mi id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.2" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.2.cmml">t</mi><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.1" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.1.cmml">+</mo><mrow id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.2" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.1.cmml"><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.2.1" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem28.p1.4.4.m4.1.1" xref="Thmtheorem28.p1.4.4.m4.1.1.cmml">ℬ</mi><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.2.2" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.3" stretchy="false" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.cmml">)</mo></mrow><mn id="Thmtheorem28.p1.4.4.m4.4.4.3.1.3" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.3.cmml">3</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem28.p1.4.4.m4.4b"><apply id="Thmtheorem28.p1.4.4.m4.4.4.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4"><and id="Thmtheorem28.p1.4.4.m4.4.4a.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4"></and><apply id="Thmtheorem28.p1.4.4.m4.4.4b.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4"><leq id="Thmtheorem28.p1.4.4.m4.4.4.5.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.5"></leq><apply id="Thmtheorem28.p1.4.4.m4.2.2.1.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1"><times id="Thmtheorem28.p1.4.4.m4.2.2.1.2.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.2"></times><apply id="Thmtheorem28.p1.4.4.m4.2.2.1.3.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="Thmtheorem28.p1.4.4.m4.2.2.1.3.1.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3">subscript</csymbol><ci id="Thmtheorem28.p1.4.4.m4.2.2.1.3.2.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3.2">𝐷</ci><union id="Thmtheorem28.p1.4.4.m4.2.2.1.3.3.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.3.3"></union></apply><apply id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1"><csymbol cd="latexml" id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.1">conditional</csymbol><ci id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.2.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.2">𝐴</ci><ci id="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.3.cmml" xref="Thmtheorem28.p1.4.4.m4.2.2.1.1.1.1.3">ℬ</ci></apply></apply><apply id="Thmtheorem28.p1.4.4.m4.3.3.2.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2"><times id="Thmtheorem28.p1.4.4.m4.3.3.2.2.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.2"></times><ci id="Thmtheorem28.p1.4.4.m4.3.3.2.3.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.3">𝐷</ci><apply id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1"><csymbol cd="latexml" id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.1">conditional</csymbol><ci id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.2.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.2">𝐴</ci><ci id="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.3.cmml" xref="Thmtheorem28.p1.4.4.m4.3.3.2.1.1.1.3">ℬ</ci></apply></apply></apply><apply id="Thmtheorem28.p1.4.4.m4.4.4c.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4"><leq id="Thmtheorem28.p1.4.4.m4.4.4.6.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.6"></leq><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem28.p1.4.4.m4.3.3.2.cmml" id="Thmtheorem28.p1.4.4.m4.4.4d.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4"></share><apply id="Thmtheorem28.p1.4.4.m4.4.4.3.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3"><times id="Thmtheorem28.p1.4.4.m4.4.4.3.2.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.2"></times><ci id="Thmtheorem28.p1.4.4.m4.4.4.3.3.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.3">𝑂</ci><apply id="Thmtheorem28.p1.4.4.m4.4.4.3.1.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1"><csymbol cd="ambiguous" id="Thmtheorem28.p1.4.4.m4.4.4.3.1.2.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1">superscript</csymbol><apply id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1"><plus id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.1"></plus><ci id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.2.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.2">𝑡</ci><apply id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.1.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.2"><abs id="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.1.1.1.3.2.1"></abs><ci id="Thmtheorem28.p1.4.4.m4.1.1.cmml" xref="Thmtheorem28.p1.4.4.m4.1.1">ℬ</ci></apply></apply><cn id="Thmtheorem28.p1.4.4.m4.4.4.3.1.3.cmml" type="integer" xref="Thmtheorem28.p1.4.4.m4.4.4.3.1.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem28.p1.4.4.m4.4c">D_{\cup}(A\mid\mathcal{B})\leq D(A\mid\mathcal{B})\leq O(t+|\mathcal{B}|)^{3}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem28.p1.4.4.m4.4d">italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_D ( italic_A ∣ caligraphic_B ) ≤ italic_O ( italic_t + | caligraphic_B | ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.14"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.14.p1"> <p class="ltx_p" id="S3.SS3.14.p1.7">If <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.14.p1.1.m1.1"><semantics id="S3.SS3.14.p1.1.m1.1a"><mi id="S3.SS3.14.p1.1.m1.1.1" xref="S3.SS3.14.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.1.m1.1b"><ci id="S3.SS3.14.p1.1.m1.1.1.cmml" xref="S3.SS3.14.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.1.m1.1d">italic_A</annotation></semantics></math> is empty and can be constructed from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.14.p1.2.m2.1"><semantics id="S3.SS3.14.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.14.p1.2.m2.1.1" xref="S3.SS3.14.p1.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.2.m2.1b"><ci id="S3.SS3.14.p1.2.m2.1.1.cmml" xref="S3.SS3.14.p1.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.2.m2.1d">caligraphic_B</annotation></semantics></math>, then it can also be constructed from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS3.14.p1.3.m3.1"><semantics id="S3.SS3.14.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.14.p1.3.m3.1.1" xref="S3.SS3.14.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.3.m3.1b"><ci id="S3.SS3.14.p1.3.m3.1.1.cmml" xref="S3.SS3.14.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.3.m3.1d">caligraphic_B</annotation></semantics></math> using <math alttext="|\mathcal{B}|" class="ltx_Math" display="inline" id="S3.SS3.14.p1.4.m4.1"><semantics id="S3.SS3.14.p1.4.m4.1a"><mrow id="S3.SS3.14.p1.4.m4.1.2.2" xref="S3.SS3.14.p1.4.m4.1.2.1.cmml"><mo id="S3.SS3.14.p1.4.m4.1.2.2.1" stretchy="false" xref="S3.SS3.14.p1.4.m4.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.14.p1.4.m4.1.1" xref="S3.SS3.14.p1.4.m4.1.1.cmml">ℬ</mi><mo id="S3.SS3.14.p1.4.m4.1.2.2.2" stretchy="false" xref="S3.SS3.14.p1.4.m4.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.4.m4.1b"><apply id="S3.SS3.14.p1.4.m4.1.2.1.cmml" xref="S3.SS3.14.p1.4.m4.1.2.2"><abs id="S3.SS3.14.p1.4.m4.1.2.1.1.cmml" xref="S3.SS3.14.p1.4.m4.1.2.2.1"></abs><ci id="S3.SS3.14.p1.4.m4.1.1.cmml" xref="S3.SS3.14.p1.4.m4.1.1">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.4.m4.1c">|\mathcal{B}|</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.4.m4.1d">| caligraphic_B |</annotation></semantics></math> intersections (and no union operation). If <math alttext="A=\Gamma" class="ltx_Math" display="inline" id="S3.SS3.14.p1.5.m5.1"><semantics id="S3.SS3.14.p1.5.m5.1a"><mrow id="S3.SS3.14.p1.5.m5.1.1" xref="S3.SS3.14.p1.5.m5.1.1.cmml"><mi id="S3.SS3.14.p1.5.m5.1.1.2" xref="S3.SS3.14.p1.5.m5.1.1.2.cmml">A</mi><mo id="S3.SS3.14.p1.5.m5.1.1.1" xref="S3.SS3.14.p1.5.m5.1.1.1.cmml">=</mo><mi id="S3.SS3.14.p1.5.m5.1.1.3" mathvariant="normal" xref="S3.SS3.14.p1.5.m5.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.5.m5.1b"><apply id="S3.SS3.14.p1.5.m5.1.1.cmml" xref="S3.SS3.14.p1.5.m5.1.1"><eq id="S3.SS3.14.p1.5.m5.1.1.1.cmml" xref="S3.SS3.14.p1.5.m5.1.1.1"></eq><ci id="S3.SS3.14.p1.5.m5.1.1.2.cmml" xref="S3.SS3.14.p1.5.m5.1.1.2">𝐴</ci><ci id="S3.SS3.14.p1.5.m5.1.1.3.cmml" xref="S3.SS3.14.p1.5.m5.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.5.m5.1c">A=\Gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.5.m5.1d">italic_A = roman_Γ</annotation></semantics></math> the same is true with respect to unions. On the other hand, for a non-trivial <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.14.p1.6.m6.1"><semantics id="S3.SS3.14.p1.6.m6.1a"><mi id="S3.SS3.14.p1.6.m6.1.1" xref="S3.SS3.14.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.6.m6.1b"><ci id="S3.SS3.14.p1.6.m6.1.1.cmml" xref="S3.SS3.14.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.6.m6.1d">italic_A</annotation></semantics></math>, the result follows from Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a>, by noticing that in the construction underlying the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a> a total of at most <math alttext="O(t+|\mathcal{B}|)^{3}" class="ltx_Math" display="inline" id="S3.SS3.14.p1.7.m7.2"><semantics id="S3.SS3.14.p1.7.m7.2a"><mrow id="S3.SS3.14.p1.7.m7.2.2" xref="S3.SS3.14.p1.7.m7.2.2.cmml"><mi id="S3.SS3.14.p1.7.m7.2.2.3" xref="S3.SS3.14.p1.7.m7.2.2.3.cmml">O</mi><mo id="S3.SS3.14.p1.7.m7.2.2.2" xref="S3.SS3.14.p1.7.m7.2.2.2.cmml">⁢</mo><msup id="S3.SS3.14.p1.7.m7.2.2.1" xref="S3.SS3.14.p1.7.m7.2.2.1.cmml"><mrow id="S3.SS3.14.p1.7.m7.2.2.1.1.1" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S3.SS3.14.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.2" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.2.cmml">t</mi><mo id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.1" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.1.cmml">+</mo><mrow id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.2" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.1.cmml"><mo id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.14.p1.7.m7.1.1" xref="S3.SS3.14.p1.7.m7.1.1.cmml">ℬ</mi><mo id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="S3.SS3.14.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow><mn id="S3.SS3.14.p1.7.m7.2.2.1.3" xref="S3.SS3.14.p1.7.m7.2.2.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.14.p1.7.m7.2b"><apply id="S3.SS3.14.p1.7.m7.2.2.cmml" xref="S3.SS3.14.p1.7.m7.2.2"><times id="S3.SS3.14.p1.7.m7.2.2.2.cmml" xref="S3.SS3.14.p1.7.m7.2.2.2"></times><ci id="S3.SS3.14.p1.7.m7.2.2.3.cmml" xref="S3.SS3.14.p1.7.m7.2.2.3">𝑂</ci><apply id="S3.SS3.14.p1.7.m7.2.2.1.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1"><csymbol cd="ambiguous" id="S3.SS3.14.p1.7.m7.2.2.1.2.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1">superscript</csymbol><apply id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1"><plus id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.1"></plus><ci id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.2">𝑡</ci><apply id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.1.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.2"><abs id="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.1.1.cmml" xref="S3.SS3.14.p1.7.m7.2.2.1.1.1.1.3.2.1"></abs><ci id="S3.SS3.14.p1.7.m7.1.1.cmml" xref="S3.SS3.14.p1.7.m7.1.1">ℬ</ci></apply></apply><cn id="S3.SS3.14.p1.7.m7.2.2.1.3.cmml" type="integer" xref="S3.SS3.14.p1.7.m7.2.2.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.14.p1.7.m7.2c">O(t+|\mathcal{B}|)^{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.14.p1.7.m7.2d">italic_O ( italic_t + | caligraphic_B | ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> operations are needed. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="Thmtheorem29"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem29.1.1.1">Remark 29</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem29.2.2"> </span>(The fusion method and complexity in Boolean algebras)<span class="ltx_text ltx_font_bold" id="Thmtheorem29.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem29.p1"> <p class="ltx_p" id="Thmtheorem29.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem29.p1.5.5">Our presentation allows us to conclude, in particular, that the fusion method provides a framework to lower bound the number of operations in any <em class="ltx_emph ltx_font_upright" id="Thmtheorem29.p1.5.5.1">(</em>finite<em class="ltx_emph ltx_font_upright" id="Thmtheorem29.p1.5.5.2">)</em> Boolean algebra <math alttext="\mathfrak{B}" class="ltx_Math" display="inline" id="Thmtheorem29.p1.1.1.m1.1"><semantics id="Thmtheorem29.p1.1.1.m1.1a"><mi id="Thmtheorem29.p1.1.1.m1.1.1" xref="Thmtheorem29.p1.1.1.m1.1.1.cmml">𝔅</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem29.p1.1.1.m1.1b"><ci id="Thmtheorem29.p1.1.1.m1.1.1.cmml" xref="Thmtheorem29.p1.1.1.m1.1.1">𝔅</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem29.p1.1.1.m1.1c">\mathfrak{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem29.p1.1.1.m1.1d">fraktur_B</annotation></semantics></math>. Indeed, by the Stone Representation Theorem <em class="ltx_emph ltx_font_upright" id="Thmtheorem29.p1.5.5.3">(</em>cf. <em class="ltx_emph ltx_font_upright" id="Thmtheorem29.p1.5.5.4"><cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib4" title="">4</a>]</cite>)</em>, any Boolean algebra is isomorphic to a field of sets. Therefore, the problem of determining the number of <math alttext="\vee_{\mathcal{B}}" class="ltx_Math" display="inline" id="Thmtheorem29.p1.2.2.m2.1"><semantics id="Thmtheorem29.p1.2.2.m2.1a"><msub id="Thmtheorem29.p1.2.2.m2.1.1" xref="Thmtheorem29.p1.2.2.m2.1.1.cmml"><mo id="Thmtheorem29.p1.2.2.m2.1.1.2" xref="Thmtheorem29.p1.2.2.m2.1.1.2.cmml">∨</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem29.p1.2.2.m2.1.1.3" xref="Thmtheorem29.p1.2.2.m2.1.1.3.cmml">ℬ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem29.p1.2.2.m2.1b"><apply id="Thmtheorem29.p1.2.2.m2.1.1.cmml" xref="Thmtheorem29.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmtheorem29.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem29.p1.2.2.m2.1.1">subscript</csymbol><or id="Thmtheorem29.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem29.p1.2.2.m2.1.1.2"></or><ci id="Thmtheorem29.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem29.p1.2.2.m2.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem29.p1.2.2.m2.1c">\vee_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem29.p1.2.2.m2.1d">∨ start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\wedge_{\mathcal{B}}" class="ltx_Math" display="inline" id="Thmtheorem29.p1.3.3.m3.1"><semantics id="Thmtheorem29.p1.3.3.m3.1a"><msub id="Thmtheorem29.p1.3.3.m3.1.1" xref="Thmtheorem29.p1.3.3.m3.1.1.cmml"><mo id="Thmtheorem29.p1.3.3.m3.1.1.2" xref="Thmtheorem29.p1.3.3.m3.1.1.2.cmml">∧</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem29.p1.3.3.m3.1.1.3" xref="Thmtheorem29.p1.3.3.m3.1.1.3.cmml">ℬ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem29.p1.3.3.m3.1b"><apply id="Thmtheorem29.p1.3.3.m3.1.1.cmml" xref="Thmtheorem29.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmtheorem29.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem29.p1.3.3.m3.1.1">subscript</csymbol><and id="Thmtheorem29.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem29.p1.3.3.m3.1.1.2"></and><ci id="Thmtheorem29.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem29.p1.3.3.m3.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem29.p1.3.3.m3.1c">\wedge_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem29.p1.3.3.m3.1d">∧ start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> operations necessary to obtain a non-trivial element <math alttext="a\in\mathfrak{B}" class="ltx_Math" display="inline" id="Thmtheorem29.p1.4.4.m4.1"><semantics id="Thmtheorem29.p1.4.4.m4.1a"><mrow id="Thmtheorem29.p1.4.4.m4.1.1" xref="Thmtheorem29.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem29.p1.4.4.m4.1.1.2" xref="Thmtheorem29.p1.4.4.m4.1.1.2.cmml">a</mi><mo id="Thmtheorem29.p1.4.4.m4.1.1.1" xref="Thmtheorem29.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="Thmtheorem29.p1.4.4.m4.1.1.3" xref="Thmtheorem29.p1.4.4.m4.1.1.3.cmml">𝔅</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem29.p1.4.4.m4.1b"><apply id="Thmtheorem29.p1.4.4.m4.1.1.cmml" xref="Thmtheorem29.p1.4.4.m4.1.1"><in id="Thmtheorem29.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem29.p1.4.4.m4.1.1.1"></in><ci id="Thmtheorem29.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem29.p1.4.4.m4.1.1.2">𝑎</ci><ci id="Thmtheorem29.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem29.p1.4.4.m4.1.1.3">𝔅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem29.p1.4.4.m4.1c">a\in\mathfrak{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem29.p1.4.4.m4.1d">italic_a ∈ fraktur_B</annotation></semantics></math> from elements <math alttext="b_{1},\ldots,b_{m}\in\mathfrak{B}" class="ltx_Math" display="inline" id="Thmtheorem29.p1.5.5.m5.3"><semantics id="Thmtheorem29.p1.5.5.m5.3a"><mrow id="Thmtheorem29.p1.5.5.m5.3.3" xref="Thmtheorem29.p1.5.5.m5.3.3.cmml"><mrow id="Thmtheorem29.p1.5.5.m5.3.3.2.2" xref="Thmtheorem29.p1.5.5.m5.3.3.2.3.cmml"><msub id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.cmml"><mi id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.2" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.2.cmml">b</mi><mn id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.3" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="Thmtheorem29.p1.5.5.m5.3.3.2.2.3" xref="Thmtheorem29.p1.5.5.m5.3.3.2.3.cmml">,</mo><mi id="Thmtheorem29.p1.5.5.m5.1.1" mathvariant="normal" xref="Thmtheorem29.p1.5.5.m5.1.1.cmml">…</mi><mo id="Thmtheorem29.p1.5.5.m5.3.3.2.2.4" xref="Thmtheorem29.p1.5.5.m5.3.3.2.3.cmml">,</mo><msub id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.cmml"><mi id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.2" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.2.cmml">b</mi><mi id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.3" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.3.cmml">m</mi></msub></mrow><mo id="Thmtheorem29.p1.5.5.m5.3.3.3" xref="Thmtheorem29.p1.5.5.m5.3.3.3.cmml">∈</mo><mi id="Thmtheorem29.p1.5.5.m5.3.3.4" xref="Thmtheorem29.p1.5.5.m5.3.3.4.cmml">𝔅</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem29.p1.5.5.m5.3b"><apply id="Thmtheorem29.p1.5.5.m5.3.3.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3"><in id="Thmtheorem29.p1.5.5.m5.3.3.3.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.3"></in><list id="Thmtheorem29.p1.5.5.m5.3.3.2.3.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2"><apply id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.cmml" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1">subscript</csymbol><ci id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.2.cmml" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.2">𝑏</ci><cn id="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="Thmtheorem29.p1.5.5.m5.2.2.1.1.1.3">1</cn></apply><ci id="Thmtheorem29.p1.5.5.m5.1.1.cmml" xref="Thmtheorem29.p1.5.5.m5.1.1">…</ci><apply id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.1.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2">subscript</csymbol><ci id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.2.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.2">𝑏</ci><ci id="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.3.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.2.2.2.3">𝑚</ci></apply></list><ci id="Thmtheorem29.p1.5.5.m5.3.3.4.cmml" xref="Thmtheorem29.p1.5.5.m5.3.3.4">𝔅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem29.p1.5.5.m5.3c">b_{1},\ldots,b_{m}\in\mathfrak{B}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem29.p1.5.5.m5.3d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ fraktur_B</annotation></semantics></math> can be captured via cover complexity by Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a>.</span></p> </div> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4 </span>An exact characterization via cyclic discrete complexity</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.1">In this section, we show that cover complexity can be <em class="ltx_emph ltx_font_italic" id="S3.SS4.p1.1.1">exactly</em> characterized using the intersection complexity variant of cyclic complexity. The tight correspondence is obtained by a simple adaptation of an idea from <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem30"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem30.1.1.1">Theorem 30</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem30.2.2"> </span>(Exact characterization of cover complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem30.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem30.p1"> <p class="ltx_p" id="Thmtheorem30.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem30.p1.2.2">Let <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem30.p1.1.1.m1.1"><semantics id="Thmtheorem30.p1.1.1.m1.1a"><mrow id="Thmtheorem30.p1.1.1.m1.1.1" xref="Thmtheorem30.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem30.p1.1.1.m1.1.1.2" xref="Thmtheorem30.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="Thmtheorem30.p1.1.1.m1.1.1.1" xref="Thmtheorem30.p1.1.1.m1.1.1.1.cmml">⊆</mo><mi id="Thmtheorem30.p1.1.1.m1.1.1.3" mathvariant="normal" xref="Thmtheorem30.p1.1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem30.p1.1.1.m1.1b"><apply id="Thmtheorem30.p1.1.1.m1.1.1.cmml" xref="Thmtheorem30.p1.1.1.m1.1.1"><subset id="Thmtheorem30.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem30.p1.1.1.m1.1.1.1"></subset><ci id="Thmtheorem30.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem30.p1.1.1.m1.1.1.2">𝐴</ci><ci id="Thmtheorem30.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem30.p1.1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem30.p1.1.1.m1.1c">A\subseteq\Gamma</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem30.p1.1.1.m1.1d">italic_A ⊆ roman_Γ</annotation></semantics></math> be non-trivial, and <math alttext="\mathcal{B}\subseteq\mathcal{P}(\Gamma)" class="ltx_Math" display="inline" id="Thmtheorem30.p1.2.2.m2.1"><semantics id="Thmtheorem30.p1.2.2.m2.1a"><mrow id="Thmtheorem30.p1.2.2.m2.1.2" xref="Thmtheorem30.p1.2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem30.p1.2.2.m2.1.2.2" xref="Thmtheorem30.p1.2.2.m2.1.2.2.cmml">ℬ</mi><mo id="Thmtheorem30.p1.2.2.m2.1.2.1" xref="Thmtheorem30.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem30.p1.2.2.m2.1.2.3" xref="Thmtheorem30.p1.2.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem30.p1.2.2.m2.1.2.3.2" xref="Thmtheorem30.p1.2.2.m2.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem30.p1.2.2.m2.1.2.3.1" xref="Thmtheorem30.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem30.p1.2.2.m2.1.2.3.3.2" xref="Thmtheorem30.p1.2.2.m2.1.2.3.cmml"><mo id="Thmtheorem30.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem30.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmtheorem30.p1.2.2.m2.1.1" mathvariant="normal" xref="Thmtheorem30.p1.2.2.m2.1.1.cmml">Γ</mi><mo id="Thmtheorem30.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem30.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem30.p1.2.2.m2.1b"><apply id="Thmtheorem30.p1.2.2.m2.1.2.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2"><subset id="Thmtheorem30.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2.1"></subset><ci id="Thmtheorem30.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2.2">ℬ</ci><apply id="Thmtheorem30.p1.2.2.m2.1.2.3.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2.3"><times id="Thmtheorem30.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2.3.1"></times><ci id="Thmtheorem30.p1.2.2.m2.1.2.3.2.cmml" xref="Thmtheorem30.p1.2.2.m2.1.2.3.2">𝒫</ci><ci id="Thmtheorem30.p1.2.2.m2.1.1.cmml" xref="Thmtheorem30.p1.2.2.m2.1.1">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem30.p1.2.2.m2.1c">\mathcal{B}\subseteq\mathcal{P}(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem30.p1.2.2.m2.1d">caligraphic_B ⊆ caligraphic_P ( roman_Γ )</annotation></semantics></math> be a non-empty family of generators. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(A,\mathcal{B})\;=\;D_{\cap}(A\mid\mathcal{B})." class="ltx_Math" display="block" id="S3.Ex25.m1.3"><semantics id="S3.Ex25.m1.3a"><mrow id="S3.Ex25.m1.3.3.1" xref="S3.Ex25.m1.3.3.1.1.cmml"><mrow id="S3.Ex25.m1.3.3.1.1" xref="S3.Ex25.m1.3.3.1.1.cmml"><mrow id="S3.Ex25.m1.3.3.1.1.3" xref="S3.Ex25.m1.3.3.1.1.3.cmml"><mi id="S3.Ex25.m1.3.3.1.1.3.2" xref="S3.Ex25.m1.3.3.1.1.3.2.cmml">ρ</mi><mo id="S3.Ex25.m1.3.3.1.1.3.1" xref="S3.Ex25.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex25.m1.3.3.1.1.3.3.2" xref="S3.Ex25.m1.3.3.1.1.3.3.1.cmml"><mo id="S3.Ex25.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.Ex25.m1.3.3.1.1.3.3.1.cmml">(</mo><mi id="S3.Ex25.m1.1.1" xref="S3.Ex25.m1.1.1.cmml">A</mi><mo id="S3.Ex25.m1.3.3.1.1.3.3.2.2" xref="S3.Ex25.m1.3.3.1.1.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex25.m1.2.2" xref="S3.Ex25.m1.2.2.cmml">ℬ</mi><mo id="S3.Ex25.m1.3.3.1.1.3.3.2.3" rspace="0.280em" stretchy="false" xref="S3.Ex25.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex25.m1.3.3.1.1.2" rspace="0.558em" xref="S3.Ex25.m1.3.3.1.1.2.cmml">=</mo><mrow id="S3.Ex25.m1.3.3.1.1.1" xref="S3.Ex25.m1.3.3.1.1.1.cmml"><msub id="S3.Ex25.m1.3.3.1.1.1.3" xref="S3.Ex25.m1.3.3.1.1.1.3.cmml"><mi id="S3.Ex25.m1.3.3.1.1.1.3.2" xref="S3.Ex25.m1.3.3.1.1.1.3.2.cmml">D</mi><mo id="S3.Ex25.m1.3.3.1.1.1.3.3" xref="S3.Ex25.m1.3.3.1.1.1.3.3.cmml">∩</mo></msub><mo id="S3.Ex25.m1.3.3.1.1.1.2" xref="S3.Ex25.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex25.m1.3.3.1.1.1.1.1" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S3.Ex25.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex25.m1.3.3.1.1.1.1.1.1" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex25.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex25.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex25.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.Ex25.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex25.m1.3.3.1.2" lspace="0em" xref="S3.Ex25.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex25.m1.3b"><apply id="S3.Ex25.m1.3.3.1.1.cmml" xref="S3.Ex25.m1.3.3.1"><eq id="S3.Ex25.m1.3.3.1.1.2.cmml" xref="S3.Ex25.m1.3.3.1.1.2"></eq><apply id="S3.Ex25.m1.3.3.1.1.3.cmml" xref="S3.Ex25.m1.3.3.1.1.3"><times id="S3.Ex25.m1.3.3.1.1.3.1.cmml" xref="S3.Ex25.m1.3.3.1.1.3.1"></times><ci id="S3.Ex25.m1.3.3.1.1.3.2.cmml" xref="S3.Ex25.m1.3.3.1.1.3.2">𝜌</ci><interval closure="open" id="S3.Ex25.m1.3.3.1.1.3.3.1.cmml" xref="S3.Ex25.m1.3.3.1.1.3.3.2"><ci id="S3.Ex25.m1.1.1.cmml" xref="S3.Ex25.m1.1.1">𝐴</ci><ci id="S3.Ex25.m1.2.2.cmml" xref="S3.Ex25.m1.2.2">ℬ</ci></interval></apply><apply id="S3.Ex25.m1.3.3.1.1.1.cmml" xref="S3.Ex25.m1.3.3.1.1.1"><times id="S3.Ex25.m1.3.3.1.1.1.2.cmml" xref="S3.Ex25.m1.3.3.1.1.1.2"></times><apply id="S3.Ex25.m1.3.3.1.1.1.3.cmml" xref="S3.Ex25.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex25.m1.3.3.1.1.1.3.1.cmml" xref="S3.Ex25.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S3.Ex25.m1.3.3.1.1.1.3.2.cmml" xref="S3.Ex25.m1.3.3.1.1.1.3.2">𝐷</ci><intersect id="S3.Ex25.m1.3.3.1.1.1.3.3.cmml" xref="S3.Ex25.m1.3.3.1.1.1.3.3"></intersect></apply><apply id="S3.Ex25.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex25.m1.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S3.Ex25.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.1">conditional</csymbol><ci id="S3.Ex25.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex25.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S3.Ex25.m1.3.3.1.1.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex25.m1.3c">\rho(A,\mathcal{B})\;=\;D_{\cap}(A\mid\mathcal{B}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex25.m1.3d">italic_ρ ( italic_A , caligraphic_B ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.SS4.13"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS4.1.p1"> <p class="ltx_p" id="S3.SS4.1.p1.6">The proof that <math alttext="D_{\cap}(A\mid\mathcal{B})\leq\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS4.1.p1.1.m1.3"><semantics id="S3.SS4.1.p1.1.m1.3a"><mrow id="S3.SS4.1.p1.1.m1.3.3" xref="S3.SS4.1.p1.1.m1.3.3.cmml"><mrow id="S3.SS4.1.p1.1.m1.3.3.1" xref="S3.SS4.1.p1.1.m1.3.3.1.cmml"><msub id="S3.SS4.1.p1.1.m1.3.3.1.3" xref="S3.SS4.1.p1.1.m1.3.3.1.3.cmml"><mi id="S3.SS4.1.p1.1.m1.3.3.1.3.2" xref="S3.SS4.1.p1.1.m1.3.3.1.3.2.cmml">D</mi><mo id="S3.SS4.1.p1.1.m1.3.3.1.3.3" xref="S3.SS4.1.p1.1.m1.3.3.1.3.3.cmml">∩</mo></msub><mo id="S3.SS4.1.p1.1.m1.3.3.1.2" xref="S3.SS4.1.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS4.1.p1.1.m1.3.3.1.1.1" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.SS4.1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.2" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.2.cmml">A</mi><mo id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.1" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.3" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS4.1.p1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.1.p1.1.m1.3.3.2" xref="S3.SS4.1.p1.1.m1.3.3.2.cmml">≤</mo><mrow id="S3.SS4.1.p1.1.m1.3.3.3" xref="S3.SS4.1.p1.1.m1.3.3.3.cmml"><mi id="S3.SS4.1.p1.1.m1.3.3.3.2" xref="S3.SS4.1.p1.1.m1.3.3.3.2.cmml">ρ</mi><mo id="S3.SS4.1.p1.1.m1.3.3.3.1" xref="S3.SS4.1.p1.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS4.1.p1.1.m1.3.3.3.3.2" xref="S3.SS4.1.p1.1.m1.3.3.3.3.1.cmml"><mo id="S3.SS4.1.p1.1.m1.3.3.3.3.2.1" stretchy="false" xref="S3.SS4.1.p1.1.m1.3.3.3.3.1.cmml">(</mo><mi id="S3.SS4.1.p1.1.m1.1.1" xref="S3.SS4.1.p1.1.m1.1.1.cmml">A</mi><mo id="S3.SS4.1.p1.1.m1.3.3.3.3.2.2" xref="S3.SS4.1.p1.1.m1.3.3.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.1.m1.2.2" xref="S3.SS4.1.p1.1.m1.2.2.cmml">ℬ</mi><mo id="S3.SS4.1.p1.1.m1.3.3.3.3.2.3" stretchy="false" xref="S3.SS4.1.p1.1.m1.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.1.m1.3b"><apply id="S3.SS4.1.p1.1.m1.3.3.cmml" xref="S3.SS4.1.p1.1.m1.3.3"><leq id="S3.SS4.1.p1.1.m1.3.3.2.cmml" xref="S3.SS4.1.p1.1.m1.3.3.2"></leq><apply id="S3.SS4.1.p1.1.m1.3.3.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1"><times id="S3.SS4.1.p1.1.m1.3.3.1.2.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.2"></times><apply id="S3.SS4.1.p1.1.m1.3.3.1.3.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.1.m1.3.3.1.3.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.3">subscript</csymbol><ci id="S3.SS4.1.p1.1.m1.3.3.1.3.2.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.3.2">𝐷</ci><intersect id="S3.SS4.1.p1.1.m1.3.3.1.3.3.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.3.3"></intersect></apply><apply id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1"><csymbol cd="latexml" id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.1">conditional</csymbol><ci id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.2">𝐴</ci><ci id="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.3.cmml" xref="S3.SS4.1.p1.1.m1.3.3.1.1.1.1.3">ℬ</ci></apply></apply><apply id="S3.SS4.1.p1.1.m1.3.3.3.cmml" xref="S3.SS4.1.p1.1.m1.3.3.3"><times id="S3.SS4.1.p1.1.m1.3.3.3.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.3.1"></times><ci id="S3.SS4.1.p1.1.m1.3.3.3.2.cmml" xref="S3.SS4.1.p1.1.m1.3.3.3.2">𝜌</ci><interval closure="open" id="S3.SS4.1.p1.1.m1.3.3.3.3.1.cmml" xref="S3.SS4.1.p1.1.m1.3.3.3.3.2"><ci id="S3.SS4.1.p1.1.m1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.1.1">𝐴</ci><ci id="S3.SS4.1.p1.1.m1.2.2.cmml" xref="S3.SS4.1.p1.1.m1.2.2">ℬ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.1.m1.3c">D_{\cap}(A\mid\mathcal{B})\leq\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.1.m1.3d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B ) ≤ italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math> is essentially immediate from the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a>. It is enough to observe that the construction of <math alttext="A" class="ltx_Math" display="inline" id="S3.SS4.1.p1.2.m2.1"><semantics id="S3.SS4.1.p1.2.m2.1a"><mi id="S3.SS4.1.p1.2.m2.1.1" xref="S3.SS4.1.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.2.m2.1b"><ci id="S3.SS4.1.p1.2.m2.1.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.2.m2.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.3.m3.1"><semantics id="S3.SS4.1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.3.m3.1.1" xref="S3.SS4.1.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.3.m3.1b"><ci id="S3.SS4.1.p1.3.m3.1.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.3.m3.1d">caligraphic_B</annotation></semantics></math> via <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS4.1.p1.4.m4.1"><semantics id="S3.SS4.1.p1.4.m4.1a"><mi id="S3.SS4.1.p1.4.m4.1.1" mathvariant="normal" xref="S3.SS4.1.p1.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.4.m4.1b"><ci id="S3.SS4.1.p1.4.m4.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.4.m4.1d">roman_Λ</annotation></semantics></math> described there can be transformed into a syntactic sequence for <math alttext="A" class="ltx_Math" display="inline" id="S3.SS4.1.p1.5.m5.1"><semantics id="S3.SS4.1.p1.5.m5.1a"><mi id="S3.SS4.1.p1.5.m5.1.1" xref="S3.SS4.1.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.5.m5.1b"><ci id="S3.SS4.1.p1.5.m5.1.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.5.m5.1d">italic_A</annotation></semantics></math> that employs at most <math alttext="|\Lambda|" class="ltx_Math" display="inline" id="S3.SS4.1.p1.6.m6.1"><semantics id="S3.SS4.1.p1.6.m6.1a"><mrow id="S3.SS4.1.p1.6.m6.1.2.2" xref="S3.SS4.1.p1.6.m6.1.2.1.cmml"><mo id="S3.SS4.1.p1.6.m6.1.2.2.1" stretchy="false" xref="S3.SS4.1.p1.6.m6.1.2.1.1.cmml">|</mo><mi id="S3.SS4.1.p1.6.m6.1.1" mathvariant="normal" xref="S3.SS4.1.p1.6.m6.1.1.cmml">Λ</mi><mo id="S3.SS4.1.p1.6.m6.1.2.2.2" stretchy="false" xref="S3.SS4.1.p1.6.m6.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.6.m6.1b"><apply id="S3.SS4.1.p1.6.m6.1.2.1.cmml" xref="S3.SS4.1.p1.6.m6.1.2.2"><abs id="S3.SS4.1.p1.6.m6.1.2.1.1.cmml" xref="S3.SS4.1.p1.6.m6.1.2.2.1"></abs><ci id="S3.SS4.1.p1.6.m6.1.1.cmml" xref="S3.SS4.1.p1.6.m6.1.1">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.6.m6.1c">|\Lambda|</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.6.m6.1d">| roman_Λ |</annotation></semantics></math> intersection operations. This is similar to the example presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS5" title="2.5 Cyclic Discrete Complexity ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.5</span></a>.</p> </div> <div class="ltx_para" id="S3.SS4.2.p2"> <p class="ltx_p" id="S3.SS4.2.p2.1">We establish next that <math alttext="\rho(A,\mathcal{B})\leq D_{\cap}(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS4.2.p2.1.m1.3"><semantics id="S3.SS4.2.p2.1.m1.3a"><mrow id="S3.SS4.2.p2.1.m1.3.3" xref="S3.SS4.2.p2.1.m1.3.3.cmml"><mrow id="S3.SS4.2.p2.1.m1.3.3.3" xref="S3.SS4.2.p2.1.m1.3.3.3.cmml"><mi id="S3.SS4.2.p2.1.m1.3.3.3.2" xref="S3.SS4.2.p2.1.m1.3.3.3.2.cmml">ρ</mi><mo id="S3.SS4.2.p2.1.m1.3.3.3.1" xref="S3.SS4.2.p2.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS4.2.p2.1.m1.3.3.3.3.2" xref="S3.SS4.2.p2.1.m1.3.3.3.3.1.cmml"><mo id="S3.SS4.2.p2.1.m1.3.3.3.3.2.1" stretchy="false" xref="S3.SS4.2.p2.1.m1.3.3.3.3.1.cmml">(</mo><mi id="S3.SS4.2.p2.1.m1.1.1" xref="S3.SS4.2.p2.1.m1.1.1.cmml">A</mi><mo id="S3.SS4.2.p2.1.m1.3.3.3.3.2.2" xref="S3.SS4.2.p2.1.m1.3.3.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.1.m1.2.2" xref="S3.SS4.2.p2.1.m1.2.2.cmml">ℬ</mi><mo id="S3.SS4.2.p2.1.m1.3.3.3.3.2.3" stretchy="false" xref="S3.SS4.2.p2.1.m1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.2.p2.1.m1.3.3.2" xref="S3.SS4.2.p2.1.m1.3.3.2.cmml">≤</mo><mrow id="S3.SS4.2.p2.1.m1.3.3.1" xref="S3.SS4.2.p2.1.m1.3.3.1.cmml"><msub id="S3.SS4.2.p2.1.m1.3.3.1.3" xref="S3.SS4.2.p2.1.m1.3.3.1.3.cmml"><mi id="S3.SS4.2.p2.1.m1.3.3.1.3.2" xref="S3.SS4.2.p2.1.m1.3.3.1.3.2.cmml">D</mi><mo id="S3.SS4.2.p2.1.m1.3.3.1.3.3" xref="S3.SS4.2.p2.1.m1.3.3.1.3.3.cmml">∩</mo></msub><mo id="S3.SS4.2.p2.1.m1.3.3.1.2" xref="S3.SS4.2.p2.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS4.2.p2.1.m1.3.3.1.1.1" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.SS4.2.p2.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml"><mi id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.2" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.2.cmml">A</mi><mo id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.1" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.3" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS4.2.p2.1.m1.3.3.1.1.1.3" stretchy="false" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.1.m1.3b"><apply id="S3.SS4.2.p2.1.m1.3.3.cmml" xref="S3.SS4.2.p2.1.m1.3.3"><leq id="S3.SS4.2.p2.1.m1.3.3.2.cmml" xref="S3.SS4.2.p2.1.m1.3.3.2"></leq><apply id="S3.SS4.2.p2.1.m1.3.3.3.cmml" xref="S3.SS4.2.p2.1.m1.3.3.3"><times id="S3.SS4.2.p2.1.m1.3.3.3.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.3.1"></times><ci id="S3.SS4.2.p2.1.m1.3.3.3.2.cmml" xref="S3.SS4.2.p2.1.m1.3.3.3.2">𝜌</ci><interval closure="open" id="S3.SS4.2.p2.1.m1.3.3.3.3.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.3.3.2"><ci id="S3.SS4.2.p2.1.m1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1">𝐴</ci><ci id="S3.SS4.2.p2.1.m1.2.2.cmml" xref="S3.SS4.2.p2.1.m1.2.2">ℬ</ci></interval></apply><apply id="S3.SS4.2.p2.1.m1.3.3.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1"><times id="S3.SS4.2.p2.1.m1.3.3.1.2.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.2"></times><apply id="S3.SS4.2.p2.1.m1.3.3.1.3.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.2.p2.1.m1.3.3.1.3.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.3">subscript</csymbol><ci id="S3.SS4.2.p2.1.m1.3.3.1.3.2.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.3.2">𝐷</ci><intersect id="S3.SS4.2.p2.1.m1.3.3.1.3.3.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.3.3"></intersect></apply><apply id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1"><csymbol cd="latexml" id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.1">conditional</csymbol><ci id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.2">𝐴</ci><ci id="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.3.cmml" xref="S3.SS4.2.p2.1.m1.3.3.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.1.m1.3c">\rho(A,\mathcal{B})\leq D_{\cap}(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.1.m1.3d">italic_ρ ( italic_A , caligraphic_B ) ≤ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_A ∣ caligraphic_B )</annotation></semantics></math>. The main difficulty here is that simply unfolding the evaluation of the syntactic sequence introduces further intersection operations (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem17" title="Corollary 17 (Cyclic discrete complexity versus discrete complexity). ‣ 2.5 Cyclic Discrete Complexity ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">17</span></a>), and we cannot rely on Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>. We argue as follows.</p> </div> <div class="ltx_para" id="S3.SS4.3.p3"> <p class="ltx_p" id="S3.SS4.3.p3.14">Let <math alttext="\mathcal{B}=\{B_{1},\ldots,B_{m}\}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.1.m1.3"><semantics id="S3.SS4.3.p3.1.m1.3a"><mrow id="S3.SS4.3.p3.1.m1.3.3" xref="S3.SS4.3.p3.1.m1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.3.3.4" xref="S3.SS4.3.p3.1.m1.3.3.4.cmml">ℬ</mi><mo id="S3.SS4.3.p3.1.m1.3.3.3" xref="S3.SS4.3.p3.1.m1.3.3.3.cmml">=</mo><mrow id="S3.SS4.3.p3.1.m1.3.3.2.2" xref="S3.SS4.3.p3.1.m1.3.3.2.3.cmml"><mo id="S3.SS4.3.p3.1.m1.3.3.2.2.3" stretchy="false" xref="S3.SS4.3.p3.1.m1.3.3.2.3.cmml">{</mo><msub id="S3.SS4.3.p3.1.m1.2.2.1.1.1" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1.cmml"><mi id="S3.SS4.3.p3.1.m1.2.2.1.1.1.2" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1.2.cmml">B</mi><mn id="S3.SS4.3.p3.1.m1.2.2.1.1.1.3" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS4.3.p3.1.m1.3.3.2.2.4" xref="S3.SS4.3.p3.1.m1.3.3.2.3.cmml">,</mo><mi id="S3.SS4.3.p3.1.m1.1.1" mathvariant="normal" xref="S3.SS4.3.p3.1.m1.1.1.cmml">…</mi><mo id="S3.SS4.3.p3.1.m1.3.3.2.2.5" xref="S3.SS4.3.p3.1.m1.3.3.2.3.cmml">,</mo><msub id="S3.SS4.3.p3.1.m1.3.3.2.2.2" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2.cmml"><mi id="S3.SS4.3.p3.1.m1.3.3.2.2.2.2" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2.2.cmml">B</mi><mi id="S3.SS4.3.p3.1.m1.3.3.2.2.2.3" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S3.SS4.3.p3.1.m1.3.3.2.2.6" stretchy="false" xref="S3.SS4.3.p3.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.1.m1.3b"><apply id="S3.SS4.3.p3.1.m1.3.3.cmml" xref="S3.SS4.3.p3.1.m1.3.3"><eq id="S3.SS4.3.p3.1.m1.3.3.3.cmml" xref="S3.SS4.3.p3.1.m1.3.3.3"></eq><ci id="S3.SS4.3.p3.1.m1.3.3.4.cmml" xref="S3.SS4.3.p3.1.m1.3.3.4">ℬ</ci><set id="S3.SS4.3.p3.1.m1.3.3.2.3.cmml" xref="S3.SS4.3.p3.1.m1.3.3.2.2"><apply id="S3.SS4.3.p3.1.m1.2.2.1.1.1.cmml" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.SS4.3.p3.1.m1.2.2.1.1.1.2.cmml" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1.2">𝐵</ci><cn id="S3.SS4.3.p3.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS4.3.p3.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS4.3.p3.1.m1.1.1.cmml" xref="S3.SS4.3.p3.1.m1.1.1">…</ci><apply id="S3.SS4.3.p3.1.m1.3.3.2.2.2.cmml" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.3.p3.1.m1.3.3.2.2.2.1.cmml" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.SS4.3.p3.1.m1.3.3.2.2.2.2.cmml" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2.2">𝐵</ci><ci id="S3.SS4.3.p3.1.m1.3.3.2.2.2.3.cmml" xref="S3.SS4.3.p3.1.m1.3.3.2.2.2.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.1.m1.3c">\mathcal{B}=\{B_{1},\ldots,B_{m}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.1.m1.3d">caligraphic_B = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math>, and <math alttext="I_{1},\ldots,I_{t}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.2.m2.3"><semantics id="S3.SS4.3.p3.2.m2.3a"><mrow id="S3.SS4.3.p3.2.m2.3.3.2" xref="S3.SS4.3.p3.2.m2.3.3.3.cmml"><msub id="S3.SS4.3.p3.2.m2.2.2.1.1" xref="S3.SS4.3.p3.2.m2.2.2.1.1.cmml"><mi id="S3.SS4.3.p3.2.m2.2.2.1.1.2" xref="S3.SS4.3.p3.2.m2.2.2.1.1.2.cmml">I</mi><mn id="S3.SS4.3.p3.2.m2.2.2.1.1.3" xref="S3.SS4.3.p3.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS4.3.p3.2.m2.3.3.2.3" xref="S3.SS4.3.p3.2.m2.3.3.3.cmml">,</mo><mi id="S3.SS4.3.p3.2.m2.1.1" mathvariant="normal" xref="S3.SS4.3.p3.2.m2.1.1.cmml">…</mi><mo id="S3.SS4.3.p3.2.m2.3.3.2.4" xref="S3.SS4.3.p3.2.m2.3.3.3.cmml">,</mo><msub id="S3.SS4.3.p3.2.m2.3.3.2.2" xref="S3.SS4.3.p3.2.m2.3.3.2.2.cmml"><mi id="S3.SS4.3.p3.2.m2.3.3.2.2.2" xref="S3.SS4.3.p3.2.m2.3.3.2.2.2.cmml">I</mi><mi id="S3.SS4.3.p3.2.m2.3.3.2.2.3" xref="S3.SS4.3.p3.2.m2.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.2.m2.3b"><list id="S3.SS4.3.p3.2.m2.3.3.3.cmml" xref="S3.SS4.3.p3.2.m2.3.3.2"><apply id="S3.SS4.3.p3.2.m2.2.2.1.1.cmml" xref="S3.SS4.3.p3.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.2.m2.2.2.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.SS4.3.p3.2.m2.2.2.1.1.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.1.1.2">𝐼</ci><cn id="S3.SS4.3.p3.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.SS4.3.p3.2.m2.2.2.1.1.3">1</cn></apply><ci id="S3.SS4.3.p3.2.m2.1.1.cmml" xref="S3.SS4.3.p3.2.m2.1.1">…</ci><apply id="S3.SS4.3.p3.2.m2.3.3.2.2.cmml" xref="S3.SS4.3.p3.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS4.3.p3.2.m2.3.3.2.2.1.cmml" xref="S3.SS4.3.p3.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.SS4.3.p3.2.m2.3.3.2.2.2.cmml" xref="S3.SS4.3.p3.2.m2.3.3.2.2.2">𝐼</ci><ci id="S3.SS4.3.p3.2.m2.3.3.2.2.3.cmml" xref="S3.SS4.3.p3.2.m2.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.2.m2.3c">I_{1},\ldots,I_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.2.m2.3d">italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> be a syntactic sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S3.SS4.3.p3.3.m3.1"><semantics id="S3.SS4.3.p3.3.m3.1a"><mi id="S3.SS4.3.p3.3.m3.1.1" xref="S3.SS4.3.p3.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.3.m3.1b"><ci id="S3.SS4.3.p3.3.m3.1.1.cmml" xref="S3.SS4.3.p3.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.3.m3.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.4.m4.1"><semantics id="S3.SS4.3.p3.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.4.m4.1.1" xref="S3.SS4.3.p3.4.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.4.m4.1b"><ci id="S3.SS4.3.p3.4.m4.1.1.cmml" xref="S3.SS4.3.p3.4.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.4.m4.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.4.m4.1d">caligraphic_B</annotation></semantics></math> using operations <math alttext="\star_{i}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.5.m5.1"><semantics id="S3.SS4.3.p3.5.m5.1a"><msub id="S3.SS4.3.p3.5.m5.1.1" xref="S3.SS4.3.p3.5.m5.1.1.cmml"><mo id="S3.SS4.3.p3.5.m5.1.1.2" xref="S3.SS4.3.p3.5.m5.1.1.2.cmml">⋆</mo><mi id="S3.SS4.3.p3.5.m5.1.1.3" xref="S3.SS4.3.p3.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.5.m5.1b"><apply id="S3.SS4.3.p3.5.m5.1.1.cmml" xref="S3.SS4.3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.5.m5.1.1.1.cmml" xref="S3.SS4.3.p3.5.m5.1.1">subscript</csymbol><ci id="S3.SS4.3.p3.5.m5.1.1.2.cmml" xref="S3.SS4.3.p3.5.m5.1.1.2">⋆</ci><ci id="S3.SS4.3.p3.5.m5.1.1.3.cmml" xref="S3.SS4.3.p3.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.5.m5.1c">\star_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.5.m5.1d">⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="t=D(A\mid\mathcal{B})" class="ltx_Math" display="inline" id="S3.SS4.3.p3.6.m6.1"><semantics id="S3.SS4.3.p3.6.m6.1a"><mrow id="S3.SS4.3.p3.6.m6.1.1" xref="S3.SS4.3.p3.6.m6.1.1.cmml"><mi id="S3.SS4.3.p3.6.m6.1.1.3" xref="S3.SS4.3.p3.6.m6.1.1.3.cmml">t</mi><mo id="S3.SS4.3.p3.6.m6.1.1.2" xref="S3.SS4.3.p3.6.m6.1.1.2.cmml">=</mo><mrow id="S3.SS4.3.p3.6.m6.1.1.1" xref="S3.SS4.3.p3.6.m6.1.1.1.cmml"><mi id="S3.SS4.3.p3.6.m6.1.1.1.3" xref="S3.SS4.3.p3.6.m6.1.1.1.3.cmml">D</mi><mo id="S3.SS4.3.p3.6.m6.1.1.1.2" xref="S3.SS4.3.p3.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.6.m6.1.1.1.1.1" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.cmml"><mo id="S3.SS4.3.p3.6.m6.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.cmml"><mi id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.2" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.1" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.1.cmml">∣</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.3" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S3.SS4.3.p3.6.m6.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.6.m6.1b"><apply id="S3.SS4.3.p3.6.m6.1.1.cmml" xref="S3.SS4.3.p3.6.m6.1.1"><eq id="S3.SS4.3.p3.6.m6.1.1.2.cmml" xref="S3.SS4.3.p3.6.m6.1.1.2"></eq><ci id="S3.SS4.3.p3.6.m6.1.1.3.cmml" xref="S3.SS4.3.p3.6.m6.1.1.3">𝑡</ci><apply id="S3.SS4.3.p3.6.m6.1.1.1.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1"><times id="S3.SS4.3.p3.6.m6.1.1.1.2.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.2"></times><ci id="S3.SS4.3.p3.6.m6.1.1.1.3.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.3">𝐷</ci><apply id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.1">conditional</csymbol><ci id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.2.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.3.cmml" xref="S3.SS4.3.p3.6.m6.1.1.1.1.1.1.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.6.m6.1c">t=D(A\mid\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.6.m6.1d">italic_t = italic_D ( italic_A ∣ caligraphic_B )</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem16" title="Lemma 16 (Convergence of the evaluation procedure). ‣ 2.5 Cyclic Discrete Complexity ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">16</span></a>, the evaluation procedure converges to a sequence <math alttext="C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}=A" class="ltx_Math" display="inline" id="S3.SS4.3.p3.7.m7.6"><semantics id="S3.SS4.3.p3.7.m7.6a"><mrow id="S3.SS4.3.p3.7.m7.6.6" xref="S3.SS4.3.p3.7.m7.6.6.cmml"><mrow id="S3.SS4.3.p3.7.m7.6.6.4.4" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml"><msup id="S3.SS4.3.p3.7.m7.3.3.1.1.1" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1.cmml"><mi id="S3.SS4.3.p3.7.m7.3.3.1.1.1.2" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1.2.cmml">C</mi><mn id="S3.SS4.3.p3.7.m7.3.3.1.1.1.3" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1.3.cmml">1</mn></msup><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.5" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml">,</mo><mi id="S3.SS4.3.p3.7.m7.1.1" mathvariant="normal" xref="S3.SS4.3.p3.7.m7.1.1.cmml">…</mi><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.6" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml">,</mo><msup id="S3.SS4.3.p3.7.m7.4.4.2.2.2" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2.cmml"><mi id="S3.SS4.3.p3.7.m7.4.4.2.2.2.2" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2.2.cmml">C</mi><mi id="S3.SS4.3.p3.7.m7.4.4.2.2.2.3" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2.3.cmml">m</mi></msup><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.7" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml">,</mo><msup id="S3.SS4.3.p3.7.m7.5.5.3.3.3" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.cmml"><mi id="S3.SS4.3.p3.7.m7.5.5.3.3.3.2" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.2.cmml">C</mi><mrow id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.cmml"><mi id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.2" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.2.cmml">m</mi><mo id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.1" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.1.cmml">+</mo><mn id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.3" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.8" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml">,</mo><mi id="S3.SS4.3.p3.7.m7.2.2" mathvariant="normal" xref="S3.SS4.3.p3.7.m7.2.2.cmml">…</mi><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.9" xref="S3.SS4.3.p3.7.m7.6.6.4.5.cmml">,</mo><msup id="S3.SS4.3.p3.7.m7.6.6.4.4.4" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.cmml"><mi id="S3.SS4.3.p3.7.m7.6.6.4.4.4.2" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.2.cmml">C</mi><mrow id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.cmml"><mi id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.2" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.2.cmml">m</mi><mo id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.1" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.1.cmml">+</mo><mi id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.3" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.3.cmml">t</mi></mrow></msup></mrow><mo id="S3.SS4.3.p3.7.m7.6.6.5" xref="S3.SS4.3.p3.7.m7.6.6.5.cmml">=</mo><mi id="S3.SS4.3.p3.7.m7.6.6.6" xref="S3.SS4.3.p3.7.m7.6.6.6.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.7.m7.6b"><apply id="S3.SS4.3.p3.7.m7.6.6.cmml" xref="S3.SS4.3.p3.7.m7.6.6"><eq id="S3.SS4.3.p3.7.m7.6.6.5.cmml" xref="S3.SS4.3.p3.7.m7.6.6.5"></eq><list id="S3.SS4.3.p3.7.m7.6.6.4.5.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4"><apply id="S3.SS4.3.p3.7.m7.3.3.1.1.1.cmml" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.7.m7.3.3.1.1.1.1.cmml" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1">superscript</csymbol><ci id="S3.SS4.3.p3.7.m7.3.3.1.1.1.2.cmml" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1.2">𝐶</ci><cn id="S3.SS4.3.p3.7.m7.3.3.1.1.1.3.cmml" type="integer" xref="S3.SS4.3.p3.7.m7.3.3.1.1.1.3">1</cn></apply><ci id="S3.SS4.3.p3.7.m7.1.1.cmml" xref="S3.SS4.3.p3.7.m7.1.1">…</ci><apply id="S3.SS4.3.p3.7.m7.4.4.2.2.2.cmml" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.3.p3.7.m7.4.4.2.2.2.1.cmml" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2">superscript</csymbol><ci id="S3.SS4.3.p3.7.m7.4.4.2.2.2.2.cmml" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2.2">𝐶</ci><ci id="S3.SS4.3.p3.7.m7.4.4.2.2.2.3.cmml" xref="S3.SS4.3.p3.7.m7.4.4.2.2.2.3">𝑚</ci></apply><apply id="S3.SS4.3.p3.7.m7.5.5.3.3.3.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.3.p3.7.m7.5.5.3.3.3.1.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3">superscript</csymbol><ci id="S3.SS4.3.p3.7.m7.5.5.3.3.3.2.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.2">𝐶</ci><apply id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3"><plus id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.1.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.1"></plus><ci id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.2.cmml" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.2">𝑚</ci><cn id="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.3.cmml" type="integer" xref="S3.SS4.3.p3.7.m7.5.5.3.3.3.3.3">1</cn></apply></apply><ci id="S3.SS4.3.p3.7.m7.2.2.cmml" xref="S3.SS4.3.p3.7.m7.2.2">…</ci><apply id="S3.SS4.3.p3.7.m7.6.6.4.4.4.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4"><csymbol cd="ambiguous" id="S3.SS4.3.p3.7.m7.6.6.4.4.4.1.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4">superscript</csymbol><ci id="S3.SS4.3.p3.7.m7.6.6.4.4.4.2.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.2">𝐶</ci><apply id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3"><plus id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.1.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.1"></plus><ci id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.2.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.2">𝑚</ci><ci id="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.3.cmml" xref="S3.SS4.3.p3.7.m7.6.6.4.4.4.3.3">𝑡</ci></apply></apply></list><ci id="S3.SS4.3.p3.7.m7.6.6.6.cmml" xref="S3.SS4.3.p3.7.m7.6.6.6">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.7.m7.6c">C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}=A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.7.m7.6d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_C start_POSTSUPERSCRIPT italic_m + 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT = italic_A</annotation></semantics></math>, where <math alttext="C^{i}=B_{i}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.8.m8.1"><semantics id="S3.SS4.3.p3.8.m8.1a"><mrow id="S3.SS4.3.p3.8.m8.1.1" xref="S3.SS4.3.p3.8.m8.1.1.cmml"><msup id="S3.SS4.3.p3.8.m8.1.1.2" xref="S3.SS4.3.p3.8.m8.1.1.2.cmml"><mi id="S3.SS4.3.p3.8.m8.1.1.2.2" xref="S3.SS4.3.p3.8.m8.1.1.2.2.cmml">C</mi><mi id="S3.SS4.3.p3.8.m8.1.1.2.3" xref="S3.SS4.3.p3.8.m8.1.1.2.3.cmml">i</mi></msup><mo id="S3.SS4.3.p3.8.m8.1.1.1" xref="S3.SS4.3.p3.8.m8.1.1.1.cmml">=</mo><msub id="S3.SS4.3.p3.8.m8.1.1.3" xref="S3.SS4.3.p3.8.m8.1.1.3.cmml"><mi id="S3.SS4.3.p3.8.m8.1.1.3.2" xref="S3.SS4.3.p3.8.m8.1.1.3.2.cmml">B</mi><mi id="S3.SS4.3.p3.8.m8.1.1.3.3" xref="S3.SS4.3.p3.8.m8.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.8.m8.1b"><apply id="S3.SS4.3.p3.8.m8.1.1.cmml" xref="S3.SS4.3.p3.8.m8.1.1"><eq id="S3.SS4.3.p3.8.m8.1.1.1.cmml" xref="S3.SS4.3.p3.8.m8.1.1.1"></eq><apply id="S3.SS4.3.p3.8.m8.1.1.2.cmml" xref="S3.SS4.3.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.3.p3.8.m8.1.1.2.1.cmml" xref="S3.SS4.3.p3.8.m8.1.1.2">superscript</csymbol><ci id="S3.SS4.3.p3.8.m8.1.1.2.2.cmml" xref="S3.SS4.3.p3.8.m8.1.1.2.2">𝐶</ci><ci id="S3.SS4.3.p3.8.m8.1.1.2.3.cmml" xref="S3.SS4.3.p3.8.m8.1.1.2.3">𝑖</ci></apply><apply id="S3.SS4.3.p3.8.m8.1.1.3.cmml" xref="S3.SS4.3.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.3.p3.8.m8.1.1.3.1.cmml" xref="S3.SS4.3.p3.8.m8.1.1.3">subscript</csymbol><ci id="S3.SS4.3.p3.8.m8.1.1.3.2.cmml" xref="S3.SS4.3.p3.8.m8.1.1.3.2">𝐵</ci><ci id="S3.SS4.3.p3.8.m8.1.1.3.3.cmml" xref="S3.SS4.3.p3.8.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.8.m8.1c">C^{i}=B_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.8.m8.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in[m]" class="ltx_Math" display="inline" id="S3.SS4.3.p3.9.m9.1"><semantics id="S3.SS4.3.p3.9.m9.1a"><mrow id="S3.SS4.3.p3.9.m9.1.2" xref="S3.SS4.3.p3.9.m9.1.2.cmml"><mi id="S3.SS4.3.p3.9.m9.1.2.2" xref="S3.SS4.3.p3.9.m9.1.2.2.cmml">i</mi><mo id="S3.SS4.3.p3.9.m9.1.2.1" xref="S3.SS4.3.p3.9.m9.1.2.1.cmml">∈</mo><mrow id="S3.SS4.3.p3.9.m9.1.2.3.2" xref="S3.SS4.3.p3.9.m9.1.2.3.1.cmml"><mo id="S3.SS4.3.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS4.3.p3.9.m9.1.2.3.1.1.cmml">[</mo><mi id="S3.SS4.3.p3.9.m9.1.1" xref="S3.SS4.3.p3.9.m9.1.1.cmml">m</mi><mo id="S3.SS4.3.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS4.3.p3.9.m9.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.9.m9.1b"><apply id="S3.SS4.3.p3.9.m9.1.2.cmml" xref="S3.SS4.3.p3.9.m9.1.2"><in id="S3.SS4.3.p3.9.m9.1.2.1.cmml" xref="S3.SS4.3.p3.9.m9.1.2.1"></in><ci id="S3.SS4.3.p3.9.m9.1.2.2.cmml" xref="S3.SS4.3.p3.9.m9.1.2.2">𝑖</ci><apply id="S3.SS4.3.p3.9.m9.1.2.3.1.cmml" xref="S3.SS4.3.p3.9.m9.1.2.3.2"><csymbol cd="latexml" id="S3.SS4.3.p3.9.m9.1.2.3.1.1.cmml" xref="S3.SS4.3.p3.9.m9.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS4.3.p3.9.m9.1.1.cmml" xref="S3.SS4.3.p3.9.m9.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.9.m9.1c">i\in[m]</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.9.m9.1d">italic_i ∈ [ italic_m ]</annotation></semantics></math>. Moreover, each set <math alttext="I_{i}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.10.m10.1"><semantics id="S3.SS4.3.p3.10.m10.1a"><msub id="S3.SS4.3.p3.10.m10.1.1" xref="S3.SS4.3.p3.10.m10.1.1.cmml"><mi id="S3.SS4.3.p3.10.m10.1.1.2" xref="S3.SS4.3.p3.10.m10.1.1.2.cmml">I</mi><mi id="S3.SS4.3.p3.10.m10.1.1.3" xref="S3.SS4.3.p3.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.10.m10.1b"><apply id="S3.SS4.3.p3.10.m10.1.1.cmml" xref="S3.SS4.3.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.10.m10.1.1.1.cmml" xref="S3.SS4.3.p3.10.m10.1.1">subscript</csymbol><ci id="S3.SS4.3.p3.10.m10.1.1.2.cmml" xref="S3.SS4.3.p3.10.m10.1.1.2">𝐼</ci><ci id="S3.SS4.3.p3.10.m10.1.1.3.cmml" xref="S3.SS4.3.p3.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.10.m10.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.10.m10.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> converges to the set <math alttext="C^{i+m}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.11.m11.1"><semantics id="S3.SS4.3.p3.11.m11.1a"><msup id="S3.SS4.3.p3.11.m11.1.1" xref="S3.SS4.3.p3.11.m11.1.1.cmml"><mi id="S3.SS4.3.p3.11.m11.1.1.2" xref="S3.SS4.3.p3.11.m11.1.1.2.cmml">C</mi><mrow id="S3.SS4.3.p3.11.m11.1.1.3" xref="S3.SS4.3.p3.11.m11.1.1.3.cmml"><mi id="S3.SS4.3.p3.11.m11.1.1.3.2" xref="S3.SS4.3.p3.11.m11.1.1.3.2.cmml">i</mi><mo id="S3.SS4.3.p3.11.m11.1.1.3.1" xref="S3.SS4.3.p3.11.m11.1.1.3.1.cmml">+</mo><mi id="S3.SS4.3.p3.11.m11.1.1.3.3" xref="S3.SS4.3.p3.11.m11.1.1.3.3.cmml">m</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.11.m11.1b"><apply id="S3.SS4.3.p3.11.m11.1.1.cmml" xref="S3.SS4.3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.11.m11.1.1.1.cmml" xref="S3.SS4.3.p3.11.m11.1.1">superscript</csymbol><ci id="S3.SS4.3.p3.11.m11.1.1.2.cmml" xref="S3.SS4.3.p3.11.m11.1.1.2">𝐶</ci><apply id="S3.SS4.3.p3.11.m11.1.1.3.cmml" xref="S3.SS4.3.p3.11.m11.1.1.3"><plus id="S3.SS4.3.p3.11.m11.1.1.3.1.cmml" xref="S3.SS4.3.p3.11.m11.1.1.3.1"></plus><ci id="S3.SS4.3.p3.11.m11.1.1.3.2.cmml" xref="S3.SS4.3.p3.11.m11.1.1.3.2">𝑖</ci><ci id="S3.SS4.3.p3.11.m11.1.1.3.3.cmml" xref="S3.SS4.3.p3.11.m11.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.11.m11.1c">C^{i+m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.11.m11.1d">italic_C start_POSTSUPERSCRIPT italic_i + italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="i\in[t]" class="ltx_Math" display="inline" id="S3.SS4.3.p3.12.m12.1"><semantics id="S3.SS4.3.p3.12.m12.1a"><mrow id="S3.SS4.3.p3.12.m12.1.2" xref="S3.SS4.3.p3.12.m12.1.2.cmml"><mi id="S3.SS4.3.p3.12.m12.1.2.2" xref="S3.SS4.3.p3.12.m12.1.2.2.cmml">i</mi><mo id="S3.SS4.3.p3.12.m12.1.2.1" xref="S3.SS4.3.p3.12.m12.1.2.1.cmml">∈</mo><mrow id="S3.SS4.3.p3.12.m12.1.2.3.2" xref="S3.SS4.3.p3.12.m12.1.2.3.1.cmml"><mo id="S3.SS4.3.p3.12.m12.1.2.3.2.1" stretchy="false" xref="S3.SS4.3.p3.12.m12.1.2.3.1.1.cmml">[</mo><mi id="S3.SS4.3.p3.12.m12.1.1" xref="S3.SS4.3.p3.12.m12.1.1.cmml">t</mi><mo id="S3.SS4.3.p3.12.m12.1.2.3.2.2" stretchy="false" xref="S3.SS4.3.p3.12.m12.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.12.m12.1b"><apply id="S3.SS4.3.p3.12.m12.1.2.cmml" xref="S3.SS4.3.p3.12.m12.1.2"><in id="S3.SS4.3.p3.12.m12.1.2.1.cmml" xref="S3.SS4.3.p3.12.m12.1.2.1"></in><ci id="S3.SS4.3.p3.12.m12.1.2.2.cmml" xref="S3.SS4.3.p3.12.m12.1.2.2">𝑖</ci><apply id="S3.SS4.3.p3.12.m12.1.2.3.1.cmml" xref="S3.SS4.3.p3.12.m12.1.2.3.2"><csymbol cd="latexml" id="S3.SS4.3.p3.12.m12.1.2.3.1.1.cmml" xref="S3.SS4.3.p3.12.m12.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS4.3.p3.12.m12.1.1.cmml" xref="S3.SS4.3.p3.12.m12.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.12.m12.1c">i\in[t]</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.12.m12.1d">italic_i ∈ [ italic_t ]</annotation></semantics></math>. (This is not an extended sequence that generates <math alttext="A" class="ltx_Math" display="inline" id="S3.SS4.3.p3.13.m13.1"><semantics id="S3.SS4.3.p3.13.m13.1a"><mi id="S3.SS4.3.p3.13.m13.1.1" xref="S3.SS4.3.p3.13.m13.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.13.m13.1b"><ci id="S3.SS4.3.p3.13.m13.1.1.cmml" xref="S3.SS4.3.p3.13.m13.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.13.m13.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.13.m13.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS4.3.p3.14.m14.1"><semantics id="S3.SS4.3.p3.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.14.m14.1.1" xref="S3.SS4.3.p3.14.m14.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.14.m14.1b"><ci id="S3.SS4.3.p3.14.m14.1.1.cmml" xref="S3.SS4.3.p3.14.m14.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.14.m14.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.14.m14.1d">caligraphic_B</annotation></semantics></math>, since the corresponding operations are not acyclic. However, the relation between the sets is clear.)</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem31"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem31.1.1.1">Claim 31</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem31.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem31.p1"> <p class="ltx_p" id="Thmtheorem31.p1.9"><span class="ltx_text ltx_font_italic" id="Thmtheorem31.p1.9.9">If <math alttext="I_{i}=K_{i_{1}}\star_{i}K_{i_{2}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.1.1.m1.1"><semantics id="Thmtheorem31.p1.1.1.m1.1a"><mrow id="Thmtheorem31.p1.1.1.m1.1.1" xref="Thmtheorem31.p1.1.1.m1.1.1.cmml"><msub id="Thmtheorem31.p1.1.1.m1.1.1.2" xref="Thmtheorem31.p1.1.1.m1.1.1.2.cmml"><mi id="Thmtheorem31.p1.1.1.m1.1.1.2.2" xref="Thmtheorem31.p1.1.1.m1.1.1.2.2.cmml">I</mi><mi id="Thmtheorem31.p1.1.1.m1.1.1.2.3" xref="Thmtheorem31.p1.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="Thmtheorem31.p1.1.1.m1.1.1.1" xref="Thmtheorem31.p1.1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmtheorem31.p1.1.1.m1.1.1.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.cmml"><msub id="Thmtheorem31.p1.1.1.m1.1.1.3.2" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.cmml"><mi id="Thmtheorem31.p1.1.1.m1.1.1.3.2.2" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.2.cmml">K</mi><msub id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.cmml"><mi id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.2" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.2.cmml">i</mi><mn id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.3.cmml">1</mn></msub></msub><msub id="Thmtheorem31.p1.1.1.m1.1.1.3.1" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1.cmml"><mo id="Thmtheorem31.p1.1.1.m1.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1.2.cmml">⋆</mo><mi id="Thmtheorem31.p1.1.1.m1.1.1.3.1.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1.3.cmml">i</mi></msub><msub id="Thmtheorem31.p1.1.1.m1.1.1.3.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.cmml"><mi id="Thmtheorem31.p1.1.1.m1.1.1.3.3.2" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.2.cmml">K</mi><msub id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.cmml"><mi id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.2" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.2.cmml">i</mi><mn id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.3" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.3.cmml">2</mn></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.1.1.m1.1b"><apply id="Thmtheorem31.p1.1.1.m1.1.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1"><eq id="Thmtheorem31.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.1"></eq><apply id="Thmtheorem31.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.2.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.2.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.2.2">𝐼</ci><ci id="Thmtheorem31.p1.1.1.m1.1.1.2.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3"><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.3.1.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.1.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1.2">⋆</ci><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.1.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.1.3">𝑖</ci></apply><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.3.2.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.2.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.2">𝐾</ci><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.2">𝑖</ci><cn id="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.3.cmml" type="integer" xref="Thmtheorem31.p1.1.1.m1.1.1.3.2.3.3">1</cn></apply></apply><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.3.3.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.3.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.2">𝐾</ci><apply id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.1.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3">subscript</csymbol><ci id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.2.cmml" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.2">𝑖</ci><cn id="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="Thmtheorem31.p1.1.1.m1.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.1.1.m1.1c">I_{i}=K_{i_{1}}\star_{i}K_{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.1.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in[t]" class="ltx_Math" display="inline" id="Thmtheorem31.p1.2.2.m2.1"><semantics id="Thmtheorem31.p1.2.2.m2.1a"><mrow id="Thmtheorem31.p1.2.2.m2.1.2" xref="Thmtheorem31.p1.2.2.m2.1.2.cmml"><mi id="Thmtheorem31.p1.2.2.m2.1.2.2" xref="Thmtheorem31.p1.2.2.m2.1.2.2.cmml">i</mi><mo id="Thmtheorem31.p1.2.2.m2.1.2.1" xref="Thmtheorem31.p1.2.2.m2.1.2.1.cmml">∈</mo><mrow id="Thmtheorem31.p1.2.2.m2.1.2.3.2" xref="Thmtheorem31.p1.2.2.m2.1.2.3.1.cmml"><mo id="Thmtheorem31.p1.2.2.m2.1.2.3.2.1" stretchy="false" xref="Thmtheorem31.p1.2.2.m2.1.2.3.1.1.cmml">[</mo><mi id="Thmtheorem31.p1.2.2.m2.1.1" xref="Thmtheorem31.p1.2.2.m2.1.1.cmml">t</mi><mo id="Thmtheorem31.p1.2.2.m2.1.2.3.2.2" stretchy="false" xref="Thmtheorem31.p1.2.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.2.2.m2.1b"><apply id="Thmtheorem31.p1.2.2.m2.1.2.cmml" xref="Thmtheorem31.p1.2.2.m2.1.2"><in id="Thmtheorem31.p1.2.2.m2.1.2.1.cmml" xref="Thmtheorem31.p1.2.2.m2.1.2.1"></in><ci id="Thmtheorem31.p1.2.2.m2.1.2.2.cmml" xref="Thmtheorem31.p1.2.2.m2.1.2.2">𝑖</ci><apply id="Thmtheorem31.p1.2.2.m2.1.2.3.1.cmml" xref="Thmtheorem31.p1.2.2.m2.1.2.3.2"><csymbol cd="latexml" id="Thmtheorem31.p1.2.2.m2.1.2.3.1.1.cmml" xref="Thmtheorem31.p1.2.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem31.p1.2.2.m2.1.1.cmml" xref="Thmtheorem31.p1.2.2.m2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.2.2.m2.1c">i\in[t]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.2.2.m2.1d">italic_i ∈ [ italic_t ]</annotation></semantics></math>, then <math alttext="C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.3.3.m3.1"><semantics id="Thmtheorem31.p1.3.3.m3.1a"><mrow id="Thmtheorem31.p1.3.3.m3.1.1" xref="Thmtheorem31.p1.3.3.m3.1.1.cmml"><msup id="Thmtheorem31.p1.3.3.m3.1.1.2" xref="Thmtheorem31.p1.3.3.m3.1.1.2.cmml"><mi id="Thmtheorem31.p1.3.3.m3.1.1.2.2" xref="Thmtheorem31.p1.3.3.m3.1.1.2.2.cmml">C</mi><mi id="Thmtheorem31.p1.3.3.m3.1.1.2.3" xref="Thmtheorem31.p1.3.3.m3.1.1.2.3.cmml">j</mi></msup><mo id="Thmtheorem31.p1.3.3.m3.1.1.1" xref="Thmtheorem31.p1.3.3.m3.1.1.1.cmml">=</mo><mrow id="Thmtheorem31.p1.3.3.m3.1.1.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.cmml"><msup id="Thmtheorem31.p1.3.3.m3.1.1.3.2" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.cmml"><mi id="Thmtheorem31.p1.3.3.m3.1.1.3.2.2" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.2.cmml">C</mi><msup id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.cmml"><mi id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.2" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.2.cmml">j</mi><mo id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.3.cmml">′</mo></msup></msup><msub id="Thmtheorem31.p1.3.3.m3.1.1.3.1" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1.cmml"><mo id="Thmtheorem31.p1.3.3.m3.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1.2.cmml">⋄</mo><mi id="Thmtheorem31.p1.3.3.m3.1.1.3.1.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1.3.cmml">j</mi></msub><msup id="Thmtheorem31.p1.3.3.m3.1.1.3.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.cmml"><mi id="Thmtheorem31.p1.3.3.m3.1.1.3.3.2" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.2.cmml">C</mi><msup id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.cmml"><mi id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.2" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.2.cmml">j</mi><mo id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.3" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.3.3.m3.1b"><apply id="Thmtheorem31.p1.3.3.m3.1.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1"><eq id="Thmtheorem31.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.1"></eq><apply id="Thmtheorem31.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.2.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.2">superscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.2.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.2.2">𝐶</ci><ci id="Thmtheorem31.p1.3.3.m3.1.1.2.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.2.3">𝑗</ci></apply><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3"><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.3.1.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1">subscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.1.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1.2">⋄</ci><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.1.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.1.3">𝑗</ci></apply><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.3.2.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2">superscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.2.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.2">𝐶</ci><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3">superscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.2">𝑗</ci><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.2.3.3">′</ci></apply></apply><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.3.3.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3">superscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.3.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.2">𝐶</ci><apply id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.1.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3">superscript</csymbol><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.2.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.2">𝑗</ci><ci id="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.3.cmml" xref="Thmtheorem31.p1.3.3.m3.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.3.3.m3.1c">C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.3.3.m3.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="j=i+m" class="ltx_Math" display="inline" id="Thmtheorem31.p1.4.4.m4.1"><semantics id="Thmtheorem31.p1.4.4.m4.1a"><mrow id="Thmtheorem31.p1.4.4.m4.1.1" xref="Thmtheorem31.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem31.p1.4.4.m4.1.1.2" xref="Thmtheorem31.p1.4.4.m4.1.1.2.cmml">j</mi><mo id="Thmtheorem31.p1.4.4.m4.1.1.1" xref="Thmtheorem31.p1.4.4.m4.1.1.1.cmml">=</mo><mrow id="Thmtheorem31.p1.4.4.m4.1.1.3" xref="Thmtheorem31.p1.4.4.m4.1.1.3.cmml"><mi id="Thmtheorem31.p1.4.4.m4.1.1.3.2" xref="Thmtheorem31.p1.4.4.m4.1.1.3.2.cmml">i</mi><mo id="Thmtheorem31.p1.4.4.m4.1.1.3.1" xref="Thmtheorem31.p1.4.4.m4.1.1.3.1.cmml">+</mo><mi id="Thmtheorem31.p1.4.4.m4.1.1.3.3" xref="Thmtheorem31.p1.4.4.m4.1.1.3.3.cmml">m</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.4.4.m4.1b"><apply id="Thmtheorem31.p1.4.4.m4.1.1.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1"><eq id="Thmtheorem31.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.1"></eq><ci id="Thmtheorem31.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.2">𝑗</ci><apply id="Thmtheorem31.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.3"><plus id="Thmtheorem31.p1.4.4.m4.1.1.3.1.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.3.1"></plus><ci id="Thmtheorem31.p1.4.4.m4.1.1.3.2.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.3.2">𝑖</ci><ci id="Thmtheorem31.p1.4.4.m4.1.1.3.3.cmml" xref="Thmtheorem31.p1.4.4.m4.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.4.4.m4.1c">j=i+m</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.4.4.m4.1d">italic_j = italic_i + italic_m</annotation></semantics></math> and <math alttext="\diamond_{j}=\star_{i}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.5.5.m5.3"><semantics id="Thmtheorem31.p1.5.5.m5.3a"><mrow id="Thmtheorem31.p1.5.5.m5.3.3.2" xref="Thmtheorem31.p1.5.5.m5.3.3.3.cmml"><msub id="Thmtheorem31.p1.5.5.m5.2.2.1.1" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1.cmml"><mo id="Thmtheorem31.p1.5.5.m5.2.2.1.1.2" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1.2.cmml">⋄</mo><mi id="Thmtheorem31.p1.5.5.m5.2.2.1.1.3" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1.3.cmml">j</mi></msub><mo id="Thmtheorem31.p1.5.5.m5.3.3.2.3" lspace="0em" xref="Thmtheorem31.p1.5.5.m5.3.3.3.cmml">⁣</mo><mo id="Thmtheorem31.p1.5.5.m5.1.1" xref="Thmtheorem31.p1.5.5.m5.1.1.cmml">=</mo><mo id="Thmtheorem31.p1.5.5.m5.3.3.2.4" lspace="0em" xref="Thmtheorem31.p1.5.5.m5.3.3.3.cmml">⁣</mo><msub id="Thmtheorem31.p1.5.5.m5.3.3.2.2" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2.cmml"><mo id="Thmtheorem31.p1.5.5.m5.3.3.2.2.2" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2.2.cmml">⋆</mo><mi id="Thmtheorem31.p1.5.5.m5.3.3.2.2.3" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.5.5.m5.3b"><list id="Thmtheorem31.p1.5.5.m5.3.3.3.cmml" xref="Thmtheorem31.p1.5.5.m5.3.3.2"><apply id="Thmtheorem31.p1.5.5.m5.2.2.1.1.cmml" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.5.5.m5.2.2.1.1.1.cmml" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1">subscript</csymbol><ci id="Thmtheorem31.p1.5.5.m5.2.2.1.1.2.cmml" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1.2">⋄</ci><ci id="Thmtheorem31.p1.5.5.m5.2.2.1.1.3.cmml" xref="Thmtheorem31.p1.5.5.m5.2.2.1.1.3">𝑗</ci></apply><eq id="Thmtheorem31.p1.5.5.m5.1.1.cmml" xref="Thmtheorem31.p1.5.5.m5.1.1"></eq><apply id="Thmtheorem31.p1.5.5.m5.3.3.2.2.cmml" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="Thmtheorem31.p1.5.5.m5.3.3.2.2.1.cmml" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2">subscript</csymbol><ci id="Thmtheorem31.p1.5.5.m5.3.3.2.2.2.cmml" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2.2">⋆</ci><ci id="Thmtheorem31.p1.5.5.m5.3.3.2.2.3.cmml" xref="Thmtheorem31.p1.5.5.m5.3.3.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.5.5.m5.3c">\diamond_{j}=\star_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.5.5.m5.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="C^{j^{\prime}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.6.6.m6.1"><semantics id="Thmtheorem31.p1.6.6.m6.1a"><msup id="Thmtheorem31.p1.6.6.m6.1.1" xref="Thmtheorem31.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem31.p1.6.6.m6.1.1.2" xref="Thmtheorem31.p1.6.6.m6.1.1.2.cmml">C</mi><msup id="Thmtheorem31.p1.6.6.m6.1.1.3" xref="Thmtheorem31.p1.6.6.m6.1.1.3.cmml"><mi id="Thmtheorem31.p1.6.6.m6.1.1.3.2" xref="Thmtheorem31.p1.6.6.m6.1.1.3.2.cmml">j</mi><mo id="Thmtheorem31.p1.6.6.m6.1.1.3.3" xref="Thmtheorem31.p1.6.6.m6.1.1.3.3.cmml">′</mo></msup></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.6.6.m6.1b"><apply id="Thmtheorem31.p1.6.6.m6.1.1.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.6.6.m6.1.1.1.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1">superscript</csymbol><ci id="Thmtheorem31.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1.2">𝐶</ci><apply id="Thmtheorem31.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.6.6.m6.1.1.3.1.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="Thmtheorem31.p1.6.6.m6.1.1.3.2.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1.3.2">𝑗</ci><ci id="Thmtheorem31.p1.6.6.m6.1.1.3.3.cmml" xref="Thmtheorem31.p1.6.6.m6.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.6.6.m6.1c">C^{j^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.6.6.m6.1d">italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.7.7.m7.1"><semantics id="Thmtheorem31.p1.7.7.m7.1a"><msup id="Thmtheorem31.p1.7.7.m7.1.1" xref="Thmtheorem31.p1.7.7.m7.1.1.cmml"><mi id="Thmtheorem31.p1.7.7.m7.1.1.2" xref="Thmtheorem31.p1.7.7.m7.1.1.2.cmml">C</mi><msup id="Thmtheorem31.p1.7.7.m7.1.1.3" xref="Thmtheorem31.p1.7.7.m7.1.1.3.cmml"><mi id="Thmtheorem31.p1.7.7.m7.1.1.3.2" xref="Thmtheorem31.p1.7.7.m7.1.1.3.2.cmml">j</mi><mo id="Thmtheorem31.p1.7.7.m7.1.1.3.3" xref="Thmtheorem31.p1.7.7.m7.1.1.3.3.cmml">′′</mo></msup></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.7.7.m7.1b"><apply id="Thmtheorem31.p1.7.7.m7.1.1.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.7.7.m7.1.1.1.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1">superscript</csymbol><ci id="Thmtheorem31.p1.7.7.m7.1.1.2.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1.2">𝐶</ci><apply id="Thmtheorem31.p1.7.7.m7.1.1.3.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.7.7.m7.1.1.3.1.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1.3">superscript</csymbol><ci id="Thmtheorem31.p1.7.7.m7.1.1.3.2.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1.3.2">𝑗</ci><ci id="Thmtheorem31.p1.7.7.m7.1.1.3.3.cmml" xref="Thmtheorem31.p1.7.7.m7.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.7.7.m7.1c">C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.7.7.m7.1d">italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> are the sets to which <math alttext="K_{i_{1}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.8.8.m8.1"><semantics id="Thmtheorem31.p1.8.8.m8.1a"><msub id="Thmtheorem31.p1.8.8.m8.1.1" xref="Thmtheorem31.p1.8.8.m8.1.1.cmml"><mi id="Thmtheorem31.p1.8.8.m8.1.1.2" xref="Thmtheorem31.p1.8.8.m8.1.1.2.cmml">K</mi><msub id="Thmtheorem31.p1.8.8.m8.1.1.3" xref="Thmtheorem31.p1.8.8.m8.1.1.3.cmml"><mi id="Thmtheorem31.p1.8.8.m8.1.1.3.2" xref="Thmtheorem31.p1.8.8.m8.1.1.3.2.cmml">i</mi><mn id="Thmtheorem31.p1.8.8.m8.1.1.3.3" xref="Thmtheorem31.p1.8.8.m8.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.8.8.m8.1b"><apply id="Thmtheorem31.p1.8.8.m8.1.1.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.8.8.m8.1.1.1.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1">subscript</csymbol><ci id="Thmtheorem31.p1.8.8.m8.1.1.2.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1.2">𝐾</ci><apply id="Thmtheorem31.p1.8.8.m8.1.1.3.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.8.8.m8.1.1.3.1.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="Thmtheorem31.p1.8.8.m8.1.1.3.2.cmml" xref="Thmtheorem31.p1.8.8.m8.1.1.3.2">𝑖</ci><cn id="Thmtheorem31.p1.8.8.m8.1.1.3.3.cmml" type="integer" xref="Thmtheorem31.p1.8.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.8.8.m8.1c">K_{i_{1}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.8.8.m8.1d">italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="K_{i_{2}}" class="ltx_Math" display="inline" id="Thmtheorem31.p1.9.9.m9.1"><semantics id="Thmtheorem31.p1.9.9.m9.1a"><msub id="Thmtheorem31.p1.9.9.m9.1.1" xref="Thmtheorem31.p1.9.9.m9.1.1.cmml"><mi id="Thmtheorem31.p1.9.9.m9.1.1.2" xref="Thmtheorem31.p1.9.9.m9.1.1.2.cmml">K</mi><msub id="Thmtheorem31.p1.9.9.m9.1.1.3" xref="Thmtheorem31.p1.9.9.m9.1.1.3.cmml"><mi id="Thmtheorem31.p1.9.9.m9.1.1.3.2" xref="Thmtheorem31.p1.9.9.m9.1.1.3.2.cmml">i</mi><mn id="Thmtheorem31.p1.9.9.m9.1.1.3.3" xref="Thmtheorem31.p1.9.9.m9.1.1.3.3.cmml">2</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem31.p1.9.9.m9.1b"><apply id="Thmtheorem31.p1.9.9.m9.1.1.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="Thmtheorem31.p1.9.9.m9.1.1.1.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1">subscript</csymbol><ci id="Thmtheorem31.p1.9.9.m9.1.1.2.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1.2">𝐾</ci><apply id="Thmtheorem31.p1.9.9.m9.1.1.3.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem31.p1.9.9.m9.1.1.3.1.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="Thmtheorem31.p1.9.9.m9.1.1.3.2.cmml" xref="Thmtheorem31.p1.9.9.m9.1.1.3.2">𝑖</ci><cn id="Thmtheorem31.p1.9.9.m9.1.1.3.3.cmml" type="integer" xref="Thmtheorem31.p1.9.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem31.p1.9.9.m9.1c">K_{i_{2}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem31.p1.9.9.m9.1d">italic_K start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> converge, respectively.</span></p> </div> </div> <div class="ltx_para" id="S3.SS4.4.p4"> <p class="ltx_p" id="S3.SS4.4.p4.4">In order to see this, recall that during the evaluation of the syntactic sequence <math alttext="I^{\ell+1}_{i}=I^{\ell}_{i}\cup(K^{\ell}_{i_{1}}\star_{i}K^{\ell}_{i_{2}})" class="ltx_Math" display="inline" id="S3.SS4.4.p4.1.m1.1"><semantics id="S3.SS4.4.p4.1.m1.1a"><mrow id="S3.SS4.4.p4.1.m1.1.1" xref="S3.SS4.4.p4.1.m1.1.1.cmml"><msubsup id="S3.SS4.4.p4.1.m1.1.1.3" xref="S3.SS4.4.p4.1.m1.1.1.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.3.2.2" xref="S3.SS4.4.p4.1.m1.1.1.3.2.2.cmml">I</mi><mi id="S3.SS4.4.p4.1.m1.1.1.3.3" xref="S3.SS4.4.p4.1.m1.1.1.3.3.cmml">i</mi><mrow id="S3.SS4.4.p4.1.m1.1.1.3.2.3" xref="S3.SS4.4.p4.1.m1.1.1.3.2.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.3.2.3.2" mathvariant="normal" xref="S3.SS4.4.p4.1.m1.1.1.3.2.3.2.cmml">ℓ</mi><mo id="S3.SS4.4.p4.1.m1.1.1.3.2.3.1" xref="S3.SS4.4.p4.1.m1.1.1.3.2.3.1.cmml">+</mo><mn id="S3.SS4.4.p4.1.m1.1.1.3.2.3.3" xref="S3.SS4.4.p4.1.m1.1.1.3.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S3.SS4.4.p4.1.m1.1.1.2" xref="S3.SS4.4.p4.1.m1.1.1.2.cmml">=</mo><mrow id="S3.SS4.4.p4.1.m1.1.1.1" xref="S3.SS4.4.p4.1.m1.1.1.1.cmml"><msubsup id="S3.SS4.4.p4.1.m1.1.1.1.3" xref="S3.SS4.4.p4.1.m1.1.1.1.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.1.3.2.2" xref="S3.SS4.4.p4.1.m1.1.1.1.3.2.2.cmml">I</mi><mi id="S3.SS4.4.p4.1.m1.1.1.1.3.3" xref="S3.SS4.4.p4.1.m1.1.1.1.3.3.cmml">i</mi><mi id="S3.SS4.4.p4.1.m1.1.1.1.3.2.3" mathvariant="normal" xref="S3.SS4.4.p4.1.m1.1.1.1.3.2.3.cmml">ℓ</mi></msubsup><mo id="S3.SS4.4.p4.1.m1.1.1.1.2" xref="S3.SS4.4.p4.1.m1.1.1.1.2.cmml">∪</mo><mrow id="S3.SS4.4.p4.1.m1.1.1.1.1.1" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.4.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.cmml"><msubsup id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.2" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.2.cmml">K</mi><msub id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.2" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.2.cmml">i</mi><mn id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.3.cmml">1</mn></msub><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.3" mathvariant="normal" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.3.cmml">ℓ</mi></msubsup><msub id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.2.cmml">⋆</mo><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><msubsup id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.2" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.2.cmml">K</mi><msub id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.cmml"><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.2" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.2.cmml">i</mi><mn id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.3" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.3.cmml">2</mn></msub><mi id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.3" mathvariant="normal" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.3.cmml">ℓ</mi></msubsup></mrow><mo id="S3.SS4.4.p4.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.4.p4.1.m1.1b"><apply id="S3.SS4.4.p4.1.m1.1.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1"><eq id="S3.SS4.4.p4.1.m1.1.1.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.2"></eq><apply id="S3.SS4.4.p4.1.m1.1.1.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.3.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3">subscript</csymbol><apply id="S3.SS4.4.p4.1.m1.1.1.3.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.3.2.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.3.2.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3.2.2">𝐼</ci><apply id="S3.SS4.4.p4.1.m1.1.1.3.2.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.3.2.3"><plus id="S3.SS4.4.p4.1.m1.1.1.3.2.3.1.cmml" 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xref="S3.SS4.4.p4.1.m1.1.1.1.3.2.3">ℓ</ci></apply><ci id="S3.SS4.4.p4.1.m1.1.1.1.3.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.3.3">𝑖</ci></apply><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1"><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.2">⋆</ci><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2">subscript</csymbol><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.2">𝐾</ci><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.2.3">ℓ</ci></apply><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3">subscript</csymbol><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.2">𝐾</ci><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.2.3">ℓ</ci></apply><apply id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.2.cmml" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.2">𝑖</ci><cn id="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.SS4.4.p4.1.m1.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.4.p4.1.m1.1c">I^{\ell+1}_{i}=I^{\ell}_{i}\cup(K^{\ell}_{i_{1}}\star_{i}K^{\ell}_{i_{2}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.4.p4.1.m1.1d">italic_I start_POSTSUPERSCRIPT roman_ℓ + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∪ ( italic_K start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. Since the evaluation is monotone, and <math alttext="C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}" class="ltx_Math" display="inline" id="S3.SS4.4.p4.2.m2.6"><semantics id="S3.SS4.4.p4.2.m2.6a"><mrow id="S3.SS4.4.p4.2.m2.6.6.4" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml"><msup id="S3.SS4.4.p4.2.m2.3.3.1.1" xref="S3.SS4.4.p4.2.m2.3.3.1.1.cmml"><mi id="S3.SS4.4.p4.2.m2.3.3.1.1.2" xref="S3.SS4.4.p4.2.m2.3.3.1.1.2.cmml">C</mi><mn id="S3.SS4.4.p4.2.m2.3.3.1.1.3" xref="S3.SS4.4.p4.2.m2.3.3.1.1.3.cmml">1</mn></msup><mo id="S3.SS4.4.p4.2.m2.6.6.4.5" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml">,</mo><mi id="S3.SS4.4.p4.2.m2.1.1" mathvariant="normal" xref="S3.SS4.4.p4.2.m2.1.1.cmml">…</mi><mo id="S3.SS4.4.p4.2.m2.6.6.4.6" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml">,</mo><msup id="S3.SS4.4.p4.2.m2.4.4.2.2" xref="S3.SS4.4.p4.2.m2.4.4.2.2.cmml"><mi id="S3.SS4.4.p4.2.m2.4.4.2.2.2" xref="S3.SS4.4.p4.2.m2.4.4.2.2.2.cmml">C</mi><mi id="S3.SS4.4.p4.2.m2.4.4.2.2.3" xref="S3.SS4.4.p4.2.m2.4.4.2.2.3.cmml">m</mi></msup><mo id="S3.SS4.4.p4.2.m2.6.6.4.7" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml">,</mo><msup id="S3.SS4.4.p4.2.m2.5.5.3.3" xref="S3.SS4.4.p4.2.m2.5.5.3.3.cmml"><mi id="S3.SS4.4.p4.2.m2.5.5.3.3.2" xref="S3.SS4.4.p4.2.m2.5.5.3.3.2.cmml">C</mi><mrow id="S3.SS4.4.p4.2.m2.5.5.3.3.3" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.cmml"><mi id="S3.SS4.4.p4.2.m2.5.5.3.3.3.2" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.2.cmml">m</mi><mo id="S3.SS4.4.p4.2.m2.5.5.3.3.3.1" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.1.cmml">+</mo><mn id="S3.SS4.4.p4.2.m2.5.5.3.3.3.3" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS4.4.p4.2.m2.6.6.4.8" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml">,</mo><mi id="S3.SS4.4.p4.2.m2.2.2" mathvariant="normal" xref="S3.SS4.4.p4.2.m2.2.2.cmml">…</mi><mo id="S3.SS4.4.p4.2.m2.6.6.4.9" xref="S3.SS4.4.p4.2.m2.6.6.5.cmml">,</mo><msup id="S3.SS4.4.p4.2.m2.6.6.4.4" xref="S3.SS4.4.p4.2.m2.6.6.4.4.cmml"><mi id="S3.SS4.4.p4.2.m2.6.6.4.4.2" xref="S3.SS4.4.p4.2.m2.6.6.4.4.2.cmml">C</mi><mrow id="S3.SS4.4.p4.2.m2.6.6.4.4.3" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.cmml"><mi id="S3.SS4.4.p4.2.m2.6.6.4.4.3.2" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.2.cmml">m</mi><mo id="S3.SS4.4.p4.2.m2.6.6.4.4.3.1" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.1.cmml">+</mo><mi id="S3.SS4.4.p4.2.m2.6.6.4.4.3.3" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.3.cmml">t</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.4.p4.2.m2.6b"><list id="S3.SS4.4.p4.2.m2.6.6.5.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4"><apply id="S3.SS4.4.p4.2.m2.3.3.1.1.cmml" xref="S3.SS4.4.p4.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.4.p4.2.m2.3.3.1.1.1.cmml" xref="S3.SS4.4.p4.2.m2.3.3.1.1">superscript</csymbol><ci id="S3.SS4.4.p4.2.m2.3.3.1.1.2.cmml" xref="S3.SS4.4.p4.2.m2.3.3.1.1.2">𝐶</ci><cn id="S3.SS4.4.p4.2.m2.3.3.1.1.3.cmml" type="integer" xref="S3.SS4.4.p4.2.m2.3.3.1.1.3">1</cn></apply><ci id="S3.SS4.4.p4.2.m2.1.1.cmml" xref="S3.SS4.4.p4.2.m2.1.1">…</ci><apply id="S3.SS4.4.p4.2.m2.4.4.2.2.cmml" xref="S3.SS4.4.p4.2.m2.4.4.2.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.2.m2.4.4.2.2.1.cmml" xref="S3.SS4.4.p4.2.m2.4.4.2.2">superscript</csymbol><ci id="S3.SS4.4.p4.2.m2.4.4.2.2.2.cmml" xref="S3.SS4.4.p4.2.m2.4.4.2.2.2">𝐶</ci><ci id="S3.SS4.4.p4.2.m2.4.4.2.2.3.cmml" xref="S3.SS4.4.p4.2.m2.4.4.2.2.3">𝑚</ci></apply><apply id="S3.SS4.4.p4.2.m2.5.5.3.3.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.2.m2.5.5.3.3.1.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3">superscript</csymbol><ci id="S3.SS4.4.p4.2.m2.5.5.3.3.2.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3.2">𝐶</ci><apply id="S3.SS4.4.p4.2.m2.5.5.3.3.3.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3"><plus id="S3.SS4.4.p4.2.m2.5.5.3.3.3.1.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.1"></plus><ci id="S3.SS4.4.p4.2.m2.5.5.3.3.3.2.cmml" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.2">𝑚</ci><cn id="S3.SS4.4.p4.2.m2.5.5.3.3.3.3.cmml" type="integer" xref="S3.SS4.4.p4.2.m2.5.5.3.3.3.3">1</cn></apply></apply><ci id="S3.SS4.4.p4.2.m2.2.2.cmml" xref="S3.SS4.4.p4.2.m2.2.2">…</ci><apply id="S3.SS4.4.p4.2.m2.6.6.4.4.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4"><csymbol cd="ambiguous" id="S3.SS4.4.p4.2.m2.6.6.4.4.1.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4">superscript</csymbol><ci id="S3.SS4.4.p4.2.m2.6.6.4.4.2.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4.2">𝐶</ci><apply id="S3.SS4.4.p4.2.m2.6.6.4.4.3.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3"><plus id="S3.SS4.4.p4.2.m2.6.6.4.4.3.1.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.1"></plus><ci id="S3.SS4.4.p4.2.m2.6.6.4.4.3.2.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.2">𝑚</ci><ci id="S3.SS4.4.p4.2.m2.6.6.4.4.3.3.cmml" xref="S3.SS4.4.p4.2.m2.6.6.4.4.3.3">𝑡</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.4.p4.2.m2.6c">C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.4.p4.2.m2.6d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_C start_POSTSUPERSCRIPT italic_m + 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> is the convergent sequence, we eventually have <math alttext="I^{\ell+1}_{i}=I^{\ell}_{i}=(K^{\ell}_{i_{1}}\star_{i}K^{\ell}_{i_{2}})" class="ltx_Math" display="inline" id="S3.SS4.4.p4.3.m3.1"><semantics id="S3.SS4.4.p4.3.m3.1a"><mrow id="S3.SS4.4.p4.3.m3.1.1" xref="S3.SS4.4.p4.3.m3.1.1.cmml"><msubsup id="S3.SS4.4.p4.3.m3.1.1.3" xref="S3.SS4.4.p4.3.m3.1.1.3.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.3.2.2" xref="S3.SS4.4.p4.3.m3.1.1.3.2.2.cmml">I</mi><mi id="S3.SS4.4.p4.3.m3.1.1.3.3" xref="S3.SS4.4.p4.3.m3.1.1.3.3.cmml">i</mi><mrow id="S3.SS4.4.p4.3.m3.1.1.3.2.3" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.3.2.3.2" mathvariant="normal" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.2.cmml">ℓ</mi><mo id="S3.SS4.4.p4.3.m3.1.1.3.2.3.1" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.1.cmml">+</mo><mn id="S3.SS4.4.p4.3.m3.1.1.3.2.3.3" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S3.SS4.4.p4.3.m3.1.1.4" xref="S3.SS4.4.p4.3.m3.1.1.4.cmml">=</mo><msubsup id="S3.SS4.4.p4.3.m3.1.1.5" xref="S3.SS4.4.p4.3.m3.1.1.5.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.5.2.2" xref="S3.SS4.4.p4.3.m3.1.1.5.2.2.cmml">I</mi><mi id="S3.SS4.4.p4.3.m3.1.1.5.3" xref="S3.SS4.4.p4.3.m3.1.1.5.3.cmml">i</mi><mi id="S3.SS4.4.p4.3.m3.1.1.5.2.3" mathvariant="normal" xref="S3.SS4.4.p4.3.m3.1.1.5.2.3.cmml">ℓ</mi></msubsup><mo id="S3.SS4.4.p4.3.m3.1.1.6" xref="S3.SS4.4.p4.3.m3.1.1.6.cmml">=</mo><mrow id="S3.SS4.4.p4.3.m3.1.1.1.1" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS4.4.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.4.p4.3.m3.1.1.1.1.1" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.cmml"><msubsup id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.2" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.2.cmml">K</mi><msub id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.2" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.2.cmml">i</mi><mn id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.3.cmml">1</mn></msub><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.3" mathvariant="normal" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.3.cmml">ℓ</mi></msubsup><msub id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.2.cmml">⋆</mo><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><msubsup id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.2" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.2.cmml">K</mi><msub id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.cmml"><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.2" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.2.cmml">i</mi><mn id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.3" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.3.cmml">2</mn></msub><mi id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.3" mathvariant="normal" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.3.cmml">ℓ</mi></msubsup></mrow><mo id="S3.SS4.4.p4.3.m3.1.1.1.1.3" stretchy="false" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.4.p4.3.m3.1b"><apply id="S3.SS4.4.p4.3.m3.1.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1"><and id="S3.SS4.4.p4.3.m3.1.1a.cmml" xref="S3.SS4.4.p4.3.m3.1.1"></and><apply id="S3.SS4.4.p4.3.m3.1.1b.cmml" xref="S3.SS4.4.p4.3.m3.1.1"><eq id="S3.SS4.4.p4.3.m3.1.1.4.cmml" xref="S3.SS4.4.p4.3.m3.1.1.4"></eq><apply id="S3.SS4.4.p4.3.m3.1.1.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.3.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3">subscript</csymbol><apply id="S3.SS4.4.p4.3.m3.1.1.3.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.3.2.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.3.2.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3.2.2">𝐼</ci><apply id="S3.SS4.4.p4.3.m3.1.1.3.2.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3"><plus id="S3.SS4.4.p4.3.m3.1.1.3.2.3.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.1"></plus><ci id="S3.SS4.4.p4.3.m3.1.1.3.2.3.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.2">ℓ</ci><cn id="S3.SS4.4.p4.3.m3.1.1.3.2.3.3.cmml" type="integer" xref="S3.SS4.4.p4.3.m3.1.1.3.2.3.3">1</cn></apply></apply><ci id="S3.SS4.4.p4.3.m3.1.1.3.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.3.3">𝑖</ci></apply><apply id="S3.SS4.4.p4.3.m3.1.1.5.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.5.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5">subscript</csymbol><apply id="S3.SS4.4.p4.3.m3.1.1.5.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.5.2.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5">superscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.5.2.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5.2.2">𝐼</ci><ci id="S3.SS4.4.p4.3.m3.1.1.5.2.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5.2.3">ℓ</ci></apply><ci id="S3.SS4.4.p4.3.m3.1.1.5.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.5.3">𝑖</ci></apply></apply><apply id="S3.SS4.4.p4.3.m3.1.1c.cmml" xref="S3.SS4.4.p4.3.m3.1.1"><eq id="S3.SS4.4.p4.3.m3.1.1.6.cmml" xref="S3.SS4.4.p4.3.m3.1.1.6"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.4.p4.3.m3.1.1.5.cmml" id="S3.SS4.4.p4.3.m3.1.1d.cmml" xref="S3.SS4.4.p4.3.m3.1.1"></share><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1"><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.2">⋆</ci><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2">subscript</csymbol><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.2">𝐾</ci><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.2.3">ℓ</ci></apply><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3">subscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3">subscript</csymbol><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.2">𝐾</ci><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.2.3">ℓ</ci></apply><apply id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3">subscript</csymbol><ci id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.2.cmml" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.2">𝑖</ci><cn id="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.SS4.4.p4.3.m3.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.4.p4.3.m3.1c">I^{\ell+1}_{i}=I^{\ell}_{i}=(K^{\ell}_{i_{1}}\star_{i}K^{\ell}_{i_{2}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.4.p4.3.m3.1d">italic_I start_POSTSUPERSCRIPT roman_ℓ + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( italic_K start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. Consequently, <math alttext="C^{j}=C^{j^{\prime}}\star_{i}C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.4.p4.4.m4.1"><semantics id="S3.SS4.4.p4.4.m4.1a"><mrow id="S3.SS4.4.p4.4.m4.1.1" xref="S3.SS4.4.p4.4.m4.1.1.cmml"><msup id="S3.SS4.4.p4.4.m4.1.1.2" xref="S3.SS4.4.p4.4.m4.1.1.2.cmml"><mi id="S3.SS4.4.p4.4.m4.1.1.2.2" xref="S3.SS4.4.p4.4.m4.1.1.2.2.cmml">C</mi><mi id="S3.SS4.4.p4.4.m4.1.1.2.3" xref="S3.SS4.4.p4.4.m4.1.1.2.3.cmml">j</mi></msup><mo id="S3.SS4.4.p4.4.m4.1.1.1" xref="S3.SS4.4.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS4.4.p4.4.m4.1.1.3" xref="S3.SS4.4.p4.4.m4.1.1.3.cmml"><msup id="S3.SS4.4.p4.4.m4.1.1.3.2" xref="S3.SS4.4.p4.4.m4.1.1.3.2.cmml"><mi id="S3.SS4.4.p4.4.m4.1.1.3.2.2" xref="S3.SS4.4.p4.4.m4.1.1.3.2.2.cmml">C</mi><msup id="S3.SS4.4.p4.4.m4.1.1.3.2.3" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3.cmml"><mi id="S3.SS4.4.p4.4.m4.1.1.3.2.3.2" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3.2.cmml">j</mi><mo id="S3.SS4.4.p4.4.m4.1.1.3.2.3.3" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3.3.cmml">′</mo></msup></msup><msub id="S3.SS4.4.p4.4.m4.1.1.3.1" xref="S3.SS4.4.p4.4.m4.1.1.3.1.cmml"><mo id="S3.SS4.4.p4.4.m4.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="S3.SS4.4.p4.4.m4.1.1.3.1.2.cmml">⋆</mo><mi id="S3.SS4.4.p4.4.m4.1.1.3.1.3" xref="S3.SS4.4.p4.4.m4.1.1.3.1.3.cmml">i</mi></msub><msup id="S3.SS4.4.p4.4.m4.1.1.3.3" xref="S3.SS4.4.p4.4.m4.1.1.3.3.cmml"><mi id="S3.SS4.4.p4.4.m4.1.1.3.3.2" xref="S3.SS4.4.p4.4.m4.1.1.3.3.2.cmml">C</mi><msup id="S3.SS4.4.p4.4.m4.1.1.3.3.3" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3.cmml"><mi id="S3.SS4.4.p4.4.m4.1.1.3.3.3.2" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3.2.cmml">j</mi><mo id="S3.SS4.4.p4.4.m4.1.1.3.3.3.3" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.4.p4.4.m4.1b"><apply id="S3.SS4.4.p4.4.m4.1.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1"><eq id="S3.SS4.4.p4.4.m4.1.1.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.1"></eq><apply id="S3.SS4.4.p4.4.m4.1.1.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.2.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.2">superscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.2.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.2.2">𝐶</ci><ci id="S3.SS4.4.p4.4.m4.1.1.2.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.2.3">𝑗</ci></apply><apply id="S3.SS4.4.p4.4.m4.1.1.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3"><apply id="S3.SS4.4.p4.4.m4.1.1.3.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.3.1.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.1">subscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.3.1.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.1.2">⋆</ci><ci id="S3.SS4.4.p4.4.m4.1.1.3.1.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.1.3">𝑖</ci></apply><apply id="S3.SS4.4.p4.4.m4.1.1.3.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.3.2.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.3.2.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2.2">𝐶</ci><apply id="S3.SS4.4.p4.4.m4.1.1.3.2.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.3.2.3.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3">superscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.3.2.3.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3.2">𝑗</ci><ci id="S3.SS4.4.p4.4.m4.1.1.3.2.3.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.2.3.3">′</ci></apply></apply><apply id="S3.SS4.4.p4.4.m4.1.1.3.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.3.3.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3">superscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.3.3.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3.2">𝐶</ci><apply id="S3.SS4.4.p4.4.m4.1.1.3.3.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.4.p4.4.m4.1.1.3.3.3.1.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3">superscript</csymbol><ci id="S3.SS4.4.p4.4.m4.1.1.3.3.3.2.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3.2">𝑗</ci><ci id="S3.SS4.4.p4.4.m4.1.1.3.3.3.3.cmml" xref="S3.SS4.4.p4.4.m4.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.4.p4.4.m4.1c">C^{j}=C^{j^{\prime}}\star_{i}C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.4.p4.4.m4.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> after the indices are appropriately renamed. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS4.5.p5"> <p class="ltx_p" id="S3.SS4.5.p5.6">For <math alttext="U=A^{c}" class="ltx_Math" display="inline" id="S3.SS4.5.p5.1.m1.1"><semantics id="S3.SS4.5.p5.1.m1.1a"><mrow id="S3.SS4.5.p5.1.m1.1.1" xref="S3.SS4.5.p5.1.m1.1.1.cmml"><mi id="S3.SS4.5.p5.1.m1.1.1.2" xref="S3.SS4.5.p5.1.m1.1.1.2.cmml">U</mi><mo id="S3.SS4.5.p5.1.m1.1.1.1" xref="S3.SS4.5.p5.1.m1.1.1.1.cmml">=</mo><msup id="S3.SS4.5.p5.1.m1.1.1.3" xref="S3.SS4.5.p5.1.m1.1.1.3.cmml"><mi id="S3.SS4.5.p5.1.m1.1.1.3.2" xref="S3.SS4.5.p5.1.m1.1.1.3.2.cmml">A</mi><mi id="S3.SS4.5.p5.1.m1.1.1.3.3" xref="S3.SS4.5.p5.1.m1.1.1.3.3.cmml">c</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p5.1.m1.1b"><apply id="S3.SS4.5.p5.1.m1.1.1.cmml" xref="S3.SS4.5.p5.1.m1.1.1"><eq id="S3.SS4.5.p5.1.m1.1.1.1.cmml" xref="S3.SS4.5.p5.1.m1.1.1.1"></eq><ci id="S3.SS4.5.p5.1.m1.1.1.2.cmml" xref="S3.SS4.5.p5.1.m1.1.1.2">𝑈</ci><apply id="S3.SS4.5.p5.1.m1.1.1.3.cmml" xref="S3.SS4.5.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p5.1.m1.1.1.3.1.cmml" xref="S3.SS4.5.p5.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS4.5.p5.1.m1.1.1.3.2.cmml" xref="S3.SS4.5.p5.1.m1.1.1.3.2">𝐴</ci><ci id="S3.SS4.5.p5.1.m1.1.1.3.3.cmml" xref="S3.SS4.5.p5.1.m1.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.1.m1.1c">U=A^{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.1.m1.1d">italic_U = italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="\Lambda\stackrel{{\scriptstyle\rm def}}{{=}}\{(C^{j^{\prime}}_{U},C^{j^{\prime% \prime}}_{U})\mid j\in\{m+1,\ldots,m+t\}~{}\text{and}~{}\diamond_{j}=\cap\}" class="ltx_Math" display="inline" id="S3.SS4.5.p5.2.m2.3"><semantics id="S3.SS4.5.p5.2.m2.3a"><mrow id="S3.SS4.5.p5.2.m2.3.3" xref="S3.SS4.5.p5.2.m2.3.3.cmml"><mi id="S3.SS4.5.p5.2.m2.3.3.4" mathvariant="normal" xref="S3.SS4.5.p5.2.m2.3.3.4.cmml">Λ</mi><mover id="S3.SS4.5.p5.2.m2.3.3.3" xref="S3.SS4.5.p5.2.m2.3.3.3.cmml"><mo id="S3.SS4.5.p5.2.m2.3.3.3.2" xref="S3.SS4.5.p5.2.m2.3.3.3.2.cmml">=</mo><mi id="S3.SS4.5.p5.2.m2.3.3.3.3" xref="S3.SS4.5.p5.2.m2.3.3.3.3.cmml">def</mi></mover><mrow id="S3.SS4.5.p5.2.m2.3.3.2.2" xref="S3.SS4.5.p5.2.m2.3.3.2.3.cmml"><mo id="S3.SS4.5.p5.2.m2.3.3.2.2.3" stretchy="false" xref="S3.SS4.5.p5.2.m2.3.3.2.3.1.cmml">{</mo><mrow id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.3.cmml"><mo id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.3" stretchy="false" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.3.cmml">(</mo><msubsup id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.3.cmml">U</mi><msup id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3.cmml"><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.4" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.3.cmml">,</mo><msubsup id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.cmml"><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.2.cmml">C</mi><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.3.cmml">U</mi><msup id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3.cmml"><mi id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3.2" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3.2.cmml">j</mi><mo id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3.3" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.2.2.3.3.cmml">′′</mo></msup></msubsup><mo id="S3.SS4.5.p5.2.m2.2.2.1.1.1.2.5" stretchy="false" xref="S3.SS4.5.p5.2.m2.2.2.1.1.1.3.cmml">)</mo></mrow><mo fence="true" id="S3.SS4.5.p5.2.m2.3.3.2.2.4" lspace="0em" rspace="0em" xref="S3.SS4.5.p5.2.m2.3.3.2.3.1.cmml">∣</mo><mrow id="S3.SS4.5.p5.2.m2.3.3.2.2.2" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.cmml"><mi id="S3.SS4.5.p5.2.m2.3.3.2.2.2.4" 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id="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4.1.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4">subscript</csymbol><ci id="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4.2.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4.2">⋄</ci><ci id="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4.3.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.2.4.3">𝑗</ci></apply></apply></apply><apply id="S3.SS4.5.p5.2.m2.3.3.2.2.2c.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2"><eq id="S3.SS4.5.p5.2.m2.3.3.2.2.2.6.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.6"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.5.p5.2.m2.3.3.2.2.2.2.cmml" id="S3.SS4.5.p5.2.m2.3.3.2.2.2d.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2"></share><intersect id="S3.SS4.5.p5.2.m2.3.3.2.2.2.7.cmml" xref="S3.SS4.5.p5.2.m2.3.3.2.2.2.7"></intersect></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.2.m2.3c">\Lambda\stackrel{{\scriptstyle\rm def}}{{=}}\{(C^{j^{\prime}}_{U},C^{j^{\prime% \prime}}_{U})\mid j\in\{m+1,\ldots,m+t\}~{}\text{and}~{}\diamond_{j}=\cap\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.2.m2.3d">roman_Λ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { ( italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ) ∣ italic_j ∈ { italic_m + 1 , … , italic_m + italic_t } and ⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∩ }</annotation></semantics></math> be a family of pairs of subsets of <math alttext="U" class="ltx_Math" display="inline" id="S3.SS4.5.p5.3.m3.1"><semantics id="S3.SS4.5.p5.3.m3.1a"><mi id="S3.SS4.5.p5.3.m3.1.1" xref="S3.SS4.5.p5.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p5.3.m3.1b"><ci id="S3.SS4.5.p5.3.m3.1.1.cmml" xref="S3.SS4.5.p5.3.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.3.m3.1d">italic_U</annotation></semantics></math>. In order to complete the proof, it is enough to show that <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS4.5.p5.4.m4.1"><semantics id="S3.SS4.5.p5.4.m4.1a"><mi id="S3.SS4.5.p5.4.m4.1.1" mathvariant="normal" xref="S3.SS4.5.p5.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p5.4.m4.1b"><ci id="S3.SS4.5.p5.4.m4.1.1.cmml" xref="S3.SS4.5.p5.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.4.m4.1d">roman_Λ</annotation></semantics></math> covers all semi-filters <math alttext="\mathcal{F}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="S3.SS4.5.p5.5.m5.1"><semantics id="S3.SS4.5.p5.5.m5.1a"><mrow id="S3.SS4.5.p5.5.m5.1.2" xref="S3.SS4.5.p5.5.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p5.5.m5.1.2.2" xref="S3.SS4.5.p5.5.m5.1.2.2.cmml">ℱ</mi><mo id="S3.SS4.5.p5.5.m5.1.2.1" xref="S3.SS4.5.p5.5.m5.1.2.1.cmml">⊆</mo><mrow id="S3.SS4.5.p5.5.m5.1.2.3" xref="S3.SS4.5.p5.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p5.5.m5.1.2.3.2" xref="S3.SS4.5.p5.5.m5.1.2.3.2.cmml">𝒫</mi><mo id="S3.SS4.5.p5.5.m5.1.2.3.1" xref="S3.SS4.5.p5.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS4.5.p5.5.m5.1.2.3.3.2" xref="S3.SS4.5.p5.5.m5.1.2.3.cmml"><mo id="S3.SS4.5.p5.5.m5.1.2.3.3.2.1" stretchy="false" xref="S3.SS4.5.p5.5.m5.1.2.3.cmml">(</mo><mi id="S3.SS4.5.p5.5.m5.1.1" xref="S3.SS4.5.p5.5.m5.1.1.cmml">U</mi><mo id="S3.SS4.5.p5.5.m5.1.2.3.3.2.2" stretchy="false" xref="S3.SS4.5.p5.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p5.5.m5.1b"><apply id="S3.SS4.5.p5.5.m5.1.2.cmml" xref="S3.SS4.5.p5.5.m5.1.2"><subset id="S3.SS4.5.p5.5.m5.1.2.1.cmml" xref="S3.SS4.5.p5.5.m5.1.2.1"></subset><ci id="S3.SS4.5.p5.5.m5.1.2.2.cmml" xref="S3.SS4.5.p5.5.m5.1.2.2">ℱ</ci><apply id="S3.SS4.5.p5.5.m5.1.2.3.cmml" xref="S3.SS4.5.p5.5.m5.1.2.3"><times id="S3.SS4.5.p5.5.m5.1.2.3.1.cmml" xref="S3.SS4.5.p5.5.m5.1.2.3.1"></times><ci id="S3.SS4.5.p5.5.m5.1.2.3.2.cmml" xref="S3.SS4.5.p5.5.m5.1.2.3.2">𝒫</ci><ci id="S3.SS4.5.p5.5.m5.1.1.cmml" xref="S3.SS4.5.p5.5.m5.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.5.m5.1c">\mathcal{F}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.5.m5.1d">caligraphic_F ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> that are above some element <math alttext="a=a(\mathcal{F})\in A" class="ltx_Math" display="inline" id="S3.SS4.5.p5.6.m6.1"><semantics id="S3.SS4.5.p5.6.m6.1a"><mrow id="S3.SS4.5.p5.6.m6.1.2" xref="S3.SS4.5.p5.6.m6.1.2.cmml"><mi id="S3.SS4.5.p5.6.m6.1.2.2" xref="S3.SS4.5.p5.6.m6.1.2.2.cmml">a</mi><mo id="S3.SS4.5.p5.6.m6.1.2.3" xref="S3.SS4.5.p5.6.m6.1.2.3.cmml">=</mo><mrow id="S3.SS4.5.p5.6.m6.1.2.4" xref="S3.SS4.5.p5.6.m6.1.2.4.cmml"><mi id="S3.SS4.5.p5.6.m6.1.2.4.2" xref="S3.SS4.5.p5.6.m6.1.2.4.2.cmml">a</mi><mo id="S3.SS4.5.p5.6.m6.1.2.4.1" xref="S3.SS4.5.p5.6.m6.1.2.4.1.cmml">⁢</mo><mrow id="S3.SS4.5.p5.6.m6.1.2.4.3.2" xref="S3.SS4.5.p5.6.m6.1.2.4.cmml"><mo id="S3.SS4.5.p5.6.m6.1.2.4.3.2.1" stretchy="false" xref="S3.SS4.5.p5.6.m6.1.2.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p5.6.m6.1.1" xref="S3.SS4.5.p5.6.m6.1.1.cmml">ℱ</mi><mo id="S3.SS4.5.p5.6.m6.1.2.4.3.2.2" stretchy="false" xref="S3.SS4.5.p5.6.m6.1.2.4.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p5.6.m6.1.2.5" xref="S3.SS4.5.p5.6.m6.1.2.5.cmml">∈</mo><mi id="S3.SS4.5.p5.6.m6.1.2.6" xref="S3.SS4.5.p5.6.m6.1.2.6.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p5.6.m6.1b"><apply id="S3.SS4.5.p5.6.m6.1.2.cmml" xref="S3.SS4.5.p5.6.m6.1.2"><and id="S3.SS4.5.p5.6.m6.1.2a.cmml" xref="S3.SS4.5.p5.6.m6.1.2"></and><apply id="S3.SS4.5.p5.6.m6.1.2b.cmml" xref="S3.SS4.5.p5.6.m6.1.2"><eq id="S3.SS4.5.p5.6.m6.1.2.3.cmml" xref="S3.SS4.5.p5.6.m6.1.2.3"></eq><ci id="S3.SS4.5.p5.6.m6.1.2.2.cmml" xref="S3.SS4.5.p5.6.m6.1.2.2">𝑎</ci><apply id="S3.SS4.5.p5.6.m6.1.2.4.cmml" xref="S3.SS4.5.p5.6.m6.1.2.4"><times id="S3.SS4.5.p5.6.m6.1.2.4.1.cmml" xref="S3.SS4.5.p5.6.m6.1.2.4.1"></times><ci id="S3.SS4.5.p5.6.m6.1.2.4.2.cmml" xref="S3.SS4.5.p5.6.m6.1.2.4.2">𝑎</ci><ci id="S3.SS4.5.p5.6.m6.1.1.cmml" xref="S3.SS4.5.p5.6.m6.1.1">ℱ</ci></apply></apply><apply id="S3.SS4.5.p5.6.m6.1.2c.cmml" xref="S3.SS4.5.p5.6.m6.1.2"><in id="S3.SS4.5.p5.6.m6.1.2.5.cmml" xref="S3.SS4.5.p5.6.m6.1.2.5"></in><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.5.p5.6.m6.1.2.4.cmml" id="S3.SS4.5.p5.6.m6.1.2d.cmml" xref="S3.SS4.5.p5.6.m6.1.2"></share><ci id="S3.SS4.5.p5.6.m6.1.2.6.cmml" xref="S3.SS4.5.p5.6.m6.1.2.6">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p5.6.m6.1c">a=a(\mathcal{F})\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p5.6.m6.1d">italic_a = italic_a ( caligraphic_F ) ∈ italic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.6.p6"> <p class="ltx_p" id="S3.SS4.6.p6.12">Suppose this is not the case, i.e., there is a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.1.m1.1"><semantics id="S3.SS4.6.p6.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p6.1.m1.1.1" xref="S3.SS4.6.p6.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.1.m1.1b"><ci id="S3.SS4.6.p6.1.m1.1.1.cmml" xref="S3.SS4.6.p6.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.1.m1.1d">caligraphic_F</annotation></semantics></math> above <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.SS4.6.p6.2.m2.1"><semantics id="S3.SS4.6.p6.2.m2.1a"><mrow id="S3.SS4.6.p6.2.m2.1.1" xref="S3.SS4.6.p6.2.m2.1.1.cmml"><mi id="S3.SS4.6.p6.2.m2.1.1.2" xref="S3.SS4.6.p6.2.m2.1.1.2.cmml">a</mi><mo id="S3.SS4.6.p6.2.m2.1.1.1" xref="S3.SS4.6.p6.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS4.6.p6.2.m2.1.1.3" xref="S3.SS4.6.p6.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.2.m2.1b"><apply id="S3.SS4.6.p6.2.m2.1.1.cmml" xref="S3.SS4.6.p6.2.m2.1.1"><in id="S3.SS4.6.p6.2.m2.1.1.1.cmml" xref="S3.SS4.6.p6.2.m2.1.1.1"></in><ci id="S3.SS4.6.p6.2.m2.1.1.2.cmml" xref="S3.SS4.6.p6.2.m2.1.1.2">𝑎</ci><ci id="S3.SS4.6.p6.2.m2.1.1.3.cmml" xref="S3.SS4.6.p6.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.2.m2.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.2.m2.1d">italic_a ∈ italic_A</annotation></semantics></math> such that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.3.m3.1"><semantics id="S3.SS4.6.p6.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p6.3.m3.1.1" xref="S3.SS4.6.p6.3.m3.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.3.m3.1b"><ci id="S3.SS4.6.p6.3.m3.1.1.cmml" xref="S3.SS4.6.p6.3.m3.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.3.m3.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.3.m3.1d">caligraphic_F</annotation></semantics></math> is not covered by <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS4.6.p6.4.m4.1"><semantics id="S3.SS4.6.p6.4.m4.1a"><mi id="S3.SS4.6.p6.4.m4.1.1" mathvariant="normal" xref="S3.SS4.6.p6.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.4.m4.1b"><ci id="S3.SS4.6.p6.4.m4.1.1.cmml" xref="S3.SS4.6.p6.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.4.m4.1d">roman_Λ</annotation></semantics></math>. We proceed in part as in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>. For each <math alttext="i\in[m+t]" class="ltx_Math" display="inline" id="S3.SS4.6.p6.5.m5.1"><semantics id="S3.SS4.6.p6.5.m5.1a"><mrow id="S3.SS4.6.p6.5.m5.1.1" xref="S3.SS4.6.p6.5.m5.1.1.cmml"><mi id="S3.SS4.6.p6.5.m5.1.1.3" xref="S3.SS4.6.p6.5.m5.1.1.3.cmml">i</mi><mo id="S3.SS4.6.p6.5.m5.1.1.2" xref="S3.SS4.6.p6.5.m5.1.1.2.cmml">∈</mo><mrow id="S3.SS4.6.p6.5.m5.1.1.1.1" xref="S3.SS4.6.p6.5.m5.1.1.1.2.cmml"><mo id="S3.SS4.6.p6.5.m5.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p6.5.m5.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS4.6.p6.5.m5.1.1.1.1.1" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.cmml"><mi id="S3.SS4.6.p6.5.m5.1.1.1.1.1.2" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.2.cmml">m</mi><mo id="S3.SS4.6.p6.5.m5.1.1.1.1.1.1" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS4.6.p6.5.m5.1.1.1.1.1.3" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.SS4.6.p6.5.m5.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p6.5.m5.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.5.m5.1b"><apply id="S3.SS4.6.p6.5.m5.1.1.cmml" xref="S3.SS4.6.p6.5.m5.1.1"><in id="S3.SS4.6.p6.5.m5.1.1.2.cmml" xref="S3.SS4.6.p6.5.m5.1.1.2"></in><ci id="S3.SS4.6.p6.5.m5.1.1.3.cmml" xref="S3.SS4.6.p6.5.m5.1.1.3">𝑖</ci><apply id="S3.SS4.6.p6.5.m5.1.1.1.2.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1"><csymbol cd="latexml" id="S3.SS4.6.p6.5.m5.1.1.1.2.1.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS4.6.p6.5.m5.1.1.1.1.1.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1"><plus id="S3.SS4.6.p6.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.1"></plus><ci id="S3.SS4.6.p6.5.m5.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.2">𝑚</ci><ci id="S3.SS4.6.p6.5.m5.1.1.1.1.1.3.cmml" xref="S3.SS4.6.p6.5.m5.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.5.m5.1c">i\in[m+t]</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.5.m5.1d">italic_i ∈ [ italic_m + italic_t ]</annotation></semantics></math>, let <math alttext="\alpha_{i}\in\{0,1\}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.6.m6.2"><semantics id="S3.SS4.6.p6.6.m6.2a"><mrow id="S3.SS4.6.p6.6.m6.2.3" xref="S3.SS4.6.p6.6.m6.2.3.cmml"><msub id="S3.SS4.6.p6.6.m6.2.3.2" xref="S3.SS4.6.p6.6.m6.2.3.2.cmml"><mi id="S3.SS4.6.p6.6.m6.2.3.2.2" xref="S3.SS4.6.p6.6.m6.2.3.2.2.cmml">α</mi><mi id="S3.SS4.6.p6.6.m6.2.3.2.3" xref="S3.SS4.6.p6.6.m6.2.3.2.3.cmml">i</mi></msub><mo id="S3.SS4.6.p6.6.m6.2.3.1" xref="S3.SS4.6.p6.6.m6.2.3.1.cmml">∈</mo><mrow id="S3.SS4.6.p6.6.m6.2.3.3.2" xref="S3.SS4.6.p6.6.m6.2.3.3.1.cmml"><mo id="S3.SS4.6.p6.6.m6.2.3.3.2.1" stretchy="false" xref="S3.SS4.6.p6.6.m6.2.3.3.1.cmml">{</mo><mn id="S3.SS4.6.p6.6.m6.1.1" xref="S3.SS4.6.p6.6.m6.1.1.cmml">0</mn><mo id="S3.SS4.6.p6.6.m6.2.3.3.2.2" xref="S3.SS4.6.p6.6.m6.2.3.3.1.cmml">,</mo><mn id="S3.SS4.6.p6.6.m6.2.2" xref="S3.SS4.6.p6.6.m6.2.2.cmml">1</mn><mo id="S3.SS4.6.p6.6.m6.2.3.3.2.3" stretchy="false" xref="S3.SS4.6.p6.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.6.m6.2b"><apply id="S3.SS4.6.p6.6.m6.2.3.cmml" xref="S3.SS4.6.p6.6.m6.2.3"><in id="S3.SS4.6.p6.6.m6.2.3.1.cmml" xref="S3.SS4.6.p6.6.m6.2.3.1"></in><apply id="S3.SS4.6.p6.6.m6.2.3.2.cmml" xref="S3.SS4.6.p6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.6.p6.6.m6.2.3.2.1.cmml" xref="S3.SS4.6.p6.6.m6.2.3.2">subscript</csymbol><ci id="S3.SS4.6.p6.6.m6.2.3.2.2.cmml" xref="S3.SS4.6.p6.6.m6.2.3.2.2">𝛼</ci><ci id="S3.SS4.6.p6.6.m6.2.3.2.3.cmml" xref="S3.SS4.6.p6.6.m6.2.3.2.3">𝑖</ci></apply><set id="S3.SS4.6.p6.6.m6.2.3.3.1.cmml" xref="S3.SS4.6.p6.6.m6.2.3.3.2"><cn id="S3.SS4.6.p6.6.m6.1.1.cmml" type="integer" xref="S3.SS4.6.p6.6.m6.1.1">0</cn><cn id="S3.SS4.6.p6.6.m6.2.2.cmml" type="integer" xref="S3.SS4.6.p6.6.m6.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.6.m6.2c">\alpha_{i}\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.6.m6.2d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { 0 , 1 }</annotation></semantics></math> be <math alttext="1" class="ltx_Math" display="inline" id="S3.SS4.6.p6.7.m7.1"><semantics id="S3.SS4.6.p6.7.m7.1a"><mn id="S3.SS4.6.p6.7.m7.1.1" xref="S3.SS4.6.p6.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.7.m7.1b"><cn id="S3.SS4.6.p6.7.m7.1.1.cmml" type="integer" xref="S3.SS4.6.p6.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.7.m7.1d">1</annotation></semantics></math> if and only if <math alttext="a\in C^{i}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.8.m8.1"><semantics id="S3.SS4.6.p6.8.m8.1a"><mrow id="S3.SS4.6.p6.8.m8.1.1" xref="S3.SS4.6.p6.8.m8.1.1.cmml"><mi id="S3.SS4.6.p6.8.m8.1.1.2" xref="S3.SS4.6.p6.8.m8.1.1.2.cmml">a</mi><mo id="S3.SS4.6.p6.8.m8.1.1.1" xref="S3.SS4.6.p6.8.m8.1.1.1.cmml">∈</mo><msup id="S3.SS4.6.p6.8.m8.1.1.3" xref="S3.SS4.6.p6.8.m8.1.1.3.cmml"><mi id="S3.SS4.6.p6.8.m8.1.1.3.2" xref="S3.SS4.6.p6.8.m8.1.1.3.2.cmml">C</mi><mi id="S3.SS4.6.p6.8.m8.1.1.3.3" xref="S3.SS4.6.p6.8.m8.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.8.m8.1b"><apply id="S3.SS4.6.p6.8.m8.1.1.cmml" xref="S3.SS4.6.p6.8.m8.1.1"><in id="S3.SS4.6.p6.8.m8.1.1.1.cmml" xref="S3.SS4.6.p6.8.m8.1.1.1"></in><ci id="S3.SS4.6.p6.8.m8.1.1.2.cmml" xref="S3.SS4.6.p6.8.m8.1.1.2">𝑎</ci><apply id="S3.SS4.6.p6.8.m8.1.1.3.cmml" xref="S3.SS4.6.p6.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p6.8.m8.1.1.3.1.cmml" xref="S3.SS4.6.p6.8.m8.1.1.3">superscript</csymbol><ci id="S3.SS4.6.p6.8.m8.1.1.3.2.cmml" xref="S3.SS4.6.p6.8.m8.1.1.3.2">𝐶</ci><ci id="S3.SS4.6.p6.8.m8.1.1.3.3.cmml" xref="S3.SS4.6.p6.8.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.8.m8.1c">a\in C^{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.8.m8.1d">italic_a ∈ italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\beta_{i}\in\{0,1\}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.9.m9.2"><semantics id="S3.SS4.6.p6.9.m9.2a"><mrow id="S3.SS4.6.p6.9.m9.2.3" xref="S3.SS4.6.p6.9.m9.2.3.cmml"><msub id="S3.SS4.6.p6.9.m9.2.3.2" xref="S3.SS4.6.p6.9.m9.2.3.2.cmml"><mi id="S3.SS4.6.p6.9.m9.2.3.2.2" xref="S3.SS4.6.p6.9.m9.2.3.2.2.cmml">β</mi><mi id="S3.SS4.6.p6.9.m9.2.3.2.3" xref="S3.SS4.6.p6.9.m9.2.3.2.3.cmml">i</mi></msub><mo id="S3.SS4.6.p6.9.m9.2.3.1" xref="S3.SS4.6.p6.9.m9.2.3.1.cmml">∈</mo><mrow id="S3.SS4.6.p6.9.m9.2.3.3.2" xref="S3.SS4.6.p6.9.m9.2.3.3.1.cmml"><mo id="S3.SS4.6.p6.9.m9.2.3.3.2.1" stretchy="false" xref="S3.SS4.6.p6.9.m9.2.3.3.1.cmml">{</mo><mn id="S3.SS4.6.p6.9.m9.1.1" xref="S3.SS4.6.p6.9.m9.1.1.cmml">0</mn><mo id="S3.SS4.6.p6.9.m9.2.3.3.2.2" xref="S3.SS4.6.p6.9.m9.2.3.3.1.cmml">,</mo><mn id="S3.SS4.6.p6.9.m9.2.2" xref="S3.SS4.6.p6.9.m9.2.2.cmml">1</mn><mo id="S3.SS4.6.p6.9.m9.2.3.3.2.3" stretchy="false" xref="S3.SS4.6.p6.9.m9.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.9.m9.2b"><apply id="S3.SS4.6.p6.9.m9.2.3.cmml" xref="S3.SS4.6.p6.9.m9.2.3"><in id="S3.SS4.6.p6.9.m9.2.3.1.cmml" xref="S3.SS4.6.p6.9.m9.2.3.1"></in><apply id="S3.SS4.6.p6.9.m9.2.3.2.cmml" xref="S3.SS4.6.p6.9.m9.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.6.p6.9.m9.2.3.2.1.cmml" xref="S3.SS4.6.p6.9.m9.2.3.2">subscript</csymbol><ci id="S3.SS4.6.p6.9.m9.2.3.2.2.cmml" xref="S3.SS4.6.p6.9.m9.2.3.2.2">𝛽</ci><ci id="S3.SS4.6.p6.9.m9.2.3.2.3.cmml" xref="S3.SS4.6.p6.9.m9.2.3.2.3">𝑖</ci></apply><set id="S3.SS4.6.p6.9.m9.2.3.3.1.cmml" xref="S3.SS4.6.p6.9.m9.2.3.3.2"><cn id="S3.SS4.6.p6.9.m9.1.1.cmml" type="integer" xref="S3.SS4.6.p6.9.m9.1.1">0</cn><cn id="S3.SS4.6.p6.9.m9.2.2.cmml" type="integer" xref="S3.SS4.6.p6.9.m9.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.9.m9.2c">\beta_{i}\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.9.m9.2d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { 0 , 1 }</annotation></semantics></math> be <math alttext="1" class="ltx_Math" display="inline" id="S3.SS4.6.p6.10.m10.1"><semantics id="S3.SS4.6.p6.10.m10.1a"><mn id="S3.SS4.6.p6.10.m10.1.1" xref="S3.SS4.6.p6.10.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.10.m10.1b"><cn id="S3.SS4.6.p6.10.m10.1.1.cmml" type="integer" xref="S3.SS4.6.p6.10.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.10.m10.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.10.m10.1d">1</annotation></semantics></math> if and only if <math alttext="C^{i}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.11.m11.1"><semantics id="S3.SS4.6.p6.11.m11.1a"><mrow id="S3.SS4.6.p6.11.m11.1.1" xref="S3.SS4.6.p6.11.m11.1.1.cmml"><msubsup id="S3.SS4.6.p6.11.m11.1.1.2" xref="S3.SS4.6.p6.11.m11.1.1.2.cmml"><mi id="S3.SS4.6.p6.11.m11.1.1.2.2.2" xref="S3.SS4.6.p6.11.m11.1.1.2.2.2.cmml">C</mi><mi id="S3.SS4.6.p6.11.m11.1.1.2.3" xref="S3.SS4.6.p6.11.m11.1.1.2.3.cmml">U</mi><mi id="S3.SS4.6.p6.11.m11.1.1.2.2.3" xref="S3.SS4.6.p6.11.m11.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S3.SS4.6.p6.11.m11.1.1.1" xref="S3.SS4.6.p6.11.m11.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p6.11.m11.1.1.3" xref="S3.SS4.6.p6.11.m11.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.11.m11.1b"><apply id="S3.SS4.6.p6.11.m11.1.1.cmml" xref="S3.SS4.6.p6.11.m11.1.1"><in id="S3.SS4.6.p6.11.m11.1.1.1.cmml" xref="S3.SS4.6.p6.11.m11.1.1.1"></in><apply id="S3.SS4.6.p6.11.m11.1.1.2.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.6.p6.11.m11.1.1.2.1.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2">subscript</csymbol><apply id="S3.SS4.6.p6.11.m11.1.1.2.2.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.6.p6.11.m11.1.1.2.2.1.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2">superscript</csymbol><ci id="S3.SS4.6.p6.11.m11.1.1.2.2.2.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2.2.2">𝐶</ci><ci id="S3.SS4.6.p6.11.m11.1.1.2.2.3.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2.2.3">𝑖</ci></apply><ci id="S3.SS4.6.p6.11.m11.1.1.2.3.cmml" xref="S3.SS4.6.p6.11.m11.1.1.2.3">𝑈</ci></apply><ci id="S3.SS4.6.p6.11.m11.1.1.3.cmml" xref="S3.SS4.6.p6.11.m11.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.11.m11.1c">C^{i}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.11.m11.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>. We obtain a contradiction by a slightly different argument, which is in analogy to the proof in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib12" title="">12</a>]</cite>. Since the operations performed over <math alttext="C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}" class="ltx_Math" display="inline" id="S3.SS4.6.p6.12.m12.6"><semantics id="S3.SS4.6.p6.12.m12.6a"><mrow id="S3.SS4.6.p6.12.m12.6.6.4" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml"><msup id="S3.SS4.6.p6.12.m12.3.3.1.1" xref="S3.SS4.6.p6.12.m12.3.3.1.1.cmml"><mi id="S3.SS4.6.p6.12.m12.3.3.1.1.2" xref="S3.SS4.6.p6.12.m12.3.3.1.1.2.cmml">C</mi><mn id="S3.SS4.6.p6.12.m12.3.3.1.1.3" xref="S3.SS4.6.p6.12.m12.3.3.1.1.3.cmml">1</mn></msup><mo id="S3.SS4.6.p6.12.m12.6.6.4.5" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml">,</mo><mi id="S3.SS4.6.p6.12.m12.1.1" mathvariant="normal" xref="S3.SS4.6.p6.12.m12.1.1.cmml">…</mi><mo id="S3.SS4.6.p6.12.m12.6.6.4.6" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml">,</mo><msup id="S3.SS4.6.p6.12.m12.4.4.2.2" xref="S3.SS4.6.p6.12.m12.4.4.2.2.cmml"><mi id="S3.SS4.6.p6.12.m12.4.4.2.2.2" xref="S3.SS4.6.p6.12.m12.4.4.2.2.2.cmml">C</mi><mi id="S3.SS4.6.p6.12.m12.4.4.2.2.3" xref="S3.SS4.6.p6.12.m12.4.4.2.2.3.cmml">m</mi></msup><mo id="S3.SS4.6.p6.12.m12.6.6.4.7" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml">,</mo><msup id="S3.SS4.6.p6.12.m12.5.5.3.3" xref="S3.SS4.6.p6.12.m12.5.5.3.3.cmml"><mi id="S3.SS4.6.p6.12.m12.5.5.3.3.2" xref="S3.SS4.6.p6.12.m12.5.5.3.3.2.cmml">C</mi><mrow id="S3.SS4.6.p6.12.m12.5.5.3.3.3" xref="S3.SS4.6.p6.12.m12.5.5.3.3.3.cmml"><mi id="S3.SS4.6.p6.12.m12.5.5.3.3.3.2" xref="S3.SS4.6.p6.12.m12.5.5.3.3.3.2.cmml">m</mi><mo id="S3.SS4.6.p6.12.m12.5.5.3.3.3.1" xref="S3.SS4.6.p6.12.m12.5.5.3.3.3.1.cmml">+</mo><mn id="S3.SS4.6.p6.12.m12.5.5.3.3.3.3" xref="S3.SS4.6.p6.12.m12.5.5.3.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS4.6.p6.12.m12.6.6.4.8" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml">,</mo><mi id="S3.SS4.6.p6.12.m12.2.2" mathvariant="normal" xref="S3.SS4.6.p6.12.m12.2.2.cmml">…</mi><mo id="S3.SS4.6.p6.12.m12.6.6.4.9" xref="S3.SS4.6.p6.12.m12.6.6.5.cmml">,</mo><msup id="S3.SS4.6.p6.12.m12.6.6.4.4" xref="S3.SS4.6.p6.12.m12.6.6.4.4.cmml"><mi id="S3.SS4.6.p6.12.m12.6.6.4.4.2" xref="S3.SS4.6.p6.12.m12.6.6.4.4.2.cmml">C</mi><mrow id="S3.SS4.6.p6.12.m12.6.6.4.4.3" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.cmml"><mi id="S3.SS4.6.p6.12.m12.6.6.4.4.3.2" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.2.cmml">m</mi><mo id="S3.SS4.6.p6.12.m12.6.6.4.4.3.1" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.1.cmml">+</mo><mi id="S3.SS4.6.p6.12.m12.6.6.4.4.3.3" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.3.cmml">t</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p6.12.m12.6b"><list id="S3.SS4.6.p6.12.m12.6.6.5.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4"><apply id="S3.SS4.6.p6.12.m12.3.3.1.1.cmml" xref="S3.SS4.6.p6.12.m12.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.6.p6.12.m12.3.3.1.1.1.cmml" xref="S3.SS4.6.p6.12.m12.3.3.1.1">superscript</csymbol><ci id="S3.SS4.6.p6.12.m12.3.3.1.1.2.cmml" xref="S3.SS4.6.p6.12.m12.3.3.1.1.2">𝐶</ci><cn id="S3.SS4.6.p6.12.m12.3.3.1.1.3.cmml" type="integer" xref="S3.SS4.6.p6.12.m12.3.3.1.1.3">1</cn></apply><ci 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id="S3.SS4.6.p6.12.m12.5.5.3.3.3.3.cmml" type="integer" xref="S3.SS4.6.p6.12.m12.5.5.3.3.3.3">1</cn></apply></apply><ci id="S3.SS4.6.p6.12.m12.2.2.cmml" xref="S3.SS4.6.p6.12.m12.2.2">…</ci><apply id="S3.SS4.6.p6.12.m12.6.6.4.4.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4"><csymbol cd="ambiguous" id="S3.SS4.6.p6.12.m12.6.6.4.4.1.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4">superscript</csymbol><ci id="S3.SS4.6.p6.12.m12.6.6.4.4.2.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4.2">𝐶</ci><apply id="S3.SS4.6.p6.12.m12.6.6.4.4.3.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3"><plus id="S3.SS4.6.p6.12.m12.6.6.4.4.3.1.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.1"></plus><ci id="S3.SS4.6.p6.12.m12.6.6.4.4.3.2.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.2">𝑚</ci><ci id="S3.SS4.6.p6.12.m12.6.6.4.4.3.3.cmml" xref="S3.SS4.6.p6.12.m12.6.6.4.4.3.3">𝑡</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p6.12.m12.6c">C^{1},\ldots,C^{m},C^{m+1},\ldots,C^{m+t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p6.12.m12.6d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_C start_POSTSUPERSCRIPT italic_m + 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> do not follow a linear order, and these sets are obtained after the convergence of the evaluation procedure, we employ a top-down approach, as opposed to the bottom-up presentation that appears in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>.</p> </div> <div class="ltx_para" id="S3.SS4.7.p7"> <p class="ltx_p" id="S3.SS4.7.p7.15">We define a partition <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S3.SS4.7.p7.1.m1.2"><semantics id="S3.SS4.7.p7.1.m1.2a"><mrow id="S3.SS4.7.p7.1.m1.2.3.2" xref="S3.SS4.7.p7.1.m1.2.3.1.cmml"><mo id="S3.SS4.7.p7.1.m1.2.3.2.1" stretchy="false" xref="S3.SS4.7.p7.1.m1.2.3.1.cmml">(</mo><mi id="S3.SS4.7.p7.1.m1.1.1" xref="S3.SS4.7.p7.1.m1.1.1.cmml">X</mi><mo id="S3.SS4.7.p7.1.m1.2.3.2.2" xref="S3.SS4.7.p7.1.m1.2.3.1.cmml">,</mo><mi id="S3.SS4.7.p7.1.m1.2.2" xref="S3.SS4.7.p7.1.m1.2.2.cmml">Y</mi><mo id="S3.SS4.7.p7.1.m1.2.3.2.3" stretchy="false" xref="S3.SS4.7.p7.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.1.m1.2b"><interval closure="open" id="S3.SS4.7.p7.1.m1.2.3.1.cmml" xref="S3.SS4.7.p7.1.m1.2.3.2"><ci id="S3.SS4.7.p7.1.m1.1.1.cmml" xref="S3.SS4.7.p7.1.m1.1.1">𝑋</ci><ci id="S3.SS4.7.p7.1.m1.2.2.cmml" xref="S3.SS4.7.p7.1.m1.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.1.m1.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.1.m1.2d">( italic_X , italic_Y )</annotation></semantics></math> of the indices of the sets <math alttext="C^{1},\ldots,C^{m+t}" class="ltx_Math" display="inline" id="S3.SS4.7.p7.2.m2.3"><semantics id="S3.SS4.7.p7.2.m2.3a"><mrow id="S3.SS4.7.p7.2.m2.3.3.2" xref="S3.SS4.7.p7.2.m2.3.3.3.cmml"><msup id="S3.SS4.7.p7.2.m2.2.2.1.1" xref="S3.SS4.7.p7.2.m2.2.2.1.1.cmml"><mi id="S3.SS4.7.p7.2.m2.2.2.1.1.2" xref="S3.SS4.7.p7.2.m2.2.2.1.1.2.cmml">C</mi><mn id="S3.SS4.7.p7.2.m2.2.2.1.1.3" xref="S3.SS4.7.p7.2.m2.2.2.1.1.3.cmml">1</mn></msup><mo id="S3.SS4.7.p7.2.m2.3.3.2.3" xref="S3.SS4.7.p7.2.m2.3.3.3.cmml">,</mo><mi id="S3.SS4.7.p7.2.m2.1.1" mathvariant="normal" xref="S3.SS4.7.p7.2.m2.1.1.cmml">…</mi><mo id="S3.SS4.7.p7.2.m2.3.3.2.4" xref="S3.SS4.7.p7.2.m2.3.3.3.cmml">,</mo><msup id="S3.SS4.7.p7.2.m2.3.3.2.2" xref="S3.SS4.7.p7.2.m2.3.3.2.2.cmml"><mi id="S3.SS4.7.p7.2.m2.3.3.2.2.2" xref="S3.SS4.7.p7.2.m2.3.3.2.2.2.cmml">C</mi><mrow id="S3.SS4.7.p7.2.m2.3.3.2.2.3" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.cmml"><mi id="S3.SS4.7.p7.2.m2.3.3.2.2.3.2" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.2.cmml">m</mi><mo id="S3.SS4.7.p7.2.m2.3.3.2.2.3.1" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.1.cmml">+</mo><mi id="S3.SS4.7.p7.2.m2.3.3.2.2.3.3" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.3.cmml">t</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.2.m2.3b"><list id="S3.SS4.7.p7.2.m2.3.3.3.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2"><apply id="S3.SS4.7.p7.2.m2.2.2.1.1.cmml" xref="S3.SS4.7.p7.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.7.p7.2.m2.2.2.1.1.1.cmml" xref="S3.SS4.7.p7.2.m2.2.2.1.1">superscript</csymbol><ci id="S3.SS4.7.p7.2.m2.2.2.1.1.2.cmml" xref="S3.SS4.7.p7.2.m2.2.2.1.1.2">𝐶</ci><cn id="S3.SS4.7.p7.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.SS4.7.p7.2.m2.2.2.1.1.3">1</cn></apply><ci id="S3.SS4.7.p7.2.m2.1.1.cmml" xref="S3.SS4.7.p7.2.m2.1.1">…</ci><apply id="S3.SS4.7.p7.2.m2.3.3.2.2.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.2.m2.3.3.2.2.1.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2">superscript</csymbol><ci id="S3.SS4.7.p7.2.m2.3.3.2.2.2.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2.2">𝐶</ci><apply id="S3.SS4.7.p7.2.m2.3.3.2.2.3.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3"><plus id="S3.SS4.7.p7.2.m2.3.3.2.2.3.1.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.1"></plus><ci id="S3.SS4.7.p7.2.m2.3.3.2.2.3.2.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.2">𝑚</ci><ci id="S3.SS4.7.p7.2.m2.3.3.2.2.3.3.cmml" xref="S3.SS4.7.p7.2.m2.3.3.2.2.3.3">𝑡</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.2.m2.3c">C^{1},\ldots,C^{m+t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.2.m2.3d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. Note that <math alttext="\alpha_{m+t}=1" class="ltx_Math" display="inline" id="S3.SS4.7.p7.3.m3.1"><semantics id="S3.SS4.7.p7.3.m3.1a"><mrow id="S3.SS4.7.p7.3.m3.1.1" xref="S3.SS4.7.p7.3.m3.1.1.cmml"><msub id="S3.SS4.7.p7.3.m3.1.1.2" xref="S3.SS4.7.p7.3.m3.1.1.2.cmml"><mi id="S3.SS4.7.p7.3.m3.1.1.2.2" xref="S3.SS4.7.p7.3.m3.1.1.2.2.cmml">α</mi><mrow id="S3.SS4.7.p7.3.m3.1.1.2.3" xref="S3.SS4.7.p7.3.m3.1.1.2.3.cmml"><mi id="S3.SS4.7.p7.3.m3.1.1.2.3.2" xref="S3.SS4.7.p7.3.m3.1.1.2.3.2.cmml">m</mi><mo id="S3.SS4.7.p7.3.m3.1.1.2.3.1" xref="S3.SS4.7.p7.3.m3.1.1.2.3.1.cmml">+</mo><mi id="S3.SS4.7.p7.3.m3.1.1.2.3.3" xref="S3.SS4.7.p7.3.m3.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS4.7.p7.3.m3.1.1.1" xref="S3.SS4.7.p7.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS4.7.p7.3.m3.1.1.3" xref="S3.SS4.7.p7.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.3.m3.1b"><apply id="S3.SS4.7.p7.3.m3.1.1.cmml" xref="S3.SS4.7.p7.3.m3.1.1"><eq id="S3.SS4.7.p7.3.m3.1.1.1.cmml" xref="S3.SS4.7.p7.3.m3.1.1.1"></eq><apply id="S3.SS4.7.p7.3.m3.1.1.2.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.3.m3.1.1.2.1.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.7.p7.3.m3.1.1.2.2.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2.2">𝛼</ci><apply id="S3.SS4.7.p7.3.m3.1.1.2.3.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2.3"><plus id="S3.SS4.7.p7.3.m3.1.1.2.3.1.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2.3.1"></plus><ci id="S3.SS4.7.p7.3.m3.1.1.2.3.2.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2.3.2">𝑚</ci><ci id="S3.SS4.7.p7.3.m3.1.1.2.3.3.cmml" xref="S3.SS4.7.p7.3.m3.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS4.7.p7.3.m3.1.1.3.cmml" type="integer" xref="S3.SS4.7.p7.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.3.m3.1c">\alpha_{m+t}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{m+t}=0" class="ltx_Math" display="inline" id="S3.SS4.7.p7.4.m4.1"><semantics id="S3.SS4.7.p7.4.m4.1a"><mrow id="S3.SS4.7.p7.4.m4.1.1" xref="S3.SS4.7.p7.4.m4.1.1.cmml"><msub id="S3.SS4.7.p7.4.m4.1.1.2" xref="S3.SS4.7.p7.4.m4.1.1.2.cmml"><mi id="S3.SS4.7.p7.4.m4.1.1.2.2" xref="S3.SS4.7.p7.4.m4.1.1.2.2.cmml">β</mi><mrow id="S3.SS4.7.p7.4.m4.1.1.2.3" xref="S3.SS4.7.p7.4.m4.1.1.2.3.cmml"><mi id="S3.SS4.7.p7.4.m4.1.1.2.3.2" xref="S3.SS4.7.p7.4.m4.1.1.2.3.2.cmml">m</mi><mo id="S3.SS4.7.p7.4.m4.1.1.2.3.1" xref="S3.SS4.7.p7.4.m4.1.1.2.3.1.cmml">+</mo><mi id="S3.SS4.7.p7.4.m4.1.1.2.3.3" xref="S3.SS4.7.p7.4.m4.1.1.2.3.3.cmml">t</mi></mrow></msub><mo id="S3.SS4.7.p7.4.m4.1.1.1" xref="S3.SS4.7.p7.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS4.7.p7.4.m4.1.1.3" xref="S3.SS4.7.p7.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.4.m4.1b"><apply id="S3.SS4.7.p7.4.m4.1.1.cmml" xref="S3.SS4.7.p7.4.m4.1.1"><eq id="S3.SS4.7.p7.4.m4.1.1.1.cmml" xref="S3.SS4.7.p7.4.m4.1.1.1"></eq><apply id="S3.SS4.7.p7.4.m4.1.1.2.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.4.m4.1.1.2.1.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.7.p7.4.m4.1.1.2.2.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2.2">𝛽</ci><apply id="S3.SS4.7.p7.4.m4.1.1.2.3.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2.3"><plus id="S3.SS4.7.p7.4.m4.1.1.2.3.1.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2.3.1"></plus><ci id="S3.SS4.7.p7.4.m4.1.1.2.3.2.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2.3.2">𝑚</ci><ci id="S3.SS4.7.p7.4.m4.1.1.2.3.3.cmml" xref="S3.SS4.7.p7.4.m4.1.1.2.3.3">𝑡</ci></apply></apply><cn id="S3.SS4.7.p7.4.m4.1.1.3.cmml" type="integer" xref="S3.SS4.7.p7.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.4.m4.1c">\beta_{m+t}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.4.m4.1d">italic_β start_POSTSUBSCRIPT italic_m + italic_t end_POSTSUBSCRIPT = 0</annotation></semantics></math> (cf. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>). Initially, <math alttext="X" class="ltx_Math" display="inline" id="S3.SS4.7.p7.5.m5.1"><semantics id="S3.SS4.7.p7.5.m5.1a"><mi id="S3.SS4.7.p7.5.m5.1.1" xref="S3.SS4.7.p7.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.5.m5.1b"><ci id="S3.SS4.7.p7.5.m5.1.1.cmml" xref="S3.SS4.7.p7.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.5.m5.1d">italic_X</annotation></semantics></math> contains only the element <math alttext="m+t" class="ltx_Math" display="inline" id="S3.SS4.7.p7.6.m6.1"><semantics id="S3.SS4.7.p7.6.m6.1a"><mrow id="S3.SS4.7.p7.6.m6.1.1" xref="S3.SS4.7.p7.6.m6.1.1.cmml"><mi id="S3.SS4.7.p7.6.m6.1.1.2" xref="S3.SS4.7.p7.6.m6.1.1.2.cmml">m</mi><mo id="S3.SS4.7.p7.6.m6.1.1.1" xref="S3.SS4.7.p7.6.m6.1.1.1.cmml">+</mo><mi id="S3.SS4.7.p7.6.m6.1.1.3" xref="S3.SS4.7.p7.6.m6.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.6.m6.1b"><apply id="S3.SS4.7.p7.6.m6.1.1.cmml" xref="S3.SS4.7.p7.6.m6.1.1"><plus id="S3.SS4.7.p7.6.m6.1.1.1.cmml" xref="S3.SS4.7.p7.6.m6.1.1.1"></plus><ci id="S3.SS4.7.p7.6.m6.1.1.2.cmml" xref="S3.SS4.7.p7.6.m6.1.1.2">𝑚</ci><ci id="S3.SS4.7.p7.6.m6.1.1.3.cmml" xref="S3.SS4.7.p7.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.6.m6.1c">m+t</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.6.m6.1d">italic_m + italic_t</annotation></semantics></math>. Now for each <math alttext="j\in X" class="ltx_Math" display="inline" id="S3.SS4.7.p7.7.m7.1"><semantics id="S3.SS4.7.p7.7.m7.1a"><mrow id="S3.SS4.7.p7.7.m7.1.1" xref="S3.SS4.7.p7.7.m7.1.1.cmml"><mi id="S3.SS4.7.p7.7.m7.1.1.2" xref="S3.SS4.7.p7.7.m7.1.1.2.cmml">j</mi><mo id="S3.SS4.7.p7.7.m7.1.1.1" xref="S3.SS4.7.p7.7.m7.1.1.1.cmml">∈</mo><mi id="S3.SS4.7.p7.7.m7.1.1.3" xref="S3.SS4.7.p7.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.7.m7.1b"><apply id="S3.SS4.7.p7.7.m7.1.1.cmml" xref="S3.SS4.7.p7.7.m7.1.1"><in id="S3.SS4.7.p7.7.m7.1.1.1.cmml" xref="S3.SS4.7.p7.7.m7.1.1.1"></in><ci id="S3.SS4.7.p7.7.m7.1.1.2.cmml" xref="S3.SS4.7.p7.7.m7.1.1.2">𝑗</ci><ci id="S3.SS4.7.p7.7.m7.1.1.3.cmml" xref="S3.SS4.7.p7.7.m7.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.7.m7.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.7.m7.1d">italic_j ∈ italic_X</annotation></semantics></math>, if <math alttext="C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.7.p7.8.m8.1"><semantics id="S3.SS4.7.p7.8.m8.1a"><mrow id="S3.SS4.7.p7.8.m8.1.1" xref="S3.SS4.7.p7.8.m8.1.1.cmml"><msup id="S3.SS4.7.p7.8.m8.1.1.2" xref="S3.SS4.7.p7.8.m8.1.1.2.cmml"><mi id="S3.SS4.7.p7.8.m8.1.1.2.2" xref="S3.SS4.7.p7.8.m8.1.1.2.2.cmml">C</mi><mi id="S3.SS4.7.p7.8.m8.1.1.2.3" xref="S3.SS4.7.p7.8.m8.1.1.2.3.cmml">j</mi></msup><mo id="S3.SS4.7.p7.8.m8.1.1.1" xref="S3.SS4.7.p7.8.m8.1.1.1.cmml">=</mo><mrow id="S3.SS4.7.p7.8.m8.1.1.3" xref="S3.SS4.7.p7.8.m8.1.1.3.cmml"><msup id="S3.SS4.7.p7.8.m8.1.1.3.2" xref="S3.SS4.7.p7.8.m8.1.1.3.2.cmml"><mi id="S3.SS4.7.p7.8.m8.1.1.3.2.2" xref="S3.SS4.7.p7.8.m8.1.1.3.2.2.cmml">C</mi><msup id="S3.SS4.7.p7.8.m8.1.1.3.2.3" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3.cmml"><mi id="S3.SS4.7.p7.8.m8.1.1.3.2.3.2" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3.2.cmml">j</mi><mo id="S3.SS4.7.p7.8.m8.1.1.3.2.3.3" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3.3.cmml">′</mo></msup></msup><msub id="S3.SS4.7.p7.8.m8.1.1.3.1" xref="S3.SS4.7.p7.8.m8.1.1.3.1.cmml"><mo id="S3.SS4.7.p7.8.m8.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="S3.SS4.7.p7.8.m8.1.1.3.1.2.cmml">⋄</mo><mi id="S3.SS4.7.p7.8.m8.1.1.3.1.3" xref="S3.SS4.7.p7.8.m8.1.1.3.1.3.cmml">j</mi></msub><msup id="S3.SS4.7.p7.8.m8.1.1.3.3" xref="S3.SS4.7.p7.8.m8.1.1.3.3.cmml"><mi id="S3.SS4.7.p7.8.m8.1.1.3.3.2" xref="S3.SS4.7.p7.8.m8.1.1.3.3.2.cmml">C</mi><msup id="S3.SS4.7.p7.8.m8.1.1.3.3.3" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3.cmml"><mi id="S3.SS4.7.p7.8.m8.1.1.3.3.3.2" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3.2.cmml">j</mi><mo id="S3.SS4.7.p7.8.m8.1.1.3.3.3.3" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.8.m8.1b"><apply id="S3.SS4.7.p7.8.m8.1.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1"><eq id="S3.SS4.7.p7.8.m8.1.1.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.1"></eq><apply id="S3.SS4.7.p7.8.m8.1.1.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.2.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.2">superscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.2.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.2.2">𝐶</ci><ci id="S3.SS4.7.p7.8.m8.1.1.2.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.2.3">𝑗</ci></apply><apply id="S3.SS4.7.p7.8.m8.1.1.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3"><apply id="S3.SS4.7.p7.8.m8.1.1.3.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.3.1.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.1">subscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.3.1.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.1.2">⋄</ci><ci id="S3.SS4.7.p7.8.m8.1.1.3.1.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.1.3">𝑗</ci></apply><apply id="S3.SS4.7.p7.8.m8.1.1.3.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.3.2.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2">superscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.3.2.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2.2">𝐶</ci><apply id="S3.SS4.7.p7.8.m8.1.1.3.2.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.3.2.3.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3">superscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.3.2.3.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3.2">𝑗</ci><ci id="S3.SS4.7.p7.8.m8.1.1.3.2.3.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.2.3.3">′</ci></apply></apply><apply id="S3.SS4.7.p7.8.m8.1.1.3.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.3.3.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3">superscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.3.3.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3.2">𝐶</ci><apply id="S3.SS4.7.p7.8.m8.1.1.3.3.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.7.p7.8.m8.1.1.3.3.3.1.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3">superscript</csymbol><ci id="S3.SS4.7.p7.8.m8.1.1.3.3.3.2.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3.2">𝑗</ci><ci id="S3.SS4.7.p7.8.m8.1.1.3.3.3.3.cmml" xref="S3.SS4.7.p7.8.m8.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.8.m8.1c">C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.8.m8.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\alpha_{j^{\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.7.p7.9.m9.1"><semantics id="S3.SS4.7.p7.9.m9.1a"><mrow id="S3.SS4.7.p7.9.m9.1.1" xref="S3.SS4.7.p7.9.m9.1.1.cmml"><msub id="S3.SS4.7.p7.9.m9.1.1.2" xref="S3.SS4.7.p7.9.m9.1.1.2.cmml"><mi id="S3.SS4.7.p7.9.m9.1.1.2.2" xref="S3.SS4.7.p7.9.m9.1.1.2.2.cmml">α</mi><msup id="S3.SS4.7.p7.9.m9.1.1.2.3" xref="S3.SS4.7.p7.9.m9.1.1.2.3.cmml"><mi id="S3.SS4.7.p7.9.m9.1.1.2.3.2" xref="S3.SS4.7.p7.9.m9.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.7.p7.9.m9.1.1.2.3.3" xref="S3.SS4.7.p7.9.m9.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.7.p7.9.m9.1.1.1" xref="S3.SS4.7.p7.9.m9.1.1.1.cmml">=</mo><mn id="S3.SS4.7.p7.9.m9.1.1.3" xref="S3.SS4.7.p7.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.9.m9.1b"><apply id="S3.SS4.7.p7.9.m9.1.1.cmml" xref="S3.SS4.7.p7.9.m9.1.1"><eq id="S3.SS4.7.p7.9.m9.1.1.1.cmml" xref="S3.SS4.7.p7.9.m9.1.1.1"></eq><apply id="S3.SS4.7.p7.9.m9.1.1.2.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.9.m9.1.1.2.1.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS4.7.p7.9.m9.1.1.2.2.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2.2">𝛼</ci><apply id="S3.SS4.7.p7.9.m9.1.1.2.3.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.7.p7.9.m9.1.1.2.3.1.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2.3">superscript</csymbol><ci id="S3.SS4.7.p7.9.m9.1.1.2.3.2.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.7.p7.9.m9.1.1.2.3.3.cmml" xref="S3.SS4.7.p7.9.m9.1.1.2.3.3">′</ci></apply></apply><cn id="S3.SS4.7.p7.9.m9.1.1.3.cmml" type="integer" xref="S3.SS4.7.p7.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.9.m9.1c">\alpha_{j^{\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.9.m9.1d">italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>, and <math alttext="\beta_{j^{\prime}}=0" class="ltx_Math" display="inline" id="S3.SS4.7.p7.10.m10.1"><semantics id="S3.SS4.7.p7.10.m10.1a"><mrow id="S3.SS4.7.p7.10.m10.1.1" xref="S3.SS4.7.p7.10.m10.1.1.cmml"><msub id="S3.SS4.7.p7.10.m10.1.1.2" xref="S3.SS4.7.p7.10.m10.1.1.2.cmml"><mi id="S3.SS4.7.p7.10.m10.1.1.2.2" xref="S3.SS4.7.p7.10.m10.1.1.2.2.cmml">β</mi><msup id="S3.SS4.7.p7.10.m10.1.1.2.3" xref="S3.SS4.7.p7.10.m10.1.1.2.3.cmml"><mi id="S3.SS4.7.p7.10.m10.1.1.2.3.2" xref="S3.SS4.7.p7.10.m10.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.7.p7.10.m10.1.1.2.3.3" xref="S3.SS4.7.p7.10.m10.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.7.p7.10.m10.1.1.1" xref="S3.SS4.7.p7.10.m10.1.1.1.cmml">=</mo><mn id="S3.SS4.7.p7.10.m10.1.1.3" xref="S3.SS4.7.p7.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.10.m10.1b"><apply id="S3.SS4.7.p7.10.m10.1.1.cmml" xref="S3.SS4.7.p7.10.m10.1.1"><eq id="S3.SS4.7.p7.10.m10.1.1.1.cmml" xref="S3.SS4.7.p7.10.m10.1.1.1"></eq><apply id="S3.SS4.7.p7.10.m10.1.1.2.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.10.m10.1.1.2.1.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS4.7.p7.10.m10.1.1.2.2.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2.2">𝛽</ci><apply id="S3.SS4.7.p7.10.m10.1.1.2.3.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.7.p7.10.m10.1.1.2.3.1.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2.3">superscript</csymbol><ci id="S3.SS4.7.p7.10.m10.1.1.2.3.2.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.7.p7.10.m10.1.1.2.3.3.cmml" xref="S3.SS4.7.p7.10.m10.1.1.2.3.3">′</ci></apply></apply><cn id="S3.SS4.7.p7.10.m10.1.1.3.cmml" type="integer" xref="S3.SS4.7.p7.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.10.m10.1c">\beta_{j^{\prime}}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.10.m10.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>, then we add the element <math alttext="j^{\prime}" class="ltx_Math" display="inline" id="S3.SS4.7.p7.11.m11.1"><semantics id="S3.SS4.7.p7.11.m11.1a"><msup id="S3.SS4.7.p7.11.m11.1.1" xref="S3.SS4.7.p7.11.m11.1.1.cmml"><mi id="S3.SS4.7.p7.11.m11.1.1.2" xref="S3.SS4.7.p7.11.m11.1.1.2.cmml">j</mi><mo id="S3.SS4.7.p7.11.m11.1.1.3" xref="S3.SS4.7.p7.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.11.m11.1b"><apply id="S3.SS4.7.p7.11.m11.1.1.cmml" xref="S3.SS4.7.p7.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS4.7.p7.11.m11.1.1.1.cmml" xref="S3.SS4.7.p7.11.m11.1.1">superscript</csymbol><ci id="S3.SS4.7.p7.11.m11.1.1.2.cmml" xref="S3.SS4.7.p7.11.m11.1.1.2">𝑗</ci><ci id="S3.SS4.7.p7.11.m11.1.1.3.cmml" xref="S3.SS4.7.p7.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.11.m11.1c">j^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.11.m11.1d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="X" class="ltx_Math" display="inline" id="S3.SS4.7.p7.12.m12.1"><semantics id="S3.SS4.7.p7.12.m12.1a"><mi id="S3.SS4.7.p7.12.m12.1.1" xref="S3.SS4.7.p7.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.12.m12.1b"><ci id="S3.SS4.7.p7.12.m12.1.1.cmml" xref="S3.SS4.7.p7.12.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.12.m12.1d">italic_X</annotation></semantics></math> (and similarly for the index <math alttext="j^{\prime\prime}" class="ltx_Math" display="inline" id="S3.SS4.7.p7.13.m13.1"><semantics id="S3.SS4.7.p7.13.m13.1a"><msup id="S3.SS4.7.p7.13.m13.1.1" xref="S3.SS4.7.p7.13.m13.1.1.cmml"><mi id="S3.SS4.7.p7.13.m13.1.1.2" xref="S3.SS4.7.p7.13.m13.1.1.2.cmml">j</mi><mo id="S3.SS4.7.p7.13.m13.1.1.3" xref="S3.SS4.7.p7.13.m13.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.13.m13.1b"><apply id="S3.SS4.7.p7.13.m13.1.1.cmml" xref="S3.SS4.7.p7.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS4.7.p7.13.m13.1.1.1.cmml" xref="S3.SS4.7.p7.13.m13.1.1">superscript</csymbol><ci id="S3.SS4.7.p7.13.m13.1.1.2.cmml" xref="S3.SS4.7.p7.13.m13.1.1.2">𝑗</ci><ci id="S3.SS4.7.p7.13.m13.1.1.3.cmml" xref="S3.SS4.7.p7.13.m13.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.13.m13.1c">j^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.13.m13.1d">italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>). We repeat this procedure until no more elements are added to <math alttext="X" class="ltx_Math" display="inline" id="S3.SS4.7.p7.14.m14.1"><semantics id="S3.SS4.7.p7.14.m14.1a"><mi id="S3.SS4.7.p7.14.m14.1.1" xref="S3.SS4.7.p7.14.m14.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.14.m14.1b"><ci id="S3.SS4.7.p7.14.m14.1.1.cmml" xref="S3.SS4.7.p7.14.m14.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.14.m14.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.14.m14.1d">italic_X</annotation></semantics></math>, and let <math alttext="Y\stackrel{{\scriptstyle\rm def}}{{=}}[m+t]\setminus X" class="ltx_Math" display="inline" id="S3.SS4.7.p7.15.m15.1"><semantics id="S3.SS4.7.p7.15.m15.1a"><mrow id="S3.SS4.7.p7.15.m15.1.1" xref="S3.SS4.7.p7.15.m15.1.1.cmml"><mi id="S3.SS4.7.p7.15.m15.1.1.3" xref="S3.SS4.7.p7.15.m15.1.1.3.cmml">Y</mi><mover id="S3.SS4.7.p7.15.m15.1.1.2" xref="S3.SS4.7.p7.15.m15.1.1.2.cmml"><mo id="S3.SS4.7.p7.15.m15.1.1.2.2" xref="S3.SS4.7.p7.15.m15.1.1.2.2.cmml">=</mo><mi id="S3.SS4.7.p7.15.m15.1.1.2.3" xref="S3.SS4.7.p7.15.m15.1.1.2.3.cmml">def</mi></mover><mrow id="S3.SS4.7.p7.15.m15.1.1.1" xref="S3.SS4.7.p7.15.m15.1.1.1.cmml"><mrow id="S3.SS4.7.p7.15.m15.1.1.1.1.1" xref="S3.SS4.7.p7.15.m15.1.1.1.1.2.cmml"><mo id="S3.SS4.7.p7.15.m15.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.7.p7.15.m15.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.cmml"><mi id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.2" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.2.cmml">m</mi><mo id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.1" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.3" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.SS4.7.p7.15.m15.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.7.p7.15.m15.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.SS4.7.p7.15.m15.1.1.1.2" xref="S3.SS4.7.p7.15.m15.1.1.1.2.cmml">∖</mo><mi id="S3.SS4.7.p7.15.m15.1.1.1.3" xref="S3.SS4.7.p7.15.m15.1.1.1.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p7.15.m15.1b"><apply id="S3.SS4.7.p7.15.m15.1.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1"><apply id="S3.SS4.7.p7.15.m15.1.1.2.cmml" xref="S3.SS4.7.p7.15.m15.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.7.p7.15.m15.1.1.2.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1.2">superscript</csymbol><eq id="S3.SS4.7.p7.15.m15.1.1.2.2.cmml" xref="S3.SS4.7.p7.15.m15.1.1.2.2"></eq><ci id="S3.SS4.7.p7.15.m15.1.1.2.3.cmml" xref="S3.SS4.7.p7.15.m15.1.1.2.3">def</ci></apply><ci id="S3.SS4.7.p7.15.m15.1.1.3.cmml" xref="S3.SS4.7.p7.15.m15.1.1.3">𝑌</ci><apply id="S3.SS4.7.p7.15.m15.1.1.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1"><setdiff id="S3.SS4.7.p7.15.m15.1.1.1.2.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.2"></setdiff><apply id="S3.SS4.7.p7.15.m15.1.1.1.1.2.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS4.7.p7.15.m15.1.1.1.1.2.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1"><plus id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.1"></plus><ci id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.2.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.2">𝑚</ci><ci id="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.3.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.1.1.1.3">𝑡</ci></apply></apply><ci id="S3.SS4.7.p7.15.m15.1.1.1.3.cmml" xref="S3.SS4.7.p7.15.m15.1.1.1.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p7.15.m15.1c">Y\stackrel{{\scriptstyle\rm def}}{{=}}[m+t]\setminus X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p7.15.m15.1d">italic_Y start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP [ italic_m + italic_t ] ∖ italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.8.p8"> <p class="ltx_p" id="S3.SS4.8.p8.1">We observe the following properties of this partition.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem32"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem32.1.1.1">Claim 32</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem32.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem32.p1"> <p class="ltx_p" id="Thmtheorem32.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem32.p1.5.5">We have <math alttext="m+t\in X" class="ltx_Math" display="inline" id="Thmtheorem32.p1.1.1.m1.1"><semantics id="Thmtheorem32.p1.1.1.m1.1a"><mrow id="Thmtheorem32.p1.1.1.m1.1.1" xref="Thmtheorem32.p1.1.1.m1.1.1.cmml"><mrow id="Thmtheorem32.p1.1.1.m1.1.1.2" xref="Thmtheorem32.p1.1.1.m1.1.1.2.cmml"><mi id="Thmtheorem32.p1.1.1.m1.1.1.2.2" xref="Thmtheorem32.p1.1.1.m1.1.1.2.2.cmml">m</mi><mo id="Thmtheorem32.p1.1.1.m1.1.1.2.1" xref="Thmtheorem32.p1.1.1.m1.1.1.2.1.cmml">+</mo><mi id="Thmtheorem32.p1.1.1.m1.1.1.2.3" xref="Thmtheorem32.p1.1.1.m1.1.1.2.3.cmml">t</mi></mrow><mo id="Thmtheorem32.p1.1.1.m1.1.1.1" xref="Thmtheorem32.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem32.p1.1.1.m1.1.1.3" xref="Thmtheorem32.p1.1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem32.p1.1.1.m1.1b"><apply id="Thmtheorem32.p1.1.1.m1.1.1.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1"><in id="Thmtheorem32.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.1"></in><apply id="Thmtheorem32.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.2"><plus id="Thmtheorem32.p1.1.1.m1.1.1.2.1.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.2.1"></plus><ci id="Thmtheorem32.p1.1.1.m1.1.1.2.2.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.2.2">𝑚</ci><ci id="Thmtheorem32.p1.1.1.m1.1.1.2.3.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="Thmtheorem32.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem32.p1.1.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem32.p1.1.1.m1.1c">m+t\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem32.p1.1.1.m1.1d">italic_m + italic_t ∈ italic_X</annotation></semantics></math> and <math alttext="\{1,\ldots,m\}\subseteq Y" class="ltx_Math" display="inline" id="Thmtheorem32.p1.2.2.m2.3"><semantics id="Thmtheorem32.p1.2.2.m2.3a"><mrow id="Thmtheorem32.p1.2.2.m2.3.4" xref="Thmtheorem32.p1.2.2.m2.3.4.cmml"><mrow id="Thmtheorem32.p1.2.2.m2.3.4.2.2" xref="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml"><mo id="Thmtheorem32.p1.2.2.m2.3.4.2.2.1" stretchy="false" xref="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml">{</mo><mn id="Thmtheorem32.p1.2.2.m2.1.1" xref="Thmtheorem32.p1.2.2.m2.1.1.cmml">1</mn><mo id="Thmtheorem32.p1.2.2.m2.3.4.2.2.2" xref="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml">,</mo><mi id="Thmtheorem32.p1.2.2.m2.2.2" mathvariant="normal" xref="Thmtheorem32.p1.2.2.m2.2.2.cmml">…</mi><mo id="Thmtheorem32.p1.2.2.m2.3.4.2.2.3" xref="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml">,</mo><mi id="Thmtheorem32.p1.2.2.m2.3.3" xref="Thmtheorem32.p1.2.2.m2.3.3.cmml">m</mi><mo id="Thmtheorem32.p1.2.2.m2.3.4.2.2.4" stretchy="false" xref="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml">}</mo></mrow><mo id="Thmtheorem32.p1.2.2.m2.3.4.1" xref="Thmtheorem32.p1.2.2.m2.3.4.1.cmml">⊆</mo><mi id="Thmtheorem32.p1.2.2.m2.3.4.3" xref="Thmtheorem32.p1.2.2.m2.3.4.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem32.p1.2.2.m2.3b"><apply id="Thmtheorem32.p1.2.2.m2.3.4.cmml" xref="Thmtheorem32.p1.2.2.m2.3.4"><subset id="Thmtheorem32.p1.2.2.m2.3.4.1.cmml" xref="Thmtheorem32.p1.2.2.m2.3.4.1"></subset><set id="Thmtheorem32.p1.2.2.m2.3.4.2.1.cmml" xref="Thmtheorem32.p1.2.2.m2.3.4.2.2"><cn id="Thmtheorem32.p1.2.2.m2.1.1.cmml" type="integer" xref="Thmtheorem32.p1.2.2.m2.1.1">1</cn><ci id="Thmtheorem32.p1.2.2.m2.2.2.cmml" xref="Thmtheorem32.p1.2.2.m2.2.2">…</ci><ci id="Thmtheorem32.p1.2.2.m2.3.3.cmml" xref="Thmtheorem32.p1.2.2.m2.3.3">𝑚</ci></set><ci id="Thmtheorem32.p1.2.2.m2.3.4.3.cmml" xref="Thmtheorem32.p1.2.2.m2.3.4.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem32.p1.2.2.m2.3c">\{1,\ldots,m\}\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem32.p1.2.2.m2.3d">{ 1 , … , italic_m } ⊆ italic_Y</annotation></semantics></math>. If an element <math alttext="j\in X" class="ltx_Math" display="inline" id="Thmtheorem32.p1.3.3.m3.1"><semantics id="Thmtheorem32.p1.3.3.m3.1a"><mrow id="Thmtheorem32.p1.3.3.m3.1.1" xref="Thmtheorem32.p1.3.3.m3.1.1.cmml"><mi id="Thmtheorem32.p1.3.3.m3.1.1.2" xref="Thmtheorem32.p1.3.3.m3.1.1.2.cmml">j</mi><mo id="Thmtheorem32.p1.3.3.m3.1.1.1" xref="Thmtheorem32.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="Thmtheorem32.p1.3.3.m3.1.1.3" xref="Thmtheorem32.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem32.p1.3.3.m3.1b"><apply id="Thmtheorem32.p1.3.3.m3.1.1.cmml" xref="Thmtheorem32.p1.3.3.m3.1.1"><in id="Thmtheorem32.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem32.p1.3.3.m3.1.1.1"></in><ci id="Thmtheorem32.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem32.p1.3.3.m3.1.1.2">𝑗</ci><ci id="Thmtheorem32.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem32.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem32.p1.3.3.m3.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem32.p1.3.3.m3.1d">italic_j ∈ italic_X</annotation></semantics></math>, then <math alttext="\alpha_{j}=1" class="ltx_Math" display="inline" id="Thmtheorem32.p1.4.4.m4.1"><semantics id="Thmtheorem32.p1.4.4.m4.1a"><mrow id="Thmtheorem32.p1.4.4.m4.1.1" xref="Thmtheorem32.p1.4.4.m4.1.1.cmml"><msub id="Thmtheorem32.p1.4.4.m4.1.1.2" xref="Thmtheorem32.p1.4.4.m4.1.1.2.cmml"><mi id="Thmtheorem32.p1.4.4.m4.1.1.2.2" xref="Thmtheorem32.p1.4.4.m4.1.1.2.2.cmml">α</mi><mi id="Thmtheorem32.p1.4.4.m4.1.1.2.3" xref="Thmtheorem32.p1.4.4.m4.1.1.2.3.cmml">j</mi></msub><mo id="Thmtheorem32.p1.4.4.m4.1.1.1" xref="Thmtheorem32.p1.4.4.m4.1.1.1.cmml">=</mo><mn id="Thmtheorem32.p1.4.4.m4.1.1.3" xref="Thmtheorem32.p1.4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem32.p1.4.4.m4.1b"><apply id="Thmtheorem32.p1.4.4.m4.1.1.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1"><eq id="Thmtheorem32.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1.1"></eq><apply id="Thmtheorem32.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem32.p1.4.4.m4.1.1.2.1.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1.2">subscript</csymbol><ci id="Thmtheorem32.p1.4.4.m4.1.1.2.2.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1.2.2">𝛼</ci><ci id="Thmtheorem32.p1.4.4.m4.1.1.2.3.cmml" xref="Thmtheorem32.p1.4.4.m4.1.1.2.3">𝑗</ci></apply><cn id="Thmtheorem32.p1.4.4.m4.1.1.3.cmml" type="integer" xref="Thmtheorem32.p1.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem32.p1.4.4.m4.1c">\alpha_{j}=1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem32.p1.4.4.m4.1d">italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{j}=0" class="ltx_Math" display="inline" id="Thmtheorem32.p1.5.5.m5.1"><semantics id="Thmtheorem32.p1.5.5.m5.1a"><mrow id="Thmtheorem32.p1.5.5.m5.1.1" xref="Thmtheorem32.p1.5.5.m5.1.1.cmml"><msub id="Thmtheorem32.p1.5.5.m5.1.1.2" xref="Thmtheorem32.p1.5.5.m5.1.1.2.cmml"><mi id="Thmtheorem32.p1.5.5.m5.1.1.2.2" xref="Thmtheorem32.p1.5.5.m5.1.1.2.2.cmml">β</mi><mi id="Thmtheorem32.p1.5.5.m5.1.1.2.3" xref="Thmtheorem32.p1.5.5.m5.1.1.2.3.cmml">j</mi></msub><mo id="Thmtheorem32.p1.5.5.m5.1.1.1" xref="Thmtheorem32.p1.5.5.m5.1.1.1.cmml">=</mo><mn id="Thmtheorem32.p1.5.5.m5.1.1.3" xref="Thmtheorem32.p1.5.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem32.p1.5.5.m5.1b"><apply id="Thmtheorem32.p1.5.5.m5.1.1.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1"><eq id="Thmtheorem32.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1.1"></eq><apply id="Thmtheorem32.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem32.p1.5.5.m5.1.1.2.1.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="Thmtheorem32.p1.5.5.m5.1.1.2.2.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1.2.2">𝛽</ci><ci id="Thmtheorem32.p1.5.5.m5.1.1.2.3.cmml" xref="Thmtheorem32.p1.5.5.m5.1.1.2.3">𝑗</ci></apply><cn id="Thmtheorem32.p1.5.5.m5.1.1.3.cmml" type="integer" xref="Thmtheorem32.p1.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem32.p1.5.5.m5.1c">\beta_{j}=0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem32.p1.5.5.m5.1d">italic_β start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS4.9.p9"> <p class="ltx_p" id="S3.SS4.9.p9.11">The only non-trivial statement is that <math alttext="\{1,\ldots,m\}\subseteq Y" class="ltx_Math" display="inline" id="S3.SS4.9.p9.1.m1.3"><semantics id="S3.SS4.9.p9.1.m1.3a"><mrow id="S3.SS4.9.p9.1.m1.3.4" xref="S3.SS4.9.p9.1.m1.3.4.cmml"><mrow id="S3.SS4.9.p9.1.m1.3.4.2.2" xref="S3.SS4.9.p9.1.m1.3.4.2.1.cmml"><mo id="S3.SS4.9.p9.1.m1.3.4.2.2.1" stretchy="false" xref="S3.SS4.9.p9.1.m1.3.4.2.1.cmml">{</mo><mn id="S3.SS4.9.p9.1.m1.1.1" xref="S3.SS4.9.p9.1.m1.1.1.cmml">1</mn><mo id="S3.SS4.9.p9.1.m1.3.4.2.2.2" xref="S3.SS4.9.p9.1.m1.3.4.2.1.cmml">,</mo><mi id="S3.SS4.9.p9.1.m1.2.2" mathvariant="normal" xref="S3.SS4.9.p9.1.m1.2.2.cmml">…</mi><mo id="S3.SS4.9.p9.1.m1.3.4.2.2.3" xref="S3.SS4.9.p9.1.m1.3.4.2.1.cmml">,</mo><mi id="S3.SS4.9.p9.1.m1.3.3" xref="S3.SS4.9.p9.1.m1.3.3.cmml">m</mi><mo id="S3.SS4.9.p9.1.m1.3.4.2.2.4" stretchy="false" xref="S3.SS4.9.p9.1.m1.3.4.2.1.cmml">}</mo></mrow><mo id="S3.SS4.9.p9.1.m1.3.4.1" xref="S3.SS4.9.p9.1.m1.3.4.1.cmml">⊆</mo><mi id="S3.SS4.9.p9.1.m1.3.4.3" xref="S3.SS4.9.p9.1.m1.3.4.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.1.m1.3b"><apply id="S3.SS4.9.p9.1.m1.3.4.cmml" xref="S3.SS4.9.p9.1.m1.3.4"><subset id="S3.SS4.9.p9.1.m1.3.4.1.cmml" xref="S3.SS4.9.p9.1.m1.3.4.1"></subset><set id="S3.SS4.9.p9.1.m1.3.4.2.1.cmml" xref="S3.SS4.9.p9.1.m1.3.4.2.2"><cn id="S3.SS4.9.p9.1.m1.1.1.cmml" type="integer" xref="S3.SS4.9.p9.1.m1.1.1">1</cn><ci id="S3.SS4.9.p9.1.m1.2.2.cmml" xref="S3.SS4.9.p9.1.m1.2.2">…</ci><ci id="S3.SS4.9.p9.1.m1.3.3.cmml" xref="S3.SS4.9.p9.1.m1.3.3">𝑚</ci></set><ci id="S3.SS4.9.p9.1.m1.3.4.3.cmml" xref="S3.SS4.9.p9.1.m1.3.4.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.1.m1.3c">\{1,\ldots,m\}\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.1.m1.3d">{ 1 , … , italic_m } ⊆ italic_Y</annotation></semantics></math>. It is enough to argue that if <math alttext="\ell\in[m]" class="ltx_Math" display="inline" id="S3.SS4.9.p9.2.m2.1"><semantics id="S3.SS4.9.p9.2.m2.1a"><mrow id="S3.SS4.9.p9.2.m2.1.2" xref="S3.SS4.9.p9.2.m2.1.2.cmml"><mi id="S3.SS4.9.p9.2.m2.1.2.2" mathvariant="normal" xref="S3.SS4.9.p9.2.m2.1.2.2.cmml">ℓ</mi><mo id="S3.SS4.9.p9.2.m2.1.2.1" xref="S3.SS4.9.p9.2.m2.1.2.1.cmml">∈</mo><mrow id="S3.SS4.9.p9.2.m2.1.2.3.2" xref="S3.SS4.9.p9.2.m2.1.2.3.1.cmml"><mo id="S3.SS4.9.p9.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS4.9.p9.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S3.SS4.9.p9.2.m2.1.1" xref="S3.SS4.9.p9.2.m2.1.1.cmml">m</mi><mo id="S3.SS4.9.p9.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS4.9.p9.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.2.m2.1b"><apply id="S3.SS4.9.p9.2.m2.1.2.cmml" xref="S3.SS4.9.p9.2.m2.1.2"><in id="S3.SS4.9.p9.2.m2.1.2.1.cmml" xref="S3.SS4.9.p9.2.m2.1.2.1"></in><ci id="S3.SS4.9.p9.2.m2.1.2.2.cmml" xref="S3.SS4.9.p9.2.m2.1.2.2">ℓ</ci><apply id="S3.SS4.9.p9.2.m2.1.2.3.1.cmml" xref="S3.SS4.9.p9.2.m2.1.2.3.2"><csymbol cd="latexml" id="S3.SS4.9.p9.2.m2.1.2.3.1.1.cmml" xref="S3.SS4.9.p9.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS4.9.p9.2.m2.1.1.cmml" xref="S3.SS4.9.p9.2.m2.1.1">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.2.m2.1c">\ell\in[m]</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.2.m2.1d">roman_ℓ ∈ [ italic_m ]</annotation></semantics></math> then it is not the case that <math alttext="\alpha_{\ell}=1" class="ltx_Math" display="inline" id="S3.SS4.9.p9.3.m3.1"><semantics id="S3.SS4.9.p9.3.m3.1a"><mrow id="S3.SS4.9.p9.3.m3.1.1" xref="S3.SS4.9.p9.3.m3.1.1.cmml"><msub id="S3.SS4.9.p9.3.m3.1.1.2" xref="S3.SS4.9.p9.3.m3.1.1.2.cmml"><mi id="S3.SS4.9.p9.3.m3.1.1.2.2" xref="S3.SS4.9.p9.3.m3.1.1.2.2.cmml">α</mi><mi id="S3.SS4.9.p9.3.m3.1.1.2.3" mathvariant="normal" xref="S3.SS4.9.p9.3.m3.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.SS4.9.p9.3.m3.1.1.1" xref="S3.SS4.9.p9.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS4.9.p9.3.m3.1.1.3" xref="S3.SS4.9.p9.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.3.m3.1b"><apply id="S3.SS4.9.p9.3.m3.1.1.cmml" xref="S3.SS4.9.p9.3.m3.1.1"><eq id="S3.SS4.9.p9.3.m3.1.1.1.cmml" xref="S3.SS4.9.p9.3.m3.1.1.1"></eq><apply id="S3.SS4.9.p9.3.m3.1.1.2.cmml" xref="S3.SS4.9.p9.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.9.p9.3.m3.1.1.2.1.cmml" xref="S3.SS4.9.p9.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.9.p9.3.m3.1.1.2.2.cmml" xref="S3.SS4.9.p9.3.m3.1.1.2.2">𝛼</ci><ci id="S3.SS4.9.p9.3.m3.1.1.2.3.cmml" xref="S3.SS4.9.p9.3.m3.1.1.2.3">ℓ</ci></apply><cn id="S3.SS4.9.p9.3.m3.1.1.3.cmml" type="integer" xref="S3.SS4.9.p9.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.3.m3.1c">\alpha_{\ell}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.3.m3.1d">italic_α start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{\ell}=0" class="ltx_Math" display="inline" id="S3.SS4.9.p9.4.m4.1"><semantics id="S3.SS4.9.p9.4.m4.1a"><mrow id="S3.SS4.9.p9.4.m4.1.1" xref="S3.SS4.9.p9.4.m4.1.1.cmml"><msub id="S3.SS4.9.p9.4.m4.1.1.2" xref="S3.SS4.9.p9.4.m4.1.1.2.cmml"><mi id="S3.SS4.9.p9.4.m4.1.1.2.2" xref="S3.SS4.9.p9.4.m4.1.1.2.2.cmml">β</mi><mi id="S3.SS4.9.p9.4.m4.1.1.2.3" mathvariant="normal" xref="S3.SS4.9.p9.4.m4.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.SS4.9.p9.4.m4.1.1.1" xref="S3.SS4.9.p9.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS4.9.p9.4.m4.1.1.3" xref="S3.SS4.9.p9.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.4.m4.1b"><apply id="S3.SS4.9.p9.4.m4.1.1.cmml" xref="S3.SS4.9.p9.4.m4.1.1"><eq id="S3.SS4.9.p9.4.m4.1.1.1.cmml" xref="S3.SS4.9.p9.4.m4.1.1.1"></eq><apply id="S3.SS4.9.p9.4.m4.1.1.2.cmml" xref="S3.SS4.9.p9.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.9.p9.4.m4.1.1.2.1.cmml" xref="S3.SS4.9.p9.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.9.p9.4.m4.1.1.2.2.cmml" xref="S3.SS4.9.p9.4.m4.1.1.2.2">𝛽</ci><ci id="S3.SS4.9.p9.4.m4.1.1.2.3.cmml" xref="S3.SS4.9.p9.4.m4.1.1.2.3">ℓ</ci></apply><cn id="S3.SS4.9.p9.4.m4.1.1.3.cmml" type="integer" xref="S3.SS4.9.p9.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.4.m4.1c">\beta_{\ell}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.4.m4.1d">italic_β start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 0</annotation></semantics></math>. But since <math alttext="C^{\ell}=B_{\ell}\in\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS4.9.p9.5.m5.1"><semantics id="S3.SS4.9.p9.5.m5.1a"><mrow id="S3.SS4.9.p9.5.m5.1.1" xref="S3.SS4.9.p9.5.m5.1.1.cmml"><msup id="S3.SS4.9.p9.5.m5.1.1.2" xref="S3.SS4.9.p9.5.m5.1.1.2.cmml"><mi id="S3.SS4.9.p9.5.m5.1.1.2.2" xref="S3.SS4.9.p9.5.m5.1.1.2.2.cmml">C</mi><mi id="S3.SS4.9.p9.5.m5.1.1.2.3" mathvariant="normal" xref="S3.SS4.9.p9.5.m5.1.1.2.3.cmml">ℓ</mi></msup><mo id="S3.SS4.9.p9.5.m5.1.1.3" xref="S3.SS4.9.p9.5.m5.1.1.3.cmml">=</mo><msub id="S3.SS4.9.p9.5.m5.1.1.4" xref="S3.SS4.9.p9.5.m5.1.1.4.cmml"><mi id="S3.SS4.9.p9.5.m5.1.1.4.2" xref="S3.SS4.9.p9.5.m5.1.1.4.2.cmml">B</mi><mi id="S3.SS4.9.p9.5.m5.1.1.4.3" mathvariant="normal" xref="S3.SS4.9.p9.5.m5.1.1.4.3.cmml">ℓ</mi></msub><mo id="S3.SS4.9.p9.5.m5.1.1.5" xref="S3.SS4.9.p9.5.m5.1.1.5.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.9.p9.5.m5.1.1.6" xref="S3.SS4.9.p9.5.m5.1.1.6.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.5.m5.1b"><apply id="S3.SS4.9.p9.5.m5.1.1.cmml" xref="S3.SS4.9.p9.5.m5.1.1"><and id="S3.SS4.9.p9.5.m5.1.1a.cmml" xref="S3.SS4.9.p9.5.m5.1.1"></and><apply id="S3.SS4.9.p9.5.m5.1.1b.cmml" xref="S3.SS4.9.p9.5.m5.1.1"><eq id="S3.SS4.9.p9.5.m5.1.1.3.cmml" xref="S3.SS4.9.p9.5.m5.1.1.3"></eq><apply id="S3.SS4.9.p9.5.m5.1.1.2.cmml" xref="S3.SS4.9.p9.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.9.p9.5.m5.1.1.2.1.cmml" xref="S3.SS4.9.p9.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS4.9.p9.5.m5.1.1.2.2.cmml" xref="S3.SS4.9.p9.5.m5.1.1.2.2">𝐶</ci><ci id="S3.SS4.9.p9.5.m5.1.1.2.3.cmml" xref="S3.SS4.9.p9.5.m5.1.1.2.3">ℓ</ci></apply><apply id="S3.SS4.9.p9.5.m5.1.1.4.cmml" xref="S3.SS4.9.p9.5.m5.1.1.4"><csymbol cd="ambiguous" id="S3.SS4.9.p9.5.m5.1.1.4.1.cmml" xref="S3.SS4.9.p9.5.m5.1.1.4">subscript</csymbol><ci id="S3.SS4.9.p9.5.m5.1.1.4.2.cmml" xref="S3.SS4.9.p9.5.m5.1.1.4.2">𝐵</ci><ci id="S3.SS4.9.p9.5.m5.1.1.4.3.cmml" xref="S3.SS4.9.p9.5.m5.1.1.4.3">ℓ</ci></apply></apply><apply id="S3.SS4.9.p9.5.m5.1.1c.cmml" xref="S3.SS4.9.p9.5.m5.1.1"><in id="S3.SS4.9.p9.5.m5.1.1.5.cmml" xref="S3.SS4.9.p9.5.m5.1.1.5"></in><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.9.p9.5.m5.1.1.4.cmml" id="S3.SS4.9.p9.5.m5.1.1d.cmml" xref="S3.SS4.9.p9.5.m5.1.1"></share><ci id="S3.SS4.9.p9.5.m5.1.1.6.cmml" xref="S3.SS4.9.p9.5.m5.1.1.6">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.5.m5.1c">C^{\ell}=B_{\ell}\in\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.5.m5.1d">italic_C start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT = italic_B start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math> and <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.9.p9.6.m6.1"><semantics id="S3.SS4.9.p9.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.9.p9.6.m6.1.1" xref="S3.SS4.9.p9.6.m6.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.6.m6.1b"><ci id="S3.SS4.9.p9.6.m6.1.1.cmml" xref="S3.SS4.9.p9.6.m6.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.6.m6.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.6.m6.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.9.p9.7.m7.1"><semantics id="S3.SS4.9.p9.7.m7.1a"><mi id="S3.SS4.9.p9.7.m7.1.1" xref="S3.SS4.9.p9.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.7.m7.1b"><ci id="S3.SS4.9.p9.7.m7.1.1.cmml" xref="S3.SS4.9.p9.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.7.m7.1d">italic_a</annotation></semantics></math>, if <math alttext="\alpha=1" class="ltx_Math" display="inline" id="S3.SS4.9.p9.8.m8.1"><semantics id="S3.SS4.9.p9.8.m8.1a"><mrow id="S3.SS4.9.p9.8.m8.1.1" xref="S3.SS4.9.p9.8.m8.1.1.cmml"><mi id="S3.SS4.9.p9.8.m8.1.1.2" xref="S3.SS4.9.p9.8.m8.1.1.2.cmml">α</mi><mo id="S3.SS4.9.p9.8.m8.1.1.1" xref="S3.SS4.9.p9.8.m8.1.1.1.cmml">=</mo><mn id="S3.SS4.9.p9.8.m8.1.1.3" xref="S3.SS4.9.p9.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.8.m8.1b"><apply id="S3.SS4.9.p9.8.m8.1.1.cmml" xref="S3.SS4.9.p9.8.m8.1.1"><eq id="S3.SS4.9.p9.8.m8.1.1.1.cmml" xref="S3.SS4.9.p9.8.m8.1.1.1"></eq><ci id="S3.SS4.9.p9.8.m8.1.1.2.cmml" xref="S3.SS4.9.p9.8.m8.1.1.2">𝛼</ci><cn id="S3.SS4.9.p9.8.m8.1.1.3.cmml" type="integer" xref="S3.SS4.9.p9.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.8.m8.1c">\alpha=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.8.m8.1d">italic_α = 1</annotation></semantics></math> (i.e., <math alttext="a\in C^{\ell}" class="ltx_Math" display="inline" id="S3.SS4.9.p9.9.m9.1"><semantics id="S3.SS4.9.p9.9.m9.1a"><mrow id="S3.SS4.9.p9.9.m9.1.1" xref="S3.SS4.9.p9.9.m9.1.1.cmml"><mi id="S3.SS4.9.p9.9.m9.1.1.2" xref="S3.SS4.9.p9.9.m9.1.1.2.cmml">a</mi><mo id="S3.SS4.9.p9.9.m9.1.1.1" xref="S3.SS4.9.p9.9.m9.1.1.1.cmml">∈</mo><msup id="S3.SS4.9.p9.9.m9.1.1.3" xref="S3.SS4.9.p9.9.m9.1.1.3.cmml"><mi id="S3.SS4.9.p9.9.m9.1.1.3.2" xref="S3.SS4.9.p9.9.m9.1.1.3.2.cmml">C</mi><mi id="S3.SS4.9.p9.9.m9.1.1.3.3" mathvariant="normal" xref="S3.SS4.9.p9.9.m9.1.1.3.3.cmml">ℓ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.9.m9.1b"><apply id="S3.SS4.9.p9.9.m9.1.1.cmml" xref="S3.SS4.9.p9.9.m9.1.1"><in id="S3.SS4.9.p9.9.m9.1.1.1.cmml" xref="S3.SS4.9.p9.9.m9.1.1.1"></in><ci id="S3.SS4.9.p9.9.m9.1.1.2.cmml" xref="S3.SS4.9.p9.9.m9.1.1.2">𝑎</ci><apply id="S3.SS4.9.p9.9.m9.1.1.3.cmml" xref="S3.SS4.9.p9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.9.p9.9.m9.1.1.3.1.cmml" xref="S3.SS4.9.p9.9.m9.1.1.3">superscript</csymbol><ci id="S3.SS4.9.p9.9.m9.1.1.3.2.cmml" xref="S3.SS4.9.p9.9.m9.1.1.3.2">𝐶</ci><ci id="S3.SS4.9.p9.9.m9.1.1.3.3.cmml" xref="S3.SS4.9.p9.9.m9.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.9.m9.1c">a\in C^{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.9.m9.1d">italic_a ∈ italic_C start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT</annotation></semantics></math>) then <math alttext="\beta=1" class="ltx_Math" display="inline" id="S3.SS4.9.p9.10.m10.1"><semantics id="S3.SS4.9.p9.10.m10.1a"><mrow id="S3.SS4.9.p9.10.m10.1.1" xref="S3.SS4.9.p9.10.m10.1.1.cmml"><mi id="S3.SS4.9.p9.10.m10.1.1.2" xref="S3.SS4.9.p9.10.m10.1.1.2.cmml">β</mi><mo id="S3.SS4.9.p9.10.m10.1.1.1" xref="S3.SS4.9.p9.10.m10.1.1.1.cmml">=</mo><mn id="S3.SS4.9.p9.10.m10.1.1.3" xref="S3.SS4.9.p9.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.10.m10.1b"><apply id="S3.SS4.9.p9.10.m10.1.1.cmml" xref="S3.SS4.9.p9.10.m10.1.1"><eq id="S3.SS4.9.p9.10.m10.1.1.1.cmml" xref="S3.SS4.9.p9.10.m10.1.1.1"></eq><ci id="S3.SS4.9.p9.10.m10.1.1.2.cmml" xref="S3.SS4.9.p9.10.m10.1.1.2">𝛽</ci><cn id="S3.SS4.9.p9.10.m10.1.1.3.cmml" type="integer" xref="S3.SS4.9.p9.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.10.m10.1c">\beta=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.10.m10.1d">italic_β = 1</annotation></semantics></math> (i.e., <math alttext="B_{\ell}\cap U\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.9.p9.11.m11.1"><semantics id="S3.SS4.9.p9.11.m11.1a"><mrow id="S3.SS4.9.p9.11.m11.1.1" xref="S3.SS4.9.p9.11.m11.1.1.cmml"><mrow id="S3.SS4.9.p9.11.m11.1.1.2" xref="S3.SS4.9.p9.11.m11.1.1.2.cmml"><msub id="S3.SS4.9.p9.11.m11.1.1.2.2" xref="S3.SS4.9.p9.11.m11.1.1.2.2.cmml"><mi id="S3.SS4.9.p9.11.m11.1.1.2.2.2" xref="S3.SS4.9.p9.11.m11.1.1.2.2.2.cmml">B</mi><mi id="S3.SS4.9.p9.11.m11.1.1.2.2.3" mathvariant="normal" xref="S3.SS4.9.p9.11.m11.1.1.2.2.3.cmml">ℓ</mi></msub><mo id="S3.SS4.9.p9.11.m11.1.1.2.1" xref="S3.SS4.9.p9.11.m11.1.1.2.1.cmml">∩</mo><mi id="S3.SS4.9.p9.11.m11.1.1.2.3" xref="S3.SS4.9.p9.11.m11.1.1.2.3.cmml">U</mi></mrow><mo id="S3.SS4.9.p9.11.m11.1.1.1" xref="S3.SS4.9.p9.11.m11.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.9.p9.11.m11.1.1.3" xref="S3.SS4.9.p9.11.m11.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.9.p9.11.m11.1b"><apply id="S3.SS4.9.p9.11.m11.1.1.cmml" xref="S3.SS4.9.p9.11.m11.1.1"><in id="S3.SS4.9.p9.11.m11.1.1.1.cmml" xref="S3.SS4.9.p9.11.m11.1.1.1"></in><apply id="S3.SS4.9.p9.11.m11.1.1.2.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2"><intersect id="S3.SS4.9.p9.11.m11.1.1.2.1.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.1"></intersect><apply id="S3.SS4.9.p9.11.m11.1.1.2.2.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS4.9.p9.11.m11.1.1.2.2.1.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.2">subscript</csymbol><ci id="S3.SS4.9.p9.11.m11.1.1.2.2.2.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.2.2">𝐵</ci><ci id="S3.SS4.9.p9.11.m11.1.1.2.2.3.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.2.3">ℓ</ci></apply><ci id="S3.SS4.9.p9.11.m11.1.1.2.3.cmml" xref="S3.SS4.9.p9.11.m11.1.1.2.3">𝑈</ci></apply><ci id="S3.SS4.9.p9.11.m11.1.1.3.cmml" xref="S3.SS4.9.p9.11.m11.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.9.p9.11.m11.1c">B_{\ell}\cap U\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.9.p9.11.m11.1d">italic_B start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∩ italic_U ∈ caligraphic_F</annotation></semantics></math>).</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem33"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem33.1.1.1">Claim 33</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem33.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem33.p1"> <p class="ltx_p" id="Thmtheorem33.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem33.p1.5.5">If <math alttext="j\in X" class="ltx_Math" display="inline" id="Thmtheorem33.p1.1.1.m1.1"><semantics id="Thmtheorem33.p1.1.1.m1.1a"><mrow id="Thmtheorem33.p1.1.1.m1.1.1" xref="Thmtheorem33.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem33.p1.1.1.m1.1.1.2" xref="Thmtheorem33.p1.1.1.m1.1.1.2.cmml">j</mi><mo id="Thmtheorem33.p1.1.1.m1.1.1.1" xref="Thmtheorem33.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem33.p1.1.1.m1.1.1.3" xref="Thmtheorem33.p1.1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem33.p1.1.1.m1.1b"><apply id="Thmtheorem33.p1.1.1.m1.1.1.cmml" xref="Thmtheorem33.p1.1.1.m1.1.1"><in id="Thmtheorem33.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem33.p1.1.1.m1.1.1.1"></in><ci id="Thmtheorem33.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem33.p1.1.1.m1.1.1.2">𝑗</ci><ci id="Thmtheorem33.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem33.p1.1.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem33.p1.1.1.m1.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem33.p1.1.1.m1.1d">italic_j ∈ italic_X</annotation></semantics></math> and <math alttext="C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="Thmtheorem33.p1.2.2.m2.1"><semantics id="Thmtheorem33.p1.2.2.m2.1a"><mrow id="Thmtheorem33.p1.2.2.m2.1.1" xref="Thmtheorem33.p1.2.2.m2.1.1.cmml"><msup id="Thmtheorem33.p1.2.2.m2.1.1.2" xref="Thmtheorem33.p1.2.2.m2.1.1.2.cmml"><mi id="Thmtheorem33.p1.2.2.m2.1.1.2.2" xref="Thmtheorem33.p1.2.2.m2.1.1.2.2.cmml">C</mi><mi id="Thmtheorem33.p1.2.2.m2.1.1.2.3" xref="Thmtheorem33.p1.2.2.m2.1.1.2.3.cmml">j</mi></msup><mo id="Thmtheorem33.p1.2.2.m2.1.1.1" xref="Thmtheorem33.p1.2.2.m2.1.1.1.cmml">=</mo><mrow id="Thmtheorem33.p1.2.2.m2.1.1.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.cmml"><msup id="Thmtheorem33.p1.2.2.m2.1.1.3.2" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.cmml"><mi id="Thmtheorem33.p1.2.2.m2.1.1.3.2.2" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.2.cmml">C</mi><msup id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.cmml"><mi id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.2" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.2.cmml">j</mi><mo id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.3.cmml">′</mo></msup></msup><msub id="Thmtheorem33.p1.2.2.m2.1.1.3.1" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1.cmml"><mo id="Thmtheorem33.p1.2.2.m2.1.1.3.1.2" lspace="0.222em" rspace="0.222em" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1.2.cmml">⋄</mo><mi id="Thmtheorem33.p1.2.2.m2.1.1.3.1.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1.3.cmml">j</mi></msub><msup id="Thmtheorem33.p1.2.2.m2.1.1.3.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.cmml"><mi id="Thmtheorem33.p1.2.2.m2.1.1.3.3.2" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.2.cmml">C</mi><msup id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.cmml"><mi id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.2" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.2.cmml">j</mi><mo id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.3" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem33.p1.2.2.m2.1b"><apply id="Thmtheorem33.p1.2.2.m2.1.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1"><eq id="Thmtheorem33.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.1"></eq><apply id="Thmtheorem33.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.2.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.2">superscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.2.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.2.2">𝐶</ci><ci id="Thmtheorem33.p1.2.2.m2.1.1.2.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.2.3">𝑗</ci></apply><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3"><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.3.1.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1">subscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.1.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1.2">⋄</ci><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.1.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.1.3">𝑗</ci></apply><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.3.2.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2">superscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.2.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.2">𝐶</ci><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3">superscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.2">𝑗</ci><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.2.3.3">′</ci></apply></apply><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.3.3.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3">superscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.3.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.2">𝐶</ci><apply id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.1.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3">superscript</csymbol><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.2.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.2">𝑗</ci><ci id="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.3.cmml" xref="Thmtheorem33.p1.2.2.m2.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem33.p1.2.2.m2.1c">C^{j}=C^{j^{\prime}}\diamond_{j}C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem33.p1.2.2.m2.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\diamond_{j}\in\{\cap,\cup\}" class="ltx_math_unparsed" display="inline" id="Thmtheorem33.p1.3.3.m3.1"><semantics id="Thmtheorem33.p1.3.3.m3.1a"><mrow id="Thmtheorem33.p1.3.3.m3.1b"><msub id="Thmtheorem33.p1.3.3.m3.1.2"><mo id="Thmtheorem33.p1.3.3.m3.1.2.2">⋄</mo><mi id="Thmtheorem33.p1.3.3.m3.1.2.3">j</mi></msub><mo id="Thmtheorem33.p1.3.3.m3.1.1" lspace="0em">∈</mo><mrow id="Thmtheorem33.p1.3.3.m3.1.3"><mo id="Thmtheorem33.p1.3.3.m3.1.3.1" stretchy="false">{</mo><mo id="Thmtheorem33.p1.3.3.m3.1.3.2" lspace="0em" rspace="0em">∩</mo><mo id="Thmtheorem33.p1.3.3.m3.1.3.3" rspace="0em">,</mo><mo id="Thmtheorem33.p1.3.3.m3.1.3.4" lspace="0em" rspace="0em">∪</mo><mo id="Thmtheorem33.p1.3.3.m3.1.3.5" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmtheorem33.p1.3.3.m3.1c">\diamond_{j}\in\{\cap,\cup\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem33.p1.3.3.m3.1d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ { ∩ , ∪ }</annotation></semantics></math> is arbitrary, then either <math alttext="j^{\prime}\in X" class="ltx_Math" display="inline" id="Thmtheorem33.p1.4.4.m4.1"><semantics id="Thmtheorem33.p1.4.4.m4.1a"><mrow id="Thmtheorem33.p1.4.4.m4.1.1" xref="Thmtheorem33.p1.4.4.m4.1.1.cmml"><msup id="Thmtheorem33.p1.4.4.m4.1.1.2" xref="Thmtheorem33.p1.4.4.m4.1.1.2.cmml"><mi id="Thmtheorem33.p1.4.4.m4.1.1.2.2" xref="Thmtheorem33.p1.4.4.m4.1.1.2.2.cmml">j</mi><mo id="Thmtheorem33.p1.4.4.m4.1.1.2.3" xref="Thmtheorem33.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="Thmtheorem33.p1.4.4.m4.1.1.1" xref="Thmtheorem33.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="Thmtheorem33.p1.4.4.m4.1.1.3" xref="Thmtheorem33.p1.4.4.m4.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem33.p1.4.4.m4.1b"><apply id="Thmtheorem33.p1.4.4.m4.1.1.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1"><in id="Thmtheorem33.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.1"></in><apply id="Thmtheorem33.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem33.p1.4.4.m4.1.1.2.1.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="Thmtheorem33.p1.4.4.m4.1.1.2.2.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.2.2">𝑗</ci><ci id="Thmtheorem33.p1.4.4.m4.1.1.2.3.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.2.3">′</ci></apply><ci id="Thmtheorem33.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem33.p1.4.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem33.p1.4.4.m4.1c">j^{\prime}\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem33.p1.4.4.m4.1d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_X</annotation></semantics></math> or <math alttext="j^{\prime\prime}\in X" class="ltx_Math" display="inline" id="Thmtheorem33.p1.5.5.m5.1"><semantics id="Thmtheorem33.p1.5.5.m5.1a"><mrow id="Thmtheorem33.p1.5.5.m5.1.1" xref="Thmtheorem33.p1.5.5.m5.1.1.cmml"><msup id="Thmtheorem33.p1.5.5.m5.1.1.2" xref="Thmtheorem33.p1.5.5.m5.1.1.2.cmml"><mi id="Thmtheorem33.p1.5.5.m5.1.1.2.2" xref="Thmtheorem33.p1.5.5.m5.1.1.2.2.cmml">j</mi><mo id="Thmtheorem33.p1.5.5.m5.1.1.2.3" xref="Thmtheorem33.p1.5.5.m5.1.1.2.3.cmml">′′</mo></msup><mo id="Thmtheorem33.p1.5.5.m5.1.1.1" xref="Thmtheorem33.p1.5.5.m5.1.1.1.cmml">∈</mo><mi id="Thmtheorem33.p1.5.5.m5.1.1.3" xref="Thmtheorem33.p1.5.5.m5.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem33.p1.5.5.m5.1b"><apply id="Thmtheorem33.p1.5.5.m5.1.1.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1"><in id="Thmtheorem33.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.1"></in><apply id="Thmtheorem33.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem33.p1.5.5.m5.1.1.2.1.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.2">superscript</csymbol><ci id="Thmtheorem33.p1.5.5.m5.1.1.2.2.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.2.2">𝑗</ci><ci id="Thmtheorem33.p1.5.5.m5.1.1.2.3.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.2.3">′′</ci></apply><ci id="Thmtheorem33.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem33.p1.5.5.m5.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem33.p1.5.5.m5.1c">j^{\prime\prime}\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem33.p1.5.5.m5.1d">italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS4.10.p10"> <p class="ltx_p" id="S3.SS4.10.p10.26">Assume contrariwise that <math alttext="j\in X" class="ltx_Math" display="inline" id="S3.SS4.10.p10.1.m1.1"><semantics id="S3.SS4.10.p10.1.m1.1a"><mrow id="S3.SS4.10.p10.1.m1.1.1" xref="S3.SS4.10.p10.1.m1.1.1.cmml"><mi id="S3.SS4.10.p10.1.m1.1.1.2" xref="S3.SS4.10.p10.1.m1.1.1.2.cmml">j</mi><mo id="S3.SS4.10.p10.1.m1.1.1.1" xref="S3.SS4.10.p10.1.m1.1.1.1.cmml">∈</mo><mi id="S3.SS4.10.p10.1.m1.1.1.3" xref="S3.SS4.10.p10.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.1.m1.1b"><apply id="S3.SS4.10.p10.1.m1.1.1.cmml" xref="S3.SS4.10.p10.1.m1.1.1"><in id="S3.SS4.10.p10.1.m1.1.1.1.cmml" xref="S3.SS4.10.p10.1.m1.1.1.1"></in><ci id="S3.SS4.10.p10.1.m1.1.1.2.cmml" xref="S3.SS4.10.p10.1.m1.1.1.2">𝑗</ci><ci id="S3.SS4.10.p10.1.m1.1.1.3.cmml" xref="S3.SS4.10.p10.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.1.m1.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.1.m1.1d">italic_j ∈ italic_X</annotation></semantics></math> and <math alttext="j^{\prime},j^{\prime\prime}\in Y" class="ltx_Math" display="inline" id="S3.SS4.10.p10.2.m2.2"><semantics id="S3.SS4.10.p10.2.m2.2a"><mrow id="S3.SS4.10.p10.2.m2.2.2" xref="S3.SS4.10.p10.2.m2.2.2.cmml"><mrow id="S3.SS4.10.p10.2.m2.2.2.2.2" xref="S3.SS4.10.p10.2.m2.2.2.2.3.cmml"><msup id="S3.SS4.10.p10.2.m2.1.1.1.1.1" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS4.10.p10.2.m2.1.1.1.1.1.2" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1.2.cmml">j</mi><mo id="S3.SS4.10.p10.2.m2.1.1.1.1.1.3" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS4.10.p10.2.m2.2.2.2.2.3" xref="S3.SS4.10.p10.2.m2.2.2.2.3.cmml">,</mo><msup id="S3.SS4.10.p10.2.m2.2.2.2.2.2" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2.cmml"><mi id="S3.SS4.10.p10.2.m2.2.2.2.2.2.2" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2.2.cmml">j</mi><mo id="S3.SS4.10.p10.2.m2.2.2.2.2.2.3" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S3.SS4.10.p10.2.m2.2.2.3" xref="S3.SS4.10.p10.2.m2.2.2.3.cmml">∈</mo><mi id="S3.SS4.10.p10.2.m2.2.2.4" xref="S3.SS4.10.p10.2.m2.2.2.4.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.2.m2.2b"><apply id="S3.SS4.10.p10.2.m2.2.2.cmml" xref="S3.SS4.10.p10.2.m2.2.2"><in id="S3.SS4.10.p10.2.m2.2.2.3.cmml" xref="S3.SS4.10.p10.2.m2.2.2.3"></in><list id="S3.SS4.10.p10.2.m2.2.2.2.3.cmml" xref="S3.SS4.10.p10.2.m2.2.2.2.2"><apply id="S3.SS4.10.p10.2.m2.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.10.p10.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1.2">𝑗</ci><ci id="S3.SS4.10.p10.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS4.10.p10.2.m2.1.1.1.1.1.3">′</ci></apply><apply id="S3.SS4.10.p10.2.m2.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.2.m2.2.2.2.2.2.1.cmml" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2">superscript</csymbol><ci id="S3.SS4.10.p10.2.m2.2.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2.2">𝑗</ci><ci id="S3.SS4.10.p10.2.m2.2.2.2.2.2.3.cmml" xref="S3.SS4.10.p10.2.m2.2.2.2.2.2.3">′′</ci></apply></list><ci id="S3.SS4.10.p10.2.m2.2.2.4.cmml" xref="S3.SS4.10.p10.2.m2.2.2.4">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.2.m2.2c">j^{\prime},j^{\prime\prime}\in Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.2.m2.2d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_Y</annotation></semantics></math>. First, suppose that <math alttext="\diamond_{j}=\cap" class="ltx_Math" display="inline" id="S3.SS4.10.p10.3.m3.3"><semantics id="S3.SS4.10.p10.3.m3.3a"><mrow id="S3.SS4.10.p10.3.m3.3.3.1" xref="S3.SS4.10.p10.3.m3.3.3.2.cmml"><msub id="S3.SS4.10.p10.3.m3.3.3.1.1" xref="S3.SS4.10.p10.3.m3.3.3.1.1.cmml"><mo id="S3.SS4.10.p10.3.m3.3.3.1.1.2" xref="S3.SS4.10.p10.3.m3.3.3.1.1.2.cmml">⋄</mo><mi id="S3.SS4.10.p10.3.m3.3.3.1.1.3" xref="S3.SS4.10.p10.3.m3.3.3.1.1.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.3.m3.3.3.1.2" lspace="0em" xref="S3.SS4.10.p10.3.m3.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.3.m3.1.1" xref="S3.SS4.10.p10.3.m3.1.1.cmml">=</mo><mo id="S3.SS4.10.p10.3.m3.3.3.1.3" lspace="0em" xref="S3.SS4.10.p10.3.m3.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.3.m3.2.2" xref="S3.SS4.10.p10.3.m3.2.2.cmml">∩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.3.m3.3b"><list id="S3.SS4.10.p10.3.m3.3.3.2.cmml" xref="S3.SS4.10.p10.3.m3.3.3.1"><apply id="S3.SS4.10.p10.3.m3.3.3.1.1.cmml" xref="S3.SS4.10.p10.3.m3.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.3.m3.3.3.1.1.1.cmml" xref="S3.SS4.10.p10.3.m3.3.3.1.1">subscript</csymbol><ci id="S3.SS4.10.p10.3.m3.3.3.1.1.2.cmml" xref="S3.SS4.10.p10.3.m3.3.3.1.1.2">⋄</ci><ci id="S3.SS4.10.p10.3.m3.3.3.1.1.3.cmml" xref="S3.SS4.10.p10.3.m3.3.3.1.1.3">𝑗</ci></apply><eq id="S3.SS4.10.p10.3.m3.1.1.cmml" xref="S3.SS4.10.p10.3.m3.1.1"></eq><intersect id="S3.SS4.10.p10.3.m3.2.2.cmml" xref="S3.SS4.10.p10.3.m3.2.2"></intersect></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.3.m3.3c">\diamond_{j}=\cap</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.3.m3.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∩</annotation></semantics></math>. Since <math alttext="\alpha_{j}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.4.m4.1"><semantics id="S3.SS4.10.p10.4.m4.1a"><mrow id="S3.SS4.10.p10.4.m4.1.1" xref="S3.SS4.10.p10.4.m4.1.1.cmml"><msub id="S3.SS4.10.p10.4.m4.1.1.2" xref="S3.SS4.10.p10.4.m4.1.1.2.cmml"><mi id="S3.SS4.10.p10.4.m4.1.1.2.2" xref="S3.SS4.10.p10.4.m4.1.1.2.2.cmml">α</mi><mi id="S3.SS4.10.p10.4.m4.1.1.2.3" xref="S3.SS4.10.p10.4.m4.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.4.m4.1.1.1" xref="S3.SS4.10.p10.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS4.10.p10.4.m4.1.1.3" xref="S3.SS4.10.p10.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.4.m4.1b"><apply id="S3.SS4.10.p10.4.m4.1.1.cmml" xref="S3.SS4.10.p10.4.m4.1.1"><eq id="S3.SS4.10.p10.4.m4.1.1.1.cmml" xref="S3.SS4.10.p10.4.m4.1.1.1"></eq><apply id="S3.SS4.10.p10.4.m4.1.1.2.cmml" xref="S3.SS4.10.p10.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.4.m4.1.1.2.1.cmml" xref="S3.SS4.10.p10.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.4.m4.1.1.2.2.cmml" xref="S3.SS4.10.p10.4.m4.1.1.2.2">𝛼</ci><ci id="S3.SS4.10.p10.4.m4.1.1.2.3.cmml" xref="S3.SS4.10.p10.4.m4.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.10.p10.4.m4.1.1.3.cmml" type="integer" xref="S3.SS4.10.p10.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.4.m4.1c">\alpha_{j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.4.m4.1d">italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="C^{j}=C^{j^{\prime}}\cap C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.5.m5.1"><semantics id="S3.SS4.10.p10.5.m5.1a"><mrow id="S3.SS4.10.p10.5.m5.1.1" xref="S3.SS4.10.p10.5.m5.1.1.cmml"><msup id="S3.SS4.10.p10.5.m5.1.1.2" xref="S3.SS4.10.p10.5.m5.1.1.2.cmml"><mi id="S3.SS4.10.p10.5.m5.1.1.2.2" xref="S3.SS4.10.p10.5.m5.1.1.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.5.m5.1.1.2.3" xref="S3.SS4.10.p10.5.m5.1.1.2.3.cmml">j</mi></msup><mo id="S3.SS4.10.p10.5.m5.1.1.1" xref="S3.SS4.10.p10.5.m5.1.1.1.cmml">=</mo><mrow id="S3.SS4.10.p10.5.m5.1.1.3" xref="S3.SS4.10.p10.5.m5.1.1.3.cmml"><msup id="S3.SS4.10.p10.5.m5.1.1.3.2" xref="S3.SS4.10.p10.5.m5.1.1.3.2.cmml"><mi id="S3.SS4.10.p10.5.m5.1.1.3.2.2" xref="S3.SS4.10.p10.5.m5.1.1.3.2.2.cmml">C</mi><msup id="S3.SS4.10.p10.5.m5.1.1.3.2.3" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3.cmml"><mi id="S3.SS4.10.p10.5.m5.1.1.3.2.3.2" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.5.m5.1.1.3.2.3.3" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3.3.cmml">′</mo></msup></msup><mo id="S3.SS4.10.p10.5.m5.1.1.3.1" xref="S3.SS4.10.p10.5.m5.1.1.3.1.cmml">∩</mo><msup id="S3.SS4.10.p10.5.m5.1.1.3.3" xref="S3.SS4.10.p10.5.m5.1.1.3.3.cmml"><mi id="S3.SS4.10.p10.5.m5.1.1.3.3.2" xref="S3.SS4.10.p10.5.m5.1.1.3.3.2.cmml">C</mi><msup id="S3.SS4.10.p10.5.m5.1.1.3.3.3" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3.cmml"><mi id="S3.SS4.10.p10.5.m5.1.1.3.3.3.2" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.5.m5.1.1.3.3.3.3" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.5.m5.1b"><apply id="S3.SS4.10.p10.5.m5.1.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1"><eq id="S3.SS4.10.p10.5.m5.1.1.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.1"></eq><apply id="S3.SS4.10.p10.5.m5.1.1.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.5.m5.1.1.2.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.5.m5.1.1.2.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.2.2">𝐶</ci><ci id="S3.SS4.10.p10.5.m5.1.1.2.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.2.3">𝑗</ci></apply><apply id="S3.SS4.10.p10.5.m5.1.1.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3"><intersect id="S3.SS4.10.p10.5.m5.1.1.3.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.1"></intersect><apply id="S3.SS4.10.p10.5.m5.1.1.3.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.5.m5.1.1.3.2.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2">superscript</csymbol><ci id="S3.SS4.10.p10.5.m5.1.1.3.2.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2.2">𝐶</ci><apply id="S3.SS4.10.p10.5.m5.1.1.3.2.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.5.m5.1.1.3.2.3.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.5.m5.1.1.3.2.3.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.5.m5.1.1.3.2.3.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.2.3.3">′</ci></apply></apply><apply id="S3.SS4.10.p10.5.m5.1.1.3.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.5.m5.1.1.3.3.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3">superscript</csymbol><ci id="S3.SS4.10.p10.5.m5.1.1.3.3.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3.2">𝐶</ci><apply id="S3.SS4.10.p10.5.m5.1.1.3.3.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.5.m5.1.1.3.3.3.1.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3">superscript</csymbol><ci id="S3.SS4.10.p10.5.m5.1.1.3.3.3.2.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3.2">𝑗</ci><ci id="S3.SS4.10.p10.5.m5.1.1.3.3.3.3.cmml" xref="S3.SS4.10.p10.5.m5.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.5.m5.1c">C^{j}=C^{j^{\prime}}\cap C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.5.m5.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∩ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, we have <math alttext="\alpha_{j^{\prime}}=\alpha_{j^{\prime\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.6.m6.1"><semantics id="S3.SS4.10.p10.6.m6.1a"><mrow id="S3.SS4.10.p10.6.m6.1.1" xref="S3.SS4.10.p10.6.m6.1.1.cmml"><msub id="S3.SS4.10.p10.6.m6.1.1.2" xref="S3.SS4.10.p10.6.m6.1.1.2.cmml"><mi id="S3.SS4.10.p10.6.m6.1.1.2.2" xref="S3.SS4.10.p10.6.m6.1.1.2.2.cmml">α</mi><msup id="S3.SS4.10.p10.6.m6.1.1.2.3" xref="S3.SS4.10.p10.6.m6.1.1.2.3.cmml"><mi id="S3.SS4.10.p10.6.m6.1.1.2.3.2" xref="S3.SS4.10.p10.6.m6.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.6.m6.1.1.2.3.3" xref="S3.SS4.10.p10.6.m6.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.10.p10.6.m6.1.1.3" xref="S3.SS4.10.p10.6.m6.1.1.3.cmml">=</mo><msub id="S3.SS4.10.p10.6.m6.1.1.4" xref="S3.SS4.10.p10.6.m6.1.1.4.cmml"><mi id="S3.SS4.10.p10.6.m6.1.1.4.2" xref="S3.SS4.10.p10.6.m6.1.1.4.2.cmml">α</mi><msup id="S3.SS4.10.p10.6.m6.1.1.4.3" xref="S3.SS4.10.p10.6.m6.1.1.4.3.cmml"><mi id="S3.SS4.10.p10.6.m6.1.1.4.3.2" xref="S3.SS4.10.p10.6.m6.1.1.4.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.6.m6.1.1.4.3.3" xref="S3.SS4.10.p10.6.m6.1.1.4.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.10.p10.6.m6.1.1.5" xref="S3.SS4.10.p10.6.m6.1.1.5.cmml">=</mo><mn id="S3.SS4.10.p10.6.m6.1.1.6" xref="S3.SS4.10.p10.6.m6.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.6.m6.1b"><apply id="S3.SS4.10.p10.6.m6.1.1.cmml" xref="S3.SS4.10.p10.6.m6.1.1"><and id="S3.SS4.10.p10.6.m6.1.1a.cmml" xref="S3.SS4.10.p10.6.m6.1.1"></and><apply id="S3.SS4.10.p10.6.m6.1.1b.cmml" xref="S3.SS4.10.p10.6.m6.1.1"><eq id="S3.SS4.10.p10.6.m6.1.1.3.cmml" xref="S3.SS4.10.p10.6.m6.1.1.3"></eq><apply id="S3.SS4.10.p10.6.m6.1.1.2.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.6.m6.1.1.2.1.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.6.m6.1.1.2.2.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2.2">𝛼</ci><apply id="S3.SS4.10.p10.6.m6.1.1.2.3.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.6.m6.1.1.2.3.1.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.6.m6.1.1.2.3.2.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.6.m6.1.1.2.3.3.cmml" xref="S3.SS4.10.p10.6.m6.1.1.2.3.3">′</ci></apply></apply><apply id="S3.SS4.10.p10.6.m6.1.1.4.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4"><csymbol cd="ambiguous" id="S3.SS4.10.p10.6.m6.1.1.4.1.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4">subscript</csymbol><ci id="S3.SS4.10.p10.6.m6.1.1.4.2.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4.2">𝛼</ci><apply id="S3.SS4.10.p10.6.m6.1.1.4.3.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.6.m6.1.1.4.3.1.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4.3">superscript</csymbol><ci id="S3.SS4.10.p10.6.m6.1.1.4.3.2.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4.3.2">𝑗</ci><ci id="S3.SS4.10.p10.6.m6.1.1.4.3.3.cmml" xref="S3.SS4.10.p10.6.m6.1.1.4.3.3">′′</ci></apply></apply></apply><apply id="S3.SS4.10.p10.6.m6.1.1c.cmml" xref="S3.SS4.10.p10.6.m6.1.1"><eq id="S3.SS4.10.p10.6.m6.1.1.5.cmml" xref="S3.SS4.10.p10.6.m6.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.10.p10.6.m6.1.1.4.cmml" id="S3.SS4.10.p10.6.m6.1.1d.cmml" xref="S3.SS4.10.p10.6.m6.1.1"></share><cn id="S3.SS4.10.p10.6.m6.1.1.6.cmml" type="integer" xref="S3.SS4.10.p10.6.m6.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.6.m6.1c">\alpha_{j^{\prime}}=\alpha_{j^{\prime\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.6.m6.1d">italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>. As <math alttext="j^{\prime},j^{\prime\prime}\in Y" class="ltx_Math" display="inline" id="S3.SS4.10.p10.7.m7.2"><semantics id="S3.SS4.10.p10.7.m7.2a"><mrow id="S3.SS4.10.p10.7.m7.2.2" xref="S3.SS4.10.p10.7.m7.2.2.cmml"><mrow id="S3.SS4.10.p10.7.m7.2.2.2.2" xref="S3.SS4.10.p10.7.m7.2.2.2.3.cmml"><msup id="S3.SS4.10.p10.7.m7.1.1.1.1.1" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1.cmml"><mi id="S3.SS4.10.p10.7.m7.1.1.1.1.1.2" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1.2.cmml">j</mi><mo id="S3.SS4.10.p10.7.m7.1.1.1.1.1.3" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS4.10.p10.7.m7.2.2.2.2.3" xref="S3.SS4.10.p10.7.m7.2.2.2.3.cmml">,</mo><msup id="S3.SS4.10.p10.7.m7.2.2.2.2.2" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2.cmml"><mi id="S3.SS4.10.p10.7.m7.2.2.2.2.2.2" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2.2.cmml">j</mi><mo id="S3.SS4.10.p10.7.m7.2.2.2.2.2.3" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S3.SS4.10.p10.7.m7.2.2.3" xref="S3.SS4.10.p10.7.m7.2.2.3.cmml">∈</mo><mi id="S3.SS4.10.p10.7.m7.2.2.4" xref="S3.SS4.10.p10.7.m7.2.2.4.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.7.m7.2b"><apply id="S3.SS4.10.p10.7.m7.2.2.cmml" xref="S3.SS4.10.p10.7.m7.2.2"><in id="S3.SS4.10.p10.7.m7.2.2.3.cmml" xref="S3.SS4.10.p10.7.m7.2.2.3"></in><list id="S3.SS4.10.p10.7.m7.2.2.2.3.cmml" xref="S3.SS4.10.p10.7.m7.2.2.2.2"><apply id="S3.SS4.10.p10.7.m7.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.7.m7.1.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.10.p10.7.m7.1.1.1.1.1.2.cmml" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1.2">𝑗</ci><ci id="S3.SS4.10.p10.7.m7.1.1.1.1.1.3.cmml" xref="S3.SS4.10.p10.7.m7.1.1.1.1.1.3">′</ci></apply><apply id="S3.SS4.10.p10.7.m7.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.7.m7.2.2.2.2.2.1.cmml" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2">superscript</csymbol><ci id="S3.SS4.10.p10.7.m7.2.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2.2">𝑗</ci><ci id="S3.SS4.10.p10.7.m7.2.2.2.2.2.3.cmml" xref="S3.SS4.10.p10.7.m7.2.2.2.2.2.3">′′</ci></apply></list><ci id="S3.SS4.10.p10.7.m7.2.2.4.cmml" xref="S3.SS4.10.p10.7.m7.2.2.4">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.7.m7.2c">j^{\prime},j^{\prime\prime}\in Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.7.m7.2d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_Y</annotation></semantics></math>, by construction, we get <math alttext="\beta_{j^{\prime}}=\beta_{j^{\prime\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.8.m8.1"><semantics id="S3.SS4.10.p10.8.m8.1a"><mrow id="S3.SS4.10.p10.8.m8.1.1" xref="S3.SS4.10.p10.8.m8.1.1.cmml"><msub id="S3.SS4.10.p10.8.m8.1.1.2" xref="S3.SS4.10.p10.8.m8.1.1.2.cmml"><mi id="S3.SS4.10.p10.8.m8.1.1.2.2" xref="S3.SS4.10.p10.8.m8.1.1.2.2.cmml">β</mi><msup id="S3.SS4.10.p10.8.m8.1.1.2.3" xref="S3.SS4.10.p10.8.m8.1.1.2.3.cmml"><mi id="S3.SS4.10.p10.8.m8.1.1.2.3.2" xref="S3.SS4.10.p10.8.m8.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.8.m8.1.1.2.3.3" xref="S3.SS4.10.p10.8.m8.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.10.p10.8.m8.1.1.3" xref="S3.SS4.10.p10.8.m8.1.1.3.cmml">=</mo><msub id="S3.SS4.10.p10.8.m8.1.1.4" xref="S3.SS4.10.p10.8.m8.1.1.4.cmml"><mi id="S3.SS4.10.p10.8.m8.1.1.4.2" xref="S3.SS4.10.p10.8.m8.1.1.4.2.cmml">β</mi><msup id="S3.SS4.10.p10.8.m8.1.1.4.3" xref="S3.SS4.10.p10.8.m8.1.1.4.3.cmml"><mi id="S3.SS4.10.p10.8.m8.1.1.4.3.2" xref="S3.SS4.10.p10.8.m8.1.1.4.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.8.m8.1.1.4.3.3" xref="S3.SS4.10.p10.8.m8.1.1.4.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.10.p10.8.m8.1.1.5" xref="S3.SS4.10.p10.8.m8.1.1.5.cmml">=</mo><mn id="S3.SS4.10.p10.8.m8.1.1.6" xref="S3.SS4.10.p10.8.m8.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.8.m8.1b"><apply id="S3.SS4.10.p10.8.m8.1.1.cmml" xref="S3.SS4.10.p10.8.m8.1.1"><and id="S3.SS4.10.p10.8.m8.1.1a.cmml" xref="S3.SS4.10.p10.8.m8.1.1"></and><apply id="S3.SS4.10.p10.8.m8.1.1b.cmml" xref="S3.SS4.10.p10.8.m8.1.1"><eq id="S3.SS4.10.p10.8.m8.1.1.3.cmml" xref="S3.SS4.10.p10.8.m8.1.1.3"></eq><apply id="S3.SS4.10.p10.8.m8.1.1.2.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.8.m8.1.1.2.1.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.8.m8.1.1.2.2.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2.2">𝛽</ci><apply id="S3.SS4.10.p10.8.m8.1.1.2.3.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.8.m8.1.1.2.3.1.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.8.m8.1.1.2.3.2.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.8.m8.1.1.2.3.3.cmml" xref="S3.SS4.10.p10.8.m8.1.1.2.3.3">′</ci></apply></apply><apply id="S3.SS4.10.p10.8.m8.1.1.4.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4"><csymbol cd="ambiguous" id="S3.SS4.10.p10.8.m8.1.1.4.1.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4">subscript</csymbol><ci id="S3.SS4.10.p10.8.m8.1.1.4.2.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4.2">𝛽</ci><apply id="S3.SS4.10.p10.8.m8.1.1.4.3.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.8.m8.1.1.4.3.1.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4.3">superscript</csymbol><ci id="S3.SS4.10.p10.8.m8.1.1.4.3.2.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4.3.2">𝑗</ci><ci id="S3.SS4.10.p10.8.m8.1.1.4.3.3.cmml" xref="S3.SS4.10.p10.8.m8.1.1.4.3.3">′′</ci></apply></apply></apply><apply id="S3.SS4.10.p10.8.m8.1.1c.cmml" xref="S3.SS4.10.p10.8.m8.1.1"><eq id="S3.SS4.10.p10.8.m8.1.1.5.cmml" xref="S3.SS4.10.p10.8.m8.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S3.SS4.10.p10.8.m8.1.1.4.cmml" id="S3.SS4.10.p10.8.m8.1.1d.cmml" xref="S3.SS4.10.p10.8.m8.1.1"></share><cn id="S3.SS4.10.p10.8.m8.1.1.6.cmml" type="integer" xref="S3.SS4.10.p10.8.m8.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.8.m8.1c">\beta_{j^{\prime}}=\beta_{j^{\prime\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.8.m8.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math> (otherwise one of the indices would be in <math alttext="X" class="ltx_Math" display="inline" id="S3.SS4.10.p10.9.m9.1"><semantics id="S3.SS4.10.p10.9.m9.1a"><mi id="S3.SS4.10.p10.9.m9.1.1" xref="S3.SS4.10.p10.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.9.m9.1b"><ci id="S3.SS4.10.p10.9.m9.1.1.cmml" xref="S3.SS4.10.p10.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.9.m9.1d">italic_X</annotation></semantics></math> and not in <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS4.10.p10.10.m10.1"><semantics id="S3.SS4.10.p10.10.m10.1a"><mi id="S3.SS4.10.p10.10.m10.1.1" xref="S3.SS4.10.p10.10.m10.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.10.m10.1b"><ci id="S3.SS4.10.p10.10.m10.1.1.cmml" xref="S3.SS4.10.p10.10.m10.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.10.m10.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.10.m10.1d">italic_Y</annotation></semantics></math>). Consequently, by the definition of the sequence <math alttext="\beta" class="ltx_Math" display="inline" id="S3.SS4.10.p10.11.m11.1"><semantics id="S3.SS4.10.p10.11.m11.1a"><mi id="S3.SS4.10.p10.11.m11.1.1" xref="S3.SS4.10.p10.11.m11.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.11.m11.1b"><ci id="S3.SS4.10.p10.11.m11.1.1.cmml" xref="S3.SS4.10.p10.11.m11.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.11.m11.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.11.m11.1d">italic_β</annotation></semantics></math>, <math alttext="C^{j}_{U}\notin\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.12.m12.1"><semantics id="S3.SS4.10.p10.12.m12.1a"><mrow id="S3.SS4.10.p10.12.m12.1.1" xref="S3.SS4.10.p10.12.m12.1.1.cmml"><msubsup id="S3.SS4.10.p10.12.m12.1.1.2" xref="S3.SS4.10.p10.12.m12.1.1.2.cmml"><mi id="S3.SS4.10.p10.12.m12.1.1.2.2.2" xref="S3.SS4.10.p10.12.m12.1.1.2.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.12.m12.1.1.2.3" xref="S3.SS4.10.p10.12.m12.1.1.2.3.cmml">U</mi><mi id="S3.SS4.10.p10.12.m12.1.1.2.2.3" xref="S3.SS4.10.p10.12.m12.1.1.2.2.3.cmml">j</mi></msubsup><mo id="S3.SS4.10.p10.12.m12.1.1.1" xref="S3.SS4.10.p10.12.m12.1.1.1.cmml">∉</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.12.m12.1.1.3" xref="S3.SS4.10.p10.12.m12.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.12.m12.1b"><apply id="S3.SS4.10.p10.12.m12.1.1.cmml" xref="S3.SS4.10.p10.12.m12.1.1"><notin id="S3.SS4.10.p10.12.m12.1.1.1.cmml" xref="S3.SS4.10.p10.12.m12.1.1.1"></notin><apply id="S3.SS4.10.p10.12.m12.1.1.2.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.12.m12.1.1.2.1.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2">subscript</csymbol><apply id="S3.SS4.10.p10.12.m12.1.1.2.2.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.12.m12.1.1.2.2.1.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.12.m12.1.1.2.2.2.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2.2.2">𝐶</ci><ci id="S3.SS4.10.p10.12.m12.1.1.2.2.3.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2.2.3">𝑗</ci></apply><ci id="S3.SS4.10.p10.12.m12.1.1.2.3.cmml" xref="S3.SS4.10.p10.12.m12.1.1.2.3">𝑈</ci></apply><ci id="S3.SS4.10.p10.12.m12.1.1.3.cmml" xref="S3.SS4.10.p10.12.m12.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.12.m12.1c">C^{j}_{U}\notin\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.12.m12.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∉ caligraphic_F</annotation></semantics></math>, while <math alttext="C^{j^{\prime}}_{U},C^{j^{\prime\prime}}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.13.m13.2"><semantics id="S3.SS4.10.p10.13.m13.2a"><mrow id="S3.SS4.10.p10.13.m13.2.2" xref="S3.SS4.10.p10.13.m13.2.2.cmml"><mrow id="S3.SS4.10.p10.13.m13.2.2.2.2" xref="S3.SS4.10.p10.13.m13.2.2.2.3.cmml"><msubsup id="S3.SS4.10.p10.13.m13.1.1.1.1.1" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.cmml"><mi id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.2" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.13.m13.1.1.1.1.1.3" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.3.cmml">U</mi><msup id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.cmml"><mi id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.2" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.3" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3.SS4.10.p10.13.m13.2.2.2.2.3" xref="S3.SS4.10.p10.13.m13.2.2.2.3.cmml">,</mo><msubsup id="S3.SS4.10.p10.13.m13.2.2.2.2.2" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.cmml"><mi id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.2" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.13.m13.2.2.2.2.2.3" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.3.cmml">U</mi><msup id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.cmml"><mi id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.2" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.3" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.3.cmml">′′</mo></msup></msubsup></mrow><mo id="S3.SS4.10.p10.13.m13.2.2.3" xref="S3.SS4.10.p10.13.m13.2.2.3.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.13.m13.2.2.4" xref="S3.SS4.10.p10.13.m13.2.2.4.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.13.m13.2b"><apply id="S3.SS4.10.p10.13.m13.2.2.cmml" xref="S3.SS4.10.p10.13.m13.2.2"><in id="S3.SS4.10.p10.13.m13.2.2.3.cmml" xref="S3.SS4.10.p10.13.m13.2.2.3"></in><list id="S3.SS4.10.p10.13.m13.2.2.2.3.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2"><apply id="S3.SS4.10.p10.13.m13.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.1.1.1.1.1.1.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1">subscript</csymbol><apply id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.1.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.2.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.2">𝐶</ci><apply id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.1.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.2.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.3.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S3.SS4.10.p10.13.m13.1.1.1.1.1.3.cmml" xref="S3.SS4.10.p10.13.m13.1.1.1.1.1.3">𝑈</ci></apply><apply id="S3.SS4.10.p10.13.m13.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.2.2.2.2.2.1.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2">subscript</csymbol><apply id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.1.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2">superscript</csymbol><ci id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.2.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.2">𝐶</ci><apply id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.1.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.2.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.3.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.2.3.3">′′</ci></apply></apply><ci id="S3.SS4.10.p10.13.m13.2.2.2.2.2.3.cmml" xref="S3.SS4.10.p10.13.m13.2.2.2.2.2.3">𝑈</ci></apply></list><ci id="S3.SS4.10.p10.13.m13.2.2.4.cmml" xref="S3.SS4.10.p10.13.m13.2.2.4">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.13.m13.2c">C^{j^{\prime}}_{U},C^{j^{\prime\prime}}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.13.m13.2d">italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>. This contradictions the assumption that <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS4.10.p10.14.m14.1"><semantics id="S3.SS4.10.p10.14.m14.1a"><mi id="S3.SS4.10.p10.14.m14.1.1" mathvariant="normal" xref="S3.SS4.10.p10.14.m14.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.14.m14.1b"><ci id="S3.SS4.10.p10.14.m14.1.1.cmml" xref="S3.SS4.10.p10.14.m14.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.14.m14.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.14.m14.1d">roman_Λ</annotation></semantics></math> does not cover <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.15.m15.1"><semantics id="S3.SS4.10.p10.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.15.m15.1.1" xref="S3.SS4.10.p10.15.m15.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.15.m15.1b"><ci id="S3.SS4.10.p10.15.m15.1.1.cmml" xref="S3.SS4.10.p10.15.m15.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.15.m15.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.15.m15.1d">caligraphic_F</annotation></semantics></math>. Assume now that <math alttext="\diamond_{j}=\cup" class="ltx_Math" display="inline" id="S3.SS4.10.p10.16.m16.3"><semantics id="S3.SS4.10.p10.16.m16.3a"><mrow id="S3.SS4.10.p10.16.m16.3.3.1" xref="S3.SS4.10.p10.16.m16.3.3.2.cmml"><msub id="S3.SS4.10.p10.16.m16.3.3.1.1" xref="S3.SS4.10.p10.16.m16.3.3.1.1.cmml"><mo id="S3.SS4.10.p10.16.m16.3.3.1.1.2" xref="S3.SS4.10.p10.16.m16.3.3.1.1.2.cmml">⋄</mo><mi id="S3.SS4.10.p10.16.m16.3.3.1.1.3" xref="S3.SS4.10.p10.16.m16.3.3.1.1.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.16.m16.3.3.1.2" lspace="0em" xref="S3.SS4.10.p10.16.m16.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.16.m16.1.1" xref="S3.SS4.10.p10.16.m16.1.1.cmml">=</mo><mo id="S3.SS4.10.p10.16.m16.3.3.1.3" lspace="0em" xref="S3.SS4.10.p10.16.m16.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.16.m16.2.2" xref="S3.SS4.10.p10.16.m16.2.2.cmml">∪</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.16.m16.3b"><list id="S3.SS4.10.p10.16.m16.3.3.2.cmml" xref="S3.SS4.10.p10.16.m16.3.3.1"><apply id="S3.SS4.10.p10.16.m16.3.3.1.1.cmml" xref="S3.SS4.10.p10.16.m16.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.16.m16.3.3.1.1.1.cmml" xref="S3.SS4.10.p10.16.m16.3.3.1.1">subscript</csymbol><ci id="S3.SS4.10.p10.16.m16.3.3.1.1.2.cmml" xref="S3.SS4.10.p10.16.m16.3.3.1.1.2">⋄</ci><ci id="S3.SS4.10.p10.16.m16.3.3.1.1.3.cmml" xref="S3.SS4.10.p10.16.m16.3.3.1.1.3">𝑗</ci></apply><eq id="S3.SS4.10.p10.16.m16.1.1.cmml" xref="S3.SS4.10.p10.16.m16.1.1"></eq><union id="S3.SS4.10.p10.16.m16.2.2.cmml" xref="S3.SS4.10.p10.16.m16.2.2"></union></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.16.m16.3c">\diamond_{j}=\cup</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.16.m16.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∪</annotation></semantics></math>. Moreover, suppose w.l.o.g. that <math alttext="\alpha_{j^{\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.17.m17.1"><semantics id="S3.SS4.10.p10.17.m17.1a"><mrow id="S3.SS4.10.p10.17.m17.1.1" xref="S3.SS4.10.p10.17.m17.1.1.cmml"><msub id="S3.SS4.10.p10.17.m17.1.1.2" xref="S3.SS4.10.p10.17.m17.1.1.2.cmml"><mi id="S3.SS4.10.p10.17.m17.1.1.2.2" xref="S3.SS4.10.p10.17.m17.1.1.2.2.cmml">α</mi><msup id="S3.SS4.10.p10.17.m17.1.1.2.3" xref="S3.SS4.10.p10.17.m17.1.1.2.3.cmml"><mi id="S3.SS4.10.p10.17.m17.1.1.2.3.2" xref="S3.SS4.10.p10.17.m17.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.17.m17.1.1.2.3.3" xref="S3.SS4.10.p10.17.m17.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.10.p10.17.m17.1.1.1" xref="S3.SS4.10.p10.17.m17.1.1.1.cmml">=</mo><mn id="S3.SS4.10.p10.17.m17.1.1.3" xref="S3.SS4.10.p10.17.m17.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.17.m17.1b"><apply id="S3.SS4.10.p10.17.m17.1.1.cmml" xref="S3.SS4.10.p10.17.m17.1.1"><eq id="S3.SS4.10.p10.17.m17.1.1.1.cmml" xref="S3.SS4.10.p10.17.m17.1.1.1"></eq><apply id="S3.SS4.10.p10.17.m17.1.1.2.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.17.m17.1.1.2.1.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.17.m17.1.1.2.2.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2.2">𝛼</ci><apply id="S3.SS4.10.p10.17.m17.1.1.2.3.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.17.m17.1.1.2.3.1.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.17.m17.1.1.2.3.2.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.17.m17.1.1.2.3.3.cmml" xref="S3.SS4.10.p10.17.m17.1.1.2.3.3">′</ci></apply></apply><cn id="S3.SS4.10.p10.17.m17.1.1.3.cmml" type="integer" xref="S3.SS4.10.p10.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.17.m17.1c">\alpha_{j^{\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.17.m17.1d">italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>, which can be done thanks to <math alttext="C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.18.m18.1"><semantics id="S3.SS4.10.p10.18.m18.1a"><mrow id="S3.SS4.10.p10.18.m18.1.1" xref="S3.SS4.10.p10.18.m18.1.1.cmml"><msup id="S3.SS4.10.p10.18.m18.1.1.2" xref="S3.SS4.10.p10.18.m18.1.1.2.cmml"><mi id="S3.SS4.10.p10.18.m18.1.1.2.2" xref="S3.SS4.10.p10.18.m18.1.1.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.18.m18.1.1.2.3" xref="S3.SS4.10.p10.18.m18.1.1.2.3.cmml">j</mi></msup><mo id="S3.SS4.10.p10.18.m18.1.1.1" xref="S3.SS4.10.p10.18.m18.1.1.1.cmml">=</mo><mrow id="S3.SS4.10.p10.18.m18.1.1.3" xref="S3.SS4.10.p10.18.m18.1.1.3.cmml"><msup id="S3.SS4.10.p10.18.m18.1.1.3.2" xref="S3.SS4.10.p10.18.m18.1.1.3.2.cmml"><mi id="S3.SS4.10.p10.18.m18.1.1.3.2.2" xref="S3.SS4.10.p10.18.m18.1.1.3.2.2.cmml">C</mi><msup id="S3.SS4.10.p10.18.m18.1.1.3.2.3" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3.cmml"><mi id="S3.SS4.10.p10.18.m18.1.1.3.2.3.2" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.18.m18.1.1.3.2.3.3" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3.3.cmml">′</mo></msup></msup><mo id="S3.SS4.10.p10.18.m18.1.1.3.1" xref="S3.SS4.10.p10.18.m18.1.1.3.1.cmml">∪</mo><msup id="S3.SS4.10.p10.18.m18.1.1.3.3" xref="S3.SS4.10.p10.18.m18.1.1.3.3.cmml"><mi id="S3.SS4.10.p10.18.m18.1.1.3.3.2" xref="S3.SS4.10.p10.18.m18.1.1.3.3.2.cmml">C</mi><msup id="S3.SS4.10.p10.18.m18.1.1.3.3.3" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3.cmml"><mi id="S3.SS4.10.p10.18.m18.1.1.3.3.3.2" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.18.m18.1.1.3.3.3.3" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.18.m18.1b"><apply id="S3.SS4.10.p10.18.m18.1.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1"><eq id="S3.SS4.10.p10.18.m18.1.1.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.1"></eq><apply id="S3.SS4.10.p10.18.m18.1.1.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.18.m18.1.1.2.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.18.m18.1.1.2.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.2.2">𝐶</ci><ci id="S3.SS4.10.p10.18.m18.1.1.2.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.2.3">𝑗</ci></apply><apply id="S3.SS4.10.p10.18.m18.1.1.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3"><union id="S3.SS4.10.p10.18.m18.1.1.3.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.1"></union><apply id="S3.SS4.10.p10.18.m18.1.1.3.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.18.m18.1.1.3.2.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2">superscript</csymbol><ci id="S3.SS4.10.p10.18.m18.1.1.3.2.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2.2">𝐶</ci><apply id="S3.SS4.10.p10.18.m18.1.1.3.2.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.18.m18.1.1.3.2.3.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.18.m18.1.1.3.2.3.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.18.m18.1.1.3.2.3.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.2.3.3">′</ci></apply></apply><apply id="S3.SS4.10.p10.18.m18.1.1.3.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.18.m18.1.1.3.3.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3">superscript</csymbol><ci id="S3.SS4.10.p10.18.m18.1.1.3.3.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3.2">𝐶</ci><apply id="S3.SS4.10.p10.18.m18.1.1.3.3.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.18.m18.1.1.3.3.3.1.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3">superscript</csymbol><ci id="S3.SS4.10.p10.18.m18.1.1.3.3.3.2.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3.2">𝑗</ci><ci id="S3.SS4.10.p10.18.m18.1.1.3.3.3.3.cmml" xref="S3.SS4.10.p10.18.m18.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.18.m18.1c">C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.18.m18.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∪ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\alpha_{j}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.19.m19.1"><semantics id="S3.SS4.10.p10.19.m19.1a"><mrow id="S3.SS4.10.p10.19.m19.1.1" xref="S3.SS4.10.p10.19.m19.1.1.cmml"><msub id="S3.SS4.10.p10.19.m19.1.1.2" xref="S3.SS4.10.p10.19.m19.1.1.2.cmml"><mi id="S3.SS4.10.p10.19.m19.1.1.2.2" xref="S3.SS4.10.p10.19.m19.1.1.2.2.cmml">α</mi><mi id="S3.SS4.10.p10.19.m19.1.1.2.3" xref="S3.SS4.10.p10.19.m19.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.19.m19.1.1.1" xref="S3.SS4.10.p10.19.m19.1.1.1.cmml">=</mo><mn id="S3.SS4.10.p10.19.m19.1.1.3" xref="S3.SS4.10.p10.19.m19.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.19.m19.1b"><apply id="S3.SS4.10.p10.19.m19.1.1.cmml" xref="S3.SS4.10.p10.19.m19.1.1"><eq id="S3.SS4.10.p10.19.m19.1.1.1.cmml" xref="S3.SS4.10.p10.19.m19.1.1.1"></eq><apply id="S3.SS4.10.p10.19.m19.1.1.2.cmml" xref="S3.SS4.10.p10.19.m19.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.19.m19.1.1.2.1.cmml" xref="S3.SS4.10.p10.19.m19.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.19.m19.1.1.2.2.cmml" xref="S3.SS4.10.p10.19.m19.1.1.2.2">𝛼</ci><ci id="S3.SS4.10.p10.19.m19.1.1.2.3.cmml" xref="S3.SS4.10.p10.19.m19.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.10.p10.19.m19.1.1.3.cmml" type="integer" xref="S3.SS4.10.p10.19.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.19.m19.1c">\alpha_{j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.19.m19.1d">italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Since <math alttext="j^{\prime}\in Y" class="ltx_Math" display="inline" id="S3.SS4.10.p10.20.m20.1"><semantics id="S3.SS4.10.p10.20.m20.1a"><mrow id="S3.SS4.10.p10.20.m20.1.1" xref="S3.SS4.10.p10.20.m20.1.1.cmml"><msup id="S3.SS4.10.p10.20.m20.1.1.2" xref="S3.SS4.10.p10.20.m20.1.1.2.cmml"><mi id="S3.SS4.10.p10.20.m20.1.1.2.2" xref="S3.SS4.10.p10.20.m20.1.1.2.2.cmml">j</mi><mo id="S3.SS4.10.p10.20.m20.1.1.2.3" xref="S3.SS4.10.p10.20.m20.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS4.10.p10.20.m20.1.1.1" xref="S3.SS4.10.p10.20.m20.1.1.1.cmml">∈</mo><mi id="S3.SS4.10.p10.20.m20.1.1.3" xref="S3.SS4.10.p10.20.m20.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.20.m20.1b"><apply id="S3.SS4.10.p10.20.m20.1.1.cmml" xref="S3.SS4.10.p10.20.m20.1.1"><in id="S3.SS4.10.p10.20.m20.1.1.1.cmml" xref="S3.SS4.10.p10.20.m20.1.1.1"></in><apply id="S3.SS4.10.p10.20.m20.1.1.2.cmml" xref="S3.SS4.10.p10.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.20.m20.1.1.2.1.cmml" xref="S3.SS4.10.p10.20.m20.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.20.m20.1.1.2.2.cmml" xref="S3.SS4.10.p10.20.m20.1.1.2.2">𝑗</ci><ci id="S3.SS4.10.p10.20.m20.1.1.2.3.cmml" xref="S3.SS4.10.p10.20.m20.1.1.2.3">′</ci></apply><ci id="S3.SS4.10.p10.20.m20.1.1.3.cmml" xref="S3.SS4.10.p10.20.m20.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.20.m20.1c">j^{\prime}\in Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.20.m20.1d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y</annotation></semantics></math>, we must have <math alttext="\beta_{j^{\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.10.p10.21.m21.1"><semantics id="S3.SS4.10.p10.21.m21.1a"><mrow id="S3.SS4.10.p10.21.m21.1.1" xref="S3.SS4.10.p10.21.m21.1.1.cmml"><msub id="S3.SS4.10.p10.21.m21.1.1.2" xref="S3.SS4.10.p10.21.m21.1.1.2.cmml"><mi id="S3.SS4.10.p10.21.m21.1.1.2.2" xref="S3.SS4.10.p10.21.m21.1.1.2.2.cmml">β</mi><msup id="S3.SS4.10.p10.21.m21.1.1.2.3" xref="S3.SS4.10.p10.21.m21.1.1.2.3.cmml"><mi id="S3.SS4.10.p10.21.m21.1.1.2.3.2" xref="S3.SS4.10.p10.21.m21.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.21.m21.1.1.2.3.3" xref="S3.SS4.10.p10.21.m21.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS4.10.p10.21.m21.1.1.1" xref="S3.SS4.10.p10.21.m21.1.1.1.cmml">=</mo><mn id="S3.SS4.10.p10.21.m21.1.1.3" xref="S3.SS4.10.p10.21.m21.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.21.m21.1b"><apply id="S3.SS4.10.p10.21.m21.1.1.cmml" xref="S3.SS4.10.p10.21.m21.1.1"><eq id="S3.SS4.10.p10.21.m21.1.1.1.cmml" xref="S3.SS4.10.p10.21.m21.1.1.1"></eq><apply id="S3.SS4.10.p10.21.m21.1.1.2.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.21.m21.1.1.2.1.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.21.m21.1.1.2.2.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2.2">𝛽</ci><apply id="S3.SS4.10.p10.21.m21.1.1.2.3.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.21.m21.1.1.2.3.1.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.21.m21.1.1.2.3.2.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.21.m21.1.1.2.3.3.cmml" xref="S3.SS4.10.p10.21.m21.1.1.2.3.3">′</ci></apply></apply><cn id="S3.SS4.10.p10.21.m21.1.1.3.cmml" type="integer" xref="S3.SS4.10.p10.21.m21.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.21.m21.1c">\beta_{j^{\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.21.m21.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>. This means that <math alttext="C^{j^{\prime}}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.22.m22.1"><semantics id="S3.SS4.10.p10.22.m22.1a"><mrow id="S3.SS4.10.p10.22.m22.1.1" xref="S3.SS4.10.p10.22.m22.1.1.cmml"><msubsup id="S3.SS4.10.p10.22.m22.1.1.2" xref="S3.SS4.10.p10.22.m22.1.1.2.cmml"><mi id="S3.SS4.10.p10.22.m22.1.1.2.2.2" xref="S3.SS4.10.p10.22.m22.1.1.2.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.22.m22.1.1.2.3" xref="S3.SS4.10.p10.22.m22.1.1.2.3.cmml">U</mi><msup id="S3.SS4.10.p10.22.m22.1.1.2.2.3" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3.cmml"><mi id="S3.SS4.10.p10.22.m22.1.1.2.2.3.2" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3.2.cmml">j</mi><mo id="S3.SS4.10.p10.22.m22.1.1.2.2.3.3" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3.SS4.10.p10.22.m22.1.1.1" xref="S3.SS4.10.p10.22.m22.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.22.m22.1.1.3" xref="S3.SS4.10.p10.22.m22.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.22.m22.1b"><apply id="S3.SS4.10.p10.22.m22.1.1.cmml" xref="S3.SS4.10.p10.22.m22.1.1"><in id="S3.SS4.10.p10.22.m22.1.1.1.cmml" xref="S3.SS4.10.p10.22.m22.1.1.1"></in><apply id="S3.SS4.10.p10.22.m22.1.1.2.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.22.m22.1.1.2.1.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2">subscript</csymbol><apply id="S3.SS4.10.p10.22.m22.1.1.2.2.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.22.m22.1.1.2.2.1.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.22.m22.1.1.2.2.2.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.2.2">𝐶</ci><apply id="S3.SS4.10.p10.22.m22.1.1.2.2.3.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS4.10.p10.22.m22.1.1.2.2.3.1.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3">superscript</csymbol><ci id="S3.SS4.10.p10.22.m22.1.1.2.2.3.2.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3.2">𝑗</ci><ci id="S3.SS4.10.p10.22.m22.1.1.2.2.3.3.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.2.3.3">′</ci></apply></apply><ci id="S3.SS4.10.p10.22.m22.1.1.2.3.cmml" xref="S3.SS4.10.p10.22.m22.1.1.2.3">𝑈</ci></apply><ci id="S3.SS4.10.p10.22.m22.1.1.3.cmml" xref="S3.SS4.10.p10.22.m22.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.22.m22.1c">C^{j^{\prime}}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.22.m22.1d">italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>, and by the monotonicity of <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.23.m23.1"><semantics id="S3.SS4.10.p10.23.m23.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.23.m23.1.1" xref="S3.SS4.10.p10.23.m23.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.23.m23.1b"><ci id="S3.SS4.10.p10.23.m23.1.1.cmml" xref="S3.SS4.10.p10.23.m23.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.23.m23.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.23.m23.1d">caligraphic_F</annotation></semantics></math> and <math alttext="\diamond_{j}=\cup" class="ltx_Math" display="inline" id="S3.SS4.10.p10.24.m24.3"><semantics id="S3.SS4.10.p10.24.m24.3a"><mrow id="S3.SS4.10.p10.24.m24.3.3.1" xref="S3.SS4.10.p10.24.m24.3.3.2.cmml"><msub id="S3.SS4.10.p10.24.m24.3.3.1.1" xref="S3.SS4.10.p10.24.m24.3.3.1.1.cmml"><mo id="S3.SS4.10.p10.24.m24.3.3.1.1.2" xref="S3.SS4.10.p10.24.m24.3.3.1.1.2.cmml">⋄</mo><mi id="S3.SS4.10.p10.24.m24.3.3.1.1.3" xref="S3.SS4.10.p10.24.m24.3.3.1.1.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.24.m24.3.3.1.2" lspace="0em" xref="S3.SS4.10.p10.24.m24.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.24.m24.1.1" xref="S3.SS4.10.p10.24.m24.1.1.cmml">=</mo><mo id="S3.SS4.10.p10.24.m24.3.3.1.3" lspace="0em" xref="S3.SS4.10.p10.24.m24.3.3.2.cmml">⁣</mo><mo id="S3.SS4.10.p10.24.m24.2.2" xref="S3.SS4.10.p10.24.m24.2.2.cmml">∪</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.24.m24.3b"><list id="S3.SS4.10.p10.24.m24.3.3.2.cmml" xref="S3.SS4.10.p10.24.m24.3.3.1"><apply id="S3.SS4.10.p10.24.m24.3.3.1.1.cmml" xref="S3.SS4.10.p10.24.m24.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.10.p10.24.m24.3.3.1.1.1.cmml" xref="S3.SS4.10.p10.24.m24.3.3.1.1">subscript</csymbol><ci id="S3.SS4.10.p10.24.m24.3.3.1.1.2.cmml" xref="S3.SS4.10.p10.24.m24.3.3.1.1.2">⋄</ci><ci id="S3.SS4.10.p10.24.m24.3.3.1.1.3.cmml" xref="S3.SS4.10.p10.24.m24.3.3.1.1.3">𝑗</ci></apply><eq id="S3.SS4.10.p10.24.m24.1.1.cmml" xref="S3.SS4.10.p10.24.m24.1.1"></eq><union id="S3.SS4.10.p10.24.m24.2.2.cmml" xref="S3.SS4.10.p10.24.m24.2.2"></union></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.24.m24.3c">\diamond_{j}=\cup</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.24.m24.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∪</annotation></semantics></math>, it follows that <math alttext="C^{j}_{U}\in\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.10.p10.25.m25.1"><semantics id="S3.SS4.10.p10.25.m25.1a"><mrow id="S3.SS4.10.p10.25.m25.1.1" xref="S3.SS4.10.p10.25.m25.1.1.cmml"><msubsup id="S3.SS4.10.p10.25.m25.1.1.2" xref="S3.SS4.10.p10.25.m25.1.1.2.cmml"><mi id="S3.SS4.10.p10.25.m25.1.1.2.2.2" xref="S3.SS4.10.p10.25.m25.1.1.2.2.2.cmml">C</mi><mi id="S3.SS4.10.p10.25.m25.1.1.2.3" xref="S3.SS4.10.p10.25.m25.1.1.2.3.cmml">U</mi><mi id="S3.SS4.10.p10.25.m25.1.1.2.2.3" xref="S3.SS4.10.p10.25.m25.1.1.2.2.3.cmml">j</mi></msubsup><mo id="S3.SS4.10.p10.25.m25.1.1.1" xref="S3.SS4.10.p10.25.m25.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.10.p10.25.m25.1.1.3" xref="S3.SS4.10.p10.25.m25.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.25.m25.1b"><apply id="S3.SS4.10.p10.25.m25.1.1.cmml" xref="S3.SS4.10.p10.25.m25.1.1"><in id="S3.SS4.10.p10.25.m25.1.1.1.cmml" xref="S3.SS4.10.p10.25.m25.1.1.1"></in><apply id="S3.SS4.10.p10.25.m25.1.1.2.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.25.m25.1.1.2.1.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2">subscript</csymbol><apply id="S3.SS4.10.p10.25.m25.1.1.2.2.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.25.m25.1.1.2.2.1.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2">superscript</csymbol><ci id="S3.SS4.10.p10.25.m25.1.1.2.2.2.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2.2.2">𝐶</ci><ci id="S3.SS4.10.p10.25.m25.1.1.2.2.3.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2.2.3">𝑗</ci></apply><ci id="S3.SS4.10.p10.25.m25.1.1.2.3.cmml" xref="S3.SS4.10.p10.25.m25.1.1.2.3">𝑈</ci></apply><ci id="S3.SS4.10.p10.25.m25.1.1.3.cmml" xref="S3.SS4.10.p10.25.m25.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.25.m25.1c">C^{j}_{U}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.25.m25.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ∈ caligraphic_F</annotation></semantics></math>. But this is in contradiction to <math alttext="\beta_{j}=0" class="ltx_Math" display="inline" id="S3.SS4.10.p10.26.m26.1"><semantics id="S3.SS4.10.p10.26.m26.1a"><mrow id="S3.SS4.10.p10.26.m26.1.1" xref="S3.SS4.10.p10.26.m26.1.1.cmml"><msub id="S3.SS4.10.p10.26.m26.1.1.2" xref="S3.SS4.10.p10.26.m26.1.1.2.cmml"><mi id="S3.SS4.10.p10.26.m26.1.1.2.2" xref="S3.SS4.10.p10.26.m26.1.1.2.2.cmml">β</mi><mi id="S3.SS4.10.p10.26.m26.1.1.2.3" xref="S3.SS4.10.p10.26.m26.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.10.p10.26.m26.1.1.1" xref="S3.SS4.10.p10.26.m26.1.1.1.cmml">=</mo><mn id="S3.SS4.10.p10.26.m26.1.1.3" xref="S3.SS4.10.p10.26.m26.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.10.p10.26.m26.1b"><apply id="S3.SS4.10.p10.26.m26.1.1.cmml" xref="S3.SS4.10.p10.26.m26.1.1"><eq id="S3.SS4.10.p10.26.m26.1.1.1.cmml" xref="S3.SS4.10.p10.26.m26.1.1.1"></eq><apply id="S3.SS4.10.p10.26.m26.1.1.2.cmml" xref="S3.SS4.10.p10.26.m26.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.10.p10.26.m26.1.1.2.1.cmml" xref="S3.SS4.10.p10.26.m26.1.1.2">subscript</csymbol><ci id="S3.SS4.10.p10.26.m26.1.1.2.2.cmml" xref="S3.SS4.10.p10.26.m26.1.1.2.2">𝛽</ci><ci id="S3.SS4.10.p10.26.m26.1.1.2.3.cmml" xref="S3.SS4.10.p10.26.m26.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.10.p10.26.m26.1.1.3.cmml" type="integer" xref="S3.SS4.10.p10.26.m26.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.10.p10.26.m26.1c">\beta_{j}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.10.p10.26.m26.1d">italic_β start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0</annotation></semantics></math>, which completes the proof of the claim.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem34"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem34.1.1.1">Claim 34</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem34.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem34.p1"> <p class="ltx_p" id="Thmtheorem34.p1.4"><span class="ltx_text ltx_font_italic" id="Thmtheorem34.p1.4.4">Suppose that <math alttext="j,j^{\prime}\in X" class="ltx_Math" display="inline" id="Thmtheorem34.p1.1.1.m1.2"><semantics id="Thmtheorem34.p1.1.1.m1.2a"><mrow id="Thmtheorem34.p1.1.1.m1.2.2" xref="Thmtheorem34.p1.1.1.m1.2.2.cmml"><mrow id="Thmtheorem34.p1.1.1.m1.2.2.1.1" xref="Thmtheorem34.p1.1.1.m1.2.2.1.2.cmml"><mi id="Thmtheorem34.p1.1.1.m1.1.1" xref="Thmtheorem34.p1.1.1.m1.1.1.cmml">j</mi><mo id="Thmtheorem34.p1.1.1.m1.2.2.1.1.2" xref="Thmtheorem34.p1.1.1.m1.2.2.1.2.cmml">,</mo><msup id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.cmml"><mi id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.2" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.2.cmml">j</mi><mo id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.3" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="Thmtheorem34.p1.1.1.m1.2.2.2" xref="Thmtheorem34.p1.1.1.m1.2.2.2.cmml">∈</mo><mi id="Thmtheorem34.p1.1.1.m1.2.2.3" xref="Thmtheorem34.p1.1.1.m1.2.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem34.p1.1.1.m1.2b"><apply id="Thmtheorem34.p1.1.1.m1.2.2.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2"><in id="Thmtheorem34.p1.1.1.m1.2.2.2.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.2"></in><list id="Thmtheorem34.p1.1.1.m1.2.2.1.2.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1"><ci id="Thmtheorem34.p1.1.1.m1.1.1.cmml" xref="Thmtheorem34.p1.1.1.m1.1.1">𝑗</ci><apply id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.1.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1">superscript</csymbol><ci id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.2.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.2">𝑗</ci><ci id="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.3.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.1.1.1.3">′</ci></apply></list><ci id="Thmtheorem34.p1.1.1.m1.2.2.3.cmml" xref="Thmtheorem34.p1.1.1.m1.2.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem34.p1.1.1.m1.2c">j,j^{\prime}\in X</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem34.p1.1.1.m1.2d">italic_j , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_X</annotation></semantics></math>, <math alttext="C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="Thmtheorem34.p1.2.2.m2.1"><semantics id="Thmtheorem34.p1.2.2.m2.1a"><mrow id="Thmtheorem34.p1.2.2.m2.1.1" xref="Thmtheorem34.p1.2.2.m2.1.1.cmml"><msup id="Thmtheorem34.p1.2.2.m2.1.1.2" xref="Thmtheorem34.p1.2.2.m2.1.1.2.cmml"><mi id="Thmtheorem34.p1.2.2.m2.1.1.2.2" xref="Thmtheorem34.p1.2.2.m2.1.1.2.2.cmml">C</mi><mi id="Thmtheorem34.p1.2.2.m2.1.1.2.3" xref="Thmtheorem34.p1.2.2.m2.1.1.2.3.cmml">j</mi></msup><mo id="Thmtheorem34.p1.2.2.m2.1.1.1" xref="Thmtheorem34.p1.2.2.m2.1.1.1.cmml">=</mo><mrow id="Thmtheorem34.p1.2.2.m2.1.1.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.cmml"><msup id="Thmtheorem34.p1.2.2.m2.1.1.3.2" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.cmml"><mi id="Thmtheorem34.p1.2.2.m2.1.1.3.2.2" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.2.cmml">C</mi><msup id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.cmml"><mi id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.2" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.2.cmml">j</mi><mo id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.3.cmml">′</mo></msup></msup><mo id="Thmtheorem34.p1.2.2.m2.1.1.3.1" xref="Thmtheorem34.p1.2.2.m2.1.1.3.1.cmml">∪</mo><msup id="Thmtheorem34.p1.2.2.m2.1.1.3.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.cmml"><mi id="Thmtheorem34.p1.2.2.m2.1.1.3.3.2" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.2.cmml">C</mi><msup id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.cmml"><mi id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.2" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.2.cmml">j</mi><mo id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.3" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem34.p1.2.2.m2.1b"><apply id="Thmtheorem34.p1.2.2.m2.1.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1"><eq id="Thmtheorem34.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.1"></eq><apply id="Thmtheorem34.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem34.p1.2.2.m2.1.1.2.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.2">superscript</csymbol><ci id="Thmtheorem34.p1.2.2.m2.1.1.2.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.2.2">𝐶</ci><ci id="Thmtheorem34.p1.2.2.m2.1.1.2.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.2.3">𝑗</ci></apply><apply id="Thmtheorem34.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3"><union id="Thmtheorem34.p1.2.2.m2.1.1.3.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.1"></union><apply id="Thmtheorem34.p1.2.2.m2.1.1.3.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="Thmtheorem34.p1.2.2.m2.1.1.3.2.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2">superscript</csymbol><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.2.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.2">𝐶</ci><apply id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3">superscript</csymbol><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.2">𝑗</ci><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.2.3.3">′</ci></apply></apply><apply id="Thmtheorem34.p1.2.2.m2.1.1.3.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem34.p1.2.2.m2.1.1.3.3.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3">superscript</csymbol><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.3.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.2">𝐶</ci><apply id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.1.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3">superscript</csymbol><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.2.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.2">𝑗</ci><ci id="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.3.cmml" xref="Thmtheorem34.p1.2.2.m2.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem34.p1.2.2.m2.1c">C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem34.p1.2.2.m2.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∪ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="j^{\prime\prime}\in Y" class="ltx_Math" display="inline" id="Thmtheorem34.p1.3.3.m3.1"><semantics id="Thmtheorem34.p1.3.3.m3.1a"><mrow id="Thmtheorem34.p1.3.3.m3.1.1" xref="Thmtheorem34.p1.3.3.m3.1.1.cmml"><msup id="Thmtheorem34.p1.3.3.m3.1.1.2" xref="Thmtheorem34.p1.3.3.m3.1.1.2.cmml"><mi id="Thmtheorem34.p1.3.3.m3.1.1.2.2" xref="Thmtheorem34.p1.3.3.m3.1.1.2.2.cmml">j</mi><mo id="Thmtheorem34.p1.3.3.m3.1.1.2.3" xref="Thmtheorem34.p1.3.3.m3.1.1.2.3.cmml">′′</mo></msup><mo id="Thmtheorem34.p1.3.3.m3.1.1.1" xref="Thmtheorem34.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="Thmtheorem34.p1.3.3.m3.1.1.3" xref="Thmtheorem34.p1.3.3.m3.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem34.p1.3.3.m3.1b"><apply id="Thmtheorem34.p1.3.3.m3.1.1.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1"><in id="Thmtheorem34.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.1"></in><apply id="Thmtheorem34.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem34.p1.3.3.m3.1.1.2.1.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.2">superscript</csymbol><ci id="Thmtheorem34.p1.3.3.m3.1.1.2.2.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.2.2">𝑗</ci><ci id="Thmtheorem34.p1.3.3.m3.1.1.2.3.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.2.3">′′</ci></apply><ci id="Thmtheorem34.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem34.p1.3.3.m3.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem34.p1.3.3.m3.1c">j^{\prime\prime}\in Y</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem34.p1.3.3.m3.1d">italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_Y</annotation></semantics></math>. Then <math alttext="a\notin C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="Thmtheorem34.p1.4.4.m4.1"><semantics id="Thmtheorem34.p1.4.4.m4.1a"><mrow id="Thmtheorem34.p1.4.4.m4.1.1" xref="Thmtheorem34.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem34.p1.4.4.m4.1.1.2" xref="Thmtheorem34.p1.4.4.m4.1.1.2.cmml">a</mi><mo id="Thmtheorem34.p1.4.4.m4.1.1.1" xref="Thmtheorem34.p1.4.4.m4.1.1.1.cmml">∉</mo><msup id="Thmtheorem34.p1.4.4.m4.1.1.3" xref="Thmtheorem34.p1.4.4.m4.1.1.3.cmml"><mi id="Thmtheorem34.p1.4.4.m4.1.1.3.2" xref="Thmtheorem34.p1.4.4.m4.1.1.3.2.cmml">C</mi><msup id="Thmtheorem34.p1.4.4.m4.1.1.3.3" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3.cmml"><mi id="Thmtheorem34.p1.4.4.m4.1.1.3.3.2" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3.2.cmml">j</mi><mo id="Thmtheorem34.p1.4.4.m4.1.1.3.3.3" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3.3.cmml">′′</mo></msup></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem34.p1.4.4.m4.1b"><apply id="Thmtheorem34.p1.4.4.m4.1.1.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1"><notin id="Thmtheorem34.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.1"></notin><ci id="Thmtheorem34.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.2">𝑎</ci><apply id="Thmtheorem34.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem34.p1.4.4.m4.1.1.3.1.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="Thmtheorem34.p1.4.4.m4.1.1.3.2.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3.2">𝐶</ci><apply id="Thmtheorem34.p1.4.4.m4.1.1.3.3.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="Thmtheorem34.p1.4.4.m4.1.1.3.3.1.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3">superscript</csymbol><ci id="Thmtheorem34.p1.4.4.m4.1.1.3.3.2.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3.2">𝑗</ci><ci id="Thmtheorem34.p1.4.4.m4.1.1.3.3.3.cmml" xref="Thmtheorem34.p1.4.4.m4.1.1.3.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem34.p1.4.4.m4.1c">a\notin C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem34.p1.4.4.m4.1d">italic_a ∉ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS4.11.p11"> <p class="ltx_p" id="S3.SS4.11.p11.11">The assumptions force <math alttext="\alpha_{j}=1" class="ltx_Math" display="inline" id="S3.SS4.11.p11.1.m1.1"><semantics id="S3.SS4.11.p11.1.m1.1a"><mrow id="S3.SS4.11.p11.1.m1.1.1" xref="S3.SS4.11.p11.1.m1.1.1.cmml"><msub id="S3.SS4.11.p11.1.m1.1.1.2" xref="S3.SS4.11.p11.1.m1.1.1.2.cmml"><mi id="S3.SS4.11.p11.1.m1.1.1.2.2" xref="S3.SS4.11.p11.1.m1.1.1.2.2.cmml">α</mi><mi id="S3.SS4.11.p11.1.m1.1.1.2.3" xref="S3.SS4.11.p11.1.m1.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.11.p11.1.m1.1.1.1" xref="S3.SS4.11.p11.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.1.m1.1.1.3" xref="S3.SS4.11.p11.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.1.m1.1b"><apply id="S3.SS4.11.p11.1.m1.1.1.cmml" xref="S3.SS4.11.p11.1.m1.1.1"><eq id="S3.SS4.11.p11.1.m1.1.1.1.cmml" xref="S3.SS4.11.p11.1.m1.1.1.1"></eq><apply id="S3.SS4.11.p11.1.m1.1.1.2.cmml" xref="S3.SS4.11.p11.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.1.m1.1.1.2.1.cmml" xref="S3.SS4.11.p11.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.1.m1.1.1.2.2.cmml" xref="S3.SS4.11.p11.1.m1.1.1.2.2">𝛼</ci><ci id="S3.SS4.11.p11.1.m1.1.1.2.3.cmml" xref="S3.SS4.11.p11.1.m1.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.11.p11.1.m1.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.1.m1.1c">\alpha_{j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{j}=0" class="ltx_Math" display="inline" id="S3.SS4.11.p11.2.m2.1"><semantics id="S3.SS4.11.p11.2.m2.1a"><mrow id="S3.SS4.11.p11.2.m2.1.1" xref="S3.SS4.11.p11.2.m2.1.1.cmml"><msub id="S3.SS4.11.p11.2.m2.1.1.2" xref="S3.SS4.11.p11.2.m2.1.1.2.cmml"><mi id="S3.SS4.11.p11.2.m2.1.1.2.2" xref="S3.SS4.11.p11.2.m2.1.1.2.2.cmml">β</mi><mi id="S3.SS4.11.p11.2.m2.1.1.2.3" xref="S3.SS4.11.p11.2.m2.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.11.p11.2.m2.1.1.1" xref="S3.SS4.11.p11.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.2.m2.1.1.3" xref="S3.SS4.11.p11.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.2.m2.1b"><apply id="S3.SS4.11.p11.2.m2.1.1.cmml" xref="S3.SS4.11.p11.2.m2.1.1"><eq id="S3.SS4.11.p11.2.m2.1.1.1.cmml" xref="S3.SS4.11.p11.2.m2.1.1.1"></eq><apply id="S3.SS4.11.p11.2.m2.1.1.2.cmml" xref="S3.SS4.11.p11.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.2.m2.1.1.2.1.cmml" xref="S3.SS4.11.p11.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.2.m2.1.1.2.2.cmml" xref="S3.SS4.11.p11.2.m2.1.1.2.2">𝛽</ci><ci id="S3.SS4.11.p11.2.m2.1.1.2.3.cmml" xref="S3.SS4.11.p11.2.m2.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.11.p11.2.m2.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.2.m2.1c">\beta_{j}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0</annotation></semantics></math>, and that it is not the case that <math alttext="\alpha_{j^{\prime\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.11.p11.3.m3.1"><semantics id="S3.SS4.11.p11.3.m3.1a"><mrow id="S3.SS4.11.p11.3.m3.1.1" xref="S3.SS4.11.p11.3.m3.1.1.cmml"><msub id="S3.SS4.11.p11.3.m3.1.1.2" xref="S3.SS4.11.p11.3.m3.1.1.2.cmml"><mi id="S3.SS4.11.p11.3.m3.1.1.2.2" xref="S3.SS4.11.p11.3.m3.1.1.2.2.cmml">α</mi><msup id="S3.SS4.11.p11.3.m3.1.1.2.3" xref="S3.SS4.11.p11.3.m3.1.1.2.3.cmml"><mi id="S3.SS4.11.p11.3.m3.1.1.2.3.2" xref="S3.SS4.11.p11.3.m3.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.3.m3.1.1.2.3.3" xref="S3.SS4.11.p11.3.m3.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.11.p11.3.m3.1.1.1" xref="S3.SS4.11.p11.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.3.m3.1.1.3" xref="S3.SS4.11.p11.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.3.m3.1b"><apply id="S3.SS4.11.p11.3.m3.1.1.cmml" xref="S3.SS4.11.p11.3.m3.1.1"><eq id="S3.SS4.11.p11.3.m3.1.1.1.cmml" xref="S3.SS4.11.p11.3.m3.1.1.1"></eq><apply id="S3.SS4.11.p11.3.m3.1.1.2.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.3.m3.1.1.2.1.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.3.m3.1.1.2.2.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2.2">𝛼</ci><apply id="S3.SS4.11.p11.3.m3.1.1.2.3.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.3.m3.1.1.2.3.1.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.3.m3.1.1.2.3.2.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.3.m3.1.1.2.3.3.cmml" xref="S3.SS4.11.p11.3.m3.1.1.2.3.3">′′</ci></apply></apply><cn id="S3.SS4.11.p11.3.m3.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.3.m3.1c">\alpha_{j^{\prime\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\beta_{j^{\prime\prime}}=0" class="ltx_Math" display="inline" id="S3.SS4.11.p11.4.m4.1"><semantics id="S3.SS4.11.p11.4.m4.1a"><mrow id="S3.SS4.11.p11.4.m4.1.1" xref="S3.SS4.11.p11.4.m4.1.1.cmml"><msub id="S3.SS4.11.p11.4.m4.1.1.2" xref="S3.SS4.11.p11.4.m4.1.1.2.cmml"><mi id="S3.SS4.11.p11.4.m4.1.1.2.2" xref="S3.SS4.11.p11.4.m4.1.1.2.2.cmml">β</mi><msup id="S3.SS4.11.p11.4.m4.1.1.2.3" xref="S3.SS4.11.p11.4.m4.1.1.2.3.cmml"><mi id="S3.SS4.11.p11.4.m4.1.1.2.3.2" xref="S3.SS4.11.p11.4.m4.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.4.m4.1.1.2.3.3" xref="S3.SS4.11.p11.4.m4.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.11.p11.4.m4.1.1.1" xref="S3.SS4.11.p11.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.4.m4.1.1.3" xref="S3.SS4.11.p11.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.4.m4.1b"><apply id="S3.SS4.11.p11.4.m4.1.1.cmml" xref="S3.SS4.11.p11.4.m4.1.1"><eq id="S3.SS4.11.p11.4.m4.1.1.1.cmml" xref="S3.SS4.11.p11.4.m4.1.1.1"></eq><apply id="S3.SS4.11.p11.4.m4.1.1.2.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.4.m4.1.1.2.1.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.4.m4.1.1.2.2.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2.2">𝛽</ci><apply id="S3.SS4.11.p11.4.m4.1.1.2.3.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.4.m4.1.1.2.3.1.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.4.m4.1.1.2.3.2.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.4.m4.1.1.2.3.3.cmml" xref="S3.SS4.11.p11.4.m4.1.1.2.3.3">′′</ci></apply></apply><cn id="S3.SS4.11.p11.4.m4.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.4.m4.1c">\beta_{j^{\prime\prime}}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.4.m4.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>. We must argue that <math alttext="\alpha_{j^{\prime\prime}}=0" class="ltx_Math" display="inline" id="S3.SS4.11.p11.5.m5.1"><semantics id="S3.SS4.11.p11.5.m5.1a"><mrow id="S3.SS4.11.p11.5.m5.1.1" xref="S3.SS4.11.p11.5.m5.1.1.cmml"><msub id="S3.SS4.11.p11.5.m5.1.1.2" xref="S3.SS4.11.p11.5.m5.1.1.2.cmml"><mi id="S3.SS4.11.p11.5.m5.1.1.2.2" xref="S3.SS4.11.p11.5.m5.1.1.2.2.cmml">α</mi><msup id="S3.SS4.11.p11.5.m5.1.1.2.3" xref="S3.SS4.11.p11.5.m5.1.1.2.3.cmml"><mi id="S3.SS4.11.p11.5.m5.1.1.2.3.2" xref="S3.SS4.11.p11.5.m5.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.5.m5.1.1.2.3.3" xref="S3.SS4.11.p11.5.m5.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.11.p11.5.m5.1.1.1" xref="S3.SS4.11.p11.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.5.m5.1.1.3" xref="S3.SS4.11.p11.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.5.m5.1b"><apply id="S3.SS4.11.p11.5.m5.1.1.cmml" xref="S3.SS4.11.p11.5.m5.1.1"><eq id="S3.SS4.11.p11.5.m5.1.1.1.cmml" xref="S3.SS4.11.p11.5.m5.1.1.1"></eq><apply id="S3.SS4.11.p11.5.m5.1.1.2.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.5.m5.1.1.2.1.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.5.m5.1.1.2.2.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2.2">𝛼</ci><apply id="S3.SS4.11.p11.5.m5.1.1.2.3.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.5.m5.1.1.2.3.1.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.5.m5.1.1.2.3.2.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.5.m5.1.1.2.3.3.cmml" xref="S3.SS4.11.p11.5.m5.1.1.2.3.3">′′</ci></apply></apply><cn id="S3.SS4.11.p11.5.m5.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.5.m5.1c">\alpha_{j^{\prime\prime}}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.5.m5.1d">italic_α start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math> (i.e., <math alttext="a\notin C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.11.p11.6.m6.1"><semantics id="S3.SS4.11.p11.6.m6.1a"><mrow id="S3.SS4.11.p11.6.m6.1.1" xref="S3.SS4.11.p11.6.m6.1.1.cmml"><mi id="S3.SS4.11.p11.6.m6.1.1.2" xref="S3.SS4.11.p11.6.m6.1.1.2.cmml">a</mi><mo id="S3.SS4.11.p11.6.m6.1.1.1" xref="S3.SS4.11.p11.6.m6.1.1.1.cmml">∉</mo><msup id="S3.SS4.11.p11.6.m6.1.1.3" xref="S3.SS4.11.p11.6.m6.1.1.3.cmml"><mi id="S3.SS4.11.p11.6.m6.1.1.3.2" xref="S3.SS4.11.p11.6.m6.1.1.3.2.cmml">C</mi><msup id="S3.SS4.11.p11.6.m6.1.1.3.3" xref="S3.SS4.11.p11.6.m6.1.1.3.3.cmml"><mi id="S3.SS4.11.p11.6.m6.1.1.3.3.2" xref="S3.SS4.11.p11.6.m6.1.1.3.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.6.m6.1.1.3.3.3" xref="S3.SS4.11.p11.6.m6.1.1.3.3.3.cmml">′′</mo></msup></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.6.m6.1b"><apply id="S3.SS4.11.p11.6.m6.1.1.cmml" xref="S3.SS4.11.p11.6.m6.1.1"><notin id="S3.SS4.11.p11.6.m6.1.1.1.cmml" xref="S3.SS4.11.p11.6.m6.1.1.1"></notin><ci id="S3.SS4.11.p11.6.m6.1.1.2.cmml" xref="S3.SS4.11.p11.6.m6.1.1.2">𝑎</ci><apply id="S3.SS4.11.p11.6.m6.1.1.3.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.6.m6.1.1.3.1.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3">superscript</csymbol><ci id="S3.SS4.11.p11.6.m6.1.1.3.2.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3.2">𝐶</ci><apply id="S3.SS4.11.p11.6.m6.1.1.3.3.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.6.m6.1.1.3.3.1.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3.3">superscript</csymbol><ci id="S3.SS4.11.p11.6.m6.1.1.3.3.2.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3.3.2">𝑗</ci><ci id="S3.SS4.11.p11.6.m6.1.1.3.3.3.cmml" xref="S3.SS4.11.p11.6.m6.1.1.3.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.6.m6.1c">a\notin C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.6.m6.1d">italic_a ∉ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>), and to do so we show that <math alttext="\beta_{j^{\prime\prime}}=0" class="ltx_Math" display="inline" id="S3.SS4.11.p11.7.m7.1"><semantics id="S3.SS4.11.p11.7.m7.1a"><mrow id="S3.SS4.11.p11.7.m7.1.1" xref="S3.SS4.11.p11.7.m7.1.1.cmml"><msub id="S3.SS4.11.p11.7.m7.1.1.2" xref="S3.SS4.11.p11.7.m7.1.1.2.cmml"><mi id="S3.SS4.11.p11.7.m7.1.1.2.2" xref="S3.SS4.11.p11.7.m7.1.1.2.2.cmml">β</mi><msup id="S3.SS4.11.p11.7.m7.1.1.2.3" xref="S3.SS4.11.p11.7.m7.1.1.2.3.cmml"><mi id="S3.SS4.11.p11.7.m7.1.1.2.3.2" xref="S3.SS4.11.p11.7.m7.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.7.m7.1.1.2.3.3" xref="S3.SS4.11.p11.7.m7.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.11.p11.7.m7.1.1.1" xref="S3.SS4.11.p11.7.m7.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.7.m7.1.1.3" xref="S3.SS4.11.p11.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.7.m7.1b"><apply id="S3.SS4.11.p11.7.m7.1.1.cmml" xref="S3.SS4.11.p11.7.m7.1.1"><eq id="S3.SS4.11.p11.7.m7.1.1.1.cmml" xref="S3.SS4.11.p11.7.m7.1.1.1"></eq><apply id="S3.SS4.11.p11.7.m7.1.1.2.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.7.m7.1.1.2.1.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.7.m7.1.1.2.2.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2.2">𝛽</ci><apply id="S3.SS4.11.p11.7.m7.1.1.2.3.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.7.m7.1.1.2.3.1.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.7.m7.1.1.2.3.2.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.7.m7.1.1.2.3.3.cmml" xref="S3.SS4.11.p11.7.m7.1.1.2.3.3">′′</ci></apply></apply><cn id="S3.SS4.11.p11.7.m7.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.7.m7.1c">\beta_{j^{\prime\prime}}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.7.m7.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>. But if <math alttext="\beta_{j^{\prime\prime}}=1" class="ltx_Math" display="inline" id="S3.SS4.11.p11.8.m8.1"><semantics id="S3.SS4.11.p11.8.m8.1a"><mrow id="S3.SS4.11.p11.8.m8.1.1" xref="S3.SS4.11.p11.8.m8.1.1.cmml"><msub id="S3.SS4.11.p11.8.m8.1.1.2" xref="S3.SS4.11.p11.8.m8.1.1.2.cmml"><mi id="S3.SS4.11.p11.8.m8.1.1.2.2" xref="S3.SS4.11.p11.8.m8.1.1.2.2.cmml">β</mi><msup id="S3.SS4.11.p11.8.m8.1.1.2.3" xref="S3.SS4.11.p11.8.m8.1.1.2.3.cmml"><mi id="S3.SS4.11.p11.8.m8.1.1.2.3.2" xref="S3.SS4.11.p11.8.m8.1.1.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.8.m8.1.1.2.3.3" xref="S3.SS4.11.p11.8.m8.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S3.SS4.11.p11.8.m8.1.1.1" xref="S3.SS4.11.p11.8.m8.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.8.m8.1.1.3" xref="S3.SS4.11.p11.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.8.m8.1b"><apply id="S3.SS4.11.p11.8.m8.1.1.cmml" xref="S3.SS4.11.p11.8.m8.1.1"><eq id="S3.SS4.11.p11.8.m8.1.1.1.cmml" xref="S3.SS4.11.p11.8.m8.1.1.1"></eq><apply id="S3.SS4.11.p11.8.m8.1.1.2.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.8.m8.1.1.2.1.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.8.m8.1.1.2.2.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2.2">𝛽</ci><apply id="S3.SS4.11.p11.8.m8.1.1.2.3.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.8.m8.1.1.2.3.1.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.8.m8.1.1.2.3.2.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.8.m8.1.1.2.3.3.cmml" xref="S3.SS4.11.p11.8.m8.1.1.2.3.3">′′</ci></apply></apply><cn id="S3.SS4.11.p11.8.m8.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.8.m8.1c">\beta_{j^{\prime\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.8.m8.1d">italic_β start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>, the monotonicity of <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.11.p11.9.m9.1"><semantics id="S3.SS4.11.p11.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.11.p11.9.m9.1.1" xref="S3.SS4.11.p11.9.m9.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.9.m9.1b"><ci id="S3.SS4.11.p11.9.m9.1.1.cmml" xref="S3.SS4.11.p11.9.m9.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.9.m9.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.9.m9.1d">caligraphic_F</annotation></semantics></math> and <math alttext="C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}" class="ltx_Math" display="inline" id="S3.SS4.11.p11.10.m10.1"><semantics id="S3.SS4.11.p11.10.m10.1a"><mrow id="S3.SS4.11.p11.10.m10.1.1" xref="S3.SS4.11.p11.10.m10.1.1.cmml"><msup id="S3.SS4.11.p11.10.m10.1.1.2" xref="S3.SS4.11.p11.10.m10.1.1.2.cmml"><mi id="S3.SS4.11.p11.10.m10.1.1.2.2" xref="S3.SS4.11.p11.10.m10.1.1.2.2.cmml">C</mi><mi id="S3.SS4.11.p11.10.m10.1.1.2.3" xref="S3.SS4.11.p11.10.m10.1.1.2.3.cmml">j</mi></msup><mo id="S3.SS4.11.p11.10.m10.1.1.1" xref="S3.SS4.11.p11.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS4.11.p11.10.m10.1.1.3" xref="S3.SS4.11.p11.10.m10.1.1.3.cmml"><msup id="S3.SS4.11.p11.10.m10.1.1.3.2" xref="S3.SS4.11.p11.10.m10.1.1.3.2.cmml"><mi id="S3.SS4.11.p11.10.m10.1.1.3.2.2" xref="S3.SS4.11.p11.10.m10.1.1.3.2.2.cmml">C</mi><msup id="S3.SS4.11.p11.10.m10.1.1.3.2.3" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3.cmml"><mi id="S3.SS4.11.p11.10.m10.1.1.3.2.3.2" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.10.m10.1.1.3.2.3.3" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3.3.cmml">′</mo></msup></msup><mo id="S3.SS4.11.p11.10.m10.1.1.3.1" xref="S3.SS4.11.p11.10.m10.1.1.3.1.cmml">∪</mo><msup id="S3.SS4.11.p11.10.m10.1.1.3.3" xref="S3.SS4.11.p11.10.m10.1.1.3.3.cmml"><mi id="S3.SS4.11.p11.10.m10.1.1.3.3.2" xref="S3.SS4.11.p11.10.m10.1.1.3.3.2.cmml">C</mi><msup id="S3.SS4.11.p11.10.m10.1.1.3.3.3" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3.cmml"><mi id="S3.SS4.11.p11.10.m10.1.1.3.3.3.2" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3.2.cmml">j</mi><mo id="S3.SS4.11.p11.10.m10.1.1.3.3.3.3" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3.3.cmml">′′</mo></msup></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.10.m10.1b"><apply id="S3.SS4.11.p11.10.m10.1.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1"><eq id="S3.SS4.11.p11.10.m10.1.1.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.1"></eq><apply id="S3.SS4.11.p11.10.m10.1.1.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.10.m10.1.1.2.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.2">superscript</csymbol><ci id="S3.SS4.11.p11.10.m10.1.1.2.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.2.2">𝐶</ci><ci id="S3.SS4.11.p11.10.m10.1.1.2.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.2.3">𝑗</ci></apply><apply id="S3.SS4.11.p11.10.m10.1.1.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3"><union id="S3.SS4.11.p11.10.m10.1.1.3.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.1"></union><apply id="S3.SS4.11.p11.10.m10.1.1.3.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.10.m10.1.1.3.2.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2">superscript</csymbol><ci id="S3.SS4.11.p11.10.m10.1.1.3.2.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2.2">𝐶</ci><apply id="S3.SS4.11.p11.10.m10.1.1.3.2.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.10.m10.1.1.3.2.3.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3">superscript</csymbol><ci id="S3.SS4.11.p11.10.m10.1.1.3.2.3.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3.2">𝑗</ci><ci id="S3.SS4.11.p11.10.m10.1.1.3.2.3.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.2.3.3">′</ci></apply></apply><apply id="S3.SS4.11.p11.10.m10.1.1.3.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.10.m10.1.1.3.3.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3">superscript</csymbol><ci id="S3.SS4.11.p11.10.m10.1.1.3.3.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3.2">𝐶</ci><apply id="S3.SS4.11.p11.10.m10.1.1.3.3.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.11.p11.10.m10.1.1.3.3.3.1.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3">superscript</csymbol><ci id="S3.SS4.11.p11.10.m10.1.1.3.3.3.2.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3.2">𝑗</ci><ci id="S3.SS4.11.p11.10.m10.1.1.3.3.3.3.cmml" xref="S3.SS4.11.p11.10.m10.1.1.3.3.3.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.10.m10.1c">C^{j}=C^{j^{\prime}}\cup C^{j^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.10.m10.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∪ italic_C start_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> imply <math alttext="\beta_{j}=1" class="ltx_Math" display="inline" id="S3.SS4.11.p11.11.m11.1"><semantics id="S3.SS4.11.p11.11.m11.1a"><mrow id="S3.SS4.11.p11.11.m11.1.1" xref="S3.SS4.11.p11.11.m11.1.1.cmml"><msub id="S3.SS4.11.p11.11.m11.1.1.2" xref="S3.SS4.11.p11.11.m11.1.1.2.cmml"><mi id="S3.SS4.11.p11.11.m11.1.1.2.2" xref="S3.SS4.11.p11.11.m11.1.1.2.2.cmml">β</mi><mi id="S3.SS4.11.p11.11.m11.1.1.2.3" xref="S3.SS4.11.p11.11.m11.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS4.11.p11.11.m11.1.1.1" xref="S3.SS4.11.p11.11.m11.1.1.1.cmml">=</mo><mn id="S3.SS4.11.p11.11.m11.1.1.3" xref="S3.SS4.11.p11.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.11.p11.11.m11.1b"><apply id="S3.SS4.11.p11.11.m11.1.1.cmml" xref="S3.SS4.11.p11.11.m11.1.1"><eq id="S3.SS4.11.p11.11.m11.1.1.1.cmml" xref="S3.SS4.11.p11.11.m11.1.1.1"></eq><apply id="S3.SS4.11.p11.11.m11.1.1.2.cmml" xref="S3.SS4.11.p11.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.11.p11.11.m11.1.1.2.1.cmml" xref="S3.SS4.11.p11.11.m11.1.1.2">subscript</csymbol><ci id="S3.SS4.11.p11.11.m11.1.1.2.2.cmml" xref="S3.SS4.11.p11.11.m11.1.1.2.2">𝛽</ci><ci id="S3.SS4.11.p11.11.m11.1.1.2.3.cmml" xref="S3.SS4.11.p11.11.m11.1.1.2.3">𝑗</ci></apply><cn id="S3.SS4.11.p11.11.m11.1.1.3.cmml" type="integer" xref="S3.SS4.11.p11.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.11.p11.11.m11.1c">\beta_{j}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.11.p11.11.m11.1d">italic_β start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, a contradiction. This completes the proof of this claim. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS4.12.p12"> <p class="ltx_p" id="S3.SS4.12.p12.8">Finally, we combine these three claims, derived from the assumption that there is a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS4.12.p12.1.m1.1"><semantics id="S3.SS4.12.p12.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.12.p12.1.m1.1.1" xref="S3.SS4.12.p12.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.1.m1.1b"><ci id="S3.SS4.12.p12.1.m1.1.1.cmml" xref="S3.SS4.12.p12.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.1.m1.1d">caligraphic_F</annotation></semantics></math> above <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.12.p12.2.m2.1"><semantics id="S3.SS4.12.p12.2.m2.1a"><mi id="S3.SS4.12.p12.2.m2.1.1" xref="S3.SS4.12.p12.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.2.m2.1b"><ci id="S3.SS4.12.p12.2.m2.1.1.cmml" xref="S3.SS4.12.p12.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.2.m2.1d">italic_a</annotation></semantics></math> that is not covered by <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS4.12.p12.3.m3.1"><semantics id="S3.SS4.12.p12.3.m3.1a"><mi id="S3.SS4.12.p12.3.m3.1.1" mathvariant="normal" xref="S3.SS4.12.p12.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.3.m3.1b"><ci id="S3.SS4.12.p12.3.m3.1.1.cmml" xref="S3.SS4.12.p12.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.3.m3.1d">roman_Λ</annotation></semantics></math>, to get a contradiction. Recall that <math alttext="C^{1},\ldots,C^{m+t}=A" class="ltx_Math" display="inline" id="S3.SS4.12.p12.4.m4.3"><semantics id="S3.SS4.12.p12.4.m4.3a"><mrow id="S3.SS4.12.p12.4.m4.3.3" xref="S3.SS4.12.p12.4.m4.3.3.cmml"><mrow id="S3.SS4.12.p12.4.m4.3.3.2.2" xref="S3.SS4.12.p12.4.m4.3.3.2.3.cmml"><msup id="S3.SS4.12.p12.4.m4.2.2.1.1.1" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS4.12.p12.4.m4.2.2.1.1.1.2" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1.2.cmml">C</mi><mn id="S3.SS4.12.p12.4.m4.2.2.1.1.1.3" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1.3.cmml">1</mn></msup><mo id="S3.SS4.12.p12.4.m4.3.3.2.2.3" xref="S3.SS4.12.p12.4.m4.3.3.2.3.cmml">,</mo><mi id="S3.SS4.12.p12.4.m4.1.1" mathvariant="normal" xref="S3.SS4.12.p12.4.m4.1.1.cmml">…</mi><mo id="S3.SS4.12.p12.4.m4.3.3.2.2.4" xref="S3.SS4.12.p12.4.m4.3.3.2.3.cmml">,</mo><msup id="S3.SS4.12.p12.4.m4.3.3.2.2.2" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.cmml"><mi id="S3.SS4.12.p12.4.m4.3.3.2.2.2.2" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.2.cmml">C</mi><mrow id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.cmml"><mi id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.2" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.2.cmml">m</mi><mo id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.1" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.1.cmml">+</mo><mi id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.3" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.3.cmml">t</mi></mrow></msup></mrow><mo id="S3.SS4.12.p12.4.m4.3.3.3" xref="S3.SS4.12.p12.4.m4.3.3.3.cmml">=</mo><mi id="S3.SS4.12.p12.4.m4.3.3.4" xref="S3.SS4.12.p12.4.m4.3.3.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.4.m4.3b"><apply id="S3.SS4.12.p12.4.m4.3.3.cmml" xref="S3.SS4.12.p12.4.m4.3.3"><eq id="S3.SS4.12.p12.4.m4.3.3.3.cmml" xref="S3.SS4.12.p12.4.m4.3.3.3"></eq><list id="S3.SS4.12.p12.4.m4.3.3.2.3.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2"><apply id="S3.SS4.12.p12.4.m4.2.2.1.1.1.cmml" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.12.p12.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S3.SS4.12.p12.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1.2">𝐶</ci><cn id="S3.SS4.12.p12.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS4.12.p12.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS4.12.p12.4.m4.1.1.cmml" xref="S3.SS4.12.p12.4.m4.1.1">…</ci><apply id="S3.SS4.12.p12.4.m4.3.3.2.2.2.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.12.p12.4.m4.3.3.2.2.2.1.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2">superscript</csymbol><ci id="S3.SS4.12.p12.4.m4.3.3.2.2.2.2.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.2">𝐶</ci><apply id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3"><plus id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.1.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.1"></plus><ci id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.2.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.2">𝑚</ci><ci id="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.3.cmml" xref="S3.SS4.12.p12.4.m4.3.3.2.2.2.3.3">𝑡</ci></apply></apply></list><ci id="S3.SS4.12.p12.4.m4.3.3.4.cmml" xref="S3.SS4.12.p12.4.m4.3.3.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.4.m4.3c">C^{1},\ldots,C^{m+t}=A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.4.m4.3d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT = italic_A</annotation></semantics></math> is the convergent sequence obtained from the syntactic sequence <math alttext="I_{1},\ldots,I_{t}" class="ltx_Math" display="inline" id="S3.SS4.12.p12.5.m5.3"><semantics id="S3.SS4.12.p12.5.m5.3a"><mrow id="S3.SS4.12.p12.5.m5.3.3.2" xref="S3.SS4.12.p12.5.m5.3.3.3.cmml"><msub id="S3.SS4.12.p12.5.m5.2.2.1.1" xref="S3.SS4.12.p12.5.m5.2.2.1.1.cmml"><mi id="S3.SS4.12.p12.5.m5.2.2.1.1.2" xref="S3.SS4.12.p12.5.m5.2.2.1.1.2.cmml">I</mi><mn id="S3.SS4.12.p12.5.m5.2.2.1.1.3" xref="S3.SS4.12.p12.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS4.12.p12.5.m5.3.3.2.3" xref="S3.SS4.12.p12.5.m5.3.3.3.cmml">,</mo><mi id="S3.SS4.12.p12.5.m5.1.1" mathvariant="normal" xref="S3.SS4.12.p12.5.m5.1.1.cmml">…</mi><mo id="S3.SS4.12.p12.5.m5.3.3.2.4" xref="S3.SS4.12.p12.5.m5.3.3.3.cmml">,</mo><msub id="S3.SS4.12.p12.5.m5.3.3.2.2" xref="S3.SS4.12.p12.5.m5.3.3.2.2.cmml"><mi id="S3.SS4.12.p12.5.m5.3.3.2.2.2" xref="S3.SS4.12.p12.5.m5.3.3.2.2.2.cmml">I</mi><mi id="S3.SS4.12.p12.5.m5.3.3.2.2.3" xref="S3.SS4.12.p12.5.m5.3.3.2.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.5.m5.3b"><list id="S3.SS4.12.p12.5.m5.3.3.3.cmml" xref="S3.SS4.12.p12.5.m5.3.3.2"><apply id="S3.SS4.12.p12.5.m5.2.2.1.1.cmml" xref="S3.SS4.12.p12.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.12.p12.5.m5.2.2.1.1.1.cmml" xref="S3.SS4.12.p12.5.m5.2.2.1.1">subscript</csymbol><ci id="S3.SS4.12.p12.5.m5.2.2.1.1.2.cmml" xref="S3.SS4.12.p12.5.m5.2.2.1.1.2">𝐼</ci><cn id="S3.SS4.12.p12.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.SS4.12.p12.5.m5.2.2.1.1.3">1</cn></apply><ci id="S3.SS4.12.p12.5.m5.1.1.cmml" xref="S3.SS4.12.p12.5.m5.1.1">…</ci><apply id="S3.SS4.12.p12.5.m5.3.3.2.2.cmml" xref="S3.SS4.12.p12.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS4.12.p12.5.m5.3.3.2.2.1.cmml" xref="S3.SS4.12.p12.5.m5.3.3.2.2">subscript</csymbol><ci id="S3.SS4.12.p12.5.m5.3.3.2.2.2.cmml" xref="S3.SS4.12.p12.5.m5.3.3.2.2.2">𝐼</ci><ci id="S3.SS4.12.p12.5.m5.3.3.2.2.3.cmml" xref="S3.SS4.12.p12.5.m5.3.3.2.2.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.5.m5.3c">I_{1},\ldots,I_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.5.m5.3d">italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and its operations <math alttext="\star_{i}" class="ltx_Math" display="inline" id="S3.SS4.12.p12.6.m6.1"><semantics id="S3.SS4.12.p12.6.m6.1a"><msub id="S3.SS4.12.p12.6.m6.1.1" xref="S3.SS4.12.p12.6.m6.1.1.cmml"><mo id="S3.SS4.12.p12.6.m6.1.1.2" xref="S3.SS4.12.p12.6.m6.1.1.2.cmml">⋆</mo><mi id="S3.SS4.12.p12.6.m6.1.1.3" xref="S3.SS4.12.p12.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.6.m6.1b"><apply id="S3.SS4.12.p12.6.m6.1.1.cmml" xref="S3.SS4.12.p12.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS4.12.p12.6.m6.1.1.1.cmml" xref="S3.SS4.12.p12.6.m6.1.1">subscript</csymbol><ci id="S3.SS4.12.p12.6.m6.1.1.2.cmml" xref="S3.SS4.12.p12.6.m6.1.1.2">⋆</ci><ci id="S3.SS4.12.p12.6.m6.1.1.3.cmml" xref="S3.SS4.12.p12.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.6.m6.1c">\star_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.6.m6.1d">⋆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and that by assumption <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.SS4.12.p12.7.m7.1"><semantics id="S3.SS4.12.p12.7.m7.1a"><mrow id="S3.SS4.12.p12.7.m7.1.1" xref="S3.SS4.12.p12.7.m7.1.1.cmml"><mi id="S3.SS4.12.p12.7.m7.1.1.2" xref="S3.SS4.12.p12.7.m7.1.1.2.cmml">a</mi><mo id="S3.SS4.12.p12.7.m7.1.1.1" xref="S3.SS4.12.p12.7.m7.1.1.1.cmml">∈</mo><mi id="S3.SS4.12.p12.7.m7.1.1.3" xref="S3.SS4.12.p12.7.m7.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.7.m7.1b"><apply id="S3.SS4.12.p12.7.m7.1.1.cmml" xref="S3.SS4.12.p12.7.m7.1.1"><in id="S3.SS4.12.p12.7.m7.1.1.1.cmml" xref="S3.SS4.12.p12.7.m7.1.1.1"></in><ci id="S3.SS4.12.p12.7.m7.1.1.2.cmml" xref="S3.SS4.12.p12.7.m7.1.1.2">𝑎</ci><ci id="S3.SS4.12.p12.7.m7.1.1.3.cmml" xref="S3.SS4.12.p12.7.m7.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.7.m7.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.7.m7.1d">italic_a ∈ italic_A</annotation></semantics></math>. Therefore, our proof will be complete if we can show that <math alttext="a\notin C^{m+t}" class="ltx_Math" display="inline" id="S3.SS4.12.p12.8.m8.1"><semantics id="S3.SS4.12.p12.8.m8.1a"><mrow id="S3.SS4.12.p12.8.m8.1.1" xref="S3.SS4.12.p12.8.m8.1.1.cmml"><mi id="S3.SS4.12.p12.8.m8.1.1.2" xref="S3.SS4.12.p12.8.m8.1.1.2.cmml">a</mi><mo id="S3.SS4.12.p12.8.m8.1.1.1" xref="S3.SS4.12.p12.8.m8.1.1.1.cmml">∉</mo><msup id="S3.SS4.12.p12.8.m8.1.1.3" xref="S3.SS4.12.p12.8.m8.1.1.3.cmml"><mi id="S3.SS4.12.p12.8.m8.1.1.3.2" xref="S3.SS4.12.p12.8.m8.1.1.3.2.cmml">C</mi><mrow id="S3.SS4.12.p12.8.m8.1.1.3.3" xref="S3.SS4.12.p12.8.m8.1.1.3.3.cmml"><mi id="S3.SS4.12.p12.8.m8.1.1.3.3.2" xref="S3.SS4.12.p12.8.m8.1.1.3.3.2.cmml">m</mi><mo id="S3.SS4.12.p12.8.m8.1.1.3.3.1" xref="S3.SS4.12.p12.8.m8.1.1.3.3.1.cmml">+</mo><mi id="S3.SS4.12.p12.8.m8.1.1.3.3.3" xref="S3.SS4.12.p12.8.m8.1.1.3.3.3.cmml">t</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.12.p12.8.m8.1b"><apply id="S3.SS4.12.p12.8.m8.1.1.cmml" xref="S3.SS4.12.p12.8.m8.1.1"><notin id="S3.SS4.12.p12.8.m8.1.1.1.cmml" xref="S3.SS4.12.p12.8.m8.1.1.1"></notin><ci id="S3.SS4.12.p12.8.m8.1.1.2.cmml" xref="S3.SS4.12.p12.8.m8.1.1.2">𝑎</ci><apply id="S3.SS4.12.p12.8.m8.1.1.3.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.12.p12.8.m8.1.1.3.1.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3">superscript</csymbol><ci id="S3.SS4.12.p12.8.m8.1.1.3.2.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3.2">𝐶</ci><apply id="S3.SS4.12.p12.8.m8.1.1.3.3.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3.3"><plus id="S3.SS4.12.p12.8.m8.1.1.3.3.1.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3.3.1"></plus><ci id="S3.SS4.12.p12.8.m8.1.1.3.3.2.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3.3.2">𝑚</ci><ci id="S3.SS4.12.p12.8.m8.1.1.3.3.3.cmml" xref="S3.SS4.12.p12.8.m8.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.12.p12.8.m8.1c">a\notin C^{m+t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.12.p12.8.m8.1d">italic_a ∉ italic_C start_POSTSUPERSCRIPT italic_m + italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.13.p13"> <p class="ltx_p" id="S3.SS4.13.p13.19">In order to establish this final implication, we show the stronger statement that the element <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.13.p13.1.m1.1"><semantics id="S3.SS4.13.p13.1.m1.1a"><mi id="S3.SS4.13.p13.1.m1.1.1" xref="S3.SS4.13.p13.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.1.m1.1b"><ci id="S3.SS4.13.p13.1.m1.1.1.cmml" xref="S3.SS4.13.p13.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.1.m1.1d">italic_a</annotation></semantics></math> is never added to a set <math alttext="C^{j}" class="ltx_Math" display="inline" id="S3.SS4.13.p13.2.m2.1"><semantics id="S3.SS4.13.p13.2.m2.1a"><msup id="S3.SS4.13.p13.2.m2.1.1" xref="S3.SS4.13.p13.2.m2.1.1.cmml"><mi id="S3.SS4.13.p13.2.m2.1.1.2" xref="S3.SS4.13.p13.2.m2.1.1.2.cmml">C</mi><mi id="S3.SS4.13.p13.2.m2.1.1.3" xref="S3.SS4.13.p13.2.m2.1.1.3.cmml">j</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.2.m2.1b"><apply id="S3.SS4.13.p13.2.m2.1.1.cmml" xref="S3.SS4.13.p13.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.2.m2.1.1.1.cmml" xref="S3.SS4.13.p13.2.m2.1.1">superscript</csymbol><ci id="S3.SS4.13.p13.2.m2.1.1.2.cmml" xref="S3.SS4.13.p13.2.m2.1.1.2">𝐶</ci><ci id="S3.SS4.13.p13.2.m2.1.1.3.cmml" xref="S3.SS4.13.p13.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.2.m2.1c">C^{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.2.m2.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math> during the update steps of the evaluation procedure if <math alttext="j\in X" class="ltx_Math" display="inline" id="S3.SS4.13.p13.3.m3.1"><semantics id="S3.SS4.13.p13.3.m3.1a"><mrow id="S3.SS4.13.p13.3.m3.1.1" xref="S3.SS4.13.p13.3.m3.1.1.cmml"><mi id="S3.SS4.13.p13.3.m3.1.1.2" xref="S3.SS4.13.p13.3.m3.1.1.2.cmml">j</mi><mo id="S3.SS4.13.p13.3.m3.1.1.1" xref="S3.SS4.13.p13.3.m3.1.1.1.cmml">∈</mo><mi id="S3.SS4.13.p13.3.m3.1.1.3" xref="S3.SS4.13.p13.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.3.m3.1b"><apply id="S3.SS4.13.p13.3.m3.1.1.cmml" xref="S3.SS4.13.p13.3.m3.1.1"><in id="S3.SS4.13.p13.3.m3.1.1.1.cmml" xref="S3.SS4.13.p13.3.m3.1.1.1"></in><ci id="S3.SS4.13.p13.3.m3.1.1.2.cmml" xref="S3.SS4.13.p13.3.m3.1.1.2">𝑗</ci><ci id="S3.SS4.13.p13.3.m3.1.1.3.cmml" xref="S3.SS4.13.p13.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.3.m3.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.3.m3.1d">italic_j ∈ italic_X</annotation></semantics></math> (since <math alttext="m+t\in X" class="ltx_Math" display="inline" id="S3.SS4.13.p13.4.m4.1"><semantics id="S3.SS4.13.p13.4.m4.1a"><mrow id="S3.SS4.13.p13.4.m4.1.1" xref="S3.SS4.13.p13.4.m4.1.1.cmml"><mrow id="S3.SS4.13.p13.4.m4.1.1.2" xref="S3.SS4.13.p13.4.m4.1.1.2.cmml"><mi id="S3.SS4.13.p13.4.m4.1.1.2.2" xref="S3.SS4.13.p13.4.m4.1.1.2.2.cmml">m</mi><mo id="S3.SS4.13.p13.4.m4.1.1.2.1" xref="S3.SS4.13.p13.4.m4.1.1.2.1.cmml">+</mo><mi id="S3.SS4.13.p13.4.m4.1.1.2.3" xref="S3.SS4.13.p13.4.m4.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS4.13.p13.4.m4.1.1.1" xref="S3.SS4.13.p13.4.m4.1.1.1.cmml">∈</mo><mi id="S3.SS4.13.p13.4.m4.1.1.3" xref="S3.SS4.13.p13.4.m4.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.4.m4.1b"><apply id="S3.SS4.13.p13.4.m4.1.1.cmml" xref="S3.SS4.13.p13.4.m4.1.1"><in id="S3.SS4.13.p13.4.m4.1.1.1.cmml" xref="S3.SS4.13.p13.4.m4.1.1.1"></in><apply id="S3.SS4.13.p13.4.m4.1.1.2.cmml" xref="S3.SS4.13.p13.4.m4.1.1.2"><plus id="S3.SS4.13.p13.4.m4.1.1.2.1.cmml" xref="S3.SS4.13.p13.4.m4.1.1.2.1"></plus><ci id="S3.SS4.13.p13.4.m4.1.1.2.2.cmml" xref="S3.SS4.13.p13.4.m4.1.1.2.2">𝑚</ci><ci id="S3.SS4.13.p13.4.m4.1.1.2.3.cmml" xref="S3.SS4.13.p13.4.m4.1.1.2.3">𝑡</ci></apply><ci id="S3.SS4.13.p13.4.m4.1.1.3.cmml" xref="S3.SS4.13.p13.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.4.m4.1c">m+t\in X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.4.m4.1d">italic_m + italic_t ∈ italic_X</annotation></semantics></math> by Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem32" title="Claim 32. ‣ Proof. ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">32</span></a>), which is a contradiction. Before the first update, each such set is empty, as the only non-empty sets are in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.SS4.13.p13.5.m5.1"><semantics id="S3.SS4.13.p13.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.13.p13.5.m5.1.1" xref="S3.SS4.13.p13.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.5.m5.1b"><ci id="S3.SS4.13.p13.5.m5.1.1.cmml" xref="S3.SS4.13.p13.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.5.m5.1d">caligraphic_B</annotation></semantics></math>, and these have indices in <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS4.13.p13.6.m6.1"><semantics id="S3.SS4.13.p13.6.m6.1a"><mi id="S3.SS4.13.p13.6.m6.1.1" xref="S3.SS4.13.p13.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.6.m6.1b"><ci id="S3.SS4.13.p13.6.m6.1.1.cmml" xref="S3.SS4.13.p13.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.6.m6.1d">italic_Y</annotation></semantics></math> (Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem32" title="Claim 32. ‣ Proof. ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">32</span></a>). During an update of the elements of a set <math alttext="C^{j}" class="ltx_Math" display="inline" id="S3.SS4.13.p13.7.m7.1"><semantics id="S3.SS4.13.p13.7.m7.1a"><msup id="S3.SS4.13.p13.7.m7.1.1" xref="S3.SS4.13.p13.7.m7.1.1.cmml"><mi id="S3.SS4.13.p13.7.m7.1.1.2" xref="S3.SS4.13.p13.7.m7.1.1.2.cmml">C</mi><mi id="S3.SS4.13.p13.7.m7.1.1.3" xref="S3.SS4.13.p13.7.m7.1.1.3.cmml">j</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.7.m7.1b"><apply id="S3.SS4.13.p13.7.m7.1.1.cmml" xref="S3.SS4.13.p13.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.7.m7.1.1.1.cmml" xref="S3.SS4.13.p13.7.m7.1.1">superscript</csymbol><ci id="S3.SS4.13.p13.7.m7.1.1.2.cmml" xref="S3.SS4.13.p13.7.m7.1.1.2">𝐶</ci><ci id="S3.SS4.13.p13.7.m7.1.1.3.cmml" xref="S3.SS4.13.p13.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.7.m7.1c">C^{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.7.m7.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="j\in X" class="ltx_Math" display="inline" id="S3.SS4.13.p13.8.m8.1"><semantics id="S3.SS4.13.p13.8.m8.1a"><mrow id="S3.SS4.13.p13.8.m8.1.1" xref="S3.SS4.13.p13.8.m8.1.1.cmml"><mi id="S3.SS4.13.p13.8.m8.1.1.2" xref="S3.SS4.13.p13.8.m8.1.1.2.cmml">j</mi><mo id="S3.SS4.13.p13.8.m8.1.1.1" xref="S3.SS4.13.p13.8.m8.1.1.1.cmml">∈</mo><mi id="S3.SS4.13.p13.8.m8.1.1.3" xref="S3.SS4.13.p13.8.m8.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.8.m8.1b"><apply id="S3.SS4.13.p13.8.m8.1.1.cmml" xref="S3.SS4.13.p13.8.m8.1.1"><in id="S3.SS4.13.p13.8.m8.1.1.1.cmml" xref="S3.SS4.13.p13.8.m8.1.1.1"></in><ci id="S3.SS4.13.p13.8.m8.1.1.2.cmml" xref="S3.SS4.13.p13.8.m8.1.1.2">𝑗</ci><ci id="S3.SS4.13.p13.8.m8.1.1.3.cmml" xref="S3.SS4.13.p13.8.m8.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.8.m8.1c">j\in X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.8.m8.1d">italic_j ∈ italic_X</annotation></semantics></math>, we consider two cases based on <math alttext="\diamond_{j}\in\{\cup,\cap\}" class="ltx_math_unparsed" display="inline" id="S3.SS4.13.p13.9.m9.1"><semantics id="S3.SS4.13.p13.9.m9.1a"><mrow id="S3.SS4.13.p13.9.m9.1b"><msub id="S3.SS4.13.p13.9.m9.1.2"><mo id="S3.SS4.13.p13.9.m9.1.2.2">⋄</mo><mi id="S3.SS4.13.p13.9.m9.1.2.3">j</mi></msub><mo id="S3.SS4.13.p13.9.m9.1.1" lspace="0em">∈</mo><mrow id="S3.SS4.13.p13.9.m9.1.3"><mo id="S3.SS4.13.p13.9.m9.1.3.1" stretchy="false">{</mo><mo id="S3.SS4.13.p13.9.m9.1.3.2" lspace="0em" rspace="0em">∪</mo><mo id="S3.SS4.13.p13.9.m9.1.3.3" rspace="0em">,</mo><mo id="S3.SS4.13.p13.9.m9.1.3.4" lspace="0em" rspace="0em">∩</mo><mo id="S3.SS4.13.p13.9.m9.1.3.5" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS4.13.p13.9.m9.1c">\diamond_{j}\in\{\cup,\cap\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.9.m9.1d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ { ∪ , ∩ }</annotation></semantics></math>. If <math alttext="\diamond_{j}=\cap" class="ltx_Math" display="inline" id="S3.SS4.13.p13.10.m10.3"><semantics id="S3.SS4.13.p13.10.m10.3a"><mrow id="S3.SS4.13.p13.10.m10.3.3.1" xref="S3.SS4.13.p13.10.m10.3.3.2.cmml"><msub id="S3.SS4.13.p13.10.m10.3.3.1.1" xref="S3.SS4.13.p13.10.m10.3.3.1.1.cmml"><mo id="S3.SS4.13.p13.10.m10.3.3.1.1.2" xref="S3.SS4.13.p13.10.m10.3.3.1.1.2.cmml">⋄</mo><mi id="S3.SS4.13.p13.10.m10.3.3.1.1.3" xref="S3.SS4.13.p13.10.m10.3.3.1.1.3.cmml">j</mi></msub><mo id="S3.SS4.13.p13.10.m10.3.3.1.2" lspace="0em" xref="S3.SS4.13.p13.10.m10.3.3.2.cmml">⁣</mo><mo id="S3.SS4.13.p13.10.m10.1.1" xref="S3.SS4.13.p13.10.m10.1.1.cmml">=</mo><mo id="S3.SS4.13.p13.10.m10.3.3.1.3" lspace="0em" xref="S3.SS4.13.p13.10.m10.3.3.2.cmml">⁣</mo><mo id="S3.SS4.13.p13.10.m10.2.2" xref="S3.SS4.13.p13.10.m10.2.2.cmml">∩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.10.m10.3b"><list id="S3.SS4.13.p13.10.m10.3.3.2.cmml" xref="S3.SS4.13.p13.10.m10.3.3.1"><apply id="S3.SS4.13.p13.10.m10.3.3.1.1.cmml" xref="S3.SS4.13.p13.10.m10.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.10.m10.3.3.1.1.1.cmml" xref="S3.SS4.13.p13.10.m10.3.3.1.1">subscript</csymbol><ci id="S3.SS4.13.p13.10.m10.3.3.1.1.2.cmml" xref="S3.SS4.13.p13.10.m10.3.3.1.1.2">⋄</ci><ci id="S3.SS4.13.p13.10.m10.3.3.1.1.3.cmml" xref="S3.SS4.13.p13.10.m10.3.3.1.1.3">𝑗</ci></apply><eq id="S3.SS4.13.p13.10.m10.1.1.cmml" xref="S3.SS4.13.p13.10.m10.1.1"></eq><intersect id="S3.SS4.13.p13.10.m10.2.2.cmml" xref="S3.SS4.13.p13.10.m10.2.2"></intersect></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.10.m10.3c">\diamond_{j}=\cap</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.10.m10.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∩</annotation></semantics></math>, Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem33" title="Claim 33. ‣ Proof. ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">33</span></a> implies that at least one of the operands comes from <math alttext="X" class="ltx_Math" display="inline" id="S3.SS4.13.p13.11.m11.1"><semantics id="S3.SS4.13.p13.11.m11.1a"><mi id="S3.SS4.13.p13.11.m11.1.1" xref="S3.SS4.13.p13.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.11.m11.1b"><ci id="S3.SS4.13.p13.11.m11.1.1.cmml" xref="S3.SS4.13.p13.11.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.11.m11.1d">italic_X</annotation></semantics></math>, and thus by induction the update step will not include <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.13.p13.12.m12.1"><semantics id="S3.SS4.13.p13.12.m12.1a"><mi id="S3.SS4.13.p13.12.m12.1.1" xref="S3.SS4.13.p13.12.m12.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.12.m12.1b"><ci id="S3.SS4.13.p13.12.m12.1.1.cmml" xref="S3.SS4.13.p13.12.m12.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.12.m12.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.12.m12.1d">italic_a</annotation></semantics></math> in <math alttext="C^{j}" class="ltx_Math" display="inline" id="S3.SS4.13.p13.13.m13.1"><semantics id="S3.SS4.13.p13.13.m13.1a"><msup id="S3.SS4.13.p13.13.m13.1.1" xref="S3.SS4.13.p13.13.m13.1.1.cmml"><mi id="S3.SS4.13.p13.13.m13.1.1.2" xref="S3.SS4.13.p13.13.m13.1.1.2.cmml">C</mi><mi id="S3.SS4.13.p13.13.m13.1.1.3" xref="S3.SS4.13.p13.13.m13.1.1.3.cmml">j</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.13.m13.1b"><apply id="S3.SS4.13.p13.13.m13.1.1.cmml" xref="S3.SS4.13.p13.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.13.m13.1.1.1.cmml" xref="S3.SS4.13.p13.13.m13.1.1">superscript</csymbol><ci id="S3.SS4.13.p13.13.m13.1.1.2.cmml" xref="S3.SS4.13.p13.13.m13.1.1.2">𝐶</ci><ci id="S3.SS4.13.p13.13.m13.1.1.3.cmml" xref="S3.SS4.13.p13.13.m13.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.13.m13.1c">C^{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.13.m13.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>. On the other hand, if <math alttext="\diamond_{j}=\cup" class="ltx_Math" display="inline" id="S3.SS4.13.p13.14.m14.3"><semantics id="S3.SS4.13.p13.14.m14.3a"><mrow id="S3.SS4.13.p13.14.m14.3.3.1" xref="S3.SS4.13.p13.14.m14.3.3.2.cmml"><msub id="S3.SS4.13.p13.14.m14.3.3.1.1" xref="S3.SS4.13.p13.14.m14.3.3.1.1.cmml"><mo id="S3.SS4.13.p13.14.m14.3.3.1.1.2" xref="S3.SS4.13.p13.14.m14.3.3.1.1.2.cmml">⋄</mo><mi id="S3.SS4.13.p13.14.m14.3.3.1.1.3" xref="S3.SS4.13.p13.14.m14.3.3.1.1.3.cmml">j</mi></msub><mo id="S3.SS4.13.p13.14.m14.3.3.1.2" lspace="0em" xref="S3.SS4.13.p13.14.m14.3.3.2.cmml">⁣</mo><mo id="S3.SS4.13.p13.14.m14.1.1" xref="S3.SS4.13.p13.14.m14.1.1.cmml">=</mo><mo id="S3.SS4.13.p13.14.m14.3.3.1.3" lspace="0em" xref="S3.SS4.13.p13.14.m14.3.3.2.cmml">⁣</mo><mo id="S3.SS4.13.p13.14.m14.2.2" xref="S3.SS4.13.p13.14.m14.2.2.cmml">∪</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.14.m14.3b"><list id="S3.SS4.13.p13.14.m14.3.3.2.cmml" xref="S3.SS4.13.p13.14.m14.3.3.1"><apply id="S3.SS4.13.p13.14.m14.3.3.1.1.cmml" xref="S3.SS4.13.p13.14.m14.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.14.m14.3.3.1.1.1.cmml" xref="S3.SS4.13.p13.14.m14.3.3.1.1">subscript</csymbol><ci id="S3.SS4.13.p13.14.m14.3.3.1.1.2.cmml" xref="S3.SS4.13.p13.14.m14.3.3.1.1.2">⋄</ci><ci id="S3.SS4.13.p13.14.m14.3.3.1.1.3.cmml" xref="S3.SS4.13.p13.14.m14.3.3.1.1.3">𝑗</ci></apply><eq id="S3.SS4.13.p13.14.m14.1.1.cmml" xref="S3.SS4.13.p13.14.m14.1.1"></eq><union id="S3.SS4.13.p13.14.m14.2.2.cmml" xref="S3.SS4.13.p13.14.m14.2.2"></union></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.14.m14.3c">\diamond_{j}=\cup</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.14.m14.3d">⋄ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∪</annotation></semantics></math>, Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem33" title="Claim 33. ‣ Proof. ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">33</span></a> shows that at most one operand comes from <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS4.13.p13.15.m15.1"><semantics id="S3.SS4.13.p13.15.m15.1a"><mi id="S3.SS4.13.p13.15.m15.1.1" xref="S3.SS4.13.p13.15.m15.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.15.m15.1b"><ci id="S3.SS4.13.p13.15.m15.1.1.cmml" xref="S3.SS4.13.p13.15.m15.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.15.m15.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.15.m15.1d">italic_Y</annotation></semantics></math>. If there is no operand from <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS4.13.p13.16.m16.1"><semantics id="S3.SS4.13.p13.16.m16.1a"><mi id="S3.SS4.13.p13.16.m16.1.1" xref="S3.SS4.13.p13.16.m16.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.16.m16.1b"><ci id="S3.SS4.13.p13.16.m16.1.1.cmml" xref="S3.SS4.13.p13.16.m16.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.16.m16.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.16.m16.1d">italic_Y</annotation></semantics></math>, we are done using the induction hypothesis. Otherwise, Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem34" title="Claim 34. ‣ Proof. ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">34</span></a> implies that <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.13.p13.17.m17.1"><semantics id="S3.SS4.13.p13.17.m17.1a"><mi id="S3.SS4.13.p13.17.m17.1.1" xref="S3.SS4.13.p13.17.m17.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.17.m17.1b"><ci id="S3.SS4.13.p13.17.m17.1.1.cmml" xref="S3.SS4.13.p13.17.m17.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.17.m17.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.17.m17.1d">italic_a</annotation></semantics></math> is not an element of this operand (as it is not in the corresponding set even after the evaluation procedure converges). By the induction hypothesis, <math alttext="a" class="ltx_Math" display="inline" id="S3.SS4.13.p13.18.m18.1"><semantics id="S3.SS4.13.p13.18.m18.1a"><mi id="S3.SS4.13.p13.18.m18.1.1" xref="S3.SS4.13.p13.18.m18.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.18.m18.1b"><ci id="S3.SS4.13.p13.18.m18.1.1.cmml" xref="S3.SS4.13.p13.18.m18.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.18.m18.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.18.m18.1d">italic_a</annotation></semantics></math> is not added to <math alttext="C^{j}" class="ltx_Math" display="inline" id="S3.SS4.13.p13.19.m19.1"><semantics id="S3.SS4.13.p13.19.m19.1a"><msup id="S3.SS4.13.p13.19.m19.1.1" xref="S3.SS4.13.p13.19.m19.1.1.cmml"><mi id="S3.SS4.13.p13.19.m19.1.1.2" xref="S3.SS4.13.p13.19.m19.1.1.2.cmml">C</mi><mi id="S3.SS4.13.p13.19.m19.1.1.3" xref="S3.SS4.13.p13.19.m19.1.1.3.cmml">j</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.13.p13.19.m19.1b"><apply id="S3.SS4.13.p13.19.m19.1.1.cmml" xref="S3.SS4.13.p13.19.m19.1.1"><csymbol cd="ambiguous" id="S3.SS4.13.p13.19.m19.1.1.1.cmml" xref="S3.SS4.13.p13.19.m19.1.1">superscript</csymbol><ci id="S3.SS4.13.p13.19.m19.1.1.2.cmml" xref="S3.SS4.13.p13.19.m19.1.1.2">𝐶</ci><ci id="S3.SS4.13.p13.19.m19.1.1.3.cmml" xref="S3.SS4.13.p13.19.m19.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.13.p13.19.m19.1c">C^{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.13.p13.19.m19.1d">italic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>. This finishes the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem30" title="Theorem 30 (Exact characterization of cover complexity). ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">30</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS4.p2"> <p class="ltx_p" id="S3.SS4.p2.5">In particular, this result shows that the <math alttext="k" class="ltx_Math" display="inline" id="S3.SS4.p2.1.m1.1"><semantics id="S3.SS4.p2.1.m1.1a"><mi id="S3.SS4.p2.1.m1.1.1" xref="S3.SS4.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.1.m1.1b"><ci id="S3.SS4.p2.1.m1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.1.m1.1d">italic_k</annotation></semantics></math>-clique lower bound discussed in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite> holds in the more general model of cyclic Boolean circuits. Indeed, Karchmer shows a lower bound for <math alttext="\rho(A,{\mathcal{B}})" class="ltx_Math" display="inline" id="S3.SS4.p2.2.m2.2"><semantics id="S3.SS4.p2.2.m2.2a"><mrow id="S3.SS4.p2.2.m2.2.3" xref="S3.SS4.p2.2.m2.2.3.cmml"><mi id="S3.SS4.p2.2.m2.2.3.2" xref="S3.SS4.p2.2.m2.2.3.2.cmml">ρ</mi><mo id="S3.SS4.p2.2.m2.2.3.1" xref="S3.SS4.p2.2.m2.2.3.1.cmml">⁢</mo><mrow id="S3.SS4.p2.2.m2.2.3.3.2" xref="S3.SS4.p2.2.m2.2.3.3.1.cmml"><mo id="S3.SS4.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.SS4.p2.2.m2.2.3.3.1.cmml">(</mo><mi id="S3.SS4.p2.2.m2.1.1" xref="S3.SS4.p2.2.m2.1.1.cmml">A</mi><mo id="S3.SS4.p2.2.m2.2.3.3.2.2" xref="S3.SS4.p2.2.m2.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.2.m2.2.2" xref="S3.SS4.p2.2.m2.2.2.cmml">ℬ</mi><mo id="S3.SS4.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.SS4.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.2.m2.2b"><apply id="S3.SS4.p2.2.m2.2.3.cmml" xref="S3.SS4.p2.2.m2.2.3"><times id="S3.SS4.p2.2.m2.2.3.1.cmml" xref="S3.SS4.p2.2.m2.2.3.1"></times><ci id="S3.SS4.p2.2.m2.2.3.2.cmml" xref="S3.SS4.p2.2.m2.2.3.2">𝜌</ci><interval closure="open" id="S3.SS4.p2.2.m2.2.3.3.1.cmml" xref="S3.SS4.p2.2.m2.2.3.3.2"><ci id="S3.SS4.p2.2.m2.1.1.cmml" xref="S3.SS4.p2.2.m2.1.1">𝐴</ci><ci id="S3.SS4.p2.2.m2.2.2.cmml" xref="S3.SS4.p2.2.m2.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.2.m2.2c">\rho(A,{\mathcal{B}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.2.m2.2d">italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math>, where <math alttext="A" class="ltx_Math" display="inline" id="S3.SS4.p2.3.m3.1"><semantics id="S3.SS4.p2.3.m3.1a"><mi id="S3.SS4.p2.3.m3.1.1" xref="S3.SS4.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.3.m3.1b"><ci id="S3.SS4.p2.3.m3.1.1.cmml" xref="S3.SS4.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.3.m3.1d">italic_A</annotation></semantics></math> is the set of graphs with <math alttext="k" class="ltx_Math" display="inline" id="S3.SS4.p2.4.m4.1"><semantics id="S3.SS4.p2.4.m4.1a"><mi id="S3.SS4.p2.4.m4.1.1" xref="S3.SS4.p2.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.4.m4.1b"><ci id="S3.SS4.p2.4.m4.1.1.cmml" xref="S3.SS4.p2.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.4.m4.1d">italic_k</annotation></semantics></math>-cliques and <math alttext="{\mathcal{B}}" class="ltx_Math" display="inline" id="S3.SS4.p2.5.m5.1"><semantics id="S3.SS4.p2.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.5.m5.1.1" xref="S3.SS4.p2.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.5.m5.1b"><ci id="S3.SS4.p2.5.m5.1.1.cmml" xref="S3.SS4.p2.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.5.m5.1c">{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.5.m5.1d">caligraphic_B</annotation></semantics></math> is the monotone Boolean basis. Combined with the previous result, this gives the following corollary.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmtheorem35"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem35.1.1.1">Corollary 35</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem35.2.2"> </span>(Consequence of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem30" title="Theorem 30 (Exact characterization of cover complexity). ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">30</span></a> and <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite>)<span class="ltx_text ltx_font_bold" id="Thmtheorem35.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem35.p1"> <p class="ltx_p" id="Thmtheorem35.p1.10"><span class="ltx_text ltx_font_italic" id="Thmtheorem35.p1.10.10">Let <math alttext="k" class="ltx_Math" display="inline" id="Thmtheorem35.p1.1.1.m1.1"><semantics id="Thmtheorem35.p1.1.1.m1.1a"><mi id="Thmtheorem35.p1.1.1.m1.1.1" xref="Thmtheorem35.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.1.1.m1.1b"><ci id="Thmtheorem35.p1.1.1.m1.1.1.cmml" xref="Thmtheorem35.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-<math alttext="\mathsf{clique}\colon\{0,1\}^{\binom{n}{2}}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem35.p1.2.2.m2.6"><semantics id="Thmtheorem35.p1.2.2.m2.6a"><mrow id="Thmtheorem35.p1.2.2.m2.6.7" xref="Thmtheorem35.p1.2.2.m2.6.7.cmml"><mi id="Thmtheorem35.p1.2.2.m2.6.7.2" xref="Thmtheorem35.p1.2.2.m2.6.7.2.cmml">𝖼𝗅𝗂𝗊𝗎𝖾</mi><mo id="Thmtheorem35.p1.2.2.m2.6.7.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem35.p1.2.2.m2.6.7.1.cmml">:</mo><mrow id="Thmtheorem35.p1.2.2.m2.6.7.3" xref="Thmtheorem35.p1.2.2.m2.6.7.3.cmml"><msup id="Thmtheorem35.p1.2.2.m2.6.7.3.2" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.cmml"><mrow id="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.2" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.1.cmml"><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.2.1" stretchy="false" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.1.cmml">{</mo><mn id="Thmtheorem35.p1.2.2.m2.3.3" xref="Thmtheorem35.p1.2.2.m2.3.3.cmml">0</mn><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.2.2" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.1.cmml">,</mo><mn id="Thmtheorem35.p1.2.2.m2.4.4" xref="Thmtheorem35.p1.2.2.m2.4.4.cmml">1</mn><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.2.3" stretchy="false" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem35.p1.2.2.m2.2.2.2.4" xref="Thmtheorem35.p1.2.2.m2.2.2.2.3.cmml"><mo id="Thmtheorem35.p1.2.2.m2.2.2.2.4.1" xref="Thmtheorem35.p1.2.2.m2.2.2.2.3.1.cmml">(</mo><mfrac id="Thmtheorem35.p1.2.2.m2.2.2.2.2.2.2" linethickness="0pt" xref="Thmtheorem35.p1.2.2.m2.2.2.2.3.cmml"><mi id="Thmtheorem35.p1.2.2.m2.1.1.1.1.1.1.1.1" xref="Thmtheorem35.p1.2.2.m2.1.1.1.1.1.1.1.1.cmml">n</mi><mn id="Thmtheorem35.p1.2.2.m2.2.2.2.2.2.2.2.1" xref="Thmtheorem35.p1.2.2.m2.2.2.2.2.2.2.2.1.cmml">2</mn></mfrac><mo id="Thmtheorem35.p1.2.2.m2.2.2.2.4.2" xref="Thmtheorem35.p1.2.2.m2.2.2.2.3.1.cmml">)</mo></mrow></msup><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.1" stretchy="false" xref="Thmtheorem35.p1.2.2.m2.6.7.3.1.cmml">→</mo><mrow id="Thmtheorem35.p1.2.2.m2.6.7.3.3.2" xref="Thmtheorem35.p1.2.2.m2.6.7.3.3.1.cmml"><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.3.2.1" stretchy="false" xref="Thmtheorem35.p1.2.2.m2.6.7.3.3.1.cmml">{</mo><mn id="Thmtheorem35.p1.2.2.m2.5.5" xref="Thmtheorem35.p1.2.2.m2.5.5.cmml">0</mn><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.3.2.2" xref="Thmtheorem35.p1.2.2.m2.6.7.3.3.1.cmml">,</mo><mn id="Thmtheorem35.p1.2.2.m2.6.6" xref="Thmtheorem35.p1.2.2.m2.6.6.cmml">1</mn><mo id="Thmtheorem35.p1.2.2.m2.6.7.3.3.2.3" stretchy="false" xref="Thmtheorem35.p1.2.2.m2.6.7.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.2.2.m2.6b"><apply id="Thmtheorem35.p1.2.2.m2.6.7.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7"><ci id="Thmtheorem35.p1.2.2.m2.6.7.1.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.1">:</ci><ci id="Thmtheorem35.p1.2.2.m2.6.7.2.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.2">𝖼𝗅𝗂𝗊𝗎𝖾</ci><apply id="Thmtheorem35.p1.2.2.m2.6.7.3.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3"><ci id="Thmtheorem35.p1.2.2.m2.6.7.3.1.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3.1">→</ci><apply id="Thmtheorem35.p1.2.2.m2.6.7.3.2.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2"><csymbol cd="ambiguous" id="Thmtheorem35.p1.2.2.m2.6.7.3.2.1.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2">superscript</csymbol><set id="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.1.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3.2.2.2"><cn id="Thmtheorem35.p1.2.2.m2.3.3.cmml" type="integer" xref="Thmtheorem35.p1.2.2.m2.3.3">0</cn><cn id="Thmtheorem35.p1.2.2.m2.4.4.cmml" type="integer" xref="Thmtheorem35.p1.2.2.m2.4.4">1</cn></set><apply id="Thmtheorem35.p1.2.2.m2.2.2.2.3.cmml" xref="Thmtheorem35.p1.2.2.m2.2.2.2.4"><csymbol cd="latexml" id="Thmtheorem35.p1.2.2.m2.2.2.2.3.1.cmml" xref="Thmtheorem35.p1.2.2.m2.2.2.2.4.1">binomial</csymbol><ci id="Thmtheorem35.p1.2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="Thmtheorem35.p1.2.2.m2.1.1.1.1.1.1.1.1">𝑛</ci><cn id="Thmtheorem35.p1.2.2.m2.2.2.2.2.2.2.2.1.cmml" type="integer" xref="Thmtheorem35.p1.2.2.m2.2.2.2.2.2.2.2.1">2</cn></apply></apply><set id="Thmtheorem35.p1.2.2.m2.6.7.3.3.1.cmml" xref="Thmtheorem35.p1.2.2.m2.6.7.3.3.2"><cn id="Thmtheorem35.p1.2.2.m2.5.5.cmml" type="integer" xref="Thmtheorem35.p1.2.2.m2.5.5">0</cn><cn id="Thmtheorem35.p1.2.2.m2.6.6.cmml" type="integer" xref="Thmtheorem35.p1.2.2.m2.6.6">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.2.2.m2.6c">\mathsf{clique}\colon\{0,1\}^{\binom{n}{2}}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.2.2.m2.6d">sansserif_clique : { 0 , 1 } start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_n end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> be the function that evaluates to <math alttext="1" class="ltx_Math" display="inline" id="Thmtheorem35.p1.3.3.m3.1"><semantics id="Thmtheorem35.p1.3.3.m3.1a"><mn id="Thmtheorem35.p1.3.3.m3.1.1" xref="Thmtheorem35.p1.3.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.3.3.m3.1b"><cn id="Thmtheorem35.p1.3.3.m3.1.1.cmml" type="integer" xref="Thmtheorem35.p1.3.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.3.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.3.3.m3.1d">1</annotation></semantics></math> on an undirected <math alttext="n" class="ltx_Math" display="inline" id="Thmtheorem35.p1.4.4.m4.1"><semantics id="Thmtheorem35.p1.4.4.m4.1a"><mi id="Thmtheorem35.p1.4.4.m4.1.1" xref="Thmtheorem35.p1.4.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.4.4.m4.1b"><ci id="Thmtheorem35.p1.4.4.m4.1.1.cmml" xref="Thmtheorem35.p1.4.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.4.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.4.4.m4.1d">italic_n</annotation></semantics></math>-vertex input graph <math alttext="G" class="ltx_Math" display="inline" id="Thmtheorem35.p1.5.5.m5.1"><semantics id="Thmtheorem35.p1.5.5.m5.1a"><mi id="Thmtheorem35.p1.5.5.m5.1.1" xref="Thmtheorem35.p1.5.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.5.5.m5.1b"><ci id="Thmtheorem35.p1.5.5.m5.1.1.cmml" xref="Thmtheorem35.p1.5.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.5.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.5.5.m5.1d">italic_G</annotation></semantics></math> if and only if <math alttext="G" class="ltx_Math" display="inline" id="Thmtheorem35.p1.6.6.m6.1"><semantics id="Thmtheorem35.p1.6.6.m6.1a"><mi id="Thmtheorem35.p1.6.6.m6.1.1" xref="Thmtheorem35.p1.6.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.6.6.m6.1b"><ci id="Thmtheorem35.p1.6.6.m6.1.1.cmml" xref="Thmtheorem35.p1.6.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.6.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.6.6.m6.1d">italic_G</annotation></semantics></math> contains a <math alttext="k" class="ltx_Math" display="inline" id="Thmtheorem35.p1.7.7.m7.1"><semantics id="Thmtheorem35.p1.7.7.m7.1a"><mi id="Thmtheorem35.p1.7.7.m7.1.1" xref="Thmtheorem35.p1.7.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.7.7.m7.1b"><ci id="Thmtheorem35.p1.7.7.m7.1.1.cmml" xref="Thmtheorem35.p1.7.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.7.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.7.7.m7.1d">italic_k</annotation></semantics></math>-clique. Then every monotone cyclic Boolean circuit that computes <math alttext="3" class="ltx_Math" display="inline" id="Thmtheorem35.p1.8.8.m8.1"><semantics id="Thmtheorem35.p1.8.8.m8.1a"><mn id="Thmtheorem35.p1.8.8.m8.1.1" xref="Thmtheorem35.p1.8.8.m8.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.8.8.m8.1b"><cn id="Thmtheorem35.p1.8.8.m8.1.1.cmml" type="integer" xref="Thmtheorem35.p1.8.8.m8.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.8.8.m8.1c">3</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.8.8.m8.1d">3</annotation></semantics></math>-<math alttext="\mathsf{clique}" class="ltx_Math" display="inline" id="Thmtheorem35.p1.9.9.m9.1"><semantics id="Thmtheorem35.p1.9.9.m9.1a"><mi id="Thmtheorem35.p1.9.9.m9.1.1" xref="Thmtheorem35.p1.9.9.m9.1.1.cmml">𝖼𝗅𝗂𝗊𝗎𝖾</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.9.9.m9.1b"><ci id="Thmtheorem35.p1.9.9.m9.1.1.cmml" xref="Thmtheorem35.p1.9.9.m9.1.1">𝖼𝗅𝗂𝗊𝗎𝖾</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.9.9.m9.1c">\mathsf{clique}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.9.9.m9.1d">sansserif_clique</annotation></semantics></math> contains at least <math alttext="\Omega(n^{3}/(\log n)^{4})" class="ltx_Math" display="inline" id="Thmtheorem35.p1.10.10.m10.1"><semantics id="Thmtheorem35.p1.10.10.m10.1a"><mrow id="Thmtheorem35.p1.10.10.m10.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.cmml"><mi id="Thmtheorem35.p1.10.10.m10.1.1.3" mathvariant="normal" xref="Thmtheorem35.p1.10.10.m10.1.1.3.cmml">Ω</mi><mo id="Thmtheorem35.p1.10.10.m10.1.1.2" xref="Thmtheorem35.p1.10.10.m10.1.1.2.cmml">⁢</mo><mrow id="Thmtheorem35.p1.10.10.m10.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.cmml"><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.2" stretchy="false" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.cmml"><msup id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.cmml"><mi id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.2" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.2.cmml">n</mi><mn id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.3" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.3.cmml">3</mn></msup><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.2" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.2.cmml">/</mo><msup id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.cmml"><mrow id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.1" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.1.cmml">log</mi><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1a" lspace="0.167em" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml">⁡</mo><mi id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.2" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.3" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.3.cmml">4</mn></msup></mrow><mo id="Thmtheorem35.p1.10.10.m10.1.1.1.1.3" stretchy="false" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem35.p1.10.10.m10.1b"><apply id="Thmtheorem35.p1.10.10.m10.1.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1"><times id="Thmtheorem35.p1.10.10.m10.1.1.2.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.2"></times><ci id="Thmtheorem35.p1.10.10.m10.1.1.3.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.3">Ω</ci><apply id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1"><divide id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.2.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.2"></divide><apply id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3">superscript</csymbol><ci id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.2.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.2">𝑛</ci><cn id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.3.3">3</cn></apply><apply id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.2.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1">superscript</csymbol><apply id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1"><log id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.1"></log><ci id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.1.1.1.2">𝑛</ci></apply><cn id="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmtheorem35.p1.10.10.m10.1.1.1.1.1.1.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem35.p1.10.10.m10.1c">\Omega(n^{3}/(\log n)^{4})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem35.p1.10.10.m10.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / ( roman_log italic_n ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math> fan-in two <em class="ltx_emph ltx_font_upright" id="Thmtheorem35.p1.10.10.1">AND</em> gates.</span></p> </div> </div> <div class="ltx_para" id="S3.SS4.p3"> <p class="ltx_p" id="S3.SS4.p3.1">This lower bound against monotone cyclic circuits does not seem to easily follow from the proofs in <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib1" title="">1</a>]</cite>.</p> </div> </section> </section> <section class="ltx_section ltx_indent_first" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Graph Complexity and Two-Dimensional Cover Problems</h2> <section class="ltx_subsection ltx_indent_first" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Basic results and connections</h3> <div class="ltx_theorem ltx_theorem_proposition" id="Thmtheorem36"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem36.1.1.1">Proposition 36</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem36.2.2"> </span>(The intersection complexity of a random graph)<span class="ltx_text ltx_font_bold" id="Thmtheorem36.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem36.p1"> <p class="ltx_p" id="Thmtheorem36.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem36.p1.1.1">Let <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem36.p1.1.1.m1.2"><semantics id="Thmtheorem36.p1.1.1.m1.2a"><mrow id="Thmtheorem36.p1.1.1.m1.2.3" xref="Thmtheorem36.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem36.p1.1.1.m1.2.3.2" xref="Thmtheorem36.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="Thmtheorem36.p1.1.1.m1.2.3.1" xref="Thmtheorem36.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem36.p1.1.1.m1.2.3.3" xref="Thmtheorem36.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem36.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem36.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem36.p1.1.1.m1.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem36.p1.1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem36.p1.1.1.m1.1.1" xref="Thmtheorem36.p1.1.1.m1.1.1.cmml">N</mi><mo id="Thmtheorem36.p1.1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem36.p1.1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem36.p1.1.1.m1.2.3.3.1" rspace="0.222em" xref="Thmtheorem36.p1.1.1.m1.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem36.p1.1.1.m1.2.3.3.3.2" xref="Thmtheorem36.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="Thmtheorem36.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem36.p1.1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem36.p1.1.1.m1.2.2" xref="Thmtheorem36.p1.1.1.m1.2.2.cmml">N</mi><mo id="Thmtheorem36.p1.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem36.p1.1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem36.p1.1.1.m1.2b"><apply id="Thmtheorem36.p1.1.1.m1.2.3.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3"><subset id="Thmtheorem36.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.1"></subset><ci id="Thmtheorem36.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.2">𝐺</ci><apply id="Thmtheorem36.p1.1.1.m1.2.3.3.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3"><times id="Thmtheorem36.p1.1.1.m1.2.3.3.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3.1"></times><apply id="Thmtheorem36.p1.1.1.m1.2.3.3.2.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem36.p1.1.1.m1.2.3.3.2.1.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem36.p1.1.1.m1.1.1.cmml" xref="Thmtheorem36.p1.1.1.m1.1.1">𝑁</ci></apply><apply id="Thmtheorem36.p1.1.1.m1.2.3.3.3.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem36.p1.1.1.m1.2.3.3.3.1.1.cmml" xref="Thmtheorem36.p1.1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem36.p1.1.1.m1.2.2.cmml" xref="Thmtheorem36.p1.1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem36.p1.1.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem36.p1.1.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> be a random bipartite graph. Then, asymptotically almost surely,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D_{\cap}(G\mid\mathcal{G}_{N,N})\;=\;\Theta(N)." class="ltx_Math" display="block" id="S4.Ex26.m1.4"><semantics id="S4.Ex26.m1.4a"><mrow id="S4.Ex26.m1.4.4.1" xref="S4.Ex26.m1.4.4.1.1.cmml"><mrow id="S4.Ex26.m1.4.4.1.1" xref="S4.Ex26.m1.4.4.1.1.cmml"><mrow id="S4.Ex26.m1.4.4.1.1.1" xref="S4.Ex26.m1.4.4.1.1.1.cmml"><msub id="S4.Ex26.m1.4.4.1.1.1.3" xref="S4.Ex26.m1.4.4.1.1.1.3.cmml"><mi id="S4.Ex26.m1.4.4.1.1.1.3.2" xref="S4.Ex26.m1.4.4.1.1.1.3.2.cmml">D</mi><mo id="S4.Ex26.m1.4.4.1.1.1.3.3" xref="S4.Ex26.m1.4.4.1.1.1.3.3.cmml">∩</mo></msub><mo id="S4.Ex26.m1.4.4.1.1.1.2" xref="S4.Ex26.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex26.m1.4.4.1.1.1.1.1" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S4.Ex26.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex26.m1.4.4.1.1.1.1.1.1" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S4.Ex26.m1.4.4.1.1.1.1.1.1.2" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.2.cmml">G</mi><mo id="S4.Ex26.m1.4.4.1.1.1.1.1.1.1" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.1.cmml">∣</mo><msub id="S4.Ex26.m1.4.4.1.1.1.1.1.1.3" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.2" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S4.Ex26.m1.2.2.2.4" xref="S4.Ex26.m1.2.2.2.3.cmml"><mi id="S4.Ex26.m1.1.1.1.1" xref="S4.Ex26.m1.1.1.1.1.cmml">N</mi><mo id="S4.Ex26.m1.2.2.2.4.1" xref="S4.Ex26.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex26.m1.2.2.2.2" xref="S4.Ex26.m1.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S4.Ex26.m1.4.4.1.1.1.1.1.3" rspace="0.280em" stretchy="false" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex26.m1.4.4.1.1.2" rspace="0.558em" xref="S4.Ex26.m1.4.4.1.1.2.cmml">=</mo><mrow id="S4.Ex26.m1.4.4.1.1.3" xref="S4.Ex26.m1.4.4.1.1.3.cmml"><mi id="S4.Ex26.m1.4.4.1.1.3.2" mathvariant="normal" xref="S4.Ex26.m1.4.4.1.1.3.2.cmml">Θ</mi><mo id="S4.Ex26.m1.4.4.1.1.3.1" xref="S4.Ex26.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S4.Ex26.m1.4.4.1.1.3.3.2" xref="S4.Ex26.m1.4.4.1.1.3.cmml"><mo id="S4.Ex26.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S4.Ex26.m1.4.4.1.1.3.cmml">(</mo><mi id="S4.Ex26.m1.3.3" xref="S4.Ex26.m1.3.3.cmml">N</mi><mo id="S4.Ex26.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S4.Ex26.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex26.m1.4.4.1.2" lspace="0em" xref="S4.Ex26.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex26.m1.4b"><apply id="S4.Ex26.m1.4.4.1.1.cmml" xref="S4.Ex26.m1.4.4.1"><eq id="S4.Ex26.m1.4.4.1.1.2.cmml" xref="S4.Ex26.m1.4.4.1.1.2"></eq><apply id="S4.Ex26.m1.4.4.1.1.1.cmml" xref="S4.Ex26.m1.4.4.1.1.1"><times id="S4.Ex26.m1.4.4.1.1.1.2.cmml" xref="S4.Ex26.m1.4.4.1.1.1.2"></times><apply id="S4.Ex26.m1.4.4.1.1.1.3.cmml" xref="S4.Ex26.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex26.m1.4.4.1.1.1.3.1.cmml" xref="S4.Ex26.m1.4.4.1.1.1.3">subscript</csymbol><ci id="S4.Ex26.m1.4.4.1.1.1.3.2.cmml" xref="S4.Ex26.m1.4.4.1.1.1.3.2">𝐷</ci><intersect id="S4.Ex26.m1.4.4.1.1.1.3.3.cmml" xref="S4.Ex26.m1.4.4.1.1.1.3.3"></intersect></apply><apply id="S4.Ex26.m1.4.4.1.1.1.1.1.1.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1"><csymbol cd="latexml" id="S4.Ex26.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.1">conditional</csymbol><ci id="S4.Ex26.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.2">𝐺</ci><apply id="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex26.m1.4.4.1.1.1.1.1.1.3.2">𝒢</ci><list id="S4.Ex26.m1.2.2.2.3.cmml" xref="S4.Ex26.m1.2.2.2.4"><ci id="S4.Ex26.m1.1.1.1.1.cmml" xref="S4.Ex26.m1.1.1.1.1">𝑁</ci><ci id="S4.Ex26.m1.2.2.2.2.cmml" xref="S4.Ex26.m1.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="S4.Ex26.m1.4.4.1.1.3.cmml" xref="S4.Ex26.m1.4.4.1.1.3"><times id="S4.Ex26.m1.4.4.1.1.3.1.cmml" xref="S4.Ex26.m1.4.4.1.1.3.1"></times><ci id="S4.Ex26.m1.4.4.1.1.3.2.cmml" xref="S4.Ex26.m1.4.4.1.1.3.2">Θ</ci><ci id="S4.Ex26.m1.3.3.cmml" xref="S4.Ex26.m1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex26.m1.4c">D_{\cap}(G\mid\mathcal{G}_{N,N})\;=\;\Theta(N).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex26.m1.4d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Θ ( italic_N ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.1.p1"> <p class="ltx_p" id="S4.SS1.1.p1.3">The upper bound is easy, and holds in the worst case as well (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S2.SS2.SSS2" title="2.2.2 Bipartite graph complexity ‣ 2.2 Examples ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">2.2.2</span></a>). For the lower bound, recall that a random graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.1.p1.1.m1.1"><semantics id="S4.SS1.1.p1.1.m1.1a"><mi id="S4.SS1.1.p1.1.m1.1.1" xref="S4.SS1.1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.1.m1.1b"><ci id="S4.SS1.1.p1.1.m1.1.1.cmml" xref="S4.SS1.1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.1.m1.1d">italic_G</annotation></semantics></math> satisfies <math alttext="D(G\mid\mathcal{G}_{N,N})=\Omega(N^{2}/\log N)" class="ltx_Math" display="inline" id="S4.SS1.1.p1.2.m2.4"><semantics id="S4.SS1.1.p1.2.m2.4a"><mrow id="S4.SS1.1.p1.2.m2.4.4" xref="S4.SS1.1.p1.2.m2.4.4.cmml"><mrow id="S4.SS1.1.p1.2.m2.3.3.1" xref="S4.SS1.1.p1.2.m2.3.3.1.cmml"><mi id="S4.SS1.1.p1.2.m2.3.3.1.3" xref="S4.SS1.1.p1.2.m2.3.3.1.3.cmml">D</mi><mo id="S4.SS1.1.p1.2.m2.3.3.1.2" 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id="S4.SS1.1.p1.2.m2.4.4.2.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2"><times id="S4.SS1.1.p1.2.m2.4.4.2.2.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.2"></times><ci id="S4.SS1.1.p1.2.m2.4.4.2.3.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.3">Ω</ci><apply id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1"><divide id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.1.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.1"></divide><apply id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.1.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2">superscript</csymbol><ci id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.2.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.2">𝑁</ci><cn id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.3.cmml" type="integer" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.2.3">2</cn></apply><apply id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3"><log id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3.1.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3.1"></log><ci id="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3.2.cmml" xref="S4.SS1.1.p1.2.m2.4.4.2.1.1.1.3.2">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.2.m2.4c">D(G\mid\mathcal{G}_{N,N})=\Omega(N^{2}/\log N)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.2.m2.4d">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Ω ( italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / roman_log italic_N )</annotation></semantics></math>, which is an immediate consequence of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem11" title="Lemma 11 (Complex sets). ‣ 2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">11</span></a>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem9" title="Lemma 9 (Immediate from [21]). ‣ 2.3 Basic lemmas and other useful results ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">9</span></a>, it must be the case that <math alttext="D_{\cap}(G\mid\mathcal{G}_{N,N})\;=\;\Omega(N)" class="ltx_Math" display="inline" id="S4.SS1.1.p1.3.m3.4"><semantics id="S4.SS1.1.p1.3.m3.4a"><mrow id="S4.SS1.1.p1.3.m3.4.4" xref="S4.SS1.1.p1.3.m3.4.4.cmml"><mrow id="S4.SS1.1.p1.3.m3.4.4.1" xref="S4.SS1.1.p1.3.m3.4.4.1.cmml"><msub id="S4.SS1.1.p1.3.m3.4.4.1.3" xref="S4.SS1.1.p1.3.m3.4.4.1.3.cmml"><mi id="S4.SS1.1.p1.3.m3.4.4.1.3.2" xref="S4.SS1.1.p1.3.m3.4.4.1.3.2.cmml">D</mi><mo id="S4.SS1.1.p1.3.m3.4.4.1.3.3" xref="S4.SS1.1.p1.3.m3.4.4.1.3.3.cmml">∩</mo></msub><mo id="S4.SS1.1.p1.3.m3.4.4.1.2" xref="S4.SS1.1.p1.3.m3.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.1.p1.3.m3.4.4.1.1.1" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.cmml"><mo id="S4.SS1.1.p1.3.m3.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.cmml"><mi id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.2" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.2.cmml">G</mi><mo id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.1" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.1.cmml">∣</mo><msub id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3.2" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3.2.cmml">𝒢</mi><mrow id="S4.SS1.1.p1.3.m3.2.2.2.4" xref="S4.SS1.1.p1.3.m3.2.2.2.3.cmml"><mi id="S4.SS1.1.p1.3.m3.1.1.1.1" xref="S4.SS1.1.p1.3.m3.1.1.1.1.cmml">N</mi><mo id="S4.SS1.1.p1.3.m3.2.2.2.4.1" xref="S4.SS1.1.p1.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS1.1.p1.3.m3.2.2.2.2" xref="S4.SS1.1.p1.3.m3.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S4.SS1.1.p1.3.m3.4.4.1.1.1.3" rspace="0.280em" stretchy="false" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.1.p1.3.m3.4.4.2" rspace="0.558em" xref="S4.SS1.1.p1.3.m3.4.4.2.cmml">=</mo><mrow id="S4.SS1.1.p1.3.m3.4.4.3" xref="S4.SS1.1.p1.3.m3.4.4.3.cmml"><mi id="S4.SS1.1.p1.3.m3.4.4.3.2" mathvariant="normal" xref="S4.SS1.1.p1.3.m3.4.4.3.2.cmml">Ω</mi><mo id="S4.SS1.1.p1.3.m3.4.4.3.1" xref="S4.SS1.1.p1.3.m3.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS1.1.p1.3.m3.4.4.3.3.2" xref="S4.SS1.1.p1.3.m3.4.4.3.cmml"><mo id="S4.SS1.1.p1.3.m3.4.4.3.3.2.1" stretchy="false" xref="S4.SS1.1.p1.3.m3.4.4.3.cmml">(</mo><mi id="S4.SS1.1.p1.3.m3.3.3" xref="S4.SS1.1.p1.3.m3.3.3.cmml">N</mi><mo id="S4.SS1.1.p1.3.m3.4.4.3.3.2.2" stretchy="false" xref="S4.SS1.1.p1.3.m3.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.3.m3.4b"><apply id="S4.SS1.1.p1.3.m3.4.4.cmml" xref="S4.SS1.1.p1.3.m3.4.4"><eq id="S4.SS1.1.p1.3.m3.4.4.2.cmml" xref="S4.SS1.1.p1.3.m3.4.4.2"></eq><apply 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xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3">subscript</csymbol><ci id="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3.2.cmml" xref="S4.SS1.1.p1.3.m3.4.4.1.1.1.1.3.2">𝒢</ci><list id="S4.SS1.1.p1.3.m3.2.2.2.3.cmml" xref="S4.SS1.1.p1.3.m3.2.2.2.4"><ci id="S4.SS1.1.p1.3.m3.1.1.1.1.cmml" xref="S4.SS1.1.p1.3.m3.1.1.1.1">𝑁</ci><ci id="S4.SS1.1.p1.3.m3.2.2.2.2.cmml" xref="S4.SS1.1.p1.3.m3.2.2.2.2">𝑁</ci></list></apply></apply></apply><apply id="S4.SS1.1.p1.3.m3.4.4.3.cmml" xref="S4.SS1.1.p1.3.m3.4.4.3"><times id="S4.SS1.1.p1.3.m3.4.4.3.1.cmml" xref="S4.SS1.1.p1.3.m3.4.4.3.1"></times><ci id="S4.SS1.1.p1.3.m3.4.4.3.2.cmml" xref="S4.SS1.1.p1.3.m3.4.4.3.2">Ω</ci><ci id="S4.SS1.1.p1.3.m3.3.3.cmml" xref="S4.SS1.1.p1.3.m3.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.3.m3.4c">D_{\cap}(G\mid\mathcal{G}_{N,N})\;=\;\Omega(N)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.3.m3.4d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Ω ( italic_N )</annotation></semantics></math>, which completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Recall the definition of cover complexity introduced in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.SS1" title="3.1 Definitions and notation ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">3.1</span></a>. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a> and Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem36" title="Proposition 36 (The intersection complexity of a random graph). ‣ 4.1 Basic results and connections ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">36</span></a> yield an <math alttext="\Omega(\sqrt{N})" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.2" xref="S4.SS1.p1.1.m1.1.2.cmml"><mi id="S4.SS1.p1.1.m1.1.2.2" mathvariant="normal" xref="S4.SS1.p1.1.m1.1.2.2.cmml">Ω</mi><mo id="S4.SS1.p1.1.m1.1.2.1" xref="S4.SS1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p1.1.m1.1.2.3.2" xref="S4.SS1.p1.1.m1.1.1.cmml"><mo id="S4.SS1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p1.1.m1.1.1.cmml">(</mo><msqrt id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">N</mi></msqrt><mo id="S4.SS1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.2"><times id="S4.SS1.p1.1.m1.1.2.1.cmml" xref="S4.SS1.p1.1.m1.1.2.1"></times><ci id="S4.SS1.p1.1.m1.1.2.2.cmml" xref="S4.SS1.p1.1.m1.1.2.2">Ω</ci><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.2.3.2"><root id="S4.SS1.p1.1.m1.1.1a.cmml" xref="S4.SS1.p1.1.m1.1.2.3.2"></root><ci id="S4.SS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.1.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\Omega(\sqrt{N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">roman_Ω ( square-root start_ARG italic_N end_ARG )</annotation></semantics></math> lower bound on the cover complexity of a random graph. It is possible to obtain a tight lower bound using a more careful argument.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem37"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem37.1.1.1">Theorem 37</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem37.2.2"> </span>(The cover complexity of a random graph)<span class="ltx_text ltx_font_bold" id="Thmtheorem37.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem37.p1"> <p class="ltx_p" id="Thmtheorem37.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem37.p1.1.1">Let <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem37.p1.1.1.m1.2"><semantics id="Thmtheorem37.p1.1.1.m1.2a"><mrow id="Thmtheorem37.p1.1.1.m1.2.3" xref="Thmtheorem37.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem37.p1.1.1.m1.2.3.2" xref="Thmtheorem37.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="Thmtheorem37.p1.1.1.m1.2.3.1" xref="Thmtheorem37.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem37.p1.1.1.m1.2.3.3" xref="Thmtheorem37.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem37.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem37.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem37.p1.1.1.m1.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem37.p1.1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem37.p1.1.1.m1.1.1" xref="Thmtheorem37.p1.1.1.m1.1.1.cmml">N</mi><mo id="Thmtheorem37.p1.1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem37.p1.1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem37.p1.1.1.m1.2.3.3.1" rspace="0.222em" xref="Thmtheorem37.p1.1.1.m1.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem37.p1.1.1.m1.2.3.3.3.2" xref="Thmtheorem37.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="Thmtheorem37.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem37.p1.1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem37.p1.1.1.m1.2.2" xref="Thmtheorem37.p1.1.1.m1.2.2.cmml">N</mi><mo id="Thmtheorem37.p1.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem37.p1.1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem37.p1.1.1.m1.2b"><apply id="Thmtheorem37.p1.1.1.m1.2.3.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3"><subset id="Thmtheorem37.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.1"></subset><ci id="Thmtheorem37.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.2">𝐺</ci><apply id="Thmtheorem37.p1.1.1.m1.2.3.3.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3"><times id="Thmtheorem37.p1.1.1.m1.2.3.3.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3.1"></times><apply id="Thmtheorem37.p1.1.1.m1.2.3.3.2.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem37.p1.1.1.m1.2.3.3.2.1.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem37.p1.1.1.m1.1.1.cmml" xref="Thmtheorem37.p1.1.1.m1.1.1">𝑁</ci></apply><apply id="Thmtheorem37.p1.1.1.m1.2.3.3.3.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem37.p1.1.1.m1.2.3.3.3.1.1.cmml" xref="Thmtheorem37.p1.1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem37.p1.1.1.m1.2.2.cmml" xref="Thmtheorem37.p1.1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem37.p1.1.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem37.p1.1.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> be a random bipartite graph. Then, asymptotically almost surely,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(G,\mathcal{G}_{N,N})\;=\;\Theta(N)." class="ltx_Math" display="block" id="S4.Ex27.m1.5"><semantics id="S4.Ex27.m1.5a"><mrow id="S4.Ex27.m1.5.5.1" xref="S4.Ex27.m1.5.5.1.1.cmml"><mrow id="S4.Ex27.m1.5.5.1.1" xref="S4.Ex27.m1.5.5.1.1.cmml"><mrow id="S4.Ex27.m1.5.5.1.1.1" xref="S4.Ex27.m1.5.5.1.1.1.cmml"><mi id="S4.Ex27.m1.5.5.1.1.1.3" xref="S4.Ex27.m1.5.5.1.1.1.3.cmml">ρ</mi><mo id="S4.Ex27.m1.5.5.1.1.1.2" xref="S4.Ex27.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex27.m1.5.5.1.1.1.1.1" xref="S4.Ex27.m1.5.5.1.1.1.1.2.cmml"><mo id="S4.Ex27.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S4.Ex27.m1.5.5.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex27.m1.3.3" xref="S4.Ex27.m1.3.3.cmml">G</mi><mo id="S4.Ex27.m1.5.5.1.1.1.1.1.3" xref="S4.Ex27.m1.5.5.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex27.m1.5.5.1.1.1.1.1.1" xref="S4.Ex27.m1.5.5.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex27.m1.5.5.1.1.1.1.1.1.2" xref="S4.Ex27.m1.5.5.1.1.1.1.1.1.2.cmml">𝒢</mi><mrow id="S4.Ex27.m1.2.2.2.4" xref="S4.Ex27.m1.2.2.2.3.cmml"><mi id="S4.Ex27.m1.1.1.1.1" xref="S4.Ex27.m1.1.1.1.1.cmml">N</mi><mo id="S4.Ex27.m1.2.2.2.4.1" xref="S4.Ex27.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex27.m1.2.2.2.2" xref="S4.Ex27.m1.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.Ex27.m1.5.5.1.1.1.1.1.4" rspace="0.280em" stretchy="false" xref="S4.Ex27.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex27.m1.5.5.1.1.2" rspace="0.558em" xref="S4.Ex27.m1.5.5.1.1.2.cmml">=</mo><mrow id="S4.Ex27.m1.5.5.1.1.3" xref="S4.Ex27.m1.5.5.1.1.3.cmml"><mi id="S4.Ex27.m1.5.5.1.1.3.2" mathvariant="normal" xref="S4.Ex27.m1.5.5.1.1.3.2.cmml">Θ</mi><mo id="S4.Ex27.m1.5.5.1.1.3.1" xref="S4.Ex27.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S4.Ex27.m1.5.5.1.1.3.3.2" xref="S4.Ex27.m1.5.5.1.1.3.cmml"><mo id="S4.Ex27.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S4.Ex27.m1.5.5.1.1.3.cmml">(</mo><mi id="S4.Ex27.m1.4.4" xref="S4.Ex27.m1.4.4.cmml">N</mi><mo id="S4.Ex27.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S4.Ex27.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex27.m1.5.5.1.2" lspace="0em" xref="S4.Ex27.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex27.m1.5b"><apply id="S4.Ex27.m1.5.5.1.1.cmml" xref="S4.Ex27.m1.5.5.1"><eq id="S4.Ex27.m1.5.5.1.1.2.cmml" xref="S4.Ex27.m1.5.5.1.1.2"></eq><apply id="S4.Ex27.m1.5.5.1.1.1.cmml" xref="S4.Ex27.m1.5.5.1.1.1"><times id="S4.Ex27.m1.5.5.1.1.1.2.cmml" xref="S4.Ex27.m1.5.5.1.1.1.2"></times><ci id="S4.Ex27.m1.5.5.1.1.1.3.cmml" xref="S4.Ex27.m1.5.5.1.1.1.3">𝜌</ci><interval closure="open" id="S4.Ex27.m1.5.5.1.1.1.1.2.cmml" xref="S4.Ex27.m1.5.5.1.1.1.1.1"><ci id="S4.Ex27.m1.3.3.cmml" xref="S4.Ex27.m1.3.3">𝐺</ci><apply id="S4.Ex27.m1.5.5.1.1.1.1.1.1.cmml" xref="S4.Ex27.m1.5.5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex27.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S4.Ex27.m1.5.5.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex27.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S4.Ex27.m1.5.5.1.1.1.1.1.1.2">𝒢</ci><list id="S4.Ex27.m1.2.2.2.3.cmml" xref="S4.Ex27.m1.2.2.2.4"><ci id="S4.Ex27.m1.1.1.1.1.cmml" xref="S4.Ex27.m1.1.1.1.1">𝑁</ci><ci id="S4.Ex27.m1.2.2.2.2.cmml" xref="S4.Ex27.m1.2.2.2.2">𝑁</ci></list></apply></interval></apply><apply id="S4.Ex27.m1.5.5.1.1.3.cmml" xref="S4.Ex27.m1.5.5.1.1.3"><times id="S4.Ex27.m1.5.5.1.1.3.1.cmml" xref="S4.Ex27.m1.5.5.1.1.3.1"></times><ci id="S4.Ex27.m1.5.5.1.1.3.2.cmml" xref="S4.Ex27.m1.5.5.1.1.3.2">Θ</ci><ci id="S4.Ex27.m1.4.4.cmml" xref="S4.Ex27.m1.4.4">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex27.m1.5c">\rho(G,\mathcal{G}_{N,N})\;=\;\Theta(N).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex27.m1.5d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Θ ( italic_N ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.2.p1"> <p class="ltx_p" id="S4.SS1.2.p1.16">The proof is based on a counting argument, and can be formalized using Kolmogorov complexity. Observe that the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem24" title="Theorem 24 (Fusion upper bound). ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">24</span></a> describes a <em class="ltx_emph ltx_font_italic" id="S4.SS1.2.p1.16.1">universal procedure</em> that generates an <em class="ltx_emph ltx_font_italic" id="S4.SS1.2.p1.16.2">arbitrary</em> set <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.2.p1.1.m1.1"><semantics id="S4.SS1.2.p1.1.m1.1a"><mi id="S4.SS1.2.p1.1.m1.1.1" xref="S4.SS1.2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.1.m1.1b"><ci id="S4.SS1.2.p1.1.m1.1.1.cmml" xref="S4.SS1.2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.1.m1.1d">italic_A</annotation></semantics></math> from <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.2.m2.1"><semantics id="S4.SS1.2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.2.m2.1.1" xref="S4.SS1.2.p1.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.2.m2.1b"><ci id="S4.SS1.2.p1.2.m2.1.1.cmml" xref="S4.SS1.2.p1.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.2.m2.1d">caligraphic_B</annotation></semantics></math> using <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS1.2.p1.3.m3.1"><semantics id="S4.SS1.2.p1.3.m3.1a"><mi id="S4.SS1.2.p1.3.m3.1.1" mathvariant="normal" xref="S4.SS1.2.p1.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.3.m3.1b"><ci id="S4.SS1.2.p1.3.m3.1.1.cmml" xref="S4.SS1.2.p1.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.3.m3.1d">roman_Λ</annotation></semantics></math>. However, for a <em class="ltx_emph ltx_font_italic" id="S4.SS1.2.p1.16.3">fixed</em> family <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.4.m4.1"><semantics id="S4.SS1.2.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.4.m4.1.1" xref="S4.SS1.2.p1.4.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.4.m4.1b"><ci id="S4.SS1.2.p1.4.m4.1.1.cmml" xref="S4.SS1.2.p1.4.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.4.m4.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.4.m4.1d">caligraphic_B</annotation></semantics></math> such as <math alttext="\mathcal{B}=\mathcal{G}_{N,N}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.5.m5.2"><semantics id="S4.SS1.2.p1.5.m5.2a"><mrow id="S4.SS1.2.p1.5.m5.2.3" xref="S4.SS1.2.p1.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.5.m5.2.3.2" xref="S4.SS1.2.p1.5.m5.2.3.2.cmml">ℬ</mi><mo id="S4.SS1.2.p1.5.m5.2.3.1" xref="S4.SS1.2.p1.5.m5.2.3.1.cmml">=</mo><msub id="S4.SS1.2.p1.5.m5.2.3.3" xref="S4.SS1.2.p1.5.m5.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.5.m5.2.3.3.2" xref="S4.SS1.2.p1.5.m5.2.3.3.2.cmml">𝒢</mi><mrow id="S4.SS1.2.p1.5.m5.2.2.2.4" xref="S4.SS1.2.p1.5.m5.2.2.2.3.cmml"><mi id="S4.SS1.2.p1.5.m5.1.1.1.1" xref="S4.SS1.2.p1.5.m5.1.1.1.1.cmml">N</mi><mo id="S4.SS1.2.p1.5.m5.2.2.2.4.1" xref="S4.SS1.2.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS1.2.p1.5.m5.2.2.2.2" xref="S4.SS1.2.p1.5.m5.2.2.2.2.cmml">N</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.5.m5.2b"><apply id="S4.SS1.2.p1.5.m5.2.3.cmml" xref="S4.SS1.2.p1.5.m5.2.3"><eq id="S4.SS1.2.p1.5.m5.2.3.1.cmml" xref="S4.SS1.2.p1.5.m5.2.3.1"></eq><ci id="S4.SS1.2.p1.5.m5.2.3.2.cmml" xref="S4.SS1.2.p1.5.m5.2.3.2">ℬ</ci><apply id="S4.SS1.2.p1.5.m5.2.3.3.cmml" xref="S4.SS1.2.p1.5.m5.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.2.p1.5.m5.2.3.3.1.cmml" xref="S4.SS1.2.p1.5.m5.2.3.3">subscript</csymbol><ci id="S4.SS1.2.p1.5.m5.2.3.3.2.cmml" xref="S4.SS1.2.p1.5.m5.2.3.3.2">𝒢</ci><list id="S4.SS1.2.p1.5.m5.2.2.2.3.cmml" xref="S4.SS1.2.p1.5.m5.2.2.2.4"><ci id="S4.SS1.2.p1.5.m5.1.1.1.1.cmml" xref="S4.SS1.2.p1.5.m5.1.1.1.1">𝑁</ci><ci id="S4.SS1.2.p1.5.m5.2.2.2.2.cmml" xref="S4.SS1.2.p1.5.m5.2.2.2.2">𝑁</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.5.m5.2c">\mathcal{B}=\mathcal{G}_{N,N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.5.m5.2d">caligraphic_B = caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT</annotation></semantics></math>, the only information the procedure needs is the inclusion relation among the sets appearing in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS1.2.p1.6.m6.1"><semantics id="S4.SS1.2.p1.6.m6.1a"><mi id="S4.SS1.2.p1.6.m6.1.1" mathvariant="normal" xref="S4.SS1.2.p1.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.6.m6.1b"><ci id="S4.SS1.2.p1.6.m6.1.1.cmml" xref="S4.SS1.2.p1.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.6.m6.1d">roman_Λ</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.7.m7.1"><semantics id="S4.SS1.2.p1.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.7.m7.1.1" xref="S4.SS1.2.p1.7.m7.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.7.m7.1b"><ci id="S4.SS1.2.p1.7.m7.1.1.cmml" xref="S4.SS1.2.p1.7.m7.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.7.m7.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.7.m7.1d">caligraphic_B</annotation></semantics></math>. Crucially, the explicit description of the sets that appear in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS1.2.p1.8.m8.1"><semantics id="S4.SS1.2.p1.8.m8.1a"><mi id="S4.SS1.2.p1.8.m8.1.1" mathvariant="normal" xref="S4.SS1.2.p1.8.m8.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.8.m8.1b"><ci id="S4.SS1.2.p1.8.m8.1.1.cmml" xref="S4.SS1.2.p1.8.m8.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.8.m8.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.8.m8.1d">roman_Λ</annotation></semantics></math> is not necessary to fully specify the corresponding set <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.2.p1.9.m9.1"><semantics id="S4.SS1.2.p1.9.m9.1a"><mi id="S4.SS1.2.p1.9.m9.1.1" xref="S4.SS1.2.p1.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.9.m9.1b"><ci id="S4.SS1.2.p1.9.m9.1.1.cmml" xref="S4.SS1.2.p1.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.9.m9.1d">italic_A</annotation></semantics></math> that is generated by the universal procedure. Indeed, observe that the core of the construction after the base case (which does not depend on <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.2.p1.10.m10.1"><semantics id="S4.SS1.2.p1.10.m10.1a"><mi id="S4.SS1.2.p1.10.m10.1.1" xref="S4.SS1.2.p1.10.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.10.m10.1b"><ci id="S4.SS1.2.p1.10.m10.1.1.cmml" xref="S4.SS1.2.p1.10.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.10.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.10.m10.1d">italic_A</annotation></semantics></math>) are the sub-indices appearing in Equations <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E6" title="Equation 6 ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">6</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E7" title="Equation 7 ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">7</span></a>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S3.E8" title="Equation 8 ‣ Proof. ‣ 3.3 Set-theoretic fusion as a complete framework for lower bounds ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">8</span></a>, which are determined by the aforementioned inclusion relations. These inclusions can be described by <math alttext="O(|\Lambda|(|\mathcal{B}|+|\Lambda|))" class="ltx_Math" display="inline" id="S4.SS1.2.p1.11.m11.4"><semantics id="S4.SS1.2.p1.11.m11.4a"><mrow id="S4.SS1.2.p1.11.m11.4.4" xref="S4.SS1.2.p1.11.m11.4.4.cmml"><mi id="S4.SS1.2.p1.11.m11.4.4.3" xref="S4.SS1.2.p1.11.m11.4.4.3.cmml">O</mi><mo id="S4.SS1.2.p1.11.m11.4.4.2" xref="S4.SS1.2.p1.11.m11.4.4.2.cmml">⁢</mo><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.cmml"><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.2" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.cmml">(</mo><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.cmml"><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.2" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.1.cmml"><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.1.1.cmml">|</mo><mi id="S4.SS1.2.p1.11.m11.1.1" mathvariant="normal" xref="S4.SS1.2.p1.11.m11.1.1.cmml">Λ</mi><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.2" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.cmml"><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.cmml"><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.2" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.1.cmml"><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.2.1" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.11.m11.2.2" xref="S4.SS1.2.p1.11.m11.2.2.cmml">ℬ</mi><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.2.2" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.1" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.2" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.1.cmml"><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.1.1.cmml">|</mo><mi id="S4.SS1.2.p1.11.m11.3.3" mathvariant="normal" xref="S4.SS1.2.p1.11.m11.3.3.cmml">Λ</mi><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.2.p1.11.m11.4.4.1.1.3" stretchy="false" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.11.m11.4b"><apply id="S4.SS1.2.p1.11.m11.4.4.cmml" xref="S4.SS1.2.p1.11.m11.4.4"><times id="S4.SS1.2.p1.11.m11.4.4.2.cmml" xref="S4.SS1.2.p1.11.m11.4.4.2"></times><ci id="S4.SS1.2.p1.11.m11.4.4.3.cmml" xref="S4.SS1.2.p1.11.m11.4.4.3">𝑂</ci><apply id="S4.SS1.2.p1.11.m11.4.4.1.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1"><times id="S4.SS1.2.p1.11.m11.4.4.1.1.1.2.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.2"></times><apply id="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.2"><abs id="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.3.2.1"></abs><ci id="S4.SS1.2.p1.11.m11.1.1.cmml" xref="S4.SS1.2.p1.11.m11.1.1">Λ</ci></apply><apply id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1"><plus id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.1"></plus><apply id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.2"><abs id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.2.2.1"></abs><ci id="S4.SS1.2.p1.11.m11.2.2.cmml" xref="S4.SS1.2.p1.11.m11.2.2">ℬ</ci></apply><apply id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.2"><abs id="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.1.1.cmml" xref="S4.SS1.2.p1.11.m11.4.4.1.1.1.1.1.1.3.2.1"></abs><ci id="S4.SS1.2.p1.11.m11.3.3.cmml" xref="S4.SS1.2.p1.11.m11.3.3">Λ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.11.m11.4c">O(|\Lambda|(|\mathcal{B}|+|\Lambda|))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.11.m11.4d">italic_O ( | roman_Λ | ( | caligraphic_B | + | roman_Λ | ) )</annotation></semantics></math> bits. Since a random graph has description complexity <math alttext="\Omega(N^{2})" class="ltx_Math" display="inline" id="S4.SS1.2.p1.12.m12.1"><semantics id="S4.SS1.2.p1.12.m12.1a"><mrow id="S4.SS1.2.p1.12.m12.1.1" xref="S4.SS1.2.p1.12.m12.1.1.cmml"><mi id="S4.SS1.2.p1.12.m12.1.1.3" mathvariant="normal" xref="S4.SS1.2.p1.12.m12.1.1.3.cmml">Ω</mi><mo id="S4.SS1.2.p1.12.m12.1.1.2" xref="S4.SS1.2.p1.12.m12.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.2.p1.12.m12.1.1.1.1" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.cmml"><mo id="S4.SS1.2.p1.12.m12.1.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.2.p1.12.m12.1.1.1.1.1" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.cmml"><mi id="S4.SS1.2.p1.12.m12.1.1.1.1.1.2" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.2.cmml">N</mi><mn id="S4.SS1.2.p1.12.m12.1.1.1.1.1.3" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.SS1.2.p1.12.m12.1.1.1.1.3" stretchy="false" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.12.m12.1b"><apply id="S4.SS1.2.p1.12.m12.1.1.cmml" xref="S4.SS1.2.p1.12.m12.1.1"><times id="S4.SS1.2.p1.12.m12.1.1.2.cmml" xref="S4.SS1.2.p1.12.m12.1.1.2"></times><ci id="S4.SS1.2.p1.12.m12.1.1.3.cmml" xref="S4.SS1.2.p1.12.m12.1.1.3">Ω</ci><apply id="S4.SS1.2.p1.12.m12.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.12.m12.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.12.m12.1.1.1.1">superscript</csymbol><ci id="S4.SS1.2.p1.12.m12.1.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.2">𝑁</ci><cn id="S4.SS1.2.p1.12.m12.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.2.p1.12.m12.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.12.m12.1c">\Omega(N^{2})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.12.m12.1d">roman_Ω ( italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="|\mathcal{G}_{N,N}|=2N" class="ltx_Math" display="inline" id="S4.SS1.2.p1.13.m13.3"><semantics id="S4.SS1.2.p1.13.m13.3a"><mrow id="S4.SS1.2.p1.13.m13.3.3" xref="S4.SS1.2.p1.13.m13.3.3.cmml"><mrow id="S4.SS1.2.p1.13.m13.3.3.1.1" xref="S4.SS1.2.p1.13.m13.3.3.1.2.cmml"><mo id="S4.SS1.2.p1.13.m13.3.3.1.1.2" stretchy="false" xref="S4.SS1.2.p1.13.m13.3.3.1.2.1.cmml">|</mo><msub id="S4.SS1.2.p1.13.m13.3.3.1.1.1" xref="S4.SS1.2.p1.13.m13.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.13.m13.3.3.1.1.1.2" xref="S4.SS1.2.p1.13.m13.3.3.1.1.1.2.cmml">𝒢</mi><mrow id="S4.SS1.2.p1.13.m13.2.2.2.4" xref="S4.SS1.2.p1.13.m13.2.2.2.3.cmml"><mi id="S4.SS1.2.p1.13.m13.1.1.1.1" xref="S4.SS1.2.p1.13.m13.1.1.1.1.cmml">N</mi><mo id="S4.SS1.2.p1.13.m13.2.2.2.4.1" xref="S4.SS1.2.p1.13.m13.2.2.2.3.cmml">,</mo><mi id="S4.SS1.2.p1.13.m13.2.2.2.2" xref="S4.SS1.2.p1.13.m13.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.SS1.2.p1.13.m13.3.3.1.1.3" stretchy="false" xref="S4.SS1.2.p1.13.m13.3.3.1.2.1.cmml">|</mo></mrow><mo id="S4.SS1.2.p1.13.m13.3.3.2" xref="S4.SS1.2.p1.13.m13.3.3.2.cmml">=</mo><mrow id="S4.SS1.2.p1.13.m13.3.3.3" xref="S4.SS1.2.p1.13.m13.3.3.3.cmml"><mn id="S4.SS1.2.p1.13.m13.3.3.3.2" xref="S4.SS1.2.p1.13.m13.3.3.3.2.cmml">2</mn><mo id="S4.SS1.2.p1.13.m13.3.3.3.1" xref="S4.SS1.2.p1.13.m13.3.3.3.1.cmml">⁢</mo><mi id="S4.SS1.2.p1.13.m13.3.3.3.3" xref="S4.SS1.2.p1.13.m13.3.3.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.13.m13.3b"><apply id="S4.SS1.2.p1.13.m13.3.3.cmml" xref="S4.SS1.2.p1.13.m13.3.3"><eq id="S4.SS1.2.p1.13.m13.3.3.2.cmml" xref="S4.SS1.2.p1.13.m13.3.3.2"></eq><apply id="S4.SS1.2.p1.13.m13.3.3.1.2.cmml" xref="S4.SS1.2.p1.13.m13.3.3.1.1"><abs id="S4.SS1.2.p1.13.m13.3.3.1.2.1.cmml" xref="S4.SS1.2.p1.13.m13.3.3.1.1.2"></abs><apply id="S4.SS1.2.p1.13.m13.3.3.1.1.1.cmml" xref="S4.SS1.2.p1.13.m13.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.13.m13.3.3.1.1.1.1.cmml" xref="S4.SS1.2.p1.13.m13.3.3.1.1.1">subscript</csymbol><ci id="S4.SS1.2.p1.13.m13.3.3.1.1.1.2.cmml" xref="S4.SS1.2.p1.13.m13.3.3.1.1.1.2">𝒢</ci><list id="S4.SS1.2.p1.13.m13.2.2.2.3.cmml" xref="S4.SS1.2.p1.13.m13.2.2.2.4"><ci id="S4.SS1.2.p1.13.m13.1.1.1.1.cmml" xref="S4.SS1.2.p1.13.m13.1.1.1.1">𝑁</ci><ci id="S4.SS1.2.p1.13.m13.2.2.2.2.cmml" xref="S4.SS1.2.p1.13.m13.2.2.2.2">𝑁</ci></list></apply></apply><apply id="S4.SS1.2.p1.13.m13.3.3.3.cmml" xref="S4.SS1.2.p1.13.m13.3.3.3"><times id="S4.SS1.2.p1.13.m13.3.3.3.1.cmml" xref="S4.SS1.2.p1.13.m13.3.3.3.1"></times><cn id="S4.SS1.2.p1.13.m13.3.3.3.2.cmml" type="integer" xref="S4.SS1.2.p1.13.m13.3.3.3.2">2</cn><ci id="S4.SS1.2.p1.13.m13.3.3.3.3.cmml" xref="S4.SS1.2.p1.13.m13.3.3.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.13.m13.3c">|\mathcal{G}_{N,N}|=2N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.13.m13.3d">| caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT | = 2 italic_N</annotation></semantics></math>, we must have <math alttext="|\Lambda|=\Omega(N)" class="ltx_Math" display="inline" id="S4.SS1.2.p1.14.m14.2"><semantics id="S4.SS1.2.p1.14.m14.2a"><mrow id="S4.SS1.2.p1.14.m14.2.3" xref="S4.SS1.2.p1.14.m14.2.3.cmml"><mrow id="S4.SS1.2.p1.14.m14.2.3.2.2" xref="S4.SS1.2.p1.14.m14.2.3.2.1.cmml"><mo id="S4.SS1.2.p1.14.m14.2.3.2.2.1" stretchy="false" xref="S4.SS1.2.p1.14.m14.2.3.2.1.1.cmml">|</mo><mi id="S4.SS1.2.p1.14.m14.1.1" mathvariant="normal" xref="S4.SS1.2.p1.14.m14.1.1.cmml">Λ</mi><mo id="S4.SS1.2.p1.14.m14.2.3.2.2.2" stretchy="false" xref="S4.SS1.2.p1.14.m14.2.3.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.2.p1.14.m14.2.3.1" xref="S4.SS1.2.p1.14.m14.2.3.1.cmml">=</mo><mrow id="S4.SS1.2.p1.14.m14.2.3.3" xref="S4.SS1.2.p1.14.m14.2.3.3.cmml"><mi id="S4.SS1.2.p1.14.m14.2.3.3.2" mathvariant="normal" xref="S4.SS1.2.p1.14.m14.2.3.3.2.cmml">Ω</mi><mo id="S4.SS1.2.p1.14.m14.2.3.3.1" xref="S4.SS1.2.p1.14.m14.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.2.p1.14.m14.2.3.3.3.2" xref="S4.SS1.2.p1.14.m14.2.3.3.cmml"><mo id="S4.SS1.2.p1.14.m14.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.2.p1.14.m14.2.3.3.cmml">(</mo><mi id="S4.SS1.2.p1.14.m14.2.2" xref="S4.SS1.2.p1.14.m14.2.2.cmml">N</mi><mo id="S4.SS1.2.p1.14.m14.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.2.p1.14.m14.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.14.m14.2b"><apply id="S4.SS1.2.p1.14.m14.2.3.cmml" xref="S4.SS1.2.p1.14.m14.2.3"><eq id="S4.SS1.2.p1.14.m14.2.3.1.cmml" xref="S4.SS1.2.p1.14.m14.2.3.1"></eq><apply id="S4.SS1.2.p1.14.m14.2.3.2.1.cmml" xref="S4.SS1.2.p1.14.m14.2.3.2.2"><abs id="S4.SS1.2.p1.14.m14.2.3.2.1.1.cmml" xref="S4.SS1.2.p1.14.m14.2.3.2.2.1"></abs><ci id="S4.SS1.2.p1.14.m14.1.1.cmml" xref="S4.SS1.2.p1.14.m14.1.1">Λ</ci></apply><apply id="S4.SS1.2.p1.14.m14.2.3.3.cmml" xref="S4.SS1.2.p1.14.m14.2.3.3"><times id="S4.SS1.2.p1.14.m14.2.3.3.1.cmml" xref="S4.SS1.2.p1.14.m14.2.3.3.1"></times><ci id="S4.SS1.2.p1.14.m14.2.3.3.2.cmml" xref="S4.SS1.2.p1.14.m14.2.3.3.2">Ω</ci><ci id="S4.SS1.2.p1.14.m14.2.2.cmml" xref="S4.SS1.2.p1.14.m14.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.14.m14.2c">|\Lambda|=\Omega(N)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.14.m14.2d">| roman_Λ | = roman_Ω ( italic_N )</annotation></semantics></math> asymptotically almost surely. In other words, <math alttext="\rho(G,\mathcal{G}_{N,N})=\Omega(N)" class="ltx_Math" display="inline" id="S4.SS1.2.p1.15.m15.5"><semantics id="S4.SS1.2.p1.15.m15.5a"><mrow id="S4.SS1.2.p1.15.m15.5.5" xref="S4.SS1.2.p1.15.m15.5.5.cmml"><mrow id="S4.SS1.2.p1.15.m15.5.5.1" xref="S4.SS1.2.p1.15.m15.5.5.1.cmml"><mi id="S4.SS1.2.p1.15.m15.5.5.1.3" xref="S4.SS1.2.p1.15.m15.5.5.1.3.cmml">ρ</mi><mo id="S4.SS1.2.p1.15.m15.5.5.1.2" xref="S4.SS1.2.p1.15.m15.5.5.1.2.cmml">⁢</mo><mrow id="S4.SS1.2.p1.15.m15.5.5.1.1.1" xref="S4.SS1.2.p1.15.m15.5.5.1.1.2.cmml"><mo id="S4.SS1.2.p1.15.m15.5.5.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.15.m15.5.5.1.1.2.cmml">(</mo><mi id="S4.SS1.2.p1.15.m15.3.3" xref="S4.SS1.2.p1.15.m15.3.3.cmml">G</mi><mo id="S4.SS1.2.p1.15.m15.5.5.1.1.1.3" xref="S4.SS1.2.p1.15.m15.5.5.1.1.2.cmml">,</mo><msub id="S4.SS1.2.p1.15.m15.5.5.1.1.1.1" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.2" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.2.cmml">𝒢</mi><mrow id="S4.SS1.2.p1.15.m15.2.2.2.4" xref="S4.SS1.2.p1.15.m15.2.2.2.3.cmml"><mi id="S4.SS1.2.p1.15.m15.1.1.1.1" xref="S4.SS1.2.p1.15.m15.1.1.1.1.cmml">N</mi><mo id="S4.SS1.2.p1.15.m15.2.2.2.4.1" xref="S4.SS1.2.p1.15.m15.2.2.2.3.cmml">,</mo><mi id="S4.SS1.2.p1.15.m15.2.2.2.2" xref="S4.SS1.2.p1.15.m15.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.SS1.2.p1.15.m15.5.5.1.1.1.4" stretchy="false" xref="S4.SS1.2.p1.15.m15.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.2.p1.15.m15.5.5.2" xref="S4.SS1.2.p1.15.m15.5.5.2.cmml">=</mo><mrow id="S4.SS1.2.p1.15.m15.5.5.3" xref="S4.SS1.2.p1.15.m15.5.5.3.cmml"><mi id="S4.SS1.2.p1.15.m15.5.5.3.2" mathvariant="normal" xref="S4.SS1.2.p1.15.m15.5.5.3.2.cmml">Ω</mi><mo id="S4.SS1.2.p1.15.m15.5.5.3.1" xref="S4.SS1.2.p1.15.m15.5.5.3.1.cmml">⁢</mo><mrow id="S4.SS1.2.p1.15.m15.5.5.3.3.2" xref="S4.SS1.2.p1.15.m15.5.5.3.cmml"><mo id="S4.SS1.2.p1.15.m15.5.5.3.3.2.1" stretchy="false" xref="S4.SS1.2.p1.15.m15.5.5.3.cmml">(</mo><mi id="S4.SS1.2.p1.15.m15.4.4" xref="S4.SS1.2.p1.15.m15.4.4.cmml">N</mi><mo id="S4.SS1.2.p1.15.m15.5.5.3.3.2.2" stretchy="false" xref="S4.SS1.2.p1.15.m15.5.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.15.m15.5b"><apply id="S4.SS1.2.p1.15.m15.5.5.cmml" xref="S4.SS1.2.p1.15.m15.5.5"><eq id="S4.SS1.2.p1.15.m15.5.5.2.cmml" xref="S4.SS1.2.p1.15.m15.5.5.2"></eq><apply id="S4.SS1.2.p1.15.m15.5.5.1.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1"><times id="S4.SS1.2.p1.15.m15.5.5.1.2.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.2"></times><ci id="S4.SS1.2.p1.15.m15.5.5.1.3.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.3">𝜌</ci><interval closure="open" id="S4.SS1.2.p1.15.m15.5.5.1.1.2.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1"><ci id="S4.SS1.2.p1.15.m15.3.3.cmml" xref="S4.SS1.2.p1.15.m15.3.3">𝐺</ci><apply id="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1.1">subscript</csymbol><ci id="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.15.m15.5.5.1.1.1.1.2">𝒢</ci><list id="S4.SS1.2.p1.15.m15.2.2.2.3.cmml" xref="S4.SS1.2.p1.15.m15.2.2.2.4"><ci id="S4.SS1.2.p1.15.m15.1.1.1.1.cmml" xref="S4.SS1.2.p1.15.m15.1.1.1.1">𝑁</ci><ci id="S4.SS1.2.p1.15.m15.2.2.2.2.cmml" xref="S4.SS1.2.p1.15.m15.2.2.2.2">𝑁</ci></list></apply></interval></apply><apply id="S4.SS1.2.p1.15.m15.5.5.3.cmml" xref="S4.SS1.2.p1.15.m15.5.5.3"><times id="S4.SS1.2.p1.15.m15.5.5.3.1.cmml" xref="S4.SS1.2.p1.15.m15.5.5.3.1"></times><ci id="S4.SS1.2.p1.15.m15.5.5.3.2.cmml" xref="S4.SS1.2.p1.15.m15.5.5.3.2">Ω</ci><ci id="S4.SS1.2.p1.15.m15.4.4.cmml" xref="S4.SS1.2.p1.15.m15.4.4">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.15.m15.5c">\rho(G,\mathcal{G}_{N,N})=\Omega(N)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.15.m15.5d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = roman_Ω ( italic_N )</annotation></semantics></math> for a typical graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS1.2.p1.16.m16.2"><semantics id="S4.SS1.2.p1.16.m16.2a"><mrow id="S4.SS1.2.p1.16.m16.2.3" xref="S4.SS1.2.p1.16.m16.2.3.cmml"><mi id="S4.SS1.2.p1.16.m16.2.3.2" xref="S4.SS1.2.p1.16.m16.2.3.2.cmml">G</mi><mo id="S4.SS1.2.p1.16.m16.2.3.1" xref="S4.SS1.2.p1.16.m16.2.3.1.cmml">⊆</mo><mrow id="S4.SS1.2.p1.16.m16.2.3.3" xref="S4.SS1.2.p1.16.m16.2.3.3.cmml"><mrow id="S4.SS1.2.p1.16.m16.2.3.3.2.2" xref="S4.SS1.2.p1.16.m16.2.3.3.2.1.cmml"><mo id="S4.SS1.2.p1.16.m16.2.3.3.2.2.1" stretchy="false" xref="S4.SS1.2.p1.16.m16.2.3.3.2.1.1.cmml">[</mo><mi id="S4.SS1.2.p1.16.m16.1.1" xref="S4.SS1.2.p1.16.m16.1.1.cmml">N</mi><mo id="S4.SS1.2.p1.16.m16.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS1.2.p1.16.m16.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.2.p1.16.m16.2.3.3.1" rspace="0.222em" xref="S4.SS1.2.p1.16.m16.2.3.3.1.cmml">×</mo><mrow id="S4.SS1.2.p1.16.m16.2.3.3.3.2" xref="S4.SS1.2.p1.16.m16.2.3.3.3.1.cmml"><mo id="S4.SS1.2.p1.16.m16.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.2.p1.16.m16.2.3.3.3.1.1.cmml">[</mo><mi id="S4.SS1.2.p1.16.m16.2.2" xref="S4.SS1.2.p1.16.m16.2.2.cmml">N</mi><mo id="S4.SS1.2.p1.16.m16.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.2.p1.16.m16.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.16.m16.2b"><apply id="S4.SS1.2.p1.16.m16.2.3.cmml" xref="S4.SS1.2.p1.16.m16.2.3"><subset id="S4.SS1.2.p1.16.m16.2.3.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.1"></subset><ci id="S4.SS1.2.p1.16.m16.2.3.2.cmml" xref="S4.SS1.2.p1.16.m16.2.3.2">𝐺</ci><apply id="S4.SS1.2.p1.16.m16.2.3.3.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3"><times id="S4.SS1.2.p1.16.m16.2.3.3.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3.1"></times><apply id="S4.SS1.2.p1.16.m16.2.3.3.2.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3.2.2"><csymbol cd="latexml" id="S4.SS1.2.p1.16.m16.2.3.3.2.1.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.2.p1.16.m16.1.1.cmml" xref="S4.SS1.2.p1.16.m16.1.1">𝑁</ci></apply><apply id="S4.SS1.2.p1.16.m16.2.3.3.3.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3.3.2"><csymbol cd="latexml" id="S4.SS1.2.p1.16.m16.2.3.3.3.1.1.cmml" xref="S4.SS1.2.p1.16.m16.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.2.p1.16.m16.2.2.cmml" xref="S4.SS1.2.p1.16.m16.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.16.m16.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.16.m16.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.5">Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mrow id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">N</mi><mo id="S4.SS1.p2.1.m1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.cmml">=</mo><msup id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml"><mn id="S4.SS1.p2.1.m1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.3.2.cmml">2</mn><mi id="S4.SS1.p2.1.m1.1.1.3.3" xref="S4.SS1.p2.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><eq id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1"></eq><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑁</ci><apply id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS1.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p2.1.m1.1.1.3.2">2</cn><ci id="S4.SS1.p2.1.m1.1.1.3.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. For a graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.2"><semantics id="S4.SS1.p2.2.m2.2a"><mrow id="S4.SS1.p2.2.m2.2.3" xref="S4.SS1.p2.2.m2.2.3.cmml"><mi id="S4.SS1.p2.2.m2.2.3.2" xref="S4.SS1.p2.2.m2.2.3.2.cmml">G</mi><mo id="S4.SS1.p2.2.m2.2.3.1" xref="S4.SS1.p2.2.m2.2.3.1.cmml">⊆</mo><mrow id="S4.SS1.p2.2.m2.2.3.3" xref="S4.SS1.p2.2.m2.2.3.3.cmml"><mrow id="S4.SS1.p2.2.m2.2.3.3.2.2" xref="S4.SS1.p2.2.m2.2.3.3.2.1.cmml"><mo id="S4.SS1.p2.2.m2.2.3.3.2.2.1" stretchy="false" xref="S4.SS1.p2.2.m2.2.3.3.2.1.1.cmml">[</mo><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">N</mi><mo id="S4.SS1.p2.2.m2.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS1.p2.2.m2.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.p2.2.m2.2.3.3.1" rspace="0.222em" xref="S4.SS1.p2.2.m2.2.3.3.1.cmml">×</mo><mrow id="S4.SS1.p2.2.m2.2.3.3.3.2" xref="S4.SS1.p2.2.m2.2.3.3.3.1.cmml"><mo id="S4.SS1.p2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.p2.2.m2.2.3.3.3.1.1.cmml">[</mo><mi id="S4.SS1.p2.2.m2.2.2" xref="S4.SS1.p2.2.m2.2.2.cmml">N</mi><mo id="S4.SS1.p2.2.m2.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.p2.2.m2.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.2b"><apply id="S4.SS1.p2.2.m2.2.3.cmml" xref="S4.SS1.p2.2.m2.2.3"><subset id="S4.SS1.p2.2.m2.2.3.1.cmml" xref="S4.SS1.p2.2.m2.2.3.1"></subset><ci id="S4.SS1.p2.2.m2.2.3.2.cmml" xref="S4.SS1.p2.2.m2.2.3.2">𝐺</ci><apply id="S4.SS1.p2.2.m2.2.3.3.cmml" xref="S4.SS1.p2.2.m2.2.3.3"><times id="S4.SS1.p2.2.m2.2.3.3.1.cmml" xref="S4.SS1.p2.2.m2.2.3.3.1"></times><apply id="S4.SS1.p2.2.m2.2.3.3.2.1.cmml" xref="S4.SS1.p2.2.m2.2.3.3.2.2"><csymbol cd="latexml" id="S4.SS1.p2.2.m2.2.3.3.2.1.1.cmml" xref="S4.SS1.p2.2.m2.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝑁</ci></apply><apply id="S4.SS1.p2.2.m2.2.3.3.3.1.cmml" xref="S4.SS1.p2.2.m2.2.3.3.3.2"><csymbol cd="latexml" id="S4.SS1.p2.2.m2.2.3.3.3.1.1.cmml" xref="S4.SS1.p2.2.m2.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.p2.2.m2.2.2.cmml" xref="S4.SS1.p2.2.m2.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>, we let <math alttext="f_{G}\colon\{0,1\}^{2n}\to\{0,1\}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.4"><semantics id="S4.SS1.p2.3.m3.4a"><mrow id="S4.SS1.p2.3.m3.4.5" xref="S4.SS1.p2.3.m3.4.5.cmml"><msub id="S4.SS1.p2.3.m3.4.5.2" xref="S4.SS1.p2.3.m3.4.5.2.cmml"><mi id="S4.SS1.p2.3.m3.4.5.2.2" xref="S4.SS1.p2.3.m3.4.5.2.2.cmml">f</mi><mi id="S4.SS1.p2.3.m3.4.5.2.3" xref="S4.SS1.p2.3.m3.4.5.2.3.cmml">G</mi></msub><mo id="S4.SS1.p2.3.m3.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.3.m3.4.5.1.cmml">:</mo><mrow id="S4.SS1.p2.3.m3.4.5.3" xref="S4.SS1.p2.3.m3.4.5.3.cmml"><msup id="S4.SS1.p2.3.m3.4.5.3.2" xref="S4.SS1.p2.3.m3.4.5.3.2.cmml"><mrow id="S4.SS1.p2.3.m3.4.5.3.2.2.2" xref="S4.SS1.p2.3.m3.4.5.3.2.2.1.cmml"><mo id="S4.SS1.p2.3.m3.4.5.3.2.2.2.1" stretchy="false" xref="S4.SS1.p2.3.m3.4.5.3.2.2.1.cmml">{</mo><mn id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.p2.3.m3.4.5.3.2.2.2.2" xref="S4.SS1.p2.3.m3.4.5.3.2.2.1.cmml">,</mo><mn id="S4.SS1.p2.3.m3.2.2" xref="S4.SS1.p2.3.m3.2.2.cmml">1</mn><mo id="S4.SS1.p2.3.m3.4.5.3.2.2.2.3" stretchy="false" xref="S4.SS1.p2.3.m3.4.5.3.2.2.1.cmml">}</mo></mrow><mrow id="S4.SS1.p2.3.m3.4.5.3.2.3" xref="S4.SS1.p2.3.m3.4.5.3.2.3.cmml"><mn id="S4.SS1.p2.3.m3.4.5.3.2.3.2" xref="S4.SS1.p2.3.m3.4.5.3.2.3.2.cmml">2</mn><mo id="S4.SS1.p2.3.m3.4.5.3.2.3.1" xref="S4.SS1.p2.3.m3.4.5.3.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.3.m3.4.5.3.2.3.3" xref="S4.SS1.p2.3.m3.4.5.3.2.3.3.cmml">n</mi></mrow></msup><mo id="S4.SS1.p2.3.m3.4.5.3.1" stretchy="false" xref="S4.SS1.p2.3.m3.4.5.3.1.cmml">→</mo><mrow id="S4.SS1.p2.3.m3.4.5.3.3.2" xref="S4.SS1.p2.3.m3.4.5.3.3.1.cmml"><mo id="S4.SS1.p2.3.m3.4.5.3.3.2.1" stretchy="false" xref="S4.SS1.p2.3.m3.4.5.3.3.1.cmml">{</mo><mn id="S4.SS1.p2.3.m3.3.3" xref="S4.SS1.p2.3.m3.3.3.cmml">0</mn><mo id="S4.SS1.p2.3.m3.4.5.3.3.2.2" xref="S4.SS1.p2.3.m3.4.5.3.3.1.cmml">,</mo><mn id="S4.SS1.p2.3.m3.4.4" xref="S4.SS1.p2.3.m3.4.4.cmml">1</mn><mo id="S4.SS1.p2.3.m3.4.5.3.3.2.3" stretchy="false" xref="S4.SS1.p2.3.m3.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.4b"><apply id="S4.SS1.p2.3.m3.4.5.cmml" xref="S4.SS1.p2.3.m3.4.5"><ci id="S4.SS1.p2.3.m3.4.5.1.cmml" xref="S4.SS1.p2.3.m3.4.5.1">:</ci><apply id="S4.SS1.p2.3.m3.4.5.2.cmml" xref="S4.SS1.p2.3.m3.4.5.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.4.5.2.1.cmml" xref="S4.SS1.p2.3.m3.4.5.2">subscript</csymbol><ci id="S4.SS1.p2.3.m3.4.5.2.2.cmml" xref="S4.SS1.p2.3.m3.4.5.2.2">𝑓</ci><ci id="S4.SS1.p2.3.m3.4.5.2.3.cmml" xref="S4.SS1.p2.3.m3.4.5.2.3">𝐺</ci></apply><apply id="S4.SS1.p2.3.m3.4.5.3.cmml" xref="S4.SS1.p2.3.m3.4.5.3"><ci id="S4.SS1.p2.3.m3.4.5.3.1.cmml" xref="S4.SS1.p2.3.m3.4.5.3.1">→</ci><apply id="S4.SS1.p2.3.m3.4.5.3.2.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.4.5.3.2.1.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2">superscript</csymbol><set id="S4.SS1.p2.3.m3.4.5.3.2.2.1.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2.2.2"><cn id="S4.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p2.3.m3.1.1">0</cn><cn id="S4.SS1.p2.3.m3.2.2.cmml" type="integer" xref="S4.SS1.p2.3.m3.2.2">1</cn></set><apply id="S4.SS1.p2.3.m3.4.5.3.2.3.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2.3"><times id="S4.SS1.p2.3.m3.4.5.3.2.3.1.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2.3.1"></times><cn id="S4.SS1.p2.3.m3.4.5.3.2.3.2.cmml" type="integer" xref="S4.SS1.p2.3.m3.4.5.3.2.3.2">2</cn><ci id="S4.SS1.p2.3.m3.4.5.3.2.3.3.cmml" xref="S4.SS1.p2.3.m3.4.5.3.2.3.3">𝑛</ci></apply></apply><set id="S4.SS1.p2.3.m3.4.5.3.3.1.cmml" xref="S4.SS1.p2.3.m3.4.5.3.3.2"><cn id="S4.SS1.p2.3.m3.3.3.cmml" type="integer" xref="S4.SS1.p2.3.m3.3.3">0</cn><cn id="S4.SS1.p2.3.m3.4.4.cmml" type="integer" xref="S4.SS1.p2.3.m3.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.4c">f_{G}\colon\{0,1\}^{2n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.4d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> be the Boolean function associated with <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_G</annotation></semantics></math>, as described in Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem13" title="Lemma 13 (Tight transference from graph complexity to circuit complexity). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">13</span></a> (in other words, <math alttext="f_{G}^{-1}(1)=\phi(G)" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.2"><semantics id="S4.SS1.p2.5.m5.2a"><mrow id="S4.SS1.p2.5.m5.2.3" xref="S4.SS1.p2.5.m5.2.3.cmml"><mrow id="S4.SS1.p2.5.m5.2.3.2" xref="S4.SS1.p2.5.m5.2.3.2.cmml"><msubsup id="S4.SS1.p2.5.m5.2.3.2.2" xref="S4.SS1.p2.5.m5.2.3.2.2.cmml"><mi id="S4.SS1.p2.5.m5.2.3.2.2.2.2" xref="S4.SS1.p2.5.m5.2.3.2.2.2.2.cmml">f</mi><mi id="S4.SS1.p2.5.m5.2.3.2.2.2.3" xref="S4.SS1.p2.5.m5.2.3.2.2.2.3.cmml">G</mi><mrow id="S4.SS1.p2.5.m5.2.3.2.2.3" xref="S4.SS1.p2.5.m5.2.3.2.2.3.cmml"><mo id="S4.SS1.p2.5.m5.2.3.2.2.3a" xref="S4.SS1.p2.5.m5.2.3.2.2.3.cmml">−</mo><mn id="S4.SS1.p2.5.m5.2.3.2.2.3.2" xref="S4.SS1.p2.5.m5.2.3.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.p2.5.m5.2.3.2.1" xref="S4.SS1.p2.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.5.m5.2.3.2.3.2" xref="S4.SS1.p2.5.m5.2.3.2.cmml"><mo id="S4.SS1.p2.5.m5.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.2.cmml">(</mo><mn id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">1</mn><mo id="S4.SS1.p2.5.m5.2.3.2.3.2.2" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.5.m5.2.3.1" xref="S4.SS1.p2.5.m5.2.3.1.cmml">=</mo><mrow id="S4.SS1.p2.5.m5.2.3.3" xref="S4.SS1.p2.5.m5.2.3.3.cmml"><mi id="S4.SS1.p2.5.m5.2.3.3.2" xref="S4.SS1.p2.5.m5.2.3.3.2.cmml">ϕ</mi><mo id="S4.SS1.p2.5.m5.2.3.3.1" xref="S4.SS1.p2.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.5.m5.2.3.3.3.2" xref="S4.SS1.p2.5.m5.2.3.3.cmml"><mo id="S4.SS1.p2.5.m5.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.3.cmml">(</mo><mi id="S4.SS1.p2.5.m5.2.2" xref="S4.SS1.p2.5.m5.2.2.cmml">G</mi><mo id="S4.SS1.p2.5.m5.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.p2.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.2b"><apply id="S4.SS1.p2.5.m5.2.3.cmml" xref="S4.SS1.p2.5.m5.2.3"><eq id="S4.SS1.p2.5.m5.2.3.1.cmml" xref="S4.SS1.p2.5.m5.2.3.1"></eq><apply id="S4.SS1.p2.5.m5.2.3.2.cmml" xref="S4.SS1.p2.5.m5.2.3.2"><times id="S4.SS1.p2.5.m5.2.3.2.1.cmml" xref="S4.SS1.p2.5.m5.2.3.2.1"></times><apply id="S4.SS1.p2.5.m5.2.3.2.2.cmml" xref="S4.SS1.p2.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m5.2.3.2.2.1.cmml" xref="S4.SS1.p2.5.m5.2.3.2.2">superscript</csymbol><apply id="S4.SS1.p2.5.m5.2.3.2.2.2.cmml" xref="S4.SS1.p2.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m5.2.3.2.2.2.1.cmml" 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id="S4.SS1.p2.5.m5.2c">f_{G}^{-1}(1)=\phi(G)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.2d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) = italic_ϕ ( italic_G )</annotation></semantics></math>).</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmtheorem38"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem38.1.1.1">Proposition 38</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem38.2.2"> </span>(Reducing circuit complexity lower bounds to two-dimensional cover problems)<span class="ltx_text ltx_font_bold" id="Thmtheorem38.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem38.p1"> <p class="ltx_p" id="Thmtheorem38.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem38.p1.1.1">For any non-trivial graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem38.p1.1.1.m1.2"><semantics id="Thmtheorem38.p1.1.1.m1.2a"><mrow id="Thmtheorem38.p1.1.1.m1.2.3" xref="Thmtheorem38.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem38.p1.1.1.m1.2.3.2" xref="Thmtheorem38.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="Thmtheorem38.p1.1.1.m1.2.3.1" xref="Thmtheorem38.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem38.p1.1.1.m1.2.3.3" xref="Thmtheorem38.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem38.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem38.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem38.p1.1.1.m1.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem38.p1.1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem38.p1.1.1.m1.1.1" xref="Thmtheorem38.p1.1.1.m1.1.1.cmml">N</mi><mo id="Thmtheorem38.p1.1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem38.p1.1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem38.p1.1.1.m1.2.3.3.1" rspace="0.222em" xref="Thmtheorem38.p1.1.1.m1.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem38.p1.1.1.m1.2.3.3.3.2" 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id="Thmtheorem38.p1.1.1.m1.2.3.3.2.1.1.cmml" xref="Thmtheorem38.p1.1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem38.p1.1.1.m1.1.1.cmml" xref="Thmtheorem38.p1.1.1.m1.1.1">𝑁</ci></apply><apply id="Thmtheorem38.p1.1.1.m1.2.3.3.3.1.cmml" xref="Thmtheorem38.p1.1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem38.p1.1.1.m1.2.3.3.3.1.1.cmml" xref="Thmtheorem38.p1.1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem38.p1.1.1.m1.2.2.cmml" xref="Thmtheorem38.p1.1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem38.p1.1.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem38.p1.1.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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xref="S4.Ex28.m1.5.5.1.1.2.1.1.1.3.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex28.m1.5c">\rho(G,\mathcal{G}_{N,N})\;\leq\;D_{\cap}(f_{G}^{-1}(1)\mid\mathcal{B}_{2n}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex28.m1.5d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) ∣ caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.3.p1"> <p class="ltx_p" id="S4.SS1.3.p1.1">This follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem13" title="Lemma 13 (Tight transference from graph complexity to circuit complexity). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">13</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.3">These results do not immediately imply that <math alttext="\rho(G,\mathcal{G}_{N,N})\leq\rho(f_{G}^{-1}(1),\mathcal{B}_{2n})" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.7"><semantics 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id="S4.SS1.p3.1.m1.7c">\rho(G,\mathcal{G}_{N,N})\leq\rho(f_{G}^{-1}(1),\mathcal{B}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.7d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_ρ ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, since the connection between <math alttext="D_{\cap}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><msub id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><mi id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml">D</mi><mo id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">∩</mo></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2">𝐷</ci><intersect id="S4.SS1.p3.2.m2.1.1.3.cmml" xref="S4.SS1.p3.2.m2.1.1.3"></intersect></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">D_{\cap}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\rho" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_ρ</annotation></semantics></math> might not be tight. This can be shown by a direct argument.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem39"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem39.1.1.1">Lemma 39</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem39.2.2"> </span>(A fusion transference lemma)<span class="ltx_text ltx_font_bold" id="Thmtheorem39.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem39.p1"> <p class="ltx_p" id="Thmtheorem39.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem39.p1.1.1">Let <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem39.p1.1.1.m1.2"><semantics id="Thmtheorem39.p1.1.1.m1.2a"><mrow id="Thmtheorem39.p1.1.1.m1.2.3" xref="Thmtheorem39.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem39.p1.1.1.m1.2.3.2" xref="Thmtheorem39.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="Thmtheorem39.p1.1.1.m1.2.3.1" xref="Thmtheorem39.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem39.p1.1.1.m1.2.3.3" xref="Thmtheorem39.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem39.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem39.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem39.p1.1.1.m1.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem39.p1.1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem39.p1.1.1.m1.1.1" xref="Thmtheorem39.p1.1.1.m1.1.1.cmml">N</mi><mo id="Thmtheorem39.p1.1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem39.p1.1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem39.p1.1.1.m1.2.3.3.1" rspace="0.222em" xref="Thmtheorem39.p1.1.1.m1.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem39.p1.1.1.m1.2.3.3.3.2" xref="Thmtheorem39.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="Thmtheorem39.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem39.p1.1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem39.p1.1.1.m1.2.2" xref="Thmtheorem39.p1.1.1.m1.2.2.cmml">N</mi><mo id="Thmtheorem39.p1.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem39.p1.1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem39.p1.1.1.m1.2b"><apply id="Thmtheorem39.p1.1.1.m1.2.3.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3"><subset id="Thmtheorem39.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.1"></subset><ci id="Thmtheorem39.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.2">𝐺</ci><apply id="Thmtheorem39.p1.1.1.m1.2.3.3.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3"><times id="Thmtheorem39.p1.1.1.m1.2.3.3.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3.1"></times><apply id="Thmtheorem39.p1.1.1.m1.2.3.3.2.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem39.p1.1.1.m1.2.3.3.2.1.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem39.p1.1.1.m1.1.1.cmml" xref="Thmtheorem39.p1.1.1.m1.1.1">𝑁</ci></apply><apply id="Thmtheorem39.p1.1.1.m1.2.3.3.3.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem39.p1.1.1.m1.2.3.3.3.1.1.cmml" xref="Thmtheorem39.p1.1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem39.p1.1.1.m1.2.2.cmml" xref="Thmtheorem39.p1.1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem39.p1.1.1.m1.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem39.p1.1.1.m1.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> be a non-trivial graph. Then,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho(G,\mathcal{G}_{N,N})\;\leq\;\rho(f_{G}^{-1}(1),\mathcal{B}_{2n})." class="ltx_Math" display="block" id="S4.Ex29.m1.5"><semantics id="S4.Ex29.m1.5a"><mrow id="S4.Ex29.m1.5.5.1" xref="S4.Ex29.m1.5.5.1.1.cmml"><mrow id="S4.Ex29.m1.5.5.1.1" xref="S4.Ex29.m1.5.5.1.1.cmml"><mrow id="S4.Ex29.m1.5.5.1.1.1" xref="S4.Ex29.m1.5.5.1.1.1.cmml"><mi id="S4.Ex29.m1.5.5.1.1.1.3" xref="S4.Ex29.m1.5.5.1.1.1.3.cmml">ρ</mi><mo id="S4.Ex29.m1.5.5.1.1.1.2" xref="S4.Ex29.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex29.m1.5.5.1.1.1.1.1" xref="S4.Ex29.m1.5.5.1.1.1.1.2.cmml"><mo id="S4.Ex29.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S4.Ex29.m1.5.5.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex29.m1.3.3" xref="S4.Ex29.m1.3.3.cmml">G</mi><mo id="S4.Ex29.m1.5.5.1.1.1.1.1.3" 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xref="S4.Ex29.m1.5.5.1.1.3.2.2.2.3"><times id="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.1.cmml" xref="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.1"></times><cn id="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.2.cmml" type="integer" xref="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.2">2</cn><ci id="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.3.cmml" xref="S4.Ex29.m1.5.5.1.1.3.2.2.2.3.3">𝑛</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex29.m1.5c">\rho(G,\mathcal{G}_{N,N})\;\leq\;\rho(f_{G}^{-1}(1),\mathcal{B}_{2n}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex29.m1.5d">italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_ρ ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 ) , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.4.p1"> <p class="ltx_p" id="S4.SS1.4.p1.20">Let <math alttext="\mathfrak{F}^{\uparrow}_{f_{G}}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.1.m1.1"><semantics id="S4.SS1.4.p1.1.m1.1a"><msubsup id="S4.SS1.4.p1.1.m1.1.1" xref="S4.SS1.4.p1.1.m1.1.1.cmml"><mi id="S4.SS1.4.p1.1.m1.1.1.2.2" xref="S4.SS1.4.p1.1.m1.1.1.2.2.cmml">𝔉</mi><msub id="S4.SS1.4.p1.1.m1.1.1.3" xref="S4.SS1.4.p1.1.m1.1.1.3.cmml"><mi id="S4.SS1.4.p1.1.m1.1.1.3.2" xref="S4.SS1.4.p1.1.m1.1.1.3.2.cmml">f</mi><mi id="S4.SS1.4.p1.1.m1.1.1.3.3" xref="S4.SS1.4.p1.1.m1.1.1.3.3.cmml">G</mi></msub><mo id="S4.SS1.4.p1.1.m1.1.1.2.3" stretchy="false" xref="S4.SS1.4.p1.1.m1.1.1.2.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.1.m1.1b"><apply id="S4.SS1.4.p1.1.m1.1.1.cmml" xref="S4.SS1.4.p1.1.m1.1.1"><csymbol cd="ambiguous" 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be the set that contains a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.2.m2.1"><semantics id="S4.SS1.4.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p1.2.m2.1.1" xref="S4.SS1.4.p1.2.m2.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.2.m2.1b"><ci id="S4.SS1.4.p1.2.m2.1.1.cmml" xref="S4.SS1.4.p1.2.m2.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.2.m2.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.2.m2.1d">caligraphic_F</annotation></semantics></math> over <math alttext="f^{-1}_{G}(0)" class="ltx_Math" display="inline" id="S4.SS1.4.p1.3.m3.1"><semantics id="S4.SS1.4.p1.3.m3.1a"><mrow id="S4.SS1.4.p1.3.m3.1.2" xref="S4.SS1.4.p1.3.m3.1.2.cmml"><msubsup id="S4.SS1.4.p1.3.m3.1.2.2" xref="S4.SS1.4.p1.3.m3.1.2.2.cmml"><mi id="S4.SS1.4.p1.3.m3.1.2.2.2.2" xref="S4.SS1.4.p1.3.m3.1.2.2.2.2.cmml">f</mi><mi id="S4.SS1.4.p1.3.m3.1.2.2.3" xref="S4.SS1.4.p1.3.m3.1.2.2.3.cmml">G</mi><mrow id="S4.SS1.4.p1.3.m3.1.2.2.2.3" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3.cmml"><mo id="S4.SS1.4.p1.3.m3.1.2.2.2.3a" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3.cmml">−</mo><mn id="S4.SS1.4.p1.3.m3.1.2.2.2.3.2" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.4.p1.3.m3.1.2.1" xref="S4.SS1.4.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.4.p1.3.m3.1.2.3.2" xref="S4.SS1.4.p1.3.m3.1.2.cmml"><mo id="S4.SS1.4.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS1.4.p1.3.m3.1.2.cmml">(</mo><mn id="S4.SS1.4.p1.3.m3.1.1" xref="S4.SS1.4.p1.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.4.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS1.4.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.3.m3.1b"><apply id="S4.SS1.4.p1.3.m3.1.2.cmml" xref="S4.SS1.4.p1.3.m3.1.2"><times id="S4.SS1.4.p1.3.m3.1.2.1.cmml" xref="S4.SS1.4.p1.3.m3.1.2.1"></times><apply id="S4.SS1.4.p1.3.m3.1.2.2.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.3.m3.1.2.2.1.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2">subscript</csymbol><apply id="S4.SS1.4.p1.3.m3.1.2.2.2.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.3.m3.1.2.2.2.1.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2">superscript</csymbol><ci id="S4.SS1.4.p1.3.m3.1.2.2.2.2.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2.2.2">𝑓</ci><apply id="S4.SS1.4.p1.3.m3.1.2.2.2.3.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3"><minus id="S4.SS1.4.p1.3.m3.1.2.2.2.3.1.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3"></minus><cn id="S4.SS1.4.p1.3.m3.1.2.2.2.3.2.cmml" type="integer" xref="S4.SS1.4.p1.3.m3.1.2.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.4.p1.3.m3.1.2.2.3.cmml" xref="S4.SS1.4.p1.3.m3.1.2.2.3">𝐺</ci></apply><cn id="S4.SS1.4.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS1.4.p1.3.m3.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.3.m3.1c">f^{-1}_{G}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.3.m3.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math> if and only if it is above some element <math alttext="a\in f^{-1}_{G}(1)" class="ltx_Math" display="inline" id="S4.SS1.4.p1.4.m4.1"><semantics id="S4.SS1.4.p1.4.m4.1a"><mrow id="S4.SS1.4.p1.4.m4.1.2" xref="S4.SS1.4.p1.4.m4.1.2.cmml"><mi id="S4.SS1.4.p1.4.m4.1.2.2" xref="S4.SS1.4.p1.4.m4.1.2.2.cmml">a</mi><mo id="S4.SS1.4.p1.4.m4.1.2.1" xref="S4.SS1.4.p1.4.m4.1.2.1.cmml">∈</mo><mrow id="S4.SS1.4.p1.4.m4.1.2.3" xref="S4.SS1.4.p1.4.m4.1.2.3.cmml"><msubsup id="S4.SS1.4.p1.4.m4.1.2.3.2" xref="S4.SS1.4.p1.4.m4.1.2.3.2.cmml"><mi id="S4.SS1.4.p1.4.m4.1.2.3.2.2.2" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.2.cmml">f</mi><mi id="S4.SS1.4.p1.4.m4.1.2.3.2.3" xref="S4.SS1.4.p1.4.m4.1.2.3.2.3.cmml">G</mi><mrow id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.cmml"><mo id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3a" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.cmml">−</mo><mn id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.2" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.4.p1.4.m4.1.2.3.1" xref="S4.SS1.4.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.4.p1.4.m4.1.2.3.3.2" xref="S4.SS1.4.p1.4.m4.1.2.3.cmml"><mo id="S4.SS1.4.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S4.SS1.4.p1.4.m4.1.2.3.cmml">(</mo><mn id="S4.SS1.4.p1.4.m4.1.1" xref="S4.SS1.4.p1.4.m4.1.1.cmml">1</mn><mo id="S4.SS1.4.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S4.SS1.4.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.4.m4.1b"><apply id="S4.SS1.4.p1.4.m4.1.2.cmml" xref="S4.SS1.4.p1.4.m4.1.2"><in id="S4.SS1.4.p1.4.m4.1.2.1.cmml" xref="S4.SS1.4.p1.4.m4.1.2.1"></in><ci id="S4.SS1.4.p1.4.m4.1.2.2.cmml" xref="S4.SS1.4.p1.4.m4.1.2.2">𝑎</ci><apply id="S4.SS1.4.p1.4.m4.1.2.3.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3"><times id="S4.SS1.4.p1.4.m4.1.2.3.1.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.1"></times><apply id="S4.SS1.4.p1.4.m4.1.2.3.2.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.4.m4.1.2.3.2.1.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2">subscript</csymbol><apply id="S4.SS1.4.p1.4.m4.1.2.3.2.2.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.4.m4.1.2.3.2.2.1.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2">superscript</csymbol><ci id="S4.SS1.4.p1.4.m4.1.2.3.2.2.2.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.2">𝑓</ci><apply id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3"><minus id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.1.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3"></minus><cn id="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.2.cmml" type="integer" xref="S4.SS1.4.p1.4.m4.1.2.3.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.4.p1.4.m4.1.2.3.2.3.cmml" xref="S4.SS1.4.p1.4.m4.1.2.3.2.3">𝐺</ci></apply><cn id="S4.SS1.4.p1.4.m4.1.1.cmml" type="integer" xref="S4.SS1.4.p1.4.m4.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.4.m4.1c">a\in f^{-1}_{G}(1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.4.m4.1d">italic_a ∈ italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 1 )</annotation></semantics></math>. Similarly, let <math alttext="\mathfrak{F}^{\uparrow}_{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.5.m5.1"><semantics id="S4.SS1.4.p1.5.m5.1a"><msubsup id="S4.SS1.4.p1.5.m5.1.1" xref="S4.SS1.4.p1.5.m5.1.1.cmml"><mi id="S4.SS1.4.p1.5.m5.1.1.2.2" xref="S4.SS1.4.p1.5.m5.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS1.4.p1.5.m5.1.1.3" xref="S4.SS1.4.p1.5.m5.1.1.3.cmml">G</mi><mo id="S4.SS1.4.p1.5.m5.1.1.2.3" stretchy="false" xref="S4.SS1.4.p1.5.m5.1.1.2.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.5.m5.1b"><apply id="S4.SS1.4.p1.5.m5.1.1.cmml" xref="S4.SS1.4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.5.m5.1.1.1.cmml" xref="S4.SS1.4.p1.5.m5.1.1">subscript</csymbol><apply id="S4.SS1.4.p1.5.m5.1.1.2.cmml" xref="S4.SS1.4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.5.m5.1.1.2.1.cmml" xref="S4.SS1.4.p1.5.m5.1.1">superscript</csymbol><ci id="S4.SS1.4.p1.5.m5.1.1.2.2.cmml" xref="S4.SS1.4.p1.5.m5.1.1.2.2">𝔉</ci><ci id="S4.SS1.4.p1.5.m5.1.1.2.3.cmml" xref="S4.SS1.4.p1.5.m5.1.1.2.3">↑</ci></apply><ci id="S4.SS1.4.p1.5.m5.1.1.3.cmml" xref="S4.SS1.4.p1.5.m5.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.5.m5.1c">\mathfrak{F}^{\uparrow}_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.5.m5.1d">fraktur_F start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> contain a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.6.m6.1"><semantics id="S4.SS1.4.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p1.6.m6.1.1" xref="S4.SS1.4.p1.6.m6.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.6.m6.1b"><ci id="S4.SS1.4.p1.6.m6.1.1.cmml" xref="S4.SS1.4.p1.6.m6.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.6.m6.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.6.m6.1d">caligraphic_F</annotation></semantics></math> over <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.7.m7.1"><semantics id="S4.SS1.4.p1.7.m7.1a"><mover accent="true" id="S4.SS1.4.p1.7.m7.1.1" xref="S4.SS1.4.p1.7.m7.1.1.cmml"><mi id="S4.SS1.4.p1.7.m7.1.1.2" xref="S4.SS1.4.p1.7.m7.1.1.2.cmml">G</mi><mo id="S4.SS1.4.p1.7.m7.1.1.1" xref="S4.SS1.4.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.7.m7.1b"><apply id="S4.SS1.4.p1.7.m7.1.1.cmml" xref="S4.SS1.4.p1.7.m7.1.1"><ci id="S4.SS1.4.p1.7.m7.1.1.1.cmml" xref="S4.SS1.4.p1.7.m7.1.1.1">¯</ci><ci id="S4.SS1.4.p1.7.m7.1.1.2.cmml" xref="S4.SS1.4.p1.7.m7.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.7.m7.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.7.m7.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> if and only if there is <math alttext="(u,v)\in G" class="ltx_Math" display="inline" id="S4.SS1.4.p1.8.m8.2"><semantics id="S4.SS1.4.p1.8.m8.2a"><mrow id="S4.SS1.4.p1.8.m8.2.3" xref="S4.SS1.4.p1.8.m8.2.3.cmml"><mrow id="S4.SS1.4.p1.8.m8.2.3.2.2" xref="S4.SS1.4.p1.8.m8.2.3.2.1.cmml"><mo id="S4.SS1.4.p1.8.m8.2.3.2.2.1" stretchy="false" xref="S4.SS1.4.p1.8.m8.2.3.2.1.cmml">(</mo><mi id="S4.SS1.4.p1.8.m8.1.1" xref="S4.SS1.4.p1.8.m8.1.1.cmml">u</mi><mo id="S4.SS1.4.p1.8.m8.2.3.2.2.2" xref="S4.SS1.4.p1.8.m8.2.3.2.1.cmml">,</mo><mi id="S4.SS1.4.p1.8.m8.2.2" xref="S4.SS1.4.p1.8.m8.2.2.cmml">v</mi><mo id="S4.SS1.4.p1.8.m8.2.3.2.2.3" stretchy="false" xref="S4.SS1.4.p1.8.m8.2.3.2.1.cmml">)</mo></mrow><mo id="S4.SS1.4.p1.8.m8.2.3.1" xref="S4.SS1.4.p1.8.m8.2.3.1.cmml">∈</mo><mi id="S4.SS1.4.p1.8.m8.2.3.3" xref="S4.SS1.4.p1.8.m8.2.3.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.8.m8.2b"><apply id="S4.SS1.4.p1.8.m8.2.3.cmml" xref="S4.SS1.4.p1.8.m8.2.3"><in id="S4.SS1.4.p1.8.m8.2.3.1.cmml" xref="S4.SS1.4.p1.8.m8.2.3.1"></in><interval closure="open" id="S4.SS1.4.p1.8.m8.2.3.2.1.cmml" xref="S4.SS1.4.p1.8.m8.2.3.2.2"><ci id="S4.SS1.4.p1.8.m8.1.1.cmml" xref="S4.SS1.4.p1.8.m8.1.1">𝑢</ci><ci id="S4.SS1.4.p1.8.m8.2.2.cmml" xref="S4.SS1.4.p1.8.m8.2.2">𝑣</ci></interval><ci id="S4.SS1.4.p1.8.m8.2.3.3.cmml" xref="S4.SS1.4.p1.8.m8.2.3.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.8.m8.2c">(u,v)\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.8.m8.2d">( italic_u , italic_v ) ∈ italic_G</annotation></semantics></math> such that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.9.m9.1"><semantics id="S4.SS1.4.p1.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p1.9.m9.1.1" xref="S4.SS1.4.p1.9.m9.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.9.m9.1b"><ci id="S4.SS1.4.p1.9.m9.1.1.cmml" xref="S4.SS1.4.p1.9.m9.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.9.m9.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.9.m9.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS1.4.p1.10.m10.2"><semantics id="S4.SS1.4.p1.10.m10.2a"><mrow id="S4.SS1.4.p1.10.m10.2.3.2" xref="S4.SS1.4.p1.10.m10.2.3.1.cmml"><mo id="S4.SS1.4.p1.10.m10.2.3.2.1" stretchy="false" xref="S4.SS1.4.p1.10.m10.2.3.1.cmml">(</mo><mi id="S4.SS1.4.p1.10.m10.1.1" xref="S4.SS1.4.p1.10.m10.1.1.cmml">u</mi><mo id="S4.SS1.4.p1.10.m10.2.3.2.2" xref="S4.SS1.4.p1.10.m10.2.3.1.cmml">,</mo><mi id="S4.SS1.4.p1.10.m10.2.2" xref="S4.SS1.4.p1.10.m10.2.2.cmml">v</mi><mo id="S4.SS1.4.p1.10.m10.2.3.2.3" stretchy="false" xref="S4.SS1.4.p1.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.10.m10.2b"><interval closure="open" id="S4.SS1.4.p1.10.m10.2.3.1.cmml" xref="S4.SS1.4.p1.10.m10.2.3.2"><ci id="S4.SS1.4.p1.10.m10.1.1.cmml" xref="S4.SS1.4.p1.10.m10.1.1">𝑢</ci><ci id="S4.SS1.4.p1.10.m10.2.2.cmml" xref="S4.SS1.4.p1.10.m10.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.10.m10.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.10.m10.2d">( italic_u , italic_v )</annotation></semantics></math>. Assume <math alttext="\Lambda_{f_{G}}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.11.m11.1"><semantics id="S4.SS1.4.p1.11.m11.1a"><msub id="S4.SS1.4.p1.11.m11.1.1" xref="S4.SS1.4.p1.11.m11.1.1.cmml"><mi id="S4.SS1.4.p1.11.m11.1.1.2" mathvariant="normal" xref="S4.SS1.4.p1.11.m11.1.1.2.cmml">Λ</mi><msub id="S4.SS1.4.p1.11.m11.1.1.3" xref="S4.SS1.4.p1.11.m11.1.1.3.cmml"><mi id="S4.SS1.4.p1.11.m11.1.1.3.2" xref="S4.SS1.4.p1.11.m11.1.1.3.2.cmml">f</mi><mi id="S4.SS1.4.p1.11.m11.1.1.3.3" xref="S4.SS1.4.p1.11.m11.1.1.3.3.cmml">G</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.11.m11.1b"><apply id="S4.SS1.4.p1.11.m11.1.1.cmml" xref="S4.SS1.4.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.11.m11.1.1.1.cmml" xref="S4.SS1.4.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS1.4.p1.11.m11.1.1.2.cmml" xref="S4.SS1.4.p1.11.m11.1.1.2">Λ</ci><apply id="S4.SS1.4.p1.11.m11.1.1.3.cmml" xref="S4.SS1.4.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p1.11.m11.1.1.3.1.cmml" xref="S4.SS1.4.p1.11.m11.1.1.3">subscript</csymbol><ci id="S4.SS1.4.p1.11.m11.1.1.3.2.cmml" xref="S4.SS1.4.p1.11.m11.1.1.3.2">𝑓</ci><ci id="S4.SS1.4.p1.11.m11.1.1.3.3.cmml" xref="S4.SS1.4.p1.11.m11.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.11.m11.1c">\Lambda_{f_{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.11.m11.1d">roman_Λ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a family of pairs of subsets of <math alttext="f^{-1}_{G}(0)" class="ltx_Math" display="inline" id="S4.SS1.4.p1.12.m12.1"><semantics id="S4.SS1.4.p1.12.m12.1a"><mrow id="S4.SS1.4.p1.12.m12.1.2" xref="S4.SS1.4.p1.12.m12.1.2.cmml"><msubsup id="S4.SS1.4.p1.12.m12.1.2.2" xref="S4.SS1.4.p1.12.m12.1.2.2.cmml"><mi id="S4.SS1.4.p1.12.m12.1.2.2.2.2" xref="S4.SS1.4.p1.12.m12.1.2.2.2.2.cmml">f</mi><mi id="S4.SS1.4.p1.12.m12.1.2.2.3" xref="S4.SS1.4.p1.12.m12.1.2.2.3.cmml">G</mi><mrow id="S4.SS1.4.p1.12.m12.1.2.2.2.3" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3.cmml"><mo id="S4.SS1.4.p1.12.m12.1.2.2.2.3a" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3.cmml">−</mo><mn id="S4.SS1.4.p1.12.m12.1.2.2.2.3.2" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.4.p1.12.m12.1.2.1" xref="S4.SS1.4.p1.12.m12.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.4.p1.12.m12.1.2.3.2" xref="S4.SS1.4.p1.12.m12.1.2.cmml"><mo id="S4.SS1.4.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S4.SS1.4.p1.12.m12.1.2.cmml">(</mo><mn id="S4.SS1.4.p1.12.m12.1.1" xref="S4.SS1.4.p1.12.m12.1.1.cmml">0</mn><mo id="S4.SS1.4.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S4.SS1.4.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.12.m12.1b"><apply id="S4.SS1.4.p1.12.m12.1.2.cmml" xref="S4.SS1.4.p1.12.m12.1.2"><times id="S4.SS1.4.p1.12.m12.1.2.1.cmml" xref="S4.SS1.4.p1.12.m12.1.2.1"></times><apply id="S4.SS1.4.p1.12.m12.1.2.2.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.12.m12.1.2.2.1.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2">subscript</csymbol><apply id="S4.SS1.4.p1.12.m12.1.2.2.2.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p1.12.m12.1.2.2.2.1.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2">superscript</csymbol><ci id="S4.SS1.4.p1.12.m12.1.2.2.2.2.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2.2.2">𝑓</ci><apply id="S4.SS1.4.p1.12.m12.1.2.2.2.3.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3"><minus id="S4.SS1.4.p1.12.m12.1.2.2.2.3.1.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3"></minus><cn id="S4.SS1.4.p1.12.m12.1.2.2.2.3.2.cmml" type="integer" xref="S4.SS1.4.p1.12.m12.1.2.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.4.p1.12.m12.1.2.2.3.cmml" xref="S4.SS1.4.p1.12.m12.1.2.2.3">𝐺</ci></apply><cn id="S4.SS1.4.p1.12.m12.1.1.cmml" type="integer" xref="S4.SS1.4.p1.12.m12.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.12.m12.1c">f^{-1}_{G}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.12.m12.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math> that cover all semi-filters in <math alttext="\mathfrak{F}^{\uparrow}_{f_{G}}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.13.m13.1"><semantics id="S4.SS1.4.p1.13.m13.1a"><msubsup id="S4.SS1.4.p1.13.m13.1.1" xref="S4.SS1.4.p1.13.m13.1.1.cmml"><mi id="S4.SS1.4.p1.13.m13.1.1.2.2" xref="S4.SS1.4.p1.13.m13.1.1.2.2.cmml">𝔉</mi><msub id="S4.SS1.4.p1.13.m13.1.1.3" xref="S4.SS1.4.p1.13.m13.1.1.3.cmml"><mi id="S4.SS1.4.p1.13.m13.1.1.3.2" xref="S4.SS1.4.p1.13.m13.1.1.3.2.cmml">f</mi><mi id="S4.SS1.4.p1.13.m13.1.1.3.3" xref="S4.SS1.4.p1.13.m13.1.1.3.3.cmml">G</mi></msub><mo id="S4.SS1.4.p1.13.m13.1.1.2.3" stretchy="false" xref="S4.SS1.4.p1.13.m13.1.1.2.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.13.m13.1b"><apply id="S4.SS1.4.p1.13.m13.1.1.cmml" xref="S4.SS1.4.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.13.m13.1.1.1.cmml" xref="S4.SS1.4.p1.13.m13.1.1">subscript</csymbol><apply id="S4.SS1.4.p1.13.m13.1.1.2.cmml" xref="S4.SS1.4.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.13.m13.1.1.2.1.cmml" xref="S4.SS1.4.p1.13.m13.1.1">superscript</csymbol><ci id="S4.SS1.4.p1.13.m13.1.1.2.2.cmml" xref="S4.SS1.4.p1.13.m13.1.1.2.2">𝔉</ci><ci id="S4.SS1.4.p1.13.m13.1.1.2.3.cmml" xref="S4.SS1.4.p1.13.m13.1.1.2.3">↑</ci></apply><apply id="S4.SS1.4.p1.13.m13.1.1.3.cmml" xref="S4.SS1.4.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p1.13.m13.1.1.3.1.cmml" xref="S4.SS1.4.p1.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS1.4.p1.13.m13.1.1.3.2.cmml" xref="S4.SS1.4.p1.13.m13.1.1.3.2">𝑓</ci><ci id="S4.SS1.4.p1.13.m13.1.1.3.3.cmml" xref="S4.SS1.4.p1.13.m13.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.13.m13.1c">\mathfrak{F}^{\uparrow}_{f_{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.13.m13.1d">fraktur_F start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Now let <math alttext="\Lambda_{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.14.m14.1"><semantics id="S4.SS1.4.p1.14.m14.1a"><msub id="S4.SS1.4.p1.14.m14.1.1" xref="S4.SS1.4.p1.14.m14.1.1.cmml"><mi id="S4.SS1.4.p1.14.m14.1.1.2" mathvariant="normal" xref="S4.SS1.4.p1.14.m14.1.1.2.cmml">Λ</mi><mi id="S4.SS1.4.p1.14.m14.1.1.3" xref="S4.SS1.4.p1.14.m14.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.14.m14.1b"><apply id="S4.SS1.4.p1.14.m14.1.1.cmml" xref="S4.SS1.4.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.14.m14.1.1.1.cmml" xref="S4.SS1.4.p1.14.m14.1.1">subscript</csymbol><ci id="S4.SS1.4.p1.14.m14.1.1.2.cmml" xref="S4.SS1.4.p1.14.m14.1.1.2">Λ</ci><ci id="S4.SS1.4.p1.14.m14.1.1.3.cmml" xref="S4.SS1.4.p1.14.m14.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.14.m14.1c">\Lambda_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.14.m14.1d">roman_Λ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> be the family of pairs of subsets of <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.15.m15.1"><semantics id="S4.SS1.4.p1.15.m15.1a"><mover accent="true" id="S4.SS1.4.p1.15.m15.1.1" xref="S4.SS1.4.p1.15.m15.1.1.cmml"><mi id="S4.SS1.4.p1.15.m15.1.1.2" xref="S4.SS1.4.p1.15.m15.1.1.2.cmml">G</mi><mo id="S4.SS1.4.p1.15.m15.1.1.1" xref="S4.SS1.4.p1.15.m15.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.15.m15.1b"><apply id="S4.SS1.4.p1.15.m15.1.1.cmml" xref="S4.SS1.4.p1.15.m15.1.1"><ci id="S4.SS1.4.p1.15.m15.1.1.1.cmml" xref="S4.SS1.4.p1.15.m15.1.1.1">¯</ci><ci id="S4.SS1.4.p1.15.m15.1.1.2.cmml" xref="S4.SS1.4.p1.15.m15.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.15.m15.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.15.m15.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> induced by the pairs in <math alttext="\Lambda_{f_{G}}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.16.m16.1"><semantics id="S4.SS1.4.p1.16.m16.1a"><msub id="S4.SS1.4.p1.16.m16.1.1" xref="S4.SS1.4.p1.16.m16.1.1.cmml"><mi id="S4.SS1.4.p1.16.m16.1.1.2" mathvariant="normal" xref="S4.SS1.4.p1.16.m16.1.1.2.cmml">Λ</mi><msub id="S4.SS1.4.p1.16.m16.1.1.3" xref="S4.SS1.4.p1.16.m16.1.1.3.cmml"><mi id="S4.SS1.4.p1.16.m16.1.1.3.2" xref="S4.SS1.4.p1.16.m16.1.1.3.2.cmml">f</mi><mi id="S4.SS1.4.p1.16.m16.1.1.3.3" xref="S4.SS1.4.p1.16.m16.1.1.3.3.cmml">G</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.16.m16.1b"><apply id="S4.SS1.4.p1.16.m16.1.1.cmml" xref="S4.SS1.4.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.16.m16.1.1.1.cmml" xref="S4.SS1.4.p1.16.m16.1.1">subscript</csymbol><ci id="S4.SS1.4.p1.16.m16.1.1.2.cmml" xref="S4.SS1.4.p1.16.m16.1.1.2">Λ</ci><apply id="S4.SS1.4.p1.16.m16.1.1.3.cmml" xref="S4.SS1.4.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p1.16.m16.1.1.3.1.cmml" xref="S4.SS1.4.p1.16.m16.1.1.3">subscript</csymbol><ci id="S4.SS1.4.p1.16.m16.1.1.3.2.cmml" xref="S4.SS1.4.p1.16.m16.1.1.3.2">𝑓</ci><ci id="S4.SS1.4.p1.16.m16.1.1.3.3.cmml" xref="S4.SS1.4.p1.16.m16.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.16.m16.1c">\Lambda_{f_{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.16.m16.1d">roman_Λ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and the bijection between <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS1.4.p1.17.m17.2"><semantics id="S4.SS1.4.p1.17.m17.2a"><mrow id="S4.SS1.4.p1.17.m17.2.3" xref="S4.SS1.4.p1.17.m17.2.3.cmml"><mrow id="S4.SS1.4.p1.17.m17.2.3.2.2" xref="S4.SS1.4.p1.17.m17.2.3.2.1.cmml"><mo id="S4.SS1.4.p1.17.m17.2.3.2.2.1" stretchy="false" xref="S4.SS1.4.p1.17.m17.2.3.2.1.1.cmml">[</mo><mi id="S4.SS1.4.p1.17.m17.1.1" xref="S4.SS1.4.p1.17.m17.1.1.cmml">N</mi><mo id="S4.SS1.4.p1.17.m17.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS1.4.p1.17.m17.2.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.4.p1.17.m17.2.3.1" rspace="0.222em" xref="S4.SS1.4.p1.17.m17.2.3.1.cmml">×</mo><mrow id="S4.SS1.4.p1.17.m17.2.3.3.2" xref="S4.SS1.4.p1.17.m17.2.3.3.1.cmml"><mo id="S4.SS1.4.p1.17.m17.2.3.3.2.1" stretchy="false" xref="S4.SS1.4.p1.17.m17.2.3.3.1.1.cmml">[</mo><mi id="S4.SS1.4.p1.17.m17.2.2" xref="S4.SS1.4.p1.17.m17.2.2.cmml">N</mi><mo id="S4.SS1.4.p1.17.m17.2.3.3.2.2" stretchy="false" xref="S4.SS1.4.p1.17.m17.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.17.m17.2b"><apply id="S4.SS1.4.p1.17.m17.2.3.cmml" xref="S4.SS1.4.p1.17.m17.2.3"><times id="S4.SS1.4.p1.17.m17.2.3.1.cmml" xref="S4.SS1.4.p1.17.m17.2.3.1"></times><apply id="S4.SS1.4.p1.17.m17.2.3.2.1.cmml" xref="S4.SS1.4.p1.17.m17.2.3.2.2"><csymbol cd="latexml" id="S4.SS1.4.p1.17.m17.2.3.2.1.1.cmml" xref="S4.SS1.4.p1.17.m17.2.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.4.p1.17.m17.1.1.cmml" xref="S4.SS1.4.p1.17.m17.1.1">𝑁</ci></apply><apply id="S4.SS1.4.p1.17.m17.2.3.3.1.cmml" xref="S4.SS1.4.p1.17.m17.2.3.3.2"><csymbol cd="latexml" id="S4.SS1.4.p1.17.m17.2.3.3.1.1.cmml" xref="S4.SS1.4.p1.17.m17.2.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.4.p1.17.m17.2.2.cmml" xref="S4.SS1.4.p1.17.m17.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.17.m17.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.17.m17.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math> and <math alttext="\{0,1\}^{2n}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.18.m18.2"><semantics id="S4.SS1.4.p1.18.m18.2a"><msup id="S4.SS1.4.p1.18.m18.2.3" xref="S4.SS1.4.p1.18.m18.2.3.cmml"><mrow id="S4.SS1.4.p1.18.m18.2.3.2.2" xref="S4.SS1.4.p1.18.m18.2.3.2.1.cmml"><mo id="S4.SS1.4.p1.18.m18.2.3.2.2.1" stretchy="false" xref="S4.SS1.4.p1.18.m18.2.3.2.1.cmml">{</mo><mn id="S4.SS1.4.p1.18.m18.1.1" xref="S4.SS1.4.p1.18.m18.1.1.cmml">0</mn><mo id="S4.SS1.4.p1.18.m18.2.3.2.2.2" xref="S4.SS1.4.p1.18.m18.2.3.2.1.cmml">,</mo><mn id="S4.SS1.4.p1.18.m18.2.2" xref="S4.SS1.4.p1.18.m18.2.2.cmml">1</mn><mo id="S4.SS1.4.p1.18.m18.2.3.2.2.3" stretchy="false" xref="S4.SS1.4.p1.18.m18.2.3.2.1.cmml">}</mo></mrow><mrow id="S4.SS1.4.p1.18.m18.2.3.3" xref="S4.SS1.4.p1.18.m18.2.3.3.cmml"><mn id="S4.SS1.4.p1.18.m18.2.3.3.2" xref="S4.SS1.4.p1.18.m18.2.3.3.2.cmml">2</mn><mo id="S4.SS1.4.p1.18.m18.2.3.3.1" xref="S4.SS1.4.p1.18.m18.2.3.3.1.cmml">⁢</mo><mi id="S4.SS1.4.p1.18.m18.2.3.3.3" xref="S4.SS1.4.p1.18.m18.2.3.3.3.cmml">n</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.18.m18.2b"><apply id="S4.SS1.4.p1.18.m18.2.3.cmml" xref="S4.SS1.4.p1.18.m18.2.3"><csymbol cd="ambiguous" id="S4.SS1.4.p1.18.m18.2.3.1.cmml" xref="S4.SS1.4.p1.18.m18.2.3">superscript</csymbol><set id="S4.SS1.4.p1.18.m18.2.3.2.1.cmml" xref="S4.SS1.4.p1.18.m18.2.3.2.2"><cn id="S4.SS1.4.p1.18.m18.1.1.cmml" type="integer" xref="S4.SS1.4.p1.18.m18.1.1">0</cn><cn id="S4.SS1.4.p1.18.m18.2.2.cmml" type="integer" xref="S4.SS1.4.p1.18.m18.2.2">1</cn></set><apply id="S4.SS1.4.p1.18.m18.2.3.3.cmml" xref="S4.SS1.4.p1.18.m18.2.3.3"><times id="S4.SS1.4.p1.18.m18.2.3.3.1.cmml" xref="S4.SS1.4.p1.18.m18.2.3.3.1"></times><cn id="S4.SS1.4.p1.18.m18.2.3.3.2.cmml" type="integer" xref="S4.SS1.4.p1.18.m18.2.3.3.2">2</cn><ci id="S4.SS1.4.p1.18.m18.2.3.3.3.cmml" xref="S4.SS1.4.p1.18.m18.2.3.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.18.m18.2c">\{0,1\}^{2n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.18.m18.2d">{ 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. We claim that <math alttext="\Lambda_{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.19.m19.1"><semantics id="S4.SS1.4.p1.19.m19.1a"><msub id="S4.SS1.4.p1.19.m19.1.1" xref="S4.SS1.4.p1.19.m19.1.1.cmml"><mi id="S4.SS1.4.p1.19.m19.1.1.2" mathvariant="normal" xref="S4.SS1.4.p1.19.m19.1.1.2.cmml">Λ</mi><mi id="S4.SS1.4.p1.19.m19.1.1.3" xref="S4.SS1.4.p1.19.m19.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.19.m19.1b"><apply id="S4.SS1.4.p1.19.m19.1.1.cmml" xref="S4.SS1.4.p1.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.19.m19.1.1.1.cmml" xref="S4.SS1.4.p1.19.m19.1.1">subscript</csymbol><ci id="S4.SS1.4.p1.19.m19.1.1.2.cmml" xref="S4.SS1.4.p1.19.m19.1.1.2">Λ</ci><ci id="S4.SS1.4.p1.19.m19.1.1.3.cmml" xref="S4.SS1.4.p1.19.m19.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.19.m19.1c">\Lambda_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.19.m19.1d">roman_Λ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> covers all semi-filters in <math alttext="\mathfrak{F}^{\uparrow}_{G}" class="ltx_Math" display="inline" id="S4.SS1.4.p1.20.m20.1"><semantics id="S4.SS1.4.p1.20.m20.1a"><msubsup id="S4.SS1.4.p1.20.m20.1.1" xref="S4.SS1.4.p1.20.m20.1.1.cmml"><mi id="S4.SS1.4.p1.20.m20.1.1.2.2" xref="S4.SS1.4.p1.20.m20.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS1.4.p1.20.m20.1.1.3" xref="S4.SS1.4.p1.20.m20.1.1.3.cmml">G</mi><mo id="S4.SS1.4.p1.20.m20.1.1.2.3" stretchy="false" xref="S4.SS1.4.p1.20.m20.1.1.2.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p1.20.m20.1b"><apply id="S4.SS1.4.p1.20.m20.1.1.cmml" xref="S4.SS1.4.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.20.m20.1.1.1.cmml" xref="S4.SS1.4.p1.20.m20.1.1">subscript</csymbol><apply id="S4.SS1.4.p1.20.m20.1.1.2.cmml" xref="S4.SS1.4.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p1.20.m20.1.1.2.1.cmml" xref="S4.SS1.4.p1.20.m20.1.1">superscript</csymbol><ci id="S4.SS1.4.p1.20.m20.1.1.2.2.cmml" xref="S4.SS1.4.p1.20.m20.1.1.2.2">𝔉</ci><ci id="S4.SS1.4.p1.20.m20.1.1.2.3.cmml" xref="S4.SS1.4.p1.20.m20.1.1.2.3">↑</ci></apply><ci id="S4.SS1.4.p1.20.m20.1.1.3.cmml" xref="S4.SS1.4.p1.20.m20.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p1.20.m20.1c">\mathfrak{F}^{\uparrow}_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p1.20.m20.1d">fraktur_F start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote11"><sup class="ltx_note_mark">11</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">11</sup><span class="ltx_tag ltx_tag_note">11</span>Note that the semi-filters in <math alttext="\mathfrak{F}_{f_{G}}^{\uparrow}" class="ltx_Math" display="inline" id="footnote11.m1.1"><semantics id="footnote11.m1.1b"><msubsup id="footnote11.m1.1.1" xref="footnote11.m1.1.1.cmml"><mi id="footnote11.m1.1.1.2.2" xref="footnote11.m1.1.1.2.2.cmml">𝔉</mi><msub id="footnote11.m1.1.1.2.3" xref="footnote11.m1.1.1.2.3.cmml"><mi id="footnote11.m1.1.1.2.3.2" xref="footnote11.m1.1.1.2.3.2.cmml">f</mi><mi id="footnote11.m1.1.1.2.3.3" xref="footnote11.m1.1.1.2.3.3.cmml">G</mi></msub><mo id="footnote11.m1.1.1.3" stretchy="false" xref="footnote11.m1.1.1.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="footnote11.m1.1c"><apply id="footnote11.m1.1.1.cmml" xref="footnote11.m1.1.1"><csymbol cd="ambiguous" id="footnote11.m1.1.1.1.cmml" xref="footnote11.m1.1.1">superscript</csymbol><apply id="footnote11.m1.1.1.2.cmml" xref="footnote11.m1.1.1"><csymbol cd="ambiguous" id="footnote11.m1.1.1.2.1.cmml" xref="footnote11.m1.1.1">subscript</csymbol><ci id="footnote11.m1.1.1.2.2.cmml" xref="footnote11.m1.1.1.2.2">𝔉</ci><apply id="footnote11.m1.1.1.2.3.cmml" xref="footnote11.m1.1.1.2.3"><csymbol cd="ambiguous" id="footnote11.m1.1.1.2.3.1.cmml" xref="footnote11.m1.1.1.2.3">subscript</csymbol><ci id="footnote11.m1.1.1.2.3.2.cmml" xref="footnote11.m1.1.1.2.3.2">𝑓</ci><ci id="footnote11.m1.1.1.2.3.3.cmml" xref="footnote11.m1.1.1.2.3.3">𝐺</ci></apply></apply><ci id="footnote11.m1.1.1.3.cmml" xref="footnote11.m1.1.1.3">↑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m1.1d">\mathfrak{F}_{f_{G}}^{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="footnote11.m1.1e">fraktur_F start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT</annotation></semantics></math> and in <math alttext="\mathfrak{F}_{G}^{\uparrow}" class="ltx_Math" display="inline" id="footnote11.m2.1"><semantics id="footnote11.m2.1b"><msubsup id="footnote11.m2.1.1" xref="footnote11.m2.1.1.cmml"><mi id="footnote11.m2.1.1.2.2" xref="footnote11.m2.1.1.2.2.cmml">𝔉</mi><mi id="footnote11.m2.1.1.2.3" xref="footnote11.m2.1.1.2.3.cmml">G</mi><mo id="footnote11.m2.1.1.3" stretchy="false" xref="footnote11.m2.1.1.3.cmml">↑</mo></msubsup><annotation-xml encoding="MathML-Content" id="footnote11.m2.1c"><apply id="footnote11.m2.1.1.cmml" xref="footnote11.m2.1.1"><csymbol cd="ambiguous" id="footnote11.m2.1.1.1.cmml" xref="footnote11.m2.1.1">superscript</csymbol><apply id="footnote11.m2.1.1.2.cmml" xref="footnote11.m2.1.1"><csymbol cd="ambiguous" id="footnote11.m2.1.1.2.1.cmml" xref="footnote11.m2.1.1">subscript</csymbol><ci id="footnote11.m2.1.1.2.2.cmml" xref="footnote11.m2.1.1.2.2">𝔉</ci><ci id="footnote11.m2.1.1.2.3.cmml" xref="footnote11.m2.1.1.2.3">𝐺</ci></apply><ci id="footnote11.m2.1.1.3.cmml" xref="footnote11.m2.1.1.3">↑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m2.1d">\mathfrak{F}_{G}^{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="footnote11.m2.1e">fraktur_F start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT</annotation></semantics></math> differ in their definitions of “above”, as they are connected to different sets of generators.</span></span></span></p> </div> <div class="ltx_para" id="S4.SS1.5.p2"> <p class="ltx_p" id="S4.SS1.5.p2.20">Recall that we identify an element <math alttext="(u,v)\in[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS1.5.p2.1.m1.4"><semantics id="S4.SS1.5.p2.1.m1.4a"><mrow id="S4.SS1.5.p2.1.m1.4.5" xref="S4.SS1.5.p2.1.m1.4.5.cmml"><mrow id="S4.SS1.5.p2.1.m1.4.5.2.2" xref="S4.SS1.5.p2.1.m1.4.5.2.1.cmml"><mo id="S4.SS1.5.p2.1.m1.4.5.2.2.1" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.2.1.cmml">(</mo><mi id="S4.SS1.5.p2.1.m1.1.1" xref="S4.SS1.5.p2.1.m1.1.1.cmml">u</mi><mo id="S4.SS1.5.p2.1.m1.4.5.2.2.2" xref="S4.SS1.5.p2.1.m1.4.5.2.1.cmml">,</mo><mi id="S4.SS1.5.p2.1.m1.2.2" xref="S4.SS1.5.p2.1.m1.2.2.cmml">v</mi><mo id="S4.SS1.5.p2.1.m1.4.5.2.2.3" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.2.1.cmml">)</mo></mrow><mo id="S4.SS1.5.p2.1.m1.4.5.1" xref="S4.SS1.5.p2.1.m1.4.5.1.cmml">∈</mo><mrow id="S4.SS1.5.p2.1.m1.4.5.3" xref="S4.SS1.5.p2.1.m1.4.5.3.cmml"><mrow id="S4.SS1.5.p2.1.m1.4.5.3.2.2" xref="S4.SS1.5.p2.1.m1.4.5.3.2.1.cmml"><mo id="S4.SS1.5.p2.1.m1.4.5.3.2.2.1" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.3.2.1.1.cmml">[</mo><mi id="S4.SS1.5.p2.1.m1.3.3" xref="S4.SS1.5.p2.1.m1.3.3.cmml">N</mi><mo id="S4.SS1.5.p2.1.m1.4.5.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.5.p2.1.m1.4.5.3.1" rspace="0.222em" xref="S4.SS1.5.p2.1.m1.4.5.3.1.cmml">×</mo><mrow id="S4.SS1.5.p2.1.m1.4.5.3.3.2" xref="S4.SS1.5.p2.1.m1.4.5.3.3.1.cmml"><mo id="S4.SS1.5.p2.1.m1.4.5.3.3.2.1" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.3.3.1.1.cmml">[</mo><mi id="S4.SS1.5.p2.1.m1.4.4" xref="S4.SS1.5.p2.1.m1.4.4.cmml">N</mi><mo id="S4.SS1.5.p2.1.m1.4.5.3.3.2.2" stretchy="false" xref="S4.SS1.5.p2.1.m1.4.5.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.1.m1.4b"><apply id="S4.SS1.5.p2.1.m1.4.5.cmml" xref="S4.SS1.5.p2.1.m1.4.5"><in id="S4.SS1.5.p2.1.m1.4.5.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.1"></in><interval closure="open" id="S4.SS1.5.p2.1.m1.4.5.2.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.2.2"><ci id="S4.SS1.5.p2.1.m1.1.1.cmml" xref="S4.SS1.5.p2.1.m1.1.1">𝑢</ci><ci id="S4.SS1.5.p2.1.m1.2.2.cmml" xref="S4.SS1.5.p2.1.m1.2.2">𝑣</ci></interval><apply id="S4.SS1.5.p2.1.m1.4.5.3.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3"><times id="S4.SS1.5.p2.1.m1.4.5.3.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3.1"></times><apply id="S4.SS1.5.p2.1.m1.4.5.3.2.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3.2.2"><csymbol cd="latexml" id="S4.SS1.5.p2.1.m1.4.5.3.2.1.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.5.p2.1.m1.3.3.cmml" xref="S4.SS1.5.p2.1.m1.3.3">𝑁</ci></apply><apply id="S4.SS1.5.p2.1.m1.4.5.3.3.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3.3.2"><csymbol cd="latexml" id="S4.SS1.5.p2.1.m1.4.5.3.3.1.1.cmml" xref="S4.SS1.5.p2.1.m1.4.5.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.5.p2.1.m1.4.4.cmml" xref="S4.SS1.5.p2.1.m1.4.4">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.1.m1.4c">(u,v)\in[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.1.m1.4d">( italic_u , italic_v ) ∈ [ italic_N ] × [ italic_N ]</annotation></semantics></math> with its corresponding input string <math alttext="\phi(u,v)=\mathsf{binary}(u)\mathsf{binary}(v)\in\{0,1\}^{2n}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.2.m2.6"><semantics id="S4.SS1.5.p2.2.m2.6a"><mrow id="S4.SS1.5.p2.2.m2.6.7" xref="S4.SS1.5.p2.2.m2.6.7.cmml"><mrow id="S4.SS1.5.p2.2.m2.6.7.2" xref="S4.SS1.5.p2.2.m2.6.7.2.cmml"><mi id="S4.SS1.5.p2.2.m2.6.7.2.2" xref="S4.SS1.5.p2.2.m2.6.7.2.2.cmml">ϕ</mi><mo id="S4.SS1.5.p2.2.m2.6.7.2.1" xref="S4.SS1.5.p2.2.m2.6.7.2.1.cmml">⁢</mo><mrow id="S4.SS1.5.p2.2.m2.6.7.2.3.2" xref="S4.SS1.5.p2.2.m2.6.7.2.3.1.cmml"><mo id="S4.SS1.5.p2.2.m2.6.7.2.3.2.1" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.2.3.1.cmml">(</mo><mi id="S4.SS1.5.p2.2.m2.1.1" xref="S4.SS1.5.p2.2.m2.1.1.cmml">u</mi><mo id="S4.SS1.5.p2.2.m2.6.7.2.3.2.2" xref="S4.SS1.5.p2.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S4.SS1.5.p2.2.m2.2.2" xref="S4.SS1.5.p2.2.m2.2.2.cmml">v</mi><mo id="S4.SS1.5.p2.2.m2.6.7.2.3.2.3" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.5.p2.2.m2.6.7.3" xref="S4.SS1.5.p2.2.m2.6.7.3.cmml">=</mo><mrow id="S4.SS1.5.p2.2.m2.6.7.4" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml"><mi id="S4.SS1.5.p2.2.m2.6.7.4.2" xref="S4.SS1.5.p2.2.m2.6.7.4.2.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo id="S4.SS1.5.p2.2.m2.6.7.4.1" xref="S4.SS1.5.p2.2.m2.6.7.4.1.cmml">⁢</mo><mrow id="S4.SS1.5.p2.2.m2.6.7.4.3.2" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml"><mo id="S4.SS1.5.p2.2.m2.6.7.4.3.2.1" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml">(</mo><mi id="S4.SS1.5.p2.2.m2.3.3" xref="S4.SS1.5.p2.2.m2.3.3.cmml">u</mi><mo id="S4.SS1.5.p2.2.m2.6.7.4.3.2.2" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml">)</mo></mrow><mo id="S4.SS1.5.p2.2.m2.6.7.4.1a" xref="S4.SS1.5.p2.2.m2.6.7.4.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.2.m2.6.7.4.4" xref="S4.SS1.5.p2.2.m2.6.7.4.4.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo id="S4.SS1.5.p2.2.m2.6.7.4.1b" xref="S4.SS1.5.p2.2.m2.6.7.4.1.cmml">⁢</mo><mrow id="S4.SS1.5.p2.2.m2.6.7.4.5.2" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml"><mo id="S4.SS1.5.p2.2.m2.6.7.4.5.2.1" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml">(</mo><mi id="S4.SS1.5.p2.2.m2.4.4" xref="S4.SS1.5.p2.2.m2.4.4.cmml">v</mi><mo id="S4.SS1.5.p2.2.m2.6.7.4.5.2.2" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.4.cmml">)</mo></mrow></mrow><mo id="S4.SS1.5.p2.2.m2.6.7.5" xref="S4.SS1.5.p2.2.m2.6.7.5.cmml">∈</mo><msup id="S4.SS1.5.p2.2.m2.6.7.6" xref="S4.SS1.5.p2.2.m2.6.7.6.cmml"><mrow id="S4.SS1.5.p2.2.m2.6.7.6.2.2" xref="S4.SS1.5.p2.2.m2.6.7.6.2.1.cmml"><mo id="S4.SS1.5.p2.2.m2.6.7.6.2.2.1" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.6.2.1.cmml">{</mo><mn id="S4.SS1.5.p2.2.m2.5.5" xref="S4.SS1.5.p2.2.m2.5.5.cmml">0</mn><mo id="S4.SS1.5.p2.2.m2.6.7.6.2.2.2" xref="S4.SS1.5.p2.2.m2.6.7.6.2.1.cmml">,</mo><mn id="S4.SS1.5.p2.2.m2.6.6" xref="S4.SS1.5.p2.2.m2.6.6.cmml">1</mn><mo id="S4.SS1.5.p2.2.m2.6.7.6.2.2.3" stretchy="false" xref="S4.SS1.5.p2.2.m2.6.7.6.2.1.cmml">}</mo></mrow><mrow id="S4.SS1.5.p2.2.m2.6.7.6.3" xref="S4.SS1.5.p2.2.m2.6.7.6.3.cmml"><mn id="S4.SS1.5.p2.2.m2.6.7.6.3.2" xref="S4.SS1.5.p2.2.m2.6.7.6.3.2.cmml">2</mn><mo id="S4.SS1.5.p2.2.m2.6.7.6.3.1" xref="S4.SS1.5.p2.2.m2.6.7.6.3.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.2.m2.6.7.6.3.3" xref="S4.SS1.5.p2.2.m2.6.7.6.3.3.cmml">n</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.2.m2.6b"><apply id="S4.SS1.5.p2.2.m2.6.7.cmml" xref="S4.SS1.5.p2.2.m2.6.7"><and id="S4.SS1.5.p2.2.m2.6.7a.cmml" xref="S4.SS1.5.p2.2.m2.6.7"></and><apply id="S4.SS1.5.p2.2.m2.6.7b.cmml" xref="S4.SS1.5.p2.2.m2.6.7"><eq id="S4.SS1.5.p2.2.m2.6.7.3.cmml" xref="S4.SS1.5.p2.2.m2.6.7.3"></eq><apply id="S4.SS1.5.p2.2.m2.6.7.2.cmml" xref="S4.SS1.5.p2.2.m2.6.7.2"><times id="S4.SS1.5.p2.2.m2.6.7.2.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.2.1"></times><ci id="S4.SS1.5.p2.2.m2.6.7.2.2.cmml" xref="S4.SS1.5.p2.2.m2.6.7.2.2">italic-ϕ</ci><interval closure="open" id="S4.SS1.5.p2.2.m2.6.7.2.3.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.2.3.2"><ci id="S4.SS1.5.p2.2.m2.1.1.cmml" xref="S4.SS1.5.p2.2.m2.1.1">𝑢</ci><ci id="S4.SS1.5.p2.2.m2.2.2.cmml" xref="S4.SS1.5.p2.2.m2.2.2">𝑣</ci></interval></apply><apply id="S4.SS1.5.p2.2.m2.6.7.4.cmml" xref="S4.SS1.5.p2.2.m2.6.7.4"><times id="S4.SS1.5.p2.2.m2.6.7.4.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.4.1"></times><ci id="S4.SS1.5.p2.2.m2.6.7.4.2.cmml" xref="S4.SS1.5.p2.2.m2.6.7.4.2">𝖻𝗂𝗇𝖺𝗋𝗒</ci><ci id="S4.SS1.5.p2.2.m2.3.3.cmml" xref="S4.SS1.5.p2.2.m2.3.3">𝑢</ci><ci id="S4.SS1.5.p2.2.m2.6.7.4.4.cmml" xref="S4.SS1.5.p2.2.m2.6.7.4.4">𝖻𝗂𝗇𝖺𝗋𝗒</ci><ci id="S4.SS1.5.p2.2.m2.4.4.cmml" xref="S4.SS1.5.p2.2.m2.4.4">𝑣</ci></apply></apply><apply id="S4.SS1.5.p2.2.m2.6.7c.cmml" xref="S4.SS1.5.p2.2.m2.6.7"><in id="S4.SS1.5.p2.2.m2.6.7.5.cmml" xref="S4.SS1.5.p2.2.m2.6.7.5"></in><share href="https://arxiv.org/html/2503.14117v1#S4.SS1.5.p2.2.m2.6.7.4.cmml" id="S4.SS1.5.p2.2.m2.6.7d.cmml" xref="S4.SS1.5.p2.2.m2.6.7"></share><apply id="S4.SS1.5.p2.2.m2.6.7.6.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6"><csymbol cd="ambiguous" id="S4.SS1.5.p2.2.m2.6.7.6.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6">superscript</csymbol><set id="S4.SS1.5.p2.2.m2.6.7.6.2.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6.2.2"><cn id="S4.SS1.5.p2.2.m2.5.5.cmml" type="integer" xref="S4.SS1.5.p2.2.m2.5.5">0</cn><cn id="S4.SS1.5.p2.2.m2.6.6.cmml" type="integer" xref="S4.SS1.5.p2.2.m2.6.6">1</cn></set><apply id="S4.SS1.5.p2.2.m2.6.7.6.3.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6.3"><times id="S4.SS1.5.p2.2.m2.6.7.6.3.1.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6.3.1"></times><cn id="S4.SS1.5.p2.2.m2.6.7.6.3.2.cmml" type="integer" xref="S4.SS1.5.p2.2.m2.6.7.6.3.2">2</cn><ci id="S4.SS1.5.p2.2.m2.6.7.6.3.3.cmml" xref="S4.SS1.5.p2.2.m2.6.7.6.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.2.m2.6c">\phi(u,v)=\mathsf{binary}(u)\mathsf{binary}(v)\in\{0,1\}^{2n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.2.m2.6d">italic_ϕ ( italic_u , italic_v ) = sansserif_binary ( italic_u ) sansserif_binary ( italic_v ) ∈ { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, which for convenience we will simply denote by <math alttext="uv" class="ltx_Math" display="inline" id="S4.SS1.5.p2.3.m3.1"><semantics id="S4.SS1.5.p2.3.m3.1a"><mrow id="S4.SS1.5.p2.3.m3.1.1" xref="S4.SS1.5.p2.3.m3.1.1.cmml"><mi id="S4.SS1.5.p2.3.m3.1.1.2" xref="S4.SS1.5.p2.3.m3.1.1.2.cmml">u</mi><mo id="S4.SS1.5.p2.3.m3.1.1.1" xref="S4.SS1.5.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.3.m3.1.1.3" xref="S4.SS1.5.p2.3.m3.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.3.m3.1b"><apply id="S4.SS1.5.p2.3.m3.1.1.cmml" xref="S4.SS1.5.p2.3.m3.1.1"><times id="S4.SS1.5.p2.3.m3.1.1.1.cmml" xref="S4.SS1.5.p2.3.m3.1.1.1"></times><ci id="S4.SS1.5.p2.3.m3.1.1.2.cmml" xref="S4.SS1.5.p2.3.m3.1.1.2">𝑢</ci><ci id="S4.SS1.5.p2.3.m3.1.1.3.cmml" xref="S4.SS1.5.p2.3.m3.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.3.m3.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.3.m3.1d">italic_u italic_v</annotation></semantics></math>. Assume this is not the case, i.e., there is a semi-filter <math alttext="\mathcal{F}\in\mathfrak{F}^{\uparrow}_{G}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.4.m4.1"><semantics id="S4.SS1.5.p2.4.m4.1a"><mrow id="S4.SS1.5.p2.4.m4.1.1" xref="S4.SS1.5.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.4.m4.1.1.2" xref="S4.SS1.5.p2.4.m4.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.5.p2.4.m4.1.1.1" xref="S4.SS1.5.p2.4.m4.1.1.1.cmml">∈</mo><msubsup id="S4.SS1.5.p2.4.m4.1.1.3" xref="S4.SS1.5.p2.4.m4.1.1.3.cmml"><mi id="S4.SS1.5.p2.4.m4.1.1.3.2.2" xref="S4.SS1.5.p2.4.m4.1.1.3.2.2.cmml">𝔉</mi><mi id="S4.SS1.5.p2.4.m4.1.1.3.3" xref="S4.SS1.5.p2.4.m4.1.1.3.3.cmml">G</mi><mo id="S4.SS1.5.p2.4.m4.1.1.3.2.3" stretchy="false" xref="S4.SS1.5.p2.4.m4.1.1.3.2.3.cmml">↑</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.4.m4.1b"><apply id="S4.SS1.5.p2.4.m4.1.1.cmml" xref="S4.SS1.5.p2.4.m4.1.1"><in id="S4.SS1.5.p2.4.m4.1.1.1.cmml" xref="S4.SS1.5.p2.4.m4.1.1.1"></in><ci id="S4.SS1.5.p2.4.m4.1.1.2.cmml" xref="S4.SS1.5.p2.4.m4.1.1.2">ℱ</ci><apply id="S4.SS1.5.p2.4.m4.1.1.3.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.5.p2.4.m4.1.1.3.1.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3">subscript</csymbol><apply id="S4.SS1.5.p2.4.m4.1.1.3.2.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.5.p2.4.m4.1.1.3.2.1.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS1.5.p2.4.m4.1.1.3.2.2.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3.2.2">𝔉</ci><ci id="S4.SS1.5.p2.4.m4.1.1.3.2.3.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3.2.3">↑</ci></apply><ci id="S4.SS1.5.p2.4.m4.1.1.3.3.cmml" xref="S4.SS1.5.p2.4.m4.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.4.m4.1c">\mathcal{F}\in\mathfrak{F}^{\uparrow}_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.4.m4.1d">caligraphic_F ∈ fraktur_F start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> that is above some edge <math alttext="(u,v)\in G" class="ltx_Math" display="inline" id="S4.SS1.5.p2.5.m5.2"><semantics id="S4.SS1.5.p2.5.m5.2a"><mrow id="S4.SS1.5.p2.5.m5.2.3" xref="S4.SS1.5.p2.5.m5.2.3.cmml"><mrow id="S4.SS1.5.p2.5.m5.2.3.2.2" xref="S4.SS1.5.p2.5.m5.2.3.2.1.cmml"><mo id="S4.SS1.5.p2.5.m5.2.3.2.2.1" stretchy="false" xref="S4.SS1.5.p2.5.m5.2.3.2.1.cmml">(</mo><mi id="S4.SS1.5.p2.5.m5.1.1" xref="S4.SS1.5.p2.5.m5.1.1.cmml">u</mi><mo id="S4.SS1.5.p2.5.m5.2.3.2.2.2" xref="S4.SS1.5.p2.5.m5.2.3.2.1.cmml">,</mo><mi id="S4.SS1.5.p2.5.m5.2.2" xref="S4.SS1.5.p2.5.m5.2.2.cmml">v</mi><mo id="S4.SS1.5.p2.5.m5.2.3.2.2.3" stretchy="false" xref="S4.SS1.5.p2.5.m5.2.3.2.1.cmml">)</mo></mrow><mo id="S4.SS1.5.p2.5.m5.2.3.1" xref="S4.SS1.5.p2.5.m5.2.3.1.cmml">∈</mo><mi id="S4.SS1.5.p2.5.m5.2.3.3" xref="S4.SS1.5.p2.5.m5.2.3.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.5.m5.2b"><apply id="S4.SS1.5.p2.5.m5.2.3.cmml" xref="S4.SS1.5.p2.5.m5.2.3"><in id="S4.SS1.5.p2.5.m5.2.3.1.cmml" xref="S4.SS1.5.p2.5.m5.2.3.1"></in><interval closure="open" id="S4.SS1.5.p2.5.m5.2.3.2.1.cmml" xref="S4.SS1.5.p2.5.m5.2.3.2.2"><ci id="S4.SS1.5.p2.5.m5.1.1.cmml" xref="S4.SS1.5.p2.5.m5.1.1">𝑢</ci><ci id="S4.SS1.5.p2.5.m5.2.2.cmml" xref="S4.SS1.5.p2.5.m5.2.2">𝑣</ci></interval><ci id="S4.SS1.5.p2.5.m5.2.3.3.cmml" xref="S4.SS1.5.p2.5.m5.2.3.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.5.m5.2c">(u,v)\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.5.m5.2d">( italic_u , italic_v ) ∈ italic_G</annotation></semantics></math> and preserves <math alttext="\Lambda_{G}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.6.m6.1"><semantics id="S4.SS1.5.p2.6.m6.1a"><msub id="S4.SS1.5.p2.6.m6.1.1" xref="S4.SS1.5.p2.6.m6.1.1.cmml"><mi id="S4.SS1.5.p2.6.m6.1.1.2" mathvariant="normal" xref="S4.SS1.5.p2.6.m6.1.1.2.cmml">Λ</mi><mi id="S4.SS1.5.p2.6.m6.1.1.3" xref="S4.SS1.5.p2.6.m6.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.6.m6.1b"><apply id="S4.SS1.5.p2.6.m6.1.1.cmml" xref="S4.SS1.5.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.6.m6.1.1.1.cmml" xref="S4.SS1.5.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.5.p2.6.m6.1.1.2.cmml" xref="S4.SS1.5.p2.6.m6.1.1.2">Λ</ci><ci id="S4.SS1.5.p2.6.m6.1.1.3.cmml" xref="S4.SS1.5.p2.6.m6.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.6.m6.1c">\Lambda_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.6.m6.1d">roman_Λ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> (in other words, it is not covered by <math alttext="\Lambda_{G}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.7.m7.1"><semantics id="S4.SS1.5.p2.7.m7.1a"><msub id="S4.SS1.5.p2.7.m7.1.1" xref="S4.SS1.5.p2.7.m7.1.1.cmml"><mi id="S4.SS1.5.p2.7.m7.1.1.2" mathvariant="normal" xref="S4.SS1.5.p2.7.m7.1.1.2.cmml">Λ</mi><mi id="S4.SS1.5.p2.7.m7.1.1.3" xref="S4.SS1.5.p2.7.m7.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.7.m7.1b"><apply id="S4.SS1.5.p2.7.m7.1.1.cmml" xref="S4.SS1.5.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.7.m7.1.1.1.cmml" xref="S4.SS1.5.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.5.p2.7.m7.1.1.2.cmml" xref="S4.SS1.5.p2.7.m7.1.1.2">Λ</ci><ci id="S4.SS1.5.p2.7.m7.1.1.3.cmml" xref="S4.SS1.5.p2.7.m7.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.7.m7.1c">\Lambda_{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.7.m7.1d">roman_Λ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math>). Let <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.8.m8.1"><semantics id="S4.SS1.5.p2.8.m8.1a"><msup id="S4.SS1.5.p2.8.m8.1.1" xref="S4.SS1.5.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.8.m8.1.1.2" xref="S4.SS1.5.p2.8.m8.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.5.p2.8.m8.1.1.3" xref="S4.SS1.5.p2.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.8.m8.1b"><apply id="S4.SS1.5.p2.8.m8.1.1.cmml" xref="S4.SS1.5.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.8.m8.1.1.1.cmml" xref="S4.SS1.5.p2.8.m8.1.1">superscript</csymbol><ci id="S4.SS1.5.p2.8.m8.1.1.2.cmml" xref="S4.SS1.5.p2.8.m8.1.1.2">ℱ</ci><ci id="S4.SS1.5.p2.8.m8.1.1.3.cmml" xref="S4.SS1.5.p2.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.8.m8.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.8.m8.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the corresponding family of subsets of <math alttext="f^{-1}_{G}(0)" class="ltx_Math" display="inline" id="S4.SS1.5.p2.9.m9.1"><semantics id="S4.SS1.5.p2.9.m9.1a"><mrow id="S4.SS1.5.p2.9.m9.1.2" xref="S4.SS1.5.p2.9.m9.1.2.cmml"><msubsup id="S4.SS1.5.p2.9.m9.1.2.2" xref="S4.SS1.5.p2.9.m9.1.2.2.cmml"><mi id="S4.SS1.5.p2.9.m9.1.2.2.2.2" xref="S4.SS1.5.p2.9.m9.1.2.2.2.2.cmml">f</mi><mi id="S4.SS1.5.p2.9.m9.1.2.2.3" xref="S4.SS1.5.p2.9.m9.1.2.2.3.cmml">G</mi><mrow id="S4.SS1.5.p2.9.m9.1.2.2.2.3" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3.cmml"><mo id="S4.SS1.5.p2.9.m9.1.2.2.2.3a" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3.cmml">−</mo><mn id="S4.SS1.5.p2.9.m9.1.2.2.2.3.2" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.5.p2.9.m9.1.2.1" xref="S4.SS1.5.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.5.p2.9.m9.1.2.3.2" xref="S4.SS1.5.p2.9.m9.1.2.cmml"><mo id="S4.SS1.5.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS1.5.p2.9.m9.1.2.cmml">(</mo><mn id="S4.SS1.5.p2.9.m9.1.1" xref="S4.SS1.5.p2.9.m9.1.1.cmml">0</mn><mo id="S4.SS1.5.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS1.5.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.9.m9.1b"><apply id="S4.SS1.5.p2.9.m9.1.2.cmml" xref="S4.SS1.5.p2.9.m9.1.2"><times id="S4.SS1.5.p2.9.m9.1.2.1.cmml" xref="S4.SS1.5.p2.9.m9.1.2.1"></times><apply id="S4.SS1.5.p2.9.m9.1.2.2.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.5.p2.9.m9.1.2.2.1.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2">subscript</csymbol><apply id="S4.SS1.5.p2.9.m9.1.2.2.2.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.5.p2.9.m9.1.2.2.2.1.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2">superscript</csymbol><ci id="S4.SS1.5.p2.9.m9.1.2.2.2.2.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2.2.2">𝑓</ci><apply id="S4.SS1.5.p2.9.m9.1.2.2.2.3.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3"><minus id="S4.SS1.5.p2.9.m9.1.2.2.2.3.1.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3"></minus><cn id="S4.SS1.5.p2.9.m9.1.2.2.2.3.2.cmml" type="integer" xref="S4.SS1.5.p2.9.m9.1.2.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.5.p2.9.m9.1.2.2.3.cmml" xref="S4.SS1.5.p2.9.m9.1.2.2.3">𝐺</ci></apply><cn id="S4.SS1.5.p2.9.m9.1.1.cmml" type="integer" xref="S4.SS1.5.p2.9.m9.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.9.m9.1c">f^{-1}_{G}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.9.m9.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math> under <math alttext="\phi" class="ltx_Math" display="inline" id="S4.SS1.5.p2.10.m10.1"><semantics id="S4.SS1.5.p2.10.m10.1a"><mi id="S4.SS1.5.p2.10.m10.1.1" xref="S4.SS1.5.p2.10.m10.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.10.m10.1b"><ci id="S4.SS1.5.p2.10.m10.1.1.cmml" xref="S4.SS1.5.p2.10.m10.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.10.m10.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.10.m10.1d">italic_ϕ</annotation></semantics></math>. Observe that <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.11.m11.1"><semantics id="S4.SS1.5.p2.11.m11.1a"><msup id="S4.SS1.5.p2.11.m11.1.1" xref="S4.SS1.5.p2.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.11.m11.1.1.2" xref="S4.SS1.5.p2.11.m11.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.5.p2.11.m11.1.1.3" xref="S4.SS1.5.p2.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.11.m11.1b"><apply id="S4.SS1.5.p2.11.m11.1.1.cmml" xref="S4.SS1.5.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.11.m11.1.1.1.cmml" xref="S4.SS1.5.p2.11.m11.1.1">superscript</csymbol><ci id="S4.SS1.5.p2.11.m11.1.1.2.cmml" xref="S4.SS1.5.p2.11.m11.1.1.2">ℱ</ci><ci id="S4.SS1.5.p2.11.m11.1.1.3.cmml" xref="S4.SS1.5.p2.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.11.m11.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.11.m11.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a semi-filter over <math alttext="f^{-1}_{G}(0)" class="ltx_Math" display="inline" id="S4.SS1.5.p2.12.m12.1"><semantics id="S4.SS1.5.p2.12.m12.1a"><mrow id="S4.SS1.5.p2.12.m12.1.2" xref="S4.SS1.5.p2.12.m12.1.2.cmml"><msubsup id="S4.SS1.5.p2.12.m12.1.2.2" xref="S4.SS1.5.p2.12.m12.1.2.2.cmml"><mi id="S4.SS1.5.p2.12.m12.1.2.2.2.2" xref="S4.SS1.5.p2.12.m12.1.2.2.2.2.cmml">f</mi><mi id="S4.SS1.5.p2.12.m12.1.2.2.3" xref="S4.SS1.5.p2.12.m12.1.2.2.3.cmml">G</mi><mrow id="S4.SS1.5.p2.12.m12.1.2.2.2.3" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3.cmml"><mo id="S4.SS1.5.p2.12.m12.1.2.2.2.3a" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3.cmml">−</mo><mn id="S4.SS1.5.p2.12.m12.1.2.2.2.3.2" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.5.p2.12.m12.1.2.1" xref="S4.SS1.5.p2.12.m12.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.5.p2.12.m12.1.2.3.2" xref="S4.SS1.5.p2.12.m12.1.2.cmml"><mo id="S4.SS1.5.p2.12.m12.1.2.3.2.1" stretchy="false" xref="S4.SS1.5.p2.12.m12.1.2.cmml">(</mo><mn id="S4.SS1.5.p2.12.m12.1.1" xref="S4.SS1.5.p2.12.m12.1.1.cmml">0</mn><mo id="S4.SS1.5.p2.12.m12.1.2.3.2.2" stretchy="false" xref="S4.SS1.5.p2.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.12.m12.1b"><apply id="S4.SS1.5.p2.12.m12.1.2.cmml" xref="S4.SS1.5.p2.12.m12.1.2"><times id="S4.SS1.5.p2.12.m12.1.2.1.cmml" xref="S4.SS1.5.p2.12.m12.1.2.1"></times><apply id="S4.SS1.5.p2.12.m12.1.2.2.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.5.p2.12.m12.1.2.2.1.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2">subscript</csymbol><apply id="S4.SS1.5.p2.12.m12.1.2.2.2.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.5.p2.12.m12.1.2.2.2.1.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2">superscript</csymbol><ci id="S4.SS1.5.p2.12.m12.1.2.2.2.2.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2.2.2">𝑓</ci><apply id="S4.SS1.5.p2.12.m12.1.2.2.2.3.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3"><minus id="S4.SS1.5.p2.12.m12.1.2.2.2.3.1.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3"></minus><cn id="S4.SS1.5.p2.12.m12.1.2.2.2.3.2.cmml" type="integer" xref="S4.SS1.5.p2.12.m12.1.2.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.5.p2.12.m12.1.2.2.3.cmml" xref="S4.SS1.5.p2.12.m12.1.2.2.3">𝐺</ci></apply><cn id="S4.SS1.5.p2.12.m12.1.1.cmml" type="integer" xref="S4.SS1.5.p2.12.m12.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.12.m12.1c">f^{-1}_{G}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.12.m12.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math>, and that it preserves <math alttext="\Lambda_{f_{G}}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.13.m13.1"><semantics id="S4.SS1.5.p2.13.m13.1a"><msub id="S4.SS1.5.p2.13.m13.1.1" xref="S4.SS1.5.p2.13.m13.1.1.cmml"><mi id="S4.SS1.5.p2.13.m13.1.1.2" mathvariant="normal" xref="S4.SS1.5.p2.13.m13.1.1.2.cmml">Λ</mi><msub id="S4.SS1.5.p2.13.m13.1.1.3" xref="S4.SS1.5.p2.13.m13.1.1.3.cmml"><mi id="S4.SS1.5.p2.13.m13.1.1.3.2" xref="S4.SS1.5.p2.13.m13.1.1.3.2.cmml">f</mi><mi id="S4.SS1.5.p2.13.m13.1.1.3.3" xref="S4.SS1.5.p2.13.m13.1.1.3.3.cmml">G</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.13.m13.1b"><apply id="S4.SS1.5.p2.13.m13.1.1.cmml" xref="S4.SS1.5.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.13.m13.1.1.1.cmml" xref="S4.SS1.5.p2.13.m13.1.1">subscript</csymbol><ci id="S4.SS1.5.p2.13.m13.1.1.2.cmml" xref="S4.SS1.5.p2.13.m13.1.1.2">Λ</ci><apply id="S4.SS1.5.p2.13.m13.1.1.3.cmml" xref="S4.SS1.5.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.5.p2.13.m13.1.1.3.1.cmml" xref="S4.SS1.5.p2.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS1.5.p2.13.m13.1.1.3.2.cmml" xref="S4.SS1.5.p2.13.m13.1.1.3.2">𝑓</ci><ci id="S4.SS1.5.p2.13.m13.1.1.3.3.cmml" xref="S4.SS1.5.p2.13.m13.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.13.m13.1c">\Lambda_{f_{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.13.m13.1d">roman_Λ start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, in order to get a contradiction it is enough to verify that <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.14.m14.1"><semantics id="S4.SS1.5.p2.14.m14.1a"><msup id="S4.SS1.5.p2.14.m14.1.1" xref="S4.SS1.5.p2.14.m14.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.14.m14.1.1.2" xref="S4.SS1.5.p2.14.m14.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.5.p2.14.m14.1.1.3" xref="S4.SS1.5.p2.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.14.m14.1b"><apply id="S4.SS1.5.p2.14.m14.1.1.cmml" xref="S4.SS1.5.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.14.m14.1.1.1.cmml" xref="S4.SS1.5.p2.14.m14.1.1">superscript</csymbol><ci id="S4.SS1.5.p2.14.m14.1.1.2.cmml" xref="S4.SS1.5.p2.14.m14.1.1.2">ℱ</ci><ci id="S4.SS1.5.p2.14.m14.1.1.3.cmml" xref="S4.SS1.5.p2.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.14.m14.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.14.m14.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is above <math alttext="uv" class="ltx_Math" display="inline" id="S4.SS1.5.p2.15.m15.1"><semantics id="S4.SS1.5.p2.15.m15.1a"><mrow id="S4.SS1.5.p2.15.m15.1.1" xref="S4.SS1.5.p2.15.m15.1.1.cmml"><mi id="S4.SS1.5.p2.15.m15.1.1.2" xref="S4.SS1.5.p2.15.m15.1.1.2.cmml">u</mi><mo id="S4.SS1.5.p2.15.m15.1.1.1" xref="S4.SS1.5.p2.15.m15.1.1.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.15.m15.1.1.3" xref="S4.SS1.5.p2.15.m15.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.15.m15.1b"><apply id="S4.SS1.5.p2.15.m15.1.1.cmml" xref="S4.SS1.5.p2.15.m15.1.1"><times id="S4.SS1.5.p2.15.m15.1.1.1.cmml" xref="S4.SS1.5.p2.15.m15.1.1.1"></times><ci id="S4.SS1.5.p2.15.m15.1.1.2.cmml" xref="S4.SS1.5.p2.15.m15.1.1.2">𝑢</ci><ci id="S4.SS1.5.p2.15.m15.1.1.3.cmml" xref="S4.SS1.5.p2.15.m15.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.15.m15.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.15.m15.1d">italic_u italic_v</annotation></semantics></math> (with respect to the family of generators <math alttext="\mathfrak{B}_{2n}\subseteq\mathcal{P}(\{0,1\}^{2n})" class="ltx_Math" display="inline" id="S4.SS1.5.p2.16.m16.3"><semantics id="S4.SS1.5.p2.16.m16.3a"><mrow id="S4.SS1.5.p2.16.m16.3.3" xref="S4.SS1.5.p2.16.m16.3.3.cmml"><msub id="S4.SS1.5.p2.16.m16.3.3.3" xref="S4.SS1.5.p2.16.m16.3.3.3.cmml"><mi id="S4.SS1.5.p2.16.m16.3.3.3.2" xref="S4.SS1.5.p2.16.m16.3.3.3.2.cmml">𝔅</mi><mrow id="S4.SS1.5.p2.16.m16.3.3.3.3" xref="S4.SS1.5.p2.16.m16.3.3.3.3.cmml"><mn id="S4.SS1.5.p2.16.m16.3.3.3.3.2" xref="S4.SS1.5.p2.16.m16.3.3.3.3.2.cmml">2</mn><mo id="S4.SS1.5.p2.16.m16.3.3.3.3.1" xref="S4.SS1.5.p2.16.m16.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.16.m16.3.3.3.3.3" xref="S4.SS1.5.p2.16.m16.3.3.3.3.3.cmml">n</mi></mrow></msub><mo id="S4.SS1.5.p2.16.m16.3.3.2" xref="S4.SS1.5.p2.16.m16.3.3.2.cmml">⊆</mo><mrow id="S4.SS1.5.p2.16.m16.3.3.1" xref="S4.SS1.5.p2.16.m16.3.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.16.m16.3.3.1.3" xref="S4.SS1.5.p2.16.m16.3.3.1.3.cmml">𝒫</mi><mo id="S4.SS1.5.p2.16.m16.3.3.1.2" xref="S4.SS1.5.p2.16.m16.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS1.5.p2.16.m16.3.3.1.1.1" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.cmml"><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.2" stretchy="false" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.cmml">(</mo><msup id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.cmml"><mrow id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.2" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.1.cmml"><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.2.1" stretchy="false" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.1.cmml">{</mo><mn id="S4.SS1.5.p2.16.m16.1.1" xref="S4.SS1.5.p2.16.m16.1.1.cmml">0</mn><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.2.2" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.1.cmml">,</mo><mn id="S4.SS1.5.p2.16.m16.2.2" xref="S4.SS1.5.p2.16.m16.2.2.cmml">1</mn><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.2.3" stretchy="false" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.1.cmml">}</mo></mrow><mrow id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.cmml"><mn id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.2" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.2.cmml">2</mn><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.1" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.3" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.3.cmml">n</mi></mrow></msup><mo id="S4.SS1.5.p2.16.m16.3.3.1.1.1.3" stretchy="false" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.16.m16.3b"><apply id="S4.SS1.5.p2.16.m16.3.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3"><subset id="S4.SS1.5.p2.16.m16.3.3.2.cmml" xref="S4.SS1.5.p2.16.m16.3.3.2"></subset><apply id="S4.SS1.5.p2.16.m16.3.3.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.5.p2.16.m16.3.3.3.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3">subscript</csymbol><ci id="S4.SS1.5.p2.16.m16.3.3.3.2.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3.2">𝔅</ci><apply id="S4.SS1.5.p2.16.m16.3.3.3.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3.3"><times id="S4.SS1.5.p2.16.m16.3.3.3.3.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3.3.1"></times><cn id="S4.SS1.5.p2.16.m16.3.3.3.3.2.cmml" type="integer" xref="S4.SS1.5.p2.16.m16.3.3.3.3.2">2</cn><ci id="S4.SS1.5.p2.16.m16.3.3.3.3.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.3.3.3">𝑛</ci></apply></apply><apply id="S4.SS1.5.p2.16.m16.3.3.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1"><times id="S4.SS1.5.p2.16.m16.3.3.1.2.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.2"></times><ci id="S4.SS1.5.p2.16.m16.3.3.1.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.3">𝒫</ci><apply id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1">superscript</csymbol><set id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.2.2"><cn id="S4.SS1.5.p2.16.m16.1.1.cmml" type="integer" xref="S4.SS1.5.p2.16.m16.1.1">0</cn><cn id="S4.SS1.5.p2.16.m16.2.2.cmml" type="integer" xref="S4.SS1.5.p2.16.m16.2.2">1</cn></set><apply id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3"><times id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.1.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.1"></times><cn id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.2.cmml" type="integer" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.2">2</cn><ci id="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.3.cmml" xref="S4.SS1.5.p2.16.m16.3.3.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.16.m16.3c">\mathfrak{B}_{2n}\subseteq\mathcal{P}(\{0,1\}^{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.16.m16.3d">fraktur_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ⊆ caligraphic_P ( { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT )</annotation></semantics></math>). This follows easily using the upward-closure of <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.17.m17.1"><semantics id="S4.SS1.5.p2.17.m17.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.17.m17.1.1" xref="S4.SS1.5.p2.17.m17.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.17.m17.1b"><ci id="S4.SS1.5.p2.17.m17.1.1.cmml" xref="S4.SS1.5.p2.17.m17.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.17.m17.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.17.m17.1d">caligraphic_F</annotation></semantics></math> and the fact that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.18.m18.1"><semantics id="S4.SS1.5.p2.18.m18.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.18.m18.1.1" xref="S4.SS1.5.p2.18.m18.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.18.m18.1b"><ci id="S4.SS1.5.p2.18.m18.1.1.cmml" xref="S4.SS1.5.p2.18.m18.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.18.m18.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.18.m18.1d">caligraphic_F</annotation></semantics></math> is above the edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS1.5.p2.19.m19.2"><semantics id="S4.SS1.5.p2.19.m19.2a"><mrow id="S4.SS1.5.p2.19.m19.2.3.2" xref="S4.SS1.5.p2.19.m19.2.3.1.cmml"><mo id="S4.SS1.5.p2.19.m19.2.3.2.1" stretchy="false" xref="S4.SS1.5.p2.19.m19.2.3.1.cmml">(</mo><mi id="S4.SS1.5.p2.19.m19.1.1" xref="S4.SS1.5.p2.19.m19.1.1.cmml">u</mi><mo id="S4.SS1.5.p2.19.m19.2.3.2.2" xref="S4.SS1.5.p2.19.m19.2.3.1.cmml">,</mo><mi id="S4.SS1.5.p2.19.m19.2.2" xref="S4.SS1.5.p2.19.m19.2.2.cmml">v</mi><mo id="S4.SS1.5.p2.19.m19.2.3.2.3" stretchy="false" xref="S4.SS1.5.p2.19.m19.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.19.m19.2b"><interval closure="open" id="S4.SS1.5.p2.19.m19.2.3.1.cmml" xref="S4.SS1.5.p2.19.m19.2.3.2"><ci id="S4.SS1.5.p2.19.m19.1.1.cmml" xref="S4.SS1.5.p2.19.m19.1.1">𝑢</ci><ci id="S4.SS1.5.p2.19.m19.2.2.cmml" xref="S4.SS1.5.p2.19.m19.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.19.m19.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.19.m19.2d">( italic_u , italic_v )</annotation></semantics></math> with respect to <math alttext="\mathcal{G}_{N,N}" class="ltx_Math" display="inline" id="S4.SS1.5.p2.20.m20.2"><semantics id="S4.SS1.5.p2.20.m20.2a"><msub id="S4.SS1.5.p2.20.m20.2.3" xref="S4.SS1.5.p2.20.m20.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p2.20.m20.2.3.2" xref="S4.SS1.5.p2.20.m20.2.3.2.cmml">𝒢</mi><mrow id="S4.SS1.5.p2.20.m20.2.2.2.4" xref="S4.SS1.5.p2.20.m20.2.2.2.3.cmml"><mi id="S4.SS1.5.p2.20.m20.1.1.1.1" xref="S4.SS1.5.p2.20.m20.1.1.1.1.cmml">N</mi><mo id="S4.SS1.5.p2.20.m20.2.2.2.4.1" xref="S4.SS1.5.p2.20.m20.2.2.2.3.cmml">,</mo><mi id="S4.SS1.5.p2.20.m20.2.2.2.2" xref="S4.SS1.5.p2.20.m20.2.2.2.2.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p2.20.m20.2b"><apply id="S4.SS1.5.p2.20.m20.2.3.cmml" xref="S4.SS1.5.p2.20.m20.2.3"><csymbol cd="ambiguous" id="S4.SS1.5.p2.20.m20.2.3.1.cmml" xref="S4.SS1.5.p2.20.m20.2.3">subscript</csymbol><ci id="S4.SS1.5.p2.20.m20.2.3.2.cmml" xref="S4.SS1.5.p2.20.m20.2.3.2">𝒢</ci><list id="S4.SS1.5.p2.20.m20.2.2.2.3.cmml" xref="S4.SS1.5.p2.20.m20.2.2.2.4"><ci id="S4.SS1.5.p2.20.m20.1.1.1.1.cmml" xref="S4.SS1.5.p2.20.m20.1.1.1.1">𝑁</ci><ci id="S4.SS1.5.p2.20.m20.2.2.2.2.cmml" xref="S4.SS1.5.p2.20.m20.2.2.2.2">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p2.20.m20.2c">\mathcal{G}_{N,N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p2.20.m20.2d">caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT</annotation></semantics></math>, as we explain next.</p> </div> <div class="ltx_para" id="S4.SS1.6.p3"> <p class="ltx_p" id="S4.SS1.6.p3.14">For instance, assume that <math alttext="u_{i}=0" class="ltx_Math" display="inline" id="S4.SS1.6.p3.1.m1.1"><semantics id="S4.SS1.6.p3.1.m1.1a"><mrow id="S4.SS1.6.p3.1.m1.1.1" xref="S4.SS1.6.p3.1.m1.1.1.cmml"><msub id="S4.SS1.6.p3.1.m1.1.1.2" xref="S4.SS1.6.p3.1.m1.1.1.2.cmml"><mi id="S4.SS1.6.p3.1.m1.1.1.2.2" xref="S4.SS1.6.p3.1.m1.1.1.2.2.cmml">u</mi><mi id="S4.SS1.6.p3.1.m1.1.1.2.3" xref="S4.SS1.6.p3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.6.p3.1.m1.1.1.1" xref="S4.SS1.6.p3.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS1.6.p3.1.m1.1.1.3" xref="S4.SS1.6.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.1.m1.1b"><apply id="S4.SS1.6.p3.1.m1.1.1.cmml" xref="S4.SS1.6.p3.1.m1.1.1"><eq id="S4.SS1.6.p3.1.m1.1.1.1.cmml" xref="S4.SS1.6.p3.1.m1.1.1.1"></eq><apply id="S4.SS1.6.p3.1.m1.1.1.2.cmml" xref="S4.SS1.6.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.1.m1.1.1.2.1.cmml" xref="S4.SS1.6.p3.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.6.p3.1.m1.1.1.2.2.cmml" xref="S4.SS1.6.p3.1.m1.1.1.2.2">𝑢</ci><ci id="S4.SS1.6.p3.1.m1.1.1.2.3.cmml" xref="S4.SS1.6.p3.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.SS1.6.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.6.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.1.m1.1c">u_{i}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.1.m1.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0</annotation></semantics></math> for some <math alttext="i\in[n]" class="ltx_Math" display="inline" id="S4.SS1.6.p3.2.m2.1"><semantics id="S4.SS1.6.p3.2.m2.1a"><mrow id="S4.SS1.6.p3.2.m2.1.2" xref="S4.SS1.6.p3.2.m2.1.2.cmml"><mi id="S4.SS1.6.p3.2.m2.1.2.2" xref="S4.SS1.6.p3.2.m2.1.2.2.cmml">i</mi><mo id="S4.SS1.6.p3.2.m2.1.2.1" xref="S4.SS1.6.p3.2.m2.1.2.1.cmml">∈</mo><mrow id="S4.SS1.6.p3.2.m2.1.2.3.2" xref="S4.SS1.6.p3.2.m2.1.2.3.1.cmml"><mo id="S4.SS1.6.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS1.6.p3.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S4.SS1.6.p3.2.m2.1.1" xref="S4.SS1.6.p3.2.m2.1.1.cmml">n</mi><mo id="S4.SS1.6.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS1.6.p3.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.2.m2.1b"><apply id="S4.SS1.6.p3.2.m2.1.2.cmml" xref="S4.SS1.6.p3.2.m2.1.2"><in id="S4.SS1.6.p3.2.m2.1.2.1.cmml" xref="S4.SS1.6.p3.2.m2.1.2.1"></in><ci id="S4.SS1.6.p3.2.m2.1.2.2.cmml" xref="S4.SS1.6.p3.2.m2.1.2.2">𝑖</ci><apply id="S4.SS1.6.p3.2.m2.1.2.3.1.cmml" xref="S4.SS1.6.p3.2.m2.1.2.3.2"><csymbol cd="latexml" id="S4.SS1.6.p3.2.m2.1.2.3.1.1.cmml" xref="S4.SS1.6.p3.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.6.p3.2.m2.1.1.cmml" xref="S4.SS1.6.p3.2.m2.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.2.m2.1c">i\in[n]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.2.m2.1d">italic_i ∈ [ italic_n ]</annotation></semantics></math>. We must prove that the corresponding set <math alttext="B_{i}^{c}\cap f^{-1}_{G}(0)\in\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.3.m3.1"><semantics id="S4.SS1.6.p3.3.m3.1a"><mrow id="S4.SS1.6.p3.3.m3.1.2" xref="S4.SS1.6.p3.3.m3.1.2.cmml"><mrow id="S4.SS1.6.p3.3.m3.1.2.2" xref="S4.SS1.6.p3.3.m3.1.2.2.cmml"><msubsup id="S4.SS1.6.p3.3.m3.1.2.2.2" xref="S4.SS1.6.p3.3.m3.1.2.2.2.cmml"><mi id="S4.SS1.6.p3.3.m3.1.2.2.2.2.2" xref="S4.SS1.6.p3.3.m3.1.2.2.2.2.2.cmml">B</mi><mi id="S4.SS1.6.p3.3.m3.1.2.2.2.2.3" xref="S4.SS1.6.p3.3.m3.1.2.2.2.2.3.cmml">i</mi><mi id="S4.SS1.6.p3.3.m3.1.2.2.2.3" xref="S4.SS1.6.p3.3.m3.1.2.2.2.3.cmml">c</mi></msubsup><mo id="S4.SS1.6.p3.3.m3.1.2.2.1" xref="S4.SS1.6.p3.3.m3.1.2.2.1.cmml">∩</mo><mrow id="S4.SS1.6.p3.3.m3.1.2.2.3" xref="S4.SS1.6.p3.3.m3.1.2.2.3.cmml"><msubsup id="S4.SS1.6.p3.3.m3.1.2.2.3.2" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.cmml"><mi id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.2" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.2.cmml">f</mi><mi id="S4.SS1.6.p3.3.m3.1.2.2.3.2.3" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.3.cmml">G</mi><mrow id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.cmml"><mo id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3a" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.cmml">−</mo><mn id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.2" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.6.p3.3.m3.1.2.2.3.1" xref="S4.SS1.6.p3.3.m3.1.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.6.p3.3.m3.1.2.2.3.3.2" xref="S4.SS1.6.p3.3.m3.1.2.2.3.cmml"><mo id="S4.SS1.6.p3.3.m3.1.2.2.3.3.2.1" stretchy="false" xref="S4.SS1.6.p3.3.m3.1.2.2.3.cmml">(</mo><mn id="S4.SS1.6.p3.3.m3.1.1" xref="S4.SS1.6.p3.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.6.p3.3.m3.1.2.2.3.3.2.2" stretchy="false" xref="S4.SS1.6.p3.3.m3.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.6.p3.3.m3.1.2.1" xref="S4.SS1.6.p3.3.m3.1.2.1.cmml">∈</mo><msup id="S4.SS1.6.p3.3.m3.1.2.3" xref="S4.SS1.6.p3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.3.m3.1.2.3.2" xref="S4.SS1.6.p3.3.m3.1.2.3.2.cmml">ℱ</mi><mo id="S4.SS1.6.p3.3.m3.1.2.3.3" xref="S4.SS1.6.p3.3.m3.1.2.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.3.m3.1b"><apply id="S4.SS1.6.p3.3.m3.1.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2"><in id="S4.SS1.6.p3.3.m3.1.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.1"></in><apply id="S4.SS1.6.p3.3.m3.1.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2"><intersect id="S4.SS1.6.p3.3.m3.1.2.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.1"></intersect><apply id="S4.SS1.6.p3.3.m3.1.2.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.3.m3.1.2.2.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2">superscript</csymbol><apply id="S4.SS1.6.p3.3.m3.1.2.2.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.3.m3.1.2.2.2.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2">subscript</csymbol><ci id="S4.SS1.6.p3.3.m3.1.2.2.2.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2.2.2">𝐵</ci><ci id="S4.SS1.6.p3.3.m3.1.2.2.2.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2.2.3">𝑖</ci></apply><ci id="S4.SS1.6.p3.3.m3.1.2.2.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.2.3">𝑐</ci></apply><apply id="S4.SS1.6.p3.3.m3.1.2.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3"><times id="S4.SS1.6.p3.3.m3.1.2.2.3.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.1"></times><apply id="S4.SS1.6.p3.3.m3.1.2.2.3.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.3.m3.1.2.2.3.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2">subscript</csymbol><apply id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2">superscript</csymbol><ci id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.2">𝑓</ci><apply id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3"><minus id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3"></minus><cn id="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.2.cmml" type="integer" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.2.3.2">1</cn></apply></apply><ci id="S4.SS1.6.p3.3.m3.1.2.2.3.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.2.3.2.3">𝐺</ci></apply><cn id="S4.SS1.6.p3.3.m3.1.1.cmml" type="integer" xref="S4.SS1.6.p3.3.m3.1.1">0</cn></apply></apply><apply id="S4.SS1.6.p3.3.m3.1.2.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.3.m3.1.2.3.1.cmml" xref="S4.SS1.6.p3.3.m3.1.2.3">superscript</csymbol><ci id="S4.SS1.6.p3.3.m3.1.2.3.2.cmml" xref="S4.SS1.6.p3.3.m3.1.2.3.2">ℱ</ci><ci id="S4.SS1.6.p3.3.m3.1.2.3.3.cmml" xref="S4.SS1.6.p3.3.m3.1.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.3.m3.1c">B_{i}^{c}\cap f^{-1}_{G}(0)\in\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 ) ∈ caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. From <math alttext="u_{i}=0" class="ltx_Math" display="inline" id="S4.SS1.6.p3.4.m4.1"><semantics id="S4.SS1.6.p3.4.m4.1a"><mrow id="S4.SS1.6.p3.4.m4.1.1" xref="S4.SS1.6.p3.4.m4.1.1.cmml"><msub id="S4.SS1.6.p3.4.m4.1.1.2" xref="S4.SS1.6.p3.4.m4.1.1.2.cmml"><mi id="S4.SS1.6.p3.4.m4.1.1.2.2" xref="S4.SS1.6.p3.4.m4.1.1.2.2.cmml">u</mi><mi id="S4.SS1.6.p3.4.m4.1.1.2.3" xref="S4.SS1.6.p3.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.6.p3.4.m4.1.1.1" xref="S4.SS1.6.p3.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS1.6.p3.4.m4.1.1.3" xref="S4.SS1.6.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.4.m4.1b"><apply id="S4.SS1.6.p3.4.m4.1.1.cmml" xref="S4.SS1.6.p3.4.m4.1.1"><eq id="S4.SS1.6.p3.4.m4.1.1.1.cmml" xref="S4.SS1.6.p3.4.m4.1.1.1"></eq><apply id="S4.SS1.6.p3.4.m4.1.1.2.cmml" xref="S4.SS1.6.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.4.m4.1.1.2.1.cmml" xref="S4.SS1.6.p3.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS1.6.p3.4.m4.1.1.2.2.cmml" xref="S4.SS1.6.p3.4.m4.1.1.2.2">𝑢</ci><ci id="S4.SS1.6.p3.4.m4.1.1.2.3.cmml" xref="S4.SS1.6.p3.4.m4.1.1.2.3">𝑖</ci></apply><cn id="S4.SS1.6.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.6.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.4.m4.1c">u_{i}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.4.m4.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0</annotation></semantics></math>, we get <math alttext="R_{u}\subseteq\phi^{-1}(B_{i}^{c})" class="ltx_Math" display="inline" id="S4.SS1.6.p3.5.m5.1"><semantics id="S4.SS1.6.p3.5.m5.1a"><mrow id="S4.SS1.6.p3.5.m5.1.1" xref="S4.SS1.6.p3.5.m5.1.1.cmml"><msub id="S4.SS1.6.p3.5.m5.1.1.3" xref="S4.SS1.6.p3.5.m5.1.1.3.cmml"><mi id="S4.SS1.6.p3.5.m5.1.1.3.2" xref="S4.SS1.6.p3.5.m5.1.1.3.2.cmml">R</mi><mi id="S4.SS1.6.p3.5.m5.1.1.3.3" xref="S4.SS1.6.p3.5.m5.1.1.3.3.cmml">u</mi></msub><mo id="S4.SS1.6.p3.5.m5.1.1.2" xref="S4.SS1.6.p3.5.m5.1.1.2.cmml">⊆</mo><mrow id="S4.SS1.6.p3.5.m5.1.1.1" xref="S4.SS1.6.p3.5.m5.1.1.1.cmml"><msup id="S4.SS1.6.p3.5.m5.1.1.1.3" xref="S4.SS1.6.p3.5.m5.1.1.1.3.cmml"><mi id="S4.SS1.6.p3.5.m5.1.1.1.3.2" xref="S4.SS1.6.p3.5.m5.1.1.1.3.2.cmml">ϕ</mi><mrow id="S4.SS1.6.p3.5.m5.1.1.1.3.3" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3.cmml"><mo id="S4.SS1.6.p3.5.m5.1.1.1.3.3a" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3.cmml">−</mo><mn id="S4.SS1.6.p3.5.m5.1.1.1.3.3.2" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS1.6.p3.5.m5.1.1.1.2" xref="S4.SS1.6.p3.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.5.m5.1.1.1.1.1" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.cmml"><mo id="S4.SS1.6.p3.5.m5.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.cmml">(</mo><msubsup id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.2" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.2.cmml">B</mi><mi id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.3" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.3" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.3.cmml">c</mi></msubsup><mo id="S4.SS1.6.p3.5.m5.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.5.m5.1b"><apply id="S4.SS1.6.p3.5.m5.1.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1"><subset id="S4.SS1.6.p3.5.m5.1.1.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.2"></subset><apply id="S4.SS1.6.p3.5.m5.1.1.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.5.m5.1.1.3.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS1.6.p3.5.m5.1.1.3.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.3.2">𝑅</ci><ci id="S4.SS1.6.p3.5.m5.1.1.3.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.3.3">𝑢</ci></apply><apply id="S4.SS1.6.p3.5.m5.1.1.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1"><times id="S4.SS1.6.p3.5.m5.1.1.1.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.2"></times><apply id="S4.SS1.6.p3.5.m5.1.1.1.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.5.m5.1.1.1.3.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.3">superscript</csymbol><ci id="S4.SS1.6.p3.5.m5.1.1.1.3.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.3.2">italic-ϕ</ci><apply id="S4.SS1.6.p3.5.m5.1.1.1.3.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3"><minus id="S4.SS1.6.p3.5.m5.1.1.1.3.3.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3"></minus><cn id="S4.SS1.6.p3.5.m5.1.1.1.3.3.2.cmml" type="integer" xref="S4.SS1.6.p3.5.m5.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1">superscript</csymbol><apply id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.2">𝐵</ci><ci id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.3.cmml" xref="S4.SS1.6.p3.5.m5.1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.5.m5.1c">R_{u}\subseteq\phi^{-1}(B_{i}^{c})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.5.m5.1d">italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ⊆ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT )</annotation></semantics></math>, and then <math alttext="R_{u}\cap\overline{G}\subseteq\phi^{-1}(B_{i}^{c})\cap\overline{G}=\phi^{-1}(B% _{i}^{c}\cap f_{G}^{-1}(0))" class="ltx_Math" display="inline" id="S4.SS1.6.p3.6.m6.3"><semantics id="S4.SS1.6.p3.6.m6.3a"><mrow id="S4.SS1.6.p3.6.m6.3.3" xref="S4.SS1.6.p3.6.m6.3.3.cmml"><mrow id="S4.SS1.6.p3.6.m6.3.3.4" xref="S4.SS1.6.p3.6.m6.3.3.4.cmml"><msub id="S4.SS1.6.p3.6.m6.3.3.4.2" xref="S4.SS1.6.p3.6.m6.3.3.4.2.cmml"><mi id="S4.SS1.6.p3.6.m6.3.3.4.2.2" xref="S4.SS1.6.p3.6.m6.3.3.4.2.2.cmml">R</mi><mi id="S4.SS1.6.p3.6.m6.3.3.4.2.3" xref="S4.SS1.6.p3.6.m6.3.3.4.2.3.cmml">u</mi></msub><mo id="S4.SS1.6.p3.6.m6.3.3.4.1" xref="S4.SS1.6.p3.6.m6.3.3.4.1.cmml">∩</mo><mover accent="true" id="S4.SS1.6.p3.6.m6.3.3.4.3" xref="S4.SS1.6.p3.6.m6.3.3.4.3.cmml"><mi id="S4.SS1.6.p3.6.m6.3.3.4.3.2" xref="S4.SS1.6.p3.6.m6.3.3.4.3.2.cmml">G</mi><mo id="S4.SS1.6.p3.6.m6.3.3.4.3.1" xref="S4.SS1.6.p3.6.m6.3.3.4.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS1.6.p3.6.m6.3.3.5" xref="S4.SS1.6.p3.6.m6.3.3.5.cmml">⊆</mo><mrow id="S4.SS1.6.p3.6.m6.2.2.1" xref="S4.SS1.6.p3.6.m6.2.2.1.cmml"><mrow id="S4.SS1.6.p3.6.m6.2.2.1.1" xref="S4.SS1.6.p3.6.m6.2.2.1.1.cmml"><msup id="S4.SS1.6.p3.6.m6.2.2.1.1.3" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.cmml"><mi id="S4.SS1.6.p3.6.m6.2.2.1.1.3.2" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.2.cmml">ϕ</mi><mrow id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.cmml"><mo id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3a" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.cmml">−</mo><mn id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.2" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS1.6.p3.6.m6.2.2.1.1.2" xref="S4.SS1.6.p3.6.m6.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.cmml"><mo id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.cmml">(</mo><msubsup id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.cmml"><mi id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.2" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.2.cmml">B</mi><mi id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.3" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.3" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.3.cmml">c</mi></msubsup><mo id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.6.p3.6.m6.2.2.1.2" xref="S4.SS1.6.p3.6.m6.2.2.1.2.cmml">∩</mo><mover accent="true" id="S4.SS1.6.p3.6.m6.2.2.1.3" xref="S4.SS1.6.p3.6.m6.2.2.1.3.cmml"><mi id="S4.SS1.6.p3.6.m6.2.2.1.3.2" xref="S4.SS1.6.p3.6.m6.2.2.1.3.2.cmml">G</mi><mo id="S4.SS1.6.p3.6.m6.2.2.1.3.1" xref="S4.SS1.6.p3.6.m6.2.2.1.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS1.6.p3.6.m6.3.3.6" xref="S4.SS1.6.p3.6.m6.3.3.6.cmml">=</mo><mrow id="S4.SS1.6.p3.6.m6.3.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.cmml"><msup id="S4.SS1.6.p3.6.m6.3.3.2.3" xref="S4.SS1.6.p3.6.m6.3.3.2.3.cmml"><mi id="S4.SS1.6.p3.6.m6.3.3.2.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.3.2.cmml">ϕ</mi><mrow id="S4.SS1.6.p3.6.m6.3.3.2.3.3" xref="S4.SS1.6.p3.6.m6.3.3.2.3.3.cmml"><mo id="S4.SS1.6.p3.6.m6.3.3.2.3.3a" xref="S4.SS1.6.p3.6.m6.3.3.2.3.3.cmml">−</mo><mn id="S4.SS1.6.p3.6.m6.3.3.2.3.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS1.6.p3.6.m6.3.3.2.2" xref="S4.SS1.6.p3.6.m6.3.3.2.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.6.m6.3.3.2.1.1" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.cmml"><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.2" stretchy="false" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.cmml"><msubsup id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.cmml"><mi id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.2.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.2.2.cmml">B</mi><mi id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.2.3" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.2.3.cmml">i</mi><mi id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.3" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.2.3.cmml">c</mi></msubsup><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.1" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.1.cmml">∩</mo><mrow id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.cmml"><msubsup id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.cmml"><mi id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.2.cmml">f</mi><mi id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.3" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.3.cmml">G</mi><mrow id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.cmml"><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3a" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.cmml">−</mo><mn id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.1" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.3.2" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.cmml"><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.3.2.1" stretchy="false" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.cmml">(</mo><mn id="S4.SS1.6.p3.6.m6.1.1" xref="S4.SS1.6.p3.6.m6.1.1.cmml">0</mn><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.3.2.2" stretchy="false" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.6.p3.6.m6.3.3.2.1.1.3" stretchy="false" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.6.m6.3b"><apply id="S4.SS1.6.p3.6.m6.3.3.cmml" xref="S4.SS1.6.p3.6.m6.3.3"><and id="S4.SS1.6.p3.6.m6.3.3a.cmml" xref="S4.SS1.6.p3.6.m6.3.3"></and><apply id="S4.SS1.6.p3.6.m6.3.3b.cmml" xref="S4.SS1.6.p3.6.m6.3.3"><subset id="S4.SS1.6.p3.6.m6.3.3.5.cmml" xref="S4.SS1.6.p3.6.m6.3.3.5"></subset><apply id="S4.SS1.6.p3.6.m6.3.3.4.cmml" xref="S4.SS1.6.p3.6.m6.3.3.4"><intersect id="S4.SS1.6.p3.6.m6.3.3.4.1.cmml" xref="S4.SS1.6.p3.6.m6.3.3.4.1"></intersect><apply id="S4.SS1.6.p3.6.m6.3.3.4.2.cmml" xref="S4.SS1.6.p3.6.m6.3.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.6.m6.3.3.4.2.1.cmml" xref="S4.SS1.6.p3.6.m6.3.3.4.2">subscript</csymbol><ci 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xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.2">italic-ϕ</ci><apply id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3"><minus id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.1.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3"></minus><cn id="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.2.cmml" type="integer" xref="S4.SS1.6.p3.6.m6.2.2.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1">superscript</csymbol><apply id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.1.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.2.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.2">𝐵</ci><ci id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.3.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.3.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.1.1.1.1.3">𝑐</ci></apply></apply><apply id="S4.SS1.6.p3.6.m6.2.2.1.3.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.3"><ci id="S4.SS1.6.p3.6.m6.2.2.1.3.1.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.3.1">¯</ci><ci id="S4.SS1.6.p3.6.m6.2.2.1.3.2.cmml" xref="S4.SS1.6.p3.6.m6.2.2.1.3.2">𝐺</ci></apply></apply></apply><apply id="S4.SS1.6.p3.6.m6.3.3c.cmml" xref="S4.SS1.6.p3.6.m6.3.3"><eq id="S4.SS1.6.p3.6.m6.3.3.6.cmml" xref="S4.SS1.6.p3.6.m6.3.3.6"></eq><share href="https://arxiv.org/html/2503.14117v1#S4.SS1.6.p3.6.m6.2.2.1.cmml" id="S4.SS1.6.p3.6.m6.3.3d.cmml" xref="S4.SS1.6.p3.6.m6.3.3"></share><apply id="S4.SS1.6.p3.6.m6.3.3.2.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2"><times id="S4.SS1.6.p3.6.m6.3.3.2.2.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.2"></times><apply id="S4.SS1.6.p3.6.m6.3.3.2.3.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.6.m6.3.3.2.3.1.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.3">superscript</csymbol><ci 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xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.2">𝑓</ci><ci id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.3.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.2.3">𝐺</ci></apply><apply id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3"><minus id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.1.cmml" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3"></minus><cn id="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.2.cmml" type="integer" xref="S4.SS1.6.p3.6.m6.3.3.2.1.1.1.3.2.3.2">1</cn></apply></apply><cn id="S4.SS1.6.p3.6.m6.1.1.cmml" type="integer" xref="S4.SS1.6.p3.6.m6.1.1">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.6.m6.3c">R_{u}\cap\overline{G}\subseteq\phi^{-1}(B_{i}^{c})\cap\overline{G}=\phi^{-1}(B% _{i}^{c}\cap f_{G}^{-1}(0))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.6.m6.3d">italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG ⊆ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) ∩ over¯ start_ARG italic_G end_ARG = italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 0 ) )</annotation></semantics></math>. Since <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.7.m7.1"><semantics id="S4.SS1.6.p3.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.7.m7.1.1" xref="S4.SS1.6.p3.7.m7.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.7.m7.1b"><ci id="S4.SS1.6.p3.7.m7.1.1.cmml" xref="S4.SS1.6.p3.7.m7.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.7.m7.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.7.m7.1d">caligraphic_F</annotation></semantics></math> is above <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS1.6.p3.8.m8.2"><semantics id="S4.SS1.6.p3.8.m8.2a"><mrow id="S4.SS1.6.p3.8.m8.2.3.2" xref="S4.SS1.6.p3.8.m8.2.3.1.cmml"><mo id="S4.SS1.6.p3.8.m8.2.3.2.1" stretchy="false" xref="S4.SS1.6.p3.8.m8.2.3.1.cmml">(</mo><mi id="S4.SS1.6.p3.8.m8.1.1" xref="S4.SS1.6.p3.8.m8.1.1.cmml">u</mi><mo id="S4.SS1.6.p3.8.m8.2.3.2.2" xref="S4.SS1.6.p3.8.m8.2.3.1.cmml">,</mo><mi id="S4.SS1.6.p3.8.m8.2.2" xref="S4.SS1.6.p3.8.m8.2.2.cmml">v</mi><mo id="S4.SS1.6.p3.8.m8.2.3.2.3" stretchy="false" xref="S4.SS1.6.p3.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.8.m8.2b"><interval closure="open" id="S4.SS1.6.p3.8.m8.2.3.1.cmml" xref="S4.SS1.6.p3.8.m8.2.3.2"><ci id="S4.SS1.6.p3.8.m8.1.1.cmml" xref="S4.SS1.6.p3.8.m8.1.1">𝑢</ci><ci id="S4.SS1.6.p3.8.m8.2.2.cmml" xref="S4.SS1.6.p3.8.m8.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.8.m8.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.8.m8.2d">( italic_u , italic_v )</annotation></semantics></math> with respect to <math alttext="\mathcal{G}_{N,N}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.9.m9.2"><semantics id="S4.SS1.6.p3.9.m9.2a"><msub id="S4.SS1.6.p3.9.m9.2.3" xref="S4.SS1.6.p3.9.m9.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.9.m9.2.3.2" xref="S4.SS1.6.p3.9.m9.2.3.2.cmml">𝒢</mi><mrow id="S4.SS1.6.p3.9.m9.2.2.2.4" xref="S4.SS1.6.p3.9.m9.2.2.2.3.cmml"><mi id="S4.SS1.6.p3.9.m9.1.1.1.1" xref="S4.SS1.6.p3.9.m9.1.1.1.1.cmml">N</mi><mo id="S4.SS1.6.p3.9.m9.2.2.2.4.1" xref="S4.SS1.6.p3.9.m9.2.2.2.3.cmml">,</mo><mi id="S4.SS1.6.p3.9.m9.2.2.2.2" xref="S4.SS1.6.p3.9.m9.2.2.2.2.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.9.m9.2b"><apply id="S4.SS1.6.p3.9.m9.2.3.cmml" xref="S4.SS1.6.p3.9.m9.2.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.9.m9.2.3.1.cmml" xref="S4.SS1.6.p3.9.m9.2.3">subscript</csymbol><ci id="S4.SS1.6.p3.9.m9.2.3.2.cmml" xref="S4.SS1.6.p3.9.m9.2.3.2">𝒢</ci><list id="S4.SS1.6.p3.9.m9.2.2.2.3.cmml" xref="S4.SS1.6.p3.9.m9.2.2.2.4"><ci id="S4.SS1.6.p3.9.m9.1.1.1.1.cmml" xref="S4.SS1.6.p3.9.m9.1.1.1.1">𝑁</ci><ci id="S4.SS1.6.p3.9.m9.2.2.2.2.cmml" xref="S4.SS1.6.p3.9.m9.2.2.2.2">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.9.m9.2c">\mathcal{G}_{N,N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.9.m9.2d">caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="R_{u}\cap\overline{G}\in\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.10.m10.1"><semantics id="S4.SS1.6.p3.10.m10.1a"><mrow id="S4.SS1.6.p3.10.m10.1.1" xref="S4.SS1.6.p3.10.m10.1.1.cmml"><mrow id="S4.SS1.6.p3.10.m10.1.1.2" xref="S4.SS1.6.p3.10.m10.1.1.2.cmml"><msub id="S4.SS1.6.p3.10.m10.1.1.2.2" xref="S4.SS1.6.p3.10.m10.1.1.2.2.cmml"><mi id="S4.SS1.6.p3.10.m10.1.1.2.2.2" xref="S4.SS1.6.p3.10.m10.1.1.2.2.2.cmml">R</mi><mi id="S4.SS1.6.p3.10.m10.1.1.2.2.3" xref="S4.SS1.6.p3.10.m10.1.1.2.2.3.cmml">u</mi></msub><mo id="S4.SS1.6.p3.10.m10.1.1.2.1" xref="S4.SS1.6.p3.10.m10.1.1.2.1.cmml">∩</mo><mover accent="true" id="S4.SS1.6.p3.10.m10.1.1.2.3" xref="S4.SS1.6.p3.10.m10.1.1.2.3.cmml"><mi id="S4.SS1.6.p3.10.m10.1.1.2.3.2" xref="S4.SS1.6.p3.10.m10.1.1.2.3.2.cmml">G</mi><mo id="S4.SS1.6.p3.10.m10.1.1.2.3.1" xref="S4.SS1.6.p3.10.m10.1.1.2.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS1.6.p3.10.m10.1.1.1" xref="S4.SS1.6.p3.10.m10.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.10.m10.1.1.3" xref="S4.SS1.6.p3.10.m10.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.10.m10.1b"><apply id="S4.SS1.6.p3.10.m10.1.1.cmml" xref="S4.SS1.6.p3.10.m10.1.1"><in id="S4.SS1.6.p3.10.m10.1.1.1.cmml" xref="S4.SS1.6.p3.10.m10.1.1.1"></in><apply id="S4.SS1.6.p3.10.m10.1.1.2.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2"><intersect id="S4.SS1.6.p3.10.m10.1.1.2.1.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.1"></intersect><apply id="S4.SS1.6.p3.10.m10.1.1.2.2.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.10.m10.1.1.2.2.1.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.2">subscript</csymbol><ci id="S4.SS1.6.p3.10.m10.1.1.2.2.2.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.2.2">𝑅</ci><ci id="S4.SS1.6.p3.10.m10.1.1.2.2.3.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.2.3">𝑢</ci></apply><apply id="S4.SS1.6.p3.10.m10.1.1.2.3.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.3"><ci id="S4.SS1.6.p3.10.m10.1.1.2.3.1.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.3.1">¯</ci><ci id="S4.SS1.6.p3.10.m10.1.1.2.3.2.cmml" xref="S4.SS1.6.p3.10.m10.1.1.2.3.2">𝐺</ci></apply></apply><ci id="S4.SS1.6.p3.10.m10.1.1.3.cmml" xref="S4.SS1.6.p3.10.m10.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.10.m10.1c">R_{u}\cap\overline{G}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.10.m10.1d">italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG ∈ caligraphic_F</annotation></semantics></math>. Consequently, <math alttext="\phi(R_{u}\cap\overline{G})\in\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.11.m11.1"><semantics id="S4.SS1.6.p3.11.m11.1a"><mrow id="S4.SS1.6.p3.11.m11.1.1" xref="S4.SS1.6.p3.11.m11.1.1.cmml"><mrow id="S4.SS1.6.p3.11.m11.1.1.1" xref="S4.SS1.6.p3.11.m11.1.1.1.cmml"><mi id="S4.SS1.6.p3.11.m11.1.1.1.3" xref="S4.SS1.6.p3.11.m11.1.1.1.3.cmml">ϕ</mi><mo id="S4.SS1.6.p3.11.m11.1.1.1.2" xref="S4.SS1.6.p3.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.11.m11.1.1.1.1.1" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.cmml"><mo id="S4.SS1.6.p3.11.m11.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.cmml"><msub id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.cmml"><mi id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.2" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.2.cmml">R</mi><mi id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.3" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.3.cmml">u</mi></msub><mo id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.1" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.1.cmml">∩</mo><mover accent="true" id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.cmml"><mi id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.2" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.2.cmml">G</mi><mo id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.1" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS1.6.p3.11.m11.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.6.p3.11.m11.1.1.2" xref="S4.SS1.6.p3.11.m11.1.1.2.cmml">∈</mo><msup id="S4.SS1.6.p3.11.m11.1.1.3" xref="S4.SS1.6.p3.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.11.m11.1.1.3.2" xref="S4.SS1.6.p3.11.m11.1.1.3.2.cmml">ℱ</mi><mo id="S4.SS1.6.p3.11.m11.1.1.3.3" xref="S4.SS1.6.p3.11.m11.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.11.m11.1b"><apply id="S4.SS1.6.p3.11.m11.1.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1"><in id="S4.SS1.6.p3.11.m11.1.1.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.2"></in><apply id="S4.SS1.6.p3.11.m11.1.1.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1"><times id="S4.SS1.6.p3.11.m11.1.1.1.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.2"></times><ci id="S4.SS1.6.p3.11.m11.1.1.1.3.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.3">italic-ϕ</ci><apply id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1"><intersect id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.1"></intersect><apply id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.2">𝑅</ci><ci id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.3.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.2.3">𝑢</ci></apply><apply id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3"><ci id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.1">¯</ci><ci id="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.1.1.1.1.3.2">𝐺</ci></apply></apply></apply><apply id="S4.SS1.6.p3.11.m11.1.1.3.cmml" xref="S4.SS1.6.p3.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.6.p3.11.m11.1.1.3.1.cmml" xref="S4.SS1.6.p3.11.m11.1.1.3">superscript</csymbol><ci id="S4.SS1.6.p3.11.m11.1.1.3.2.cmml" xref="S4.SS1.6.p3.11.m11.1.1.3.2">ℱ</ci><ci id="S4.SS1.6.p3.11.m11.1.1.3.3.cmml" xref="S4.SS1.6.p3.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.11.m11.1c">\phi(R_{u}\cap\overline{G})\in\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.11.m11.1d">italic_ϕ ( italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG ) ∈ caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Now <math alttext="\phi(R_{u}\cap\overline{G})\subseteq\phi(\phi^{-1}(B_{i}^{c}\cap f_{G}^{-1}(0)% ))=B_{i}^{c}\cap f^{-1}_{G}(0)" class="ltx_Math" display="inline" id="S4.SS1.6.p3.12.m12.4"><semantics id="S4.SS1.6.p3.12.m12.4a"><mrow id="S4.SS1.6.p3.12.m12.4.4" xref="S4.SS1.6.p3.12.m12.4.4.cmml"><mrow id="S4.SS1.6.p3.12.m12.3.3.1" xref="S4.SS1.6.p3.12.m12.3.3.1.cmml"><mi id="S4.SS1.6.p3.12.m12.3.3.1.3" xref="S4.SS1.6.p3.12.m12.3.3.1.3.cmml">ϕ</mi><mo id="S4.SS1.6.p3.12.m12.3.3.1.2" xref="S4.SS1.6.p3.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.12.m12.3.3.1.1.1" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.cmml"><mo id="S4.SS1.6.p3.12.m12.3.3.1.1.1.2" stretchy="false" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.cmml"><msub id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2.cmml"><mi id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2.2" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2.2.cmml">R</mi><mi id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2.3" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.2.3.cmml">u</mi></msub><mo id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.1" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.1.cmml">∩</mo><mover accent="true" id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3.cmml"><mi id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3.2" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3.2.cmml">G</mi><mo id="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3.1" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS1.6.p3.12.m12.3.3.1.1.1.3" stretchy="false" xref="S4.SS1.6.p3.12.m12.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.6.p3.12.m12.4.4.4" xref="S4.SS1.6.p3.12.m12.4.4.4.cmml">⊆</mo><mrow id="S4.SS1.6.p3.12.m12.4.4.2" xref="S4.SS1.6.p3.12.m12.4.4.2.cmml"><mi id="S4.SS1.6.p3.12.m12.4.4.2.3" xref="S4.SS1.6.p3.12.m12.4.4.2.3.cmml">ϕ</mi><mo id="S4.SS1.6.p3.12.m12.4.4.2.2" xref="S4.SS1.6.p3.12.m12.4.4.2.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.12.m12.4.4.2.1.1" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.cmml"><mo id="S4.SS1.6.p3.12.m12.4.4.2.1.1.2" stretchy="false" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.cmml"><msup id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.cmml"><mi id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.2" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.2.cmml">ϕ</mi><mrow id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3.cmml"><mo id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3a" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3.cmml">−</mo><mn id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3.2" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.2" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.1.1" xref="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.1.1.1.cmml"><mo id="S4.SS1.6.p3.12.m12.4.4.2.1.1.1.1.1.2" stretchy="false" 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))=B_{i}^{c}\cap f^{-1}_{G}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.12.m12.4d">italic_ϕ ( italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG ) ⊆ italic_ϕ ( italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 0 ) ) ) = italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math>, and from the upward-closure of <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.13.m13.1"><semantics id="S4.SS1.6.p3.13.m13.1a"><msup id="S4.SS1.6.p3.13.m13.1.1" xref="S4.SS1.6.p3.13.m13.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.13.m13.1.1.2" xref="S4.SS1.6.p3.13.m13.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.6.p3.13.m13.1.1.3" xref="S4.SS1.6.p3.13.m13.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.13.m13.1b"><apply id="S4.SS1.6.p3.13.m13.1.1.cmml" xref="S4.SS1.6.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.13.m13.1.1.1.cmml" xref="S4.SS1.6.p3.13.m13.1.1">superscript</csymbol><ci id="S4.SS1.6.p3.13.m13.1.1.2.cmml" xref="S4.SS1.6.p3.13.m13.1.1.2">ℱ</ci><ci id="S4.SS1.6.p3.13.m13.1.1.3.cmml" xref="S4.SS1.6.p3.13.m13.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.13.m13.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.13.m13.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, the latter set is in <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.6.p3.14.m14.1"><semantics id="S4.SS1.6.p3.14.m14.1a"><msup id="S4.SS1.6.p3.14.m14.1.1" xref="S4.SS1.6.p3.14.m14.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.6.p3.14.m14.1.1.2" xref="S4.SS1.6.p3.14.m14.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.6.p3.14.m14.1.1.3" xref="S4.SS1.6.p3.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p3.14.m14.1b"><apply id="S4.SS1.6.p3.14.m14.1.1.cmml" xref="S4.SS1.6.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p3.14.m14.1.1.1.cmml" xref="S4.SS1.6.p3.14.m14.1.1">superscript</csymbol><ci id="S4.SS1.6.p3.14.m14.1.1.2.cmml" xref="S4.SS1.6.p3.14.m14.1.1.2">ℱ</ci><ci id="S4.SS1.6.p3.14.m14.1.1.3.cmml" xref="S4.SS1.6.p3.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p3.14.m14.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p3.14.m14.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. The remaining cases are similar. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1">This result and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a> provide an alternative proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem38" title="Proposition 38 (Reducing circuit complexity lower bounds to two-dimensional cover problems). ‣ 4.1 Basic results and connections ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">38</span></a>. As we will see later in this section, establishing a direct connection among cover problems can have further benefits (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS3" title="4.3 Nondeterministic graph complexity ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4.3</span></a>).</p> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>A simple lower bound example</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.7">Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">N</mi><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml">=</mo><msup id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml"><mn id="S4.SS2.p1.1.m1.1.1.3.2" xref="S4.SS2.p1.1.m1.1.1.3.2.cmml">2</mn><mi id="S4.SS2.p1.1.m1.1.1.3.3" xref="S4.SS2.p1.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><eq id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1"></eq><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑁</ci><apply id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.3.1.cmml" xref="S4.SS2.p1.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3.2">2</cn><ci id="S4.SS2.p1.1.m1.1.1.3.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Consider the graph <math alttext="G_{\mathsf{NEQ}}\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.2"><semantics id="S4.SS2.p1.2.m2.2a"><mrow id="S4.SS2.p1.2.m2.2.3" xref="S4.SS2.p1.2.m2.2.3.cmml"><msub id="S4.SS2.p1.2.m2.2.3.2" xref="S4.SS2.p1.2.m2.2.3.2.cmml"><mi id="S4.SS2.p1.2.m2.2.3.2.2" xref="S4.SS2.p1.2.m2.2.3.2.2.cmml">G</mi><mi id="S4.SS2.p1.2.m2.2.3.2.3" xref="S4.SS2.p1.2.m2.2.3.2.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="S4.SS2.p1.2.m2.2.3.1" xref="S4.SS2.p1.2.m2.2.3.1.cmml">⊆</mo><mrow id="S4.SS2.p1.2.m2.2.3.3" xref="S4.SS2.p1.2.m2.2.3.3.cmml"><mrow id="S4.SS2.p1.2.m2.2.3.3.2.2" xref="S4.SS2.p1.2.m2.2.3.3.2.1.cmml"><mo id="S4.SS2.p1.2.m2.2.3.3.2.2.1" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.3.2.1.1.cmml">[</mo><mi id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">N</mi><mo id="S4.SS2.p1.2.m2.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS2.p1.2.m2.2.3.3.1" rspace="0.222em" xref="S4.SS2.p1.2.m2.2.3.3.1.cmml">×</mo><mrow id="S4.SS2.p1.2.m2.2.3.3.3.2" xref="S4.SS2.p1.2.m2.2.3.3.3.1.cmml"><mo id="S4.SS2.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.3.3.1.1.cmml">[</mo><mi id="S4.SS2.p1.2.m2.2.2" xref="S4.SS2.p1.2.m2.2.2.cmml">N</mi><mo id="S4.SS2.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.2b"><apply id="S4.SS2.p1.2.m2.2.3.cmml" xref="S4.SS2.p1.2.m2.2.3"><subset id="S4.SS2.p1.2.m2.2.3.1.cmml" xref="S4.SS2.p1.2.m2.2.3.1"></subset><apply id="S4.SS2.p1.2.m2.2.3.2.cmml" xref="S4.SS2.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.2.3.2.1.cmml" xref="S4.SS2.p1.2.m2.2.3.2">subscript</csymbol><ci id="S4.SS2.p1.2.m2.2.3.2.2.cmml" xref="S4.SS2.p1.2.m2.2.3.2.2">𝐺</ci><ci id="S4.SS2.p1.2.m2.2.3.2.3.cmml" xref="S4.SS2.p1.2.m2.2.3.2.3">𝖭𝖤𝖰</ci></apply><apply id="S4.SS2.p1.2.m2.2.3.3.cmml" xref="S4.SS2.p1.2.m2.2.3.3"><times id="S4.SS2.p1.2.m2.2.3.3.1.cmml" xref="S4.SS2.p1.2.m2.2.3.3.1"></times><apply id="S4.SS2.p1.2.m2.2.3.3.2.1.cmml" xref="S4.SS2.p1.2.m2.2.3.3.2.2"><csymbol cd="latexml" id="S4.SS2.p1.2.m2.2.3.3.2.1.1.cmml" xref="S4.SS2.p1.2.m2.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">𝑁</ci></apply><apply id="S4.SS2.p1.2.m2.2.3.3.3.1.cmml" xref="S4.SS2.p1.2.m2.2.3.3.3.2"><csymbol cd="latexml" id="S4.SS2.p1.2.m2.2.3.3.3.1.1.cmml" xref="S4.SS2.p1.2.m2.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS2.p1.2.m2.2.2.cmml" xref="S4.SS2.p1.2.m2.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.2c">G_{\mathsf{NEQ}}\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.2d">italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>, where <math alttext="(u,v)\in G_{\mathsf{NEQ}}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.2"><semantics id="S4.SS2.p1.3.m3.2a"><mrow id="S4.SS2.p1.3.m3.2.3" xref="S4.SS2.p1.3.m3.2.3.cmml"><mrow id="S4.SS2.p1.3.m3.2.3.2.2" xref="S4.SS2.p1.3.m3.2.3.2.1.cmml"><mo id="S4.SS2.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S4.SS2.p1.3.m3.2.3.2.1.cmml">(</mo><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">u</mi><mo id="S4.SS2.p1.3.m3.2.3.2.2.2" xref="S4.SS2.p1.3.m3.2.3.2.1.cmml">,</mo><mi id="S4.SS2.p1.3.m3.2.2" xref="S4.SS2.p1.3.m3.2.2.cmml">v</mi><mo id="S4.SS2.p1.3.m3.2.3.2.2.3" stretchy="false" xref="S4.SS2.p1.3.m3.2.3.2.1.cmml">)</mo></mrow><mo id="S4.SS2.p1.3.m3.2.3.1" xref="S4.SS2.p1.3.m3.2.3.1.cmml">∈</mo><msub id="S4.SS2.p1.3.m3.2.3.3" xref="S4.SS2.p1.3.m3.2.3.3.cmml"><mi id="S4.SS2.p1.3.m3.2.3.3.2" xref="S4.SS2.p1.3.m3.2.3.3.2.cmml">G</mi><mi id="S4.SS2.p1.3.m3.2.3.3.3" xref="S4.SS2.p1.3.m3.2.3.3.3.cmml">𝖭𝖤𝖰</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.2b"><apply id="S4.SS2.p1.3.m3.2.3.cmml" xref="S4.SS2.p1.3.m3.2.3"><in id="S4.SS2.p1.3.m3.2.3.1.cmml" xref="S4.SS2.p1.3.m3.2.3.1"></in><interval closure="open" id="S4.SS2.p1.3.m3.2.3.2.1.cmml" xref="S4.SS2.p1.3.m3.2.3.2.2"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝑢</ci><ci id="S4.SS2.p1.3.m3.2.2.cmml" xref="S4.SS2.p1.3.m3.2.2">𝑣</ci></interval><apply id="S4.SS2.p1.3.m3.2.3.3.cmml" xref="S4.SS2.p1.3.m3.2.3.3"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.2.3.3.1.cmml" xref="S4.SS2.p1.3.m3.2.3.3">subscript</csymbol><ci id="S4.SS2.p1.3.m3.2.3.3.2.cmml" xref="S4.SS2.p1.3.m3.2.3.3.2">𝐺</ci><ci id="S4.SS2.p1.3.m3.2.3.3.3.cmml" xref="S4.SS2.p1.3.m3.2.3.3.3">𝖭𝖤𝖰</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.2c">(u,v)\in G_{\mathsf{NEQ}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.2d">( italic_u , italic_v ) ∈ italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="u\neq v" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mrow id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">u</mi><mo id="S4.SS2.p1.4.m4.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.cmml">≠</mo><mi id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><neq id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1"></neq><ci id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2">𝑢</ci><ci id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">u\neq v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">italic_u ≠ italic_v</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.F3" title="In 4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a> below describes the <math alttext="N=8" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><mrow id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.p1.5.m5.1.1.2" xref="S4.SS2.p1.5.m5.1.1.2.cmml">N</mi><mo id="S4.SS2.p1.5.m5.1.1.1" xref="S4.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.5.m5.1.1.3" xref="S4.SS2.p1.5.m5.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><apply id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1"><eq id="S4.SS2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1.1"></eq><ci id="S4.SS2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.p1.5.m5.1.1.2">𝑁</ci><cn id="S4.SS2.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.p1.5.m5.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">N=8</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_N = 8</annotation></semantics></math> case. We show a tight lower bound on <math alttext="\rho(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.4"><semantics id="S4.SS2.p1.6.m6.4a"><mrow id="S4.SS2.p1.6.m6.4.4" xref="S4.SS2.p1.6.m6.4.4.cmml"><mi id="S4.SS2.p1.6.m6.4.4.4" xref="S4.SS2.p1.6.m6.4.4.4.cmml">ρ</mi><mo id="S4.SS2.p1.6.m6.4.4.3" xref="S4.SS2.p1.6.m6.4.4.3.cmml">⁢</mo><mrow id="S4.SS2.p1.6.m6.4.4.2.2" xref="S4.SS2.p1.6.m6.4.4.2.3.cmml"><mo id="S4.SS2.p1.6.m6.4.4.2.2.3" stretchy="false" xref="S4.SS2.p1.6.m6.4.4.2.3.cmml">(</mo><msub id="S4.SS2.p1.6.m6.3.3.1.1.1" xref="S4.SS2.p1.6.m6.3.3.1.1.1.cmml"><mi id="S4.SS2.p1.6.m6.3.3.1.1.1.2" xref="S4.SS2.p1.6.m6.3.3.1.1.1.2.cmml">G</mi><mi id="S4.SS2.p1.6.m6.3.3.1.1.1.3" xref="S4.SS2.p1.6.m6.3.3.1.1.1.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="S4.SS2.p1.6.m6.4.4.2.2.4" xref="S4.SS2.p1.6.m6.4.4.2.3.cmml">,</mo><msub id="S4.SS2.p1.6.m6.4.4.2.2.2" xref="S4.SS2.p1.6.m6.4.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.6.m6.4.4.2.2.2.2" xref="S4.SS2.p1.6.m6.4.4.2.2.2.2.cmml">𝒢</mi><mrow id="S4.SS2.p1.6.m6.2.2.2.4" xref="S4.SS2.p1.6.m6.2.2.2.3.cmml"><mi id="S4.SS2.p1.6.m6.1.1.1.1" xref="S4.SS2.p1.6.m6.1.1.1.1.cmml">N</mi><mo id="S4.SS2.p1.6.m6.2.2.2.4.1" xref="S4.SS2.p1.6.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p1.6.m6.2.2.2.2" xref="S4.SS2.p1.6.m6.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.SS2.p1.6.m6.4.4.2.2.5" stretchy="false" xref="S4.SS2.p1.6.m6.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.4b"><apply id="S4.SS2.p1.6.m6.4.4.cmml" xref="S4.SS2.p1.6.m6.4.4"><times id="S4.SS2.p1.6.m6.4.4.3.cmml" xref="S4.SS2.p1.6.m6.4.4.3"></times><ci id="S4.SS2.p1.6.m6.4.4.4.cmml" xref="S4.SS2.p1.6.m6.4.4.4">𝜌</ci><interval closure="open" id="S4.SS2.p1.6.m6.4.4.2.3.cmml" xref="S4.SS2.p1.6.m6.4.4.2.2"><apply id="S4.SS2.p1.6.m6.3.3.1.1.1.cmml" xref="S4.SS2.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.3.3.1.1.1.1.cmml" xref="S4.SS2.p1.6.m6.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.p1.6.m6.3.3.1.1.1.2.cmml" xref="S4.SS2.p1.6.m6.3.3.1.1.1.2">𝐺</ci><ci id="S4.SS2.p1.6.m6.3.3.1.1.1.3.cmml" xref="S4.SS2.p1.6.m6.3.3.1.1.1.3">𝖭𝖤𝖰</ci></apply><apply id="S4.SS2.p1.6.m6.4.4.2.2.2.cmml" xref="S4.SS2.p1.6.m6.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.4.4.2.2.2.1.cmml" xref="S4.SS2.p1.6.m6.4.4.2.2.2">subscript</csymbol><ci id="S4.SS2.p1.6.m6.4.4.2.2.2.2.cmml" xref="S4.SS2.p1.6.m6.4.4.2.2.2.2">𝒢</ci><list id="S4.SS2.p1.6.m6.2.2.2.3.cmml" xref="S4.SS2.p1.6.m6.2.2.2.4"><ci id="S4.SS2.p1.6.m6.1.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1.1">𝑁</ci><ci id="S4.SS2.p1.6.m6.2.2.2.2.cmml" xref="S4.SS2.p1.6.m6.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.4c">\rho(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.4d">italic_ρ ( italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>. To prove this result, we focus on a particular set of semi-filters. For convenience, we write <math alttext="G=G_{\mathsf{NEQ}}" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><mrow id="S4.SS2.p1.7.m7.1.1" xref="S4.SS2.p1.7.m7.1.1.cmml"><mi id="S4.SS2.p1.7.m7.1.1.2" xref="S4.SS2.p1.7.m7.1.1.2.cmml">G</mi><mo id="S4.SS2.p1.7.m7.1.1.1" xref="S4.SS2.p1.7.m7.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.7.m7.1.1.3" xref="S4.SS2.p1.7.m7.1.1.3.cmml"><mi id="S4.SS2.p1.7.m7.1.1.3.2" xref="S4.SS2.p1.7.m7.1.1.3.2.cmml">G</mi><mi id="S4.SS2.p1.7.m7.1.1.3.3" xref="S4.SS2.p1.7.m7.1.1.3.3.cmml">𝖭𝖤𝖰</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1"><eq id="S4.SS2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1"></eq><ci id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.2">𝐺</ci><apply id="S4.SS2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.3.1.cmml" xref="S4.SS2.p1.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.3.2.cmml" xref="S4.SS2.p1.7.m7.1.1.3.2">𝐺</ci><ci id="S4.SS2.p1.7.m7.1.1.3.3.cmml" xref="S4.SS2.p1.7.m7.1.1.3.3">𝖭𝖤𝖰</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">G=G_{\mathsf{NEQ}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">italic_G = italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F3"><svg class="ltx_picture ltx_centering" height="159.14" id="S4.F3.pic1" overflow="visible" version="1.1" width="159.14"><g fill="#000000" stroke="#000000" transform="translate(0,159.14) matrix(1 0 0 -1 0 0) translate(0.83,0) translate(0,158.31)"><g stroke-width="0.4pt"><g color="#FFFFFF" fill="#FFFFFF" stroke="#FFFFFF"><path d="M 0 0 M 0 0 L 0 -19.69 L 19.69 -19.69 L 19.69 0 Z M 19.69 -19.69" style="stroke:none"></path></g><g color="#BFBFBF" 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style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 98.43 -137.8 M 98.43 -137.8 L 98.43 -157.48 L 118.11 -157.48 L 118.11 -137.8 Z M 118.11 -157.48" style="stroke:none"></path></g><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF"><path d="M 118.11 -137.8 M 118.11 -137.8 L 118.11 -157.48 L 137.8 -157.48 L 137.8 -137.8 Z M 137.8 -157.48" style="stroke:none"></path></g><g color="#FFFFFF" fill="#FFFFFF" stroke="#FFFFFF"><path d="M 137.8 -137.8 M 137.8 -137.8 L 137.8 -157.48 L 157.48 -157.48 L 157.48 -137.8 Z M 157.48 -157.48" style="stroke:none"></path></g></g><g stroke-width="0.8pt"><path d="M 0 0 M 0 -157.48 L 157.48 -157.48 M 0 -137.8 L 157.48 -137.8 M 0 -118.11 L 157.48 -118.11 M 0 -98.43 L 157.48 -98.43 M 0 -78.74 L 157.48 -78.74 M 0 -59.06 L 157.48 -59.06 M 0 -39.37 L 157.48 -39.37 M 0 -19.69 L 157.48 -19.69 M 0 -0.01 L 157.48 -0.01 M 0 -157.48 L 0 0 M 19.69 -157.48 L 19.69 0 M 39.37 -157.48 L 39.37 0 M 59.06 -157.48 L 59.06 0 M 78.74 -157.48 L 78.74 0 M 98.43 -157.48 L 98.43 0 M 118.11 -157.48 L 118.11 0 M 137.8 -157.48 L 137.8 0 M 157.47 -157.48 L 157.47 0 M 157.48 -157.48" style="fill:none"></path></g><g stroke-width="1.2pt"><path d="M 0 0 M 0 0 L 0 -157.48 L 157.48 -157.48 L 157.48 0 Z M 157.48 -157.48" style="fill:none"></path></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>A graphical representation of <math alttext="G_{\mathsf{NEQ}}\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S4.F3.5.m1.2"><semantics id="S4.F3.5.m1.2b"><mrow id="S4.F3.5.m1.2.3" xref="S4.F3.5.m1.2.3.cmml"><msub id="S4.F3.5.m1.2.3.2" xref="S4.F3.5.m1.2.3.2.cmml"><mi id="S4.F3.5.m1.2.3.2.2" xref="S4.F3.5.m1.2.3.2.2.cmml">G</mi><mi id="S4.F3.5.m1.2.3.2.3" xref="S4.F3.5.m1.2.3.2.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="S4.F3.5.m1.2.3.1" xref="S4.F3.5.m1.2.3.1.cmml">⊆</mo><mrow id="S4.F3.5.m1.2.3.3" xref="S4.F3.5.m1.2.3.3.cmml"><mrow id="S4.F3.5.m1.2.3.3.2.2" xref="S4.F3.5.m1.2.3.3.2.1.cmml"><mo id="S4.F3.5.m1.2.3.3.2.2.1" stretchy="false" xref="S4.F3.5.m1.2.3.3.2.1.1.cmml">[</mo><mi id="S4.F3.5.m1.1.1" xref="S4.F3.5.m1.1.1.cmml">N</mi><mo id="S4.F3.5.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.F3.5.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S4.F3.5.m1.2.3.3.1" rspace="0.222em" xref="S4.F3.5.m1.2.3.3.1.cmml">×</mo><mrow id="S4.F3.5.m1.2.3.3.3.2" xref="S4.F3.5.m1.2.3.3.3.1.cmml"><mo id="S4.F3.5.m1.2.3.3.3.2.1" stretchy="false" xref="S4.F3.5.m1.2.3.3.3.1.1.cmml">[</mo><mi id="S4.F3.5.m1.2.2" xref="S4.F3.5.m1.2.2.cmml">N</mi><mo id="S4.F3.5.m1.2.3.3.3.2.2" stretchy="false" xref="S4.F3.5.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.5.m1.2c"><apply id="S4.F3.5.m1.2.3.cmml" xref="S4.F3.5.m1.2.3"><subset id="S4.F3.5.m1.2.3.1.cmml" xref="S4.F3.5.m1.2.3.1"></subset><apply id="S4.F3.5.m1.2.3.2.cmml" xref="S4.F3.5.m1.2.3.2"><csymbol cd="ambiguous" id="S4.F3.5.m1.2.3.2.1.cmml" xref="S4.F3.5.m1.2.3.2">subscript</csymbol><ci id="S4.F3.5.m1.2.3.2.2.cmml" xref="S4.F3.5.m1.2.3.2.2">𝐺</ci><ci id="S4.F3.5.m1.2.3.2.3.cmml" xref="S4.F3.5.m1.2.3.2.3">𝖭𝖤𝖰</ci></apply><apply id="S4.F3.5.m1.2.3.3.cmml" xref="S4.F3.5.m1.2.3.3"><times id="S4.F3.5.m1.2.3.3.1.cmml" xref="S4.F3.5.m1.2.3.3.1"></times><apply id="S4.F3.5.m1.2.3.3.2.1.cmml" xref="S4.F3.5.m1.2.3.3.2.2"><csymbol cd="latexml" id="S4.F3.5.m1.2.3.3.2.1.1.cmml" xref="S4.F3.5.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S4.F3.5.m1.1.1.cmml" xref="S4.F3.5.m1.1.1">𝑁</ci></apply><apply id="S4.F3.5.m1.2.3.3.3.1.cmml" xref="S4.F3.5.m1.2.3.3.3.2"><csymbol cd="latexml" id="S4.F3.5.m1.2.3.3.3.1.1.cmml" xref="S4.F3.5.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S4.F3.5.m1.2.2.cmml" xref="S4.F3.5.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.5.m1.2d">G_{\mathsf{NEQ}}\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.F3.5.m1.2e">italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> for <math alttext="N=8" class="ltx_Math" display="inline" id="S4.F3.6.m2.1"><semantics id="S4.F3.6.m2.1b"><mrow id="S4.F3.6.m2.1.1" xref="S4.F3.6.m2.1.1.cmml"><mi id="S4.F3.6.m2.1.1.2" xref="S4.F3.6.m2.1.1.2.cmml">N</mi><mo id="S4.F3.6.m2.1.1.1" xref="S4.F3.6.m2.1.1.1.cmml">=</mo><mn id="S4.F3.6.m2.1.1.3" xref="S4.F3.6.m2.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.6.m2.1c"><apply id="S4.F3.6.m2.1.1.cmml" xref="S4.F3.6.m2.1.1"><eq id="S4.F3.6.m2.1.1.1.cmml" xref="S4.F3.6.m2.1.1.1"></eq><ci id="S4.F3.6.m2.1.1.2.cmml" xref="S4.F3.6.m2.1.1.2">𝑁</ci><cn id="S4.F3.6.m2.1.1.3.cmml" type="integer" xref="S4.F3.6.m2.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.6.m2.1d">N=8</annotation><annotation encoding="application/x-llamapun" id="S4.F3.6.m2.1e">italic_N = 8</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem40" title="Proposition 40. ‣ 4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">40</span></a> shows that for this value of <math alttext="N" class="ltx_Math" display="inline" id="S4.F3.7.m3.1"><semantics id="S4.F3.7.m3.1b"><mi id="S4.F3.7.m3.1.1" xref="S4.F3.7.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.F3.7.m3.1c"><ci id="S4.F3.7.m3.1.1.cmml" xref="S4.F3.7.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.7.m3.1d">N</annotation><annotation encoding="application/x-llamapun" id="S4.F3.7.m3.1e">italic_N</annotation></semantics></math> the intersection complexity is <math alttext="3" class="ltx_Math" display="inline" id="S4.F3.8.m4.1"><semantics id="S4.F3.8.m4.1b"><mn id="S4.F3.8.m4.1.1" xref="S4.F3.8.m4.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S4.F3.8.m4.1c"><cn id="S4.F3.8.m4.1.1.cmml" type="integer" xref="S4.F3.8.m4.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.8.m4.1d">3</annotation><annotation encoding="application/x-llamapun" id="S4.F3.8.m4.1e">3</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.14">For <math alttext="e\in G" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">e</mi><mo id="S4.SS2.p2.1.m1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.cmml">∈</mo><mi id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><in id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></in><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑒</ci><ci id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">e\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_e ∈ italic_G</annotation></semantics></math>, where <math alttext="e=(u,v)" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.2"><semantics id="S4.SS2.p2.2.m2.2a"><mrow id="S4.SS2.p2.2.m2.2.3" xref="S4.SS2.p2.2.m2.2.3.cmml"><mi id="S4.SS2.p2.2.m2.2.3.2" xref="S4.SS2.p2.2.m2.2.3.2.cmml">e</mi><mo id="S4.SS2.p2.2.m2.2.3.1" xref="S4.SS2.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS2.p2.2.m2.2.3.3.2" xref="S4.SS2.p2.2.m2.2.3.3.1.cmml"><mo id="S4.SS2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS2.p2.2.m2.2.3.3.1.cmml">(</mo><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">u</mi><mo id="S4.SS2.p2.2.m2.2.3.3.2.2" xref="S4.SS2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.SS2.p2.2.m2.2.2" xref="S4.SS2.p2.2.m2.2.2.cmml">v</mi><mo id="S4.SS2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS2.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.2b"><apply id="S4.SS2.p2.2.m2.2.3.cmml" xref="S4.SS2.p2.2.m2.2.3"><eq id="S4.SS2.p2.2.m2.2.3.1.cmml" xref="S4.SS2.p2.2.m2.2.3.1"></eq><ci id="S4.SS2.p2.2.m2.2.3.2.cmml" xref="S4.SS2.p2.2.m2.2.3.2">𝑒</ci><interval closure="open" id="S4.SS2.p2.2.m2.2.3.3.1.cmml" xref="S4.SS2.p2.2.m2.2.3.3.2"><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑢</ci><ci id="S4.SS2.p2.2.m2.2.2.cmml" xref="S4.SS2.p2.2.m2.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.2c">e=(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.2d">italic_e = ( italic_u , italic_v )</annotation></semantics></math>, we let <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><msub id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.3.m3.1.1.2" xref="S4.SS2.p2.3.m3.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.p2.3.m3.1.1.3" xref="S4.SS2.p2.3.m3.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2">ℱ</ci><ci id="S4.SS2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.p2.3.m3.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> be the upward closure (with respect to <math alttext="\overline{G})" class="ltx_math_unparsed" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mrow id="S4.SS2.p2.4.m4.1b"><mover accent="true" id="S4.SS2.p2.4.m4.1.1"><mi id="S4.SS2.p2.4.m4.1.1.2">G</mi><mo id="S4.SS2.p2.4.m4.1.1.1">¯</mo></mover><mo id="S4.SS2.p2.4.m4.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\overline{G})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">over¯ start_ARG italic_G end_ARG )</annotation></semantics></math> of the family that contains the sets <math alttext="R^{u}_{\overline{G}}" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.1"><semantics id="S4.SS2.p2.5.m5.1a"><msubsup id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml"><mi id="S4.SS2.p2.5.m5.1.1.2.2" xref="S4.SS2.p2.5.m5.1.1.2.2.cmml">R</mi><mover accent="true" id="S4.SS2.p2.5.m5.1.1.3" xref="S4.SS2.p2.5.m5.1.1.3.cmml"><mi id="S4.SS2.p2.5.m5.1.1.3.2" xref="S4.SS2.p2.5.m5.1.1.3.2.cmml">G</mi><mo id="S4.SS2.p2.5.m5.1.1.3.1" xref="S4.SS2.p2.5.m5.1.1.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.5.m5.1.1.2.3" xref="S4.SS2.p2.5.m5.1.1.2.3.cmml">u</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.1b"><apply id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m5.1.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1">subscript</csymbol><apply id="S4.SS2.p2.5.m5.1.1.2.cmml" xref="S4.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m5.1.1.2.1.cmml" xref="S4.SS2.p2.5.m5.1.1">superscript</csymbol><ci id="S4.SS2.p2.5.m5.1.1.2.2.cmml" xref="S4.SS2.p2.5.m5.1.1.2.2">𝑅</ci><ci id="S4.SS2.p2.5.m5.1.1.2.3.cmml" xref="S4.SS2.p2.5.m5.1.1.2.3">𝑢</ci></apply><apply id="S4.SS2.p2.5.m5.1.1.3.cmml" xref="S4.SS2.p2.5.m5.1.1.3"><ci id="S4.SS2.p2.5.m5.1.1.3.1.cmml" xref="S4.SS2.p2.5.m5.1.1.3.1">¯</ci><ci id="S4.SS2.p2.5.m5.1.1.3.2.cmml" xref="S4.SS2.p2.5.m5.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.1c">R^{u}_{\overline{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="C^{v}_{\overline{G}}" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.1"><semantics id="S4.SS2.p2.6.m6.1a"><msubsup id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml"><mi id="S4.SS2.p2.6.m6.1.1.2.2" xref="S4.SS2.p2.6.m6.1.1.2.2.cmml">C</mi><mover accent="true" id="S4.SS2.p2.6.m6.1.1.3" xref="S4.SS2.p2.6.m6.1.1.3.cmml"><mi id="S4.SS2.p2.6.m6.1.1.3.2" xref="S4.SS2.p2.6.m6.1.1.3.2.cmml">G</mi><mo id="S4.SS2.p2.6.m6.1.1.3.1" xref="S4.SS2.p2.6.m6.1.1.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.6.m6.1.1.2.3" xref="S4.SS2.p2.6.m6.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.1b"><apply id="S4.SS2.p2.6.m6.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1">subscript</csymbol><apply id="S4.SS2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m6.1.1.2.1.cmml" xref="S4.SS2.p2.6.m6.1.1">superscript</csymbol><ci id="S4.SS2.p2.6.m6.1.1.2.2.cmml" xref="S4.SS2.p2.6.m6.1.1.2.2">𝐶</ci><ci id="S4.SS2.p2.6.m6.1.1.2.3.cmml" xref="S4.SS2.p2.6.m6.1.1.2.3">𝑣</ci></apply><apply id="S4.SS2.p2.6.m6.1.1.3.cmml" xref="S4.SS2.p2.6.m6.1.1.3"><ci id="S4.SS2.p2.6.m6.1.1.3.1.cmml" xref="S4.SS2.p2.6.m6.1.1.3.1">¯</ci><ci id="S4.SS2.p2.6.m6.1.1.3.2.cmml" xref="S4.SS2.p2.6.m6.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.1c">C^{v}_{\overline{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="R^{u}_{\overline{G}}=R_{u}\cap\overline{G}" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><mrow id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml"><msubsup id="S4.SS2.p2.7.m7.1.1.2" xref="S4.SS2.p2.7.m7.1.1.2.cmml"><mi id="S4.SS2.p2.7.m7.1.1.2.2.2" xref="S4.SS2.p2.7.m7.1.1.2.2.2.cmml">R</mi><mover accent="true" id="S4.SS2.p2.7.m7.1.1.2.3" xref="S4.SS2.p2.7.m7.1.1.2.3.cmml"><mi id="S4.SS2.p2.7.m7.1.1.2.3.2" xref="S4.SS2.p2.7.m7.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.p2.7.m7.1.1.2.3.1" xref="S4.SS2.p2.7.m7.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.7.m7.1.1.2.2.3" xref="S4.SS2.p2.7.m7.1.1.2.2.3.cmml">u</mi></msubsup><mo id="S4.SS2.p2.7.m7.1.1.1" xref="S4.SS2.p2.7.m7.1.1.1.cmml">=</mo><mrow id="S4.SS2.p2.7.m7.1.1.3" xref="S4.SS2.p2.7.m7.1.1.3.cmml"><msub id="S4.SS2.p2.7.m7.1.1.3.2" xref="S4.SS2.p2.7.m7.1.1.3.2.cmml"><mi id="S4.SS2.p2.7.m7.1.1.3.2.2" xref="S4.SS2.p2.7.m7.1.1.3.2.2.cmml">R</mi><mi id="S4.SS2.p2.7.m7.1.1.3.2.3" xref="S4.SS2.p2.7.m7.1.1.3.2.3.cmml">u</mi></msub><mo id="S4.SS2.p2.7.m7.1.1.3.1" xref="S4.SS2.p2.7.m7.1.1.3.1.cmml">∩</mo><mover accent="true" id="S4.SS2.p2.7.m7.1.1.3.3" xref="S4.SS2.p2.7.m7.1.1.3.3.cmml"><mi id="S4.SS2.p2.7.m7.1.1.3.3.2" xref="S4.SS2.p2.7.m7.1.1.3.3.2.cmml">G</mi><mo id="S4.SS2.p2.7.m7.1.1.3.3.1" xref="S4.SS2.p2.7.m7.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><apply id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1"><eq id="S4.SS2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1.1"></eq><apply id="S4.SS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.1.1.2.1.cmml" xref="S4.SS2.p2.7.m7.1.1.2">subscript</csymbol><apply id="S4.SS2.p2.7.m7.1.1.2.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.1.1.2.2.1.cmml" xref="S4.SS2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.7.m7.1.1.2.2.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2.2.2">𝑅</ci><ci id="S4.SS2.p2.7.m7.1.1.2.2.3.cmml" xref="S4.SS2.p2.7.m7.1.1.2.2.3">𝑢</ci></apply><apply id="S4.SS2.p2.7.m7.1.1.2.3.cmml" xref="S4.SS2.p2.7.m7.1.1.2.3"><ci id="S4.SS2.p2.7.m7.1.1.2.3.1.cmml" xref="S4.SS2.p2.7.m7.1.1.2.3.1">¯</ci><ci id="S4.SS2.p2.7.m7.1.1.2.3.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2.3.2">𝐺</ci></apply></apply><apply id="S4.SS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3"><intersect id="S4.SS2.p2.7.m7.1.1.3.1.cmml" xref="S4.SS2.p2.7.m7.1.1.3.1"></intersect><apply id="S4.SS2.p2.7.m7.1.1.3.2.cmml" xref="S4.SS2.p2.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.1.1.3.2.1.cmml" xref="S4.SS2.p2.7.m7.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p2.7.m7.1.1.3.2.2.cmml" xref="S4.SS2.p2.7.m7.1.1.3.2.2">𝑅</ci><ci id="S4.SS2.p2.7.m7.1.1.3.2.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3.2.3">𝑢</ci></apply><apply id="S4.SS2.p2.7.m7.1.1.3.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3.3"><ci id="S4.SS2.p2.7.m7.1.1.3.3.1.cmml" xref="S4.SS2.p2.7.m7.1.1.3.3.1">¯</ci><ci id="S4.SS2.p2.7.m7.1.1.3.3.2.cmml" xref="S4.SS2.p2.7.m7.1.1.3.3.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">R^{u}_{\overline{G}}=R_{u}\cap\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG</annotation></semantics></math> and <math alttext="C^{v}_{\overline{G}}=C_{v}\cap\overline{G}" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.1"><semantics id="S4.SS2.p2.8.m8.1a"><mrow id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml"><msubsup id="S4.SS2.p2.8.m8.1.1.2" xref="S4.SS2.p2.8.m8.1.1.2.cmml"><mi id="S4.SS2.p2.8.m8.1.1.2.2.2" xref="S4.SS2.p2.8.m8.1.1.2.2.2.cmml">C</mi><mover accent="true" id="S4.SS2.p2.8.m8.1.1.2.3" xref="S4.SS2.p2.8.m8.1.1.2.3.cmml"><mi id="S4.SS2.p2.8.m8.1.1.2.3.2" xref="S4.SS2.p2.8.m8.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.p2.8.m8.1.1.2.3.1" xref="S4.SS2.p2.8.m8.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.8.m8.1.1.2.2.3" xref="S4.SS2.p2.8.m8.1.1.2.2.3.cmml">v</mi></msubsup><mo id="S4.SS2.p2.8.m8.1.1.1" xref="S4.SS2.p2.8.m8.1.1.1.cmml">=</mo><mrow id="S4.SS2.p2.8.m8.1.1.3" xref="S4.SS2.p2.8.m8.1.1.3.cmml"><msub id="S4.SS2.p2.8.m8.1.1.3.2" xref="S4.SS2.p2.8.m8.1.1.3.2.cmml"><mi id="S4.SS2.p2.8.m8.1.1.3.2.2" xref="S4.SS2.p2.8.m8.1.1.3.2.2.cmml">C</mi><mi id="S4.SS2.p2.8.m8.1.1.3.2.3" xref="S4.SS2.p2.8.m8.1.1.3.2.3.cmml">v</mi></msub><mo id="S4.SS2.p2.8.m8.1.1.3.1" xref="S4.SS2.p2.8.m8.1.1.3.1.cmml">∩</mo><mover accent="true" id="S4.SS2.p2.8.m8.1.1.3.3" xref="S4.SS2.p2.8.m8.1.1.3.3.cmml"><mi id="S4.SS2.p2.8.m8.1.1.3.3.2" xref="S4.SS2.p2.8.m8.1.1.3.3.2.cmml">G</mi><mo id="S4.SS2.p2.8.m8.1.1.3.3.1" xref="S4.SS2.p2.8.m8.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.1b"><apply id="S4.SS2.p2.8.m8.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1"><eq id="S4.SS2.p2.8.m8.1.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1.1"></eq><apply id="S4.SS2.p2.8.m8.1.1.2.cmml" xref="S4.SS2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.1.1.2.1.cmml" xref="S4.SS2.p2.8.m8.1.1.2">subscript</csymbol><apply id="S4.SS2.p2.8.m8.1.1.2.2.cmml" xref="S4.SS2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.1.1.2.2.1.cmml" xref="S4.SS2.p2.8.m8.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.8.m8.1.1.2.2.2.cmml" xref="S4.SS2.p2.8.m8.1.1.2.2.2">𝐶</ci><ci id="S4.SS2.p2.8.m8.1.1.2.2.3.cmml" xref="S4.SS2.p2.8.m8.1.1.2.2.3">𝑣</ci></apply><apply id="S4.SS2.p2.8.m8.1.1.2.3.cmml" xref="S4.SS2.p2.8.m8.1.1.2.3"><ci id="S4.SS2.p2.8.m8.1.1.2.3.1.cmml" xref="S4.SS2.p2.8.m8.1.1.2.3.1">¯</ci><ci id="S4.SS2.p2.8.m8.1.1.2.3.2.cmml" xref="S4.SS2.p2.8.m8.1.1.2.3.2">𝐺</ci></apply></apply><apply id="S4.SS2.p2.8.m8.1.1.3.cmml" xref="S4.SS2.p2.8.m8.1.1.3"><intersect id="S4.SS2.p2.8.m8.1.1.3.1.cmml" xref="S4.SS2.p2.8.m8.1.1.3.1"></intersect><apply id="S4.SS2.p2.8.m8.1.1.3.2.cmml" xref="S4.SS2.p2.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.1.1.3.2.1.cmml" xref="S4.SS2.p2.8.m8.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p2.8.m8.1.1.3.2.2.cmml" xref="S4.SS2.p2.8.m8.1.1.3.2.2">𝐶</ci><ci id="S4.SS2.p2.8.m8.1.1.3.2.3.cmml" xref="S4.SS2.p2.8.m8.1.1.3.2.3">𝑣</ci></apply><apply id="S4.SS2.p2.8.m8.1.1.3.3.cmml" xref="S4.SS2.p2.8.m8.1.1.3.3"><ci id="S4.SS2.p2.8.m8.1.1.3.3.1.cmml" xref="S4.SS2.p2.8.m8.1.1.3.3.1">¯</ci><ci id="S4.SS2.p2.8.m8.1.1.3.3.2.cmml" xref="S4.SS2.p2.8.m8.1.1.3.3.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.1c">C^{v}_{\overline{G}}=C_{v}\cap\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_G end_ARG</annotation></semantics></math>. More explicitly, a set <math alttext="W" class="ltx_Math" display="inline" id="S4.SS2.p2.9.m9.1"><semantics id="S4.SS2.p2.9.m9.1a"><mi id="S4.SS2.p2.9.m9.1.1" xref="S4.SS2.p2.9.m9.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.9.m9.1b"><ci id="S4.SS2.p2.9.m9.1.1.cmml" xref="S4.SS2.p2.9.m9.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.1d">italic_W</annotation></semantics></math> is in <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.p2.10.m10.1"><semantics id="S4.SS2.p2.10.m10.1a"><msub id="S4.SS2.p2.10.m10.1.1" xref="S4.SS2.p2.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.10.m10.1.1.2" xref="S4.SS2.p2.10.m10.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.p2.10.m10.1.1.3" xref="S4.SS2.p2.10.m10.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.10.m10.1b"><apply id="S4.SS2.p2.10.m10.1.1.cmml" xref="S4.SS2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.10.m10.1.1.1.cmml" xref="S4.SS2.p2.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.p2.10.m10.1.1.2.cmml" xref="S4.SS2.p2.10.m10.1.1.2">ℱ</ci><ci id="S4.SS2.p2.10.m10.1.1.3.cmml" xref="S4.SS2.p2.10.m10.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.10.m10.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.10.m10.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="R^{u}_{\overline{G}}\subseteq W" class="ltx_Math" display="inline" id="S4.SS2.p2.11.m11.1"><semantics id="S4.SS2.p2.11.m11.1a"><mrow id="S4.SS2.p2.11.m11.1.1" xref="S4.SS2.p2.11.m11.1.1.cmml"><msubsup id="S4.SS2.p2.11.m11.1.1.2" xref="S4.SS2.p2.11.m11.1.1.2.cmml"><mi id="S4.SS2.p2.11.m11.1.1.2.2.2" xref="S4.SS2.p2.11.m11.1.1.2.2.2.cmml">R</mi><mover accent="true" id="S4.SS2.p2.11.m11.1.1.2.3" xref="S4.SS2.p2.11.m11.1.1.2.3.cmml"><mi id="S4.SS2.p2.11.m11.1.1.2.3.2" xref="S4.SS2.p2.11.m11.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.p2.11.m11.1.1.2.3.1" xref="S4.SS2.p2.11.m11.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.11.m11.1.1.2.2.3" xref="S4.SS2.p2.11.m11.1.1.2.2.3.cmml">u</mi></msubsup><mo id="S4.SS2.p2.11.m11.1.1.1" xref="S4.SS2.p2.11.m11.1.1.1.cmml">⊆</mo><mi id="S4.SS2.p2.11.m11.1.1.3" xref="S4.SS2.p2.11.m11.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.11.m11.1b"><apply id="S4.SS2.p2.11.m11.1.1.cmml" xref="S4.SS2.p2.11.m11.1.1"><subset id="S4.SS2.p2.11.m11.1.1.1.cmml" xref="S4.SS2.p2.11.m11.1.1.1"></subset><apply id="S4.SS2.p2.11.m11.1.1.2.cmml" xref="S4.SS2.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.11.m11.1.1.2.1.cmml" xref="S4.SS2.p2.11.m11.1.1.2">subscript</csymbol><apply id="S4.SS2.p2.11.m11.1.1.2.2.cmml" xref="S4.SS2.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.11.m11.1.1.2.2.1.cmml" xref="S4.SS2.p2.11.m11.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.11.m11.1.1.2.2.2.cmml" xref="S4.SS2.p2.11.m11.1.1.2.2.2">𝑅</ci><ci id="S4.SS2.p2.11.m11.1.1.2.2.3.cmml" xref="S4.SS2.p2.11.m11.1.1.2.2.3">𝑢</ci></apply><apply id="S4.SS2.p2.11.m11.1.1.2.3.cmml" xref="S4.SS2.p2.11.m11.1.1.2.3"><ci id="S4.SS2.p2.11.m11.1.1.2.3.1.cmml" xref="S4.SS2.p2.11.m11.1.1.2.3.1">¯</ci><ci id="S4.SS2.p2.11.m11.1.1.2.3.2.cmml" xref="S4.SS2.p2.11.m11.1.1.2.3.2">𝐺</ci></apply></apply><ci id="S4.SS2.p2.11.m11.1.1.3.cmml" xref="S4.SS2.p2.11.m11.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.11.m11.1c">R^{u}_{\overline{G}}\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.11.m11.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊆ italic_W</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4.SS2.p2.14.1">or</em> <math alttext="C^{v}_{\overline{G}}\subseteq W" class="ltx_Math" display="inline" id="S4.SS2.p2.12.m12.1"><semantics id="S4.SS2.p2.12.m12.1a"><mrow id="S4.SS2.p2.12.m12.1.1" xref="S4.SS2.p2.12.m12.1.1.cmml"><msubsup id="S4.SS2.p2.12.m12.1.1.2" xref="S4.SS2.p2.12.m12.1.1.2.cmml"><mi id="S4.SS2.p2.12.m12.1.1.2.2.2" xref="S4.SS2.p2.12.m12.1.1.2.2.2.cmml">C</mi><mover accent="true" id="S4.SS2.p2.12.m12.1.1.2.3" xref="S4.SS2.p2.12.m12.1.1.2.3.cmml"><mi id="S4.SS2.p2.12.m12.1.1.2.3.2" xref="S4.SS2.p2.12.m12.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.p2.12.m12.1.1.2.3.1" xref="S4.SS2.p2.12.m12.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.p2.12.m12.1.1.2.2.3" xref="S4.SS2.p2.12.m12.1.1.2.2.3.cmml">v</mi></msubsup><mo id="S4.SS2.p2.12.m12.1.1.1" xref="S4.SS2.p2.12.m12.1.1.1.cmml">⊆</mo><mi id="S4.SS2.p2.12.m12.1.1.3" xref="S4.SS2.p2.12.m12.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.12.m12.1b"><apply id="S4.SS2.p2.12.m12.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1"><subset id="S4.SS2.p2.12.m12.1.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1.1"></subset><apply id="S4.SS2.p2.12.m12.1.1.2.cmml" xref="S4.SS2.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.12.m12.1.1.2.1.cmml" xref="S4.SS2.p2.12.m12.1.1.2">subscript</csymbol><apply id="S4.SS2.p2.12.m12.1.1.2.2.cmml" xref="S4.SS2.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.12.m12.1.1.2.2.1.cmml" xref="S4.SS2.p2.12.m12.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.12.m12.1.1.2.2.2.cmml" xref="S4.SS2.p2.12.m12.1.1.2.2.2">𝐶</ci><ci id="S4.SS2.p2.12.m12.1.1.2.2.3.cmml" xref="S4.SS2.p2.12.m12.1.1.2.2.3">𝑣</ci></apply><apply id="S4.SS2.p2.12.m12.1.1.2.3.cmml" xref="S4.SS2.p2.12.m12.1.1.2.3"><ci id="S4.SS2.p2.12.m12.1.1.2.3.1.cmml" xref="S4.SS2.p2.12.m12.1.1.2.3.1">¯</ci><ci id="S4.SS2.p2.12.m12.1.1.2.3.2.cmml" xref="S4.SS2.p2.12.m12.1.1.2.3.2">𝐺</ci></apply></apply><ci id="S4.SS2.p2.12.m12.1.1.3.cmml" xref="S4.SS2.p2.12.m12.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.12.m12.1c">C^{v}_{\overline{G}}\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.12.m12.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊆ italic_W</annotation></semantics></math>. Notice that, in general (i.e., for an arbitrary graph), this might not be a semi-filter, as one of the sets might be empty. But for our choice of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.p2.13.m13.1"><semantics id="S4.SS2.p2.13.m13.1a"><mi id="S4.SS2.p2.13.m13.1.1" xref="S4.SS2.p2.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.13.m13.1b"><ci id="S4.SS2.p2.13.m13.1.1.cmml" xref="S4.SS2.p2.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.13.m13.1d">italic_G</annotation></semantics></math>, this is a semi-filter above <math alttext="e" class="ltx_Math" display="inline" id="S4.SS2.p2.14.m14.1"><semantics id="S4.SS2.p2.14.m14.1a"><mi id="S4.SS2.p2.14.m14.1.1" xref="S4.SS2.p2.14.m14.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.14.m14.1b"><ci id="S4.SS2.p2.14.m14.1.1.cmml" xref="S4.SS2.p2.14.m14.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.14.m14.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.14.m14.1d">italic_e</annotation></semantics></math>. We let</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathfrak{F}_{\mathsf{can}}^{G}\stackrel{{\scriptstyle\rm def}}{{=}}\{\mathcal% {F}_{e}\mid e\in G~{}\text{and}~{}\mathcal{F}_{e}~{}\text{is a semi-filter}\;\}." class="ltx_Math" display="block" id="S4.Ex30.m1.1"><semantics id="S4.Ex30.m1.1a"><mrow id="S4.Ex30.m1.1.1.1" xref="S4.Ex30.m1.1.1.1.1.cmml"><mrow id="S4.Ex30.m1.1.1.1.1" xref="S4.Ex30.m1.1.1.1.1.cmml"><msubsup id="S4.Ex30.m1.1.1.1.1.4" xref="S4.Ex30.m1.1.1.1.1.4.cmml"><mi id="S4.Ex30.m1.1.1.1.1.4.2.2" xref="S4.Ex30.m1.1.1.1.1.4.2.2.cmml">𝔉</mi><mi id="S4.Ex30.m1.1.1.1.1.4.2.3" xref="S4.Ex30.m1.1.1.1.1.4.2.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.Ex30.m1.1.1.1.1.4.3" xref="S4.Ex30.m1.1.1.1.1.4.3.cmml">G</mi></msubsup><mover id="S4.Ex30.m1.1.1.1.1.3" xref="S4.Ex30.m1.1.1.1.1.3.cmml"><mo id="S4.Ex30.m1.1.1.1.1.3.2" xref="S4.Ex30.m1.1.1.1.1.3.2.cmml">=</mo><mi id="S4.Ex30.m1.1.1.1.1.3.3" xref="S4.Ex30.m1.1.1.1.1.3.3.cmml">def</mi></mover><mrow id="S4.Ex30.m1.1.1.1.1.2.2" xref="S4.Ex30.m1.1.1.1.1.2.3.cmml"><mo id="S4.Ex30.m1.1.1.1.1.2.2.3" stretchy="false" xref="S4.Ex30.m1.1.1.1.1.2.3.1.cmml">{</mo><msub id="S4.Ex30.m1.1.1.1.1.1.1.1" xref="S4.Ex30.m1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex30.m1.1.1.1.1.1.1.1.2" xref="S4.Ex30.m1.1.1.1.1.1.1.1.2.cmml">ℱ</mi><mi id="S4.Ex30.m1.1.1.1.1.1.1.1.3" xref="S4.Ex30.m1.1.1.1.1.1.1.1.3.cmml">e</mi></msub><mo fence="true" id="S4.Ex30.m1.1.1.1.1.2.2.4" lspace="0em" rspace="0em" xref="S4.Ex30.m1.1.1.1.1.2.3.1.cmml">∣</mo><mrow id="S4.Ex30.m1.1.1.1.1.2.2.2" xref="S4.Ex30.m1.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex30.m1.1.1.1.1.2.2.2.2" xref="S4.Ex30.m1.1.1.1.1.2.2.2.2.cmml">e</mi><mo id="S4.Ex30.m1.1.1.1.1.2.2.2.1" xref="S4.Ex30.m1.1.1.1.1.2.2.2.1.cmml">∈</mo><mrow id="S4.Ex30.m1.1.1.1.1.2.2.2.3" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.cmml"><mi id="S4.Ex30.m1.1.1.1.1.2.2.2.3.2" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.2.cmml">G</mi><mo id="S4.Ex30.m1.1.1.1.1.2.2.2.3.1" lspace="0.330em" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.1.cmml">⁢</mo><mtext id="S4.Ex30.m1.1.1.1.1.2.2.2.3.3" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.3a.cmml">and</mtext><mo id="S4.Ex30.m1.1.1.1.1.2.2.2.3.1a" lspace="0.330em" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.1.cmml">⁢</mo><msub id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.2" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.2.cmml">ℱ</mi><mi id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.3" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.3.cmml">e</mi></msub><mo id="S4.Ex30.m1.1.1.1.1.2.2.2.3.1b" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.1.cmml">⁢</mo><mtext id="S4.Ex30.m1.1.1.1.1.2.2.2.3.5" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.5a.cmml">is a semi-filter</mtext></mrow></mrow><mo id="S4.Ex30.m1.1.1.1.1.2.2.5" lspace="0.280em" stretchy="false" xref="S4.Ex30.m1.1.1.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S4.Ex30.m1.1.1.1.2" lspace="0em" xref="S4.Ex30.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex30.m1.1b"><apply id="S4.Ex30.m1.1.1.1.1.cmml" xref="S4.Ex30.m1.1.1.1"><apply id="S4.Ex30.m1.1.1.1.1.3.cmml" xref="S4.Ex30.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex30.m1.1.1.1.1.3.1.cmml" xref="S4.Ex30.m1.1.1.1.1.3">superscript</csymbol><eq id="S4.Ex30.m1.1.1.1.1.3.2.cmml" xref="S4.Ex30.m1.1.1.1.1.3.2"></eq><ci id="S4.Ex30.m1.1.1.1.1.3.3.cmml" xref="S4.Ex30.m1.1.1.1.1.3.3">def</ci></apply><apply id="S4.Ex30.m1.1.1.1.1.4.cmml" xref="S4.Ex30.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.Ex30.m1.1.1.1.1.4.1.cmml" xref="S4.Ex30.m1.1.1.1.1.4">superscript</csymbol><apply id="S4.Ex30.m1.1.1.1.1.4.2.cmml" xref="S4.Ex30.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.Ex30.m1.1.1.1.1.4.2.1.cmml" xref="S4.Ex30.m1.1.1.1.1.4">subscript</csymbol><ci id="S4.Ex30.m1.1.1.1.1.4.2.2.cmml" xref="S4.Ex30.m1.1.1.1.1.4.2.2">𝔉</ci><ci id="S4.Ex30.m1.1.1.1.1.4.2.3.cmml" xref="S4.Ex30.m1.1.1.1.1.4.2.3">𝖼𝖺𝗇</ci></apply><ci id="S4.Ex30.m1.1.1.1.1.4.3.cmml" xref="S4.Ex30.m1.1.1.1.1.4.3">𝐺</ci></apply><apply id="S4.Ex30.m1.1.1.1.1.2.3.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2"><csymbol cd="latexml" id="S4.Ex30.m1.1.1.1.1.2.3.1.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S4.Ex30.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex30.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex30.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex30.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex30.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex30.m1.1.1.1.1.1.1.1.2">ℱ</ci><ci id="S4.Ex30.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex30.m1.1.1.1.1.1.1.1.3">𝑒</ci></apply><apply id="S4.Ex30.m1.1.1.1.1.2.2.2.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2"><in id="S4.Ex30.m1.1.1.1.1.2.2.2.1.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.1"></in><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.2">𝑒</ci><apply id="S4.Ex30.m1.1.1.1.1.2.2.2.3.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3"><times id="S4.Ex30.m1.1.1.1.1.2.2.2.3.1.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.1"></times><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.3.2.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.2">𝐺</ci><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.3.3a.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.3"><mtext id="S4.Ex30.m1.1.1.1.1.2.2.2.3.3.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.3">and</mtext></ci><apply id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4"><csymbol cd="ambiguous" id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.1.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4">subscript</csymbol><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.2.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.2">ℱ</ci><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.3.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.4.3">𝑒</ci></apply><ci id="S4.Ex30.m1.1.1.1.1.2.2.2.3.5a.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.5"><mtext id="S4.Ex30.m1.1.1.1.1.2.2.2.3.5.cmml" xref="S4.Ex30.m1.1.1.1.1.2.2.2.3.5">is a semi-filter</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex30.m1.1c">\mathfrak{F}_{\mathsf{can}}^{G}\stackrel{{\scriptstyle\rm def}}{{=}}\{\mathcal% {F}_{e}\mid e\in G~{}\text{and}~{}\mathcal{F}_{e}~{}\text{is a semi-filter}\;\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex30.m1.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP { caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∣ italic_e ∈ italic_G and caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT is a semi-filter } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.24">We say that <math alttext="\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="S4.SS2.p2.15.m1.1"><semantics id="S4.SS2.p2.15.m1.1a"><msubsup id="S4.SS2.p2.15.m1.1.1" xref="S4.SS2.p2.15.m1.1.1.cmml"><mi id="S4.SS2.p2.15.m1.1.1.2.2" xref="S4.SS2.p2.15.m1.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS2.p2.15.m1.1.1.2.3" xref="S4.SS2.p2.15.m1.1.1.2.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.SS2.p2.15.m1.1.1.3" xref="S4.SS2.p2.15.m1.1.1.3.cmml">G</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.15.m1.1b"><apply id="S4.SS2.p2.15.m1.1.1.cmml" xref="S4.SS2.p2.15.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.15.m1.1.1.1.cmml" xref="S4.SS2.p2.15.m1.1.1">superscript</csymbol><apply id="S4.SS2.p2.15.m1.1.1.2.cmml" xref="S4.SS2.p2.15.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.15.m1.1.1.2.1.cmml" xref="S4.SS2.p2.15.m1.1.1">subscript</csymbol><ci id="S4.SS2.p2.15.m1.1.1.2.2.cmml" xref="S4.SS2.p2.15.m1.1.1.2.2">𝔉</ci><ci id="S4.SS2.p2.15.m1.1.1.2.3.cmml" xref="S4.SS2.p2.15.m1.1.1.2.3">𝖼𝖺𝗇</ci></apply><ci id="S4.SS2.p2.15.m1.1.1.3.cmml" xref="S4.SS2.p2.15.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.15.m1.1c">\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.15.m1.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math> is the set of <em class="ltx_emph ltx_font_italic" id="S4.SS2.p2.24.1">canonical semi-filters</em> of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.p2.16.m2.1"><semantics id="S4.SS2.p2.16.m2.1a"><mi id="S4.SS2.p2.16.m2.1.1" xref="S4.SS2.p2.16.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.16.m2.1b"><ci id="S4.SS2.p2.16.m2.1.1.cmml" xref="S4.SS2.p2.16.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.16.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.16.m2.1d">italic_G</annotation></semantics></math> (above an edge of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.p2.17.m3.1"><semantics id="S4.SS2.p2.17.m3.1a"><mi id="S4.SS2.p2.17.m3.1.1" xref="S4.SS2.p2.17.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.17.m3.1b"><ci id="S4.SS2.p2.17.m3.1.1.cmml" xref="S4.SS2.p2.17.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.17.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.17.m3.1d">italic_G</annotation></semantics></math>). Note that <math alttext="\mathfrak{F}_{\mathsf{can}}^{G^{*}}" class="ltx_Math" display="inline" id="S4.SS2.p2.18.m4.1"><semantics id="S4.SS2.p2.18.m4.1a"><msubsup id="S4.SS2.p2.18.m4.1.1" xref="S4.SS2.p2.18.m4.1.1.cmml"><mi id="S4.SS2.p2.18.m4.1.1.2.2" xref="S4.SS2.p2.18.m4.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS2.p2.18.m4.1.1.2.3" xref="S4.SS2.p2.18.m4.1.1.2.3.cmml">𝖼𝖺𝗇</mi><msup id="S4.SS2.p2.18.m4.1.1.3" xref="S4.SS2.p2.18.m4.1.1.3.cmml"><mi id="S4.SS2.p2.18.m4.1.1.3.2" xref="S4.SS2.p2.18.m4.1.1.3.2.cmml">G</mi><mo id="S4.SS2.p2.18.m4.1.1.3.3" xref="S4.SS2.p2.18.m4.1.1.3.3.cmml">∗</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.18.m4.1b"><apply id="S4.SS2.p2.18.m4.1.1.cmml" xref="S4.SS2.p2.18.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.18.m4.1.1.1.cmml" xref="S4.SS2.p2.18.m4.1.1">superscript</csymbol><apply id="S4.SS2.p2.18.m4.1.1.2.cmml" xref="S4.SS2.p2.18.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.18.m4.1.1.2.1.cmml" xref="S4.SS2.p2.18.m4.1.1">subscript</csymbol><ci id="S4.SS2.p2.18.m4.1.1.2.2.cmml" xref="S4.SS2.p2.18.m4.1.1.2.2">𝔉</ci><ci id="S4.SS2.p2.18.m4.1.1.2.3.cmml" xref="S4.SS2.p2.18.m4.1.1.2.3">𝖼𝖺𝗇</ci></apply><apply id="S4.SS2.p2.18.m4.1.1.3.cmml" xref="S4.SS2.p2.18.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.18.m4.1.1.3.1.cmml" xref="S4.SS2.p2.18.m4.1.1.3">superscript</csymbol><ci id="S4.SS2.p2.18.m4.1.1.3.2.cmml" xref="S4.SS2.p2.18.m4.1.1.3.2">𝐺</ci><times id="S4.SS2.p2.18.m4.1.1.3.3.cmml" xref="S4.SS2.p2.18.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.18.m4.1c">\mathfrak{F}_{\mathsf{can}}^{G^{*}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.18.m4.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is well-defined for any bipartite graph <math alttext="G^{*}\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS2.p2.19.m5.2"><semantics id="S4.SS2.p2.19.m5.2a"><mrow id="S4.SS2.p2.19.m5.2.3" xref="S4.SS2.p2.19.m5.2.3.cmml"><msup id="S4.SS2.p2.19.m5.2.3.2" xref="S4.SS2.p2.19.m5.2.3.2.cmml"><mi id="S4.SS2.p2.19.m5.2.3.2.2" xref="S4.SS2.p2.19.m5.2.3.2.2.cmml">G</mi><mo id="S4.SS2.p2.19.m5.2.3.2.3" xref="S4.SS2.p2.19.m5.2.3.2.3.cmml">∗</mo></msup><mo id="S4.SS2.p2.19.m5.2.3.1" xref="S4.SS2.p2.19.m5.2.3.1.cmml">⊆</mo><mrow id="S4.SS2.p2.19.m5.2.3.3" xref="S4.SS2.p2.19.m5.2.3.3.cmml"><mrow id="S4.SS2.p2.19.m5.2.3.3.2.2" xref="S4.SS2.p2.19.m5.2.3.3.2.1.cmml"><mo id="S4.SS2.p2.19.m5.2.3.3.2.2.1" stretchy="false" xref="S4.SS2.p2.19.m5.2.3.3.2.1.1.cmml">[</mo><mi id="S4.SS2.p2.19.m5.1.1" xref="S4.SS2.p2.19.m5.1.1.cmml">N</mi><mo id="S4.SS2.p2.19.m5.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS2.p2.19.m5.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS2.p2.19.m5.2.3.3.1" rspace="0.222em" xref="S4.SS2.p2.19.m5.2.3.3.1.cmml">×</mo><mrow id="S4.SS2.p2.19.m5.2.3.3.3.2" xref="S4.SS2.p2.19.m5.2.3.3.3.1.cmml"><mo id="S4.SS2.p2.19.m5.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.p2.19.m5.2.3.3.3.1.1.cmml">[</mo><mi id="S4.SS2.p2.19.m5.2.2" xref="S4.SS2.p2.19.m5.2.2.cmml">N</mi><mo id="S4.SS2.p2.19.m5.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.p2.19.m5.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.19.m5.2b"><apply id="S4.SS2.p2.19.m5.2.3.cmml" xref="S4.SS2.p2.19.m5.2.3"><subset id="S4.SS2.p2.19.m5.2.3.1.cmml" xref="S4.SS2.p2.19.m5.2.3.1"></subset><apply id="S4.SS2.p2.19.m5.2.3.2.cmml" xref="S4.SS2.p2.19.m5.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.19.m5.2.3.2.1.cmml" xref="S4.SS2.p2.19.m5.2.3.2">superscript</csymbol><ci id="S4.SS2.p2.19.m5.2.3.2.2.cmml" xref="S4.SS2.p2.19.m5.2.3.2.2">𝐺</ci><times id="S4.SS2.p2.19.m5.2.3.2.3.cmml" xref="S4.SS2.p2.19.m5.2.3.2.3"></times></apply><apply id="S4.SS2.p2.19.m5.2.3.3.cmml" xref="S4.SS2.p2.19.m5.2.3.3"><times id="S4.SS2.p2.19.m5.2.3.3.1.cmml" xref="S4.SS2.p2.19.m5.2.3.3.1"></times><apply id="S4.SS2.p2.19.m5.2.3.3.2.1.cmml" xref="S4.SS2.p2.19.m5.2.3.3.2.2"><csymbol cd="latexml" id="S4.SS2.p2.19.m5.2.3.3.2.1.1.cmml" xref="S4.SS2.p2.19.m5.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS2.p2.19.m5.1.1.cmml" xref="S4.SS2.p2.19.m5.1.1">𝑁</ci></apply><apply id="S4.SS2.p2.19.m5.2.3.3.3.1.cmml" xref="S4.SS2.p2.19.m5.2.3.3.3.2"><csymbol cd="latexml" id="S4.SS2.p2.19.m5.2.3.3.3.1.1.cmml" xref="S4.SS2.p2.19.m5.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS2.p2.19.m5.2.2.cmml" xref="S4.SS2.p2.19.m5.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.19.m5.2c">G^{*}\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.19.m5.2d">italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>. We can thus ask, for a general bipartite graph <math alttext="G^{*}" class="ltx_Math" display="inline" id="S4.SS2.p2.20.m6.1"><semantics id="S4.SS2.p2.20.m6.1a"><msup id="S4.SS2.p2.20.m6.1.1" xref="S4.SS2.p2.20.m6.1.1.cmml"><mi id="S4.SS2.p2.20.m6.1.1.2" xref="S4.SS2.p2.20.m6.1.1.2.cmml">G</mi><mo id="S4.SS2.p2.20.m6.1.1.3" xref="S4.SS2.p2.20.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.20.m6.1b"><apply id="S4.SS2.p2.20.m6.1.1.cmml" xref="S4.SS2.p2.20.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.20.m6.1.1.1.cmml" xref="S4.SS2.p2.20.m6.1.1">superscript</csymbol><ci id="S4.SS2.p2.20.m6.1.1.2.cmml" xref="S4.SS2.p2.20.m6.1.1.2">𝐺</ci><times id="S4.SS2.p2.20.m6.1.1.3.cmml" xref="S4.SS2.p2.20.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.20.m6.1c">G^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.20.m6.1d">italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, how many pairs of subsets of <math alttext="\overline{G^{*}}" class="ltx_Math" display="inline" id="S4.SS2.p2.21.m7.1"><semantics id="S4.SS2.p2.21.m7.1a"><mover accent="true" id="S4.SS2.p2.21.m7.1.1" xref="S4.SS2.p2.21.m7.1.1.cmml"><msup id="S4.SS2.p2.21.m7.1.1.2" xref="S4.SS2.p2.21.m7.1.1.2.cmml"><mi id="S4.SS2.p2.21.m7.1.1.2.2" xref="S4.SS2.p2.21.m7.1.1.2.2.cmml">G</mi><mo id="S4.SS2.p2.21.m7.1.1.2.3" xref="S4.SS2.p2.21.m7.1.1.2.3.cmml">∗</mo></msup><mo id="S4.SS2.p2.21.m7.1.1.1" xref="S4.SS2.p2.21.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.21.m7.1b"><apply id="S4.SS2.p2.21.m7.1.1.cmml" xref="S4.SS2.p2.21.m7.1.1"><ci id="S4.SS2.p2.21.m7.1.1.1.cmml" xref="S4.SS2.p2.21.m7.1.1.1">¯</ci><apply id="S4.SS2.p2.21.m7.1.1.2.cmml" xref="S4.SS2.p2.21.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.21.m7.1.1.2.1.cmml" xref="S4.SS2.p2.21.m7.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.21.m7.1.1.2.2.cmml" xref="S4.SS2.p2.21.m7.1.1.2.2">𝐺</ci><times id="S4.SS2.p2.21.m7.1.1.2.3.cmml" xref="S4.SS2.p2.21.m7.1.1.2.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.21.m7.1c">\overline{G^{*}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.21.m7.1d">over¯ start_ARG italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> are needed to cover all semi-filters in <math alttext="\mathfrak{F}_{\mathsf{can}}^{G^{*}}" class="ltx_Math" display="inline" id="S4.SS2.p2.22.m8.1"><semantics id="S4.SS2.p2.22.m8.1a"><msubsup id="S4.SS2.p2.22.m8.1.1" xref="S4.SS2.p2.22.m8.1.1.cmml"><mi id="S4.SS2.p2.22.m8.1.1.2.2" xref="S4.SS2.p2.22.m8.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS2.p2.22.m8.1.1.2.3" xref="S4.SS2.p2.22.m8.1.1.2.3.cmml">𝖼𝖺𝗇</mi><msup id="S4.SS2.p2.22.m8.1.1.3" xref="S4.SS2.p2.22.m8.1.1.3.cmml"><mi id="S4.SS2.p2.22.m8.1.1.3.2" xref="S4.SS2.p2.22.m8.1.1.3.2.cmml">G</mi><mo id="S4.SS2.p2.22.m8.1.1.3.3" xref="S4.SS2.p2.22.m8.1.1.3.3.cmml">∗</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.22.m8.1b"><apply id="S4.SS2.p2.22.m8.1.1.cmml" xref="S4.SS2.p2.22.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.22.m8.1.1.1.cmml" xref="S4.SS2.p2.22.m8.1.1">superscript</csymbol><apply id="S4.SS2.p2.22.m8.1.1.2.cmml" xref="S4.SS2.p2.22.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.22.m8.1.1.2.1.cmml" xref="S4.SS2.p2.22.m8.1.1">subscript</csymbol><ci id="S4.SS2.p2.22.m8.1.1.2.2.cmml" xref="S4.SS2.p2.22.m8.1.1.2.2">𝔉</ci><ci id="S4.SS2.p2.22.m8.1.1.2.3.cmml" xref="S4.SS2.p2.22.m8.1.1.2.3">𝖼𝖺𝗇</ci></apply><apply id="S4.SS2.p2.22.m8.1.1.3.cmml" xref="S4.SS2.p2.22.m8.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.22.m8.1.1.3.1.cmml" xref="S4.SS2.p2.22.m8.1.1.3">superscript</csymbol><ci id="S4.SS2.p2.22.m8.1.1.3.2.cmml" xref="S4.SS2.p2.22.m8.1.1.3.2">𝐺</ci><times id="S4.SS2.p2.22.m8.1.1.3.3.cmml" xref="S4.SS2.p2.22.m8.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.22.m8.1c">\mathfrak{F}_{\mathsf{can}}^{G^{*}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.22.m8.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>? Let us denote this quantity by <math alttext="\rho_{\mathsf{can}}(G^{*},\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.SS2.p2.23.m9.4"><semantics id="S4.SS2.p2.23.m9.4a"><mrow id="S4.SS2.p2.23.m9.4.4" xref="S4.SS2.p2.23.m9.4.4.cmml"><msub id="S4.SS2.p2.23.m9.4.4.4" xref="S4.SS2.p2.23.m9.4.4.4.cmml"><mi id="S4.SS2.p2.23.m9.4.4.4.2" xref="S4.SS2.p2.23.m9.4.4.4.2.cmml">ρ</mi><mi id="S4.SS2.p2.23.m9.4.4.4.3" xref="S4.SS2.p2.23.m9.4.4.4.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.SS2.p2.23.m9.4.4.3" xref="S4.SS2.p2.23.m9.4.4.3.cmml">⁢</mo><mrow id="S4.SS2.p2.23.m9.4.4.2.2" xref="S4.SS2.p2.23.m9.4.4.2.3.cmml"><mo id="S4.SS2.p2.23.m9.4.4.2.2.3" stretchy="false" xref="S4.SS2.p2.23.m9.4.4.2.3.cmml">(</mo><msup id="S4.SS2.p2.23.m9.3.3.1.1.1" xref="S4.SS2.p2.23.m9.3.3.1.1.1.cmml"><mi id="S4.SS2.p2.23.m9.3.3.1.1.1.2" xref="S4.SS2.p2.23.m9.3.3.1.1.1.2.cmml">G</mi><mo id="S4.SS2.p2.23.m9.3.3.1.1.1.3" xref="S4.SS2.p2.23.m9.3.3.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.p2.23.m9.4.4.2.2.4" xref="S4.SS2.p2.23.m9.4.4.2.3.cmml">,</mo><msub id="S4.SS2.p2.23.m9.4.4.2.2.2" xref="S4.SS2.p2.23.m9.4.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.23.m9.4.4.2.2.2.2" xref="S4.SS2.p2.23.m9.4.4.2.2.2.2.cmml">𝒢</mi><mrow id="S4.SS2.p2.23.m9.2.2.2.4" xref="S4.SS2.p2.23.m9.2.2.2.3.cmml"><mi id="S4.SS2.p2.23.m9.1.1.1.1" xref="S4.SS2.p2.23.m9.1.1.1.1.cmml">N</mi><mo id="S4.SS2.p2.23.m9.2.2.2.4.1" xref="S4.SS2.p2.23.m9.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.23.m9.2.2.2.2" xref="S4.SS2.p2.23.m9.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.SS2.p2.23.m9.4.4.2.2.5" stretchy="false" xref="S4.SS2.p2.23.m9.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.23.m9.4b"><apply id="S4.SS2.p2.23.m9.4.4.cmml" xref="S4.SS2.p2.23.m9.4.4"><times id="S4.SS2.p2.23.m9.4.4.3.cmml" xref="S4.SS2.p2.23.m9.4.4.3"></times><apply id="S4.SS2.p2.23.m9.4.4.4.cmml" xref="S4.SS2.p2.23.m9.4.4.4"><csymbol cd="ambiguous" id="S4.SS2.p2.23.m9.4.4.4.1.cmml" xref="S4.SS2.p2.23.m9.4.4.4">subscript</csymbol><ci id="S4.SS2.p2.23.m9.4.4.4.2.cmml" xref="S4.SS2.p2.23.m9.4.4.4.2">𝜌</ci><ci id="S4.SS2.p2.23.m9.4.4.4.3.cmml" xref="S4.SS2.p2.23.m9.4.4.4.3">𝖼𝖺𝗇</ci></apply><interval closure="open" id="S4.SS2.p2.23.m9.4.4.2.3.cmml" xref="S4.SS2.p2.23.m9.4.4.2.2"><apply id="S4.SS2.p2.23.m9.3.3.1.1.1.cmml" xref="S4.SS2.p2.23.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.23.m9.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.23.m9.3.3.1.1.1">superscript</csymbol><ci id="S4.SS2.p2.23.m9.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.23.m9.3.3.1.1.1.2">𝐺</ci><times id="S4.SS2.p2.23.m9.3.3.1.1.1.3.cmml" xref="S4.SS2.p2.23.m9.3.3.1.1.1.3"></times></apply><apply id="S4.SS2.p2.23.m9.4.4.2.2.2.cmml" xref="S4.SS2.p2.23.m9.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.23.m9.4.4.2.2.2.1.cmml" xref="S4.SS2.p2.23.m9.4.4.2.2.2">subscript</csymbol><ci id="S4.SS2.p2.23.m9.4.4.2.2.2.2.cmml" xref="S4.SS2.p2.23.m9.4.4.2.2.2.2">𝒢</ci><list id="S4.SS2.p2.23.m9.2.2.2.3.cmml" xref="S4.SS2.p2.23.m9.2.2.2.4"><ci id="S4.SS2.p2.23.m9.1.1.1.1.cmml" xref="S4.SS2.p2.23.m9.1.1.1.1">𝑁</ci><ci id="S4.SS2.p2.23.m9.2.2.2.2.cmml" xref="S4.SS2.p2.23.m9.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.23.m9.4c">\rho_{\mathsf{can}}(G^{*},\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.23.m9.4d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>, i.e., the <em class="ltx_emph ltx_font_italic" id="S4.SS2.p2.24.2">canonical cover complexity</em> of <math alttext="G^{*}" class="ltx_Math" display="inline" id="S4.SS2.p2.24.m10.1"><semantics id="S4.SS2.p2.24.m10.1a"><msup id="S4.SS2.p2.24.m10.1.1" xref="S4.SS2.p2.24.m10.1.1.cmml"><mi id="S4.SS2.p2.24.m10.1.1.2" xref="S4.SS2.p2.24.m10.1.1.2.cmml">G</mi><mo id="S4.SS2.p2.24.m10.1.1.3" xref="S4.SS2.p2.24.m10.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.24.m10.1b"><apply id="S4.SS2.p2.24.m10.1.1.cmml" xref="S4.SS2.p2.24.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.24.m10.1.1.1.cmml" xref="S4.SS2.p2.24.m10.1.1">superscript</csymbol><ci id="S4.SS2.p2.24.m10.1.1.2.cmml" xref="S4.SS2.p2.24.m10.1.1.2">𝐺</ci><times id="S4.SS2.p2.24.m10.1.1.3.cmml" xref="S4.SS2.p2.24.m10.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.24.m10.1c">G^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.24.m10.1d">italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. Clearly, this quantity lower bounds cover complexity.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmtheorem40"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem40.1.1.1">Proposition 40</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem40.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem40.p1"> <p class="ltx_p" id="Thmtheorem40.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem40.p1.1.1">For the graph <math alttext="G=G_{\mathsf{NEQ}}" class="ltx_Math" display="inline" id="Thmtheorem40.p1.1.1.m1.1"><semantics id="Thmtheorem40.p1.1.1.m1.1a"><mrow id="Thmtheorem40.p1.1.1.m1.1.1" xref="Thmtheorem40.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem40.p1.1.1.m1.1.1.2" xref="Thmtheorem40.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="Thmtheorem40.p1.1.1.m1.1.1.1" xref="Thmtheorem40.p1.1.1.m1.1.1.1.cmml">=</mo><msub id="Thmtheorem40.p1.1.1.m1.1.1.3" xref="Thmtheorem40.p1.1.1.m1.1.1.3.cmml"><mi id="Thmtheorem40.p1.1.1.m1.1.1.3.2" xref="Thmtheorem40.p1.1.1.m1.1.1.3.2.cmml">G</mi><mi id="Thmtheorem40.p1.1.1.m1.1.1.3.3" xref="Thmtheorem40.p1.1.1.m1.1.1.3.3.cmml">𝖭𝖤𝖰</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem40.p1.1.1.m1.1b"><apply id="Thmtheorem40.p1.1.1.m1.1.1.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1"><eq id="Thmtheorem40.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.1"></eq><ci id="Thmtheorem40.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.2">𝐺</ci><apply id="Thmtheorem40.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem40.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="Thmtheorem40.p1.1.1.m1.1.1.3.2.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.3.2">𝐺</ci><ci id="Thmtheorem40.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem40.p1.1.1.m1.1.1.3.3">𝖭𝖤𝖰</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem40.p1.1.1.m1.1c">G=G_{\mathsf{NEQ}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem40.p1.1.1.m1.1d">italic_G = italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT</annotation></semantics></math> defined above,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})=\rho(G,\mathcal{G}_{N,N})=D_{\cap}(G% \mid\mathcal{G}_{N,N})=n=\log N." class="ltx_Math" display="block" id="S4.Ex31.m1.9"><semantics id="S4.Ex31.m1.9a"><mrow id="S4.Ex31.m1.9.9.1" xref="S4.Ex31.m1.9.9.1.1.cmml"><mrow id="S4.Ex31.m1.9.9.1.1" xref="S4.Ex31.m1.9.9.1.1.cmml"><mrow id="S4.Ex31.m1.9.9.1.1.1" xref="S4.Ex31.m1.9.9.1.1.1.cmml"><msub id="S4.Ex31.m1.9.9.1.1.1.3" xref="S4.Ex31.m1.9.9.1.1.1.3.cmml"><mi id="S4.Ex31.m1.9.9.1.1.1.3.2" xref="S4.Ex31.m1.9.9.1.1.1.3.2.cmml">ρ</mi><mi id="S4.Ex31.m1.9.9.1.1.1.3.3" xref="S4.Ex31.m1.9.9.1.1.1.3.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.Ex31.m1.9.9.1.1.1.2" xref="S4.Ex31.m1.9.9.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex31.m1.9.9.1.1.1.1.1" xref="S4.Ex31.m1.9.9.1.1.1.1.2.cmml"><mo id="S4.Ex31.m1.9.9.1.1.1.1.1.2" stretchy="false" xref="S4.Ex31.m1.9.9.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex31.m1.7.7" xref="S4.Ex31.m1.7.7.cmml">G</mi><mo id="S4.Ex31.m1.9.9.1.1.1.1.1.3" xref="S4.Ex31.m1.9.9.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex31.m1.9.9.1.1.1.1.1.1" xref="S4.Ex31.m1.9.9.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex31.m1.9.9.1.1.1.1.1.1.2" xref="S4.Ex31.m1.9.9.1.1.1.1.1.1.2.cmml">𝒢</mi><mrow id="S4.Ex31.m1.2.2.2.4" xref="S4.Ex31.m1.2.2.2.3.cmml"><mi id="S4.Ex31.m1.1.1.1.1" xref="S4.Ex31.m1.1.1.1.1.cmml">N</mi><mo id="S4.Ex31.m1.2.2.2.4.1" xref="S4.Ex31.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex31.m1.2.2.2.2" 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id="S4.Ex31.m1.9.9.1.1h.cmml" xref="S4.Ex31.m1.9.9.1"></share><apply id="S4.Ex31.m1.9.9.1.1.10.cmml" xref="S4.Ex31.m1.9.9.1.1.10"><log id="S4.Ex31.m1.9.9.1.1.10.1.cmml" xref="S4.Ex31.m1.9.9.1.1.10.1"></log><ci id="S4.Ex31.m1.9.9.1.1.10.2.cmml" xref="S4.Ex31.m1.9.9.1.1.10.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex31.m1.9c">\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})=\rho(G,\mathcal{G}_{N,N})=D_{\cap}(G% \mid\mathcal{G}_{N,N})=n=\log N.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex31.m1.9d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) = italic_n = roman_log italic_N .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS2.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.1.p1"> <p class="ltx_p" id="S4.SS2.1.p1.8">The upper bound follows by transforming a circuit for the corresponding Boolean function <math alttext="f_{G}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.1.m1.6"><semantics id="S4.SS2.1.p1.1.m1.6a"><mrow id="S4.SS2.1.p1.1.m1.6.7" xref="S4.SS2.1.p1.1.m1.6.7.cmml"><msub id="S4.SS2.1.p1.1.m1.6.7.2" xref="S4.SS2.1.p1.1.m1.6.7.2.cmml"><mi id="S4.SS2.1.p1.1.m1.6.7.2.2" xref="S4.SS2.1.p1.1.m1.6.7.2.2.cmml">f</mi><mi id="S4.SS2.1.p1.1.m1.6.7.2.3" xref="S4.SS2.1.p1.1.m1.6.7.2.3.cmml">G</mi></msub><mo id="S4.SS2.1.p1.1.m1.6.7.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.1.p1.1.m1.6.7.1.cmml">:</mo><mrow 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xref="S4.SS2.1.p1.1.m1.5.5.cmml">0</mn><mo id="S4.SS2.1.p1.1.m1.6.7.3.3.2.2" xref="S4.SS2.1.p1.1.m1.6.7.3.3.1.cmml">,</mo><mn id="S4.SS2.1.p1.1.m1.6.6" xref="S4.SS2.1.p1.1.m1.6.6.cmml">1</mn><mo id="S4.SS2.1.p1.1.m1.6.7.3.3.2.3" stretchy="false" xref="S4.SS2.1.p1.1.m1.6.7.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.1.m1.6b"><apply id="S4.SS2.1.p1.1.m1.6.7.cmml" xref="S4.SS2.1.p1.1.m1.6.7"><ci id="S4.SS2.1.p1.1.m1.6.7.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.1">:</ci><apply id="S4.SS2.1.p1.1.m1.6.7.2.cmml" xref="S4.SS2.1.p1.1.m1.6.7.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.1.m1.6.7.2.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.2">subscript</csymbol><ci id="S4.SS2.1.p1.1.m1.6.7.2.2.cmml" xref="S4.SS2.1.p1.1.m1.6.7.2.2">𝑓</ci><ci id="S4.SS2.1.p1.1.m1.6.7.2.3.cmml" xref="S4.SS2.1.p1.1.m1.6.7.2.3">𝐺</ci></apply><apply id="S4.SS2.1.p1.1.m1.6.7.3.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3"><ci id="S4.SS2.1.p1.1.m1.6.7.3.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.1">→</ci><apply id="S4.SS2.1.p1.1.m1.6.7.3.2.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2"><times id="S4.SS2.1.p1.1.m1.6.7.3.2.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.1"></times><apply id="S4.SS2.1.p1.1.m1.6.7.3.2.2.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.1.m1.6.7.3.2.2.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.2">superscript</csymbol><set id="S4.SS2.1.p1.1.m1.6.7.3.2.2.2.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.2.2.2"><cn id="S4.SS2.1.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.1.1">0</cn><cn id="S4.SS2.1.p1.1.m1.2.2.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.2.2">1</cn></set><ci id="S4.SS2.1.p1.1.m1.6.7.3.2.2.3.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.2.3">𝑛</ci></apply><apply id="S4.SS2.1.p1.1.m1.6.7.3.2.3.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.1.p1.1.m1.6.7.3.2.3.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.3">superscript</csymbol><set id="S4.SS2.1.p1.1.m1.6.7.3.2.3.2.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.3.2.2"><cn id="S4.SS2.1.p1.1.m1.3.3.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.3.3">0</cn><cn id="S4.SS2.1.p1.1.m1.4.4.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.4.4">1</cn></set><ci id="S4.SS2.1.p1.1.m1.6.7.3.2.3.3.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.2.3.3">𝑛</ci></apply></apply><set id="S4.SS2.1.p1.1.m1.6.7.3.3.1.cmml" xref="S4.SS2.1.p1.1.m1.6.7.3.3.2"><cn id="S4.SS2.1.p1.1.m1.5.5.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.5.5">0</cn><cn id="S4.SS2.1.p1.1.m1.6.6.cmml" type="integer" xref="S4.SS2.1.p1.1.m1.6.6">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.1.m1.6c">f_{G}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.1.m1.6d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> into a construction of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.1.p1.2.m2.1"><semantics id="S4.SS2.1.p1.2.m2.1a"><mi id="S4.SS2.1.p1.2.m2.1.1" xref="S4.SS2.1.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.2.m2.1b"><ci id="S4.SS2.1.p1.2.m2.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.2.m2.1d">italic_G</annotation></semantics></math>. Observe that <math alttext="f_{G}(u,v)=\bigvee_{i\in[n]}u_{i}\oplus v_{i}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.3.m3.3"><semantics id="S4.SS2.1.p1.3.m3.3a"><mrow id="S4.SS2.1.p1.3.m3.3.4" xref="S4.SS2.1.p1.3.m3.3.4.cmml"><mrow id="S4.SS2.1.p1.3.m3.3.4.2" xref="S4.SS2.1.p1.3.m3.3.4.2.cmml"><msub id="S4.SS2.1.p1.3.m3.3.4.2.2" xref="S4.SS2.1.p1.3.m3.3.4.2.2.cmml"><mi id="S4.SS2.1.p1.3.m3.3.4.2.2.2" xref="S4.SS2.1.p1.3.m3.3.4.2.2.2.cmml">f</mi><mi id="S4.SS2.1.p1.3.m3.3.4.2.2.3" xref="S4.SS2.1.p1.3.m3.3.4.2.2.3.cmml">G</mi></msub><mo id="S4.SS2.1.p1.3.m3.3.4.2.1" xref="S4.SS2.1.p1.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.3.m3.3.4.2.3.2" xref="S4.SS2.1.p1.3.m3.3.4.2.3.1.cmml"><mo id="S4.SS2.1.p1.3.m3.3.4.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.3.m3.3.4.2.3.1.cmml">(</mo><mi id="S4.SS2.1.p1.3.m3.2.2" xref="S4.SS2.1.p1.3.m3.2.2.cmml">u</mi><mo id="S4.SS2.1.p1.3.m3.3.4.2.3.2.2" xref="S4.SS2.1.p1.3.m3.3.4.2.3.1.cmml">,</mo><mi id="S4.SS2.1.p1.3.m3.3.3" 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xref="S4.SS2.1.p1.3.m3.1.1.1.1.cmml">n</mi><mo id="S4.SS2.1.p1.3.m3.1.1.1.4.2.2" stretchy="false" xref="S4.SS2.1.p1.3.m3.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msub id="S4.SS2.1.p1.3.m3.3.4.3.2.2" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2.cmml"><mi id="S4.SS2.1.p1.3.m3.3.4.3.2.2.2" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2.2.cmml">u</mi><mi id="S4.SS2.1.p1.3.m3.3.4.3.2.2.3" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2.3.cmml">i</mi></msub></mrow><mo id="S4.SS2.1.p1.3.m3.3.4.3.1" xref="S4.SS2.1.p1.3.m3.3.4.3.1.cmml">⊕</mo><msub id="S4.SS2.1.p1.3.m3.3.4.3.3" xref="S4.SS2.1.p1.3.m3.3.4.3.3.cmml"><mi id="S4.SS2.1.p1.3.m3.3.4.3.3.2" xref="S4.SS2.1.p1.3.m3.3.4.3.3.2.cmml">v</mi><mi id="S4.SS2.1.p1.3.m3.3.4.3.3.3" xref="S4.SS2.1.p1.3.m3.3.4.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.3.m3.3b"><apply id="S4.SS2.1.p1.3.m3.3.4.cmml" xref="S4.SS2.1.p1.3.m3.3.4"><eq id="S4.SS2.1.p1.3.m3.3.4.1.cmml" xref="S4.SS2.1.p1.3.m3.3.4.1"></eq><apply 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xref="S4.SS2.1.p1.3.m3.3.4.3.2"><apply id="S4.SS2.1.p1.3.m3.3.4.3.2.1.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.1"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.3.4.3.2.1.1.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.1">subscript</csymbol><or id="S4.SS2.1.p1.3.m3.3.4.3.2.1.2.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.1.2"></or><apply id="S4.SS2.1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1"><in id="S4.SS2.1.p1.3.m3.1.1.1.2.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1.2"></in><ci id="S4.SS2.1.p1.3.m3.1.1.1.3.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1.3">𝑖</ci><apply id="S4.SS2.1.p1.3.m3.1.1.1.4.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1.4.2"><csymbol cd="latexml" id="S4.SS2.1.p1.3.m3.1.1.1.4.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.SS2.1.p1.3.m3.1.1.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1.1.1">𝑛</ci></apply></apply></apply><apply id="S4.SS2.1.p1.3.m3.3.4.3.2.2.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.3.4.3.2.2.1.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2">subscript</csymbol><ci id="S4.SS2.1.p1.3.m3.3.4.3.2.2.2.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2.2">𝑢</ci><ci id="S4.SS2.1.p1.3.m3.3.4.3.2.2.3.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.2.2.3">𝑖</ci></apply></apply><apply id="S4.SS2.1.p1.3.m3.3.4.3.3.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.3"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.3.4.3.3.1.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.3">subscript</csymbol><ci id="S4.SS2.1.p1.3.m3.3.4.3.3.2.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.3.2">𝑣</ci><ci id="S4.SS2.1.p1.3.m3.3.4.3.3.3.cmml" xref="S4.SS2.1.p1.3.m3.3.4.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.3.m3.3c">f_{G}(u,v)=\bigvee_{i\in[n]}u_{i}\oplus v_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.3.m3.3d">italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_u , italic_v ) = ⋁ start_POSTSUBSCRIPT italic_i ∈ [ italic_n ] end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\oplus" class="ltx_Math" display="inline" id="S4.SS2.1.p1.4.m4.1"><semantics id="S4.SS2.1.p1.4.m4.1a"><mo id="S4.SS2.1.p1.4.m4.1.1" xref="S4.SS2.1.p1.4.m4.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.4.m4.1b"><csymbol cd="latexml" id="S4.SS2.1.p1.4.m4.1.1.cmml" xref="S4.SS2.1.p1.4.m4.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.4.m4.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.4.m4.1d">⊕</annotation></semantics></math> denotes the parity operation, and that each <math alttext="\oplus" class="ltx_Math" display="inline" id="S4.SS2.1.p1.5.m5.1"><semantics id="S4.SS2.1.p1.5.m5.1a"><mo id="S4.SS2.1.p1.5.m5.1.1" xref="S4.SS2.1.p1.5.m5.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.5.m5.1b"><csymbol cd="latexml" id="S4.SS2.1.p1.5.m5.1.1.cmml" xref="S4.SS2.1.p1.5.m5.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.5.m5.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.5.m5.1d">⊕</annotation></semantics></math>-gate can be implemented using a single <math alttext="\wedge" class="ltx_Math" display="inline" id="S4.SS2.1.p1.6.m6.1"><semantics id="S4.SS2.1.p1.6.m6.1a"><mo id="S4.SS2.1.p1.6.m6.1.1" xref="S4.SS2.1.p1.6.m6.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.6.m6.1b"><and id="S4.SS2.1.p1.6.m6.1.1.cmml" xref="S4.SS2.1.p1.6.m6.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.6.m6.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.6.m6.1d">∧</annotation></semantics></math>-gate via <math alttext="a\oplus b=(a\vee b)\wedge(\overline{a}\vee\overline{b})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.7.m7.2"><semantics id="S4.SS2.1.p1.7.m7.2a"><mrow id="S4.SS2.1.p1.7.m7.2.2" xref="S4.SS2.1.p1.7.m7.2.2.cmml"><mrow id="S4.SS2.1.p1.7.m7.2.2.4" xref="S4.SS2.1.p1.7.m7.2.2.4.cmml"><mi id="S4.SS2.1.p1.7.m7.2.2.4.2" xref="S4.SS2.1.p1.7.m7.2.2.4.2.cmml">a</mi><mo id="S4.SS2.1.p1.7.m7.2.2.4.1" xref="S4.SS2.1.p1.7.m7.2.2.4.1.cmml">⊕</mo><mi id="S4.SS2.1.p1.7.m7.2.2.4.3" xref="S4.SS2.1.p1.7.m7.2.2.4.3.cmml">b</mi></mrow><mo id="S4.SS2.1.p1.7.m7.2.2.3" xref="S4.SS2.1.p1.7.m7.2.2.3.cmml">=</mo><mrow id="S4.SS2.1.p1.7.m7.2.2.2" xref="S4.SS2.1.p1.7.m7.2.2.2.cmml"><mrow id="S4.SS2.1.p1.7.m7.1.1.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S4.SS2.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.2" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.1.cmml">∨</mo><mi id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.3" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.SS2.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.1.p1.7.m7.2.2.2.3" xref="S4.SS2.1.p1.7.m7.2.2.2.3.cmml">∧</mo><mrow id="S4.SS2.1.p1.7.m7.2.2.2.2.1" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.cmml"><mo id="S4.SS2.1.p1.7.m7.2.2.2.2.1.2" stretchy="false" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.cmml">(</mo><mrow id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.cmml"><mover accent="true" id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.cmml"><mi id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.2" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.2.cmml">a</mi><mo id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.1" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.1.cmml">¯</mo></mover><mo id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.1" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.1.cmml">∨</mo><mover accent="true" id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.cmml"><mi id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.2" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.2.cmml">b</mi><mo id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.1" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS2.1.p1.7.m7.2.2.2.2.1.3" stretchy="false" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.7.m7.2b"><apply id="S4.SS2.1.p1.7.m7.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2"><eq id="S4.SS2.1.p1.7.m7.2.2.3.cmml" xref="S4.SS2.1.p1.7.m7.2.2.3"></eq><apply id="S4.SS2.1.p1.7.m7.2.2.4.cmml" xref="S4.SS2.1.p1.7.m7.2.2.4"><csymbol cd="latexml" id="S4.SS2.1.p1.7.m7.2.2.4.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.4.1">direct-sum</csymbol><ci id="S4.SS2.1.p1.7.m7.2.2.4.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2.4.2">𝑎</ci><ci id="S4.SS2.1.p1.7.m7.2.2.4.3.cmml" xref="S4.SS2.1.p1.7.m7.2.2.4.3">𝑏</ci></apply><apply id="S4.SS2.1.p1.7.m7.2.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2"><and id="S4.SS2.1.p1.7.m7.2.2.2.3.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.3"></and><apply id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1"><or id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.1"></or><ci id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1.1.1.1.3">𝑏</ci></apply><apply id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1"><or id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.1"></or><apply id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2"><ci id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.1">¯</ci><ci id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.2.2">𝑎</ci></apply><apply id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3"><ci id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.1.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.1">¯</ci><ci id="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2.2.2.1.1.3.2">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.7.m7.2c">a\oplus b=(a\vee b)\wedge(\overline{a}\vee\overline{b})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.7.m7.2d">italic_a ⊕ italic_b = ( italic_a ∨ italic_b ) ∧ ( over¯ start_ARG italic_a end_ARG ∨ over¯ start_ARG italic_b end_ARG )</annotation></semantics></math>. Therefore, <math alttext="\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})\leq\rho(G,\mathcal{G}_{N,N})\leq D_{% \cap}(G\mid\mathcal{G}_{N,N})\leq n" class="ltx_Math" display="inline" id="S4.SS2.1.p1.8.m8.11"><semantics id="S4.SS2.1.p1.8.m8.11a"><mrow id="S4.SS2.1.p1.8.m8.11.11" xref="S4.SS2.1.p1.8.m8.11.11.cmml"><mrow id="S4.SS2.1.p1.8.m8.9.9.1" xref="S4.SS2.1.p1.8.m8.9.9.1.cmml"><msub id="S4.SS2.1.p1.8.m8.9.9.1.3" xref="S4.SS2.1.p1.8.m8.9.9.1.3.cmml"><mi id="S4.SS2.1.p1.8.m8.9.9.1.3.2" xref="S4.SS2.1.p1.8.m8.9.9.1.3.2.cmml">ρ</mi><mi id="S4.SS2.1.p1.8.m8.9.9.1.3.3" xref="S4.SS2.1.p1.8.m8.9.9.1.3.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.SS2.1.p1.8.m8.9.9.1.2" xref="S4.SS2.1.p1.8.m8.9.9.1.2.cmml">⁢</mo><mrow id="S4.SS2.1.p1.8.m8.9.9.1.1.1" xref="S4.SS2.1.p1.8.m8.9.9.1.1.2.cmml"><mo id="S4.SS2.1.p1.8.m8.9.9.1.1.1.2" stretchy="false" xref="S4.SS2.1.p1.8.m8.9.9.1.1.2.cmml">(</mo><mi id="S4.SS2.1.p1.8.m8.7.7" xref="S4.SS2.1.p1.8.m8.7.7.cmml">G</mi><mo id="S4.SS2.1.p1.8.m8.9.9.1.1.1.3" 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xref="S4.SS2.1.p1.8.m8.6.6.2.4"><ci id="S4.SS2.1.p1.8.m8.5.5.1.1.cmml" xref="S4.SS2.1.p1.8.m8.5.5.1.1">𝑁</ci><ci id="S4.SS2.1.p1.8.m8.6.6.2.2.cmml" xref="S4.SS2.1.p1.8.m8.6.6.2.2">𝑁</ci></list></apply></apply></apply></apply><apply id="S4.SS2.1.p1.8.m8.11.11e.cmml" xref="S4.SS2.1.p1.8.m8.11.11"><leq id="S4.SS2.1.p1.8.m8.11.11.7.cmml" xref="S4.SS2.1.p1.8.m8.11.11.7"></leq><share href="https://arxiv.org/html/2503.14117v1#S4.SS2.1.p1.8.m8.11.11.3.cmml" id="S4.SS2.1.p1.8.m8.11.11f.cmml" xref="S4.SS2.1.p1.8.m8.11.11"></share><ci id="S4.SS2.1.p1.8.m8.11.11.8.cmml" xref="S4.SS2.1.p1.8.m8.11.11.8">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.8.m8.11c">\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})\leq\rho(G,\mathcal{G}_{N,N})\leq D_{% \cap}(G\mid\mathcal{G}_{N,N})\leq n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.8.m8.11d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_ρ ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_n</annotation></semantics></math> via Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem13" title="Lemma 13 (Tight transference from graph complexity to circuit complexity). ‣ 2.4 Transference of lower bounds ‣ 2 Discrete Complexity ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">13</span></a> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem22" title="Theorem 22 (Fusion lower bound). ‣ 3.2 Discrete complexity lower bounds using the fusion method ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">22</span></a>.</p> </div> <div class="ltx_para" id="S4.SS2.2.p2"> <p class="ltx_p" id="S4.SS2.2.p2.12">For the lower bound on <math alttext="\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.SS2.2.p2.1.m1.4"><semantics id="S4.SS2.2.p2.1.m1.4a"><mrow id="S4.SS2.2.p2.1.m1.4.4" xref="S4.SS2.2.p2.1.m1.4.4.cmml"><msub id="S4.SS2.2.p2.1.m1.4.4.3" xref="S4.SS2.2.p2.1.m1.4.4.3.cmml"><mi id="S4.SS2.2.p2.1.m1.4.4.3.2" xref="S4.SS2.2.p2.1.m1.4.4.3.2.cmml">ρ</mi><mi id="S4.SS2.2.p2.1.m1.4.4.3.3" xref="S4.SS2.2.p2.1.m1.4.4.3.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.SS2.2.p2.1.m1.4.4.2" xref="S4.SS2.2.p2.1.m1.4.4.2.cmml">⁢</mo><mrow id="S4.SS2.2.p2.1.m1.4.4.1.1" xref="S4.SS2.2.p2.1.m1.4.4.1.2.cmml"><mo id="S4.SS2.2.p2.1.m1.4.4.1.1.2" stretchy="false" xref="S4.SS2.2.p2.1.m1.4.4.1.2.cmml">(</mo><mi id="S4.SS2.2.p2.1.m1.3.3" xref="S4.SS2.2.p2.1.m1.3.3.cmml">G</mi><mo id="S4.SS2.2.p2.1.m1.4.4.1.1.3" xref="S4.SS2.2.p2.1.m1.4.4.1.2.cmml">,</mo><msub id="S4.SS2.2.p2.1.m1.4.4.1.1.1" xref="S4.SS2.2.p2.1.m1.4.4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p2.1.m1.4.4.1.1.1.2" xref="S4.SS2.2.p2.1.m1.4.4.1.1.1.2.cmml">𝒢</mi><mrow id="S4.SS2.2.p2.1.m1.2.2.2.4" xref="S4.SS2.2.p2.1.m1.2.2.2.3.cmml"><mi id="S4.SS2.2.p2.1.m1.1.1.1.1" xref="S4.SS2.2.p2.1.m1.1.1.1.1.cmml">N</mi><mo id="S4.SS2.2.p2.1.m1.2.2.2.4.1" xref="S4.SS2.2.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS2.2.p2.1.m1.2.2.2.2" xref="S4.SS2.2.p2.1.m1.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.SS2.2.p2.1.m1.4.4.1.1.4" stretchy="false" xref="S4.SS2.2.p2.1.m1.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.1.m1.4b"><apply id="S4.SS2.2.p2.1.m1.4.4.cmml" xref="S4.SS2.2.p2.1.m1.4.4"><times id="S4.SS2.2.p2.1.m1.4.4.2.cmml" xref="S4.SS2.2.p2.1.m1.4.4.2"></times><apply id="S4.SS2.2.p2.1.m1.4.4.3.cmml" xref="S4.SS2.2.p2.1.m1.4.4.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.1.m1.4.4.3.1.cmml" xref="S4.SS2.2.p2.1.m1.4.4.3">subscript</csymbol><ci id="S4.SS2.2.p2.1.m1.4.4.3.2.cmml" xref="S4.SS2.2.p2.1.m1.4.4.3.2">𝜌</ci><ci id="S4.SS2.2.p2.1.m1.4.4.3.3.cmml" xref="S4.SS2.2.p2.1.m1.4.4.3.3">𝖼𝖺𝗇</ci></apply><interval closure="open" id="S4.SS2.2.p2.1.m1.4.4.1.2.cmml" xref="S4.SS2.2.p2.1.m1.4.4.1.1"><ci id="S4.SS2.2.p2.1.m1.3.3.cmml" xref="S4.SS2.2.p2.1.m1.3.3">𝐺</ci><apply id="S4.SS2.2.p2.1.m1.4.4.1.1.1.cmml" xref="S4.SS2.2.p2.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.1.m1.4.4.1.1.1.1.cmml" xref="S4.SS2.2.p2.1.m1.4.4.1.1.1">subscript</csymbol><ci id="S4.SS2.2.p2.1.m1.4.4.1.1.1.2.cmml" xref="S4.SS2.2.p2.1.m1.4.4.1.1.1.2">𝒢</ci><list id="S4.SS2.2.p2.1.m1.2.2.2.3.cmml" xref="S4.SS2.2.p2.1.m1.2.2.2.4"><ci id="S4.SS2.2.p2.1.m1.1.1.1.1.cmml" xref="S4.SS2.2.p2.1.m1.1.1.1.1">𝑁</ci><ci id="S4.SS2.2.p2.1.m1.2.2.2.2.cmml" xref="S4.SS2.2.p2.1.m1.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.1.m1.4c">\rho_{\mathsf{can}}(G,\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.1.m1.4d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>, let <math alttext="\Lambda=\{(E_{1},H_{1}),\ldots,(E_{k},H_{k})\}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.2.m2.3"><semantics id="S4.SS2.2.p2.2.m2.3a"><mrow id="S4.SS2.2.p2.2.m2.3.3" xref="S4.SS2.2.p2.2.m2.3.3.cmml"><mi id="S4.SS2.2.p2.2.m2.3.3.4" mathvariant="normal" xref="S4.SS2.2.p2.2.m2.3.3.4.cmml">Λ</mi><mo id="S4.SS2.2.p2.2.m2.3.3.3" xref="S4.SS2.2.p2.2.m2.3.3.3.cmml">=</mo><mrow id="S4.SS2.2.p2.2.m2.3.3.2.2" xref="S4.SS2.2.p2.2.m2.3.3.2.3.cmml"><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.3" stretchy="false" xref="S4.SS2.2.p2.2.m2.3.3.2.3.cmml">{</mo><mrow id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.3.cmml"><mo id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.3" stretchy="false" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.3.cmml">(</mo><msub id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.2" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.3" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.4" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.3.cmml">,</mo><msub id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.cmml"><mi id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.2" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.2.cmml">H</mi><mn id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.3" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.5" stretchy="false" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.3.cmml">)</mo></mrow><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.4" xref="S4.SS2.2.p2.2.m2.3.3.2.3.cmml">,</mo><mi id="S4.SS2.2.p2.2.m2.1.1" mathvariant="normal" xref="S4.SS2.2.p2.2.m2.1.1.cmml">…</mi><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.5" xref="S4.SS2.2.p2.2.m2.3.3.2.3.cmml">,</mo><mrow id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.3.cmml"><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.3" stretchy="false" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.3.cmml">(</mo><msub id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.cmml"><mi id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.2" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.2.cmml">E</mi><mi id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.3" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.4" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.3.cmml">,</mo><msub id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.cmml"><mi id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.2" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.2.cmml">H</mi><mi id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.3" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.5" stretchy="false" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.3.cmml">)</mo></mrow><mo id="S4.SS2.2.p2.2.m2.3.3.2.2.6" stretchy="false" xref="S4.SS2.2.p2.2.m2.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.2.m2.3b"><apply id="S4.SS2.2.p2.2.m2.3.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3"><eq id="S4.SS2.2.p2.2.m2.3.3.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3.3"></eq><ci id="S4.SS2.2.p2.2.m2.3.3.4.cmml" xref="S4.SS2.2.p2.2.m2.3.3.4">Λ</ci><set id="S4.SS2.2.p2.2.m2.3.3.2.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2"><interval closure="open" id="S4.SS2.2.p2.2.m2.2.2.1.1.1.3.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2"><apply id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.1.1.3">1</cn></apply><apply id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.1.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2">subscript</csymbol><ci id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.2.cmml" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.2">𝐻</ci><cn id="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.3.cmml" type="integer" xref="S4.SS2.2.p2.2.m2.2.2.1.1.1.2.2.3">1</cn></apply></interval><ci id="S4.SS2.2.p2.2.m2.1.1.cmml" xref="S4.SS2.2.p2.2.m2.1.1">…</ci><interval closure="open" id="S4.SS2.2.p2.2.m2.3.3.2.2.2.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2"><apply id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.1.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1">subscript</csymbol><ci id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.2.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.2">𝐸</ci><ci id="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.1.1.3">𝑘</ci></apply><apply id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.1.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.2.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.2">𝐻</ci><ci id="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.3.cmml" xref="S4.SS2.2.p2.2.m2.3.3.2.2.2.2.2.3">𝑘</ci></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.2.m2.3c">\Lambda=\{(E_{1},H_{1}),\ldots,(E_{k},H_{k})\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.2.m2.3d">roman_Λ = { ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , ( italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) }</annotation></semantics></math> be a family of <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.2.p2.3.m3.1"><semantics id="S4.SS2.2.p2.3.m3.1a"><mi id="S4.SS2.2.p2.3.m3.1.1" xref="S4.SS2.2.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.3.m3.1b"><ci id="S4.SS2.2.p2.3.m3.1.1.cmml" xref="S4.SS2.2.p2.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.3.m3.1d">italic_k</annotation></semantics></math> pairs of subsets of <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.4.m4.1"><semantics id="S4.SS2.2.p2.4.m4.1a"><mover accent="true" id="S4.SS2.2.p2.4.m4.1.1" xref="S4.SS2.2.p2.4.m4.1.1.cmml"><mi id="S4.SS2.2.p2.4.m4.1.1.2" xref="S4.SS2.2.p2.4.m4.1.1.2.cmml">G</mi><mo id="S4.SS2.2.p2.4.m4.1.1.1" xref="S4.SS2.2.p2.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.4.m4.1b"><apply id="S4.SS2.2.p2.4.m4.1.1.cmml" xref="S4.SS2.2.p2.4.m4.1.1"><ci id="S4.SS2.2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.2.p2.4.m4.1.1.1">¯</ci><ci id="S4.SS2.2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.2.p2.4.m4.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.4.m4.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.4.m4.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math>. We argue that if <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS2.2.p2.5.m5.1"><semantics id="S4.SS2.2.p2.5.m5.1a"><mi id="S4.SS2.2.p2.5.m5.1.1" mathvariant="normal" xref="S4.SS2.2.p2.5.m5.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.5.m5.1b"><ci id="S4.SS2.2.p2.5.m5.1.1.cmml" xref="S4.SS2.2.p2.5.m5.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.5.m5.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.5.m5.1d">roman_Λ</annotation></semantics></math> covers all semi-filters in <math alttext="\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.6.m6.1"><semantics id="S4.SS2.2.p2.6.m6.1a"><msubsup id="S4.SS2.2.p2.6.m6.1.1" xref="S4.SS2.2.p2.6.m6.1.1.cmml"><mi id="S4.SS2.2.p2.6.m6.1.1.2.2" xref="S4.SS2.2.p2.6.m6.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS2.2.p2.6.m6.1.1.2.3" xref="S4.SS2.2.p2.6.m6.1.1.2.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.SS2.2.p2.6.m6.1.1.3" xref="S4.SS2.2.p2.6.m6.1.1.3.cmml">G</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.6.m6.1b"><apply id="S4.SS2.2.p2.6.m6.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1">superscript</csymbol><apply id="S4.SS2.2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.6.m6.1.1.2.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.2.p2.6.m6.1.1.2.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.2.2">𝔉</ci><ci id="S4.SS2.2.p2.6.m6.1.1.2.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.2.3">𝖼𝖺𝗇</ci></apply><ci id="S4.SS2.2.p2.6.m6.1.1.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.6.m6.1c">\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.6.m6.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math> then <math alttext="k\geq n" class="ltx_Math" display="inline" id="S4.SS2.2.p2.7.m7.1"><semantics id="S4.SS2.2.p2.7.m7.1a"><mrow id="S4.SS2.2.p2.7.m7.1.1" xref="S4.SS2.2.p2.7.m7.1.1.cmml"><mi id="S4.SS2.2.p2.7.m7.1.1.2" xref="S4.SS2.2.p2.7.m7.1.1.2.cmml">k</mi><mo id="S4.SS2.2.p2.7.m7.1.1.1" xref="S4.SS2.2.p2.7.m7.1.1.1.cmml">≥</mo><mi id="S4.SS2.2.p2.7.m7.1.1.3" xref="S4.SS2.2.p2.7.m7.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.7.m7.1b"><apply id="S4.SS2.2.p2.7.m7.1.1.cmml" xref="S4.SS2.2.p2.7.m7.1.1"><geq id="S4.SS2.2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.2.p2.7.m7.1.1.1"></geq><ci id="S4.SS2.2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.2.p2.7.m7.1.1.2">𝑘</ci><ci id="S4.SS2.2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.2.p2.7.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.7.m7.1c">k\geq n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.7.m7.1d">italic_k ≥ italic_n</annotation></semantics></math>. Recall that, for every <math alttext="e\in G" class="ltx_Math" display="inline" id="S4.SS2.2.p2.8.m8.1"><semantics id="S4.SS2.2.p2.8.m8.1a"><mrow id="S4.SS2.2.p2.8.m8.1.1" xref="S4.SS2.2.p2.8.m8.1.1.cmml"><mi id="S4.SS2.2.p2.8.m8.1.1.2" xref="S4.SS2.2.p2.8.m8.1.1.2.cmml">e</mi><mo id="S4.SS2.2.p2.8.m8.1.1.1" xref="S4.SS2.2.p2.8.m8.1.1.1.cmml">∈</mo><mi id="S4.SS2.2.p2.8.m8.1.1.3" xref="S4.SS2.2.p2.8.m8.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.8.m8.1b"><apply id="S4.SS2.2.p2.8.m8.1.1.cmml" xref="S4.SS2.2.p2.8.m8.1.1"><in id="S4.SS2.2.p2.8.m8.1.1.1.cmml" xref="S4.SS2.2.p2.8.m8.1.1.1"></in><ci id="S4.SS2.2.p2.8.m8.1.1.2.cmml" xref="S4.SS2.2.p2.8.m8.1.1.2">𝑒</ci><ci id="S4.SS2.2.p2.8.m8.1.1.3.cmml" xref="S4.SS2.2.p2.8.m8.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.8.m8.1c">e\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.8.m8.1d">italic_e ∈ italic_G</annotation></semantics></math>, <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.9.m9.1"><semantics id="S4.SS2.2.p2.9.m9.1a"><msub id="S4.SS2.2.p2.9.m9.1.1" xref="S4.SS2.2.p2.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p2.9.m9.1.1.2" xref="S4.SS2.2.p2.9.m9.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.2.p2.9.m9.1.1.3" xref="S4.SS2.2.p2.9.m9.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.9.m9.1b"><apply id="S4.SS2.2.p2.9.m9.1.1.cmml" xref="S4.SS2.2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.9.m9.1.1.1.cmml" xref="S4.SS2.2.p2.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.2.p2.9.m9.1.1.2.cmml" xref="S4.SS2.2.p2.9.m9.1.1.2">ℱ</ci><ci id="S4.SS2.2.p2.9.m9.1.1.3.cmml" xref="S4.SS2.2.p2.9.m9.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.9.m9.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.9.m9.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> is a semi-filter above <math alttext="e" class="ltx_Math" display="inline" id="S4.SS2.2.p2.10.m10.1"><semantics id="S4.SS2.2.p2.10.m10.1a"><mi id="S4.SS2.2.p2.10.m10.1.1" xref="S4.SS2.2.p2.10.m10.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.10.m10.1b"><ci id="S4.SS2.2.p2.10.m10.1.1.cmml" xref="S4.SS2.2.p2.10.m10.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.10.m10.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.10.m10.1d">italic_e</annotation></semantics></math>, i.e., <math alttext="\mathcal{F}_{e}\in\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.11.m11.1"><semantics id="S4.SS2.2.p2.11.m11.1a"><mrow id="S4.SS2.2.p2.11.m11.1.1" xref="S4.SS2.2.p2.11.m11.1.1.cmml"><msub id="S4.SS2.2.p2.11.m11.1.1.2" xref="S4.SS2.2.p2.11.m11.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p2.11.m11.1.1.2.2" xref="S4.SS2.2.p2.11.m11.1.1.2.2.cmml">ℱ</mi><mi id="S4.SS2.2.p2.11.m11.1.1.2.3" xref="S4.SS2.2.p2.11.m11.1.1.2.3.cmml">e</mi></msub><mo id="S4.SS2.2.p2.11.m11.1.1.1" xref="S4.SS2.2.p2.11.m11.1.1.1.cmml">∈</mo><msubsup id="S4.SS2.2.p2.11.m11.1.1.3" xref="S4.SS2.2.p2.11.m11.1.1.3.cmml"><mi id="S4.SS2.2.p2.11.m11.1.1.3.2.2" xref="S4.SS2.2.p2.11.m11.1.1.3.2.2.cmml">𝔉</mi><mi id="S4.SS2.2.p2.11.m11.1.1.3.2.3" xref="S4.SS2.2.p2.11.m11.1.1.3.2.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.SS2.2.p2.11.m11.1.1.3.3" xref="S4.SS2.2.p2.11.m11.1.1.3.3.cmml">G</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.11.m11.1b"><apply id="S4.SS2.2.p2.11.m11.1.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1"><in id="S4.SS2.2.p2.11.m11.1.1.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1.1"></in><apply id="S4.SS2.2.p2.11.m11.1.1.2.cmml" xref="S4.SS2.2.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.2.p2.11.m11.1.1.2.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1.2">subscript</csymbol><ci id="S4.SS2.2.p2.11.m11.1.1.2.2.cmml" xref="S4.SS2.2.p2.11.m11.1.1.2.2">ℱ</ci><ci id="S4.SS2.2.p2.11.m11.1.1.2.3.cmml" xref="S4.SS2.2.p2.11.m11.1.1.2.3">𝑒</ci></apply><apply id="S4.SS2.2.p2.11.m11.1.1.3.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.11.m11.1.1.3.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3">superscript</csymbol><apply id="S4.SS2.2.p2.11.m11.1.1.3.2.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.11.m11.1.1.3.2.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3">subscript</csymbol><ci id="S4.SS2.2.p2.11.m11.1.1.3.2.2.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3.2.2">𝔉</ci><ci id="S4.SS2.2.p2.11.m11.1.1.3.2.3.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3.2.3">𝖼𝖺𝗇</ci></apply><ci id="S4.SS2.2.p2.11.m11.1.1.3.3.cmml" xref="S4.SS2.2.p2.11.m11.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.11.m11.1c">\mathcal{F}_{e}\in\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.11.m11.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∈ fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math>. Fix a pair <math alttext="(E,H)\in\Lambda" class="ltx_Math" display="inline" id="S4.SS2.2.p2.12.m12.2"><semantics id="S4.SS2.2.p2.12.m12.2a"><mrow id="S4.SS2.2.p2.12.m12.2.3" xref="S4.SS2.2.p2.12.m12.2.3.cmml"><mrow id="S4.SS2.2.p2.12.m12.2.3.2.2" xref="S4.SS2.2.p2.12.m12.2.3.2.1.cmml"><mo id="S4.SS2.2.p2.12.m12.2.3.2.2.1" stretchy="false" xref="S4.SS2.2.p2.12.m12.2.3.2.1.cmml">(</mo><mi id="S4.SS2.2.p2.12.m12.1.1" xref="S4.SS2.2.p2.12.m12.1.1.cmml">E</mi><mo id="S4.SS2.2.p2.12.m12.2.3.2.2.2" xref="S4.SS2.2.p2.12.m12.2.3.2.1.cmml">,</mo><mi id="S4.SS2.2.p2.12.m12.2.2" xref="S4.SS2.2.p2.12.m12.2.2.cmml">H</mi><mo id="S4.SS2.2.p2.12.m12.2.3.2.2.3" stretchy="false" xref="S4.SS2.2.p2.12.m12.2.3.2.1.cmml">)</mo></mrow><mo id="S4.SS2.2.p2.12.m12.2.3.1" xref="S4.SS2.2.p2.12.m12.2.3.1.cmml">∈</mo><mi id="S4.SS2.2.p2.12.m12.2.3.3" mathvariant="normal" xref="S4.SS2.2.p2.12.m12.2.3.3.cmml">Λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.12.m12.2b"><apply id="S4.SS2.2.p2.12.m12.2.3.cmml" xref="S4.SS2.2.p2.12.m12.2.3"><in id="S4.SS2.2.p2.12.m12.2.3.1.cmml" xref="S4.SS2.2.p2.12.m12.2.3.1"></in><interval closure="open" id="S4.SS2.2.p2.12.m12.2.3.2.1.cmml" xref="S4.SS2.2.p2.12.m12.2.3.2.2"><ci id="S4.SS2.2.p2.12.m12.1.1.cmml" xref="S4.SS2.2.p2.12.m12.1.1">𝐸</ci><ci id="S4.SS2.2.p2.12.m12.2.2.cmml" xref="S4.SS2.2.p2.12.m12.2.2">𝐻</ci></interval><ci id="S4.SS2.2.p2.12.m12.2.3.3.cmml" xref="S4.SS2.2.p2.12.m12.2.3.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.12.m12.2c">(E,H)\in\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.12.m12.2d">( italic_E , italic_H ) ∈ roman_Λ</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="Thmtheorem41"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem41.1.1.1">Claim 41</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem41.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem41.p1"> <p class="ltx_p" id="Thmtheorem41.p1.8"><span class="ltx_text ltx_font_italic" id="Thmtheorem41.p1.8.8">Let <math alttext="e=(u,v)\in G" class="ltx_Math" display="inline" id="Thmtheorem41.p1.1.1.m1.2"><semantics id="Thmtheorem41.p1.1.1.m1.2a"><mrow id="Thmtheorem41.p1.1.1.m1.2.3" xref="Thmtheorem41.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem41.p1.1.1.m1.2.3.2" xref="Thmtheorem41.p1.1.1.m1.2.3.2.cmml">e</mi><mo id="Thmtheorem41.p1.1.1.m1.2.3.3" xref="Thmtheorem41.p1.1.1.m1.2.3.3.cmml">=</mo><mrow id="Thmtheorem41.p1.1.1.m1.2.3.4.2" xref="Thmtheorem41.p1.1.1.m1.2.3.4.1.cmml"><mo id="Thmtheorem41.p1.1.1.m1.2.3.4.2.1" stretchy="false" xref="Thmtheorem41.p1.1.1.m1.2.3.4.1.cmml">(</mo><mi id="Thmtheorem41.p1.1.1.m1.1.1" xref="Thmtheorem41.p1.1.1.m1.1.1.cmml">u</mi><mo id="Thmtheorem41.p1.1.1.m1.2.3.4.2.2" xref="Thmtheorem41.p1.1.1.m1.2.3.4.1.cmml">,</mo><mi id="Thmtheorem41.p1.1.1.m1.2.2" xref="Thmtheorem41.p1.1.1.m1.2.2.cmml">v</mi><mo id="Thmtheorem41.p1.1.1.m1.2.3.4.2.3" stretchy="false" xref="Thmtheorem41.p1.1.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="Thmtheorem41.p1.1.1.m1.2.3.5" xref="Thmtheorem41.p1.1.1.m1.2.3.5.cmml">∈</mo><mi id="Thmtheorem41.p1.1.1.m1.2.3.6" xref="Thmtheorem41.p1.1.1.m1.2.3.6.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.1.1.m1.2b"><apply id="Thmtheorem41.p1.1.1.m1.2.3.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3"><and id="Thmtheorem41.p1.1.1.m1.2.3a.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3"></and><apply id="Thmtheorem41.p1.1.1.m1.2.3b.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3"><eq id="Thmtheorem41.p1.1.1.m1.2.3.3.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3.3"></eq><ci id="Thmtheorem41.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3.2">𝑒</ci><interval closure="open" id="Thmtheorem41.p1.1.1.m1.2.3.4.1.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3.4.2"><ci id="Thmtheorem41.p1.1.1.m1.1.1.cmml" xref="Thmtheorem41.p1.1.1.m1.1.1">𝑢</ci><ci id="Thmtheorem41.p1.1.1.m1.2.2.cmml" xref="Thmtheorem41.p1.1.1.m1.2.2">𝑣</ci></interval></apply><apply id="Thmtheorem41.p1.1.1.m1.2.3c.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3"><in id="Thmtheorem41.p1.1.1.m1.2.3.5.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem41.p1.1.1.m1.2.3.4.cmml" id="Thmtheorem41.p1.1.1.m1.2.3d.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3"></share><ci id="Thmtheorem41.p1.1.1.m1.2.3.6.cmml" xref="Thmtheorem41.p1.1.1.m1.2.3.6">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.1.1.m1.2c">e=(u,v)\in G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.1.1.m1.2d">italic_e = ( italic_u , italic_v ) ∈ italic_G</annotation></semantics></math>, and <math alttext="\mathcal{F}_{e}\in\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="Thmtheorem41.p1.2.2.m2.1"><semantics id="Thmtheorem41.p1.2.2.m2.1a"><mrow id="Thmtheorem41.p1.2.2.m2.1.1" xref="Thmtheorem41.p1.2.2.m2.1.1.cmml"><msub id="Thmtheorem41.p1.2.2.m2.1.1.2" xref="Thmtheorem41.p1.2.2.m2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem41.p1.2.2.m2.1.1.2.2" xref="Thmtheorem41.p1.2.2.m2.1.1.2.2.cmml">ℱ</mi><mi id="Thmtheorem41.p1.2.2.m2.1.1.2.3" xref="Thmtheorem41.p1.2.2.m2.1.1.2.3.cmml">e</mi></msub><mo id="Thmtheorem41.p1.2.2.m2.1.1.1" xref="Thmtheorem41.p1.2.2.m2.1.1.1.cmml">∈</mo><msubsup id="Thmtheorem41.p1.2.2.m2.1.1.3" xref="Thmtheorem41.p1.2.2.m2.1.1.3.cmml"><mi id="Thmtheorem41.p1.2.2.m2.1.1.3.2.2" xref="Thmtheorem41.p1.2.2.m2.1.1.3.2.2.cmml">𝔉</mi><mi id="Thmtheorem41.p1.2.2.m2.1.1.3.2.3" xref="Thmtheorem41.p1.2.2.m2.1.1.3.2.3.cmml">𝖼𝖺𝗇</mi><mi id="Thmtheorem41.p1.2.2.m2.1.1.3.3" xref="Thmtheorem41.p1.2.2.m2.1.1.3.3.cmml">G</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.2.2.m2.1b"><apply id="Thmtheorem41.p1.2.2.m2.1.1.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1"><in id="Thmtheorem41.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.1"></in><apply id="Thmtheorem41.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem41.p1.2.2.m2.1.1.2.1.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="Thmtheorem41.p1.2.2.m2.1.1.2.2.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.2.2">ℱ</ci><ci id="Thmtheorem41.p1.2.2.m2.1.1.2.3.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.2.3">𝑒</ci></apply><apply id="Thmtheorem41.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem41.p1.2.2.m2.1.1.3.1.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3">superscript</csymbol><apply id="Thmtheorem41.p1.2.2.m2.1.1.3.2.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem41.p1.2.2.m2.1.1.3.2.1.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="Thmtheorem41.p1.2.2.m2.1.1.3.2.2.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3.2.2">𝔉</ci><ci id="Thmtheorem41.p1.2.2.m2.1.1.3.2.3.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3.2.3">𝖼𝖺𝗇</ci></apply><ci id="Thmtheorem41.p1.2.2.m2.1.1.3.3.cmml" xref="Thmtheorem41.p1.2.2.m2.1.1.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.2.2.m2.1c">\mathcal{F}_{e}\in\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.2.2.m2.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∈ fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="Thmtheorem41.p1.3.3.m3.1"><semantics id="Thmtheorem41.p1.3.3.m3.1a"><msub id="Thmtheorem41.p1.3.3.m3.1.1" xref="Thmtheorem41.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem41.p1.3.3.m3.1.1.2" xref="Thmtheorem41.p1.3.3.m3.1.1.2.cmml">ℱ</mi><mi id="Thmtheorem41.p1.3.3.m3.1.1.3" xref="Thmtheorem41.p1.3.3.m3.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.3.3.m3.1b"><apply id="Thmtheorem41.p1.3.3.m3.1.1.cmml" xref="Thmtheorem41.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmtheorem41.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem41.p1.3.3.m3.1.1">subscript</csymbol><ci id="Thmtheorem41.p1.3.3.m3.1.1.2.cmml" xref="Thmtheorem41.p1.3.3.m3.1.1.2">ℱ</ci><ci id="Thmtheorem41.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem41.p1.3.3.m3.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.3.3.m3.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.3.3.m3.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> is covered by <math alttext="(E,H)" class="ltx_Math" display="inline" id="Thmtheorem41.p1.4.4.m4.2"><semantics id="Thmtheorem41.p1.4.4.m4.2a"><mrow id="Thmtheorem41.p1.4.4.m4.2.3.2" xref="Thmtheorem41.p1.4.4.m4.2.3.1.cmml"><mo id="Thmtheorem41.p1.4.4.m4.2.3.2.1" stretchy="false" xref="Thmtheorem41.p1.4.4.m4.2.3.1.cmml">(</mo><mi id="Thmtheorem41.p1.4.4.m4.1.1" xref="Thmtheorem41.p1.4.4.m4.1.1.cmml">E</mi><mo id="Thmtheorem41.p1.4.4.m4.2.3.2.2" xref="Thmtheorem41.p1.4.4.m4.2.3.1.cmml">,</mo><mi id="Thmtheorem41.p1.4.4.m4.2.2" xref="Thmtheorem41.p1.4.4.m4.2.2.cmml">H</mi><mo id="Thmtheorem41.p1.4.4.m4.2.3.2.3" stretchy="false" xref="Thmtheorem41.p1.4.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.4.4.m4.2b"><interval closure="open" id="Thmtheorem41.p1.4.4.m4.2.3.1.cmml" xref="Thmtheorem41.p1.4.4.m4.2.3.2"><ci id="Thmtheorem41.p1.4.4.m4.1.1.cmml" xref="Thmtheorem41.p1.4.4.m4.1.1">𝐸</ci><ci id="Thmtheorem41.p1.4.4.m4.2.2.cmml" xref="Thmtheorem41.p1.4.4.m4.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.4.4.m4.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.4.4.m4.2d">( italic_E , italic_H )</annotation></semantics></math> if and only if each singleton set <math alttext="R^{u}_{\overline{G}}" class="ltx_Math" display="inline" id="Thmtheorem41.p1.5.5.m5.1"><semantics id="Thmtheorem41.p1.5.5.m5.1a"><msubsup id="Thmtheorem41.p1.5.5.m5.1.1" xref="Thmtheorem41.p1.5.5.m5.1.1.cmml"><mi id="Thmtheorem41.p1.5.5.m5.1.1.2.2" xref="Thmtheorem41.p1.5.5.m5.1.1.2.2.cmml">R</mi><mover accent="true" id="Thmtheorem41.p1.5.5.m5.1.1.3" xref="Thmtheorem41.p1.5.5.m5.1.1.3.cmml"><mi id="Thmtheorem41.p1.5.5.m5.1.1.3.2" xref="Thmtheorem41.p1.5.5.m5.1.1.3.2.cmml">G</mi><mo id="Thmtheorem41.p1.5.5.m5.1.1.3.1" xref="Thmtheorem41.p1.5.5.m5.1.1.3.1.cmml">¯</mo></mover><mi id="Thmtheorem41.p1.5.5.m5.1.1.2.3" xref="Thmtheorem41.p1.5.5.m5.1.1.2.3.cmml">u</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.5.5.m5.1b"><apply id="Thmtheorem41.p1.5.5.m5.1.1.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="Thmtheorem41.p1.5.5.m5.1.1.1.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1">subscript</csymbol><apply id="Thmtheorem41.p1.5.5.m5.1.1.2.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="Thmtheorem41.p1.5.5.m5.1.1.2.1.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1">superscript</csymbol><ci id="Thmtheorem41.p1.5.5.m5.1.1.2.2.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1.2.2">𝑅</ci><ci id="Thmtheorem41.p1.5.5.m5.1.1.2.3.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1.2.3">𝑢</ci></apply><apply id="Thmtheorem41.p1.5.5.m5.1.1.3.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1.3"><ci id="Thmtheorem41.p1.5.5.m5.1.1.3.1.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1.3.1">¯</ci><ci id="Thmtheorem41.p1.5.5.m5.1.1.3.2.cmml" xref="Thmtheorem41.p1.5.5.m5.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.5.5.m5.1c">R^{u}_{\overline{G}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.5.5.m5.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="C^{v}_{\overline{G}}" class="ltx_Math" display="inline" id="Thmtheorem41.p1.6.6.m6.1"><semantics id="Thmtheorem41.p1.6.6.m6.1a"><msubsup id="Thmtheorem41.p1.6.6.m6.1.1" xref="Thmtheorem41.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem41.p1.6.6.m6.1.1.2.2" xref="Thmtheorem41.p1.6.6.m6.1.1.2.2.cmml">C</mi><mover accent="true" id="Thmtheorem41.p1.6.6.m6.1.1.3" xref="Thmtheorem41.p1.6.6.m6.1.1.3.cmml"><mi id="Thmtheorem41.p1.6.6.m6.1.1.3.2" xref="Thmtheorem41.p1.6.6.m6.1.1.3.2.cmml">G</mi><mo id="Thmtheorem41.p1.6.6.m6.1.1.3.1" xref="Thmtheorem41.p1.6.6.m6.1.1.3.1.cmml">¯</mo></mover><mi id="Thmtheorem41.p1.6.6.m6.1.1.2.3" xref="Thmtheorem41.p1.6.6.m6.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.6.6.m6.1b"><apply id="Thmtheorem41.p1.6.6.m6.1.1.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="Thmtheorem41.p1.6.6.m6.1.1.1.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1">subscript</csymbol><apply id="Thmtheorem41.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="Thmtheorem41.p1.6.6.m6.1.1.2.1.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1">superscript</csymbol><ci id="Thmtheorem41.p1.6.6.m6.1.1.2.2.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1.2.2">𝐶</ci><ci id="Thmtheorem41.p1.6.6.m6.1.1.2.3.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1.2.3">𝑣</ci></apply><apply id="Thmtheorem41.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1.3"><ci id="Thmtheorem41.p1.6.6.m6.1.1.3.1.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1.3.1">¯</ci><ci id="Thmtheorem41.p1.6.6.m6.1.1.3.2.cmml" xref="Thmtheorem41.p1.6.6.m6.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.6.6.m6.1c">C^{v}_{\overline{G}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.6.6.m6.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> is contained in precisely one of <math alttext="E" class="ltx_Math" display="inline" id="Thmtheorem41.p1.7.7.m7.1"><semantics id="Thmtheorem41.p1.7.7.m7.1a"><mi id="Thmtheorem41.p1.7.7.m7.1.1" xref="Thmtheorem41.p1.7.7.m7.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.7.7.m7.1b"><ci id="Thmtheorem41.p1.7.7.m7.1.1.cmml" xref="Thmtheorem41.p1.7.7.m7.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.7.7.m7.1c">E</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.7.7.m7.1d">italic_E</annotation></semantics></math> and <math alttext="H" class="ltx_Math" display="inline" id="Thmtheorem41.p1.8.8.m8.1"><semantics id="Thmtheorem41.p1.8.8.m8.1a"><mi id="Thmtheorem41.p1.8.8.m8.1.1" xref="Thmtheorem41.p1.8.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem41.p1.8.8.m8.1b"><ci id="Thmtheorem41.p1.8.8.m8.1.1.cmml" xref="Thmtheorem41.p1.8.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem41.p1.8.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem41.p1.8.8.m8.1d">italic_H</annotation></semantics></math>, and none of the latter sets contains both of them.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.3.p3"> <p class="ltx_p" id="S4.SS2.3.p3.11">First, we argue that <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.1.m1.1"><semantics id="S4.SS2.3.p3.1.m1.1a"><msub id="S4.SS2.3.p3.1.m1.1.1" xref="S4.SS2.3.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.1.m1.1.1.2" xref="S4.SS2.3.p3.1.m1.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.1.m1.1.1.3" xref="S4.SS2.3.p3.1.m1.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.1.m1.1b"><apply id="S4.SS2.3.p3.1.m1.1.1.cmml" xref="S4.SS2.3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p3.1.m1.1.1.1.cmml" xref="S4.SS2.3.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.3.p3.1.m1.1.1.2.cmml" xref="S4.SS2.3.p3.1.m1.1.1.2">ℱ</ci><ci id="S4.SS2.3.p3.1.m1.1.1.3.cmml" xref="S4.SS2.3.p3.1.m1.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.1.m1.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.1.m1.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> is covered under the condition in the claim. Assume without loss of generality that <math alttext="R^{u}_{\overline{G}}\subseteq E" class="ltx_Math" display="inline" id="S4.SS2.3.p3.2.m2.1"><semantics id="S4.SS2.3.p3.2.m2.1a"><mrow id="S4.SS2.3.p3.2.m2.1.1" xref="S4.SS2.3.p3.2.m2.1.1.cmml"><msubsup id="S4.SS2.3.p3.2.m2.1.1.2" xref="S4.SS2.3.p3.2.m2.1.1.2.cmml"><mi id="S4.SS2.3.p3.2.m2.1.1.2.2.2" xref="S4.SS2.3.p3.2.m2.1.1.2.2.2.cmml">R</mi><mover accent="true" id="S4.SS2.3.p3.2.m2.1.1.2.3" xref="S4.SS2.3.p3.2.m2.1.1.2.3.cmml"><mi id="S4.SS2.3.p3.2.m2.1.1.2.3.2" xref="S4.SS2.3.p3.2.m2.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.3.p3.2.m2.1.1.2.3.1" xref="S4.SS2.3.p3.2.m2.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.3.p3.2.m2.1.1.2.2.3" xref="S4.SS2.3.p3.2.m2.1.1.2.2.3.cmml">u</mi></msubsup><mo id="S4.SS2.3.p3.2.m2.1.1.1" xref="S4.SS2.3.p3.2.m2.1.1.1.cmml">⊆</mo><mi id="S4.SS2.3.p3.2.m2.1.1.3" xref="S4.SS2.3.p3.2.m2.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.2.m2.1b"><apply id="S4.SS2.3.p3.2.m2.1.1.cmml" xref="S4.SS2.3.p3.2.m2.1.1"><subset id="S4.SS2.3.p3.2.m2.1.1.1.cmml" xref="S4.SS2.3.p3.2.m2.1.1.1"></subset><apply id="S4.SS2.3.p3.2.m2.1.1.2.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.2.m2.1.1.2.1.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2">subscript</csymbol><apply id="S4.SS2.3.p3.2.m2.1.1.2.2.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.2.m2.1.1.2.2.1.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2">superscript</csymbol><ci id="S4.SS2.3.p3.2.m2.1.1.2.2.2.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2.2.2">𝑅</ci><ci id="S4.SS2.3.p3.2.m2.1.1.2.2.3.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2.2.3">𝑢</ci></apply><apply id="S4.SS2.3.p3.2.m2.1.1.2.3.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2.3"><ci id="S4.SS2.3.p3.2.m2.1.1.2.3.1.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2.3.1">¯</ci><ci id="S4.SS2.3.p3.2.m2.1.1.2.3.2.cmml" xref="S4.SS2.3.p3.2.m2.1.1.2.3.2">𝐺</ci></apply></apply><ci id="S4.SS2.3.p3.2.m2.1.1.3.cmml" xref="S4.SS2.3.p3.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.2.m2.1c">R^{u}_{\overline{G}}\subseteq E</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.2.m2.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊆ italic_E</annotation></semantics></math> and <math alttext="C^{v}_{\overline{G}}\subseteq H" class="ltx_Math" display="inline" id="S4.SS2.3.p3.3.m3.1"><semantics id="S4.SS2.3.p3.3.m3.1a"><mrow id="S4.SS2.3.p3.3.m3.1.1" xref="S4.SS2.3.p3.3.m3.1.1.cmml"><msubsup id="S4.SS2.3.p3.3.m3.1.1.2" xref="S4.SS2.3.p3.3.m3.1.1.2.cmml"><mi id="S4.SS2.3.p3.3.m3.1.1.2.2.2" xref="S4.SS2.3.p3.3.m3.1.1.2.2.2.cmml">C</mi><mover accent="true" id="S4.SS2.3.p3.3.m3.1.1.2.3" xref="S4.SS2.3.p3.3.m3.1.1.2.3.cmml"><mi id="S4.SS2.3.p3.3.m3.1.1.2.3.2" xref="S4.SS2.3.p3.3.m3.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.3.p3.3.m3.1.1.2.3.1" xref="S4.SS2.3.p3.3.m3.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.3.p3.3.m3.1.1.2.2.3" xref="S4.SS2.3.p3.3.m3.1.1.2.2.3.cmml">v</mi></msubsup><mo id="S4.SS2.3.p3.3.m3.1.1.1" xref="S4.SS2.3.p3.3.m3.1.1.1.cmml">⊆</mo><mi id="S4.SS2.3.p3.3.m3.1.1.3" xref="S4.SS2.3.p3.3.m3.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.3.m3.1b"><apply id="S4.SS2.3.p3.3.m3.1.1.cmml" xref="S4.SS2.3.p3.3.m3.1.1"><subset id="S4.SS2.3.p3.3.m3.1.1.1.cmml" xref="S4.SS2.3.p3.3.m3.1.1.1"></subset><apply id="S4.SS2.3.p3.3.m3.1.1.2.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.3.m3.1.1.2.1.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2">subscript</csymbol><apply id="S4.SS2.3.p3.3.m3.1.1.2.2.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.3.m3.1.1.2.2.1.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2">superscript</csymbol><ci id="S4.SS2.3.p3.3.m3.1.1.2.2.2.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2.2.2">𝐶</ci><ci id="S4.SS2.3.p3.3.m3.1.1.2.2.3.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2.2.3">𝑣</ci></apply><apply id="S4.SS2.3.p3.3.m3.1.1.2.3.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2.3"><ci id="S4.SS2.3.p3.3.m3.1.1.2.3.1.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2.3.1">¯</ci><ci id="S4.SS2.3.p3.3.m3.1.1.2.3.2.cmml" xref="S4.SS2.3.p3.3.m3.1.1.2.3.2">𝐺</ci></apply></apply><ci id="S4.SS2.3.p3.3.m3.1.1.3.cmml" xref="S4.SS2.3.p3.3.m3.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.3.m3.1c">C^{v}_{\overline{G}}\subseteq H</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.3.m3.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊆ italic_H</annotation></semantics></math>. Then, using the definition of <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.4.m4.1"><semantics id="S4.SS2.3.p3.4.m4.1a"><msub id="S4.SS2.3.p3.4.m4.1.1" xref="S4.SS2.3.p3.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.4.m4.1.1.2" xref="S4.SS2.3.p3.4.m4.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.4.m4.1.1.3" xref="S4.SS2.3.p3.4.m4.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.4.m4.1b"><apply id="S4.SS2.3.p3.4.m4.1.1.cmml" xref="S4.SS2.3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p3.4.m4.1.1.1.cmml" xref="S4.SS2.3.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.3.p3.4.m4.1.1.2.cmml" xref="S4.SS2.3.p3.4.m4.1.1.2">ℱ</ci><ci id="S4.SS2.3.p3.4.m4.1.1.3.cmml" xref="S4.SS2.3.p3.4.m4.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.4.m4.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.4.m4.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>, we get that <math alttext="E\in\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.5.m5.1"><semantics id="S4.SS2.3.p3.5.m5.1a"><mrow id="S4.SS2.3.p3.5.m5.1.1" xref="S4.SS2.3.p3.5.m5.1.1.cmml"><mi id="S4.SS2.3.p3.5.m5.1.1.2" xref="S4.SS2.3.p3.5.m5.1.1.2.cmml">E</mi><mo id="S4.SS2.3.p3.5.m5.1.1.1" xref="S4.SS2.3.p3.5.m5.1.1.1.cmml">∈</mo><msub id="S4.SS2.3.p3.5.m5.1.1.3" xref="S4.SS2.3.p3.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.5.m5.1.1.3.2" xref="S4.SS2.3.p3.5.m5.1.1.3.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.5.m5.1.1.3.3" xref="S4.SS2.3.p3.5.m5.1.1.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.5.m5.1b"><apply id="S4.SS2.3.p3.5.m5.1.1.cmml" xref="S4.SS2.3.p3.5.m5.1.1"><in id="S4.SS2.3.p3.5.m5.1.1.1.cmml" xref="S4.SS2.3.p3.5.m5.1.1.1"></in><ci id="S4.SS2.3.p3.5.m5.1.1.2.cmml" xref="S4.SS2.3.p3.5.m5.1.1.2">𝐸</ci><apply id="S4.SS2.3.p3.5.m5.1.1.3.cmml" xref="S4.SS2.3.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.3.p3.5.m5.1.1.3.1.cmml" xref="S4.SS2.3.p3.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS2.3.p3.5.m5.1.1.3.2.cmml" xref="S4.SS2.3.p3.5.m5.1.1.3.2">ℱ</ci><ci id="S4.SS2.3.p3.5.m5.1.1.3.3.cmml" xref="S4.SS2.3.p3.5.m5.1.1.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.5.m5.1c">E\in\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.5.m5.1d">italic_E ∈ caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H\in\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.6.m6.1"><semantics id="S4.SS2.3.p3.6.m6.1a"><mrow id="S4.SS2.3.p3.6.m6.1.1" xref="S4.SS2.3.p3.6.m6.1.1.cmml"><mi id="S4.SS2.3.p3.6.m6.1.1.2" xref="S4.SS2.3.p3.6.m6.1.1.2.cmml">H</mi><mo id="S4.SS2.3.p3.6.m6.1.1.1" xref="S4.SS2.3.p3.6.m6.1.1.1.cmml">∈</mo><msub id="S4.SS2.3.p3.6.m6.1.1.3" xref="S4.SS2.3.p3.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.6.m6.1.1.3.2" xref="S4.SS2.3.p3.6.m6.1.1.3.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.6.m6.1.1.3.3" xref="S4.SS2.3.p3.6.m6.1.1.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.6.m6.1b"><apply id="S4.SS2.3.p3.6.m6.1.1.cmml" xref="S4.SS2.3.p3.6.m6.1.1"><in id="S4.SS2.3.p3.6.m6.1.1.1.cmml" xref="S4.SS2.3.p3.6.m6.1.1.1"></in><ci id="S4.SS2.3.p3.6.m6.1.1.2.cmml" xref="S4.SS2.3.p3.6.m6.1.1.2">𝐻</ci><apply id="S4.SS2.3.p3.6.m6.1.1.3.cmml" xref="S4.SS2.3.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.3.p3.6.m6.1.1.3.1.cmml" xref="S4.SS2.3.p3.6.m6.1.1.3">subscript</csymbol><ci id="S4.SS2.3.p3.6.m6.1.1.3.2.cmml" xref="S4.SS2.3.p3.6.m6.1.1.3.2">ℱ</ci><ci id="S4.SS2.3.p3.6.m6.1.1.3.3.cmml" xref="S4.SS2.3.p3.6.m6.1.1.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.6.m6.1c">H\in\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.6.m6.1d">italic_H ∈ caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>. On the other hand, by assumption, <math alttext="R^{u}_{\overline{G}}\nsubseteq E\cap H" class="ltx_Math" display="inline" id="S4.SS2.3.p3.7.m7.1"><semantics id="S4.SS2.3.p3.7.m7.1a"><mrow id="S4.SS2.3.p3.7.m7.1.1" xref="S4.SS2.3.p3.7.m7.1.1.cmml"><msubsup id="S4.SS2.3.p3.7.m7.1.1.2" xref="S4.SS2.3.p3.7.m7.1.1.2.cmml"><mi id="S4.SS2.3.p3.7.m7.1.1.2.2.2" xref="S4.SS2.3.p3.7.m7.1.1.2.2.2.cmml">R</mi><mover accent="true" id="S4.SS2.3.p3.7.m7.1.1.2.3" xref="S4.SS2.3.p3.7.m7.1.1.2.3.cmml"><mi id="S4.SS2.3.p3.7.m7.1.1.2.3.2" xref="S4.SS2.3.p3.7.m7.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.3.p3.7.m7.1.1.2.3.1" xref="S4.SS2.3.p3.7.m7.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.3.p3.7.m7.1.1.2.2.3" xref="S4.SS2.3.p3.7.m7.1.1.2.2.3.cmml">u</mi></msubsup><mo id="S4.SS2.3.p3.7.m7.1.1.1" xref="S4.SS2.3.p3.7.m7.1.1.1.cmml">⊈</mo><mrow id="S4.SS2.3.p3.7.m7.1.1.3" xref="S4.SS2.3.p3.7.m7.1.1.3.cmml"><mi id="S4.SS2.3.p3.7.m7.1.1.3.2" xref="S4.SS2.3.p3.7.m7.1.1.3.2.cmml">E</mi><mo id="S4.SS2.3.p3.7.m7.1.1.3.1" xref="S4.SS2.3.p3.7.m7.1.1.3.1.cmml">∩</mo><mi id="S4.SS2.3.p3.7.m7.1.1.3.3" xref="S4.SS2.3.p3.7.m7.1.1.3.3.cmml">H</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.7.m7.1b"><apply id="S4.SS2.3.p3.7.m7.1.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1"><csymbol cd="latexml" id="S4.SS2.3.p3.7.m7.1.1.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1.1">not-subset-of-nor-equals</csymbol><apply id="S4.SS2.3.p3.7.m7.1.1.2.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.7.m7.1.1.2.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2">subscript</csymbol><apply id="S4.SS2.3.p3.7.m7.1.1.2.2.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.7.m7.1.1.2.2.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2">superscript</csymbol><ci id="S4.SS2.3.p3.7.m7.1.1.2.2.2.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2.2.2">𝑅</ci><ci id="S4.SS2.3.p3.7.m7.1.1.2.2.3.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2.2.3">𝑢</ci></apply><apply id="S4.SS2.3.p3.7.m7.1.1.2.3.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2.3"><ci id="S4.SS2.3.p3.7.m7.1.1.2.3.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2.3.1">¯</ci><ci id="S4.SS2.3.p3.7.m7.1.1.2.3.2.cmml" xref="S4.SS2.3.p3.7.m7.1.1.2.3.2">𝐺</ci></apply></apply><apply id="S4.SS2.3.p3.7.m7.1.1.3.cmml" xref="S4.SS2.3.p3.7.m7.1.1.3"><intersect id="S4.SS2.3.p3.7.m7.1.1.3.1.cmml" xref="S4.SS2.3.p3.7.m7.1.1.3.1"></intersect><ci id="S4.SS2.3.p3.7.m7.1.1.3.2.cmml" xref="S4.SS2.3.p3.7.m7.1.1.3.2">𝐸</ci><ci id="S4.SS2.3.p3.7.m7.1.1.3.3.cmml" xref="S4.SS2.3.p3.7.m7.1.1.3.3">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.7.m7.1c">R^{u}_{\overline{G}}\nsubseteq E\cap H</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.7.m7.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊈ italic_E ∩ italic_H</annotation></semantics></math> and <math alttext="C^{v}_{\overline{G}}\nsubseteq E\cap H" class="ltx_Math" display="inline" id="S4.SS2.3.p3.8.m8.1"><semantics id="S4.SS2.3.p3.8.m8.1a"><mrow id="S4.SS2.3.p3.8.m8.1.1" xref="S4.SS2.3.p3.8.m8.1.1.cmml"><msubsup id="S4.SS2.3.p3.8.m8.1.1.2" xref="S4.SS2.3.p3.8.m8.1.1.2.cmml"><mi id="S4.SS2.3.p3.8.m8.1.1.2.2.2" xref="S4.SS2.3.p3.8.m8.1.1.2.2.2.cmml">C</mi><mover accent="true" id="S4.SS2.3.p3.8.m8.1.1.2.3" xref="S4.SS2.3.p3.8.m8.1.1.2.3.cmml"><mi id="S4.SS2.3.p3.8.m8.1.1.2.3.2" xref="S4.SS2.3.p3.8.m8.1.1.2.3.2.cmml">G</mi><mo id="S4.SS2.3.p3.8.m8.1.1.2.3.1" xref="S4.SS2.3.p3.8.m8.1.1.2.3.1.cmml">¯</mo></mover><mi id="S4.SS2.3.p3.8.m8.1.1.2.2.3" xref="S4.SS2.3.p3.8.m8.1.1.2.2.3.cmml">v</mi></msubsup><mo id="S4.SS2.3.p3.8.m8.1.1.1" xref="S4.SS2.3.p3.8.m8.1.1.1.cmml">⊈</mo><mrow id="S4.SS2.3.p3.8.m8.1.1.3" xref="S4.SS2.3.p3.8.m8.1.1.3.cmml"><mi id="S4.SS2.3.p3.8.m8.1.1.3.2" xref="S4.SS2.3.p3.8.m8.1.1.3.2.cmml">E</mi><mo id="S4.SS2.3.p3.8.m8.1.1.3.1" xref="S4.SS2.3.p3.8.m8.1.1.3.1.cmml">∩</mo><mi id="S4.SS2.3.p3.8.m8.1.1.3.3" xref="S4.SS2.3.p3.8.m8.1.1.3.3.cmml">H</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.8.m8.1b"><apply id="S4.SS2.3.p3.8.m8.1.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1"><csymbol cd="latexml" id="S4.SS2.3.p3.8.m8.1.1.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1.1">not-subset-of-nor-equals</csymbol><apply id="S4.SS2.3.p3.8.m8.1.1.2.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.8.m8.1.1.2.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2">subscript</csymbol><apply id="S4.SS2.3.p3.8.m8.1.1.2.2.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p3.8.m8.1.1.2.2.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2">superscript</csymbol><ci id="S4.SS2.3.p3.8.m8.1.1.2.2.2.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2.2.2">𝐶</ci><ci id="S4.SS2.3.p3.8.m8.1.1.2.2.3.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2.2.3">𝑣</ci></apply><apply id="S4.SS2.3.p3.8.m8.1.1.2.3.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2.3"><ci id="S4.SS2.3.p3.8.m8.1.1.2.3.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2.3.1">¯</ci><ci id="S4.SS2.3.p3.8.m8.1.1.2.3.2.cmml" xref="S4.SS2.3.p3.8.m8.1.1.2.3.2">𝐺</ci></apply></apply><apply id="S4.SS2.3.p3.8.m8.1.1.3.cmml" xref="S4.SS2.3.p3.8.m8.1.1.3"><intersect id="S4.SS2.3.p3.8.m8.1.1.3.1.cmml" xref="S4.SS2.3.p3.8.m8.1.1.3.1"></intersect><ci id="S4.SS2.3.p3.8.m8.1.1.3.2.cmml" xref="S4.SS2.3.p3.8.m8.1.1.3.2">𝐸</ci><ci id="S4.SS2.3.p3.8.m8.1.1.3.3.cmml" xref="S4.SS2.3.p3.8.m8.1.1.3.3">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.8.m8.1c">C^{v}_{\overline{G}}\nsubseteq E\cap H</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.8.m8.1d">italic_C start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ⊈ italic_E ∩ italic_H</annotation></semantics></math>. This implies that <math alttext="E\cap H\notin\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.9.m9.1"><semantics id="S4.SS2.3.p3.9.m9.1a"><mrow id="S4.SS2.3.p3.9.m9.1.1" xref="S4.SS2.3.p3.9.m9.1.1.cmml"><mrow id="S4.SS2.3.p3.9.m9.1.1.2" xref="S4.SS2.3.p3.9.m9.1.1.2.cmml"><mi id="S4.SS2.3.p3.9.m9.1.1.2.2" xref="S4.SS2.3.p3.9.m9.1.1.2.2.cmml">E</mi><mo id="S4.SS2.3.p3.9.m9.1.1.2.1" xref="S4.SS2.3.p3.9.m9.1.1.2.1.cmml">∩</mo><mi id="S4.SS2.3.p3.9.m9.1.1.2.3" xref="S4.SS2.3.p3.9.m9.1.1.2.3.cmml">H</mi></mrow><mo id="S4.SS2.3.p3.9.m9.1.1.1" xref="S4.SS2.3.p3.9.m9.1.1.1.cmml">∉</mo><msub id="S4.SS2.3.p3.9.m9.1.1.3" xref="S4.SS2.3.p3.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.9.m9.1.1.3.2" xref="S4.SS2.3.p3.9.m9.1.1.3.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.9.m9.1.1.3.3" xref="S4.SS2.3.p3.9.m9.1.1.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.9.m9.1b"><apply id="S4.SS2.3.p3.9.m9.1.1.cmml" xref="S4.SS2.3.p3.9.m9.1.1"><notin id="S4.SS2.3.p3.9.m9.1.1.1.cmml" xref="S4.SS2.3.p3.9.m9.1.1.1"></notin><apply id="S4.SS2.3.p3.9.m9.1.1.2.cmml" xref="S4.SS2.3.p3.9.m9.1.1.2"><intersect id="S4.SS2.3.p3.9.m9.1.1.2.1.cmml" xref="S4.SS2.3.p3.9.m9.1.1.2.1"></intersect><ci id="S4.SS2.3.p3.9.m9.1.1.2.2.cmml" xref="S4.SS2.3.p3.9.m9.1.1.2.2">𝐸</ci><ci id="S4.SS2.3.p3.9.m9.1.1.2.3.cmml" xref="S4.SS2.3.p3.9.m9.1.1.2.3">𝐻</ci></apply><apply id="S4.SS2.3.p3.9.m9.1.1.3.cmml" xref="S4.SS2.3.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.3.p3.9.m9.1.1.3.1.cmml" xref="S4.SS2.3.p3.9.m9.1.1.3">subscript</csymbol><ci id="S4.SS2.3.p3.9.m9.1.1.3.2.cmml" xref="S4.SS2.3.p3.9.m9.1.1.3.2">ℱ</ci><ci id="S4.SS2.3.p3.9.m9.1.1.3.3.cmml" xref="S4.SS2.3.p3.9.m9.1.1.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.9.m9.1c">E\cap H\notin\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.9.m9.1d">italic_E ∩ italic_H ∉ caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>. In other words, <math alttext="(E,H)" class="ltx_Math" display="inline" id="S4.SS2.3.p3.10.m10.2"><semantics id="S4.SS2.3.p3.10.m10.2a"><mrow id="S4.SS2.3.p3.10.m10.2.3.2" xref="S4.SS2.3.p3.10.m10.2.3.1.cmml"><mo id="S4.SS2.3.p3.10.m10.2.3.2.1" stretchy="false" xref="S4.SS2.3.p3.10.m10.2.3.1.cmml">(</mo><mi id="S4.SS2.3.p3.10.m10.1.1" xref="S4.SS2.3.p3.10.m10.1.1.cmml">E</mi><mo id="S4.SS2.3.p3.10.m10.2.3.2.2" xref="S4.SS2.3.p3.10.m10.2.3.1.cmml">,</mo><mi id="S4.SS2.3.p3.10.m10.2.2" xref="S4.SS2.3.p3.10.m10.2.2.cmml">H</mi><mo id="S4.SS2.3.p3.10.m10.2.3.2.3" stretchy="false" xref="S4.SS2.3.p3.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.10.m10.2b"><interval closure="open" id="S4.SS2.3.p3.10.m10.2.3.1.cmml" xref="S4.SS2.3.p3.10.m10.2.3.2"><ci id="S4.SS2.3.p3.10.m10.1.1.cmml" xref="S4.SS2.3.p3.10.m10.1.1">𝐸</ci><ci id="S4.SS2.3.p3.10.m10.2.2.cmml" xref="S4.SS2.3.p3.10.m10.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.10.m10.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.10.m10.2d">( italic_E , italic_H )</annotation></semantics></math> covers <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.3.p3.11.m11.1"><semantics id="S4.SS2.3.p3.11.m11.1a"><msub id="S4.SS2.3.p3.11.m11.1.1" xref="S4.SS2.3.p3.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p3.11.m11.1.1.2" xref="S4.SS2.3.p3.11.m11.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.3.p3.11.m11.1.1.3" xref="S4.SS2.3.p3.11.m11.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.11.m11.1b"><apply id="S4.SS2.3.p3.11.m11.1.1.cmml" xref="S4.SS2.3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p3.11.m11.1.1.1.cmml" xref="S4.SS2.3.p3.11.m11.1.1">subscript</csymbol><ci id="S4.SS2.3.p3.11.m11.1.1.2.cmml" xref="S4.SS2.3.p3.11.m11.1.1.2">ℱ</ci><ci id="S4.SS2.3.p3.11.m11.1.1.3.cmml" xref="S4.SS2.3.p3.11.m11.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.11.m11.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.11.m11.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.4.p4"> <p class="ltx_p" id="S4.SS2.4.p4.4">Suppose now that <math alttext="(E,H)" class="ltx_Math" display="inline" id="S4.SS2.4.p4.1.m1.2"><semantics id="S4.SS2.4.p4.1.m1.2a"><mrow id="S4.SS2.4.p4.1.m1.2.3.2" xref="S4.SS2.4.p4.1.m1.2.3.1.cmml"><mo id="S4.SS2.4.p4.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.4.p4.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS2.4.p4.1.m1.1.1" xref="S4.SS2.4.p4.1.m1.1.1.cmml">E</mi><mo id="S4.SS2.4.p4.1.m1.2.3.2.2" xref="S4.SS2.4.p4.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.4.p4.1.m1.2.2" xref="S4.SS2.4.p4.1.m1.2.2.cmml">H</mi><mo id="S4.SS2.4.p4.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.4.p4.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p4.1.m1.2b"><interval closure="open" id="S4.SS2.4.p4.1.m1.2.3.1.cmml" xref="S4.SS2.4.p4.1.m1.2.3.2"><ci id="S4.SS2.4.p4.1.m1.1.1.cmml" xref="S4.SS2.4.p4.1.m1.1.1">𝐸</ci><ci id="S4.SS2.4.p4.1.m1.2.2.cmml" xref="S4.SS2.4.p4.1.m1.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p4.1.m1.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p4.1.m1.2d">( italic_E , italic_H )</annotation></semantics></math> covers <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.4.p4.2.m2.1"><semantics id="S4.SS2.4.p4.2.m2.1a"><msub id="S4.SS2.4.p4.2.m2.1.1" xref="S4.SS2.4.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.4.p4.2.m2.1.1.2" xref="S4.SS2.4.p4.2.m2.1.1.2.cmml">ℱ</mi><mi id="S4.SS2.4.p4.2.m2.1.1.3" xref="S4.SS2.4.p4.2.m2.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p4.2.m2.1b"><apply id="S4.SS2.4.p4.2.m2.1.1.cmml" xref="S4.SS2.4.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.4.p4.2.m2.1.1.1.cmml" xref="S4.SS2.4.p4.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.4.p4.2.m2.1.1.2.cmml" xref="S4.SS2.4.p4.2.m2.1.1.2">ℱ</ci><ci id="S4.SS2.4.p4.2.m2.1.1.3.cmml" xref="S4.SS2.4.p4.2.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p4.2.m2.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p4.2.m2.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="E,H\in\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS2.4.p4.3.m3.2"><semantics id="S4.SS2.4.p4.3.m3.2a"><mrow id="S4.SS2.4.p4.3.m3.2.3" xref="S4.SS2.4.p4.3.m3.2.3.cmml"><mrow id="S4.SS2.4.p4.3.m3.2.3.2.2" xref="S4.SS2.4.p4.3.m3.2.3.2.1.cmml"><mi id="S4.SS2.4.p4.3.m3.1.1" xref="S4.SS2.4.p4.3.m3.1.1.cmml">E</mi><mo id="S4.SS2.4.p4.3.m3.2.3.2.2.1" xref="S4.SS2.4.p4.3.m3.2.3.2.1.cmml">,</mo><mi id="S4.SS2.4.p4.3.m3.2.2" xref="S4.SS2.4.p4.3.m3.2.2.cmml">H</mi></mrow><mo id="S4.SS2.4.p4.3.m3.2.3.1" xref="S4.SS2.4.p4.3.m3.2.3.1.cmml">∈</mo><msub id="S4.SS2.4.p4.3.m3.2.3.3" xref="S4.SS2.4.p4.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.4.p4.3.m3.2.3.3.2" xref="S4.SS2.4.p4.3.m3.2.3.3.2.cmml">ℱ</mi><mi id="S4.SS2.4.p4.3.m3.2.3.3.3" xref="S4.SS2.4.p4.3.m3.2.3.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p4.3.m3.2b"><apply id="S4.SS2.4.p4.3.m3.2.3.cmml" xref="S4.SS2.4.p4.3.m3.2.3"><in id="S4.SS2.4.p4.3.m3.2.3.1.cmml" xref="S4.SS2.4.p4.3.m3.2.3.1"></in><list id="S4.SS2.4.p4.3.m3.2.3.2.1.cmml" xref="S4.SS2.4.p4.3.m3.2.3.2.2"><ci id="S4.SS2.4.p4.3.m3.1.1.cmml" xref="S4.SS2.4.p4.3.m3.1.1">𝐸</ci><ci id="S4.SS2.4.p4.3.m3.2.2.cmml" xref="S4.SS2.4.p4.3.m3.2.2">𝐻</ci></list><apply id="S4.SS2.4.p4.3.m3.2.3.3.cmml" xref="S4.SS2.4.p4.3.m3.2.3.3"><csymbol cd="ambiguous" id="S4.SS2.4.p4.3.m3.2.3.3.1.cmml" xref="S4.SS2.4.p4.3.m3.2.3.3">subscript</csymbol><ci id="S4.SS2.4.p4.3.m3.2.3.3.2.cmml" xref="S4.SS2.4.p4.3.m3.2.3.3.2">ℱ</ci><ci id="S4.SS2.4.p4.3.m3.2.3.3.3.cmml" xref="S4.SS2.4.p4.3.m3.2.3.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p4.3.m3.2c">E,H\in\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p4.3.m3.2d">italic_E , italic_H ∈ caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> but <math alttext="E\cap H\notin\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS2.4.p4.4.m4.1"><semantics id="S4.SS2.4.p4.4.m4.1a"><mrow id="S4.SS2.4.p4.4.m4.1.1" xref="S4.SS2.4.p4.4.m4.1.1.cmml"><mrow id="S4.SS2.4.p4.4.m4.1.1.2" xref="S4.SS2.4.p4.4.m4.1.1.2.cmml"><mi id="S4.SS2.4.p4.4.m4.1.1.2.2" xref="S4.SS2.4.p4.4.m4.1.1.2.2.cmml">E</mi><mo id="S4.SS2.4.p4.4.m4.1.1.2.1" xref="S4.SS2.4.p4.4.m4.1.1.2.1.cmml">∩</mo><mi id="S4.SS2.4.p4.4.m4.1.1.2.3" xref="S4.SS2.4.p4.4.m4.1.1.2.3.cmml">H</mi></mrow><mo id="S4.SS2.4.p4.4.m4.1.1.1" xref="S4.SS2.4.p4.4.m4.1.1.1.cmml">∉</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.4.p4.4.m4.1.1.3" xref="S4.SS2.4.p4.4.m4.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p4.4.m4.1b"><apply id="S4.SS2.4.p4.4.m4.1.1.cmml" xref="S4.SS2.4.p4.4.m4.1.1"><notin id="S4.SS2.4.p4.4.m4.1.1.1.cmml" xref="S4.SS2.4.p4.4.m4.1.1.1"></notin><apply id="S4.SS2.4.p4.4.m4.1.1.2.cmml" xref="S4.SS2.4.p4.4.m4.1.1.2"><intersect id="S4.SS2.4.p4.4.m4.1.1.2.1.cmml" xref="S4.SS2.4.p4.4.m4.1.1.2.1"></intersect><ci id="S4.SS2.4.p4.4.m4.1.1.2.2.cmml" xref="S4.SS2.4.p4.4.m4.1.1.2.2">𝐸</ci><ci id="S4.SS2.4.p4.4.m4.1.1.2.3.cmml" xref="S4.SS2.4.p4.4.m4.1.1.2.3">𝐻</ci></apply><ci id="S4.SS2.4.p4.4.m4.1.1.3.cmml" xref="S4.SS2.4.p4.4.m4.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p4.4.m4.1c">E\cap H\notin\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p4.4.m4.1d">italic_E ∩ italic_H ∉ caligraphic_F</annotation></semantics></math>. It is easy to check that this implies the condition in the statement of Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem41" title="Claim 41. ‣ Proof. ‣ 4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">41</span></a>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S4.SS2.5.p5"> <p class="ltx_p" id="S4.SS2.5.p5.1">Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem41" title="Claim 41. ‣ Proof. ‣ 4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">41</span></a> immediately implies the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem42"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem42.1.1.1">Lemma 42</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem42.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem42.p1"> <p class="ltx_p" id="Thmtheorem42.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem42.p1.3.3">Every semi-filter in <math alttext="\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="Thmtheorem42.p1.1.1.m1.1"><semantics id="Thmtheorem42.p1.1.1.m1.1a"><msubsup id="Thmtheorem42.p1.1.1.m1.1.1" xref="Thmtheorem42.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem42.p1.1.1.m1.1.1.2.2" xref="Thmtheorem42.p1.1.1.m1.1.1.2.2.cmml">𝔉</mi><mi id="Thmtheorem42.p1.1.1.m1.1.1.2.3" xref="Thmtheorem42.p1.1.1.m1.1.1.2.3.cmml">𝖼𝖺𝗇</mi><mi id="Thmtheorem42.p1.1.1.m1.1.1.3" xref="Thmtheorem42.p1.1.1.m1.1.1.3.cmml">G</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmtheorem42.p1.1.1.m1.1b"><apply id="Thmtheorem42.p1.1.1.m1.1.1.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem42.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1">superscript</csymbol><apply id="Thmtheorem42.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem42.p1.1.1.m1.1.1.2.1.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1">subscript</csymbol><ci id="Thmtheorem42.p1.1.1.m1.1.1.2.2.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1.2.2">𝔉</ci><ci id="Thmtheorem42.p1.1.1.m1.1.1.2.3.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1.2.3">𝖼𝖺𝗇</ci></apply><ci id="Thmtheorem42.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem42.p1.1.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem42.p1.1.1.m1.1c">\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem42.p1.1.1.m1.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math> covered by <math alttext="(E,H)" class="ltx_Math" display="inline" id="Thmtheorem42.p1.2.2.m2.2"><semantics id="Thmtheorem42.p1.2.2.m2.2a"><mrow id="Thmtheorem42.p1.2.2.m2.2.3.2" xref="Thmtheorem42.p1.2.2.m2.2.3.1.cmml"><mo id="Thmtheorem42.p1.2.2.m2.2.3.2.1" stretchy="false" xref="Thmtheorem42.p1.2.2.m2.2.3.1.cmml">(</mo><mi id="Thmtheorem42.p1.2.2.m2.1.1" xref="Thmtheorem42.p1.2.2.m2.1.1.cmml">E</mi><mo id="Thmtheorem42.p1.2.2.m2.2.3.2.2" xref="Thmtheorem42.p1.2.2.m2.2.3.1.cmml">,</mo><mi id="Thmtheorem42.p1.2.2.m2.2.2" xref="Thmtheorem42.p1.2.2.m2.2.2.cmml">H</mi><mo id="Thmtheorem42.p1.2.2.m2.2.3.2.3" stretchy="false" xref="Thmtheorem42.p1.2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem42.p1.2.2.m2.2b"><interval closure="open" id="Thmtheorem42.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem42.p1.2.2.m2.2.3.2"><ci id="Thmtheorem42.p1.2.2.m2.1.1.cmml" xref="Thmtheorem42.p1.2.2.m2.1.1">𝐸</ci><ci id="Thmtheorem42.p1.2.2.m2.2.2.cmml" xref="Thmtheorem42.p1.2.2.m2.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem42.p1.2.2.m2.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem42.p1.2.2.m2.2d">( italic_E , italic_H )</annotation></semantics></math> is also covered by <math alttext="(E\setminus H,H\setminus E)" class="ltx_Math" display="inline" id="Thmtheorem42.p1.3.3.m3.2"><semantics id="Thmtheorem42.p1.3.3.m3.2a"><mrow id="Thmtheorem42.p1.3.3.m3.2.2.2" xref="Thmtheorem42.p1.3.3.m3.2.2.3.cmml"><mo id="Thmtheorem42.p1.3.3.m3.2.2.2.3" stretchy="false" xref="Thmtheorem42.p1.3.3.m3.2.2.3.cmml">(</mo><mrow id="Thmtheorem42.p1.3.3.m3.1.1.1.1" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.cmml"><mi id="Thmtheorem42.p1.3.3.m3.1.1.1.1.2" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.2.cmml">E</mi><mo id="Thmtheorem42.p1.3.3.m3.1.1.1.1.1" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.1.cmml">∖</mo><mi id="Thmtheorem42.p1.3.3.m3.1.1.1.1.3" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.3.cmml">H</mi></mrow><mo id="Thmtheorem42.p1.3.3.m3.2.2.2.4" xref="Thmtheorem42.p1.3.3.m3.2.2.3.cmml">,</mo><mrow id="Thmtheorem42.p1.3.3.m3.2.2.2.2" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.cmml"><mi id="Thmtheorem42.p1.3.3.m3.2.2.2.2.2" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.2.cmml">H</mi><mo id="Thmtheorem42.p1.3.3.m3.2.2.2.2.1" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.1.cmml">∖</mo><mi id="Thmtheorem42.p1.3.3.m3.2.2.2.2.3" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.3.cmml">E</mi></mrow><mo id="Thmtheorem42.p1.3.3.m3.2.2.2.5" stretchy="false" xref="Thmtheorem42.p1.3.3.m3.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem42.p1.3.3.m3.2b"><interval closure="open" id="Thmtheorem42.p1.3.3.m3.2.2.3.cmml" xref="Thmtheorem42.p1.3.3.m3.2.2.2"><apply id="Thmtheorem42.p1.3.3.m3.1.1.1.1.cmml" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1"><setdiff id="Thmtheorem42.p1.3.3.m3.1.1.1.1.1.cmml" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.1"></setdiff><ci id="Thmtheorem42.p1.3.3.m3.1.1.1.1.2.cmml" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.2">𝐸</ci><ci id="Thmtheorem42.p1.3.3.m3.1.1.1.1.3.cmml" xref="Thmtheorem42.p1.3.3.m3.1.1.1.1.3">𝐻</ci></apply><apply id="Thmtheorem42.p1.3.3.m3.2.2.2.2.cmml" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2"><setdiff id="Thmtheorem42.p1.3.3.m3.2.2.2.2.1.cmml" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.1"></setdiff><ci id="Thmtheorem42.p1.3.3.m3.2.2.2.2.2.cmml" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.2">𝐻</ci><ci id="Thmtheorem42.p1.3.3.m3.2.2.2.2.3.cmml" xref="Thmtheorem42.p1.3.3.m3.2.2.2.2.3">𝐸</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem42.p1.3.3.m3.2c">(E\setminus H,H\setminus E)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem42.p1.3.3.m3.2d">( italic_E ∖ italic_H , italic_H ∖ italic_E )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.6.p6"> <p class="ltx_p" id="S4.SS2.6.p6.3">Thus we can and will assume w.l.o.g. that all pairs appearing in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS2.6.p6.1.m1.1"><semantics id="S4.SS2.6.p6.1.m1.1a"><mi id="S4.SS2.6.p6.1.m1.1.1" mathvariant="normal" xref="S4.SS2.6.p6.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p6.1.m1.1b"><ci id="S4.SS2.6.p6.1.m1.1.1.cmml" xref="S4.SS2.6.p6.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p6.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p6.1.m1.1d">roman_Λ</annotation></semantics></math> have disjoint sets <math alttext="E_{i}" class="ltx_Math" display="inline" id="S4.SS2.6.p6.2.m2.1"><semantics id="S4.SS2.6.p6.2.m2.1a"><msub id="S4.SS2.6.p6.2.m2.1.1" xref="S4.SS2.6.p6.2.m2.1.1.cmml"><mi id="S4.SS2.6.p6.2.m2.1.1.2" xref="S4.SS2.6.p6.2.m2.1.1.2.cmml">E</mi><mi id="S4.SS2.6.p6.2.m2.1.1.3" xref="S4.SS2.6.p6.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p6.2.m2.1b"><apply id="S4.SS2.6.p6.2.m2.1.1.cmml" xref="S4.SS2.6.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p6.2.m2.1.1.1.cmml" xref="S4.SS2.6.p6.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.6.p6.2.m2.1.1.2.cmml" xref="S4.SS2.6.p6.2.m2.1.1.2">𝐸</ci><ci id="S4.SS2.6.p6.2.m2.1.1.3.cmml" xref="S4.SS2.6.p6.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p6.2.m2.1c">E_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p6.2.m2.1d">italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{i}" class="ltx_Math" display="inline" id="S4.SS2.6.p6.3.m3.1"><semantics id="S4.SS2.6.p6.3.m3.1a"><msub id="S4.SS2.6.p6.3.m3.1.1" xref="S4.SS2.6.p6.3.m3.1.1.cmml"><mi id="S4.SS2.6.p6.3.m3.1.1.2" xref="S4.SS2.6.p6.3.m3.1.1.2.cmml">H</mi><mi id="S4.SS2.6.p6.3.m3.1.1.3" xref="S4.SS2.6.p6.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p6.3.m3.1b"><apply id="S4.SS2.6.p6.3.m3.1.1.cmml" xref="S4.SS2.6.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p6.3.m3.1.1.1.cmml" xref="S4.SS2.6.p6.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.6.p6.3.m3.1.1.2.cmml" xref="S4.SS2.6.p6.3.m3.1.1.2">𝐻</ci><ci id="S4.SS2.6.p6.3.m3.1.1.3.cmml" xref="S4.SS2.6.p6.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p6.3.m3.1c">H_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p6.3.m3.1d">italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Using Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem41" title="Claim 41. ‣ Proof. ‣ 4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">41</span></a> again, we obtain the following additional consequence.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem43"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem43.1.1.1">Lemma 43</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem43.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem43.p1"> <p class="ltx_p" id="Thmtheorem43.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem43.p1.3.3">Every semi-filter in <math alttext="\mathfrak{F}_{\mathsf{can}}^{G}" class="ltx_Math" display="inline" id="Thmtheorem43.p1.1.1.m1.1"><semantics id="Thmtheorem43.p1.1.1.m1.1a"><msubsup id="Thmtheorem43.p1.1.1.m1.1.1" xref="Thmtheorem43.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem43.p1.1.1.m1.1.1.2.2" xref="Thmtheorem43.p1.1.1.m1.1.1.2.2.cmml">𝔉</mi><mi id="Thmtheorem43.p1.1.1.m1.1.1.2.3" xref="Thmtheorem43.p1.1.1.m1.1.1.2.3.cmml">𝖼𝖺𝗇</mi><mi id="Thmtheorem43.p1.1.1.m1.1.1.3" xref="Thmtheorem43.p1.1.1.m1.1.1.3.cmml">G</mi></msubsup><annotation-xml encoding="MathML-Content" id="Thmtheorem43.p1.1.1.m1.1b"><apply id="Thmtheorem43.p1.1.1.m1.1.1.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem43.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1">superscript</csymbol><apply id="Thmtheorem43.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem43.p1.1.1.m1.1.1.2.1.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1">subscript</csymbol><ci id="Thmtheorem43.p1.1.1.m1.1.1.2.2.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1.2.2">𝔉</ci><ci id="Thmtheorem43.p1.1.1.m1.1.1.2.3.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1.2.3">𝖼𝖺𝗇</ci></apply><ci id="Thmtheorem43.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem43.p1.1.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem43.p1.1.1.m1.1c">\mathfrak{F}_{\mathsf{can}}^{G}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem43.p1.1.1.m1.1d">fraktur_F start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT</annotation></semantics></math> covered by a disjoint pair <math alttext="(E,H)" class="ltx_Math" display="inline" id="Thmtheorem43.p1.2.2.m2.2"><semantics id="Thmtheorem43.p1.2.2.m2.2a"><mrow id="Thmtheorem43.p1.2.2.m2.2.3.2" xref="Thmtheorem43.p1.2.2.m2.2.3.1.cmml"><mo id="Thmtheorem43.p1.2.2.m2.2.3.2.1" stretchy="false" xref="Thmtheorem43.p1.2.2.m2.2.3.1.cmml">(</mo><mi id="Thmtheorem43.p1.2.2.m2.1.1" xref="Thmtheorem43.p1.2.2.m2.1.1.cmml">E</mi><mo id="Thmtheorem43.p1.2.2.m2.2.3.2.2" xref="Thmtheorem43.p1.2.2.m2.2.3.1.cmml">,</mo><mi id="Thmtheorem43.p1.2.2.m2.2.2" xref="Thmtheorem43.p1.2.2.m2.2.2.cmml">H</mi><mo id="Thmtheorem43.p1.2.2.m2.2.3.2.3" stretchy="false" xref="Thmtheorem43.p1.2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem43.p1.2.2.m2.2b"><interval closure="open" id="Thmtheorem43.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem43.p1.2.2.m2.2.3.2"><ci id="Thmtheorem43.p1.2.2.m2.1.1.cmml" xref="Thmtheorem43.p1.2.2.m2.1.1">𝐸</ci><ci id="Thmtheorem43.p1.2.2.m2.2.2.cmml" xref="Thmtheorem43.p1.2.2.m2.2.2">𝐻</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem43.p1.2.2.m2.2c">(E,H)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem43.p1.2.2.m2.2d">( italic_E , italic_H )</annotation></semantics></math> is also covered by the pair <math alttext="(E,\overline{G}\setminus E)" class="ltx_Math" display="inline" id="Thmtheorem43.p1.3.3.m3.2"><semantics id="Thmtheorem43.p1.3.3.m3.2a"><mrow id="Thmtheorem43.p1.3.3.m3.2.2.1" xref="Thmtheorem43.p1.3.3.m3.2.2.2.cmml"><mo id="Thmtheorem43.p1.3.3.m3.2.2.1.2" stretchy="false" xref="Thmtheorem43.p1.3.3.m3.2.2.2.cmml">(</mo><mi id="Thmtheorem43.p1.3.3.m3.1.1" xref="Thmtheorem43.p1.3.3.m3.1.1.cmml">E</mi><mo id="Thmtheorem43.p1.3.3.m3.2.2.1.3" xref="Thmtheorem43.p1.3.3.m3.2.2.2.cmml">,</mo><mrow id="Thmtheorem43.p1.3.3.m3.2.2.1.1" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.cmml"><mover accent="true" id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.cmml"><mi id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.2" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.2.cmml">G</mi><mo id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.1" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.1.cmml">¯</mo></mover><mo id="Thmtheorem43.p1.3.3.m3.2.2.1.1.1" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.1.cmml">∖</mo><mi id="Thmtheorem43.p1.3.3.m3.2.2.1.1.3" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.3.cmml">E</mi></mrow><mo id="Thmtheorem43.p1.3.3.m3.2.2.1.4" stretchy="false" xref="Thmtheorem43.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem43.p1.3.3.m3.2b"><interval closure="open" id="Thmtheorem43.p1.3.3.m3.2.2.2.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1"><ci id="Thmtheorem43.p1.3.3.m3.1.1.cmml" xref="Thmtheorem43.p1.3.3.m3.1.1">𝐸</ci><apply id="Thmtheorem43.p1.3.3.m3.2.2.1.1.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1"><setdiff id="Thmtheorem43.p1.3.3.m3.2.2.1.1.1.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.1"></setdiff><apply id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2"><ci id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.1.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.1">¯</ci><ci id="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.2.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.2.2">𝐺</ci></apply><ci id="Thmtheorem43.p1.3.3.m3.2.2.1.1.3.cmml" xref="Thmtheorem43.p1.3.3.m3.2.2.1.1.3">𝐸</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem43.p1.3.3.m3.2c">(E,\overline{G}\setminus E)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem43.p1.3.3.m3.2d">( italic_E , over¯ start_ARG italic_G end_ARG ∖ italic_E )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.7.p7"> <p class="ltx_p" id="S4.SS2.7.p7.15">Consequently, we will further assume that all pairs appearing in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.SS2.7.p7.1.m1.1"><semantics id="S4.SS2.7.p7.1.m1.1a"><mi id="S4.SS2.7.p7.1.m1.1.1" mathvariant="normal" xref="S4.SS2.7.p7.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.1.m1.1b"><ci id="S4.SS2.7.p7.1.m1.1.1.cmml" xref="S4.SS2.7.p7.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.1.m1.1d">roman_Λ</annotation></semantics></math> form a partition of <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.SS2.7.p7.2.m2.1"><semantics id="S4.SS2.7.p7.2.m2.1a"><mover accent="true" id="S4.SS2.7.p7.2.m2.1.1" xref="S4.SS2.7.p7.2.m2.1.1.cmml"><mi id="S4.SS2.7.p7.2.m2.1.1.2" xref="S4.SS2.7.p7.2.m2.1.1.2.cmml">G</mi><mo id="S4.SS2.7.p7.2.m2.1.1.1" xref="S4.SS2.7.p7.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.2.m2.1b"><apply id="S4.SS2.7.p7.2.m2.1.1.cmml" xref="S4.SS2.7.p7.2.m2.1.1"><ci id="S4.SS2.7.p7.2.m2.1.1.1.cmml" xref="S4.SS2.7.p7.2.m2.1.1.1">¯</ci><ci id="S4.SS2.7.p7.2.m2.1.1.2.cmml" xref="S4.SS2.7.p7.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.2.m2.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.2.m2.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math>. Let <math alttext="(E_{1},H_{1})\in\Lambda" class="ltx_Math" display="inline" id="S4.SS2.7.p7.3.m3.2"><semantics id="S4.SS2.7.p7.3.m3.2a"><mrow id="S4.SS2.7.p7.3.m3.2.2" xref="S4.SS2.7.p7.3.m3.2.2.cmml"><mrow id="S4.SS2.7.p7.3.m3.2.2.2.2" xref="S4.SS2.7.p7.3.m3.2.2.2.3.cmml"><mo id="S4.SS2.7.p7.3.m3.2.2.2.2.3" stretchy="false" xref="S4.SS2.7.p7.3.m3.2.2.2.3.cmml">(</mo><msub id="S4.SS2.7.p7.3.m3.1.1.1.1.1" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.3.m3.1.1.1.1.1.2" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.3.m3.1.1.1.1.1.3" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.3.m3.2.2.2.2.4" xref="S4.SS2.7.p7.3.m3.2.2.2.3.cmml">,</mo><msub id="S4.SS2.7.p7.3.m3.2.2.2.2.2" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2.cmml"><mi id="S4.SS2.7.p7.3.m3.2.2.2.2.2.2" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2.2.cmml">H</mi><mn id="S4.SS2.7.p7.3.m3.2.2.2.2.2.3" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.3.m3.2.2.2.2.5" stretchy="false" xref="S4.SS2.7.p7.3.m3.2.2.2.3.cmml">)</mo></mrow><mo id="S4.SS2.7.p7.3.m3.2.2.3" xref="S4.SS2.7.p7.3.m3.2.2.3.cmml">∈</mo><mi id="S4.SS2.7.p7.3.m3.2.2.4" mathvariant="normal" xref="S4.SS2.7.p7.3.m3.2.2.4.cmml">Λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.3.m3.2b"><apply id="S4.SS2.7.p7.3.m3.2.2.cmml" xref="S4.SS2.7.p7.3.m3.2.2"><in id="S4.SS2.7.p7.3.m3.2.2.3.cmml" xref="S4.SS2.7.p7.3.m3.2.2.3"></in><interval closure="open" id="S4.SS2.7.p7.3.m3.2.2.2.3.cmml" xref="S4.SS2.7.p7.3.m3.2.2.2.2"><apply id="S4.SS2.7.p7.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.3.m3.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.3.m3.1.1.1.1.1.3">1</cn></apply><apply id="S4.SS2.7.p7.3.m3.2.2.2.2.2.cmml" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.7.p7.3.m3.2.2.2.2.2.1.cmml" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.7.p7.3.m3.2.2.2.2.2.2.cmml" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2.2">𝐻</ci><cn id="S4.SS2.7.p7.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.7.p7.3.m3.2.2.2.2.2.3">1</cn></apply></interval><ci id="S4.SS2.7.p7.3.m3.2.2.4.cmml" xref="S4.SS2.7.p7.3.m3.2.2.4">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.3.m3.2c">(E_{1},H_{1})\in\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.3.m3.2d">( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∈ roman_Λ</annotation></semantics></math> be one such pair. Since <math alttext="E_{1}" class="ltx_Math" display="inline" id="S4.SS2.7.p7.4.m4.1"><semantics id="S4.SS2.7.p7.4.m4.1a"><msub id="S4.SS2.7.p7.4.m4.1.1" xref="S4.SS2.7.p7.4.m4.1.1.cmml"><mi id="S4.SS2.7.p7.4.m4.1.1.2" xref="S4.SS2.7.p7.4.m4.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.4.m4.1.1.3" xref="S4.SS2.7.p7.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.4.m4.1b"><apply id="S4.SS2.7.p7.4.m4.1.1.cmml" xref="S4.SS2.7.p7.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.4.m4.1.1.1.cmml" xref="S4.SS2.7.p7.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.4.m4.1.1.2.cmml" xref="S4.SS2.7.p7.4.m4.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.4.m4.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.4.m4.1c">E_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.4.m4.1d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{1}" class="ltx_Math" display="inline" id="S4.SS2.7.p7.5.m5.1"><semantics id="S4.SS2.7.p7.5.m5.1a"><msub id="S4.SS2.7.p7.5.m5.1.1" xref="S4.SS2.7.p7.5.m5.1.1.cmml"><mi id="S4.SS2.7.p7.5.m5.1.1.2" xref="S4.SS2.7.p7.5.m5.1.1.2.cmml">H</mi><mn id="S4.SS2.7.p7.5.m5.1.1.3" xref="S4.SS2.7.p7.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.5.m5.1b"><apply id="S4.SS2.7.p7.5.m5.1.1.cmml" xref="S4.SS2.7.p7.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.5.m5.1.1.1.cmml" xref="S4.SS2.7.p7.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.5.m5.1.1.2.cmml" xref="S4.SS2.7.p7.5.m5.1.1.2">𝐻</ci><cn id="S4.SS2.7.p7.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.5.m5.1c">H_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.5.m5.1d">italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> form a partition of <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.SS2.7.p7.6.m6.1"><semantics id="S4.SS2.7.p7.6.m6.1a"><mover accent="true" id="S4.SS2.7.p7.6.m6.1.1" xref="S4.SS2.7.p7.6.m6.1.1.cmml"><mi id="S4.SS2.7.p7.6.m6.1.1.2" xref="S4.SS2.7.p7.6.m6.1.1.2.cmml">G</mi><mo id="S4.SS2.7.p7.6.m6.1.1.1" xref="S4.SS2.7.p7.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.6.m6.1b"><apply id="S4.SS2.7.p7.6.m6.1.1.cmml" xref="S4.SS2.7.p7.6.m6.1.1"><ci id="S4.SS2.7.p7.6.m6.1.1.1.cmml" xref="S4.SS2.7.p7.6.m6.1.1.1">¯</ci><ci id="S4.SS2.7.p7.6.m6.1.1.2.cmml" xref="S4.SS2.7.p7.6.m6.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.6.m6.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.6.m6.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math>, either <math alttext="|E_{1}|\geq N/2" class="ltx_Math" display="inline" id="S4.SS2.7.p7.7.m7.1"><semantics id="S4.SS2.7.p7.7.m7.1a"><mrow id="S4.SS2.7.p7.7.m7.1.1" xref="S4.SS2.7.p7.7.m7.1.1.cmml"><mrow id="S4.SS2.7.p7.7.m7.1.1.1.1" xref="S4.SS2.7.p7.7.m7.1.1.1.2.cmml"><mo id="S4.SS2.7.p7.7.m7.1.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.7.m7.1.1.1.2.1.cmml">|</mo><msub id="S4.SS2.7.p7.7.m7.1.1.1.1.1" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.7.m7.1.1.1.1.1.2" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.7.m7.1.1.1.1.1.3" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.7.m7.1.1.1.1.3" stretchy="false" xref="S4.SS2.7.p7.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.7.p7.7.m7.1.1.2" xref="S4.SS2.7.p7.7.m7.1.1.2.cmml">≥</mo><mrow id="S4.SS2.7.p7.7.m7.1.1.3" xref="S4.SS2.7.p7.7.m7.1.1.3.cmml"><mi id="S4.SS2.7.p7.7.m7.1.1.3.2" xref="S4.SS2.7.p7.7.m7.1.1.3.2.cmml">N</mi><mo id="S4.SS2.7.p7.7.m7.1.1.3.1" xref="S4.SS2.7.p7.7.m7.1.1.3.1.cmml">/</mo><mn id="S4.SS2.7.p7.7.m7.1.1.3.3" xref="S4.SS2.7.p7.7.m7.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.7.m7.1b"><apply id="S4.SS2.7.p7.7.m7.1.1.cmml" xref="S4.SS2.7.p7.7.m7.1.1"><geq id="S4.SS2.7.p7.7.m7.1.1.2.cmml" xref="S4.SS2.7.p7.7.m7.1.1.2"></geq><apply id="S4.SS2.7.p7.7.m7.1.1.1.2.cmml" xref="S4.SS2.7.p7.7.m7.1.1.1.1"><abs id="S4.SS2.7.p7.7.m7.1.1.1.2.1.cmml" xref="S4.SS2.7.p7.7.m7.1.1.1.1.2"></abs><apply id="S4.SS2.7.p7.7.m7.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.7.m7.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.7.m7.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.SS2.7.p7.7.m7.1.1.3.cmml" xref="S4.SS2.7.p7.7.m7.1.1.3"><divide id="S4.SS2.7.p7.7.m7.1.1.3.1.cmml" xref="S4.SS2.7.p7.7.m7.1.1.3.1"></divide><ci id="S4.SS2.7.p7.7.m7.1.1.3.2.cmml" xref="S4.SS2.7.p7.7.m7.1.1.3.2">𝑁</ci><cn id="S4.SS2.7.p7.7.m7.1.1.3.3.cmml" type="integer" xref="S4.SS2.7.p7.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.7.m7.1c">|E_{1}|\geq N/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.7.m7.1d">| italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ italic_N / 2</annotation></semantics></math> or <math alttext="|H_{1}|\geq N/2" class="ltx_Math" display="inline" id="S4.SS2.7.p7.8.m8.1"><semantics id="S4.SS2.7.p7.8.m8.1a"><mrow id="S4.SS2.7.p7.8.m8.1.1" xref="S4.SS2.7.p7.8.m8.1.1.cmml"><mrow id="S4.SS2.7.p7.8.m8.1.1.1.1" xref="S4.SS2.7.p7.8.m8.1.1.1.2.cmml"><mo id="S4.SS2.7.p7.8.m8.1.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.8.m8.1.1.1.2.1.cmml">|</mo><msub id="S4.SS2.7.p7.8.m8.1.1.1.1.1" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.8.m8.1.1.1.1.1.2" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1.2.cmml">H</mi><mn id="S4.SS2.7.p7.8.m8.1.1.1.1.1.3" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.8.m8.1.1.1.1.3" stretchy="false" xref="S4.SS2.7.p7.8.m8.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.7.p7.8.m8.1.1.2" xref="S4.SS2.7.p7.8.m8.1.1.2.cmml">≥</mo><mrow id="S4.SS2.7.p7.8.m8.1.1.3" xref="S4.SS2.7.p7.8.m8.1.1.3.cmml"><mi id="S4.SS2.7.p7.8.m8.1.1.3.2" xref="S4.SS2.7.p7.8.m8.1.1.3.2.cmml">N</mi><mo id="S4.SS2.7.p7.8.m8.1.1.3.1" xref="S4.SS2.7.p7.8.m8.1.1.3.1.cmml">/</mo><mn id="S4.SS2.7.p7.8.m8.1.1.3.3" xref="S4.SS2.7.p7.8.m8.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.8.m8.1b"><apply id="S4.SS2.7.p7.8.m8.1.1.cmml" xref="S4.SS2.7.p7.8.m8.1.1"><geq id="S4.SS2.7.p7.8.m8.1.1.2.cmml" xref="S4.SS2.7.p7.8.m8.1.1.2"></geq><apply id="S4.SS2.7.p7.8.m8.1.1.1.2.cmml" xref="S4.SS2.7.p7.8.m8.1.1.1.1"><abs id="S4.SS2.7.p7.8.m8.1.1.1.2.1.cmml" xref="S4.SS2.7.p7.8.m8.1.1.1.1.2"></abs><apply id="S4.SS2.7.p7.8.m8.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.8.m8.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1.2">𝐻</ci><cn id="S4.SS2.7.p7.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.8.m8.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.SS2.7.p7.8.m8.1.1.3.cmml" xref="S4.SS2.7.p7.8.m8.1.1.3"><divide id="S4.SS2.7.p7.8.m8.1.1.3.1.cmml" xref="S4.SS2.7.p7.8.m8.1.1.3.1"></divide><ci id="S4.SS2.7.p7.8.m8.1.1.3.2.cmml" xref="S4.SS2.7.p7.8.m8.1.1.3.2">𝑁</ci><cn id="S4.SS2.7.p7.8.m8.1.1.3.3.cmml" type="integer" xref="S4.SS2.7.p7.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.8.m8.1c">|H_{1}|\geq N/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.8.m8.1d">| italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ italic_N / 2</annotation></semantics></math>. Assume w.l.o.g that <math alttext="|E_{1}|\geq N/2" class="ltx_Math" display="inline" id="S4.SS2.7.p7.9.m9.1"><semantics id="S4.SS2.7.p7.9.m9.1a"><mrow id="S4.SS2.7.p7.9.m9.1.1" xref="S4.SS2.7.p7.9.m9.1.1.cmml"><mrow id="S4.SS2.7.p7.9.m9.1.1.1.1" xref="S4.SS2.7.p7.9.m9.1.1.1.2.cmml"><mo id="S4.SS2.7.p7.9.m9.1.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.9.m9.1.1.1.2.1.cmml">|</mo><msub id="S4.SS2.7.p7.9.m9.1.1.1.1.1" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.9.m9.1.1.1.1.1.2" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.9.m9.1.1.1.1.1.3" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.9.m9.1.1.1.1.3" stretchy="false" xref="S4.SS2.7.p7.9.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.7.p7.9.m9.1.1.2" xref="S4.SS2.7.p7.9.m9.1.1.2.cmml">≥</mo><mrow id="S4.SS2.7.p7.9.m9.1.1.3" xref="S4.SS2.7.p7.9.m9.1.1.3.cmml"><mi id="S4.SS2.7.p7.9.m9.1.1.3.2" xref="S4.SS2.7.p7.9.m9.1.1.3.2.cmml">N</mi><mo id="S4.SS2.7.p7.9.m9.1.1.3.1" xref="S4.SS2.7.p7.9.m9.1.1.3.1.cmml">/</mo><mn id="S4.SS2.7.p7.9.m9.1.1.3.3" xref="S4.SS2.7.p7.9.m9.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.9.m9.1b"><apply id="S4.SS2.7.p7.9.m9.1.1.cmml" xref="S4.SS2.7.p7.9.m9.1.1"><geq id="S4.SS2.7.p7.9.m9.1.1.2.cmml" xref="S4.SS2.7.p7.9.m9.1.1.2"></geq><apply id="S4.SS2.7.p7.9.m9.1.1.1.2.cmml" xref="S4.SS2.7.p7.9.m9.1.1.1.1"><abs id="S4.SS2.7.p7.9.m9.1.1.1.2.1.cmml" xref="S4.SS2.7.p7.9.m9.1.1.1.1.2"></abs><apply id="S4.SS2.7.p7.9.m9.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.9.m9.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.9.m9.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.9.m9.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.9.m9.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.SS2.7.p7.9.m9.1.1.3.cmml" xref="S4.SS2.7.p7.9.m9.1.1.3"><divide id="S4.SS2.7.p7.9.m9.1.1.3.1.cmml" xref="S4.SS2.7.p7.9.m9.1.1.3.1"></divide><ci id="S4.SS2.7.p7.9.m9.1.1.3.2.cmml" xref="S4.SS2.7.p7.9.m9.1.1.3.2">𝑁</ci><cn id="S4.SS2.7.p7.9.m9.1.1.3.3.cmml" type="integer" xref="S4.SS2.7.p7.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.9.m9.1c">|E_{1}|\geq N/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.9.m9.1d">| italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ italic_N / 2</annotation></semantics></math>. Let <math alttext="G_{1}\subseteq G" class="ltx_Math" display="inline" id="S4.SS2.7.p7.10.m10.1"><semantics id="S4.SS2.7.p7.10.m10.1a"><mrow id="S4.SS2.7.p7.10.m10.1.1" xref="S4.SS2.7.p7.10.m10.1.1.cmml"><msub id="S4.SS2.7.p7.10.m10.1.1.2" xref="S4.SS2.7.p7.10.m10.1.1.2.cmml"><mi id="S4.SS2.7.p7.10.m10.1.1.2.2" xref="S4.SS2.7.p7.10.m10.1.1.2.2.cmml">G</mi><mn id="S4.SS2.7.p7.10.m10.1.1.2.3" xref="S4.SS2.7.p7.10.m10.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.10.m10.1.1.1" xref="S4.SS2.7.p7.10.m10.1.1.1.cmml">⊆</mo><mi id="S4.SS2.7.p7.10.m10.1.1.3" xref="S4.SS2.7.p7.10.m10.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.10.m10.1b"><apply id="S4.SS2.7.p7.10.m10.1.1.cmml" xref="S4.SS2.7.p7.10.m10.1.1"><subset id="S4.SS2.7.p7.10.m10.1.1.1.cmml" xref="S4.SS2.7.p7.10.m10.1.1.1"></subset><apply id="S4.SS2.7.p7.10.m10.1.1.2.cmml" xref="S4.SS2.7.p7.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.7.p7.10.m10.1.1.2.1.cmml" xref="S4.SS2.7.p7.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS2.7.p7.10.m10.1.1.2.2.cmml" xref="S4.SS2.7.p7.10.m10.1.1.2.2">𝐺</ci><cn id="S4.SS2.7.p7.10.m10.1.1.2.3.cmml" type="integer" xref="S4.SS2.7.p7.10.m10.1.1.2.3">1</cn></apply><ci id="S4.SS2.7.p7.10.m10.1.1.3.cmml" xref="S4.SS2.7.p7.10.m10.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.10.m10.1c">G_{1}\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.10.m10.1d">italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ italic_G</annotation></semantics></math> be the subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.7.p7.11.m11.1"><semantics id="S4.SS2.7.p7.11.m11.1a"><mi id="S4.SS2.7.p7.11.m11.1.1" xref="S4.SS2.7.p7.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.11.m11.1b"><ci id="S4.SS2.7.p7.11.m11.1.1.cmml" xref="S4.SS2.7.p7.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.11.m11.1d">italic_G</annotation></semantics></math> obtained when the ambient space <math alttext="[N]\times[N]" class="ltx_Math" display="inline" id="S4.SS2.7.p7.12.m12.2"><semantics id="S4.SS2.7.p7.12.m12.2a"><mrow id="S4.SS2.7.p7.12.m12.2.3" xref="S4.SS2.7.p7.12.m12.2.3.cmml"><mrow id="S4.SS2.7.p7.12.m12.2.3.2.2" xref="S4.SS2.7.p7.12.m12.2.3.2.1.cmml"><mo id="S4.SS2.7.p7.12.m12.2.3.2.2.1" stretchy="false" xref="S4.SS2.7.p7.12.m12.2.3.2.1.1.cmml">[</mo><mi id="S4.SS2.7.p7.12.m12.1.1" xref="S4.SS2.7.p7.12.m12.1.1.cmml">N</mi><mo id="S4.SS2.7.p7.12.m12.2.3.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS2.7.p7.12.m12.2.3.2.1.1.cmml">]</mo></mrow><mo id="S4.SS2.7.p7.12.m12.2.3.1" rspace="0.222em" xref="S4.SS2.7.p7.12.m12.2.3.1.cmml">×</mo><mrow id="S4.SS2.7.p7.12.m12.2.3.3.2" xref="S4.SS2.7.p7.12.m12.2.3.3.1.cmml"><mo id="S4.SS2.7.p7.12.m12.2.3.3.2.1" stretchy="false" xref="S4.SS2.7.p7.12.m12.2.3.3.1.1.cmml">[</mo><mi id="S4.SS2.7.p7.12.m12.2.2" xref="S4.SS2.7.p7.12.m12.2.2.cmml">N</mi><mo id="S4.SS2.7.p7.12.m12.2.3.3.2.2" stretchy="false" xref="S4.SS2.7.p7.12.m12.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.12.m12.2b"><apply id="S4.SS2.7.p7.12.m12.2.3.cmml" xref="S4.SS2.7.p7.12.m12.2.3"><times id="S4.SS2.7.p7.12.m12.2.3.1.cmml" xref="S4.SS2.7.p7.12.m12.2.3.1"></times><apply id="S4.SS2.7.p7.12.m12.2.3.2.1.cmml" xref="S4.SS2.7.p7.12.m12.2.3.2.2"><csymbol cd="latexml" id="S4.SS2.7.p7.12.m12.2.3.2.1.1.cmml" xref="S4.SS2.7.p7.12.m12.2.3.2.2.1">delimited-[]</csymbol><ci id="S4.SS2.7.p7.12.m12.1.1.cmml" xref="S4.SS2.7.p7.12.m12.1.1">𝑁</ci></apply><apply id="S4.SS2.7.p7.12.m12.2.3.3.1.cmml" xref="S4.SS2.7.p7.12.m12.2.3.3.2"><csymbol cd="latexml" id="S4.SS2.7.p7.12.m12.2.3.3.1.1.cmml" xref="S4.SS2.7.p7.12.m12.2.3.3.2.1">delimited-[]</csymbol><ci id="S4.SS2.7.p7.12.m12.2.2.cmml" xref="S4.SS2.7.p7.12.m12.2.2">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.12.m12.2c">[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.12.m12.2d">[ italic_N ] × [ italic_N ]</annotation></semantics></math> is restricted to <math alttext="\mathsf{Rows}(E_{1})\times\mathsf{Columns}(E_{1})" class="ltx_Math" display="inline" id="S4.SS2.7.p7.13.m13.2"><semantics id="S4.SS2.7.p7.13.m13.2a"><mrow id="S4.SS2.7.p7.13.m13.2.2" xref="S4.SS2.7.p7.13.m13.2.2.cmml"><mrow id="S4.SS2.7.p7.13.m13.1.1.1" xref="S4.SS2.7.p7.13.m13.1.1.1.cmml"><mrow id="S4.SS2.7.p7.13.m13.1.1.1.1" xref="S4.SS2.7.p7.13.m13.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.13.m13.1.1.1.1.3" xref="S4.SS2.7.p7.13.m13.1.1.1.1.3.cmml">𝖱𝗈𝗐𝗌</mi><mo id="S4.SS2.7.p7.13.m13.1.1.1.1.2" xref="S4.SS2.7.p7.13.m13.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.2" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.3" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.7.p7.13.m13.1.1.1.2" rspace="0.222em" xref="S4.SS2.7.p7.13.m13.1.1.1.2.cmml">×</mo><mi id="S4.SS2.7.p7.13.m13.1.1.1.3" xref="S4.SS2.7.p7.13.m13.1.1.1.3.cmml">𝖢𝗈𝗅𝗎𝗆𝗇𝗌</mi></mrow><mo id="S4.SS2.7.p7.13.m13.2.2.3" xref="S4.SS2.7.p7.13.m13.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.7.p7.13.m13.2.2.2.1" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.cmml"><mo id="S4.SS2.7.p7.13.m13.2.2.2.1.2" stretchy="false" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.cmml">(</mo><msub id="S4.SS2.7.p7.13.m13.2.2.2.1.1" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.cmml"><mi id="S4.SS2.7.p7.13.m13.2.2.2.1.1.2" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.13.m13.2.2.2.1.1.3" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.13.m13.2.2.2.1.3" stretchy="false" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.13.m13.2b"><apply id="S4.SS2.7.p7.13.m13.2.2.cmml" xref="S4.SS2.7.p7.13.m13.2.2"><times id="S4.SS2.7.p7.13.m13.2.2.3.cmml" xref="S4.SS2.7.p7.13.m13.2.2.3"></times><apply id="S4.SS2.7.p7.13.m13.1.1.1.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1"><times id="S4.SS2.7.p7.13.m13.1.1.1.2.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.2"></times><apply id="S4.SS2.7.p7.13.m13.1.1.1.1.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1"><times id="S4.SS2.7.p7.13.m13.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1.2"></times><ci id="S4.SS2.7.p7.13.m13.1.1.1.1.3.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1.3">𝖱𝗈𝗐𝗌</ci><apply id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.13.m13.1.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="S4.SS2.7.p7.13.m13.1.1.1.3.cmml" xref="S4.SS2.7.p7.13.m13.1.1.1.3">𝖢𝗈𝗅𝗎𝗆𝗇𝗌</ci></apply><apply id="S4.SS2.7.p7.13.m13.2.2.2.1.1.cmml" xref="S4.SS2.7.p7.13.m13.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.13.m13.2.2.2.1.1.1.cmml" xref="S4.SS2.7.p7.13.m13.2.2.2.1">subscript</csymbol><ci id="S4.SS2.7.p7.13.m13.2.2.2.1.1.2.cmml" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.13.m13.2.2.2.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.13.m13.2.2.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.13.m13.2c">\mathsf{Rows}(E_{1})\times\mathsf{Columns}(E_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.13.m13.2d">sansserif_Rows ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) × sansserif_Columns ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, where <math alttext="\mathsf{Rows}(E_{1})=\{a\in[N]\mid(a,b)\in E_{1}~{}\text{for some}~{}b\in[N]\}" class="ltx_Math" display="inline" id="S4.SS2.7.p7.14.m14.7"><semantics id="S4.SS2.7.p7.14.m14.7a"><mrow id="S4.SS2.7.p7.14.m14.7.7" xref="S4.SS2.7.p7.14.m14.7.7.cmml"><mrow id="S4.SS2.7.p7.14.m14.5.5.1" xref="S4.SS2.7.p7.14.m14.5.5.1.cmml"><mi id="S4.SS2.7.p7.14.m14.5.5.1.3" xref="S4.SS2.7.p7.14.m14.5.5.1.3.cmml">𝖱𝗈𝗐𝗌</mi><mo id="S4.SS2.7.p7.14.m14.5.5.1.2" xref="S4.SS2.7.p7.14.m14.5.5.1.2.cmml">⁢</mo><mrow id="S4.SS2.7.p7.14.m14.5.5.1.1.1" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.cmml"><mo id="S4.SS2.7.p7.14.m14.5.5.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.cmml">(</mo><msub id="S4.SS2.7.p7.14.m14.5.5.1.1.1.1" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.2" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.3" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.14.m14.5.5.1.1.1.3" stretchy="false" xref="S4.SS2.7.p7.14.m14.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.7.p7.14.m14.7.7.4" xref="S4.SS2.7.p7.14.m14.7.7.4.cmml">=</mo><mrow id="S4.SS2.7.p7.14.m14.7.7.3.2" xref="S4.SS2.7.p7.14.m14.7.7.3.3.cmml"><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.3" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.3.1.cmml">{</mo><mrow id="S4.SS2.7.p7.14.m14.6.6.2.1.1" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.cmml"><mi id="S4.SS2.7.p7.14.m14.6.6.2.1.1.2" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.2.cmml">a</mi><mo id="S4.SS2.7.p7.14.m14.6.6.2.1.1.1" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.1.cmml">∈</mo><mrow id="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.2" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.1.cmml"><mo id="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.2.1" stretchy="false" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.1.1.cmml">[</mo><mi id="S4.SS2.7.p7.14.m14.1.1" xref="S4.SS2.7.p7.14.m14.1.1.cmml">N</mi><mo id="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.2.2" stretchy="false" xref="S4.SS2.7.p7.14.m14.6.6.2.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo fence="true" id="S4.SS2.7.p7.14.m14.7.7.3.2.4" lspace="0em" rspace="0em" xref="S4.SS2.7.p7.14.m14.7.7.3.3.1.cmml">∣</mo><mrow id="S4.SS2.7.p7.14.m14.7.7.3.2.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.cmml"><mrow id="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.1.cmml"><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.2.1" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.1.cmml">(</mo><mi id="S4.SS2.7.p7.14.m14.2.2" xref="S4.SS2.7.p7.14.m14.2.2.cmml">a</mi><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.2.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.1.cmml">,</mo><mi id="S4.SS2.7.p7.14.m14.3.3" xref="S4.SS2.7.p7.14.m14.3.3.cmml">b</mi><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.2.3" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.2.1.cmml">)</mo></mrow><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.3" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.3.cmml">∈</mo><mrow id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.cmml"><msub id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.cmml"><mi id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.2.cmml">E</mi><mn id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.3" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1.cmml">⁢</mo><mtext id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3a.cmml">for some</mtext><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1a" lspace="0.330em" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1.cmml">⁢</mo><mi id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.4" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.4.cmml">b</mi></mrow><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.5" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.5.cmml">∈</mo><mrow id="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.2" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.1.cmml"><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.2.1" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.1.1.cmml">[</mo><mi id="S4.SS2.7.p7.14.m14.4.4" xref="S4.SS2.7.p7.14.m14.4.4.cmml">N</mi><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.2.2" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.1.1.cmml">]</mo></mrow></mrow><mo id="S4.SS2.7.p7.14.m14.7.7.3.2.5" stretchy="false" xref="S4.SS2.7.p7.14.m14.7.7.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.14.m14.7b"><apply id="S4.SS2.7.p7.14.m14.7.7.cmml" xref="S4.SS2.7.p7.14.m14.7.7"><eq id="S4.SS2.7.p7.14.m14.7.7.4.cmml" xref="S4.SS2.7.p7.14.m14.7.7.4"></eq><apply id="S4.SS2.7.p7.14.m14.5.5.1.cmml" xref="S4.SS2.7.p7.14.m14.5.5.1"><times 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id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4"><times id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.1"></times><apply id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2"><csymbol cd="ambiguous" id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.1.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2">subscript</csymbol><ci id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.2.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.2">𝐸</ci><cn id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.3.cmml" type="integer" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.2.3">1</cn></apply><ci id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3a.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3"><mtext id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.3">for some</mtext></ci><ci id="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.4.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.4.4">𝑏</ci></apply></apply><apply id="S4.SS2.7.p7.14.m14.7.7.3.2.2c.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2"><in id="S4.SS2.7.p7.14.m14.7.7.3.2.2.5.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.5"></in><share href="https://arxiv.org/html/2503.14117v1#S4.SS2.7.p7.14.m14.7.7.3.2.2.4.cmml" id="S4.SS2.7.p7.14.m14.7.7.3.2.2d.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2"></share><apply id="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.1.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.2"><csymbol cd="latexml" id="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.1.1.cmml" xref="S4.SS2.7.p7.14.m14.7.7.3.2.2.6.2.1">delimited-[]</csymbol><ci id="S4.SS2.7.p7.14.m14.4.4.cmml" xref="S4.SS2.7.p7.14.m14.4.4">𝑁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.14.m14.7c">\mathsf{Rows}(E_{1})=\{a\in[N]\mid(a,b)\in E_{1}~{}\text{for some}~{}b\in[N]\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.14.m14.7d">sansserif_Rows ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = { italic_a ∈ [ italic_N ] ∣ ( italic_a , italic_b ) ∈ italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT for some italic_b ∈ [ italic_N ] }</annotation></semantics></math>, and <math alttext="\mathsf{Columns}(E_{1})" class="ltx_Math" display="inline" id="S4.SS2.7.p7.15.m15.1"><semantics id="S4.SS2.7.p7.15.m15.1a"><mrow id="S4.SS2.7.p7.15.m15.1.1" xref="S4.SS2.7.p7.15.m15.1.1.cmml"><mi id="S4.SS2.7.p7.15.m15.1.1.3" xref="S4.SS2.7.p7.15.m15.1.1.3.cmml">𝖢𝗈𝗅𝗎𝗆𝗇𝗌</mi><mo id="S4.SS2.7.p7.15.m15.1.1.2" xref="S4.SS2.7.p7.15.m15.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.7.p7.15.m15.1.1.1.1" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.cmml"><mo id="S4.SS2.7.p7.15.m15.1.1.1.1.2" stretchy="false" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.7.p7.15.m15.1.1.1.1.1" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.cmml"><mi id="S4.SS2.7.p7.15.m15.1.1.1.1.1.2" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.7.p7.15.m15.1.1.1.1.1.3" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.7.p7.15.m15.1.1.1.1.3" stretchy="false" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p7.15.m15.1b"><apply id="S4.SS2.7.p7.15.m15.1.1.cmml" xref="S4.SS2.7.p7.15.m15.1.1"><times id="S4.SS2.7.p7.15.m15.1.1.2.cmml" xref="S4.SS2.7.p7.15.m15.1.1.2"></times><ci id="S4.SS2.7.p7.15.m15.1.1.3.cmml" xref="S4.SS2.7.p7.15.m15.1.1.3">𝖢𝗈𝗅𝗎𝗆𝗇𝗌</ci><apply id="S4.SS2.7.p7.15.m15.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.7.p7.15.m15.1.1.1.1.1.1.cmml" xref="S4.SS2.7.p7.15.m15.1.1.1.1">subscript</csymbol><ci id="S4.SS2.7.p7.15.m15.1.1.1.1.1.2.cmml" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.7.p7.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.7.p7.15.m15.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p7.15.m15.1c">\mathsf{Columns}(E_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p7.15.m15.1d">sansserif_Columns ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is defined analogously.</p> </div> <div class="ltx_para" id="S4.SS2.8.p8"> <p class="ltx_p" id="S4.SS2.8.p8.12">Observe that for no element <math alttext="e_{1}\in G_{1}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.1.m1.1"><semantics id="S4.SS2.8.p8.1.m1.1a"><mrow id="S4.SS2.8.p8.1.m1.1.1" xref="S4.SS2.8.p8.1.m1.1.1.cmml"><msub id="S4.SS2.8.p8.1.m1.1.1.2" xref="S4.SS2.8.p8.1.m1.1.1.2.cmml"><mi id="S4.SS2.8.p8.1.m1.1.1.2.2" xref="S4.SS2.8.p8.1.m1.1.1.2.2.cmml">e</mi><mn id="S4.SS2.8.p8.1.m1.1.1.2.3" xref="S4.SS2.8.p8.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.1.m1.1.1.1" xref="S4.SS2.8.p8.1.m1.1.1.1.cmml">∈</mo><msub id="S4.SS2.8.p8.1.m1.1.1.3" xref="S4.SS2.8.p8.1.m1.1.1.3.cmml"><mi id="S4.SS2.8.p8.1.m1.1.1.3.2" xref="S4.SS2.8.p8.1.m1.1.1.3.2.cmml">G</mi><mn id="S4.SS2.8.p8.1.m1.1.1.3.3" xref="S4.SS2.8.p8.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.1.m1.1b"><apply id="S4.SS2.8.p8.1.m1.1.1.cmml" xref="S4.SS2.8.p8.1.m1.1.1"><in id="S4.SS2.8.p8.1.m1.1.1.1.cmml" xref="S4.SS2.8.p8.1.m1.1.1.1"></in><apply id="S4.SS2.8.p8.1.m1.1.1.2.cmml" xref="S4.SS2.8.p8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.8.p8.1.m1.1.1.2.1.cmml" xref="S4.SS2.8.p8.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS2.8.p8.1.m1.1.1.2.2.cmml" xref="S4.SS2.8.p8.1.m1.1.1.2.2">𝑒</ci><cn id="S4.SS2.8.p8.1.m1.1.1.2.3.cmml" type="integer" xref="S4.SS2.8.p8.1.m1.1.1.2.3">1</cn></apply><apply id="S4.SS2.8.p8.1.m1.1.1.3.cmml" xref="S4.SS2.8.p8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.1.m1.1.1.3.1.cmml" xref="S4.SS2.8.p8.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS2.8.p8.1.m1.1.1.3.2.cmml" xref="S4.SS2.8.p8.1.m1.1.1.3.2">𝐺</ci><cn id="S4.SS2.8.p8.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.8.p8.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.1.m1.1c">e_{1}\in G_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.1.m1.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathcal{F}_{e_{1}}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.2.m2.1"><semantics id="S4.SS2.8.p8.2.m2.1a"><msub id="S4.SS2.8.p8.2.m2.1.1" xref="S4.SS2.8.p8.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p8.2.m2.1.1.2" xref="S4.SS2.8.p8.2.m2.1.1.2.cmml">ℱ</mi><msub id="S4.SS2.8.p8.2.m2.1.1.3" xref="S4.SS2.8.p8.2.m2.1.1.3.cmml"><mi id="S4.SS2.8.p8.2.m2.1.1.3.2" xref="S4.SS2.8.p8.2.m2.1.1.3.2.cmml">e</mi><mn id="S4.SS2.8.p8.2.m2.1.1.3.3" xref="S4.SS2.8.p8.2.m2.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.2.m2.1b"><apply id="S4.SS2.8.p8.2.m2.1.1.cmml" xref="S4.SS2.8.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.2.m2.1.1.1.cmml" xref="S4.SS2.8.p8.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.8.p8.2.m2.1.1.2.cmml" xref="S4.SS2.8.p8.2.m2.1.1.2">ℱ</ci><apply id="S4.SS2.8.p8.2.m2.1.1.3.cmml" xref="S4.SS2.8.p8.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.2.m2.1.1.3.1.cmml" xref="S4.SS2.8.p8.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.8.p8.2.m2.1.1.3.2.cmml" xref="S4.SS2.8.p8.2.m2.1.1.3.2">𝑒</ci><cn id="S4.SS2.8.p8.2.m2.1.1.3.3.cmml" type="integer" xref="S4.SS2.8.p8.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.2.m2.1c">\mathcal{F}_{e_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.2.m2.1d">caligraphic_F start_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is covered by <math alttext="(E_{1},H_{1})" class="ltx_Math" display="inline" id="S4.SS2.8.p8.3.m3.2"><semantics id="S4.SS2.8.p8.3.m3.2a"><mrow id="S4.SS2.8.p8.3.m3.2.2.2" xref="S4.SS2.8.p8.3.m3.2.2.3.cmml"><mo id="S4.SS2.8.p8.3.m3.2.2.2.3" stretchy="false" xref="S4.SS2.8.p8.3.m3.2.2.3.cmml">(</mo><msub id="S4.SS2.8.p8.3.m3.1.1.1.1" xref="S4.SS2.8.p8.3.m3.1.1.1.1.cmml"><mi id="S4.SS2.8.p8.3.m3.1.1.1.1.2" xref="S4.SS2.8.p8.3.m3.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.8.p8.3.m3.1.1.1.1.3" xref="S4.SS2.8.p8.3.m3.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.3.m3.2.2.2.4" xref="S4.SS2.8.p8.3.m3.2.2.3.cmml">,</mo><msub id="S4.SS2.8.p8.3.m3.2.2.2.2" xref="S4.SS2.8.p8.3.m3.2.2.2.2.cmml"><mi id="S4.SS2.8.p8.3.m3.2.2.2.2.2" xref="S4.SS2.8.p8.3.m3.2.2.2.2.2.cmml">H</mi><mn id="S4.SS2.8.p8.3.m3.2.2.2.2.3" xref="S4.SS2.8.p8.3.m3.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.3.m3.2.2.2.5" stretchy="false" xref="S4.SS2.8.p8.3.m3.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.3.m3.2b"><interval closure="open" id="S4.SS2.8.p8.3.m3.2.2.3.cmml" xref="S4.SS2.8.p8.3.m3.2.2.2"><apply id="S4.SS2.8.p8.3.m3.1.1.1.1.cmml" xref="S4.SS2.8.p8.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.8.p8.3.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS2.8.p8.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.8.p8.3.m3.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.8.p8.3.m3.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.8.p8.3.m3.1.1.1.1.3">1</cn></apply><apply id="S4.SS2.8.p8.3.m3.2.2.2.2.cmml" xref="S4.SS2.8.p8.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.8.p8.3.m3.2.2.2.2.1.cmml" xref="S4.SS2.8.p8.3.m3.2.2.2.2">subscript</csymbol><ci id="S4.SS2.8.p8.3.m3.2.2.2.2.2.cmml" xref="S4.SS2.8.p8.3.m3.2.2.2.2.2">𝐻</ci><cn id="S4.SS2.8.p8.3.m3.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.8.p8.3.m3.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.3.m3.2c">(E_{1},H_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.3.m3.2d">( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Furthermore, the elements in <math alttext="G_{1}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.4.m4.1"><semantics id="S4.SS2.8.p8.4.m4.1a"><msub id="S4.SS2.8.p8.4.m4.1.1" xref="S4.SS2.8.p8.4.m4.1.1.cmml"><mi id="S4.SS2.8.p8.4.m4.1.1.2" xref="S4.SS2.8.p8.4.m4.1.1.2.cmml">G</mi><mn id="S4.SS2.8.p8.4.m4.1.1.3" xref="S4.SS2.8.p8.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.4.m4.1b"><apply id="S4.SS2.8.p8.4.m4.1.1.cmml" xref="S4.SS2.8.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.4.m4.1.1.1.cmml" xref="S4.SS2.8.p8.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.8.p8.4.m4.1.1.2.cmml" xref="S4.SS2.8.p8.4.m4.1.1.2">𝐺</ci><cn id="S4.SS2.8.p8.4.m4.1.1.3.cmml" type="integer" xref="S4.SS2.8.p8.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.4.m4.1c">G_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.4.m4.1d">italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> span at least <math alttext="2^{n-1}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.5.m5.1"><semantics id="S4.SS2.8.p8.5.m5.1a"><msup id="S4.SS2.8.p8.5.m5.1.1" xref="S4.SS2.8.p8.5.m5.1.1.cmml"><mn id="S4.SS2.8.p8.5.m5.1.1.2" xref="S4.SS2.8.p8.5.m5.1.1.2.cmml">2</mn><mrow id="S4.SS2.8.p8.5.m5.1.1.3" xref="S4.SS2.8.p8.5.m5.1.1.3.cmml"><mi id="S4.SS2.8.p8.5.m5.1.1.3.2" xref="S4.SS2.8.p8.5.m5.1.1.3.2.cmml">n</mi><mo id="S4.SS2.8.p8.5.m5.1.1.3.1" xref="S4.SS2.8.p8.5.m5.1.1.3.1.cmml">−</mo><mn id="S4.SS2.8.p8.5.m5.1.1.3.3" xref="S4.SS2.8.p8.5.m5.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.5.m5.1b"><apply id="S4.SS2.8.p8.5.m5.1.1.cmml" xref="S4.SS2.8.p8.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.5.m5.1.1.1.cmml" xref="S4.SS2.8.p8.5.m5.1.1">superscript</csymbol><cn id="S4.SS2.8.p8.5.m5.1.1.2.cmml" type="integer" xref="S4.SS2.8.p8.5.m5.1.1.2">2</cn><apply id="S4.SS2.8.p8.5.m5.1.1.3.cmml" xref="S4.SS2.8.p8.5.m5.1.1.3"><minus id="S4.SS2.8.p8.5.m5.1.1.3.1.cmml" xref="S4.SS2.8.p8.5.m5.1.1.3.1"></minus><ci id="S4.SS2.8.p8.5.m5.1.1.3.2.cmml" xref="S4.SS2.8.p8.5.m5.1.1.3.2">𝑛</ci><cn id="S4.SS2.8.p8.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.8.p8.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.5.m5.1c">2^{n-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.5.m5.1d">2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> different rows and at least <math alttext="2^{n-1}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.6.m6.1"><semantics id="S4.SS2.8.p8.6.m6.1a"><msup id="S4.SS2.8.p8.6.m6.1.1" xref="S4.SS2.8.p8.6.m6.1.1.cmml"><mn id="S4.SS2.8.p8.6.m6.1.1.2" xref="S4.SS2.8.p8.6.m6.1.1.2.cmml">2</mn><mrow id="S4.SS2.8.p8.6.m6.1.1.3" xref="S4.SS2.8.p8.6.m6.1.1.3.cmml"><mi id="S4.SS2.8.p8.6.m6.1.1.3.2" xref="S4.SS2.8.p8.6.m6.1.1.3.2.cmml">n</mi><mo id="S4.SS2.8.p8.6.m6.1.1.3.1" xref="S4.SS2.8.p8.6.m6.1.1.3.1.cmml">−</mo><mn id="S4.SS2.8.p8.6.m6.1.1.3.3" xref="S4.SS2.8.p8.6.m6.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.6.m6.1b"><apply id="S4.SS2.8.p8.6.m6.1.1.cmml" xref="S4.SS2.8.p8.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.6.m6.1.1.1.cmml" xref="S4.SS2.8.p8.6.m6.1.1">superscript</csymbol><cn id="S4.SS2.8.p8.6.m6.1.1.2.cmml" type="integer" xref="S4.SS2.8.p8.6.m6.1.1.2">2</cn><apply id="S4.SS2.8.p8.6.m6.1.1.3.cmml" xref="S4.SS2.8.p8.6.m6.1.1.3"><minus id="S4.SS2.8.p8.6.m6.1.1.3.1.cmml" xref="S4.SS2.8.p8.6.m6.1.1.3.1"></minus><ci id="S4.SS2.8.p8.6.m6.1.1.3.2.cmml" xref="S4.SS2.8.p8.6.m6.1.1.3.2">𝑛</ci><cn id="S4.SS2.8.p8.6.m6.1.1.3.3.cmml" type="integer" xref="S4.SS2.8.p8.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.6.m6.1c">2^{n-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.6.m6.1d">2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> different columns of <math alttext="[N]" class="ltx_Math" display="inline" id="S4.SS2.8.p8.7.m7.1"><semantics id="S4.SS2.8.p8.7.m7.1a"><mrow id="S4.SS2.8.p8.7.m7.1.2.2" xref="S4.SS2.8.p8.7.m7.1.2.1.cmml"><mo id="S4.SS2.8.p8.7.m7.1.2.2.1" stretchy="false" xref="S4.SS2.8.p8.7.m7.1.2.1.1.cmml">[</mo><mi id="S4.SS2.8.p8.7.m7.1.1" xref="S4.SS2.8.p8.7.m7.1.1.cmml">N</mi><mo id="S4.SS2.8.p8.7.m7.1.2.2.2" stretchy="false" xref="S4.SS2.8.p8.7.m7.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.7.m7.1b"><apply id="S4.SS2.8.p8.7.m7.1.2.1.cmml" xref="S4.SS2.8.p8.7.m7.1.2.2"><csymbol cd="latexml" id="S4.SS2.8.p8.7.m7.1.2.1.1.cmml" xref="S4.SS2.8.p8.7.m7.1.2.2.1">delimited-[]</csymbol><ci id="S4.SS2.8.p8.7.m7.1.1.cmml" xref="S4.SS2.8.p8.7.m7.1.1">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.7.m7.1c">[N]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.7.m7.1d">[ italic_N ]</annotation></semantics></math>. Finally, each semi-filter <math alttext="\mathcal{F}_{e_{1}}\in\mathfrak{F}^{G}_{\mathsf{can}}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.8.m8.1"><semantics id="S4.SS2.8.p8.8.m8.1a"><mrow id="S4.SS2.8.p8.8.m8.1.1" xref="S4.SS2.8.p8.8.m8.1.1.cmml"><msub id="S4.SS2.8.p8.8.m8.1.1.2" xref="S4.SS2.8.p8.8.m8.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p8.8.m8.1.1.2.2" xref="S4.SS2.8.p8.8.m8.1.1.2.2.cmml">ℱ</mi><msub id="S4.SS2.8.p8.8.m8.1.1.2.3" xref="S4.SS2.8.p8.8.m8.1.1.2.3.cmml"><mi id="S4.SS2.8.p8.8.m8.1.1.2.3.2" xref="S4.SS2.8.p8.8.m8.1.1.2.3.2.cmml">e</mi><mn id="S4.SS2.8.p8.8.m8.1.1.2.3.3" xref="S4.SS2.8.p8.8.m8.1.1.2.3.3.cmml">1</mn></msub></msub><mo id="S4.SS2.8.p8.8.m8.1.1.1" xref="S4.SS2.8.p8.8.m8.1.1.1.cmml">∈</mo><msubsup id="S4.SS2.8.p8.8.m8.1.1.3" xref="S4.SS2.8.p8.8.m8.1.1.3.cmml"><mi id="S4.SS2.8.p8.8.m8.1.1.3.2.2" xref="S4.SS2.8.p8.8.m8.1.1.3.2.2.cmml">𝔉</mi><mi id="S4.SS2.8.p8.8.m8.1.1.3.3" xref="S4.SS2.8.p8.8.m8.1.1.3.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.SS2.8.p8.8.m8.1.1.3.2.3" xref="S4.SS2.8.p8.8.m8.1.1.3.2.3.cmml">G</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.8.m8.1b"><apply id="S4.SS2.8.p8.8.m8.1.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1"><in id="S4.SS2.8.p8.8.m8.1.1.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1.1"></in><apply id="S4.SS2.8.p8.8.m8.1.1.2.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.8.p8.8.m8.1.1.2.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2">subscript</csymbol><ci id="S4.SS2.8.p8.8.m8.1.1.2.2.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2.2">ℱ</ci><apply id="S4.SS2.8.p8.8.m8.1.1.2.3.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.8.m8.1.1.2.3.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2.3">subscript</csymbol><ci id="S4.SS2.8.p8.8.m8.1.1.2.3.2.cmml" xref="S4.SS2.8.p8.8.m8.1.1.2.3.2">𝑒</ci><cn id="S4.SS2.8.p8.8.m8.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.8.p8.8.m8.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS2.8.p8.8.m8.1.1.3.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.8.m8.1.1.3.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3">subscript</csymbol><apply id="S4.SS2.8.p8.8.m8.1.1.3.2.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.8.m8.1.1.3.2.1.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3">superscript</csymbol><ci id="S4.SS2.8.p8.8.m8.1.1.3.2.2.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3.2.2">𝔉</ci><ci id="S4.SS2.8.p8.8.m8.1.1.3.2.3.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3.2.3">𝐺</ci></apply><ci id="S4.SS2.8.p8.8.m8.1.1.3.3.cmml" xref="S4.SS2.8.p8.8.m8.1.1.3.3">𝖼𝖺𝗇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.8.m8.1c">\mathcal{F}_{e_{1}}\in\mathfrak{F}^{G}_{\mathsf{can}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.8.m8.1d">caligraphic_F start_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ fraktur_F start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="e_{1}\in G_{1}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.9.m9.1"><semantics id="S4.SS2.8.p8.9.m9.1a"><mrow id="S4.SS2.8.p8.9.m9.1.1" xref="S4.SS2.8.p8.9.m9.1.1.cmml"><msub id="S4.SS2.8.p8.9.m9.1.1.2" xref="S4.SS2.8.p8.9.m9.1.1.2.cmml"><mi id="S4.SS2.8.p8.9.m9.1.1.2.2" xref="S4.SS2.8.p8.9.m9.1.1.2.2.cmml">e</mi><mn id="S4.SS2.8.p8.9.m9.1.1.2.3" xref="S4.SS2.8.p8.9.m9.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.9.m9.1.1.1" xref="S4.SS2.8.p8.9.m9.1.1.1.cmml">∈</mo><msub id="S4.SS2.8.p8.9.m9.1.1.3" xref="S4.SS2.8.p8.9.m9.1.1.3.cmml"><mi id="S4.SS2.8.p8.9.m9.1.1.3.2" xref="S4.SS2.8.p8.9.m9.1.1.3.2.cmml">G</mi><mn id="S4.SS2.8.p8.9.m9.1.1.3.3" xref="S4.SS2.8.p8.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.9.m9.1b"><apply id="S4.SS2.8.p8.9.m9.1.1.cmml" xref="S4.SS2.8.p8.9.m9.1.1"><in id="S4.SS2.8.p8.9.m9.1.1.1.cmml" xref="S4.SS2.8.p8.9.m9.1.1.1"></in><apply id="S4.SS2.8.p8.9.m9.1.1.2.cmml" xref="S4.SS2.8.p8.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.8.p8.9.m9.1.1.2.1.cmml" xref="S4.SS2.8.p8.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS2.8.p8.9.m9.1.1.2.2.cmml" xref="S4.SS2.8.p8.9.m9.1.1.2.2">𝑒</ci><cn id="S4.SS2.8.p8.9.m9.1.1.2.3.cmml" type="integer" xref="S4.SS2.8.p8.9.m9.1.1.2.3">1</cn></apply><apply id="S4.SS2.8.p8.9.m9.1.1.3.cmml" xref="S4.SS2.8.p8.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.8.p8.9.m9.1.1.3.1.cmml" xref="S4.SS2.8.p8.9.m9.1.1.3">subscript</csymbol><ci id="S4.SS2.8.p8.9.m9.1.1.3.2.cmml" xref="S4.SS2.8.p8.9.m9.1.1.3.2">𝐺</ci><cn id="S4.SS2.8.p8.9.m9.1.1.3.3.cmml" type="integer" xref="S4.SS2.8.p8.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.9.m9.1c">e_{1}\in G_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.9.m9.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> must be covered by some pair in <math alttext="\Lambda\setminus\{(E_{1},H_{1})\}" class="ltx_Math" display="inline" id="S4.SS2.8.p8.10.m10.1"><semantics id="S4.SS2.8.p8.10.m10.1a"><mrow id="S4.SS2.8.p8.10.m10.1.1" xref="S4.SS2.8.p8.10.m10.1.1.cmml"><mi id="S4.SS2.8.p8.10.m10.1.1.3" mathvariant="normal" xref="S4.SS2.8.p8.10.m10.1.1.3.cmml">Λ</mi><mo id="S4.SS2.8.p8.10.m10.1.1.2" xref="S4.SS2.8.p8.10.m10.1.1.2.cmml">∖</mo><mrow id="S4.SS2.8.p8.10.m10.1.1.1.1" xref="S4.SS2.8.p8.10.m10.1.1.1.2.cmml"><mo id="S4.SS2.8.p8.10.m10.1.1.1.1.2" stretchy="false" xref="S4.SS2.8.p8.10.m10.1.1.1.2.cmml">{</mo><mrow id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.3.cmml"><mo id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.3" stretchy="false" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.3.cmml">(</mo><msub id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.2" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.2.cmml">E</mi><mn id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.3" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.4" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.3.cmml">,</mo><msub id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.cmml"><mi id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.2" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.2.cmml">H</mi><mn id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.3" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.5" stretchy="false" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="S4.SS2.8.p8.10.m10.1.1.1.1.3" stretchy="false" xref="S4.SS2.8.p8.10.m10.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.10.m10.1b"><apply id="S4.SS2.8.p8.10.m10.1.1.cmml" xref="S4.SS2.8.p8.10.m10.1.1"><setdiff id="S4.SS2.8.p8.10.m10.1.1.2.cmml" xref="S4.SS2.8.p8.10.m10.1.1.2"></setdiff><ci id="S4.SS2.8.p8.10.m10.1.1.3.cmml" xref="S4.SS2.8.p8.10.m10.1.1.3">Λ</ci><set id="S4.SS2.8.p8.10.m10.1.1.1.2.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1"><interval closure="open" id="S4.SS2.8.p8.10.m10.1.1.1.1.1.3.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2"><apply id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.2">𝐸</ci><cn id="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.1.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2">subscript</csymbol><ci id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.2.cmml" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.2">𝐻</ci><cn id="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S4.SS2.8.p8.10.m10.1.1.1.1.1.2.2.3">1</cn></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.10.m10.1c">\Lambda\setminus\{(E_{1},H_{1})\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.10.m10.1d">roman_Λ ∖ { ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) }</annotation></semantics></math>. By a recursive application of the previous argument, and using that in the base case <math alttext="n=1" class="ltx_Math" display="inline" id="S4.SS2.8.p8.11.m11.1"><semantics id="S4.SS2.8.p8.11.m11.1a"><mrow id="S4.SS2.8.p8.11.m11.1.1" xref="S4.SS2.8.p8.11.m11.1.1.cmml"><mi id="S4.SS2.8.p8.11.m11.1.1.2" xref="S4.SS2.8.p8.11.m11.1.1.2.cmml">n</mi><mo id="S4.SS2.8.p8.11.m11.1.1.1" xref="S4.SS2.8.p8.11.m11.1.1.1.cmml">=</mo><mn id="S4.SS2.8.p8.11.m11.1.1.3" xref="S4.SS2.8.p8.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.11.m11.1b"><apply id="S4.SS2.8.p8.11.m11.1.1.cmml" xref="S4.SS2.8.p8.11.m11.1.1"><eq id="S4.SS2.8.p8.11.m11.1.1.1.cmml" xref="S4.SS2.8.p8.11.m11.1.1.1"></eq><ci id="S4.SS2.8.p8.11.m11.1.1.2.cmml" xref="S4.SS2.8.p8.11.m11.1.1.2">𝑛</ci><cn id="S4.SS2.8.p8.11.m11.1.1.3.cmml" type="integer" xref="S4.SS2.8.p8.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.11.m11.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.11.m11.1d">italic_n = 1</annotation></semantics></math> at least one pair of sets is necessary, it is easy to see <math alttext="|\Lambda|\geq n=\log N" class="ltx_Math" display="inline" id="S4.SS2.8.p8.12.m12.1"><semantics id="S4.SS2.8.p8.12.m12.1a"><mrow id="S4.SS2.8.p8.12.m12.1.2" xref="S4.SS2.8.p8.12.m12.1.2.cmml"><mrow id="S4.SS2.8.p8.12.m12.1.2.2.2" xref="S4.SS2.8.p8.12.m12.1.2.2.1.cmml"><mo id="S4.SS2.8.p8.12.m12.1.2.2.2.1" stretchy="false" xref="S4.SS2.8.p8.12.m12.1.2.2.1.1.cmml">|</mo><mi id="S4.SS2.8.p8.12.m12.1.1" mathvariant="normal" xref="S4.SS2.8.p8.12.m12.1.1.cmml">Λ</mi><mo id="S4.SS2.8.p8.12.m12.1.2.2.2.2" stretchy="false" xref="S4.SS2.8.p8.12.m12.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS2.8.p8.12.m12.1.2.3" xref="S4.SS2.8.p8.12.m12.1.2.3.cmml">≥</mo><mi id="S4.SS2.8.p8.12.m12.1.2.4" xref="S4.SS2.8.p8.12.m12.1.2.4.cmml">n</mi><mo id="S4.SS2.8.p8.12.m12.1.2.5" xref="S4.SS2.8.p8.12.m12.1.2.5.cmml">=</mo><mrow id="S4.SS2.8.p8.12.m12.1.2.6" xref="S4.SS2.8.p8.12.m12.1.2.6.cmml"><mi id="S4.SS2.8.p8.12.m12.1.2.6.1" xref="S4.SS2.8.p8.12.m12.1.2.6.1.cmml">log</mi><mo id="S4.SS2.8.p8.12.m12.1.2.6a" lspace="0.167em" xref="S4.SS2.8.p8.12.m12.1.2.6.cmml">⁡</mo><mi id="S4.SS2.8.p8.12.m12.1.2.6.2" xref="S4.SS2.8.p8.12.m12.1.2.6.2.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p8.12.m12.1b"><apply id="S4.SS2.8.p8.12.m12.1.2.cmml" xref="S4.SS2.8.p8.12.m12.1.2"><and id="S4.SS2.8.p8.12.m12.1.2a.cmml" xref="S4.SS2.8.p8.12.m12.1.2"></and><apply id="S4.SS2.8.p8.12.m12.1.2b.cmml" xref="S4.SS2.8.p8.12.m12.1.2"><geq id="S4.SS2.8.p8.12.m12.1.2.3.cmml" xref="S4.SS2.8.p8.12.m12.1.2.3"></geq><apply id="S4.SS2.8.p8.12.m12.1.2.2.1.cmml" xref="S4.SS2.8.p8.12.m12.1.2.2.2"><abs id="S4.SS2.8.p8.12.m12.1.2.2.1.1.cmml" xref="S4.SS2.8.p8.12.m12.1.2.2.2.1"></abs><ci id="S4.SS2.8.p8.12.m12.1.1.cmml" xref="S4.SS2.8.p8.12.m12.1.1">Λ</ci></apply><ci id="S4.SS2.8.p8.12.m12.1.2.4.cmml" xref="S4.SS2.8.p8.12.m12.1.2.4">𝑛</ci></apply><apply id="S4.SS2.8.p8.12.m12.1.2c.cmml" xref="S4.SS2.8.p8.12.m12.1.2"><eq id="S4.SS2.8.p8.12.m12.1.2.5.cmml" xref="S4.SS2.8.p8.12.m12.1.2.5"></eq><share href="https://arxiv.org/html/2503.14117v1#S4.SS2.8.p8.12.m12.1.2.4.cmml" id="S4.SS2.8.p8.12.m12.1.2d.cmml" xref="S4.SS2.8.p8.12.m12.1.2"></share><apply id="S4.SS2.8.p8.12.m12.1.2.6.cmml" xref="S4.SS2.8.p8.12.m12.1.2.6"><log id="S4.SS2.8.p8.12.m12.1.2.6.1.cmml" xref="S4.SS2.8.p8.12.m12.1.2.6.1"></log><ci id="S4.SS2.8.p8.12.m12.1.2.6.2.cmml" xref="S4.SS2.8.p8.12.m12.1.2.6.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p8.12.m12.1c">|\Lambda|\geq n=\log N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p8.12.m12.1d">| roman_Λ | ≥ italic_n = roman_log italic_N</annotation></semantics></math>. This completes the proof. ∎</p> </div> </div> </section> <section class="ltx_subsection ltx_indent_first" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Nondeterministic graph complexity</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.8">Given a Boolean function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.4"><semantics id="S4.SS3.p1.1.m1.4a"><mrow id="S4.SS3.p1.1.m1.4.5" xref="S4.SS3.p1.1.m1.4.5.cmml"><mi id="S4.SS3.p1.1.m1.4.5.2" xref="S4.SS3.p1.1.m1.4.5.2.cmml">f</mi><mo id="S4.SS3.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p1.1.m1.4.5.1.cmml">:</mo><mrow id="S4.SS3.p1.1.m1.4.5.3" xref="S4.SS3.p1.1.m1.4.5.3.cmml"><msup id="S4.SS3.p1.1.m1.4.5.3.2" xref="S4.SS3.p1.1.m1.4.5.3.2.cmml"><mrow id="S4.SS3.p1.1.m1.4.5.3.2.2.2" xref="S4.SS3.p1.1.m1.4.5.3.2.2.1.cmml"><mo id="S4.SS3.p1.1.m1.4.5.3.2.2.2.1" stretchy="false" xref="S4.SS3.p1.1.m1.4.5.3.2.2.1.cmml">{</mo><mn id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">0</mn><mo id="S4.SS3.p1.1.m1.4.5.3.2.2.2.2" xref="S4.SS3.p1.1.m1.4.5.3.2.2.1.cmml">,</mo><mn id="S4.SS3.p1.1.m1.2.2" xref="S4.SS3.p1.1.m1.2.2.cmml">1</mn><mo id="S4.SS3.p1.1.m1.4.5.3.2.2.2.3" stretchy="false" xref="S4.SS3.p1.1.m1.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S4.SS3.p1.1.m1.4.5.3.2.3" xref="S4.SS3.p1.1.m1.4.5.3.2.3.cmml">n</mi></msup><mo id="S4.SS3.p1.1.m1.4.5.3.1" stretchy="false" xref="S4.SS3.p1.1.m1.4.5.3.1.cmml">→</mo><mrow id="S4.SS3.p1.1.m1.4.5.3.3.2" xref="S4.SS3.p1.1.m1.4.5.3.3.1.cmml"><mo id="S4.SS3.p1.1.m1.4.5.3.3.2.1" stretchy="false" xref="S4.SS3.p1.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S4.SS3.p1.1.m1.3.3" xref="S4.SS3.p1.1.m1.3.3.cmml">0</mn><mo id="S4.SS3.p1.1.m1.4.5.3.3.2.2" xref="S4.SS3.p1.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S4.SS3.p1.1.m1.4.4" xref="S4.SS3.p1.1.m1.4.4.cmml">1</mn><mo id="S4.SS3.p1.1.m1.4.5.3.3.2.3" stretchy="false" xref="S4.SS3.p1.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.4b"><apply id="S4.SS3.p1.1.m1.4.5.cmml" xref="S4.SS3.p1.1.m1.4.5"><ci id="S4.SS3.p1.1.m1.4.5.1.cmml" xref="S4.SS3.p1.1.m1.4.5.1">:</ci><ci id="S4.SS3.p1.1.m1.4.5.2.cmml" xref="S4.SS3.p1.1.m1.4.5.2">𝑓</ci><apply id="S4.SS3.p1.1.m1.4.5.3.cmml" xref="S4.SS3.p1.1.m1.4.5.3"><ci id="S4.SS3.p1.1.m1.4.5.3.1.cmml" xref="S4.SS3.p1.1.m1.4.5.3.1">→</ci><apply id="S4.SS3.p1.1.m1.4.5.3.2.cmml" xref="S4.SS3.p1.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.4.5.3.2.1.cmml" xref="S4.SS3.p1.1.m1.4.5.3.2">superscript</csymbol><set id="S4.SS3.p1.1.m1.4.5.3.2.2.1.cmml" xref="S4.SS3.p1.1.m1.4.5.3.2.2.2"><cn id="S4.SS3.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS3.p1.1.m1.1.1">0</cn><cn id="S4.SS3.p1.1.m1.2.2.cmml" type="integer" xref="S4.SS3.p1.1.m1.2.2">1</cn></set><ci id="S4.SS3.p1.1.m1.4.5.3.2.3.cmml" xref="S4.SS3.p1.1.m1.4.5.3.2.3">𝑛</ci></apply><set id="S4.SS3.p1.1.m1.4.5.3.3.1.cmml" xref="S4.SS3.p1.1.m1.4.5.3.3.2"><cn id="S4.SS3.p1.1.m1.3.3.cmml" type="integer" xref="S4.SS3.p1.1.m1.3.3">0</cn><cn id="S4.SS3.p1.1.m1.4.4.cmml" type="integer" xref="S4.SS3.p1.1.m1.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>, we let <math alttext="\mathsf{size}(f)" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mrow id="S4.SS3.p1.2.m2.1.2" xref="S4.SS3.p1.2.m2.1.2.cmml"><mi id="S4.SS3.p1.2.m2.1.2.2" xref="S4.SS3.p1.2.m2.1.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.2.m2.1.2.1" xref="S4.SS3.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.2.m2.1.2.3.2" xref="S4.SS3.p1.2.m2.1.2.cmml"><mo id="S4.SS3.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.2.m2.1.2.cmml">(</mo><mi id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">f</mi><mo id="S4.SS3.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.2.cmml" xref="S4.SS3.p1.2.m2.1.2"><times id="S4.SS3.p1.2.m2.1.2.1.cmml" xref="S4.SS3.p1.2.m2.1.2.1"></times><ci id="S4.SS3.p1.2.m2.1.2.2.cmml" xref="S4.SS3.p1.2.m2.1.2.2">𝗌𝗂𝗓𝖾</ci><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\mathsf{size}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">sansserif_size ( italic_f )</annotation></semantics></math> be the minimum number of fan-in two AND/OR gates in a DeMorgan Boolean circuit computing <math alttext="f" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mi id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><ci id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">italic_f</annotation></semantics></math> (we assume negations appear only at the input level). We can define <math alttext="\mathsf{size}_{\vee}(f)" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mrow id="S4.SS3.p1.4.m4.1.2" xref="S4.SS3.p1.4.m4.1.2.cmml"><msub id="S4.SS3.p1.4.m4.1.2.2" xref="S4.SS3.p1.4.m4.1.2.2.cmml"><mi id="S4.SS3.p1.4.m4.1.2.2.2" xref="S4.SS3.p1.4.m4.1.2.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.4.m4.1.2.2.3" xref="S4.SS3.p1.4.m4.1.2.2.3.cmml">∨</mo></msub><mo id="S4.SS3.p1.4.m4.1.2.1" xref="S4.SS3.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.4.m4.1.2.3.2" xref="S4.SS3.p1.4.m4.1.2.cmml"><mo id="S4.SS3.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.4.m4.1.2.cmml">(</mo><mi id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml">f</mi><mo id="S4.SS3.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.2.cmml" xref="S4.SS3.p1.4.m4.1.2"><times id="S4.SS3.p1.4.m4.1.2.1.cmml" xref="S4.SS3.p1.4.m4.1.2.1"></times><apply id="S4.SS3.p1.4.m4.1.2.2.cmml" xref="S4.SS3.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.2.2.1.cmml" xref="S4.SS3.p1.4.m4.1.2.2">subscript</csymbol><ci id="S4.SS3.p1.4.m4.1.2.2.2.cmml" xref="S4.SS3.p1.4.m4.1.2.2.2">𝗌𝗂𝗓𝖾</ci><or id="S4.SS3.p1.4.m4.1.2.2.3.cmml" xref="S4.SS3.p1.4.m4.1.2.2.3"></or></apply><ci id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">\mathsf{size}_{\vee}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( italic_f )</annotation></semantics></math> and <math alttext="\mathsf{size}_{\wedge}(f)" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><mrow id="S4.SS3.p1.5.m5.1.2" xref="S4.SS3.p1.5.m5.1.2.cmml"><msub id="S4.SS3.p1.5.m5.1.2.2" xref="S4.SS3.p1.5.m5.1.2.2.cmml"><mi id="S4.SS3.p1.5.m5.1.2.2.2" xref="S4.SS3.p1.5.m5.1.2.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.5.m5.1.2.2.3" xref="S4.SS3.p1.5.m5.1.2.2.3.cmml">∧</mo></msub><mo id="S4.SS3.p1.5.m5.1.2.1" xref="S4.SS3.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.5.m5.1.2.3.2" xref="S4.SS3.p1.5.m5.1.2.cmml"><mo id="S4.SS3.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.5.m5.1.2.cmml">(</mo><mi id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml">f</mi><mo id="S4.SS3.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><apply id="S4.SS3.p1.5.m5.1.2.cmml" xref="S4.SS3.p1.5.m5.1.2"><times id="S4.SS3.p1.5.m5.1.2.1.cmml" xref="S4.SS3.p1.5.m5.1.2.1"></times><apply id="S4.SS3.p1.5.m5.1.2.2.cmml" xref="S4.SS3.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.5.m5.1.2.2.1.cmml" xref="S4.SS3.p1.5.m5.1.2.2">subscript</csymbol><ci id="S4.SS3.p1.5.m5.1.2.2.2.cmml" xref="S4.SS3.p1.5.m5.1.2.2.2">𝗌𝗂𝗓𝖾</ci><and id="S4.SS3.p1.5.m5.1.2.2.3.cmml" xref="S4.SS3.p1.5.m5.1.2.2.3"></and></apply><ci id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">\mathsf{size}_{\wedge}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( italic_f )</annotation></semantics></math> in a similar way. Using our notation, <math alttext="\mathsf{size}(f)=D(f\mid\mathcal{B}_{n})" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.2"><semantics id="S4.SS3.p1.6.m6.2a"><mrow id="S4.SS3.p1.6.m6.2.2" xref="S4.SS3.p1.6.m6.2.2.cmml"><mrow id="S4.SS3.p1.6.m6.2.2.3" xref="S4.SS3.p1.6.m6.2.2.3.cmml"><mi id="S4.SS3.p1.6.m6.2.2.3.2" xref="S4.SS3.p1.6.m6.2.2.3.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.6.m6.2.2.3.1" xref="S4.SS3.p1.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p1.6.m6.2.2.3.3.2" xref="S4.SS3.p1.6.m6.2.2.3.cmml"><mo id="S4.SS3.p1.6.m6.2.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.6.m6.2.2.3.cmml">(</mo><mi id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml">f</mi><mo id="S4.SS3.p1.6.m6.2.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.6.m6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.6.m6.2.2.2" xref="S4.SS3.p1.6.m6.2.2.2.cmml">=</mo><mrow id="S4.SS3.p1.6.m6.2.2.1" xref="S4.SS3.p1.6.m6.2.2.1.cmml"><mi id="S4.SS3.p1.6.m6.2.2.1.3" xref="S4.SS3.p1.6.m6.2.2.1.3.cmml">D</mi><mo id="S4.SS3.p1.6.m6.2.2.1.2" xref="S4.SS3.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS3.p1.6.m6.2.2.1.1.1" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.cmml"><mo id="S4.SS3.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.6.m6.2.2.1.1.1.1" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S4.SS3.p1.6.m6.2.2.1.1.1.1.2" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.2.cmml">f</mi><mo id="S4.SS3.p1.6.m6.2.2.1.1.1.1.1" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.1.cmml">∣</mo><msub id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.2" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.3" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.SS3.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.2b"><apply id="S4.SS3.p1.6.m6.2.2.cmml" xref="S4.SS3.p1.6.m6.2.2"><eq id="S4.SS3.p1.6.m6.2.2.2.cmml" xref="S4.SS3.p1.6.m6.2.2.2"></eq><apply id="S4.SS3.p1.6.m6.2.2.3.cmml" xref="S4.SS3.p1.6.m6.2.2.3"><times id="S4.SS3.p1.6.m6.2.2.3.1.cmml" xref="S4.SS3.p1.6.m6.2.2.3.1"></times><ci id="S4.SS3.p1.6.m6.2.2.3.2.cmml" xref="S4.SS3.p1.6.m6.2.2.3.2">𝗌𝗂𝗓𝖾</ci><ci id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1">𝑓</ci></apply><apply id="S4.SS3.p1.6.m6.2.2.1.cmml" xref="S4.SS3.p1.6.m6.2.2.1"><times id="S4.SS3.p1.6.m6.2.2.1.2.cmml" xref="S4.SS3.p1.6.m6.2.2.1.2"></times><ci id="S4.SS3.p1.6.m6.2.2.1.3.cmml" xref="S4.SS3.p1.6.m6.2.2.1.3">𝐷</ci><apply id="S4.SS3.p1.6.m6.2.2.1.1.1.1.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS3.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.1">conditional</csymbol><ci id="S4.SS3.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.2">𝑓</ci><apply id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.1.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.2.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.2">ℬ</ci><ci id="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.3.cmml" xref="S4.SS3.p1.6.m6.2.2.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.2c">\mathsf{size}(f)=D(f\mid\mathcal{B}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.2d">sansserif_size ( italic_f ) = italic_D ( italic_f ∣ caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\mathsf{size}_{\vee}(f)=D_{\cup}(f\mid\mathcal{B}_{n})" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.2"><semantics id="S4.SS3.p1.7.m7.2a"><mrow id="S4.SS3.p1.7.m7.2.2" xref="S4.SS3.p1.7.m7.2.2.cmml"><mrow id="S4.SS3.p1.7.m7.2.2.3" xref="S4.SS3.p1.7.m7.2.2.3.cmml"><msub id="S4.SS3.p1.7.m7.2.2.3.2" xref="S4.SS3.p1.7.m7.2.2.3.2.cmml"><mi id="S4.SS3.p1.7.m7.2.2.3.2.2" xref="S4.SS3.p1.7.m7.2.2.3.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.7.m7.2.2.3.2.3" xref="S4.SS3.p1.7.m7.2.2.3.2.3.cmml">∨</mo></msub><mo id="S4.SS3.p1.7.m7.2.2.3.1" xref="S4.SS3.p1.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p1.7.m7.2.2.3.3.2" xref="S4.SS3.p1.7.m7.2.2.3.cmml"><mo id="S4.SS3.p1.7.m7.2.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.7.m7.2.2.3.cmml">(</mo><mi id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml">f</mi><mo id="S4.SS3.p1.7.m7.2.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.7.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.7.m7.2.2.2" xref="S4.SS3.p1.7.m7.2.2.2.cmml">=</mo><mrow id="S4.SS3.p1.7.m7.2.2.1" xref="S4.SS3.p1.7.m7.2.2.1.cmml"><msub id="S4.SS3.p1.7.m7.2.2.1.3" xref="S4.SS3.p1.7.m7.2.2.1.3.cmml"><mi id="S4.SS3.p1.7.m7.2.2.1.3.2" xref="S4.SS3.p1.7.m7.2.2.1.3.2.cmml">D</mi><mo id="S4.SS3.p1.7.m7.2.2.1.3.3" xref="S4.SS3.p1.7.m7.2.2.1.3.3.cmml">∪</mo></msub><mo id="S4.SS3.p1.7.m7.2.2.1.2" xref="S4.SS3.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS3.p1.7.m7.2.2.1.1.1" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.SS3.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.7.m7.2.2.1.1.1.1" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.SS3.p1.7.m7.2.2.1.1.1.1.2" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.2.cmml">f</mi><mo id="S4.SS3.p1.7.m7.2.2.1.1.1.1.1" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.1.cmml">∣</mo><msub id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.2" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.3" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.SS3.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.2b"><apply id="S4.SS3.p1.7.m7.2.2.cmml" xref="S4.SS3.p1.7.m7.2.2"><eq id="S4.SS3.p1.7.m7.2.2.2.cmml" xref="S4.SS3.p1.7.m7.2.2.2"></eq><apply id="S4.SS3.p1.7.m7.2.2.3.cmml" xref="S4.SS3.p1.7.m7.2.2.3"><times id="S4.SS3.p1.7.m7.2.2.3.1.cmml" xref="S4.SS3.p1.7.m7.2.2.3.1"></times><apply id="S4.SS3.p1.7.m7.2.2.3.2.cmml" xref="S4.SS3.p1.7.m7.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.2.2.3.2.1.cmml" xref="S4.SS3.p1.7.m7.2.2.3.2">subscript</csymbol><ci id="S4.SS3.p1.7.m7.2.2.3.2.2.cmml" xref="S4.SS3.p1.7.m7.2.2.3.2.2">𝗌𝗂𝗓𝖾</ci><or id="S4.SS3.p1.7.m7.2.2.3.2.3.cmml" xref="S4.SS3.p1.7.m7.2.2.3.2.3"></or></apply><ci id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">𝑓</ci></apply><apply id="S4.SS3.p1.7.m7.2.2.1.cmml" xref="S4.SS3.p1.7.m7.2.2.1"><times id="S4.SS3.p1.7.m7.2.2.1.2.cmml" xref="S4.SS3.p1.7.m7.2.2.1.2"></times><apply id="S4.SS3.p1.7.m7.2.2.1.3.cmml" xref="S4.SS3.p1.7.m7.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.2.2.1.3.1.cmml" xref="S4.SS3.p1.7.m7.2.2.1.3">subscript</csymbol><ci id="S4.SS3.p1.7.m7.2.2.1.3.2.cmml" xref="S4.SS3.p1.7.m7.2.2.1.3.2">𝐷</ci><union id="S4.SS3.p1.7.m7.2.2.1.3.3.cmml" xref="S4.SS3.p1.7.m7.2.2.1.3.3"></union></apply><apply id="S4.SS3.p1.7.m7.2.2.1.1.1.1.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS3.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.1">conditional</csymbol><ci id="S4.SS3.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.2">𝑓</ci><apply id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.1.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.2.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.2">ℬ</ci><ci id="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.3.cmml" xref="S4.SS3.p1.7.m7.2.2.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.2c">\mathsf{size}_{\vee}(f)=D_{\cup}(f\mid\mathcal{B}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.2d">sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( italic_f ) = italic_D start_POSTSUBSCRIPT ∪ end_POSTSUBSCRIPT ( italic_f ∣ caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="\mathsf{size}_{\wedge}(f)=D_{\cap}(f\mid\mathcal{B}_{n})" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.2"><semantics id="S4.SS3.p1.8.m8.2a"><mrow id="S4.SS3.p1.8.m8.2.2" xref="S4.SS3.p1.8.m8.2.2.cmml"><mrow id="S4.SS3.p1.8.m8.2.2.3" xref="S4.SS3.p1.8.m8.2.2.3.cmml"><msub id="S4.SS3.p1.8.m8.2.2.3.2" xref="S4.SS3.p1.8.m8.2.2.3.2.cmml"><mi id="S4.SS3.p1.8.m8.2.2.3.2.2" xref="S4.SS3.p1.8.m8.2.2.3.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p1.8.m8.2.2.3.2.3" xref="S4.SS3.p1.8.m8.2.2.3.2.3.cmml">∧</mo></msub><mo id="S4.SS3.p1.8.m8.2.2.3.1" xref="S4.SS3.p1.8.m8.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p1.8.m8.2.2.3.3.2" xref="S4.SS3.p1.8.m8.2.2.3.cmml"><mo id="S4.SS3.p1.8.m8.2.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.8.m8.2.2.3.cmml">(</mo><mi id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml">f</mi><mo id="S4.SS3.p1.8.m8.2.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.8.m8.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.8.m8.2.2.2" xref="S4.SS3.p1.8.m8.2.2.2.cmml">=</mo><mrow id="S4.SS3.p1.8.m8.2.2.1" xref="S4.SS3.p1.8.m8.2.2.1.cmml"><msub id="S4.SS3.p1.8.m8.2.2.1.3" xref="S4.SS3.p1.8.m8.2.2.1.3.cmml"><mi id="S4.SS3.p1.8.m8.2.2.1.3.2" xref="S4.SS3.p1.8.m8.2.2.1.3.2.cmml">D</mi><mo id="S4.SS3.p1.8.m8.2.2.1.3.3" xref="S4.SS3.p1.8.m8.2.2.1.3.3.cmml">∩</mo></msub><mo id="S4.SS3.p1.8.m8.2.2.1.2" xref="S4.SS3.p1.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS3.p1.8.m8.2.2.1.1.1" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.cmml"><mo id="S4.SS3.p1.8.m8.2.2.1.1.1.2" stretchy="false" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.8.m8.2.2.1.1.1.1" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.cmml"><mi id="S4.SS3.p1.8.m8.2.2.1.1.1.1.2" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.2.cmml">f</mi><mo id="S4.SS3.p1.8.m8.2.2.1.1.1.1.1" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.1.cmml">∣</mo><msub id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.2" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.3" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.SS3.p1.8.m8.2.2.1.1.1.3" stretchy="false" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.2b"><apply id="S4.SS3.p1.8.m8.2.2.cmml" xref="S4.SS3.p1.8.m8.2.2"><eq id="S4.SS3.p1.8.m8.2.2.2.cmml" xref="S4.SS3.p1.8.m8.2.2.2"></eq><apply id="S4.SS3.p1.8.m8.2.2.3.cmml" xref="S4.SS3.p1.8.m8.2.2.3"><times id="S4.SS3.p1.8.m8.2.2.3.1.cmml" xref="S4.SS3.p1.8.m8.2.2.3.1"></times><apply id="S4.SS3.p1.8.m8.2.2.3.2.cmml" xref="S4.SS3.p1.8.m8.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m8.2.2.3.2.1.cmml" xref="S4.SS3.p1.8.m8.2.2.3.2">subscript</csymbol><ci id="S4.SS3.p1.8.m8.2.2.3.2.2.cmml" xref="S4.SS3.p1.8.m8.2.2.3.2.2">𝗌𝗂𝗓𝖾</ci><and id="S4.SS3.p1.8.m8.2.2.3.2.3.cmml" xref="S4.SS3.p1.8.m8.2.2.3.2.3"></and></apply><ci id="S4.SS3.p1.8.m8.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1">𝑓</ci></apply><apply id="S4.SS3.p1.8.m8.2.2.1.cmml" xref="S4.SS3.p1.8.m8.2.2.1"><times id="S4.SS3.p1.8.m8.2.2.1.2.cmml" xref="S4.SS3.p1.8.m8.2.2.1.2"></times><apply id="S4.SS3.p1.8.m8.2.2.1.3.cmml" xref="S4.SS3.p1.8.m8.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m8.2.2.1.3.1.cmml" xref="S4.SS3.p1.8.m8.2.2.1.3">subscript</csymbol><ci id="S4.SS3.p1.8.m8.2.2.1.3.2.cmml" xref="S4.SS3.p1.8.m8.2.2.1.3.2">𝐷</ci><intersect id="S4.SS3.p1.8.m8.2.2.1.3.3.cmml" xref="S4.SS3.p1.8.m8.2.2.1.3.3"></intersect></apply><apply id="S4.SS3.p1.8.m8.2.2.1.1.1.1.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS3.p1.8.m8.2.2.1.1.1.1.1.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.1">conditional</csymbol><ci id="S4.SS3.p1.8.m8.2.2.1.1.1.1.2.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.2">𝑓</ci><apply id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.1.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.2.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.2">ℬ</ci><ci id="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.3.cmml" xref="S4.SS3.p1.8.m8.2.2.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.2c">\mathsf{size}_{\wedge}(f)=D_{\cap}(f\mid\mathcal{B}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.2d">sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( italic_f ) = italic_D start_POSTSUBSCRIPT ∩ end_POSTSUBSCRIPT ( italic_f ∣ caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.18">We also define <math alttext="\mathsf{conondet}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\mathsf{conondet}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">sansserif_conondet</annotation></semantics></math>-<math alttext="\mathsf{size}_{\wedge}(f)" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mrow id="S4.SS3.p2.2.m2.1.2" xref="S4.SS3.p2.2.m2.1.2.cmml"><msub id="S4.SS3.p2.2.m2.1.2.2" xref="S4.SS3.p2.2.m2.1.2.2.cmml"><mi id="S4.SS3.p2.2.m2.1.2.2.2" xref="S4.SS3.p2.2.m2.1.2.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p2.2.m2.1.2.2.3" xref="S4.SS3.p2.2.m2.1.2.2.3.cmml">∧</mo></msub><mo id="S4.SS3.p2.2.m2.1.2.1" xref="S4.SS3.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.2.m2.1.2.3.2" xref="S4.SS3.p2.2.m2.1.2.cmml"><mo id="S4.SS3.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS3.p2.2.m2.1.2.cmml">(</mo><mi id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml">f</mi><mo id="S4.SS3.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS3.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.2.cmml" xref="S4.SS3.p2.2.m2.1.2"><times id="S4.SS3.p2.2.m2.1.2.1.cmml" xref="S4.SS3.p2.2.m2.1.2.1"></times><apply id="S4.SS3.p2.2.m2.1.2.2.cmml" xref="S4.SS3.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.2.2.1.cmml" xref="S4.SS3.p2.2.m2.1.2.2">subscript</csymbol><ci id="S4.SS3.p2.2.m2.1.2.2.2.cmml" xref="S4.SS3.p2.2.m2.1.2.2.2">𝗌𝗂𝗓𝖾</ci><and id="S4.SS3.p2.2.m2.1.2.2.3.cmml" xref="S4.SS3.p2.2.m2.1.2.2.3"></and></apply><ci id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">\mathsf{size}_{\wedge}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( italic_f )</annotation></semantics></math> to be the minimum number of <math alttext="\wedge" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><mo id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><and id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">∧</annotation></semantics></math>-gates in a circuit <math alttext="D(x,y)" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.2"><semantics id="S4.SS3.p2.4.m4.2a"><mrow id="S4.SS3.p2.4.m4.2.3" xref="S4.SS3.p2.4.m4.2.3.cmml"><mi id="S4.SS3.p2.4.m4.2.3.2" xref="S4.SS3.p2.4.m4.2.3.2.cmml">D</mi><mo id="S4.SS3.p2.4.m4.2.3.1" xref="S4.SS3.p2.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p2.4.m4.2.3.3.2" xref="S4.SS3.p2.4.m4.2.3.3.1.cmml"><mo id="S4.SS3.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S4.SS3.p2.4.m4.2.3.3.1.cmml">(</mo><mi id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">x</mi><mo id="S4.SS3.p2.4.m4.2.3.3.2.2" xref="S4.SS3.p2.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.SS3.p2.4.m4.2.2" xref="S4.SS3.p2.4.m4.2.2.cmml">y</mi><mo id="S4.SS3.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S4.SS3.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.2b"><apply id="S4.SS3.p2.4.m4.2.3.cmml" xref="S4.SS3.p2.4.m4.2.3"><times id="S4.SS3.p2.4.m4.2.3.1.cmml" xref="S4.SS3.p2.4.m4.2.3.1"></times><ci id="S4.SS3.p2.4.m4.2.3.2.cmml" xref="S4.SS3.p2.4.m4.2.3.2">𝐷</ci><interval closure="open" id="S4.SS3.p2.4.m4.2.3.3.1.cmml" xref="S4.SS3.p2.4.m4.2.3.3.2"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝑥</ci><ci id="S4.SS3.p2.4.m4.2.2.cmml" xref="S4.SS3.p2.4.m4.2.2">𝑦</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.2c">D(x,y)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.2d">italic_D ( italic_x , italic_y )</annotation></semantics></math> such that <math alttext="f(x)=1" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m5.1"><semantics id="S4.SS3.p2.5.m5.1a"><mrow id="S4.SS3.p2.5.m5.1.2" xref="S4.SS3.p2.5.m5.1.2.cmml"><mrow id="S4.SS3.p2.5.m5.1.2.2" xref="S4.SS3.p2.5.m5.1.2.2.cmml"><mi id="S4.SS3.p2.5.m5.1.2.2.2" xref="S4.SS3.p2.5.m5.1.2.2.2.cmml">f</mi><mo id="S4.SS3.p2.5.m5.1.2.2.1" xref="S4.SS3.p2.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.5.m5.1.2.2.3.2" xref="S4.SS3.p2.5.m5.1.2.2.cmml"><mo id="S4.SS3.p2.5.m5.1.2.2.3.2.1" stretchy="false" xref="S4.SS3.p2.5.m5.1.2.2.cmml">(</mo><mi id="S4.SS3.p2.5.m5.1.1" xref="S4.SS3.p2.5.m5.1.1.cmml">x</mi><mo id="S4.SS3.p2.5.m5.1.2.2.3.2.2" stretchy="false" xref="S4.SS3.p2.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.5.m5.1.2.1" xref="S4.SS3.p2.5.m5.1.2.1.cmml">=</mo><mn id="S4.SS3.p2.5.m5.1.2.3" xref="S4.SS3.p2.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m5.1b"><apply id="S4.SS3.p2.5.m5.1.2.cmml" xref="S4.SS3.p2.5.m5.1.2"><eq id="S4.SS3.p2.5.m5.1.2.1.cmml" xref="S4.SS3.p2.5.m5.1.2.1"></eq><apply id="S4.SS3.p2.5.m5.1.2.2.cmml" xref="S4.SS3.p2.5.m5.1.2.2"><times id="S4.SS3.p2.5.m5.1.2.2.1.cmml" xref="S4.SS3.p2.5.m5.1.2.2.1"></times><ci id="S4.SS3.p2.5.m5.1.2.2.2.cmml" xref="S4.SS3.p2.5.m5.1.2.2.2">𝑓</ci><ci id="S4.SS3.p2.5.m5.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1">𝑥</ci></apply><cn id="S4.SS3.p2.5.m5.1.2.3.cmml" type="integer" xref="S4.SS3.p2.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m5.1c">f(x)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m5.1d">italic_f ( italic_x ) = 1</annotation></semantics></math> if and only if for all <math alttext="y" class="ltx_Math" display="inline" id="S4.SS3.p2.6.m6.1"><semantics id="S4.SS3.p2.6.m6.1a"><mi id="S4.SS3.p2.6.m6.1.1" xref="S4.SS3.p2.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.6.m6.1b"><ci id="S4.SS3.p2.6.m6.1.1.cmml" xref="S4.SS3.p2.6.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m6.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m6.1d">italic_y</annotation></semantics></math> we have <math alttext="D(x,y)=1" class="ltx_Math" display="inline" id="S4.SS3.p2.7.m7.2"><semantics id="S4.SS3.p2.7.m7.2a"><mrow id="S4.SS3.p2.7.m7.2.3" xref="S4.SS3.p2.7.m7.2.3.cmml"><mrow id="S4.SS3.p2.7.m7.2.3.2" xref="S4.SS3.p2.7.m7.2.3.2.cmml"><mi id="S4.SS3.p2.7.m7.2.3.2.2" xref="S4.SS3.p2.7.m7.2.3.2.2.cmml">D</mi><mo id="S4.SS3.p2.7.m7.2.3.2.1" xref="S4.SS3.p2.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.7.m7.2.3.2.3.2" xref="S4.SS3.p2.7.m7.2.3.2.3.1.cmml"><mo id="S4.SS3.p2.7.m7.2.3.2.3.2.1" stretchy="false" xref="S4.SS3.p2.7.m7.2.3.2.3.1.cmml">(</mo><mi id="S4.SS3.p2.7.m7.1.1" xref="S4.SS3.p2.7.m7.1.1.cmml">x</mi><mo id="S4.SS3.p2.7.m7.2.3.2.3.2.2" xref="S4.SS3.p2.7.m7.2.3.2.3.1.cmml">,</mo><mi id="S4.SS3.p2.7.m7.2.2" xref="S4.SS3.p2.7.m7.2.2.cmml">y</mi><mo id="S4.SS3.p2.7.m7.2.3.2.3.2.3" stretchy="false" xref="S4.SS3.p2.7.m7.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.7.m7.2.3.1" xref="S4.SS3.p2.7.m7.2.3.1.cmml">=</mo><mn id="S4.SS3.p2.7.m7.2.3.3" xref="S4.SS3.p2.7.m7.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.7.m7.2b"><apply id="S4.SS3.p2.7.m7.2.3.cmml" xref="S4.SS3.p2.7.m7.2.3"><eq id="S4.SS3.p2.7.m7.2.3.1.cmml" xref="S4.SS3.p2.7.m7.2.3.1"></eq><apply id="S4.SS3.p2.7.m7.2.3.2.cmml" xref="S4.SS3.p2.7.m7.2.3.2"><times id="S4.SS3.p2.7.m7.2.3.2.1.cmml" xref="S4.SS3.p2.7.m7.2.3.2.1"></times><ci id="S4.SS3.p2.7.m7.2.3.2.2.cmml" xref="S4.SS3.p2.7.m7.2.3.2.2">𝐷</ci><interval closure="open" id="S4.SS3.p2.7.m7.2.3.2.3.1.cmml" xref="S4.SS3.p2.7.m7.2.3.2.3.2"><ci id="S4.SS3.p2.7.m7.1.1.cmml" xref="S4.SS3.p2.7.m7.1.1">𝑥</ci><ci id="S4.SS3.p2.7.m7.2.2.cmml" xref="S4.SS3.p2.7.m7.2.2">𝑦</ci></interval></apply><cn id="S4.SS3.p2.7.m7.2.3.3.cmml" type="integer" xref="S4.SS3.p2.7.m7.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.7.m7.2c">D(x,y)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.7.m7.2d">italic_D ( italic_x , italic_y ) = 1</annotation></semantics></math>. Similarly, <math alttext="\mathsf{nondet}" class="ltx_Math" display="inline" id="S4.SS3.p2.8.m8.1"><semantics id="S4.SS3.p2.8.m8.1a"><mi id="S4.SS3.p2.8.m8.1.1" xref="S4.SS3.p2.8.m8.1.1.cmml">𝗇𝗈𝗇𝖽𝖾𝗍</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.8.m8.1b"><ci id="S4.SS3.p2.8.m8.1.1.cmml" xref="S4.SS3.p2.8.m8.1.1">𝗇𝗈𝗇𝖽𝖾𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.8.m8.1c">\mathsf{nondet}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.8.m8.1d">sansserif_nondet</annotation></semantics></math>-<math alttext="\mathsf{size}_{\vee}(g)" class="ltx_Math" display="inline" id="S4.SS3.p2.9.m9.1"><semantics id="S4.SS3.p2.9.m9.1a"><mrow id="S4.SS3.p2.9.m9.1.2" xref="S4.SS3.p2.9.m9.1.2.cmml"><msub id="S4.SS3.p2.9.m9.1.2.2" xref="S4.SS3.p2.9.m9.1.2.2.cmml"><mi id="S4.SS3.p2.9.m9.1.2.2.2" xref="S4.SS3.p2.9.m9.1.2.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p2.9.m9.1.2.2.3" xref="S4.SS3.p2.9.m9.1.2.2.3.cmml">∨</mo></msub><mo id="S4.SS3.p2.9.m9.1.2.1" xref="S4.SS3.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.9.m9.1.2.3.2" xref="S4.SS3.p2.9.m9.1.2.cmml"><mo id="S4.SS3.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS3.p2.9.m9.1.2.cmml">(</mo><mi id="S4.SS3.p2.9.m9.1.1" xref="S4.SS3.p2.9.m9.1.1.cmml">g</mi><mo id="S4.SS3.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS3.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.9.m9.1b"><apply id="S4.SS3.p2.9.m9.1.2.cmml" xref="S4.SS3.p2.9.m9.1.2"><times id="S4.SS3.p2.9.m9.1.2.1.cmml" xref="S4.SS3.p2.9.m9.1.2.1"></times><apply id="S4.SS3.p2.9.m9.1.2.2.cmml" xref="S4.SS3.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.9.m9.1.2.2.1.cmml" xref="S4.SS3.p2.9.m9.1.2.2">subscript</csymbol><ci id="S4.SS3.p2.9.m9.1.2.2.2.cmml" xref="S4.SS3.p2.9.m9.1.2.2.2">𝗌𝗂𝗓𝖾</ci><or id="S4.SS3.p2.9.m9.1.2.2.3.cmml" xref="S4.SS3.p2.9.m9.1.2.2.3"></or></apply><ci id="S4.SS3.p2.9.m9.1.1.cmml" xref="S4.SS3.p2.9.m9.1.1">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.9.m9.1c">\mathsf{size}_{\vee}(g)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.9.m9.1d">sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( italic_g )</annotation></semantics></math> is the minimum number of <math alttext="\vee" class="ltx_Math" display="inline" id="S4.SS3.p2.10.m10.1"><semantics id="S4.SS3.p2.10.m10.1a"><mo id="S4.SS3.p2.10.m10.1.1" xref="S4.SS3.p2.10.m10.1.1.cmml">∨</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.10.m10.1b"><or id="S4.SS3.p2.10.m10.1.1.cmml" xref="S4.SS3.p2.10.m10.1.1"></or></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.10.m10.1c">\vee</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.10.m10.1d">∨</annotation></semantics></math>-gates in a circuit <math alttext="C(x,y)" class="ltx_Math" display="inline" id="S4.SS3.p2.11.m11.2"><semantics id="S4.SS3.p2.11.m11.2a"><mrow id="S4.SS3.p2.11.m11.2.3" xref="S4.SS3.p2.11.m11.2.3.cmml"><mi id="S4.SS3.p2.11.m11.2.3.2" xref="S4.SS3.p2.11.m11.2.3.2.cmml">C</mi><mo id="S4.SS3.p2.11.m11.2.3.1" xref="S4.SS3.p2.11.m11.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p2.11.m11.2.3.3.2" xref="S4.SS3.p2.11.m11.2.3.3.1.cmml"><mo id="S4.SS3.p2.11.m11.2.3.3.2.1" stretchy="false" xref="S4.SS3.p2.11.m11.2.3.3.1.cmml">(</mo><mi id="S4.SS3.p2.11.m11.1.1" xref="S4.SS3.p2.11.m11.1.1.cmml">x</mi><mo id="S4.SS3.p2.11.m11.2.3.3.2.2" xref="S4.SS3.p2.11.m11.2.3.3.1.cmml">,</mo><mi id="S4.SS3.p2.11.m11.2.2" xref="S4.SS3.p2.11.m11.2.2.cmml">y</mi><mo id="S4.SS3.p2.11.m11.2.3.3.2.3" stretchy="false" xref="S4.SS3.p2.11.m11.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.11.m11.2b"><apply id="S4.SS3.p2.11.m11.2.3.cmml" xref="S4.SS3.p2.11.m11.2.3"><times id="S4.SS3.p2.11.m11.2.3.1.cmml" xref="S4.SS3.p2.11.m11.2.3.1"></times><ci id="S4.SS3.p2.11.m11.2.3.2.cmml" xref="S4.SS3.p2.11.m11.2.3.2">𝐶</ci><interval closure="open" id="S4.SS3.p2.11.m11.2.3.3.1.cmml" xref="S4.SS3.p2.11.m11.2.3.3.2"><ci id="S4.SS3.p2.11.m11.1.1.cmml" xref="S4.SS3.p2.11.m11.1.1">𝑥</ci><ci id="S4.SS3.p2.11.m11.2.2.cmml" xref="S4.SS3.p2.11.m11.2.2">𝑦</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.11.m11.2c">C(x,y)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.11.m11.2d">italic_C ( italic_x , italic_y )</annotation></semantics></math> such that <math alttext="g(x)=1" class="ltx_Math" display="inline" id="S4.SS3.p2.12.m12.1"><semantics id="S4.SS3.p2.12.m12.1a"><mrow id="S4.SS3.p2.12.m12.1.2" xref="S4.SS3.p2.12.m12.1.2.cmml"><mrow id="S4.SS3.p2.12.m12.1.2.2" xref="S4.SS3.p2.12.m12.1.2.2.cmml"><mi id="S4.SS3.p2.12.m12.1.2.2.2" xref="S4.SS3.p2.12.m12.1.2.2.2.cmml">g</mi><mo id="S4.SS3.p2.12.m12.1.2.2.1" xref="S4.SS3.p2.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.12.m12.1.2.2.3.2" xref="S4.SS3.p2.12.m12.1.2.2.cmml"><mo id="S4.SS3.p2.12.m12.1.2.2.3.2.1" stretchy="false" xref="S4.SS3.p2.12.m12.1.2.2.cmml">(</mo><mi id="S4.SS3.p2.12.m12.1.1" xref="S4.SS3.p2.12.m12.1.1.cmml">x</mi><mo id="S4.SS3.p2.12.m12.1.2.2.3.2.2" stretchy="false" xref="S4.SS3.p2.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.12.m12.1.2.1" xref="S4.SS3.p2.12.m12.1.2.1.cmml">=</mo><mn id="S4.SS3.p2.12.m12.1.2.3" xref="S4.SS3.p2.12.m12.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.12.m12.1b"><apply id="S4.SS3.p2.12.m12.1.2.cmml" xref="S4.SS3.p2.12.m12.1.2"><eq id="S4.SS3.p2.12.m12.1.2.1.cmml" xref="S4.SS3.p2.12.m12.1.2.1"></eq><apply id="S4.SS3.p2.12.m12.1.2.2.cmml" xref="S4.SS3.p2.12.m12.1.2.2"><times id="S4.SS3.p2.12.m12.1.2.2.1.cmml" xref="S4.SS3.p2.12.m12.1.2.2.1"></times><ci id="S4.SS3.p2.12.m12.1.2.2.2.cmml" xref="S4.SS3.p2.12.m12.1.2.2.2">𝑔</ci><ci id="S4.SS3.p2.12.m12.1.1.cmml" xref="S4.SS3.p2.12.m12.1.1">𝑥</ci></apply><cn id="S4.SS3.p2.12.m12.1.2.3.cmml" type="integer" xref="S4.SS3.p2.12.m12.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.12.m12.1c">g(x)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.12.m12.1d">italic_g ( italic_x ) = 1</annotation></semantics></math> if and only if there exists <math alttext="y" class="ltx_Math" display="inline" id="S4.SS3.p2.13.m13.1"><semantics id="S4.SS3.p2.13.m13.1a"><mi id="S4.SS3.p2.13.m13.1.1" xref="S4.SS3.p2.13.m13.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.13.m13.1b"><ci id="S4.SS3.p2.13.m13.1.1.cmml" xref="S4.SS3.p2.13.m13.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.13.m13.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.13.m13.1d">italic_y</annotation></semantics></math> such that <math alttext="C(x,y)=1" class="ltx_Math" display="inline" id="S4.SS3.p2.14.m14.2"><semantics id="S4.SS3.p2.14.m14.2a"><mrow id="S4.SS3.p2.14.m14.2.3" xref="S4.SS3.p2.14.m14.2.3.cmml"><mrow id="S4.SS3.p2.14.m14.2.3.2" xref="S4.SS3.p2.14.m14.2.3.2.cmml"><mi id="S4.SS3.p2.14.m14.2.3.2.2" xref="S4.SS3.p2.14.m14.2.3.2.2.cmml">C</mi><mo id="S4.SS3.p2.14.m14.2.3.2.1" xref="S4.SS3.p2.14.m14.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.14.m14.2.3.2.3.2" xref="S4.SS3.p2.14.m14.2.3.2.3.1.cmml"><mo id="S4.SS3.p2.14.m14.2.3.2.3.2.1" stretchy="false" xref="S4.SS3.p2.14.m14.2.3.2.3.1.cmml">(</mo><mi id="S4.SS3.p2.14.m14.1.1" xref="S4.SS3.p2.14.m14.1.1.cmml">x</mi><mo id="S4.SS3.p2.14.m14.2.3.2.3.2.2" xref="S4.SS3.p2.14.m14.2.3.2.3.1.cmml">,</mo><mi id="S4.SS3.p2.14.m14.2.2" xref="S4.SS3.p2.14.m14.2.2.cmml">y</mi><mo id="S4.SS3.p2.14.m14.2.3.2.3.2.3" stretchy="false" xref="S4.SS3.p2.14.m14.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.14.m14.2.3.1" xref="S4.SS3.p2.14.m14.2.3.1.cmml">=</mo><mn id="S4.SS3.p2.14.m14.2.3.3" xref="S4.SS3.p2.14.m14.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.14.m14.2b"><apply id="S4.SS3.p2.14.m14.2.3.cmml" xref="S4.SS3.p2.14.m14.2.3"><eq id="S4.SS3.p2.14.m14.2.3.1.cmml" xref="S4.SS3.p2.14.m14.2.3.1"></eq><apply id="S4.SS3.p2.14.m14.2.3.2.cmml" xref="S4.SS3.p2.14.m14.2.3.2"><times id="S4.SS3.p2.14.m14.2.3.2.1.cmml" xref="S4.SS3.p2.14.m14.2.3.2.1"></times><ci id="S4.SS3.p2.14.m14.2.3.2.2.cmml" xref="S4.SS3.p2.14.m14.2.3.2.2">𝐶</ci><interval closure="open" id="S4.SS3.p2.14.m14.2.3.2.3.1.cmml" xref="S4.SS3.p2.14.m14.2.3.2.3.2"><ci id="S4.SS3.p2.14.m14.1.1.cmml" xref="S4.SS3.p2.14.m14.1.1">𝑥</ci><ci id="S4.SS3.p2.14.m14.2.2.cmml" xref="S4.SS3.p2.14.m14.2.2">𝑦</ci></interval></apply><cn id="S4.SS3.p2.14.m14.2.3.3.cmml" type="integer" xref="S4.SS3.p2.14.m14.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.14.m14.2c">C(x,y)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.14.m14.2d">italic_C ( italic_x , italic_y ) = 1</annotation></semantics></math>. Observe that for every Boolean function <math alttext="h" class="ltx_Math" display="inline" id="S4.SS3.p2.15.m15.1"><semantics id="S4.SS3.p2.15.m15.1a"><mi id="S4.SS3.p2.15.m15.1.1" xref="S4.SS3.p2.15.m15.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.15.m15.1b"><ci id="S4.SS3.p2.15.m15.1.1.cmml" xref="S4.SS3.p2.15.m15.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.15.m15.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.15.m15.1d">italic_h</annotation></semantics></math>, <math alttext="\mathsf{conondet}" class="ltx_Math" display="inline" id="S4.SS3.p2.16.m16.1"><semantics id="S4.SS3.p2.16.m16.1a"><mi id="S4.SS3.p2.16.m16.1.1" xref="S4.SS3.p2.16.m16.1.1.cmml">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.16.m16.1b"><ci id="S4.SS3.p2.16.m16.1.1.cmml" xref="S4.SS3.p2.16.m16.1.1">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.16.m16.1c">\mathsf{conondet}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.16.m16.1d">sansserif_conondet</annotation></semantics></math>-<math alttext="\mathsf{size}_{\wedge}(h)=\mathsf{nondet}" class="ltx_Math" display="inline" id="S4.SS3.p2.17.m17.1"><semantics id="S4.SS3.p2.17.m17.1a"><mrow id="S4.SS3.p2.17.m17.1.2" xref="S4.SS3.p2.17.m17.1.2.cmml"><mrow id="S4.SS3.p2.17.m17.1.2.2" xref="S4.SS3.p2.17.m17.1.2.2.cmml"><msub id="S4.SS3.p2.17.m17.1.2.2.2" xref="S4.SS3.p2.17.m17.1.2.2.2.cmml"><mi id="S4.SS3.p2.17.m17.1.2.2.2.2" xref="S4.SS3.p2.17.m17.1.2.2.2.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p2.17.m17.1.2.2.2.3" xref="S4.SS3.p2.17.m17.1.2.2.2.3.cmml">∧</mo></msub><mo id="S4.SS3.p2.17.m17.1.2.2.1" xref="S4.SS3.p2.17.m17.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p2.17.m17.1.2.2.3.2" xref="S4.SS3.p2.17.m17.1.2.2.cmml"><mo id="S4.SS3.p2.17.m17.1.2.2.3.2.1" stretchy="false" xref="S4.SS3.p2.17.m17.1.2.2.cmml">(</mo><mi id="S4.SS3.p2.17.m17.1.1" xref="S4.SS3.p2.17.m17.1.1.cmml">h</mi><mo id="S4.SS3.p2.17.m17.1.2.2.3.2.2" stretchy="false" xref="S4.SS3.p2.17.m17.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.17.m17.1.2.1" xref="S4.SS3.p2.17.m17.1.2.1.cmml">=</mo><mi id="S4.SS3.p2.17.m17.1.2.3" xref="S4.SS3.p2.17.m17.1.2.3.cmml">𝗇𝗈𝗇𝖽𝖾𝗍</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.17.m17.1b"><apply id="S4.SS3.p2.17.m17.1.2.cmml" xref="S4.SS3.p2.17.m17.1.2"><eq id="S4.SS3.p2.17.m17.1.2.1.cmml" xref="S4.SS3.p2.17.m17.1.2.1"></eq><apply id="S4.SS3.p2.17.m17.1.2.2.cmml" xref="S4.SS3.p2.17.m17.1.2.2"><times id="S4.SS3.p2.17.m17.1.2.2.1.cmml" xref="S4.SS3.p2.17.m17.1.2.2.1"></times><apply id="S4.SS3.p2.17.m17.1.2.2.2.cmml" xref="S4.SS3.p2.17.m17.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.17.m17.1.2.2.2.1.cmml" xref="S4.SS3.p2.17.m17.1.2.2.2">subscript</csymbol><ci id="S4.SS3.p2.17.m17.1.2.2.2.2.cmml" xref="S4.SS3.p2.17.m17.1.2.2.2.2">𝗌𝗂𝗓𝖾</ci><and id="S4.SS3.p2.17.m17.1.2.2.2.3.cmml" xref="S4.SS3.p2.17.m17.1.2.2.2.3"></and></apply><ci id="S4.SS3.p2.17.m17.1.1.cmml" xref="S4.SS3.p2.17.m17.1.1">ℎ</ci></apply><ci id="S4.SS3.p2.17.m17.1.2.3.cmml" xref="S4.SS3.p2.17.m17.1.2.3">𝗇𝗈𝗇𝖽𝖾𝗍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.17.m17.1c">\mathsf{size}_{\wedge}(h)=\mathsf{nondet}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.17.m17.1d">sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( italic_h ) = sansserif_nondet</annotation></semantics></math>-<math alttext="\mathsf{size}_{\vee}(\neg h)" class="ltx_Math" display="inline" id="S4.SS3.p2.18.m18.1"><semantics id="S4.SS3.p2.18.m18.1a"><mrow id="S4.SS3.p2.18.m18.1.1" xref="S4.SS3.p2.18.m18.1.1.cmml"><msub id="S4.SS3.p2.18.m18.1.1.3" xref="S4.SS3.p2.18.m18.1.1.3.cmml"><mi id="S4.SS3.p2.18.m18.1.1.3.2" xref="S4.SS3.p2.18.m18.1.1.3.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.SS3.p2.18.m18.1.1.3.3" xref="S4.SS3.p2.18.m18.1.1.3.3.cmml">∨</mo></msub><mo id="S4.SS3.p2.18.m18.1.1.2" xref="S4.SS3.p2.18.m18.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.p2.18.m18.1.1.1.1" xref="S4.SS3.p2.18.m18.1.1.1.1.1.cmml"><mo id="S4.SS3.p2.18.m18.1.1.1.1.2" stretchy="false" xref="S4.SS3.p2.18.m18.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p2.18.m18.1.1.1.1.1" xref="S4.SS3.p2.18.m18.1.1.1.1.1.cmml"><mo id="S4.SS3.p2.18.m18.1.1.1.1.1.1" rspace="0.167em" xref="S4.SS3.p2.18.m18.1.1.1.1.1.1.cmml">¬</mo><mi id="S4.SS3.p2.18.m18.1.1.1.1.1.2" xref="S4.SS3.p2.18.m18.1.1.1.1.1.2.cmml">h</mi></mrow><mo id="S4.SS3.p2.18.m18.1.1.1.1.3" stretchy="false" xref="S4.SS3.p2.18.m18.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.18.m18.1b"><apply id="S4.SS3.p2.18.m18.1.1.cmml" xref="S4.SS3.p2.18.m18.1.1"><times id="S4.SS3.p2.18.m18.1.1.2.cmml" xref="S4.SS3.p2.18.m18.1.1.2"></times><apply id="S4.SS3.p2.18.m18.1.1.3.cmml" xref="S4.SS3.p2.18.m18.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.18.m18.1.1.3.1.cmml" xref="S4.SS3.p2.18.m18.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.18.m18.1.1.3.2.cmml" xref="S4.SS3.p2.18.m18.1.1.3.2">𝗌𝗂𝗓𝖾</ci><or id="S4.SS3.p2.18.m18.1.1.3.3.cmml" xref="S4.SS3.p2.18.m18.1.1.3.3"></or></apply><apply id="S4.SS3.p2.18.m18.1.1.1.1.1.cmml" xref="S4.SS3.p2.18.m18.1.1.1.1"><not id="S4.SS3.p2.18.m18.1.1.1.1.1.1.cmml" xref="S4.SS3.p2.18.m18.1.1.1.1.1.1"></not><ci id="S4.SS3.p2.18.m18.1.1.1.1.1.2.cmml" xref="S4.SS3.p2.18.m18.1.1.1.1.1.2">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.18.m18.1c">\mathsf{size}_{\vee}(\neg h)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.18.m18.1d">sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( ¬ italic_h )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.1">Observe that the definition of nondeterministic complexity for Boolean functions relies on Boolean circuits extended with extra input variables. It is not entirely clear how to introduce a natural similar definition in the context of graph complexity, i.e, a nondeterministic version of <math alttext="D(G\mid\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.3"><semantics id="S4.SS3.p3.1.m1.3a"><mrow id="S4.SS3.p3.1.m1.3.3" xref="S4.SS3.p3.1.m1.3.3.cmml"><mi id="S4.SS3.p3.1.m1.3.3.3" xref="S4.SS3.p3.1.m1.3.3.3.cmml">D</mi><mo id="S4.SS3.p3.1.m1.3.3.2" xref="S4.SS3.p3.1.m1.3.3.2.cmml">⁢</mo><mrow id="S4.SS3.p3.1.m1.3.3.1.1" xref="S4.SS3.p3.1.m1.3.3.1.1.1.cmml"><mo id="S4.SS3.p3.1.m1.3.3.1.1.2" stretchy="false" xref="S4.SS3.p3.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS3.p3.1.m1.3.3.1.1.1" xref="S4.SS3.p3.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS3.p3.1.m1.3.3.1.1.1.2" xref="S4.SS3.p3.1.m1.3.3.1.1.1.2.cmml">G</mi><mo id="S4.SS3.p3.1.m1.3.3.1.1.1.1" xref="S4.SS3.p3.1.m1.3.3.1.1.1.1.cmml">∣</mo><msub id="S4.SS3.p3.1.m1.3.3.1.1.1.3" xref="S4.SS3.p3.1.m1.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p3.1.m1.3.3.1.1.1.3.2" xref="S4.SS3.p3.1.m1.3.3.1.1.1.3.2.cmml">𝒢</mi><mrow id="S4.SS3.p3.1.m1.2.2.2.4" xref="S4.SS3.p3.1.m1.2.2.2.3.cmml"><mi id="S4.SS3.p3.1.m1.1.1.1.1" xref="S4.SS3.p3.1.m1.1.1.1.1.cmml">N</mi><mo id="S4.SS3.p3.1.m1.2.2.2.4.1" xref="S4.SS3.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS3.p3.1.m1.2.2.2.2" xref="S4.SS3.p3.1.m1.2.2.2.2.cmml">N</mi></mrow></msub></mrow><mo id="S4.SS3.p3.1.m1.3.3.1.1.3" stretchy="false" xref="S4.SS3.p3.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.3b"><apply id="S4.SS3.p3.1.m1.3.3.cmml" xref="S4.SS3.p3.1.m1.3.3"><times id="S4.SS3.p3.1.m1.3.3.2.cmml" xref="S4.SS3.p3.1.m1.3.3.2"></times><ci id="S4.SS3.p3.1.m1.3.3.3.cmml" xref="S4.SS3.p3.1.m1.3.3.3">𝐷</ci><apply id="S4.SS3.p3.1.m1.3.3.1.1.1.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1"><csymbol cd="latexml" id="S4.SS3.p3.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1.1.1">conditional</csymbol><ci id="S4.SS3.p3.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1.1.2">𝐺</ci><apply id="S4.SS3.p3.1.m1.3.3.1.1.1.3.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.1.m1.3.3.1.1.1.3.1.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p3.1.m1.3.3.1.1.1.3.2.cmml" xref="S4.SS3.p3.1.m1.3.3.1.1.1.3.2">𝒢</ci><list id="S4.SS3.p3.1.m1.2.2.2.3.cmml" xref="S4.SS3.p3.1.m1.2.2.2.4"><ci id="S4.SS3.p3.1.m1.1.1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1.1.1">𝑁</ci><ci id="S4.SS3.p3.1.m1.2.2.2.2.cmml" xref="S4.SS3.p3.1.m1.2.2.2.2">𝑁</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.3c">D(G\mid\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.3d">italic_D ( italic_G ∣ caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>. We take a different path, and translate an alternative characterization of nondeterministic complexity in the Boolean function setting (based on the fusion method) to the graph complexity setting. First, we review the necessary concepts.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem44"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem44.1.1.1">Definition 44</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem44.2.2"> </span>(Semi-ultra-filter)<span class="ltx_text ltx_font_bold" id="Thmtheorem44.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem44.p1"> <p class="ltx_p" id="Thmtheorem44.p1.5"><span class="ltx_text ltx_font_italic" id="Thmtheorem44.p1.5.5">We say that a semi-filter <math alttext="\mathcal{F}\subseteq\mathcal{P}(U)" class="ltx_Math" display="inline" id="Thmtheorem44.p1.1.1.m1.1"><semantics id="Thmtheorem44.p1.1.1.m1.1a"><mrow id="Thmtheorem44.p1.1.1.m1.1.2" xref="Thmtheorem44.p1.1.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem44.p1.1.1.m1.1.2.2" xref="Thmtheorem44.p1.1.1.m1.1.2.2.cmml">ℱ</mi><mo id="Thmtheorem44.p1.1.1.m1.1.2.1" xref="Thmtheorem44.p1.1.1.m1.1.2.1.cmml">⊆</mo><mrow id="Thmtheorem44.p1.1.1.m1.1.2.3" xref="Thmtheorem44.p1.1.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem44.p1.1.1.m1.1.2.3.2" xref="Thmtheorem44.p1.1.1.m1.1.2.3.2.cmml">𝒫</mi><mo id="Thmtheorem44.p1.1.1.m1.1.2.3.1" xref="Thmtheorem44.p1.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem44.p1.1.1.m1.1.2.3.3.2" xref="Thmtheorem44.p1.1.1.m1.1.2.3.cmml"><mo id="Thmtheorem44.p1.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmtheorem44.p1.1.1.m1.1.2.3.cmml">(</mo><mi id="Thmtheorem44.p1.1.1.m1.1.1" xref="Thmtheorem44.p1.1.1.m1.1.1.cmml">U</mi><mo id="Thmtheorem44.p1.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmtheorem44.p1.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem44.p1.1.1.m1.1b"><apply id="Thmtheorem44.p1.1.1.m1.1.2.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2"><subset id="Thmtheorem44.p1.1.1.m1.1.2.1.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2.1"></subset><ci id="Thmtheorem44.p1.1.1.m1.1.2.2.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2.2">ℱ</ci><apply id="Thmtheorem44.p1.1.1.m1.1.2.3.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2.3"><times id="Thmtheorem44.p1.1.1.m1.1.2.3.1.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2.3.1"></times><ci id="Thmtheorem44.p1.1.1.m1.1.2.3.2.cmml" xref="Thmtheorem44.p1.1.1.m1.1.2.3.2">𝒫</ci><ci id="Thmtheorem44.p1.1.1.m1.1.1.cmml" xref="Thmtheorem44.p1.1.1.m1.1.1">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem44.p1.1.1.m1.1c">\mathcal{F}\subseteq\mathcal{P}(U)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem44.p1.1.1.m1.1d">caligraphic_F ⊆ caligraphic_P ( italic_U )</annotation></semantics></math> is a <em class="ltx_emph ltx_font_upright" id="Thmtheorem44.p1.5.5.1">semi-ultra-filter</em> if for every set <math alttext="A\subseteq U" class="ltx_Math" display="inline" id="Thmtheorem44.p1.2.2.m2.1"><semantics id="Thmtheorem44.p1.2.2.m2.1a"><mrow id="Thmtheorem44.p1.2.2.m2.1.1" xref="Thmtheorem44.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem44.p1.2.2.m2.1.1.2" xref="Thmtheorem44.p1.2.2.m2.1.1.2.cmml">A</mi><mo id="Thmtheorem44.p1.2.2.m2.1.1.1" xref="Thmtheorem44.p1.2.2.m2.1.1.1.cmml">⊆</mo><mi id="Thmtheorem44.p1.2.2.m2.1.1.3" xref="Thmtheorem44.p1.2.2.m2.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem44.p1.2.2.m2.1b"><apply id="Thmtheorem44.p1.2.2.m2.1.1.cmml" xref="Thmtheorem44.p1.2.2.m2.1.1"><subset id="Thmtheorem44.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem44.p1.2.2.m2.1.1.1"></subset><ci id="Thmtheorem44.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem44.p1.2.2.m2.1.1.2">𝐴</ci><ci id="Thmtheorem44.p1.2.2.m2.1.1.3.cmml" xref="Thmtheorem44.p1.2.2.m2.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem44.p1.2.2.m2.1c">A\subseteq U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem44.p1.2.2.m2.1d">italic_A ⊆ italic_U</annotation></semantics></math>, at least one of <math alttext="A" class="ltx_Math" display="inline" id="Thmtheorem44.p1.3.3.m3.1"><semantics id="Thmtheorem44.p1.3.3.m3.1a"><mi id="Thmtheorem44.p1.3.3.m3.1.1" xref="Thmtheorem44.p1.3.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem44.p1.3.3.m3.1b"><ci id="Thmtheorem44.p1.3.3.m3.1.1.cmml" xref="Thmtheorem44.p1.3.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem44.p1.3.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem44.p1.3.3.m3.1d">italic_A</annotation></semantics></math> or <math alttext="U\setminus A" class="ltx_Math" display="inline" id="Thmtheorem44.p1.4.4.m4.1"><semantics id="Thmtheorem44.p1.4.4.m4.1a"><mrow id="Thmtheorem44.p1.4.4.m4.1.1" xref="Thmtheorem44.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem44.p1.4.4.m4.1.1.2" xref="Thmtheorem44.p1.4.4.m4.1.1.2.cmml">U</mi><mo id="Thmtheorem44.p1.4.4.m4.1.1.1" xref="Thmtheorem44.p1.4.4.m4.1.1.1.cmml">∖</mo><mi id="Thmtheorem44.p1.4.4.m4.1.1.3" xref="Thmtheorem44.p1.4.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem44.p1.4.4.m4.1b"><apply id="Thmtheorem44.p1.4.4.m4.1.1.cmml" xref="Thmtheorem44.p1.4.4.m4.1.1"><setdiff id="Thmtheorem44.p1.4.4.m4.1.1.1.cmml" xref="Thmtheorem44.p1.4.4.m4.1.1.1"></setdiff><ci id="Thmtheorem44.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem44.p1.4.4.m4.1.1.2">𝑈</ci><ci id="Thmtheorem44.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem44.p1.4.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem44.p1.4.4.m4.1c">U\setminus A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem44.p1.4.4.m4.1d">italic_U ∖ italic_A</annotation></semantics></math> is in <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="Thmtheorem44.p1.5.5.m5.1"><semantics id="Thmtheorem44.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem44.p1.5.5.m5.1.1" xref="Thmtheorem44.p1.5.5.m5.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem44.p1.5.5.m5.1b"><ci id="Thmtheorem44.p1.5.5.m5.1.1.cmml" xref="Thmtheorem44.p1.5.5.m5.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem44.p1.5.5.m5.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem44.p1.5.5.m5.1d">caligraphic_F</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p4"> <p class="ltx_p" id="S4.SS3.p4.5">For a function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S4.SS3.p4.1.m1.4"><semantics id="S4.SS3.p4.1.m1.4a"><mrow id="S4.SS3.p4.1.m1.4.5" xref="S4.SS3.p4.1.m1.4.5.cmml"><mi id="S4.SS3.p4.1.m1.4.5.2" xref="S4.SS3.p4.1.m1.4.5.2.cmml">f</mi><mo id="S4.SS3.p4.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p4.1.m1.4.5.1.cmml">:</mo><mrow id="S4.SS3.p4.1.m1.4.5.3" xref="S4.SS3.p4.1.m1.4.5.3.cmml"><msup id="S4.SS3.p4.1.m1.4.5.3.2" xref="S4.SS3.p4.1.m1.4.5.3.2.cmml"><mrow id="S4.SS3.p4.1.m1.4.5.3.2.2.2" xref="S4.SS3.p4.1.m1.4.5.3.2.2.1.cmml"><mo id="S4.SS3.p4.1.m1.4.5.3.2.2.2.1" stretchy="false" xref="S4.SS3.p4.1.m1.4.5.3.2.2.1.cmml">{</mo><mn id="S4.SS3.p4.1.m1.1.1" xref="S4.SS3.p4.1.m1.1.1.cmml">0</mn><mo id="S4.SS3.p4.1.m1.4.5.3.2.2.2.2" xref="S4.SS3.p4.1.m1.4.5.3.2.2.1.cmml">,</mo><mn id="S4.SS3.p4.1.m1.2.2" xref="S4.SS3.p4.1.m1.2.2.cmml">1</mn><mo id="S4.SS3.p4.1.m1.4.5.3.2.2.2.3" stretchy="false" xref="S4.SS3.p4.1.m1.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="S4.SS3.p4.1.m1.4.5.3.2.3" xref="S4.SS3.p4.1.m1.4.5.3.2.3.cmml">n</mi></msup><mo id="S4.SS3.p4.1.m1.4.5.3.1" stretchy="false" xref="S4.SS3.p4.1.m1.4.5.3.1.cmml">→</mo><mrow id="S4.SS3.p4.1.m1.4.5.3.3.2" xref="S4.SS3.p4.1.m1.4.5.3.3.1.cmml"><mo id="S4.SS3.p4.1.m1.4.5.3.3.2.1" stretchy="false" xref="S4.SS3.p4.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S4.SS3.p4.1.m1.3.3" xref="S4.SS3.p4.1.m1.3.3.cmml">0</mn><mo id="S4.SS3.p4.1.m1.4.5.3.3.2.2" xref="S4.SS3.p4.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S4.SS3.p4.1.m1.4.4" xref="S4.SS3.p4.1.m1.4.4.cmml">1</mn><mo id="S4.SS3.p4.1.m1.4.5.3.3.2.3" stretchy="false" xref="S4.SS3.p4.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.1.m1.4b"><apply id="S4.SS3.p4.1.m1.4.5.cmml" xref="S4.SS3.p4.1.m1.4.5"><ci id="S4.SS3.p4.1.m1.4.5.1.cmml" xref="S4.SS3.p4.1.m1.4.5.1">:</ci><ci id="S4.SS3.p4.1.m1.4.5.2.cmml" xref="S4.SS3.p4.1.m1.4.5.2">𝑓</ci><apply id="S4.SS3.p4.1.m1.4.5.3.cmml" xref="S4.SS3.p4.1.m1.4.5.3"><ci id="S4.SS3.p4.1.m1.4.5.3.1.cmml" xref="S4.SS3.p4.1.m1.4.5.3.1">→</ci><apply id="S4.SS3.p4.1.m1.4.5.3.2.cmml" xref="S4.SS3.p4.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS3.p4.1.m1.4.5.3.2.1.cmml" xref="S4.SS3.p4.1.m1.4.5.3.2">superscript</csymbol><set id="S4.SS3.p4.1.m1.4.5.3.2.2.1.cmml" xref="S4.SS3.p4.1.m1.4.5.3.2.2.2"><cn id="S4.SS3.p4.1.m1.1.1.cmml" type="integer" xref="S4.SS3.p4.1.m1.1.1">0</cn><cn id="S4.SS3.p4.1.m1.2.2.cmml" type="integer" xref="S4.SS3.p4.1.m1.2.2">1</cn></set><ci id="S4.SS3.p4.1.m1.4.5.3.2.3.cmml" xref="S4.SS3.p4.1.m1.4.5.3.2.3">𝑛</ci></apply><set id="S4.SS3.p4.1.m1.4.5.3.3.1.cmml" xref="S4.SS3.p4.1.m1.4.5.3.3.2"><cn id="S4.SS3.p4.1.m1.3.3.cmml" type="integer" xref="S4.SS3.p4.1.m1.3.3">0</cn><cn id="S4.SS3.p4.1.m1.4.4.cmml" type="integer" xref="S4.SS3.p4.1.m1.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.1.m1.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.1.m1.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>, let <math alttext="\rho_{\mathsf{ultra}}(f,\mathcal{B}_{n})" class="ltx_Math" display="inline" id="S4.SS3.p4.2.m2.2"><semantics id="S4.SS3.p4.2.m2.2a"><mrow id="S4.SS3.p4.2.m2.2.2" xref="S4.SS3.p4.2.m2.2.2.cmml"><msub 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xref="S4.SS3.p4.2.m2.2.2.1.1.1.3">𝑛</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.2.m2.2c">\rho_{\mathsf{ultra}}(f,\mathcal{B}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.2.m2.2d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_f , caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> denote the minimum number of pairs of subsets of <math alttext="f^{-1}(0)" class="ltx_Math" display="inline" id="S4.SS3.p4.3.m3.1"><semantics id="S4.SS3.p4.3.m3.1a"><mrow id="S4.SS3.p4.3.m3.1.2" xref="S4.SS3.p4.3.m3.1.2.cmml"><msup id="S4.SS3.p4.3.m3.1.2.2" xref="S4.SS3.p4.3.m3.1.2.2.cmml"><mi id="S4.SS3.p4.3.m3.1.2.2.2" xref="S4.SS3.p4.3.m3.1.2.2.2.cmml">f</mi><mrow id="S4.SS3.p4.3.m3.1.2.2.3" xref="S4.SS3.p4.3.m3.1.2.2.3.cmml"><mo id="S4.SS3.p4.3.m3.1.2.2.3a" xref="S4.SS3.p4.3.m3.1.2.2.3.cmml">−</mo><mn id="S4.SS3.p4.3.m3.1.2.2.3.2" xref="S4.SS3.p4.3.m3.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS3.p4.3.m3.1.2.1" xref="S4.SS3.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p4.3.m3.1.2.3.2" xref="S4.SS3.p4.3.m3.1.2.cmml"><mo id="S4.SS3.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS3.p4.3.m3.1.2.cmml">(</mo><mn id="S4.SS3.p4.3.m3.1.1" xref="S4.SS3.p4.3.m3.1.1.cmml">0</mn><mo id="S4.SS3.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS3.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.3.m3.1b"><apply id="S4.SS3.p4.3.m3.1.2.cmml" xref="S4.SS3.p4.3.m3.1.2"><times id="S4.SS3.p4.3.m3.1.2.1.cmml" xref="S4.SS3.p4.3.m3.1.2.1"></times><apply id="S4.SS3.p4.3.m3.1.2.2.cmml" xref="S4.SS3.p4.3.m3.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p4.3.m3.1.2.2.1.cmml" xref="S4.SS3.p4.3.m3.1.2.2">superscript</csymbol><ci id="S4.SS3.p4.3.m3.1.2.2.2.cmml" xref="S4.SS3.p4.3.m3.1.2.2.2">𝑓</ci><apply id="S4.SS3.p4.3.m3.1.2.2.3.cmml" xref="S4.SS3.p4.3.m3.1.2.2.3"><minus 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xref="S4.SS3.p4.4.m4.1.2.2.3.cmml">−</mo><mn id="S4.SS3.p4.4.m4.1.2.2.3.2" xref="S4.SS3.p4.4.m4.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS3.p4.4.m4.1.2.1" xref="S4.SS3.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p4.4.m4.1.2.3.2" xref="S4.SS3.p4.4.m4.1.2.cmml"><mo id="S4.SS3.p4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS3.p4.4.m4.1.2.cmml">(</mo><mn id="S4.SS3.p4.4.m4.1.1" xref="S4.SS3.p4.4.m4.1.1.cmml">0</mn><mo id="S4.SS3.p4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS3.p4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.4.m4.1b"><apply id="S4.SS3.p4.4.m4.1.2.cmml" xref="S4.SS3.p4.4.m4.1.2"><times id="S4.SS3.p4.4.m4.1.2.1.cmml" xref="S4.SS3.p4.4.m4.1.2.1"></times><apply id="S4.SS3.p4.4.m4.1.2.2.cmml" xref="S4.SS3.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p4.4.m4.1.2.2.1.cmml" xref="S4.SS3.p4.4.m4.1.2.2">superscript</csymbol><ci id="S4.SS3.p4.4.m4.1.2.2.2.cmml" xref="S4.SS3.p4.4.m4.1.2.2.2">𝑓</ci><apply id="S4.SS3.p4.4.m4.1.2.2.3.cmml" xref="S4.SS3.p4.4.m4.1.2.2.3"><minus id="S4.SS3.p4.4.m4.1.2.2.3.1.cmml" xref="S4.SS3.p4.4.m4.1.2.2.3"></minus><cn id="S4.SS3.p4.4.m4.1.2.2.3.2.cmml" type="integer" xref="S4.SS3.p4.4.m4.1.2.2.3.2">1</cn></apply></apply><cn id="S4.SS3.p4.4.m4.1.1.cmml" type="integer" xref="S4.SS3.p4.4.m4.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.4.m4.1c">f^{-1}(0)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.4.m4.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 0 )</annotation></semantics></math> that are above an input in <math alttext="f^{-1}(1)" class="ltx_Math" display="inline" id="S4.SS3.p4.5.m5.1"><semantics id="S4.SS3.p4.5.m5.1a"><mrow id="S4.SS3.p4.5.m5.1.2" xref="S4.SS3.p4.5.m5.1.2.cmml"><msup id="S4.SS3.p4.5.m5.1.2.2" xref="S4.SS3.p4.5.m5.1.2.2.cmml"><mi id="S4.SS3.p4.5.m5.1.2.2.2" xref="S4.SS3.p4.5.m5.1.2.2.2.cmml">f</mi><mrow id="S4.SS3.p4.5.m5.1.2.2.3" xref="S4.SS3.p4.5.m5.1.2.2.3.cmml"><mo id="S4.SS3.p4.5.m5.1.2.2.3a" xref="S4.SS3.p4.5.m5.1.2.2.3.cmml">−</mo><mn id="S4.SS3.p4.5.m5.1.2.2.3.2" xref="S4.SS3.p4.5.m5.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS3.p4.5.m5.1.2.1" xref="S4.SS3.p4.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p4.5.m5.1.2.3.2" xref="S4.SS3.p4.5.m5.1.2.cmml"><mo id="S4.SS3.p4.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS3.p4.5.m5.1.2.cmml">(</mo><mn id="S4.SS3.p4.5.m5.1.1" xref="S4.SS3.p4.5.m5.1.1.cmml">1</mn><mo id="S4.SS3.p4.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS3.p4.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.5.m5.1b"><apply id="S4.SS3.p4.5.m5.1.2.cmml" xref="S4.SS3.p4.5.m5.1.2"><times id="S4.SS3.p4.5.m5.1.2.1.cmml" xref="S4.SS3.p4.5.m5.1.2.1"></times><apply id="S4.SS3.p4.5.m5.1.2.2.cmml" xref="S4.SS3.p4.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p4.5.m5.1.2.2.1.cmml" xref="S4.SS3.p4.5.m5.1.2.2">superscript</csymbol><ci id="S4.SS3.p4.5.m5.1.2.2.2.cmml" xref="S4.SS3.p4.5.m5.1.2.2.2">𝑓</ci><apply id="S4.SS3.p4.5.m5.1.2.2.3.cmml" xref="S4.SS3.p4.5.m5.1.2.2.3"><minus id="S4.SS3.p4.5.m5.1.2.2.3.1.cmml" xref="S4.SS3.p4.5.m5.1.2.2.3"></minus><cn id="S4.SS3.p4.5.m5.1.2.2.3.2.cmml" type="integer" xref="S4.SS3.p4.5.m5.1.2.2.3.2">1</cn></apply></apply><cn id="S4.SS3.p4.5.m5.1.1.cmml" type="integer" xref="S4.SS3.p4.5.m5.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.5.m5.1c">f^{-1}(1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.5.m5.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 )</annotation></semantics></math>. <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#bib.bib9" title="">9</a>]</cite> established the following result.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem45"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem45.1.1.1">Theorem 45</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem45.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem45.p1"> <p class="ltx_p" id="Thmtheorem45.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem45.p1.2.2">There exists a constant <math alttext="c\geq 1" class="ltx_Math" display="inline" id="Thmtheorem45.p1.1.1.m1.1"><semantics id="Thmtheorem45.p1.1.1.m1.1a"><mrow id="Thmtheorem45.p1.1.1.m1.1.1" xref="Thmtheorem45.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem45.p1.1.1.m1.1.1.2" xref="Thmtheorem45.p1.1.1.m1.1.1.2.cmml">c</mi><mo id="Thmtheorem45.p1.1.1.m1.1.1.1" xref="Thmtheorem45.p1.1.1.m1.1.1.1.cmml">≥</mo><mn id="Thmtheorem45.p1.1.1.m1.1.1.3" xref="Thmtheorem45.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem45.p1.1.1.m1.1b"><apply id="Thmtheorem45.p1.1.1.m1.1.1.cmml" xref="Thmtheorem45.p1.1.1.m1.1.1"><geq id="Thmtheorem45.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem45.p1.1.1.m1.1.1.1"></geq><ci id="Thmtheorem45.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem45.p1.1.1.m1.1.1.2">𝑐</ci><cn id="Thmtheorem45.p1.1.1.m1.1.1.3.cmml" type="integer" xref="Thmtheorem45.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem45.p1.1.1.m1.1c">c\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem45.p1.1.1.m1.1d">italic_c ≥ 1</annotation></semantics></math> such that for every function <math alttext="f\colon\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem45.p1.2.2.m2.4"><semantics id="Thmtheorem45.p1.2.2.m2.4a"><mrow id="Thmtheorem45.p1.2.2.m2.4.5" xref="Thmtheorem45.p1.2.2.m2.4.5.cmml"><mi id="Thmtheorem45.p1.2.2.m2.4.5.2" xref="Thmtheorem45.p1.2.2.m2.4.5.2.cmml">f</mi><mo id="Thmtheorem45.p1.2.2.m2.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem45.p1.2.2.m2.4.5.1.cmml">:</mo><mrow id="Thmtheorem45.p1.2.2.m2.4.5.3" xref="Thmtheorem45.p1.2.2.m2.4.5.3.cmml"><msup id="Thmtheorem45.p1.2.2.m2.4.5.3.2" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.cmml"><mrow id="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.2" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.1.cmml"><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.1.cmml">{</mo><mn id="Thmtheorem45.p1.2.2.m2.1.1" xref="Thmtheorem45.p1.2.2.m2.1.1.cmml">0</mn><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.2.2" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.1.cmml">,</mo><mn id="Thmtheorem45.p1.2.2.m2.2.2" xref="Thmtheorem45.p1.2.2.m2.2.2.cmml">1</mn><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.2.3" stretchy="false" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.1.cmml">}</mo></mrow><mi id="Thmtheorem45.p1.2.2.m2.4.5.3.2.3" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.3.cmml">n</mi></msup><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.1" stretchy="false" xref="Thmtheorem45.p1.2.2.m2.4.5.3.1.cmml">→</mo><mrow id="Thmtheorem45.p1.2.2.m2.4.5.3.3.2" xref="Thmtheorem45.p1.2.2.m2.4.5.3.3.1.cmml"><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem45.p1.2.2.m2.4.5.3.3.1.cmml">{</mo><mn id="Thmtheorem45.p1.2.2.m2.3.3" xref="Thmtheorem45.p1.2.2.m2.3.3.cmml">0</mn><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.3.2.2" xref="Thmtheorem45.p1.2.2.m2.4.5.3.3.1.cmml">,</mo><mn id="Thmtheorem45.p1.2.2.m2.4.4" xref="Thmtheorem45.p1.2.2.m2.4.4.cmml">1</mn><mo id="Thmtheorem45.p1.2.2.m2.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem45.p1.2.2.m2.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem45.p1.2.2.m2.4b"><apply id="Thmtheorem45.p1.2.2.m2.4.5.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5"><ci id="Thmtheorem45.p1.2.2.m2.4.5.1.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.1">:</ci><ci id="Thmtheorem45.p1.2.2.m2.4.5.2.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.2">𝑓</ci><apply id="Thmtheorem45.p1.2.2.m2.4.5.3.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3"><ci id="Thmtheorem45.p1.2.2.m2.4.5.3.1.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.1">→</ci><apply id="Thmtheorem45.p1.2.2.m2.4.5.3.2.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="Thmtheorem45.p1.2.2.m2.4.5.3.2.1.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2">superscript</csymbol><set id="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.1.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.2.2"><cn id="Thmtheorem45.p1.2.2.m2.1.1.cmml" type="integer" xref="Thmtheorem45.p1.2.2.m2.1.1">0</cn><cn id="Thmtheorem45.p1.2.2.m2.2.2.cmml" type="integer" xref="Thmtheorem45.p1.2.2.m2.2.2">1</cn></set><ci id="Thmtheorem45.p1.2.2.m2.4.5.3.2.3.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.2.3">𝑛</ci></apply><set id="Thmtheorem45.p1.2.2.m2.4.5.3.3.1.cmml" xref="Thmtheorem45.p1.2.2.m2.4.5.3.3.2"><cn id="Thmtheorem45.p1.2.2.m2.3.3.cmml" type="integer" xref="Thmtheorem45.p1.2.2.m2.3.3">0</cn><cn id="Thmtheorem45.p1.2.2.m2.4.4.cmml" type="integer" xref="Thmtheorem45.p1.2.2.m2.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem45.p1.2.2.m2.4c">f\colon\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem45.p1.2.2.m2.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex32"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho_{\mathsf{ultra}}(f,\mathcal{B}_{n})\;\leq\;\mathsf{conondet}\text{-}% \mathsf{size}_{\wedge}(f)\;=\;\mathsf{nondet}\text{-}\mathsf{size}_{\vee}(\neg f% )\;\leq\;c\cdot\rho_{\mathsf{ultra}}(f,\mathcal{B}_{n})." class="ltx_Math" display="block" id="S4.Ex32.m1.4"><semantics id="S4.Ex32.m1.4a"><mrow id="S4.Ex32.m1.4.4.1" xref="S4.Ex32.m1.4.4.1.1.cmml"><mrow id="S4.Ex32.m1.4.4.1.1" xref="S4.Ex32.m1.4.4.1.1.cmml"><mrow id="S4.Ex32.m1.4.4.1.1.1" xref="S4.Ex32.m1.4.4.1.1.1.cmml"><msub id="S4.Ex32.m1.4.4.1.1.1.3" xref="S4.Ex32.m1.4.4.1.1.1.3.cmml"><mi id="S4.Ex32.m1.4.4.1.1.1.3.2" xref="S4.Ex32.m1.4.4.1.1.1.3.2.cmml">ρ</mi><mi id="S4.Ex32.m1.4.4.1.1.1.3.3" xref="S4.Ex32.m1.4.4.1.1.1.3.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.Ex32.m1.4.4.1.1.1.2" xref="S4.Ex32.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex32.m1.4.4.1.1.1.1.1" xref="S4.Ex32.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.Ex32.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex32.m1.1.1" xref="S4.Ex32.m1.1.1.cmml">f</mi><mo id="S4.Ex32.m1.4.4.1.1.1.1.1.3" xref="S4.Ex32.m1.4.4.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex32.m1.4.4.1.1.1.1.1.1" xref="S4.Ex32.m1.4.4.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex32.m1.4.4.1.1.1.1.1.1.2" xref="S4.Ex32.m1.4.4.1.1.1.1.1.1.2.cmml">ℬ</mi><mi id="S4.Ex32.m1.4.4.1.1.1.1.1.1.3" xref="S4.Ex32.m1.4.4.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.Ex32.m1.4.4.1.1.1.1.1.4" rspace="0.280em" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex32.m1.4.4.1.1.5" rspace="0.558em" xref="S4.Ex32.m1.4.4.1.1.5.cmml">≤</mo><mrow id="S4.Ex32.m1.4.4.1.1.6" xref="S4.Ex32.m1.4.4.1.1.6.cmml"><mi id="S4.Ex32.m1.4.4.1.1.6.2" xref="S4.Ex32.m1.4.4.1.1.6.2.cmml">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</mi><mo id="S4.Ex32.m1.4.4.1.1.6.1" xref="S4.Ex32.m1.4.4.1.1.6.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S4.Ex32.m1.4.4.1.1.6.3" xref="S4.Ex32.m1.4.4.1.1.6.3a.cmml">-</mtext><mo id="S4.Ex32.m1.4.4.1.1.6.1a" xref="S4.Ex32.m1.4.4.1.1.6.1.cmml">⁢</mo><msub id="S4.Ex32.m1.4.4.1.1.6.4" xref="S4.Ex32.m1.4.4.1.1.6.4.cmml"><mi id="S4.Ex32.m1.4.4.1.1.6.4.2" xref="S4.Ex32.m1.4.4.1.1.6.4.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex32.m1.4.4.1.1.6.4.3" xref="S4.Ex32.m1.4.4.1.1.6.4.3.cmml">∧</mo></msub><mo id="S4.Ex32.m1.4.4.1.1.6.1b" xref="S4.Ex32.m1.4.4.1.1.6.1.cmml">⁢</mo><mrow id="S4.Ex32.m1.4.4.1.1.6.5.2" xref="S4.Ex32.m1.4.4.1.1.6.cmml"><mo id="S4.Ex32.m1.4.4.1.1.6.5.2.1" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.6.cmml">(</mo><mi id="S4.Ex32.m1.2.2" xref="S4.Ex32.m1.2.2.cmml">f</mi><mo id="S4.Ex32.m1.4.4.1.1.6.5.2.2" rspace="0.280em" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.6.cmml">)</mo></mrow></mrow><mo id="S4.Ex32.m1.4.4.1.1.7" rspace="0.558em" xref="S4.Ex32.m1.4.4.1.1.7.cmml">=</mo><mrow id="S4.Ex32.m1.4.4.1.1.2" xref="S4.Ex32.m1.4.4.1.1.2.cmml"><mi id="S4.Ex32.m1.4.4.1.1.2.3" xref="S4.Ex32.m1.4.4.1.1.2.3.cmml">𝗇𝗈𝗇𝖽𝖾𝗍</mi><mo id="S4.Ex32.m1.4.4.1.1.2.2" xref="S4.Ex32.m1.4.4.1.1.2.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S4.Ex32.m1.4.4.1.1.2.4" xref="S4.Ex32.m1.4.4.1.1.2.4a.cmml">-</mtext><mo id="S4.Ex32.m1.4.4.1.1.2.2a" xref="S4.Ex32.m1.4.4.1.1.2.2.cmml">⁢</mo><msub id="S4.Ex32.m1.4.4.1.1.2.5" xref="S4.Ex32.m1.4.4.1.1.2.5.cmml"><mi id="S4.Ex32.m1.4.4.1.1.2.5.2" xref="S4.Ex32.m1.4.4.1.1.2.5.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex32.m1.4.4.1.1.2.5.3" xref="S4.Ex32.m1.4.4.1.1.2.5.3.cmml">∨</mo></msub><mo id="S4.Ex32.m1.4.4.1.1.2.2b" xref="S4.Ex32.m1.4.4.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex32.m1.4.4.1.1.2.1.1" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.cmml"><mo id="S4.Ex32.m1.4.4.1.1.2.1.1.2" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.Ex32.m1.4.4.1.1.2.1.1.1" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.cmml"><mo id="S4.Ex32.m1.4.4.1.1.2.1.1.1.1" rspace="0.167em" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.1.cmml">¬</mo><mi id="S4.Ex32.m1.4.4.1.1.2.1.1.1.2" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.2.cmml">f</mi></mrow><mo id="S4.Ex32.m1.4.4.1.1.2.1.1.3" rspace="0.280em" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex32.m1.4.4.1.1.8" rspace="0.558em" xref="S4.Ex32.m1.4.4.1.1.8.cmml">≤</mo><mrow id="S4.Ex32.m1.4.4.1.1.3" xref="S4.Ex32.m1.4.4.1.1.3.cmml"><mrow id="S4.Ex32.m1.4.4.1.1.3.3" xref="S4.Ex32.m1.4.4.1.1.3.3.cmml"><mi id="S4.Ex32.m1.4.4.1.1.3.3.2" xref="S4.Ex32.m1.4.4.1.1.3.3.2.cmml">c</mi><mo id="S4.Ex32.m1.4.4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex32.m1.4.4.1.1.3.3.1.cmml">⋅</mo><msub id="S4.Ex32.m1.4.4.1.1.3.3.3" xref="S4.Ex32.m1.4.4.1.1.3.3.3.cmml"><mi id="S4.Ex32.m1.4.4.1.1.3.3.3.2" xref="S4.Ex32.m1.4.4.1.1.3.3.3.2.cmml">ρ</mi><mi id="S4.Ex32.m1.4.4.1.1.3.3.3.3" xref="S4.Ex32.m1.4.4.1.1.3.3.3.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub></mrow><mo id="S4.Ex32.m1.4.4.1.1.3.2" xref="S4.Ex32.m1.4.4.1.1.3.2.cmml">⁢</mo><mrow id="S4.Ex32.m1.4.4.1.1.3.1.1" xref="S4.Ex32.m1.4.4.1.1.3.1.2.cmml"><mo id="S4.Ex32.m1.4.4.1.1.3.1.1.2" stretchy="false" xref="S4.Ex32.m1.4.4.1.1.3.1.2.cmml">(</mo><mi id="S4.Ex32.m1.3.3" xref="S4.Ex32.m1.3.3.cmml">f</mi><mo id="S4.Ex32.m1.4.4.1.1.3.1.1.3" xref="S4.Ex32.m1.4.4.1.1.3.1.2.cmml">,</mo><msub id="S4.Ex32.m1.4.4.1.1.3.1.1.1" xref="S4.Ex32.m1.4.4.1.1.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex32.m1.4.4.1.1.3.1.1.1.2" xref="S4.Ex32.m1.4.4.1.1.3.1.1.1.2.cmml">ℬ</mi><mi id="S4.Ex32.m1.4.4.1.1.3.1.1.1.3" xref="S4.Ex32.m1.4.4.1.1.3.1.1.1.3.cmml">n</mi></msub><mo id="S4.Ex32.m1.4.4.1.1.3.1.1.4" 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sansserif_ultra end_POSTSUBSCRIPT ( italic_f , caligraphic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS3.p5"> <p class="ltx_p" id="S4.SS3.p5.1">Roughly speaking, a variation of cover complexity can be used to characterize conondeterministic circuit complexity. This motivates the following definition, which provides a notion of nondeterministic complexity in arbitrary discrete spaces.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmtheorem46"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem46.1.1.1">Definition 46</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem46.2.2"> </span>(Conondeterministic cover complexity)<span class="ltx_text ltx_font_bold" id="Thmtheorem46.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem46.p1"> <p class="ltx_p" id="Thmtheorem46.p1.6"><span class="ltx_text ltx_font_italic" id="Thmtheorem46.p1.6.6">Given a discrete space <math alttext="\langle\Gamma,\mathcal{B}\rangle" class="ltx_Math" display="inline" id="Thmtheorem46.p1.1.1.m1.2"><semantics id="Thmtheorem46.p1.1.1.m1.2a"><mrow id="Thmtheorem46.p1.1.1.m1.2.3.2" xref="Thmtheorem46.p1.1.1.m1.2.3.1.cmml"><mo id="Thmtheorem46.p1.1.1.m1.2.3.2.1" stretchy="false" xref="Thmtheorem46.p1.1.1.m1.2.3.1.cmml">⟨</mo><mi id="Thmtheorem46.p1.1.1.m1.1.1" mathvariant="normal" xref="Thmtheorem46.p1.1.1.m1.1.1.cmml">Γ</mi><mo id="Thmtheorem46.p1.1.1.m1.2.3.2.2" xref="Thmtheorem46.p1.1.1.m1.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem46.p1.1.1.m1.2.2" xref="Thmtheorem46.p1.1.1.m1.2.2.cmml">ℬ</mi><mo id="Thmtheorem46.p1.1.1.m1.2.3.2.3" stretchy="false" xref="Thmtheorem46.p1.1.1.m1.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.1.1.m1.2b"><list id="Thmtheorem46.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem46.p1.1.1.m1.2.3.2"><ci id="Thmtheorem46.p1.1.1.m1.1.1.cmml" xref="Thmtheorem46.p1.1.1.m1.1.1">Γ</ci><ci id="Thmtheorem46.p1.1.1.m1.2.2.cmml" xref="Thmtheorem46.p1.1.1.m1.2.2">ℬ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem46.p1.1.1.m1.2c">\langle\Gamma,\mathcal{B}\rangle</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem46.p1.1.1.m1.2d">⟨ roman_Γ , caligraphic_B ⟩</annotation></semantics></math> and a set <math alttext="A\subseteq\Gamma" class="ltx_Math" display="inline" id="Thmtheorem46.p1.2.2.m2.1"><semantics id="Thmtheorem46.p1.2.2.m2.1a"><mrow id="Thmtheorem46.p1.2.2.m2.1.1" xref="Thmtheorem46.p1.2.2.m2.1.1.cmml"><mi id="Thmtheorem46.p1.2.2.m2.1.1.2" xref="Thmtheorem46.p1.2.2.m2.1.1.2.cmml">A</mi><mo id="Thmtheorem46.p1.2.2.m2.1.1.1" xref="Thmtheorem46.p1.2.2.m2.1.1.1.cmml">⊆</mo><mi id="Thmtheorem46.p1.2.2.m2.1.1.3" mathvariant="normal" xref="Thmtheorem46.p1.2.2.m2.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.2.2.m2.1b"><apply id="Thmtheorem46.p1.2.2.m2.1.1.cmml" xref="Thmtheorem46.p1.2.2.m2.1.1"><subset id="Thmtheorem46.p1.2.2.m2.1.1.1.cmml" xref="Thmtheorem46.p1.2.2.m2.1.1.1"></subset><ci id="Thmtheorem46.p1.2.2.m2.1.1.2.cmml" xref="Thmtheorem46.p1.2.2.m2.1.1.2">𝐴</ci><ci 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xref="Thmtheorem46.p1.3.3.m3.2.3.3.1.cmml"><mo id="Thmtheorem46.p1.3.3.m3.2.3.3.2.1" stretchy="false" xref="Thmtheorem46.p1.3.3.m3.2.3.3.1.cmml">(</mo><mi id="Thmtheorem46.p1.3.3.m3.1.1" xref="Thmtheorem46.p1.3.3.m3.1.1.cmml">A</mi><mo id="Thmtheorem46.p1.3.3.m3.2.3.3.2.2" xref="Thmtheorem46.p1.3.3.m3.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="Thmtheorem46.p1.3.3.m3.2.2" xref="Thmtheorem46.p1.3.3.m3.2.2.cmml">ℬ</mi><mo id="Thmtheorem46.p1.3.3.m3.2.3.3.2.3" stretchy="false" xref="Thmtheorem46.p1.3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.3.3.m3.2b"><apply id="Thmtheorem46.p1.3.3.m3.2.3.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3"><times id="Thmtheorem46.p1.3.3.m3.2.3.1.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.1"></times><apply id="Thmtheorem46.p1.3.3.m3.2.3.2.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.2"><csymbol cd="ambiguous" id="Thmtheorem46.p1.3.3.m3.2.3.2.1.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.2">subscript</csymbol><ci id="Thmtheorem46.p1.3.3.m3.2.3.2.2.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.2.2">𝜌</ci><ci id="Thmtheorem46.p1.3.3.m3.2.3.2.3.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.2.3">𝗎𝗅𝗍𝗋𝖺</ci></apply><interval closure="open" id="Thmtheorem46.p1.3.3.m3.2.3.3.1.cmml" xref="Thmtheorem46.p1.3.3.m3.2.3.3.2"><ci id="Thmtheorem46.p1.3.3.m3.1.1.cmml" xref="Thmtheorem46.p1.3.3.m3.1.1">𝐴</ci><ci id="Thmtheorem46.p1.3.3.m3.2.2.cmml" xref="Thmtheorem46.p1.3.3.m3.2.2">ℬ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem46.p1.3.3.m3.2c">\rho_{\mathsf{ultra}}(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem46.p1.3.3.m3.2d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_A , caligraphic_B )</annotation></semantics></math> denote the minimum number of pairs of subsets of <math alttext="U=A^{c}=\Gamma\setminus A" class="ltx_Math" display="inline" id="Thmtheorem46.p1.4.4.m4.1"><semantics id="Thmtheorem46.p1.4.4.m4.1a"><mrow id="Thmtheorem46.p1.4.4.m4.1.1" xref="Thmtheorem46.p1.4.4.m4.1.1.cmml"><mi id="Thmtheorem46.p1.4.4.m4.1.1.2" xref="Thmtheorem46.p1.4.4.m4.1.1.2.cmml">U</mi><mo id="Thmtheorem46.p1.4.4.m4.1.1.3" xref="Thmtheorem46.p1.4.4.m4.1.1.3.cmml">=</mo><msup id="Thmtheorem46.p1.4.4.m4.1.1.4" xref="Thmtheorem46.p1.4.4.m4.1.1.4.cmml"><mi id="Thmtheorem46.p1.4.4.m4.1.1.4.2" xref="Thmtheorem46.p1.4.4.m4.1.1.4.2.cmml">A</mi><mi id="Thmtheorem46.p1.4.4.m4.1.1.4.3" xref="Thmtheorem46.p1.4.4.m4.1.1.4.3.cmml">c</mi></msup><mo id="Thmtheorem46.p1.4.4.m4.1.1.5" xref="Thmtheorem46.p1.4.4.m4.1.1.5.cmml">=</mo><mrow id="Thmtheorem46.p1.4.4.m4.1.1.6" xref="Thmtheorem46.p1.4.4.m4.1.1.6.cmml"><mi id="Thmtheorem46.p1.4.4.m4.1.1.6.2" mathvariant="normal" xref="Thmtheorem46.p1.4.4.m4.1.1.6.2.cmml">Γ</mi><mo id="Thmtheorem46.p1.4.4.m4.1.1.6.1" xref="Thmtheorem46.p1.4.4.m4.1.1.6.1.cmml">∖</mo><mi id="Thmtheorem46.p1.4.4.m4.1.1.6.3" xref="Thmtheorem46.p1.4.4.m4.1.1.6.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.4.4.m4.1b"><apply id="Thmtheorem46.p1.4.4.m4.1.1.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1"><and id="Thmtheorem46.p1.4.4.m4.1.1a.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1"></and><apply id="Thmtheorem46.p1.4.4.m4.1.1b.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1"><eq id="Thmtheorem46.p1.4.4.m4.1.1.3.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.3"></eq><ci id="Thmtheorem46.p1.4.4.m4.1.1.2.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.2">𝑈</ci><apply id="Thmtheorem46.p1.4.4.m4.1.1.4.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.4"><csymbol cd="ambiguous" id="Thmtheorem46.p1.4.4.m4.1.1.4.1.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.4">superscript</csymbol><ci id="Thmtheorem46.p1.4.4.m4.1.1.4.2.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.4.2">𝐴</ci><ci id="Thmtheorem46.p1.4.4.m4.1.1.4.3.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.4.3">𝑐</ci></apply></apply><apply id="Thmtheorem46.p1.4.4.m4.1.1c.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1"><eq id="Thmtheorem46.p1.4.4.m4.1.1.5.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.5"></eq><share href="https://arxiv.org/html/2503.14117v1#Thmtheorem46.p1.4.4.m4.1.1.4.cmml" id="Thmtheorem46.p1.4.4.m4.1.1d.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1"></share><apply id="Thmtheorem46.p1.4.4.m4.1.1.6.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.6"><setdiff id="Thmtheorem46.p1.4.4.m4.1.1.6.1.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.6.1"></setdiff><ci id="Thmtheorem46.p1.4.4.m4.1.1.6.2.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.6.2">Γ</ci><ci id="Thmtheorem46.p1.4.4.m4.1.1.6.3.cmml" xref="Thmtheorem46.p1.4.4.m4.1.1.6.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem46.p1.4.4.m4.1c">U=A^{c}=\Gamma\setminus A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem46.p1.4.4.m4.1d">italic_U = italic_A start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = roman_Γ ∖ italic_A</annotation></semantics></math> that cover all semi-ultra-filters over <math alttext="U" class="ltx_Math" display="inline" id="Thmtheorem46.p1.5.5.m5.1"><semantics id="Thmtheorem46.p1.5.5.m5.1a"><mi id="Thmtheorem46.p1.5.5.m5.1.1" xref="Thmtheorem46.p1.5.5.m5.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.5.5.m5.1b"><ci id="Thmtheorem46.p1.5.5.m5.1.1.cmml" xref="Thmtheorem46.p1.5.5.m5.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem46.p1.5.5.m5.1c">U</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem46.p1.5.5.m5.1d">italic_U</annotation></semantics></math> that are above an element <math alttext="a\in A" class="ltx_Math" display="inline" id="Thmtheorem46.p1.6.6.m6.1"><semantics id="Thmtheorem46.p1.6.6.m6.1a"><mrow id="Thmtheorem46.p1.6.6.m6.1.1" xref="Thmtheorem46.p1.6.6.m6.1.1.cmml"><mi id="Thmtheorem46.p1.6.6.m6.1.1.2" xref="Thmtheorem46.p1.6.6.m6.1.1.2.cmml">a</mi><mo id="Thmtheorem46.p1.6.6.m6.1.1.1" xref="Thmtheorem46.p1.6.6.m6.1.1.1.cmml">∈</mo><mi id="Thmtheorem46.p1.6.6.m6.1.1.3" xref="Thmtheorem46.p1.6.6.m6.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem46.p1.6.6.m6.1b"><apply id="Thmtheorem46.p1.6.6.m6.1.1.cmml" xref="Thmtheorem46.p1.6.6.m6.1.1"><in id="Thmtheorem46.p1.6.6.m6.1.1.1.cmml" xref="Thmtheorem46.p1.6.6.m6.1.1.1"></in><ci id="Thmtheorem46.p1.6.6.m6.1.1.2.cmml" xref="Thmtheorem46.p1.6.6.m6.1.1.2">𝑎</ci><ci id="Thmtheorem46.p1.6.6.m6.1.1.3.cmml" xref="Thmtheorem46.p1.6.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem46.p1.6.6.m6.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem46.p1.6.6.m6.1d">italic_a ∈ italic_A</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p6"> <p class="ltx_p" id="S4.SS3.p6.1">Observe that <math alttext="\rho_{\mathsf{ultra}}(A,\mathcal{B})\leq\rho(A,\mathcal{B})" class="ltx_Math" display="inline" id="S4.SS3.p6.1.m1.4"><semantics id="S4.SS3.p6.1.m1.4a"><mrow id="S4.SS3.p6.1.m1.4.5" xref="S4.SS3.p6.1.m1.4.5.cmml"><mrow id="S4.SS3.p6.1.m1.4.5.2" xref="S4.SS3.p6.1.m1.4.5.2.cmml"><msub id="S4.SS3.p6.1.m1.4.5.2.2" xref="S4.SS3.p6.1.m1.4.5.2.2.cmml"><mi id="S4.SS3.p6.1.m1.4.5.2.2.2" xref="S4.SS3.p6.1.m1.4.5.2.2.2.cmml">ρ</mi><mi id="S4.SS3.p6.1.m1.4.5.2.2.3" xref="S4.SS3.p6.1.m1.4.5.2.2.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.SS3.p6.1.m1.4.5.2.1" xref="S4.SS3.p6.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S4.SS3.p6.1.m1.4.5.2.3.2" xref="S4.SS3.p6.1.m1.4.5.2.3.1.cmml"><mo id="S4.SS3.p6.1.m1.4.5.2.3.2.1" stretchy="false" xref="S4.SS3.p6.1.m1.4.5.2.3.1.cmml">(</mo><mi id="S4.SS3.p6.1.m1.1.1" xref="S4.SS3.p6.1.m1.1.1.cmml">A</mi><mo id="S4.SS3.p6.1.m1.4.5.2.3.2.2" xref="S4.SS3.p6.1.m1.4.5.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p6.1.m1.2.2" xref="S4.SS3.p6.1.m1.2.2.cmml">ℬ</mi><mo id="S4.SS3.p6.1.m1.4.5.2.3.2.3" stretchy="false" xref="S4.SS3.p6.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p6.1.m1.4.5.1" xref="S4.SS3.p6.1.m1.4.5.1.cmml">≤</mo><mrow id="S4.SS3.p6.1.m1.4.5.3" xref="S4.SS3.p6.1.m1.4.5.3.cmml"><mi id="S4.SS3.p6.1.m1.4.5.3.2" xref="S4.SS3.p6.1.m1.4.5.3.2.cmml">ρ</mi><mo id="S4.SS3.p6.1.m1.4.5.3.1" xref="S4.SS3.p6.1.m1.4.5.3.1.cmml">⁢</mo><mrow id="S4.SS3.p6.1.m1.4.5.3.3.2" xref="S4.SS3.p6.1.m1.4.5.3.3.1.cmml"><mo id="S4.SS3.p6.1.m1.4.5.3.3.2.1" stretchy="false" xref="S4.SS3.p6.1.m1.4.5.3.3.1.cmml">(</mo><mi id="S4.SS3.p6.1.m1.3.3" xref="S4.SS3.p6.1.m1.3.3.cmml">A</mi><mo id="S4.SS3.p6.1.m1.4.5.3.3.2.2" xref="S4.SS3.p6.1.m1.4.5.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p6.1.m1.4.4" xref="S4.SS3.p6.1.m1.4.4.cmml">ℬ</mi><mo id="S4.SS3.p6.1.m1.4.5.3.3.2.3" stretchy="false" xref="S4.SS3.p6.1.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.1.m1.4b"><apply id="S4.SS3.p6.1.m1.4.5.cmml" xref="S4.SS3.p6.1.m1.4.5"><leq id="S4.SS3.p6.1.m1.4.5.1.cmml" xref="S4.SS3.p6.1.m1.4.5.1"></leq><apply id="S4.SS3.p6.1.m1.4.5.2.cmml" xref="S4.SS3.p6.1.m1.4.5.2"><times id="S4.SS3.p6.1.m1.4.5.2.1.cmml" xref="S4.SS3.p6.1.m1.4.5.2.1"></times><apply id="S4.SS3.p6.1.m1.4.5.2.2.cmml" xref="S4.SS3.p6.1.m1.4.5.2.2"><csymbol cd="ambiguous" id="S4.SS3.p6.1.m1.4.5.2.2.1.cmml" xref="S4.SS3.p6.1.m1.4.5.2.2">subscript</csymbol><ci id="S4.SS3.p6.1.m1.4.5.2.2.2.cmml" xref="S4.SS3.p6.1.m1.4.5.2.2.2">𝜌</ci><ci id="S4.SS3.p6.1.m1.4.5.2.2.3.cmml" xref="S4.SS3.p6.1.m1.4.5.2.2.3">𝗎𝗅𝗍𝗋𝖺</ci></apply><interval closure="open" id="S4.SS3.p6.1.m1.4.5.2.3.1.cmml" xref="S4.SS3.p6.1.m1.4.5.2.3.2"><ci id="S4.SS3.p6.1.m1.1.1.cmml" xref="S4.SS3.p6.1.m1.1.1">𝐴</ci><ci id="S4.SS3.p6.1.m1.2.2.cmml" xref="S4.SS3.p6.1.m1.2.2">ℬ</ci></interval></apply><apply id="S4.SS3.p6.1.m1.4.5.3.cmml" xref="S4.SS3.p6.1.m1.4.5.3"><times id="S4.SS3.p6.1.m1.4.5.3.1.cmml" xref="S4.SS3.p6.1.m1.4.5.3.1"></times><ci id="S4.SS3.p6.1.m1.4.5.3.2.cmml" xref="S4.SS3.p6.1.m1.4.5.3.2">𝜌</ci><interval closure="open" id="S4.SS3.p6.1.m1.4.5.3.3.1.cmml" xref="S4.SS3.p6.1.m1.4.5.3.3.2"><ci id="S4.SS3.p6.1.m1.3.3.cmml" xref="S4.SS3.p6.1.m1.3.3">𝐴</ci><ci id="S4.SS3.p6.1.m1.4.4.cmml" xref="S4.SS3.p6.1.m1.4.4">ℬ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.1.m1.4c">\rho_{\mathsf{ultra}}(A,\mathcal{B})\leq\rho(A,\mathcal{B})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.1.m1.4d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_A , caligraphic_B ) ≤ italic_ρ ( italic_A , caligraphic_B )</annotation></semantics></math>, since every semi-ultra-filter is a semi-filter. Conondeterministic cover complexity sheds light into the strength of the simple lower bound argument presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmtheorem47"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem47.1.1.1">Proposition 47</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem47.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem47.p1"> <p class="ltx_p" id="Thmtheorem47.p1.1"><span class="ltx_text ltx_font_italic" id="Thmtheorem47.p1.1.1">Let <math alttext="G_{\mathsf{NEQ}}\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem47.p1.1.1.m1.2"><semantics id="Thmtheorem47.p1.1.1.m1.2a"><mrow id="Thmtheorem47.p1.1.1.m1.2.3" xref="Thmtheorem47.p1.1.1.m1.2.3.cmml"><msub id="Thmtheorem47.p1.1.1.m1.2.3.2" xref="Thmtheorem47.p1.1.1.m1.2.3.2.cmml"><mi id="Thmtheorem47.p1.1.1.m1.2.3.2.2" xref="Thmtheorem47.p1.1.1.m1.2.3.2.2.cmml">G</mi><mi id="Thmtheorem47.p1.1.1.m1.2.3.2.3" xref="Thmtheorem47.p1.1.1.m1.2.3.2.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="Thmtheorem47.p1.1.1.m1.2.3.1" xref="Thmtheorem47.p1.1.1.m1.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem47.p1.1.1.m1.2.3.3" xref="Thmtheorem47.p1.1.1.m1.2.3.3.cmml"><mrow id="Thmtheorem47.p1.1.1.m1.2.3.3.2.2" xref="Thmtheorem47.p1.1.1.m1.2.3.3.2.1.cmml"><mo id="Thmtheorem47.p1.1.1.m1.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem47.p1.1.1.m1.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem47.p1.1.1.m1.1.1" xref="Thmtheorem47.p1.1.1.m1.1.1.cmml">N</mi><mo id="Thmtheorem47.p1.1.1.m1.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem47.p1.1.1.m1.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem47.p1.1.1.m1.2.3.3.1" rspace="0.222em" xref="Thmtheorem47.p1.1.1.m1.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem47.p1.1.1.m1.2.3.3.3.2" xref="Thmtheorem47.p1.1.1.m1.2.3.3.3.1.cmml"><mo id="Thmtheorem47.p1.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem47.p1.1.1.m1.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem47.p1.1.1.m1.2.2" xref="Thmtheorem47.p1.1.1.m1.2.2.cmml">N</mi><mo id="Thmtheorem47.p1.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem47.p1.1.1.m1.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem47.p1.1.1.m1.2b"><apply id="Thmtheorem47.p1.1.1.m1.2.3.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3"><subset id="Thmtheorem47.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.1"></subset><apply id="Thmtheorem47.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.2"><csymbol cd="ambiguous" id="Thmtheorem47.p1.1.1.m1.2.3.2.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.2">subscript</csymbol><ci id="Thmtheorem47.p1.1.1.m1.2.3.2.2.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.2.2">𝐺</ci><ci id="Thmtheorem47.p1.1.1.m1.2.3.2.3.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.2.3">𝖭𝖤𝖰</ci></apply><apply id="Thmtheorem47.p1.1.1.m1.2.3.3.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3"><times id="Thmtheorem47.p1.1.1.m1.2.3.3.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3.1"></times><apply id="Thmtheorem47.p1.1.1.m1.2.3.3.2.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem47.p1.1.1.m1.2.3.3.2.1.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem47.p1.1.1.m1.1.1.cmml" xref="Thmtheorem47.p1.1.1.m1.1.1">𝑁</ci></apply><apply id="Thmtheorem47.p1.1.1.m1.2.3.3.3.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem47.p1.1.1.m1.2.3.3.3.1.1.cmml" xref="Thmtheorem47.p1.1.1.m1.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem47.p1.1.1.m1.2.2.cmml" xref="Thmtheorem47.p1.1.1.m1.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem47.p1.1.1.m1.2c">G_{\mathsf{NEQ}}\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem47.p1.1.1.m1.2d">italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math> be the graph defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4.2</span></a>. Then,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho_{\mathsf{can}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})\;\leq\;\rho_{\mathsf{% ultra}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})." class="ltx_Math" display="block" id="S4.Ex33.m1.5"><semantics id="S4.Ex33.m1.5a"><mrow id="S4.Ex33.m1.5.5.1" xref="S4.Ex33.m1.5.5.1.1.cmml"><mrow id="S4.Ex33.m1.5.5.1.1" xref="S4.Ex33.m1.5.5.1.1.cmml"><mrow id="S4.Ex33.m1.5.5.1.1.2" xref="S4.Ex33.m1.5.5.1.1.2.cmml"><msub id="S4.Ex33.m1.5.5.1.1.2.4" xref="S4.Ex33.m1.5.5.1.1.2.4.cmml"><mi id="S4.Ex33.m1.5.5.1.1.2.4.2" xref="S4.Ex33.m1.5.5.1.1.2.4.2.cmml">ρ</mi><mi id="S4.Ex33.m1.5.5.1.1.2.4.3" xref="S4.Ex33.m1.5.5.1.1.2.4.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.Ex33.m1.5.5.1.1.2.3" xref="S4.Ex33.m1.5.5.1.1.2.3.cmml">⁢</mo><mrow id="S4.Ex33.m1.5.5.1.1.2.2.2" 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id="S4.Ex33.m1.5.5.1.1.3.1.1.1.cmml" xref="S4.Ex33.m1.5.5.1.1.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex33.m1.5.5.1.1.3.1.1.1.1.cmml" xref="S4.Ex33.m1.5.5.1.1.3.1.1.1">subscript</csymbol><ci id="S4.Ex33.m1.5.5.1.1.3.1.1.1.2.cmml" xref="S4.Ex33.m1.5.5.1.1.3.1.1.1.2">𝐺</ci><ci id="S4.Ex33.m1.5.5.1.1.3.1.1.1.3.cmml" xref="S4.Ex33.m1.5.5.1.1.3.1.1.1.3">𝖭𝖤𝖰</ci></apply><apply id="S4.Ex33.m1.5.5.1.1.4.2.2.2.cmml" xref="S4.Ex33.m1.5.5.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S4.Ex33.m1.5.5.1.1.4.2.2.2.1.cmml" xref="S4.Ex33.m1.5.5.1.1.4.2.2.2">subscript</csymbol><ci id="S4.Ex33.m1.5.5.1.1.4.2.2.2.2.cmml" xref="S4.Ex33.m1.5.5.1.1.4.2.2.2.2">𝒢</ci><list id="S4.Ex33.m1.4.4.2.3.cmml" xref="S4.Ex33.m1.4.4.2.4"><ci id="S4.Ex33.m1.3.3.1.1.cmml" xref="S4.Ex33.m1.3.3.1.1">𝑁</ci><ci id="S4.Ex33.m1.4.4.2.2.cmml" xref="S4.Ex33.m1.4.4.2.2">𝑁</ci></list></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex33.m1.5c">\rho_{\mathsf{can}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})\;\leq\;\rho_{\mathsf{% ultra}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex33.m1.5d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS3.1.p1"> <p class="ltx_p" id="S4.SS3.1.p1.9">For convenience, let <math alttext="G=G_{\mathsf{NEQ}}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.1.m1.1"><semantics id="S4.SS3.1.p1.1.m1.1a"><mrow id="S4.SS3.1.p1.1.m1.1.1" xref="S4.SS3.1.p1.1.m1.1.1.cmml"><mi id="S4.SS3.1.p1.1.m1.1.1.2" xref="S4.SS3.1.p1.1.m1.1.1.2.cmml">G</mi><mo id="S4.SS3.1.p1.1.m1.1.1.1" xref="S4.SS3.1.p1.1.m1.1.1.1.cmml">=</mo><msub id="S4.SS3.1.p1.1.m1.1.1.3" xref="S4.SS3.1.p1.1.m1.1.1.3.cmml"><mi id="S4.SS3.1.p1.1.m1.1.1.3.2" xref="S4.SS3.1.p1.1.m1.1.1.3.2.cmml">G</mi><mi id="S4.SS3.1.p1.1.m1.1.1.3.3" xref="S4.SS3.1.p1.1.m1.1.1.3.3.cmml">𝖭𝖤𝖰</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.1.m1.1b"><apply id="S4.SS3.1.p1.1.m1.1.1.cmml" xref="S4.SS3.1.p1.1.m1.1.1"><eq id="S4.SS3.1.p1.1.m1.1.1.1.cmml" xref="S4.SS3.1.p1.1.m1.1.1.1"></eq><ci id="S4.SS3.1.p1.1.m1.1.1.2.cmml" xref="S4.SS3.1.p1.1.m1.1.1.2">𝐺</ci><apply id="S4.SS3.1.p1.1.m1.1.1.3.cmml" xref="S4.SS3.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.1.p1.1.m1.1.1.3.1.cmml" xref="S4.SS3.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS3.1.p1.1.m1.1.1.3.2.cmml" xref="S4.SS3.1.p1.1.m1.1.1.3.2">𝐺</ci><ci id="S4.SS3.1.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.1.p1.1.m1.1.1.3.3">𝖭𝖤𝖰</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.1.m1.1c">G=G_{\mathsf{NEQ}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.1.m1.1d">italic_G = italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT</annotation></semantics></math>. Simply observe that every semi-filter <math alttext="\mathcal{F}_{e}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.2.m2.1"><semantics id="S4.SS3.1.p1.2.m2.1a"><msub id="S4.SS3.1.p1.2.m2.1.1" xref="S4.SS3.1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.1.p1.2.m2.1.1.2" xref="S4.SS3.1.p1.2.m2.1.1.2.cmml">ℱ</mi><mi id="S4.SS3.1.p1.2.m2.1.1.3" xref="S4.SS3.1.p1.2.m2.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.2.m2.1b"><apply id="S4.SS3.1.p1.2.m2.1.1.cmml" xref="S4.SS3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.2.m2.1.1.1.cmml" xref="S4.SS3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.1.p1.2.m2.1.1.2.cmml" xref="S4.SS3.1.p1.2.m2.1.1.2">ℱ</ci><ci id="S4.SS3.1.p1.2.m2.1.1.3.cmml" xref="S4.SS3.1.p1.2.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.2.m2.1c">\mathcal{F}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.2.m2.1d">caligraphic_F start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathfrak{F}^{G}_{\mathsf{can}}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.3.m3.1"><semantics id="S4.SS3.1.p1.3.m3.1a"><msubsup id="S4.SS3.1.p1.3.m3.1.1" xref="S4.SS3.1.p1.3.m3.1.1.cmml"><mi id="S4.SS3.1.p1.3.m3.1.1.2.2" xref="S4.SS3.1.p1.3.m3.1.1.2.2.cmml">𝔉</mi><mi id="S4.SS3.1.p1.3.m3.1.1.3" xref="S4.SS3.1.p1.3.m3.1.1.3.cmml">𝖼𝖺𝗇</mi><mi id="S4.SS3.1.p1.3.m3.1.1.2.3" xref="S4.SS3.1.p1.3.m3.1.1.2.3.cmml">G</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.3.m3.1b"><apply id="S4.SS3.1.p1.3.m3.1.1.cmml" xref="S4.SS3.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.3.m3.1.1.1.cmml" xref="S4.SS3.1.p1.3.m3.1.1">subscript</csymbol><apply id="S4.SS3.1.p1.3.m3.1.1.2.cmml" xref="S4.SS3.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.3.m3.1.1.2.1.cmml" xref="S4.SS3.1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS3.1.p1.3.m3.1.1.2.2.cmml" xref="S4.SS3.1.p1.3.m3.1.1.2.2">𝔉</ci><ci id="S4.SS3.1.p1.3.m3.1.1.2.3.cmml" xref="S4.SS3.1.p1.3.m3.1.1.2.3">𝐺</ci></apply><ci id="S4.SS3.1.p1.3.m3.1.1.3.cmml" xref="S4.SS3.1.p1.3.m3.1.1.3">𝖼𝖺𝗇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.3.m3.1c">\mathfrak{F}^{G}_{\mathsf{can}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.3.m3.1d">fraktur_F start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT</annotation></semantics></math> is a semi-ultra-filter. Indeed, for <math alttext="e=(u,v)\in G" class="ltx_Math" display="inline" id="S4.SS3.1.p1.4.m4.2"><semantics id="S4.SS3.1.p1.4.m4.2a"><mrow id="S4.SS3.1.p1.4.m4.2.3" xref="S4.SS3.1.p1.4.m4.2.3.cmml"><mi id="S4.SS3.1.p1.4.m4.2.3.2" xref="S4.SS3.1.p1.4.m4.2.3.2.cmml">e</mi><mo id="S4.SS3.1.p1.4.m4.2.3.3" xref="S4.SS3.1.p1.4.m4.2.3.3.cmml">=</mo><mrow id="S4.SS3.1.p1.4.m4.2.3.4.2" xref="S4.SS3.1.p1.4.m4.2.3.4.1.cmml"><mo id="S4.SS3.1.p1.4.m4.2.3.4.2.1" stretchy="false" xref="S4.SS3.1.p1.4.m4.2.3.4.1.cmml">(</mo><mi id="S4.SS3.1.p1.4.m4.1.1" xref="S4.SS3.1.p1.4.m4.1.1.cmml">u</mi><mo id="S4.SS3.1.p1.4.m4.2.3.4.2.2" xref="S4.SS3.1.p1.4.m4.2.3.4.1.cmml">,</mo><mi id="S4.SS3.1.p1.4.m4.2.2" xref="S4.SS3.1.p1.4.m4.2.2.cmml">v</mi><mo id="S4.SS3.1.p1.4.m4.2.3.4.2.3" stretchy="false" xref="S4.SS3.1.p1.4.m4.2.3.4.1.cmml">)</mo></mrow><mo id="S4.SS3.1.p1.4.m4.2.3.5" xref="S4.SS3.1.p1.4.m4.2.3.5.cmml">∈</mo><mi id="S4.SS3.1.p1.4.m4.2.3.6" xref="S4.SS3.1.p1.4.m4.2.3.6.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.4.m4.2b"><apply id="S4.SS3.1.p1.4.m4.2.3.cmml" xref="S4.SS3.1.p1.4.m4.2.3"><and id="S4.SS3.1.p1.4.m4.2.3a.cmml" xref="S4.SS3.1.p1.4.m4.2.3"></and><apply id="S4.SS3.1.p1.4.m4.2.3b.cmml" xref="S4.SS3.1.p1.4.m4.2.3"><eq id="S4.SS3.1.p1.4.m4.2.3.3.cmml" xref="S4.SS3.1.p1.4.m4.2.3.3"></eq><ci id="S4.SS3.1.p1.4.m4.2.3.2.cmml" xref="S4.SS3.1.p1.4.m4.2.3.2">𝑒</ci><interval closure="open" id="S4.SS3.1.p1.4.m4.2.3.4.1.cmml" xref="S4.SS3.1.p1.4.m4.2.3.4.2"><ci id="S4.SS3.1.p1.4.m4.1.1.cmml" xref="S4.SS3.1.p1.4.m4.1.1">𝑢</ci><ci id="S4.SS3.1.p1.4.m4.2.2.cmml" xref="S4.SS3.1.p1.4.m4.2.2">𝑣</ci></interval></apply><apply id="S4.SS3.1.p1.4.m4.2.3c.cmml" xref="S4.SS3.1.p1.4.m4.2.3"><in id="S4.SS3.1.p1.4.m4.2.3.5.cmml" xref="S4.SS3.1.p1.4.m4.2.3.5"></in><share href="https://arxiv.org/html/2503.14117v1#S4.SS3.1.p1.4.m4.2.3.4.cmml" id="S4.SS3.1.p1.4.m4.2.3d.cmml" xref="S4.SS3.1.p1.4.m4.2.3"></share><ci id="S4.SS3.1.p1.4.m4.2.3.6.cmml" xref="S4.SS3.1.p1.4.m4.2.3.6">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.4.m4.2c">e=(u,v)\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.4.m4.2d">italic_e = ( italic_u , italic_v ) ∈ italic_G</annotation></semantics></math> and an arbitrary set <math alttext="W\subseteq\overline{G}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.5.m5.1"><semantics id="S4.SS3.1.p1.5.m5.1a"><mrow id="S4.SS3.1.p1.5.m5.1.1" xref="S4.SS3.1.p1.5.m5.1.1.cmml"><mi id="S4.SS3.1.p1.5.m5.1.1.2" xref="S4.SS3.1.p1.5.m5.1.1.2.cmml">W</mi><mo id="S4.SS3.1.p1.5.m5.1.1.1" xref="S4.SS3.1.p1.5.m5.1.1.1.cmml">⊆</mo><mover accent="true" id="S4.SS3.1.p1.5.m5.1.1.3" xref="S4.SS3.1.p1.5.m5.1.1.3.cmml"><mi id="S4.SS3.1.p1.5.m5.1.1.3.2" xref="S4.SS3.1.p1.5.m5.1.1.3.2.cmml">G</mi><mo id="S4.SS3.1.p1.5.m5.1.1.3.1" xref="S4.SS3.1.p1.5.m5.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.5.m5.1b"><apply id="S4.SS3.1.p1.5.m5.1.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1"><subset id="S4.SS3.1.p1.5.m5.1.1.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.1"></subset><ci id="S4.SS3.1.p1.5.m5.1.1.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.2">𝑊</ci><apply id="S4.SS3.1.p1.5.m5.1.1.3.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3"><ci id="S4.SS3.1.p1.5.m5.1.1.3.1.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.1">¯</ci><ci id="S4.SS3.1.p1.5.m5.1.1.3.2.cmml" xref="S4.SS3.1.p1.5.m5.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.5.m5.1c">W\subseteq\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.5.m5.1d">italic_W ⊆ over¯ start_ARG italic_G end_ARG</annotation></semantics></math>, either <math alttext="W" class="ltx_Math" display="inline" id="S4.SS3.1.p1.6.m6.1"><semantics id="S4.SS3.1.p1.6.m6.1a"><mi id="S4.SS3.1.p1.6.m6.1.1" xref="S4.SS3.1.p1.6.m6.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.6.m6.1b"><ci id="S4.SS3.1.p1.6.m6.1.1.cmml" xref="S4.SS3.1.p1.6.m6.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.6.m6.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.6.m6.1d">italic_W</annotation></semantics></math> or <math alttext="\overline{G}\setminus W" class="ltx_Math" display="inline" id="S4.SS3.1.p1.7.m7.1"><semantics id="S4.SS3.1.p1.7.m7.1a"><mrow id="S4.SS3.1.p1.7.m7.1.1" xref="S4.SS3.1.p1.7.m7.1.1.cmml"><mover accent="true" id="S4.SS3.1.p1.7.m7.1.1.2" xref="S4.SS3.1.p1.7.m7.1.1.2.cmml"><mi id="S4.SS3.1.p1.7.m7.1.1.2.2" xref="S4.SS3.1.p1.7.m7.1.1.2.2.cmml">G</mi><mo id="S4.SS3.1.p1.7.m7.1.1.2.1" xref="S4.SS3.1.p1.7.m7.1.1.2.1.cmml">¯</mo></mover><mo id="S4.SS3.1.p1.7.m7.1.1.1" xref="S4.SS3.1.p1.7.m7.1.1.1.cmml">∖</mo><mi id="S4.SS3.1.p1.7.m7.1.1.3" xref="S4.SS3.1.p1.7.m7.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.7.m7.1b"><apply id="S4.SS3.1.p1.7.m7.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1"><setdiff id="S4.SS3.1.p1.7.m7.1.1.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1.1"></setdiff><apply id="S4.SS3.1.p1.7.m7.1.1.2.cmml" xref="S4.SS3.1.p1.7.m7.1.1.2"><ci id="S4.SS3.1.p1.7.m7.1.1.2.1.cmml" xref="S4.SS3.1.p1.7.m7.1.1.2.1">¯</ci><ci id="S4.SS3.1.p1.7.m7.1.1.2.2.cmml" xref="S4.SS3.1.p1.7.m7.1.1.2.2">𝐺</ci></apply><ci id="S4.SS3.1.p1.7.m7.1.1.3.cmml" xref="S4.SS3.1.p1.7.m7.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.7.m7.1c">\overline{G}\setminus W</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.7.m7.1d">over¯ start_ARG italic_G end_ARG ∖ italic_W</annotation></semantics></math> contains <math alttext="R^{u}_{\overline{G}}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.8.m8.1"><semantics id="S4.SS3.1.p1.8.m8.1a"><msubsup id="S4.SS3.1.p1.8.m8.1.1" xref="S4.SS3.1.p1.8.m8.1.1.cmml"><mi id="S4.SS3.1.p1.8.m8.1.1.2.2" xref="S4.SS3.1.p1.8.m8.1.1.2.2.cmml">R</mi><mover accent="true" id="S4.SS3.1.p1.8.m8.1.1.3" xref="S4.SS3.1.p1.8.m8.1.1.3.cmml"><mi id="S4.SS3.1.p1.8.m8.1.1.3.2" xref="S4.SS3.1.p1.8.m8.1.1.3.2.cmml">G</mi><mo id="S4.SS3.1.p1.8.m8.1.1.3.1" xref="S4.SS3.1.p1.8.m8.1.1.3.1.cmml">¯</mo></mover><mi id="S4.SS3.1.p1.8.m8.1.1.2.3" xref="S4.SS3.1.p1.8.m8.1.1.2.3.cmml">u</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.8.m8.1b"><apply id="S4.SS3.1.p1.8.m8.1.1.cmml" xref="S4.SS3.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.8.m8.1.1.1.cmml" xref="S4.SS3.1.p1.8.m8.1.1">subscript</csymbol><apply id="S4.SS3.1.p1.8.m8.1.1.2.cmml" xref="S4.SS3.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.8.m8.1.1.2.1.cmml" xref="S4.SS3.1.p1.8.m8.1.1">superscript</csymbol><ci id="S4.SS3.1.p1.8.m8.1.1.2.2.cmml" xref="S4.SS3.1.p1.8.m8.1.1.2.2">𝑅</ci><ci id="S4.SS3.1.p1.8.m8.1.1.2.3.cmml" xref="S4.SS3.1.p1.8.m8.1.1.2.3">𝑢</ci></apply><apply id="S4.SS3.1.p1.8.m8.1.1.3.cmml" xref="S4.SS3.1.p1.8.m8.1.1.3"><ci id="S4.SS3.1.p1.8.m8.1.1.3.1.cmml" xref="S4.SS3.1.p1.8.m8.1.1.3.1">¯</ci><ci id="S4.SS3.1.p1.8.m8.1.1.3.2.cmml" xref="S4.SS3.1.p1.8.m8.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.8.m8.1c">R^{u}_{\overline{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.8.m8.1d">italic_R start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG italic_G end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>, since the latter is a singleton set due to our choice of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS3.1.p1.9.m9.1"><semantics id="S4.SS3.1.p1.9.m9.1a"><mi id="S4.SS3.1.p1.9.m9.1.1" xref="S4.SS3.1.p1.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.9.m9.1b"><ci id="S4.SS3.1.p1.9.m9.1.1.cmml" xref="S4.SS3.1.p1.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.9.m9.1d">italic_G</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS3.p7"> <p class="ltx_p" id="S4.SS3.p7.1">Now we translate this result into a stronger lower bound in Boolean function complexity. This will be a consequence of the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmtheorem48"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem48.1.1.1">Lemma 48</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem48.2.2"> </span>(A nondeterministic fusion transference lemma)<span class="ltx_text ltx_font_bold" id="Thmtheorem48.3.3">.</span> </h6> <div class="ltx_para" id="Thmtheorem48.p1"> <p class="ltx_p" id="Thmtheorem48.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem48.p1.2.2">Let <math alttext="N=2^{n}" class="ltx_Math" display="inline" id="Thmtheorem48.p1.1.1.m1.1"><semantics id="Thmtheorem48.p1.1.1.m1.1a"><mrow id="Thmtheorem48.p1.1.1.m1.1.1" xref="Thmtheorem48.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem48.p1.1.1.m1.1.1.2" xref="Thmtheorem48.p1.1.1.m1.1.1.2.cmml">N</mi><mo id="Thmtheorem48.p1.1.1.m1.1.1.1" xref="Thmtheorem48.p1.1.1.m1.1.1.1.cmml">=</mo><msup id="Thmtheorem48.p1.1.1.m1.1.1.3" xref="Thmtheorem48.p1.1.1.m1.1.1.3.cmml"><mn id="Thmtheorem48.p1.1.1.m1.1.1.3.2" xref="Thmtheorem48.p1.1.1.m1.1.1.3.2.cmml">2</mn><mi id="Thmtheorem48.p1.1.1.m1.1.1.3.3" xref="Thmtheorem48.p1.1.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem48.p1.1.1.m1.1b"><apply id="Thmtheorem48.p1.1.1.m1.1.1.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1"><eq id="Thmtheorem48.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1.1"></eq><ci id="Thmtheorem48.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1.2">𝑁</ci><apply id="Thmtheorem48.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmtheorem48.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1.3">superscript</csymbol><cn id="Thmtheorem48.p1.1.1.m1.1.1.3.2.cmml" type="integer" xref="Thmtheorem48.p1.1.1.m1.1.1.3.2">2</cn><ci id="Thmtheorem48.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem48.p1.1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem48.p1.1.1.m1.1c">N=2^{n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem48.p1.1.1.m1.1d">italic_N = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. For every graph <math alttext="G\subseteq[N]\times[N]" class="ltx_Math" display="inline" id="Thmtheorem48.p1.2.2.m2.2"><semantics id="Thmtheorem48.p1.2.2.m2.2a"><mrow id="Thmtheorem48.p1.2.2.m2.2.3" xref="Thmtheorem48.p1.2.2.m2.2.3.cmml"><mi id="Thmtheorem48.p1.2.2.m2.2.3.2" xref="Thmtheorem48.p1.2.2.m2.2.3.2.cmml">G</mi><mo id="Thmtheorem48.p1.2.2.m2.2.3.1" xref="Thmtheorem48.p1.2.2.m2.2.3.1.cmml">⊆</mo><mrow id="Thmtheorem48.p1.2.2.m2.2.3.3" xref="Thmtheorem48.p1.2.2.m2.2.3.3.cmml"><mrow id="Thmtheorem48.p1.2.2.m2.2.3.3.2.2" xref="Thmtheorem48.p1.2.2.m2.2.3.3.2.1.cmml"><mo id="Thmtheorem48.p1.2.2.m2.2.3.3.2.2.1" stretchy="false" xref="Thmtheorem48.p1.2.2.m2.2.3.3.2.1.1.cmml">[</mo><mi id="Thmtheorem48.p1.2.2.m2.1.1" xref="Thmtheorem48.p1.2.2.m2.1.1.cmml">N</mi><mo id="Thmtheorem48.p1.2.2.m2.2.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="Thmtheorem48.p1.2.2.m2.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="Thmtheorem48.p1.2.2.m2.2.3.3.1" rspace="0.222em" xref="Thmtheorem48.p1.2.2.m2.2.3.3.1.cmml">×</mo><mrow id="Thmtheorem48.p1.2.2.m2.2.3.3.3.2" xref="Thmtheorem48.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="Thmtheorem48.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem48.p1.2.2.m2.2.3.3.3.1.1.cmml">[</mo><mi id="Thmtheorem48.p1.2.2.m2.2.2" xref="Thmtheorem48.p1.2.2.m2.2.2.cmml">N</mi><mo id="Thmtheorem48.p1.2.2.m2.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem48.p1.2.2.m2.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem48.p1.2.2.m2.2b"><apply id="Thmtheorem48.p1.2.2.m2.2.3.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3"><subset id="Thmtheorem48.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.1"></subset><ci id="Thmtheorem48.p1.2.2.m2.2.3.2.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.2">𝐺</ci><apply id="Thmtheorem48.p1.2.2.m2.2.3.3.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3"><times id="Thmtheorem48.p1.2.2.m2.2.3.3.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3.1"></times><apply id="Thmtheorem48.p1.2.2.m2.2.3.3.2.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3.2.2"><csymbol cd="latexml" id="Thmtheorem48.p1.2.2.m2.2.3.3.2.1.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3.2.2.1">delimited-[]</csymbol><ci id="Thmtheorem48.p1.2.2.m2.1.1.cmml" xref="Thmtheorem48.p1.2.2.m2.1.1">𝑁</ci></apply><apply id="Thmtheorem48.p1.2.2.m2.2.3.3.3.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3.3.2"><csymbol cd="latexml" id="Thmtheorem48.p1.2.2.m2.2.3.3.3.1.1.cmml" xref="Thmtheorem48.p1.2.2.m2.2.3.3.3.2.1">delimited-[]</csymbol><ci id="Thmtheorem48.p1.2.2.m2.2.2.cmml" xref="Thmtheorem48.p1.2.2.m2.2.2">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem48.p1.2.2.m2.2c">G\subseteq[N]\times[N]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem48.p1.2.2.m2.2d">italic_G ⊆ [ italic_N ] × [ italic_N ]</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\rho_{\mathsf{ultra}}(G,\mathcal{G}_{N,N})\;\leq\;\rho_{\mathsf{ultra}}(f_{G},% \mathcal{B}_{2n})," class="ltx_Math" display="block" id="S4.Ex34.m1.4"><semantics id="S4.Ex34.m1.4a"><mrow id="S4.Ex34.m1.4.4.1" xref="S4.Ex34.m1.4.4.1.1.cmml"><mrow id="S4.Ex34.m1.4.4.1.1" xref="S4.Ex34.m1.4.4.1.1.cmml"><mrow id="S4.Ex34.m1.4.4.1.1.1" xref="S4.Ex34.m1.4.4.1.1.1.cmml"><msub id="S4.Ex34.m1.4.4.1.1.1.3" xref="S4.Ex34.m1.4.4.1.1.1.3.cmml"><mi id="S4.Ex34.m1.4.4.1.1.1.3.2" xref="S4.Ex34.m1.4.4.1.1.1.3.2.cmml">ρ</mi><mi id="S4.Ex34.m1.4.4.1.1.1.3.3" xref="S4.Ex34.m1.4.4.1.1.1.3.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.Ex34.m1.4.4.1.1.1.2" xref="S4.Ex34.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex34.m1.4.4.1.1.1.1.1" xref="S4.Ex34.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.Ex34.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.Ex34.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex34.m1.3.3" xref="S4.Ex34.m1.3.3.cmml">G</mi><mo id="S4.Ex34.m1.4.4.1.1.1.1.1.3" xref="S4.Ex34.m1.4.4.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex34.m1.4.4.1.1.1.1.1.1" xref="S4.Ex34.m1.4.4.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex34.m1.4.4.1.1.1.1.1.1.2" xref="S4.Ex34.m1.4.4.1.1.1.1.1.1.2.cmml">𝒢</mi><mrow id="S4.Ex34.m1.2.2.2.4" xref="S4.Ex34.m1.2.2.2.3.cmml"><mi id="S4.Ex34.m1.1.1.1.1" xref="S4.Ex34.m1.1.1.1.1.cmml">N</mi><mo id="S4.Ex34.m1.2.2.2.4.1" xref="S4.Ex34.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex34.m1.2.2.2.2" xref="S4.Ex34.m1.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.Ex34.m1.4.4.1.1.1.1.1.4" rspace="0.280em" stretchy="false" xref="S4.Ex34.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex34.m1.4.4.1.1.4" rspace="0.558em" xref="S4.Ex34.m1.4.4.1.1.4.cmml">≤</mo><mrow id="S4.Ex34.m1.4.4.1.1.3" xref="S4.Ex34.m1.4.4.1.1.3.cmml"><msub id="S4.Ex34.m1.4.4.1.1.3.4" xref="S4.Ex34.m1.4.4.1.1.3.4.cmml"><mi id="S4.Ex34.m1.4.4.1.1.3.4.2" xref="S4.Ex34.m1.4.4.1.1.3.4.2.cmml">ρ</mi><mi id="S4.Ex34.m1.4.4.1.1.3.4.3" xref="S4.Ex34.m1.4.4.1.1.3.4.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.Ex34.m1.4.4.1.1.3.3" xref="S4.Ex34.m1.4.4.1.1.3.3.cmml">⁢</mo><mrow id="S4.Ex34.m1.4.4.1.1.3.2.2" xref="S4.Ex34.m1.4.4.1.1.3.2.3.cmml"><mo id="S4.Ex34.m1.4.4.1.1.3.2.2.3" stretchy="false" xref="S4.Ex34.m1.4.4.1.1.3.2.3.cmml">(</mo><msub id="S4.Ex34.m1.4.4.1.1.2.1.1.1" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1.cmml"><mi id="S4.Ex34.m1.4.4.1.1.2.1.1.1.2" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1.2.cmml">f</mi><mi id="S4.Ex34.m1.4.4.1.1.2.1.1.1.3" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1.3.cmml">G</mi></msub><mo id="S4.Ex34.m1.4.4.1.1.3.2.2.4" xref="S4.Ex34.m1.4.4.1.1.3.2.3.cmml">,</mo><msub id="S4.Ex34.m1.4.4.1.1.3.2.2.2" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex34.m1.4.4.1.1.3.2.2.2.2" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.2.cmml">ℬ</mi><mrow id="S4.Ex34.m1.4.4.1.1.3.2.2.2.3" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.cmml"><mn 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xref="S4.Ex34.m1.4.4.1.1.3.4"><csymbol cd="ambiguous" id="S4.Ex34.m1.4.4.1.1.3.4.1.cmml" xref="S4.Ex34.m1.4.4.1.1.3.4">subscript</csymbol><ci id="S4.Ex34.m1.4.4.1.1.3.4.2.cmml" xref="S4.Ex34.m1.4.4.1.1.3.4.2">𝜌</ci><ci id="S4.Ex34.m1.4.4.1.1.3.4.3.cmml" xref="S4.Ex34.m1.4.4.1.1.3.4.3">𝗎𝗅𝗍𝗋𝖺</ci></apply><interval closure="open" id="S4.Ex34.m1.4.4.1.1.3.2.3.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2"><apply id="S4.Ex34.m1.4.4.1.1.2.1.1.1.cmml" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S4.Ex34.m1.4.4.1.1.2.1.1.1.1.cmml" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1">subscript</csymbol><ci id="S4.Ex34.m1.4.4.1.1.2.1.1.1.2.cmml" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1.2">𝑓</ci><ci id="S4.Ex34.m1.4.4.1.1.2.1.1.1.3.cmml" xref="S4.Ex34.m1.4.4.1.1.2.1.1.1.3">𝐺</ci></apply><apply id="S4.Ex34.m1.4.4.1.1.3.2.2.2.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2"><csymbol cd="ambiguous" id="S4.Ex34.m1.4.4.1.1.3.2.2.2.1.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2">subscript</csymbol><ci id="S4.Ex34.m1.4.4.1.1.3.2.2.2.2.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.2">ℬ</ci><apply id="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.3"><times id="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.1.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.1"></times><cn id="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.2.cmml" type="integer" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.2">2</cn><ci id="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.3.cmml" xref="S4.Ex34.m1.4.4.1.1.3.2.2.2.3.3">𝑛</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex34.m1.4c">\rho_{\mathsf{ultra}}(G,\mathcal{G}_{N,N})\;\leq\;\rho_{\mathsf{ultra}}(f_{G},% \mathcal{B}_{2n}),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex34.m1.4d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_G , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT ) ≤ italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmtheorem48.p1.4"><span class="ltx_text ltx_font_italic" id="Thmtheorem48.p1.4.2">where <math alttext="f\colon\{0,1\}^{2n}\to\{0,1\}" class="ltx_Math" display="inline" id="Thmtheorem48.p1.3.1.m1.4"><semantics id="Thmtheorem48.p1.3.1.m1.4a"><mrow id="Thmtheorem48.p1.3.1.m1.4.5" xref="Thmtheorem48.p1.3.1.m1.4.5.cmml"><mi id="Thmtheorem48.p1.3.1.m1.4.5.2" xref="Thmtheorem48.p1.3.1.m1.4.5.2.cmml">f</mi><mo id="Thmtheorem48.p1.3.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="Thmtheorem48.p1.3.1.m1.4.5.1.cmml">:</mo><mrow id="Thmtheorem48.p1.3.1.m1.4.5.3" xref="Thmtheorem48.p1.3.1.m1.4.5.3.cmml"><msup id="Thmtheorem48.p1.3.1.m1.4.5.3.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.cmml"><mrow id="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.1.cmml"><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.2.1" stretchy="false" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.1.cmml">{</mo><mn id="Thmtheorem48.p1.3.1.m1.1.1" xref="Thmtheorem48.p1.3.1.m1.1.1.cmml">0</mn><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.2.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.1.cmml">,</mo><mn id="Thmtheorem48.p1.3.1.m1.2.2" xref="Thmtheorem48.p1.3.1.m1.2.2.cmml">1</mn><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.2.3" stretchy="false" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.1.cmml">}</mo></mrow><mrow id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.cmml"><mn id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.2.cmml">2</mn><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.1" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.1.cmml">⁢</mo><mi id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.3" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.3.cmml">n</mi></mrow></msup><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.1" stretchy="false" xref="Thmtheorem48.p1.3.1.m1.4.5.3.1.cmml">→</mo><mrow id="Thmtheorem48.p1.3.1.m1.4.5.3.3.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.3.1.cmml"><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem48.p1.3.1.m1.4.5.3.3.1.cmml">{</mo><mn id="Thmtheorem48.p1.3.1.m1.3.3" xref="Thmtheorem48.p1.3.1.m1.3.3.cmml">0</mn><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.3.2.2" xref="Thmtheorem48.p1.3.1.m1.4.5.3.3.1.cmml">,</mo><mn id="Thmtheorem48.p1.3.1.m1.4.4" xref="Thmtheorem48.p1.3.1.m1.4.4.cmml">1</mn><mo id="Thmtheorem48.p1.3.1.m1.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem48.p1.3.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem48.p1.3.1.m1.4b"><apply id="Thmtheorem48.p1.3.1.m1.4.5.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5"><ci id="Thmtheorem48.p1.3.1.m1.4.5.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.1">:</ci><ci id="Thmtheorem48.p1.3.1.m1.4.5.2.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.2">𝑓</ci><apply id="Thmtheorem48.p1.3.1.m1.4.5.3.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3"><ci id="Thmtheorem48.p1.3.1.m1.4.5.3.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.1">→</ci><apply id="Thmtheorem48.p1.3.1.m1.4.5.3.2.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="Thmtheorem48.p1.3.1.m1.4.5.3.2.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2">superscript</csymbol><set id="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.2.2"><cn id="Thmtheorem48.p1.3.1.m1.1.1.cmml" type="integer" xref="Thmtheorem48.p1.3.1.m1.1.1">0</cn><cn id="Thmtheorem48.p1.3.1.m1.2.2.cmml" type="integer" xref="Thmtheorem48.p1.3.1.m1.2.2">1</cn></set><apply id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3"><times id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.1"></times><cn id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.2.cmml" type="integer" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.2">2</cn><ci id="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.3.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.2.3.3">𝑛</ci></apply></apply><set id="Thmtheorem48.p1.3.1.m1.4.5.3.3.1.cmml" xref="Thmtheorem48.p1.3.1.m1.4.5.3.3.2"><cn id="Thmtheorem48.p1.3.1.m1.3.3.cmml" type="integer" xref="Thmtheorem48.p1.3.1.m1.3.3">0</cn><cn id="Thmtheorem48.p1.3.1.m1.4.4.cmml" type="integer" xref="Thmtheorem48.p1.3.1.m1.4.4">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem48.p1.3.1.m1.4c">f\colon\{0,1\}^{2n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem48.p1.3.1.m1.4d">italic_f : { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> is the Boolean function associated with <math alttext="G" class="ltx_Math" display="inline" id="Thmtheorem48.p1.4.2.m2.1"><semantics id="Thmtheorem48.p1.4.2.m2.1a"><mi id="Thmtheorem48.p1.4.2.m2.1.1" xref="Thmtheorem48.p1.4.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem48.p1.4.2.m2.1b"><ci id="Thmtheorem48.p1.4.2.m2.1.1.cmml" xref="Thmtheorem48.p1.4.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem48.p1.4.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem48.p1.4.2.m2.1d">italic_G</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS3.2.p1"> <p class="ltx_p" id="S4.SS3.2.p1.3">Recall that, in the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem39" title="Lemma 39 (A fusion transference lemma). ‣ 4.1 Basic results and connections ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">39</span></a> (fusion transference lemma), if a semi-filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS3.2.p1.1.m1.1"><semantics id="S4.SS3.2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.2.p1.1.m1.1.1" xref="S4.SS3.2.p1.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.2.p1.1.m1.1b"><ci id="S4.SS3.2.p1.1.m1.1.1.cmml" xref="S4.SS3.2.p1.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.2.p1.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.2.p1.1.m1.1d">caligraphic_F</annotation></semantics></math> in the graph setting is not covered, then it gives rise to a semi-filter <math alttext="\mathcal{F}^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.2.p1.2.m2.1"><semantics id="S4.SS3.2.p1.2.m2.1a"><msup id="S4.SS3.2.p1.2.m2.1.1" xref="S4.SS3.2.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.2.p1.2.m2.1.1.2" xref="S4.SS3.2.p1.2.m2.1.1.2.cmml">ℱ</mi><mo id="S4.SS3.2.p1.2.m2.1.1.3" xref="S4.SS3.2.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.2.p1.2.m2.1b"><apply id="S4.SS3.2.p1.2.m2.1.1.cmml" xref="S4.SS3.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.2.p1.2.m2.1.1.1.cmml" xref="S4.SS3.2.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.2.p1.2.m2.1.1.2.cmml" xref="S4.SS3.2.p1.2.m2.1.1.2">ℱ</ci><ci id="S4.SS3.2.p1.2.m2.1.1.3.cmml" xref="S4.SS3.2.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.2.p1.2.m2.1c">\mathcal{F}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.2.p1.2.m2.1d">caligraphic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in the Boolean function setting that is not covered. Crucially, if the original semi-filter is a semi-ultra-filter, so is the resulting semi-filter. The proof of this fact is obvious, since <math alttext="\phi\colon[N]\times[N]\to\{0,1\}^{2n}" class="ltx_Math" display="inline" id="S4.SS3.2.p1.3.m3.4"><semantics id="S4.SS3.2.p1.3.m3.4a"><mrow id="S4.SS3.2.p1.3.m3.4.5" xref="S4.SS3.2.p1.3.m3.4.5.cmml"><mi id="S4.SS3.2.p1.3.m3.4.5.2" xref="S4.SS3.2.p1.3.m3.4.5.2.cmml">ϕ</mi><mo id="S4.SS3.2.p1.3.m3.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.2.p1.3.m3.4.5.1.cmml">:</mo><mrow id="S4.SS3.2.p1.3.m3.4.5.3" xref="S4.SS3.2.p1.3.m3.4.5.3.cmml"><mrow id="S4.SS3.2.p1.3.m3.4.5.3.2" xref="S4.SS3.2.p1.3.m3.4.5.3.2.cmml"><mrow id="S4.SS3.2.p1.3.m3.4.5.3.2.2.2" xref="S4.SS3.2.p1.3.m3.4.5.3.2.2.1.cmml"><mo id="S4.SS3.2.p1.3.m3.4.5.3.2.2.2.1" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.2.2.1.1.cmml">[</mo><mi id="S4.SS3.2.p1.3.m3.1.1" xref="S4.SS3.2.p1.3.m3.1.1.cmml">N</mi><mo id="S4.SS3.2.p1.3.m3.4.5.3.2.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.2.2.1.1.cmml">]</mo></mrow><mo id="S4.SS3.2.p1.3.m3.4.5.3.2.1" rspace="0.222em" xref="S4.SS3.2.p1.3.m3.4.5.3.2.1.cmml">×</mo><mrow id="S4.SS3.2.p1.3.m3.4.5.3.2.3.2" xref="S4.SS3.2.p1.3.m3.4.5.3.2.3.1.cmml"><mo id="S4.SS3.2.p1.3.m3.4.5.3.2.3.2.1" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.2.3.1.1.cmml">[</mo><mi id="S4.SS3.2.p1.3.m3.2.2" xref="S4.SS3.2.p1.3.m3.2.2.cmml">N</mi><mo id="S4.SS3.2.p1.3.m3.4.5.3.2.3.2.2" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S4.SS3.2.p1.3.m3.4.5.3.1" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.1.cmml">→</mo><msup id="S4.SS3.2.p1.3.m3.4.5.3.3" xref="S4.SS3.2.p1.3.m3.4.5.3.3.cmml"><mrow id="S4.SS3.2.p1.3.m3.4.5.3.3.2.2" xref="S4.SS3.2.p1.3.m3.4.5.3.3.2.1.cmml"><mo id="S4.SS3.2.p1.3.m3.4.5.3.3.2.2.1" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.3.2.1.cmml">{</mo><mn id="S4.SS3.2.p1.3.m3.3.3" xref="S4.SS3.2.p1.3.m3.3.3.cmml">0</mn><mo id="S4.SS3.2.p1.3.m3.4.5.3.3.2.2.2" xref="S4.SS3.2.p1.3.m3.4.5.3.3.2.1.cmml">,</mo><mn id="S4.SS3.2.p1.3.m3.4.4" xref="S4.SS3.2.p1.3.m3.4.4.cmml">1</mn><mo id="S4.SS3.2.p1.3.m3.4.5.3.3.2.2.3" stretchy="false" xref="S4.SS3.2.p1.3.m3.4.5.3.3.2.1.cmml">}</mo></mrow><mrow id="S4.SS3.2.p1.3.m3.4.5.3.3.3" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.cmml"><mn id="S4.SS3.2.p1.3.m3.4.5.3.3.3.2" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.2.cmml">2</mn><mo id="S4.SS3.2.p1.3.m3.4.5.3.3.3.1" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.2.p1.3.m3.4.5.3.3.3.3" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.3.cmml">n</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.2.p1.3.m3.4b"><apply id="S4.SS3.2.p1.3.m3.4.5.cmml" xref="S4.SS3.2.p1.3.m3.4.5"><ci id="S4.SS3.2.p1.3.m3.4.5.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.1">:</ci><ci id="S4.SS3.2.p1.3.m3.4.5.2.cmml" xref="S4.SS3.2.p1.3.m3.4.5.2">italic-ϕ</ci><apply id="S4.SS3.2.p1.3.m3.4.5.3.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3"><ci id="S4.SS3.2.p1.3.m3.4.5.3.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.1">→</ci><apply id="S4.SS3.2.p1.3.m3.4.5.3.2.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2"><times id="S4.SS3.2.p1.3.m3.4.5.3.2.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2.1"></times><apply id="S4.SS3.2.p1.3.m3.4.5.3.2.2.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2.2.2"><csymbol cd="latexml" id="S4.SS3.2.p1.3.m3.4.5.3.2.2.1.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2.2.2.1">delimited-[]</csymbol><ci id="S4.SS3.2.p1.3.m3.1.1.cmml" xref="S4.SS3.2.p1.3.m3.1.1">𝑁</ci></apply><apply id="S4.SS3.2.p1.3.m3.4.5.3.2.3.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2.3.2"><csymbol cd="latexml" id="S4.SS3.2.p1.3.m3.4.5.3.2.3.1.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.2.3.2.1">delimited-[]</csymbol><ci id="S4.SS3.2.p1.3.m3.2.2.cmml" xref="S4.SS3.2.p1.3.m3.2.2">𝑁</ci></apply></apply><apply id="S4.SS3.2.p1.3.m3.4.5.3.3.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3"><csymbol cd="ambiguous" id="S4.SS3.2.p1.3.m3.4.5.3.3.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3">superscript</csymbol><set id="S4.SS3.2.p1.3.m3.4.5.3.3.2.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3.2.2"><cn id="S4.SS3.2.p1.3.m3.3.3.cmml" type="integer" xref="S4.SS3.2.p1.3.m3.3.3">0</cn><cn id="S4.SS3.2.p1.3.m3.4.4.cmml" type="integer" xref="S4.SS3.2.p1.3.m3.4.4">1</cn></set><apply id="S4.SS3.2.p1.3.m3.4.5.3.3.3.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3"><times id="S4.SS3.2.p1.3.m3.4.5.3.3.3.1.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.1"></times><cn id="S4.SS3.2.p1.3.m3.4.5.3.3.3.2.cmml" type="integer" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.2">2</cn><ci id="S4.SS3.2.p1.3.m3.4.5.3.3.3.3.cmml" xref="S4.SS3.2.p1.3.m3.4.5.3.3.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.2.p1.3.m3.4c">\phi\colon[N]\times[N]\to\{0,1\}^{2n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.2.p1.3.m3.4d">italic_ϕ : [ italic_N ] × [ italic_N ] → { 0 , 1 } start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a bijection. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS3.p8"> <p class="ltx_p" id="S4.SS3.p8.4">Let <math alttext="\mathsf{NEQ}_{2n}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}" class="ltx_Math" display="inline" id="S4.SS3.p8.1.m1.6"><semantics id="S4.SS3.p8.1.m1.6a"><mrow id="S4.SS3.p8.1.m1.6.7" xref="S4.SS3.p8.1.m1.6.7.cmml"><msub id="S4.SS3.p8.1.m1.6.7.2" xref="S4.SS3.p8.1.m1.6.7.2.cmml"><mi id="S4.SS3.p8.1.m1.6.7.2.2" xref="S4.SS3.p8.1.m1.6.7.2.2.cmml">𝖭𝖤𝖰</mi><mrow id="S4.SS3.p8.1.m1.6.7.2.3" xref="S4.SS3.p8.1.m1.6.7.2.3.cmml"><mn id="S4.SS3.p8.1.m1.6.7.2.3.2" xref="S4.SS3.p8.1.m1.6.7.2.3.2.cmml">2</mn><mo id="S4.SS3.p8.1.m1.6.7.2.3.1" xref="S4.SS3.p8.1.m1.6.7.2.3.1.cmml">⁢</mo><mi id="S4.SS3.p8.1.m1.6.7.2.3.3" xref="S4.SS3.p8.1.m1.6.7.2.3.3.cmml">n</mi></mrow></msub><mo id="S4.SS3.p8.1.m1.6.7.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p8.1.m1.6.7.1.cmml">:</mo><mrow id="S4.SS3.p8.1.m1.6.7.3" xref="S4.SS3.p8.1.m1.6.7.3.cmml"><mrow id="S4.SS3.p8.1.m1.6.7.3.2" xref="S4.SS3.p8.1.m1.6.7.3.2.cmml"><msup id="S4.SS3.p8.1.m1.6.7.3.2.2" xref="S4.SS3.p8.1.m1.6.7.3.2.2.cmml"><mrow id="S4.SS3.p8.1.m1.6.7.3.2.2.2.2" xref="S4.SS3.p8.1.m1.6.7.3.2.2.2.1.cmml"><mo id="S4.SS3.p8.1.m1.6.7.3.2.2.2.2.1" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.2.2.2.1.cmml">{</mo><mn id="S4.SS3.p8.1.m1.1.1" xref="S4.SS3.p8.1.m1.1.1.cmml">0</mn><mo id="S4.SS3.p8.1.m1.6.7.3.2.2.2.2.2" xref="S4.SS3.p8.1.m1.6.7.3.2.2.2.1.cmml">,</mo><mn id="S4.SS3.p8.1.m1.2.2" xref="S4.SS3.p8.1.m1.2.2.cmml">1</mn><mo id="S4.SS3.p8.1.m1.6.7.3.2.2.2.2.3" rspace="0.055em" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.2.2.2.1.cmml">}</mo></mrow><mi id="S4.SS3.p8.1.m1.6.7.3.2.2.3" xref="S4.SS3.p8.1.m1.6.7.3.2.2.3.cmml">n</mi></msup><mo id="S4.SS3.p8.1.m1.6.7.3.2.1" rspace="0.222em" xref="S4.SS3.p8.1.m1.6.7.3.2.1.cmml">×</mo><msup id="S4.SS3.p8.1.m1.6.7.3.2.3" xref="S4.SS3.p8.1.m1.6.7.3.2.3.cmml"><mrow id="S4.SS3.p8.1.m1.6.7.3.2.3.2.2" xref="S4.SS3.p8.1.m1.6.7.3.2.3.2.1.cmml"><mo id="S4.SS3.p8.1.m1.6.7.3.2.3.2.2.1" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.2.3.2.1.cmml">{</mo><mn id="S4.SS3.p8.1.m1.3.3" xref="S4.SS3.p8.1.m1.3.3.cmml">0</mn><mo id="S4.SS3.p8.1.m1.6.7.3.2.3.2.2.2" xref="S4.SS3.p8.1.m1.6.7.3.2.3.2.1.cmml">,</mo><mn id="S4.SS3.p8.1.m1.4.4" xref="S4.SS3.p8.1.m1.4.4.cmml">1</mn><mo id="S4.SS3.p8.1.m1.6.7.3.2.3.2.2.3" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.2.3.2.1.cmml">}</mo></mrow><mi id="S4.SS3.p8.1.m1.6.7.3.2.3.3" xref="S4.SS3.p8.1.m1.6.7.3.2.3.3.cmml">n</mi></msup></mrow><mo id="S4.SS3.p8.1.m1.6.7.3.1" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.1.cmml">→</mo><mrow id="S4.SS3.p8.1.m1.6.7.3.3.2" xref="S4.SS3.p8.1.m1.6.7.3.3.1.cmml"><mo id="S4.SS3.p8.1.m1.6.7.3.3.2.1" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.3.1.cmml">{</mo><mn id="S4.SS3.p8.1.m1.5.5" xref="S4.SS3.p8.1.m1.5.5.cmml">0</mn><mo id="S4.SS3.p8.1.m1.6.7.3.3.2.2" xref="S4.SS3.p8.1.m1.6.7.3.3.1.cmml">,</mo><mn id="S4.SS3.p8.1.m1.6.6" xref="S4.SS3.p8.1.m1.6.6.cmml">1</mn><mo id="S4.SS3.p8.1.m1.6.7.3.3.2.3" stretchy="false" xref="S4.SS3.p8.1.m1.6.7.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml 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xref="S4.SS3.p8.1.m1.4.4">1</cn></set><ci id="S4.SS3.p8.1.m1.6.7.3.2.3.3.cmml" xref="S4.SS3.p8.1.m1.6.7.3.2.3.3">𝑛</ci></apply></apply><set id="S4.SS3.p8.1.m1.6.7.3.3.1.cmml" xref="S4.SS3.p8.1.m1.6.7.3.3.2"><cn id="S4.SS3.p8.1.m1.5.5.cmml" type="integer" xref="S4.SS3.p8.1.m1.5.5">0</cn><cn id="S4.SS3.p8.1.m1.6.6.cmml" type="integer" xref="S4.SS3.p8.1.m1.6.6">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.1.m1.6c">\mathsf{NEQ}_{2n}\colon\{0,1\}^{n}\times\{0,1\}^{n}\to\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.1.m1.6d">sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT : { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → { 0 , 1 }</annotation></semantics></math> be the function such that <math alttext="\mathsf{NEQ}_{2n}(x,y)=1" class="ltx_Math" display="inline" id="S4.SS3.p8.2.m2.2"><semantics id="S4.SS3.p8.2.m2.2a"><mrow id="S4.SS3.p8.2.m2.2.3" xref="S4.SS3.p8.2.m2.2.3.cmml"><mrow id="S4.SS3.p8.2.m2.2.3.2" xref="S4.SS3.p8.2.m2.2.3.2.cmml"><msub id="S4.SS3.p8.2.m2.2.3.2.2" xref="S4.SS3.p8.2.m2.2.3.2.2.cmml"><mi id="S4.SS3.p8.2.m2.2.3.2.2.2" xref="S4.SS3.p8.2.m2.2.3.2.2.2.cmml">𝖭𝖤𝖰</mi><mrow id="S4.SS3.p8.2.m2.2.3.2.2.3" xref="S4.SS3.p8.2.m2.2.3.2.2.3.cmml"><mn id="S4.SS3.p8.2.m2.2.3.2.2.3.2" xref="S4.SS3.p8.2.m2.2.3.2.2.3.2.cmml">2</mn><mo id="S4.SS3.p8.2.m2.2.3.2.2.3.1" xref="S4.SS3.p8.2.m2.2.3.2.2.3.1.cmml">⁢</mo><mi id="S4.SS3.p8.2.m2.2.3.2.2.3.3" xref="S4.SS3.p8.2.m2.2.3.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S4.SS3.p8.2.m2.2.3.2.1" xref="S4.SS3.p8.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS3.p8.2.m2.2.3.2.3.2" xref="S4.SS3.p8.2.m2.2.3.2.3.1.cmml"><mo id="S4.SS3.p8.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.SS3.p8.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.SS3.p8.2.m2.1.1" xref="S4.SS3.p8.2.m2.1.1.cmml">x</mi><mo id="S4.SS3.p8.2.m2.2.3.2.3.2.2" xref="S4.SS3.p8.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.SS3.p8.2.m2.2.2" xref="S4.SS3.p8.2.m2.2.2.cmml">y</mi><mo id="S4.SS3.p8.2.m2.2.3.2.3.2.3" stretchy="false" xref="S4.SS3.p8.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p8.2.m2.2.3.1" xref="S4.SS3.p8.2.m2.2.3.1.cmml">=</mo><mn id="S4.SS3.p8.2.m2.2.3.3" xref="S4.SS3.p8.2.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p8.2.m2.2b"><apply id="S4.SS3.p8.2.m2.2.3.cmml" xref="S4.SS3.p8.2.m2.2.3"><eq id="S4.SS3.p8.2.m2.2.3.1.cmml" xref="S4.SS3.p8.2.m2.2.3.1"></eq><apply id="S4.SS3.p8.2.m2.2.3.2.cmml" xref="S4.SS3.p8.2.m2.2.3.2"><times id="S4.SS3.p8.2.m2.2.3.2.1.cmml" xref="S4.SS3.p8.2.m2.2.3.2.1"></times><apply id="S4.SS3.p8.2.m2.2.3.2.2.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS3.p8.2.m2.2.3.2.2.1.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2">subscript</csymbol><ci id="S4.SS3.p8.2.m2.2.3.2.2.2.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2.2">𝖭𝖤𝖰</ci><apply id="S4.SS3.p8.2.m2.2.3.2.2.3.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2.3"><times id="S4.SS3.p8.2.m2.2.3.2.2.3.1.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2.3.1"></times><cn id="S4.SS3.p8.2.m2.2.3.2.2.3.2.cmml" type="integer" xref="S4.SS3.p8.2.m2.2.3.2.2.3.2">2</cn><ci id="S4.SS3.p8.2.m2.2.3.2.2.3.3.cmml" xref="S4.SS3.p8.2.m2.2.3.2.2.3.3">𝑛</ci></apply></apply><interval closure="open" id="S4.SS3.p8.2.m2.2.3.2.3.1.cmml" xref="S4.SS3.p8.2.m2.2.3.2.3.2"><ci id="S4.SS3.p8.2.m2.1.1.cmml" xref="S4.SS3.p8.2.m2.1.1">𝑥</ci><ci id="S4.SS3.p8.2.m2.2.2.cmml" xref="S4.SS3.p8.2.m2.2.2">𝑦</ci></interval></apply><cn id="S4.SS3.p8.2.m2.2.3.3.cmml" type="integer" xref="S4.SS3.p8.2.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.2.m2.2c">\mathsf{NEQ}_{2n}(x,y)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.2.m2.2d">sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ( italic_x , italic_y ) = 1</annotation></semantics></math> if and only if <math alttext="x\neq y" class="ltx_Math" display="inline" id="S4.SS3.p8.3.m3.1"><semantics id="S4.SS3.p8.3.m3.1a"><mrow id="S4.SS3.p8.3.m3.1.1" xref="S4.SS3.p8.3.m3.1.1.cmml"><mi id="S4.SS3.p8.3.m3.1.1.2" xref="S4.SS3.p8.3.m3.1.1.2.cmml">x</mi><mo id="S4.SS3.p8.3.m3.1.1.1" xref="S4.SS3.p8.3.m3.1.1.1.cmml">≠</mo><mi id="S4.SS3.p8.3.m3.1.1.3" xref="S4.SS3.p8.3.m3.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p8.3.m3.1b"><apply id="S4.SS3.p8.3.m3.1.1.cmml" xref="S4.SS3.p8.3.m3.1.1"><neq id="S4.SS3.p8.3.m3.1.1.1.cmml" xref="S4.SS3.p8.3.m3.1.1.1"></neq><ci id="S4.SS3.p8.3.m3.1.1.2.cmml" xref="S4.SS3.p8.3.m3.1.1.2">𝑥</ci><ci id="S4.SS3.p8.3.m3.1.1.3.cmml" xref="S4.SS3.p8.3.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.3.m3.1c">x\neq y</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.3.m3.1d">italic_x ≠ italic_y</annotation></semantics></math>, and <math alttext="\mathsf{EQ}_{2n}" class="ltx_Math" display="inline" id="S4.SS3.p8.4.m4.1"><semantics id="S4.SS3.p8.4.m4.1a"><msub id="S4.SS3.p8.4.m4.1.1" xref="S4.SS3.p8.4.m4.1.1.cmml"><mi id="S4.SS3.p8.4.m4.1.1.2" xref="S4.SS3.p8.4.m4.1.1.2.cmml">𝖤𝖰</mi><mrow id="S4.SS3.p8.4.m4.1.1.3" xref="S4.SS3.p8.4.m4.1.1.3.cmml"><mn id="S4.SS3.p8.4.m4.1.1.3.2" xref="S4.SS3.p8.4.m4.1.1.3.2.cmml">2</mn><mo id="S4.SS3.p8.4.m4.1.1.3.1" xref="S4.SS3.p8.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p8.4.m4.1.1.3.3" xref="S4.SS3.p8.4.m4.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p8.4.m4.1b"><apply id="S4.SS3.p8.4.m4.1.1.cmml" xref="S4.SS3.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p8.4.m4.1.1.1.cmml" xref="S4.SS3.p8.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p8.4.m4.1.1.2.cmml" xref="S4.SS3.p8.4.m4.1.1.2">𝖤𝖰</ci><apply id="S4.SS3.p8.4.m4.1.1.3.cmml" xref="S4.SS3.p8.4.m4.1.1.3"><times id="S4.SS3.p8.4.m4.1.1.3.1.cmml" xref="S4.SS3.p8.4.m4.1.1.3.1"></times><cn id="S4.SS3.p8.4.m4.1.1.3.2.cmml" type="integer" xref="S4.SS3.p8.4.m4.1.1.3.2">2</cn><ci id="S4.SS3.p8.4.m4.1.1.3.3.cmml" xref="S4.SS3.p8.4.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.4.m4.1c">\mathsf{EQ}_{2n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.4.m4.1d">sansserif_EQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math> be its negation. By combining the ideas of this section and Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#S4.SS2" title="4.2 A simple lower bound example ‣ 4 Graph Complexity and Two-Dimensional Cover Problems ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">4.2</span></a>, we get the following tight inequalities.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmtheorem49"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem49.1.1.1">Corollary 49</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem49.2.2"> </span>(A simple nondeterministic lower bound via graph complexity + fusion)<span class="ltx_text ltx_font_bold" id="Thmtheorem49.3.3">.</span> </h6> <div class="ltx_para ltx_noindent" id="Thmtheorem49.p1"> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S4.EGx6"> <tbody id="S4.Ex42"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle n" class="ltx_Math" display="inline" id="S4.Ex35.m1.1"><semantics id="S4.Ex35.m1.1a"><mi id="S4.Ex35.m1.1.1" xref="S4.Ex35.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.Ex35.m1.1b"><ci id="S4.Ex35.m1.1.1.cmml" xref="S4.Ex35.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex35.m1.1c">\displaystyle n</annotation><annotation encoding="application/x-llamapun" id="S4.Ex35.m1.1d">italic_n</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex35.m2.1"><semantics id="S4.Ex35.m2.1a"><mo id="S4.Ex35.m2.1.1" xref="S4.Ex35.m2.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex35.m2.1b"><leq id="S4.Ex35.m2.1.1.cmml" xref="S4.Ex35.m2.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex35.m2.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex35.m2.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rho_{\mathsf{can}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.Ex35.m3.4"><semantics id="S4.Ex35.m3.4a"><mrow id="S4.Ex35.m3.4.4" xref="S4.Ex35.m3.4.4.cmml"><msub id="S4.Ex35.m3.4.4.4" xref="S4.Ex35.m3.4.4.4.cmml"><mi id="S4.Ex35.m3.4.4.4.2" xref="S4.Ex35.m3.4.4.4.2.cmml">ρ</mi><mi id="S4.Ex35.m3.4.4.4.3" xref="S4.Ex35.m3.4.4.4.3.cmml">𝖼𝖺𝗇</mi></msub><mo id="S4.Ex35.m3.4.4.3" xref="S4.Ex35.m3.4.4.3.cmml">⁢</mo><mrow id="S4.Ex35.m3.4.4.2.2" xref="S4.Ex35.m3.4.4.2.3.cmml"><mo id="S4.Ex35.m3.4.4.2.2.3" stretchy="false" xref="S4.Ex35.m3.4.4.2.3.cmml">(</mo><msub id="S4.Ex35.m3.3.3.1.1.1" xref="S4.Ex35.m3.3.3.1.1.1.cmml"><mi id="S4.Ex35.m3.3.3.1.1.1.2" xref="S4.Ex35.m3.3.3.1.1.1.2.cmml">G</mi><mi id="S4.Ex35.m3.3.3.1.1.1.3" xref="S4.Ex35.m3.3.3.1.1.1.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="S4.Ex35.m3.4.4.2.2.4" xref="S4.Ex35.m3.4.4.2.3.cmml">,</mo><msub id="S4.Ex35.m3.4.4.2.2.2" xref="S4.Ex35.m3.4.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex35.m3.4.4.2.2.2.2" xref="S4.Ex35.m3.4.4.2.2.2.2.cmml">𝒢</mi><mrow id="S4.Ex35.m3.2.2.2.4" xref="S4.Ex35.m3.2.2.2.3.cmml"><mi id="S4.Ex35.m3.1.1.1.1" xref="S4.Ex35.m3.1.1.1.1.cmml">N</mi><mo id="S4.Ex35.m3.2.2.2.4.1" xref="S4.Ex35.m3.2.2.2.3.cmml">,</mo><mi id="S4.Ex35.m3.2.2.2.2" xref="S4.Ex35.m3.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.Ex35.m3.4.4.2.2.5" stretchy="false" xref="S4.Ex35.m3.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex35.m3.4b"><apply id="S4.Ex35.m3.4.4.cmml" xref="S4.Ex35.m3.4.4"><times id="S4.Ex35.m3.4.4.3.cmml" xref="S4.Ex35.m3.4.4.3"></times><apply id="S4.Ex35.m3.4.4.4.cmml" xref="S4.Ex35.m3.4.4.4"><csymbol cd="ambiguous" id="S4.Ex35.m3.4.4.4.1.cmml" xref="S4.Ex35.m3.4.4.4">subscript</csymbol><ci id="S4.Ex35.m3.4.4.4.2.cmml" xref="S4.Ex35.m3.4.4.4.2">𝜌</ci><ci id="S4.Ex35.m3.4.4.4.3.cmml" xref="S4.Ex35.m3.4.4.4.3">𝖼𝖺𝗇</ci></apply><interval closure="open" id="S4.Ex35.m3.4.4.2.3.cmml" xref="S4.Ex35.m3.4.4.2.2"><apply id="S4.Ex35.m3.3.3.1.1.1.cmml" xref="S4.Ex35.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex35.m3.3.3.1.1.1.1.cmml" xref="S4.Ex35.m3.3.3.1.1.1">subscript</csymbol><ci id="S4.Ex35.m3.3.3.1.1.1.2.cmml" xref="S4.Ex35.m3.3.3.1.1.1.2">𝐺</ci><ci id="S4.Ex35.m3.3.3.1.1.1.3.cmml" xref="S4.Ex35.m3.3.3.1.1.1.3">𝖭𝖤𝖰</ci></apply><apply id="S4.Ex35.m3.4.4.2.2.2.cmml" xref="S4.Ex35.m3.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.Ex35.m3.4.4.2.2.2.1.cmml" xref="S4.Ex35.m3.4.4.2.2.2">subscript</csymbol><ci id="S4.Ex35.m3.4.4.2.2.2.2.cmml" xref="S4.Ex35.m3.4.4.2.2.2.2">𝒢</ci><list id="S4.Ex35.m3.2.2.2.3.cmml" xref="S4.Ex35.m3.2.2.2.4"><ci id="S4.Ex35.m3.1.1.1.1.cmml" xref="S4.Ex35.m3.1.1.1.1">𝑁</ci><ci id="S4.Ex35.m3.2.2.2.2.cmml" xref="S4.Ex35.m3.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex35.m3.4c">\displaystyle\rho_{\mathsf{can}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex35.m3.4d">italic_ρ start_POSTSUBSCRIPT sansserif_can end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex36.m1.1"><semantics id="S4.Ex36.m1.1a"><mo id="S4.Ex36.m1.1.1" xref="S4.Ex36.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex36.m1.1b"><leq id="S4.Ex36.m1.1.1.cmml" xref="S4.Ex36.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex36.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex36.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rho_{\mathsf{ultra}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})" class="ltx_Math" display="inline" id="S4.Ex36.m2.4"><semantics id="S4.Ex36.m2.4a"><mrow id="S4.Ex36.m2.4.4" xref="S4.Ex36.m2.4.4.cmml"><msub id="S4.Ex36.m2.4.4.4" xref="S4.Ex36.m2.4.4.4.cmml"><mi id="S4.Ex36.m2.4.4.4.2" xref="S4.Ex36.m2.4.4.4.2.cmml">ρ</mi><mi id="S4.Ex36.m2.4.4.4.3" xref="S4.Ex36.m2.4.4.4.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.Ex36.m2.4.4.3" xref="S4.Ex36.m2.4.4.3.cmml">⁢</mo><mrow id="S4.Ex36.m2.4.4.2.2" xref="S4.Ex36.m2.4.4.2.3.cmml"><mo id="S4.Ex36.m2.4.4.2.2.3" stretchy="false" xref="S4.Ex36.m2.4.4.2.3.cmml">(</mo><msub id="S4.Ex36.m2.3.3.1.1.1" xref="S4.Ex36.m2.3.3.1.1.1.cmml"><mi id="S4.Ex36.m2.3.3.1.1.1.2" xref="S4.Ex36.m2.3.3.1.1.1.2.cmml">G</mi><mi id="S4.Ex36.m2.3.3.1.1.1.3" xref="S4.Ex36.m2.3.3.1.1.1.3.cmml">𝖭𝖤𝖰</mi></msub><mo id="S4.Ex36.m2.4.4.2.2.4" xref="S4.Ex36.m2.4.4.2.3.cmml">,</mo><msub id="S4.Ex36.m2.4.4.2.2.2" xref="S4.Ex36.m2.4.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex36.m2.4.4.2.2.2.2" xref="S4.Ex36.m2.4.4.2.2.2.2.cmml">𝒢</mi><mrow id="S4.Ex36.m2.2.2.2.4" xref="S4.Ex36.m2.2.2.2.3.cmml"><mi id="S4.Ex36.m2.1.1.1.1" xref="S4.Ex36.m2.1.1.1.1.cmml">N</mi><mo id="S4.Ex36.m2.2.2.2.4.1" xref="S4.Ex36.m2.2.2.2.3.cmml">,</mo><mi id="S4.Ex36.m2.2.2.2.2" xref="S4.Ex36.m2.2.2.2.2.cmml">N</mi></mrow></msub><mo id="S4.Ex36.m2.4.4.2.2.5" stretchy="false" xref="S4.Ex36.m2.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex36.m2.4b"><apply id="S4.Ex36.m2.4.4.cmml" xref="S4.Ex36.m2.4.4"><times id="S4.Ex36.m2.4.4.3.cmml" xref="S4.Ex36.m2.4.4.3"></times><apply id="S4.Ex36.m2.4.4.4.cmml" xref="S4.Ex36.m2.4.4.4"><csymbol cd="ambiguous" id="S4.Ex36.m2.4.4.4.1.cmml" xref="S4.Ex36.m2.4.4.4">subscript</csymbol><ci id="S4.Ex36.m2.4.4.4.2.cmml" xref="S4.Ex36.m2.4.4.4.2">𝜌</ci><ci id="S4.Ex36.m2.4.4.4.3.cmml" xref="S4.Ex36.m2.4.4.4.3">𝗎𝗅𝗍𝗋𝖺</ci></apply><interval closure="open" id="S4.Ex36.m2.4.4.2.3.cmml" xref="S4.Ex36.m2.4.4.2.2"><apply id="S4.Ex36.m2.3.3.1.1.1.cmml" xref="S4.Ex36.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex36.m2.3.3.1.1.1.1.cmml" xref="S4.Ex36.m2.3.3.1.1.1">subscript</csymbol><ci id="S4.Ex36.m2.3.3.1.1.1.2.cmml" xref="S4.Ex36.m2.3.3.1.1.1.2">𝐺</ci><ci id="S4.Ex36.m2.3.3.1.1.1.3.cmml" xref="S4.Ex36.m2.3.3.1.1.1.3">𝖭𝖤𝖰</ci></apply><apply id="S4.Ex36.m2.4.4.2.2.2.cmml" xref="S4.Ex36.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.Ex36.m2.4.4.2.2.2.1.cmml" xref="S4.Ex36.m2.4.4.2.2.2">subscript</csymbol><ci id="S4.Ex36.m2.4.4.2.2.2.2.cmml" xref="S4.Ex36.m2.4.4.2.2.2.2">𝒢</ci><list id="S4.Ex36.m2.2.2.2.3.cmml" xref="S4.Ex36.m2.2.2.2.4"><ci id="S4.Ex36.m2.1.1.1.1.cmml" xref="S4.Ex36.m2.1.1.1.1">𝑁</ci><ci id="S4.Ex36.m2.2.2.2.2.cmml" xref="S4.Ex36.m2.2.2.2.2">𝑁</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex36.m2.4c">\displaystyle\rho_{\mathsf{ultra}}(G_{\mathsf{NEQ}},\mathcal{G}_{N,N})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex36.m2.4d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( italic_G start_POSTSUBSCRIPT sansserif_NEQ end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT italic_N , italic_N end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex37.m1.1"><semantics id="S4.Ex37.m1.1a"><mo id="S4.Ex37.m1.1.1" xref="S4.Ex37.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex37.m1.1b"><leq id="S4.Ex37.m1.1.1.cmml" xref="S4.Ex37.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex37.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex37.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rho_{\mathsf{ultra}}(\mathsf{NEQ}_{2n},\mathcal{B}_{2n})" class="ltx_Math" display="inline" id="S4.Ex37.m2.2"><semantics id="S4.Ex37.m2.2a"><mrow id="S4.Ex37.m2.2.2" xref="S4.Ex37.m2.2.2.cmml"><msub id="S4.Ex37.m2.2.2.4" xref="S4.Ex37.m2.2.2.4.cmml"><mi id="S4.Ex37.m2.2.2.4.2" xref="S4.Ex37.m2.2.2.4.2.cmml">ρ</mi><mi id="S4.Ex37.m2.2.2.4.3" xref="S4.Ex37.m2.2.2.4.3.cmml">𝗎𝗅𝗍𝗋𝖺</mi></msub><mo id="S4.Ex37.m2.2.2.3" 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xref="S4.Ex37.m2.2.2.2.2.2.3.2.cmml">2</mn><mo id="S4.Ex37.m2.2.2.2.2.2.3.1" xref="S4.Ex37.m2.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex37.m2.2.2.2.2.2.3.3" xref="S4.Ex37.m2.2.2.2.2.2.3.3.cmml">n</mi></mrow></msub><mo id="S4.Ex37.m2.2.2.2.2.5" stretchy="false" xref="S4.Ex37.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex37.m2.2b"><apply id="S4.Ex37.m2.2.2.cmml" xref="S4.Ex37.m2.2.2"><times id="S4.Ex37.m2.2.2.3.cmml" xref="S4.Ex37.m2.2.2.3"></times><apply id="S4.Ex37.m2.2.2.4.cmml" xref="S4.Ex37.m2.2.2.4"><csymbol cd="ambiguous" id="S4.Ex37.m2.2.2.4.1.cmml" xref="S4.Ex37.m2.2.2.4">subscript</csymbol><ci id="S4.Ex37.m2.2.2.4.2.cmml" xref="S4.Ex37.m2.2.2.4.2">𝜌</ci><ci id="S4.Ex37.m2.2.2.4.3.cmml" xref="S4.Ex37.m2.2.2.4.3">𝗎𝗅𝗍𝗋𝖺</ci></apply><interval closure="open" id="S4.Ex37.m2.2.2.2.3.cmml" xref="S4.Ex37.m2.2.2.2.2"><apply id="S4.Ex37.m2.1.1.1.1.1.cmml" xref="S4.Ex37.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex37.m2.1.1.1.1.1.1.cmml" xref="S4.Ex37.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex37.m2.1.1.1.1.1.2.cmml" xref="S4.Ex37.m2.1.1.1.1.1.2">𝖭𝖤𝖰</ci><apply id="S4.Ex37.m2.1.1.1.1.1.3.cmml" xref="S4.Ex37.m2.1.1.1.1.1.3"><times id="S4.Ex37.m2.1.1.1.1.1.3.1.cmml" xref="S4.Ex37.m2.1.1.1.1.1.3.1"></times><cn id="S4.Ex37.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.Ex37.m2.1.1.1.1.1.3.2">2</cn><ci id="S4.Ex37.m2.1.1.1.1.1.3.3.cmml" xref="S4.Ex37.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="S4.Ex37.m2.2.2.2.2.2.cmml" xref="S4.Ex37.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex37.m2.2.2.2.2.2.1.cmml" xref="S4.Ex37.m2.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex37.m2.2.2.2.2.2.2.cmml" xref="S4.Ex37.m2.2.2.2.2.2.2">ℬ</ci><apply id="S4.Ex37.m2.2.2.2.2.2.3.cmml" xref="S4.Ex37.m2.2.2.2.2.2.3"><times id="S4.Ex37.m2.2.2.2.2.2.3.1.cmml" xref="S4.Ex37.m2.2.2.2.2.2.3.1"></times><cn id="S4.Ex37.m2.2.2.2.2.2.3.2.cmml" type="integer" xref="S4.Ex37.m2.2.2.2.2.2.3.2">2</cn><ci id="S4.Ex37.m2.2.2.2.2.2.3.3.cmml" xref="S4.Ex37.m2.2.2.2.2.2.3.3">𝑛</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex37.m2.2c">\displaystyle\rho_{\mathsf{ultra}}(\mathsf{NEQ}_{2n},\mathcal{B}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex37.m2.2d">italic_ρ start_POSTSUBSCRIPT sansserif_ultra end_POSTSUBSCRIPT ( sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT , caligraphic_B start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex38.m1.1"><semantics id="S4.Ex38.m1.1a"><mo id="S4.Ex38.m1.1.1" xref="S4.Ex38.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex38.m1.1b"><leq id="S4.Ex38.m1.1.1.cmml" xref="S4.Ex38.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex38.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex38.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathsf{conondet}\text{-}\mathsf{size}_{\wedge}(\mathsf{NEQ}_{2n})" class="ltx_Math" display="inline" id="S4.Ex38.m2.1"><semantics id="S4.Ex38.m2.1a"><mrow id="S4.Ex38.m2.1.1" xref="S4.Ex38.m2.1.1.cmml"><mi id="S4.Ex38.m2.1.1.3" xref="S4.Ex38.m2.1.1.3.cmml">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</mi><mo id="S4.Ex38.m2.1.1.2" xref="S4.Ex38.m2.1.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S4.Ex38.m2.1.1.4" xref="S4.Ex38.m2.1.1.4a.cmml">-</mtext><mo id="S4.Ex38.m2.1.1.2a" xref="S4.Ex38.m2.1.1.2.cmml">⁢</mo><msub id="S4.Ex38.m2.1.1.5" xref="S4.Ex38.m2.1.1.5.cmml"><mi id="S4.Ex38.m2.1.1.5.2" xref="S4.Ex38.m2.1.1.5.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex38.m2.1.1.5.3" xref="S4.Ex38.m2.1.1.5.3.cmml">∧</mo></msub><mo id="S4.Ex38.m2.1.1.2b" xref="S4.Ex38.m2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex38.m2.1.1.1.1" xref="S4.Ex38.m2.1.1.1.1.1.cmml"><mo id="S4.Ex38.m2.1.1.1.1.2" stretchy="false" xref="S4.Ex38.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex38.m2.1.1.1.1.1" xref="S4.Ex38.m2.1.1.1.1.1.cmml"><mi id="S4.Ex38.m2.1.1.1.1.1.2" xref="S4.Ex38.m2.1.1.1.1.1.2.cmml">𝖭𝖤𝖰</mi><mrow id="S4.Ex38.m2.1.1.1.1.1.3" xref="S4.Ex38.m2.1.1.1.1.1.3.cmml"><mn id="S4.Ex38.m2.1.1.1.1.1.3.2" xref="S4.Ex38.m2.1.1.1.1.1.3.2.cmml">2</mn><mo id="S4.Ex38.m2.1.1.1.1.1.3.1" xref="S4.Ex38.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex38.m2.1.1.1.1.1.3.3" xref="S4.Ex38.m2.1.1.1.1.1.3.3.cmml">n</mi></mrow></msub><mo id="S4.Ex38.m2.1.1.1.1.3" stretchy="false" xref="S4.Ex38.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex38.m2.1b"><apply id="S4.Ex38.m2.1.1.cmml" xref="S4.Ex38.m2.1.1"><times id="S4.Ex38.m2.1.1.2.cmml" xref="S4.Ex38.m2.1.1.2"></times><ci id="S4.Ex38.m2.1.1.3.cmml" xref="S4.Ex38.m2.1.1.3">𝖼𝗈𝗇𝗈𝗇𝖽𝖾𝗍</ci><ci id="S4.Ex38.m2.1.1.4a.cmml" xref="S4.Ex38.m2.1.1.4"><mtext class="ltx_mathvariant_italic" id="S4.Ex38.m2.1.1.4.cmml" xref="S4.Ex38.m2.1.1.4">-</mtext></ci><apply id="S4.Ex38.m2.1.1.5.cmml" xref="S4.Ex38.m2.1.1.5"><csymbol cd="ambiguous" id="S4.Ex38.m2.1.1.5.1.cmml" xref="S4.Ex38.m2.1.1.5">subscript</csymbol><ci id="S4.Ex38.m2.1.1.5.2.cmml" xref="S4.Ex38.m2.1.1.5.2">𝗌𝗂𝗓𝖾</ci><and id="S4.Ex38.m2.1.1.5.3.cmml" xref="S4.Ex38.m2.1.1.5.3"></and></apply><apply id="S4.Ex38.m2.1.1.1.1.1.cmml" xref="S4.Ex38.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex38.m2.1.1.1.1.1.1.cmml" xref="S4.Ex38.m2.1.1.1.1">subscript</csymbol><ci id="S4.Ex38.m2.1.1.1.1.1.2.cmml" xref="S4.Ex38.m2.1.1.1.1.1.2">𝖭𝖤𝖰</ci><apply id="S4.Ex38.m2.1.1.1.1.1.3.cmml" xref="S4.Ex38.m2.1.1.1.1.1.3"><times id="S4.Ex38.m2.1.1.1.1.1.3.1.cmml" xref="S4.Ex38.m2.1.1.1.1.1.3.1"></times><cn id="S4.Ex38.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.Ex38.m2.1.1.1.1.1.3.2">2</cn><ci id="S4.Ex38.m2.1.1.1.1.1.3.3.cmml" xref="S4.Ex38.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex38.m2.1c">\displaystyle\mathsf{conondet}\text{-}\mathsf{size}_{\wedge}(\mathsf{NEQ}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex38.m2.1d">sansserif_conondet - sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex39.m1.1"><semantics id="S4.Ex39.m1.1a"><mo id="S4.Ex39.m1.1.1" xref="S4.Ex39.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex39.m1.1b"><leq id="S4.Ex39.m1.1.1.cmml" xref="S4.Ex39.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex39.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex39.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathsf{nondet}\text{-}\mathsf{size}_{\vee}(\mathsf{EQ}_{2n})" class="ltx_Math" display="inline" id="S4.Ex39.m2.1"><semantics id="S4.Ex39.m2.1a"><mrow id="S4.Ex39.m2.1.1" xref="S4.Ex39.m2.1.1.cmml"><mi id="S4.Ex39.m2.1.1.3" xref="S4.Ex39.m2.1.1.3.cmml">𝗇𝗈𝗇𝖽𝖾𝗍</mi><mo id="S4.Ex39.m2.1.1.2" xref="S4.Ex39.m2.1.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S4.Ex39.m2.1.1.4" xref="S4.Ex39.m2.1.1.4a.cmml">-</mtext><mo id="S4.Ex39.m2.1.1.2a" xref="S4.Ex39.m2.1.1.2.cmml">⁢</mo><msub id="S4.Ex39.m2.1.1.5" xref="S4.Ex39.m2.1.1.5.cmml"><mi id="S4.Ex39.m2.1.1.5.2" xref="S4.Ex39.m2.1.1.5.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex39.m2.1.1.5.3" xref="S4.Ex39.m2.1.1.5.3.cmml">∨</mo></msub><mo id="S4.Ex39.m2.1.1.2b" xref="S4.Ex39.m2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex39.m2.1.1.1.1" xref="S4.Ex39.m2.1.1.1.1.1.cmml"><mo id="S4.Ex39.m2.1.1.1.1.2" stretchy="false" xref="S4.Ex39.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex39.m2.1.1.1.1.1" xref="S4.Ex39.m2.1.1.1.1.1.cmml"><mi id="S4.Ex39.m2.1.1.1.1.1.2" xref="S4.Ex39.m2.1.1.1.1.1.2.cmml">𝖤𝖰</mi><mrow id="S4.Ex39.m2.1.1.1.1.1.3" xref="S4.Ex39.m2.1.1.1.1.1.3.cmml"><mn id="S4.Ex39.m2.1.1.1.1.1.3.2" xref="S4.Ex39.m2.1.1.1.1.1.3.2.cmml">2</mn><mo id="S4.Ex39.m2.1.1.1.1.1.3.1" xref="S4.Ex39.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex39.m2.1.1.1.1.1.3.3" xref="S4.Ex39.m2.1.1.1.1.1.3.3.cmml">n</mi></mrow></msub><mo id="S4.Ex39.m2.1.1.1.1.3" stretchy="false" xref="S4.Ex39.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex39.m2.1b"><apply id="S4.Ex39.m2.1.1.cmml" xref="S4.Ex39.m2.1.1"><times id="S4.Ex39.m2.1.1.2.cmml" xref="S4.Ex39.m2.1.1.2"></times><ci id="S4.Ex39.m2.1.1.3.cmml" xref="S4.Ex39.m2.1.1.3">𝗇𝗈𝗇𝖽𝖾𝗍</ci><ci id="S4.Ex39.m2.1.1.4a.cmml" xref="S4.Ex39.m2.1.1.4"><mtext class="ltx_mathvariant_italic" id="S4.Ex39.m2.1.1.4.cmml" xref="S4.Ex39.m2.1.1.4">-</mtext></ci><apply id="S4.Ex39.m2.1.1.5.cmml" xref="S4.Ex39.m2.1.1.5"><csymbol cd="ambiguous" id="S4.Ex39.m2.1.1.5.1.cmml" xref="S4.Ex39.m2.1.1.5">subscript</csymbol><ci id="S4.Ex39.m2.1.1.5.2.cmml" xref="S4.Ex39.m2.1.1.5.2">𝗌𝗂𝗓𝖾</ci><or id="S4.Ex39.m2.1.1.5.3.cmml" xref="S4.Ex39.m2.1.1.5.3"></or></apply><apply id="S4.Ex39.m2.1.1.1.1.1.cmml" xref="S4.Ex39.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex39.m2.1.1.1.1.1.1.cmml" xref="S4.Ex39.m2.1.1.1.1">subscript</csymbol><ci id="S4.Ex39.m2.1.1.1.1.1.2.cmml" xref="S4.Ex39.m2.1.1.1.1.1.2">𝖤𝖰</ci><apply id="S4.Ex39.m2.1.1.1.1.1.3.cmml" xref="S4.Ex39.m2.1.1.1.1.1.3"><times id="S4.Ex39.m2.1.1.1.1.1.3.1.cmml" xref="S4.Ex39.m2.1.1.1.1.1.3.1"></times><cn id="S4.Ex39.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.Ex39.m2.1.1.1.1.1.3.2">2</cn><ci id="S4.Ex39.m2.1.1.1.1.1.3.3.cmml" xref="S4.Ex39.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex39.m2.1c">\displaystyle\mathsf{nondet}\text{-}\mathsf{size}_{\vee}(\mathsf{EQ}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex39.m2.1d">sansserif_nondet - sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( sansserif_EQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex40.m1.1"><semantics id="S4.Ex40.m1.1a"><mo id="S4.Ex40.m1.1.1" xref="S4.Ex40.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex40.m1.1b"><leq id="S4.Ex40.m1.1.1.cmml" xref="S4.Ex40.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex40.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex40.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathsf{size}_{\vee}(\mathsf{EQ}_{2n})" class="ltx_Math" display="inline" id="S4.Ex40.m2.1"><semantics id="S4.Ex40.m2.1a"><mrow id="S4.Ex40.m2.1.1" xref="S4.Ex40.m2.1.1.cmml"><msub id="S4.Ex40.m2.1.1.3" xref="S4.Ex40.m2.1.1.3.cmml"><mi id="S4.Ex40.m2.1.1.3.2" xref="S4.Ex40.m2.1.1.3.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex40.m2.1.1.3.3" xref="S4.Ex40.m2.1.1.3.3.cmml">∨</mo></msub><mo id="S4.Ex40.m2.1.1.2" xref="S4.Ex40.m2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex40.m2.1.1.1.1" xref="S4.Ex40.m2.1.1.1.1.1.cmml"><mo id="S4.Ex40.m2.1.1.1.1.2" stretchy="false" xref="S4.Ex40.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex40.m2.1.1.1.1.1" xref="S4.Ex40.m2.1.1.1.1.1.cmml"><mi id="S4.Ex40.m2.1.1.1.1.1.2" xref="S4.Ex40.m2.1.1.1.1.1.2.cmml">𝖤𝖰</mi><mrow id="S4.Ex40.m2.1.1.1.1.1.3" xref="S4.Ex40.m2.1.1.1.1.1.3.cmml"><mn id="S4.Ex40.m2.1.1.1.1.1.3.2" xref="S4.Ex40.m2.1.1.1.1.1.3.2.cmml">2</mn><mo id="S4.Ex40.m2.1.1.1.1.1.3.1" xref="S4.Ex40.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex40.m2.1.1.1.1.1.3.3" xref="S4.Ex40.m2.1.1.1.1.1.3.3.cmml">n</mi></mrow></msub><mo id="S4.Ex40.m2.1.1.1.1.3" stretchy="false" xref="S4.Ex40.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex40.m2.1b"><apply id="S4.Ex40.m2.1.1.cmml" xref="S4.Ex40.m2.1.1"><times id="S4.Ex40.m2.1.1.2.cmml" xref="S4.Ex40.m2.1.1.2"></times><apply id="S4.Ex40.m2.1.1.3.cmml" xref="S4.Ex40.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex40.m2.1.1.3.1.cmml" xref="S4.Ex40.m2.1.1.3">subscript</csymbol><ci id="S4.Ex40.m2.1.1.3.2.cmml" xref="S4.Ex40.m2.1.1.3.2">𝗌𝗂𝗓𝖾</ci><or id="S4.Ex40.m2.1.1.3.3.cmml" xref="S4.Ex40.m2.1.1.3.3"></or></apply><apply id="S4.Ex40.m2.1.1.1.1.1.cmml" xref="S4.Ex40.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex40.m2.1.1.1.1.1.1.cmml" xref="S4.Ex40.m2.1.1.1.1">subscript</csymbol><ci id="S4.Ex40.m2.1.1.1.1.1.2.cmml" xref="S4.Ex40.m2.1.1.1.1.1.2">𝖤𝖰</ci><apply id="S4.Ex40.m2.1.1.1.1.1.3.cmml" xref="S4.Ex40.m2.1.1.1.1.1.3"><times id="S4.Ex40.m2.1.1.1.1.1.3.1.cmml" xref="S4.Ex40.m2.1.1.1.1.1.3.1"></times><cn id="S4.Ex40.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.Ex40.m2.1.1.1.1.1.3.2">2</cn><ci id="S4.Ex40.m2.1.1.1.1.1.3.3.cmml" xref="S4.Ex40.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex40.m2.1c">\displaystyle\mathsf{size}_{\vee}(\mathsf{EQ}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex40.m2.1d">sansserif_size start_POSTSUBSCRIPT ∨ end_POSTSUBSCRIPT ( sansserif_EQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex41.m1.1"><semantics id="S4.Ex41.m1.1a"><mo id="S4.Ex41.m1.1.1" xref="S4.Ex41.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex41.m1.1b"><leq id="S4.Ex41.m1.1.1.cmml" xref="S4.Ex41.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex41.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex41.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathsf{size}_{\wedge}(\mathsf{NEQ}_{2n})" class="ltx_Math" display="inline" id="S4.Ex41.m2.1"><semantics id="S4.Ex41.m2.1a"><mrow id="S4.Ex41.m2.1.1" xref="S4.Ex41.m2.1.1.cmml"><msub id="S4.Ex41.m2.1.1.3" xref="S4.Ex41.m2.1.1.3.cmml"><mi id="S4.Ex41.m2.1.1.3.2" xref="S4.Ex41.m2.1.1.3.2.cmml">𝗌𝗂𝗓𝖾</mi><mo id="S4.Ex41.m2.1.1.3.3" xref="S4.Ex41.m2.1.1.3.3.cmml">∧</mo></msub><mo id="S4.Ex41.m2.1.1.2" xref="S4.Ex41.m2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex41.m2.1.1.1.1" xref="S4.Ex41.m2.1.1.1.1.1.cmml"><mo id="S4.Ex41.m2.1.1.1.1.2" stretchy="false" xref="S4.Ex41.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex41.m2.1.1.1.1.1" xref="S4.Ex41.m2.1.1.1.1.1.cmml"><mi id="S4.Ex41.m2.1.1.1.1.1.2" xref="S4.Ex41.m2.1.1.1.1.1.2.cmml">𝖭𝖤𝖰</mi><mrow id="S4.Ex41.m2.1.1.1.1.1.3" xref="S4.Ex41.m2.1.1.1.1.1.3.cmml"><mn id="S4.Ex41.m2.1.1.1.1.1.3.2" xref="S4.Ex41.m2.1.1.1.1.1.3.2.cmml">2</mn><mo id="S4.Ex41.m2.1.1.1.1.1.3.1" xref="S4.Ex41.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex41.m2.1.1.1.1.1.3.3" xref="S4.Ex41.m2.1.1.1.1.1.3.3.cmml">n</mi></mrow></msub><mo id="S4.Ex41.m2.1.1.1.1.3" stretchy="false" xref="S4.Ex41.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex41.m2.1b"><apply id="S4.Ex41.m2.1.1.cmml" xref="S4.Ex41.m2.1.1"><times id="S4.Ex41.m2.1.1.2.cmml" xref="S4.Ex41.m2.1.1.2"></times><apply id="S4.Ex41.m2.1.1.3.cmml" xref="S4.Ex41.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex41.m2.1.1.3.1.cmml" xref="S4.Ex41.m2.1.1.3">subscript</csymbol><ci id="S4.Ex41.m2.1.1.3.2.cmml" xref="S4.Ex41.m2.1.1.3.2">𝗌𝗂𝗓𝖾</ci><and id="S4.Ex41.m2.1.1.3.3.cmml" xref="S4.Ex41.m2.1.1.3.3"></and></apply><apply id="S4.Ex41.m2.1.1.1.1.1.cmml" xref="S4.Ex41.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex41.m2.1.1.1.1.1.1.cmml" xref="S4.Ex41.m2.1.1.1.1">subscript</csymbol><ci id="S4.Ex41.m2.1.1.1.1.1.2.cmml" xref="S4.Ex41.m2.1.1.1.1.1.2">𝖭𝖤𝖰</ci><apply id="S4.Ex41.m2.1.1.1.1.1.3.cmml" xref="S4.Ex41.m2.1.1.1.1.1.3"><times id="S4.Ex41.m2.1.1.1.1.1.3.1.cmml" xref="S4.Ex41.m2.1.1.1.1.1.3.1"></times><cn id="S4.Ex41.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.Ex41.m2.1.1.1.1.1.3.2">2</cn><ci id="S4.Ex41.m2.1.1.1.1.1.3.3.cmml" xref="S4.Ex41.m2.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex41.m2.1c">\displaystyle\mathsf{size}_{\wedge}(\mathsf{NEQ}_{2n})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex41.m2.1d">sansserif_size start_POSTSUBSCRIPT ∧ end_POSTSUBSCRIPT ( sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex42.m1.1"><semantics id="S4.Ex42.m1.1a"><mo id="S4.Ex42.m1.1.1" xref="S4.Ex42.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex42.m1.1b"><leq id="S4.Ex42.m1.1.1.cmml" xref="S4.Ex42.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex42.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex42.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle n." class="ltx_Math" display="inline" id="S4.Ex42.m2.1"><semantics id="S4.Ex42.m2.1a"><mrow id="S4.Ex42.m2.1.2.2"><mi id="S4.Ex42.m2.1.1" xref="S4.Ex42.m2.1.1.cmml">n</mi><mo id="S4.Ex42.m2.1.2.2.1" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex42.m2.1b"><ci id="S4.Ex42.m2.1.1.cmml" xref="S4.Ex42.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex42.m2.1c">\displaystyle n.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex42.m2.1d">italic_n .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="Thmtheorem49.p1.2"><span class="ltx_text ltx_font_italic" id="Thmtheorem49.p1.2.2">In particular, the nondeterministic union complexity of the Boolean function <math alttext="\mathsf{EQ}_{2n}" class="ltx_Math" display="inline" id="Thmtheorem49.p1.1.1.m1.1"><semantics id="Thmtheorem49.p1.1.1.m1.1a"><msub id="Thmtheorem49.p1.1.1.m1.1.1" xref="Thmtheorem49.p1.1.1.m1.1.1.cmml"><mi id="Thmtheorem49.p1.1.1.m1.1.1.2" xref="Thmtheorem49.p1.1.1.m1.1.1.2.cmml">𝖤𝖰</mi><mrow id="Thmtheorem49.p1.1.1.m1.1.1.3" xref="Thmtheorem49.p1.1.1.m1.1.1.3.cmml"><mn id="Thmtheorem49.p1.1.1.m1.1.1.3.2" xref="Thmtheorem49.p1.1.1.m1.1.1.3.2.cmml">2</mn><mo id="Thmtheorem49.p1.1.1.m1.1.1.3.1" xref="Thmtheorem49.p1.1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="Thmtheorem49.p1.1.1.m1.1.1.3.3" xref="Thmtheorem49.p1.1.1.m1.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem49.p1.1.1.m1.1b"><apply id="Thmtheorem49.p1.1.1.m1.1.1.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmtheorem49.p1.1.1.m1.1.1.1.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1">subscript</csymbol><ci id="Thmtheorem49.p1.1.1.m1.1.1.2.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1.2">𝖤𝖰</ci><apply id="Thmtheorem49.p1.1.1.m1.1.1.3.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1.3"><times id="Thmtheorem49.p1.1.1.m1.1.1.3.1.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1.3.1"></times><cn id="Thmtheorem49.p1.1.1.m1.1.1.3.2.cmml" type="integer" xref="Thmtheorem49.p1.1.1.m1.1.1.3.2">2</cn><ci id="Thmtheorem49.p1.1.1.m1.1.1.3.3.cmml" xref="Thmtheorem49.p1.1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem49.p1.1.1.m1.1c">\mathsf{EQ}_{2n}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem49.p1.1.1.m1.1d">sansserif_EQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is precisely <math alttext="n" class="ltx_Math" display="inline" id="Thmtheorem49.p1.2.2.m2.1"><semantics id="Thmtheorem49.p1.2.2.m2.1a"><mi id="Thmtheorem49.p1.2.2.m2.1.1" xref="Thmtheorem49.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem49.p1.2.2.m2.1b"><ci id="Thmtheorem49.p1.2.2.m2.1.1.cmml" xref="Thmtheorem49.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem49.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem49.p1.2.2.m2.1d">italic_n</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p9"> <p class="ltx_p" id="S4.SS3.p9.2">Observe that, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14117v1#Thmtheorem30" title="Theorem 30 (Exact characterization of cover complexity). ‣ 3.4 An exact characterization via cyclic discrete complexity ‣ 3 Characterizations of Discrete Complexity via Set-Theoretic Fusion ‣ Boolean Circuit Complexity and Two-Dimensional Cover Problems"><span class="ltx_text ltx_ref_tag">30</span></a>, a cyclic circuit computing <math alttext="\mathsf{NEQ}_{2n}" class="ltx_Math" display="inline" id="S4.SS3.p9.1.m1.1"><semantics id="S4.SS3.p9.1.m1.1a"><msub id="S4.SS3.p9.1.m1.1.1" xref="S4.SS3.p9.1.m1.1.1.cmml"><mi id="S4.SS3.p9.1.m1.1.1.2" xref="S4.SS3.p9.1.m1.1.1.2.cmml">𝖭𝖤𝖰</mi><mrow id="S4.SS3.p9.1.m1.1.1.3" xref="S4.SS3.p9.1.m1.1.1.3.cmml"><mn id="S4.SS3.p9.1.m1.1.1.3.2" xref="S4.SS3.p9.1.m1.1.1.3.2.cmml">2</mn><mo id="S4.SS3.p9.1.m1.1.1.3.1" xref="S4.SS3.p9.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.p9.1.m1.1.1.3.3" xref="S4.SS3.p9.1.m1.1.1.3.3.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p9.1.m1.1b"><apply id="S4.SS3.p9.1.m1.1.1.cmml" xref="S4.SS3.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p9.1.m1.1.1.1.cmml" xref="S4.SS3.p9.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p9.1.m1.1.1.2.cmml" xref="S4.SS3.p9.1.m1.1.1.2">𝖭𝖤𝖰</ci><apply id="S4.SS3.p9.1.m1.1.1.3.cmml" xref="S4.SS3.p9.1.m1.1.1.3"><times id="S4.SS3.p9.1.m1.1.1.3.1.cmml" xref="S4.SS3.p9.1.m1.1.1.3.1"></times><cn id="S4.SS3.p9.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS3.p9.1.m1.1.1.3.2">2</cn><ci id="S4.SS3.p9.1.m1.1.1.3.3.cmml" xref="S4.SS3.p9.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p9.1.m1.1c">\mathsf{NEQ}_{2n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p9.1.m1.1d">sansserif_NEQ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT</annotation></semantics></math> also requires <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p9.2.m2.1"><semantics id="S4.SS3.p9.2.m2.1a"><mi id="S4.SS3.p9.2.m2.1.1" xref="S4.SS3.p9.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p9.2.m2.1b"><ci id="S4.SS3.p9.2.m2.1.1.cmml" xref="S4.SS3.p9.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p9.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p9.2.m2.1d">italic_n</annotation></semantics></math> fan-in two AND gates.</p> </div> </section> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">AB [87]</span> <span class="ltx_bibblock"> Noga Alon and Ravi B. 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