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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>The Misinterpretable Evidence Conveyed by Arbitrary Codes</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/2503.18984v1/"/></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/2503.18984v1#S1" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Ambiguous Ancestral Genetic Codes</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Anticipatory Brains</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>What do They Think about Me?</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Evolutionary Pressures on Communication Codes</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S6" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusions</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#A1" title="In The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Lyapunov Functions</span></a></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">The Misinterpretable Evidence <br class="ltx_break"/>Conveyed by Arbitrary Codes</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Guido Fioretti <br class="ltx_break"/>University of Bologna <br class="ltx_break"/>Contact address: guido.fioretti@unibo.it </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Evidence Theory is a mathematical framework for handling imprecise reasoning in the context of a judge evaluating testimonies or a detective evaluating cues, rather than a gambler playing games of chance. In comparison to Probability Theory, it is better equipped to deal with ambiguous information and novel possibilities. Furthermore, arrival and evaluation of testimonies implies a communication channel.</p> <p class="ltx_p" id="id2.id2">This paper explores the possibility of employing Evidence Theory to represent arbitrary communication codes between and within living organisms. In this paper, different schemes are explored for living organisms incapable of anticipation, animals sufficiently sophisticated to be capable of extrapolation, and humans capable of reading one other’s minds.</p> </div> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1"><span class="ltx_text ltx_font_bold" id="p1.1.1">Keywords:</span> Ambiguous Communication, Evidence Theory, Semantic Information, Origin of Life, Abduction</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">This essay explores the possibility of making use of Evidence Theory (ET) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib51" title="">51</a>]</cite> in order to represent communication between and within living organisms ranging from humans to bacteria. ET, also known as “Dempster-Shafer Theory” or “Belief Functions Theory,” is a mathematical theory of uncertain reasoning that takes as prototypical situation a judge evaluating testimonies, or a detective examining cues, rather than a gambler playing dice <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib48" title="">48</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib52" title="">52</a>]</cite>. This marks a sharp difference with Probability Theory (PT) because, albeit fundamental constructs such as Bayes’ Theorem can be obtained from the corresponding expressions of ET as special cases, gamblers know the faces of a die or the numbers on a roulette — they assume to live in a closed world — whereas judges and detectives are aware that unexpected clues and testimonies may open up novel possibilities — they are aware of living in an open world <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib23" title="">23</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">I submit that ET is more appropriate than PT to represent information transmission through arbitrary codes that multiply the generation of novelties. Furthermore, its paradigmatic situation of judges listening to testimonies is structurally similar to information communication, whereas the paradigmatic situation of gamblers playing games of chance is not <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib52" title="">52</a>]</cite>. Since ET has been conceived for humans, in this essay I shall take steps to adapt it to simpler organisms.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Specifically, in § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2" title="2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> I shall introduce concepts that are relevant even for organisms that have an extremely simple nervous system, or no nervous system at all, such as bacteria. In particular, I shall outline an application to the transmission of information through arbitrary and ambiguous ancestral genetic codes, pointing to further adaptations of ET for this class of organisms. In § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> I shall introduce basic ET concepts that require the ability to anticipate future states, for instance by extrapolation. This applies to animals endowed with a sufficiently sophisticated nervous system, not necessarily humans <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib8" title="">8</a>]</cite>. In § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4" title="4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> I shall consider the case of decision-makers so complex to imagine what others are thinking. This is evidently the province of humans, though certain primates appear to share this capability to some degree <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib11" title="">11</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Albeit the transitions between species and organisms that pertain to the aforementioned partition may be fuzzy to some extent, its divides mark substantial qualitative differences. Specifically, living organisms that are incapable of anticipation can only react to possibilities that are out of their control, whereas organisms capable of anticipation are able to conceive possibilities on their own. The second transition is just as substantial because the ability to think what others think can induce potentially infinite regressions on what possibilities are being conceived, making social codes inherently unstable <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib42" title="">42</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Evolutionary pressures act upon the redundancy and the ambiguity of biological communication codes, as well as their ability to generate novelties. In § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5" title="5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a> I illustrate one among several generalized entropy functions that are being proposed for ET. In particular, I show that this functional captures the evolutionary trends of biological communication.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Henceforth, I shall assume that living organisms attach <em class="ltx_emph ltx_font_italic" id="S1.p6.1.1">meaning</em> to the information that they receive if the environment provides any relevant feedback <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib45" title="">45</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib36" title="">36</a>]</cite>. For instance, the location of nutrients or poisonous substances is meaningful for bacteria, whereas the colour of the surface on which they rest is generally irrelevant for them.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">I shall use the term <em class="ltx_emph ltx_font_italic" id="S1.p7.1.1">interpretation</em> for the process of attaching a meaning to a piece of information received by a living organism. This concept is useful if the code is ambiguous and several interpretations are possible or, by adapting a metaphor that is often quoted in Code Biology, if a key opens several, yet not all doors. Note that this definition is sufficiently general to apply to living organisms of any sort.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">By contrast, semiotics generally employs definitions that require an interpretant, a requirement that unicellular organisms hardly meet. There are two solutions to this conundrum: Either one does not ascribe interpretation to the simplest life forms, or one adopts a different definition <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib7" title="">7</a>]</cite>. I opted for the second possibility.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">The relations between my understanding of “meaning” and “interpretation” and the logics of deduction, induction and abduction are discussed in the concluding § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S6" title="6 Conclusions ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a> in the light of the usage of these terms in practical domains such as legal adjudications and medical diagnoses. Furthermore, this section frames the contents of the ensuing § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2" title="2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a>, § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> and § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4" title="4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> with respect to one another, as well as § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5" title="5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Ambiguous Ancestral Genetic Codes</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">In this section I focus on the information expressed by triplets of nucleotides (codons) to generate amino acids, comparing the current, unambiguous genetic code to the likely state of the ancestral, ambiguous code <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib7" title="">7</a>]</cite>. This example is emblematic of all biological codes whose receiver is unable to conceive additional possibilities beyond those entailed in the code, such as bacteria, plants, as well as animals with an extremely simple nervous systems. Nevertheless, a degree of freedom exists insofar the possibilities conveyed by an arbitrary code can be coupled to different meanings — specifically, different amino acids.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.2">Let <math alttext="\Theta" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" mathvariant="normal" xref="S2.p2.1.m1.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">roman_Θ</annotation></semantics></math> denote the set of meaningful possibilities envisaged by the receiving organism, which in ET takes the name of <em class="ltx_emph ltx_font_italic" id="S2.p2.2.1">frame of discernment</em> (FoD). In our example, the possibilities envisaged in <math alttext="\Theta" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mi id="S2.p2.2.m2.1.1" mathvariant="normal" xref="S2.p2.2.m2.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">roman_Θ</annotation></semantics></math> represent the amino acids accepted by ribosomal-RNAs in order to be added to proteins. In their turn, these sets entail the triplets of nucleotides that code for it. In the juridical metaphor employed by ET, the nucleotides are testimonies that support interpretations represented by amino acids. Unambiguous codes correspond to triplets of testimonies that identify one and only one amino acid. By contrast, ambiguous codes correspond to triplets of testimonies that can be interpreted in different ways, i.e., they can generate several amino acids. Hence the statistical proteins of the ancestral, ambiguous code.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1">The left portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> illustrates the possibility set in PT. It is made of singletons that can either be distinct or coincide, but cannot accomodate partial intersections. This is all what a gambler needs in order to represent a game of chance, where singletons may represent the faces of a die or the numbers on a roulette. This scheme can be used to represent the genetic code only in the case that each amino acid is coded by exactly one codon. It is the hypothetical case of a non-redundant, non-degenerate genetic code.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">The central portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> illustrates two amino acids as they are actually coded by the genetic code, where most amino acids correspond to several codons. Thus, the genetic code is redundant, or degenerate. However, the current genetic code is also unambiguous in the sense that each codon corresponds to one and only one amino acid. Correspondingly, the sets representing amino acids in the central portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> do not intersect one another.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.2">However, in the early stages of life the genetic code is likely to have been ambiguous, i.e., one and the same codon could produce several amino acids <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib7" title="">7</a>]</cite>. This is represented in the right portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> where the sets representing amino acids can eventually intersect one another. Specifically, one and the same codon is hypothesized to enable two interpretations, namely amino acid <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><msub id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml"><mi id="S2.p5.1.m1.1.1.2" xref="S2.p5.1.m1.1.1.2.cmml">A</mi><mn id="S2.p5.1.m1.1.1.3" xref="S2.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.1.cmml" xref="S2.p5.1.m1.1.1">subscript</csymbol><ci id="S2.p5.1.m1.1.1.2.cmml" xref="S2.p5.1.m1.1.1.2">𝐴</ci><cn id="S2.p5.1.m1.1.1.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and amino acid <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><msub id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><mi id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml">A</mi><mn id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p5.2.m2.1.1.1.cmml" xref="S2.p5.2.m2.1.1">subscript</csymbol><ci id="S2.p5.2.m2.1.1.2.cmml" xref="S2.p5.2.m2.1.1.2">𝐴</ci><cn id="S2.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S2.F1"> <p class="ltx_p ltx_align_center" id="S2.F1.1"> <span class="ltx_inline-block ltx_framed ltx_framed_rectangle ltx_transformed_outer" id="S2.F1.1.1.1" style="width:346.9pt;height:86.9pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(3.3pt,-0.8pt) scale(1.01911214452631,1.01911214452631) ;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="164" id="S2.F1.1.1.1.g1" src="x1.png" width="654"/> </span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>A simplified picture of a FoD with two amino acids, <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F1.10.m1.1"><semantics id="S2.F1.10.m1.1b"><msub id="S2.F1.10.m1.1.1" xref="S2.F1.10.m1.1.1.cmml"><mi id="S2.F1.10.m1.1.1.2" xref="S2.F1.10.m1.1.1.2.cmml">A</mi><mn id="S2.F1.10.m1.1.1.3" xref="S2.F1.10.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.10.m1.1c"><apply id="S2.F1.10.m1.1.1.cmml" xref="S2.F1.10.m1.1.1"><csymbol cd="ambiguous" id="S2.F1.10.m1.1.1.1.cmml" xref="S2.F1.10.m1.1.1">subscript</csymbol><ci id="S2.F1.10.m1.1.1.2.cmml" xref="S2.F1.10.m1.1.1.2">𝐴</ci><cn id="S2.F1.10.m1.1.1.3.cmml" type="integer" xref="S2.F1.10.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.10.m1.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.10.m1.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F1.11.m2.1"><semantics id="S2.F1.11.m2.1b"><msub id="S2.F1.11.m2.1.1" xref="S2.F1.11.m2.1.1.cmml"><mi id="S2.F1.11.m2.1.1.2" xref="S2.F1.11.m2.1.1.2.cmml">A</mi><mn id="S2.F1.11.m2.1.1.3" xref="S2.F1.11.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.11.m2.1c"><apply id="S2.F1.11.m2.1.1.cmml" xref="S2.F1.11.m2.1.1"><csymbol cd="ambiguous" id="S2.F1.11.m2.1.1.1.cmml" xref="S2.F1.11.m2.1.1">subscript</csymbol><ci id="S2.F1.11.m2.1.1.2.cmml" xref="S2.F1.11.m2.1.1.2">𝐴</ci><cn id="S2.F1.11.m2.1.1.3.cmml" type="integer" xref="S2.F1.11.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.11.m2.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.11.m2.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, produced by codons represented by dark dots. Left, a hypothetical state where <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F1.12.m3.1"><semantics id="S2.F1.12.m3.1b"><msub id="S2.F1.12.m3.1.1" xref="S2.F1.12.m3.1.1.cmml"><mi id="S2.F1.12.m3.1.1.2" xref="S2.F1.12.m3.1.1.2.cmml">A</mi><mn id="S2.F1.12.m3.1.1.3" xref="S2.F1.12.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.12.m3.1c"><apply id="S2.F1.12.m3.1.1.cmml" xref="S2.F1.12.m3.1.1"><csymbol cd="ambiguous" id="S2.F1.12.m3.1.1.1.cmml" xref="S2.F1.12.m3.1.1">subscript</csymbol><ci id="S2.F1.12.m3.1.1.2.cmml" xref="S2.F1.12.m3.1.1.2">𝐴</ci><cn id="S2.F1.12.m3.1.1.3.cmml" type="integer" xref="S2.F1.12.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.12.m3.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.12.m3.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F1.13.m4.1"><semantics id="S2.F1.13.m4.1b"><msub id="S2.F1.13.m4.1.1" xref="S2.F1.13.m4.1.1.cmml"><mi id="S2.F1.13.m4.1.1.2" xref="S2.F1.13.m4.1.1.2.cmml">A</mi><mn id="S2.F1.13.m4.1.1.3" xref="S2.F1.13.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.13.m4.1c"><apply id="S2.F1.13.m4.1.1.cmml" xref="S2.F1.13.m4.1.1"><csymbol cd="ambiguous" id="S2.F1.13.m4.1.1.1.cmml" xref="S2.F1.13.m4.1.1">subscript</csymbol><ci id="S2.F1.13.m4.1.1.2.cmml" xref="S2.F1.13.m4.1.1.2">𝐴</ci><cn id="S2.F1.13.m4.1.1.3.cmml" type="integer" xref="S2.F1.13.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.13.m4.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.13.m4.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are coded by one specific codon each. Centre, a more realistic state of affairs where <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F1.14.m5.1"><semantics id="S2.F1.14.m5.1b"><msub id="S2.F1.14.m5.1.1" xref="S2.F1.14.m5.1.1.cmml"><mi id="S2.F1.14.m5.1.1.2" xref="S2.F1.14.m5.1.1.2.cmml">A</mi><mn id="S2.F1.14.m5.1.1.3" xref="S2.F1.14.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.14.m5.1c"><apply id="S2.F1.14.m5.1.1.cmml" xref="S2.F1.14.m5.1.1"><csymbol cd="ambiguous" id="S2.F1.14.m5.1.1.1.cmml" xref="S2.F1.14.m5.1.1">subscript</csymbol><ci id="S2.F1.14.m5.1.1.2.cmml" xref="S2.F1.14.m5.1.1.2">𝐴</ci><cn id="S2.F1.14.m5.1.1.3.cmml" type="integer" xref="S2.F1.14.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.14.m5.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.14.m5.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is coded by four codons whereas <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F1.15.m6.1"><semantics id="S2.F1.15.m6.1b"><msub id="S2.F1.15.m6.1.1" xref="S2.F1.15.m6.1.1.cmml"><mi id="S2.F1.15.m6.1.1.2" xref="S2.F1.15.m6.1.1.2.cmml">A</mi><mn id="S2.F1.15.m6.1.1.3" xref="S2.F1.15.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.15.m6.1c"><apply id="S2.F1.15.m6.1.1.cmml" xref="S2.F1.15.m6.1.1"><csymbol cd="ambiguous" id="S2.F1.15.m6.1.1.1.cmml" xref="S2.F1.15.m6.1.1">subscript</csymbol><ci id="S2.F1.15.m6.1.1.2.cmml" xref="S2.F1.15.m6.1.1.2">𝐴</ci><cn id="S2.F1.15.m6.1.1.3.cmml" type="integer" xref="S2.F1.15.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.15.m6.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.15.m6.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is coded by two codons. Right, one possible ancestral genetic code where one and the same codon could produce either <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F1.16.m7.1"><semantics id="S2.F1.16.m7.1b"><msub id="S2.F1.16.m7.1.1" xref="S2.F1.16.m7.1.1.cmml"><mi id="S2.F1.16.m7.1.1.2" xref="S2.F1.16.m7.1.1.2.cmml">A</mi><mn id="S2.F1.16.m7.1.1.3" xref="S2.F1.16.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.16.m7.1c"><apply id="S2.F1.16.m7.1.1.cmml" xref="S2.F1.16.m7.1.1"><csymbol cd="ambiguous" id="S2.F1.16.m7.1.1.1.cmml" xref="S2.F1.16.m7.1.1">subscript</csymbol><ci id="S2.F1.16.m7.1.1.2.cmml" xref="S2.F1.16.m7.1.1.2">𝐴</ci><cn id="S2.F1.16.m7.1.1.3.cmml" type="integer" xref="S2.F1.16.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.16.m7.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.16.m7.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F1.17.m8.1"><semantics id="S2.F1.17.m8.1b"><msub id="S2.F1.17.m8.1.1" xref="S2.F1.17.m8.1.1.cmml"><mi id="S2.F1.17.m8.1.1.2" xref="S2.F1.17.m8.1.1.2.cmml">A</mi><mn id="S2.F1.17.m8.1.1.3" xref="S2.F1.17.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F1.17.m8.1c"><apply id="S2.F1.17.m8.1.1.cmml" xref="S2.F1.17.m8.1.1"><csymbol cd="ambiguous" id="S2.F1.17.m8.1.1.1.cmml" xref="S2.F1.17.m8.1.1">subscript</csymbol><ci id="S2.F1.17.m8.1.1.2.cmml" xref="S2.F1.17.m8.1.1.2">𝐴</ci><cn id="S2.F1.17.m8.1.1.3.cmml" type="integer" xref="S2.F1.17.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.17.m8.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.17.m8.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.2">The situation illustrated in the right portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> is indeterminate. Either <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><msub id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml"><mi id="S2.p6.1.m1.1.1.2" xref="S2.p6.1.m1.1.1.2.cmml">A</mi><mn id="S2.p6.1.m1.1.1.3" xref="S2.p6.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.1.cmml" xref="S2.p6.1.m1.1.1">subscript</csymbol><ci id="S2.p6.1.m1.1.1.2.cmml" xref="S2.p6.1.m1.1.1.2">𝐴</ci><cn id="S2.p6.1.m1.1.1.3.cmml" type="integer" xref="S2.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p6.2.m2.1"><semantics id="S2.p6.2.m2.1a"><msub id="S2.p6.2.m2.1.1" xref="S2.p6.2.m2.1.1.cmml"><mi id="S2.p6.2.m2.1.1.2" xref="S2.p6.2.m2.1.1.2.cmml">A</mi><mn id="S2.p6.2.m2.1.1.3" xref="S2.p6.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p6.2.m2.1b"><apply id="S2.p6.2.m2.1.1.cmml" xref="S2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p6.2.m2.1.1.1.cmml" xref="S2.p6.2.m2.1.1">subscript</csymbol><ci id="S2.p6.2.m2.1.1.2.cmml" xref="S2.p6.2.m2.1.1.2">𝐴</ci><cn id="S2.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.p6.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m2.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m2.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> can be added to the protein under construction. Therefore, in this case one speaks of <em class="ltx_emph ltx_font_italic" id="S2.p6.2.1">statistical proteins</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib7" title="">7</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1">Let us observe in greater detail how this may happen. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F2" title="Figure 2 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> explodes the ambiguous codon of the right portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> into its component nucleic acids.</p> </div> <figure class="ltx_figure" id="S2.F2"> <p class="ltx_p ltx_align_center" id="S2.F2.1"> <span class="ltx_inline-block ltx_framed ltx_framed_rectangle ltx_transformed_outer" id="S2.F2.1.1.1" style="width:346.9pt;height:86pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-1.4pt,0.4pt) scale(0.991739265704364,0.991739265704364) ;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="167" id="S2.F2.1.1.1.g1" src="x2.png" width="672"/> </span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>The codon (black circle) that can either correspond to amino acid <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F2.17.m1.1"><semantics id="S2.F2.17.m1.1b"><msub id="S2.F2.17.m1.1.1" xref="S2.F2.17.m1.1.1.cmml"><mi id="S2.F2.17.m1.1.1.2" xref="S2.F2.17.m1.1.1.2.cmml">A</mi><mn id="S2.F2.17.m1.1.1.3" xref="S2.F2.17.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.17.m1.1c"><apply id="S2.F2.17.m1.1.1.cmml" xref="S2.F2.17.m1.1.1"><csymbol cd="ambiguous" id="S2.F2.17.m1.1.1.1.cmml" xref="S2.F2.17.m1.1.1">subscript</csymbol><ci id="S2.F2.17.m1.1.1.2.cmml" xref="S2.F2.17.m1.1.1.2">𝐴</ci><cn id="S2.F2.17.m1.1.1.3.cmml" type="integer" xref="S2.F2.17.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.17.m1.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.17.m1.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F2.18.m2.1"><semantics id="S2.F2.18.m2.1b"><msub id="S2.F2.18.m2.1.1" xref="S2.F2.18.m2.1.1.cmml"><mi id="S2.F2.18.m2.1.1.2" xref="S2.F2.18.m2.1.1.2.cmml">A</mi><mn id="S2.F2.18.m2.1.1.3" xref="S2.F2.18.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.18.m2.1c"><apply id="S2.F2.18.m2.1.1.cmml" xref="S2.F2.18.m2.1.1"><csymbol cd="ambiguous" id="S2.F2.18.m2.1.1.1.cmml" xref="S2.F2.18.m2.1.1">subscript</csymbol><ci id="S2.F2.18.m2.1.1.2.cmml" xref="S2.F2.18.m2.1.1.2">𝐴</ci><cn id="S2.F2.18.m2.1.1.3.cmml" type="integer" xref="S2.F2.18.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.18.m2.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.18.m2.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is exploded into its component nucleotides <math alttext="N_{1}" class="ltx_Math" display="inline" id="S2.F2.19.m3.1"><semantics id="S2.F2.19.m3.1b"><msub id="S2.F2.19.m3.1.1" xref="S2.F2.19.m3.1.1.cmml"><mi id="S2.F2.19.m3.1.1.2" xref="S2.F2.19.m3.1.1.2.cmml">N</mi><mn id="S2.F2.19.m3.1.1.3" xref="S2.F2.19.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.19.m3.1c"><apply id="S2.F2.19.m3.1.1.cmml" xref="S2.F2.19.m3.1.1"><csymbol cd="ambiguous" id="S2.F2.19.m3.1.1.1.cmml" xref="S2.F2.19.m3.1.1">subscript</csymbol><ci id="S2.F2.19.m3.1.1.2.cmml" xref="S2.F2.19.m3.1.1.2">𝑁</ci><cn id="S2.F2.19.m3.1.1.3.cmml" type="integer" xref="S2.F2.19.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.19.m3.1d">N_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.19.m3.1e">italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="N_{2}" class="ltx_Math" display="inline" id="S2.F2.20.m4.1"><semantics id="S2.F2.20.m4.1b"><msub id="S2.F2.20.m4.1.1" xref="S2.F2.20.m4.1.1.cmml"><mi id="S2.F2.20.m4.1.1.2" xref="S2.F2.20.m4.1.1.2.cmml">N</mi><mn id="S2.F2.20.m4.1.1.3" xref="S2.F2.20.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.20.m4.1c"><apply id="S2.F2.20.m4.1.1.cmml" xref="S2.F2.20.m4.1.1"><csymbol cd="ambiguous" id="S2.F2.20.m4.1.1.1.cmml" xref="S2.F2.20.m4.1.1">subscript</csymbol><ci id="S2.F2.20.m4.1.1.2.cmml" xref="S2.F2.20.m4.1.1.2">𝑁</ci><cn id="S2.F2.20.m4.1.1.3.cmml" type="integer" xref="S2.F2.20.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.20.m4.1d">N_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.20.m4.1e">italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="N_{3}" class="ltx_Math" display="inline" id="S2.F2.21.m5.1"><semantics id="S2.F2.21.m5.1b"><msub id="S2.F2.21.m5.1.1" xref="S2.F2.21.m5.1.1.cmml"><mi id="S2.F2.21.m5.1.1.2" xref="S2.F2.21.m5.1.1.2.cmml">N</mi><mn id="S2.F2.21.m5.1.1.3" xref="S2.F2.21.m5.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.21.m5.1c"><apply id="S2.F2.21.m5.1.1.cmml" xref="S2.F2.21.m5.1.1"><csymbol cd="ambiguous" id="S2.F2.21.m5.1.1.1.cmml" xref="S2.F2.21.m5.1.1">subscript</csymbol><ci id="S2.F2.21.m5.1.1.2.cmml" xref="S2.F2.21.m5.1.1.2">𝑁</ci><cn id="S2.F2.21.m5.1.1.3.cmml" type="integer" xref="S2.F2.21.m5.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.21.m5.1d">N_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.21.m5.1e">italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. It appears that <math alttext="N_{1}" class="ltx_Math" display="inline" id="S2.F2.22.m6.1"><semantics id="S2.F2.22.m6.1b"><msub id="S2.F2.22.m6.1.1" xref="S2.F2.22.m6.1.1.cmml"><mi id="S2.F2.22.m6.1.1.2" xref="S2.F2.22.m6.1.1.2.cmml">N</mi><mn id="S2.F2.22.m6.1.1.3" xref="S2.F2.22.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.22.m6.1c"><apply id="S2.F2.22.m6.1.1.cmml" xref="S2.F2.22.m6.1.1"><csymbol cd="ambiguous" id="S2.F2.22.m6.1.1.1.cmml" xref="S2.F2.22.m6.1.1">subscript</csymbol><ci id="S2.F2.22.m6.1.1.2.cmml" xref="S2.F2.22.m6.1.1.2">𝑁</ci><cn id="S2.F2.22.m6.1.1.3.cmml" type="integer" xref="S2.F2.22.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.22.m6.1d">N_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.22.m6.1e">italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="N_{2}" class="ltx_Math" display="inline" id="S2.F2.23.m7.1"><semantics id="S2.F2.23.m7.1b"><msub id="S2.F2.23.m7.1.1" xref="S2.F2.23.m7.1.1.cmml"><mi id="S2.F2.23.m7.1.1.2" xref="S2.F2.23.m7.1.1.2.cmml">N</mi><mn id="S2.F2.23.m7.1.1.3" xref="S2.F2.23.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.23.m7.1c"><apply id="S2.F2.23.m7.1.1.cmml" xref="S2.F2.23.m7.1.1"><csymbol cd="ambiguous" id="S2.F2.23.m7.1.1.1.cmml" xref="S2.F2.23.m7.1.1">subscript</csymbol><ci id="S2.F2.23.m7.1.1.2.cmml" xref="S2.F2.23.m7.1.1.2">𝑁</ci><cn id="S2.F2.23.m7.1.1.3.cmml" type="integer" xref="S2.F2.23.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.23.m7.1d">N_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.23.m7.1e">italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> pertain to both <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F2.24.m8.1"><semantics id="S2.F2.24.m8.1b"><msub id="S2.F2.24.m8.1.1" xref="S2.F2.24.m8.1.1.cmml"><mi id="S2.F2.24.m8.1.1.2" xref="S2.F2.24.m8.1.1.2.cmml">A</mi><mn id="S2.F2.24.m8.1.1.3" xref="S2.F2.24.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.24.m8.1c"><apply id="S2.F2.24.m8.1.1.cmml" xref="S2.F2.24.m8.1.1"><csymbol cd="ambiguous" id="S2.F2.24.m8.1.1.1.cmml" xref="S2.F2.24.m8.1.1">subscript</csymbol><ci id="S2.F2.24.m8.1.1.2.cmml" xref="S2.F2.24.m8.1.1.2">𝐴</ci><cn id="S2.F2.24.m8.1.1.3.cmml" type="integer" xref="S2.F2.24.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.24.m8.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.24.m8.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F2.25.m9.1"><semantics id="S2.F2.25.m9.1b"><msub id="S2.F2.25.m9.1.1" xref="S2.F2.25.m9.1.1.cmml"><mi id="S2.F2.25.m9.1.1.2" xref="S2.F2.25.m9.1.1.2.cmml">A</mi><mn id="S2.F2.25.m9.1.1.3" xref="S2.F2.25.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.25.m9.1c"><apply id="S2.F2.25.m9.1.1.cmml" xref="S2.F2.25.m9.1.1"><csymbol cd="ambiguous" id="S2.F2.25.m9.1.1.1.cmml" xref="S2.F2.25.m9.1.1">subscript</csymbol><ci id="S2.F2.25.m9.1.1.2.cmml" xref="S2.F2.25.m9.1.1.2">𝐴</ci><cn id="S2.F2.25.m9.1.1.3.cmml" type="integer" xref="S2.F2.25.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.25.m9.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.25.m9.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, whereas <math alttext="N_{3}" class="ltx_Math" display="inline" id="S2.F2.26.m10.1"><semantics id="S2.F2.26.m10.1b"><msub id="S2.F2.26.m10.1.1" xref="S2.F2.26.m10.1.1.cmml"><mi id="S2.F2.26.m10.1.1.2" xref="S2.F2.26.m10.1.1.2.cmml">N</mi><mn id="S2.F2.26.m10.1.1.3" xref="S2.F2.26.m10.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.26.m10.1c"><apply id="S2.F2.26.m10.1.1.cmml" xref="S2.F2.26.m10.1.1"><csymbol cd="ambiguous" id="S2.F2.26.m10.1.1.1.cmml" xref="S2.F2.26.m10.1.1">subscript</csymbol><ci id="S2.F2.26.m10.1.1.2.cmml" xref="S2.F2.26.m10.1.1.2">𝑁</ci><cn id="S2.F2.26.m10.1.1.3.cmml" type="integer" xref="S2.F2.26.m10.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.26.m10.1d">N_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.26.m10.1e">italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> pertains to <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F2.27.m11.1"><semantics id="S2.F2.27.m11.1b"><msub id="S2.F2.27.m11.1.1" xref="S2.F2.27.m11.1.1.cmml"><mi id="S2.F2.27.m11.1.1.2" xref="S2.F2.27.m11.1.1.2.cmml">A</mi><mn id="S2.F2.27.m11.1.1.3" xref="S2.F2.27.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.27.m11.1c"><apply id="S2.F2.27.m11.1.1.cmml" xref="S2.F2.27.m11.1.1"><csymbol cd="ambiguous" id="S2.F2.27.m11.1.1.1.cmml" xref="S2.F2.27.m11.1.1">subscript</csymbol><ci id="S2.F2.27.m11.1.1.2.cmml" xref="S2.F2.27.m11.1.1.2">𝐴</ci><cn id="S2.F2.27.m11.1.1.3.cmml" type="integer" xref="S2.F2.27.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.27.m11.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.27.m11.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> only. If this codon is captured by a transfer-RNA that transports <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.F2.28.m12.1"><semantics id="S2.F2.28.m12.1b"><msub id="S2.F2.28.m12.1.1" xref="S2.F2.28.m12.1.1.cmml"><mi id="S2.F2.28.m12.1.1.2" xref="S2.F2.28.m12.1.1.2.cmml">A</mi><mn id="S2.F2.28.m12.1.1.3" xref="S2.F2.28.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.28.m12.1c"><apply id="S2.F2.28.m12.1.1.cmml" xref="S2.F2.28.m12.1.1"><csymbol cd="ambiguous" id="S2.F2.28.m12.1.1.1.cmml" xref="S2.F2.28.m12.1.1">subscript</csymbol><ci id="S2.F2.28.m12.1.1.2.cmml" xref="S2.F2.28.m12.1.1.2">𝐴</ci><cn id="S2.F2.28.m12.1.1.3.cmml" type="integer" xref="S2.F2.28.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.28.m12.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.28.m12.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, this codon is interpret correctly. If this codon is captured by a transfer-RNA that transports <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F2.29.m13.1"><semantics id="S2.F2.29.m13.1b"><msub id="S2.F2.29.m13.1.1" xref="S2.F2.29.m13.1.1.cmml"><mi id="S2.F2.29.m13.1.1.2" xref="S2.F2.29.m13.1.1.2.cmml">A</mi><mn id="S2.F2.29.m13.1.1.3" xref="S2.F2.29.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.29.m13.1c"><apply id="S2.F2.29.m13.1.1.cmml" xref="S2.F2.29.m13.1.1"><csymbol cd="ambiguous" id="S2.F2.29.m13.1.1.1.cmml" xref="S2.F2.29.m13.1.1">subscript</csymbol><ci id="S2.F2.29.m13.1.1.2.cmml" xref="S2.F2.29.m13.1.1.2">𝐴</ci><cn id="S2.F2.29.m13.1.1.3.cmml" type="integer" xref="S2.F2.29.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.29.m13.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.29.m13.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and if its third anti-codon is sufficiently similar to <math alttext="N_{3}" class="ltx_Math" display="inline" id="S2.F2.30.m14.1"><semantics id="S2.F2.30.m14.1b"><msub id="S2.F2.30.m14.1.1" xref="S2.F2.30.m14.1.1.cmml"><mi id="S2.F2.30.m14.1.1.2" xref="S2.F2.30.m14.1.1.2.cmml">N</mi><mn id="S2.F2.30.m14.1.1.3" xref="S2.F2.30.m14.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.30.m14.1c"><apply id="S2.F2.30.m14.1.1.cmml" xref="S2.F2.30.m14.1.1"><csymbol cd="ambiguous" id="S2.F2.30.m14.1.1.1.cmml" xref="S2.F2.30.m14.1.1">subscript</csymbol><ci id="S2.F2.30.m14.1.1.2.cmml" xref="S2.F2.30.m14.1.1.2">𝑁</ci><cn id="S2.F2.30.m14.1.1.3.cmml" type="integer" xref="S2.F2.30.m14.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.30.m14.1d">N_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.30.m14.1e">italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> to “misinterpret” it, then it is translated into <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.F2.31.m15.1"><semantics id="S2.F2.31.m15.1b"><msub id="S2.F2.31.m15.1.1" xref="S2.F2.31.m15.1.1.cmml"><mi id="S2.F2.31.m15.1.1.2" xref="S2.F2.31.m15.1.1.2.cmml">A</mi><mn id="S2.F2.31.m15.1.1.3" xref="S2.F2.31.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.F2.31.m15.1c"><apply id="S2.F2.31.m15.1.1.cmml" xref="S2.F2.31.m15.1.1"><csymbol cd="ambiguous" id="S2.F2.31.m15.1.1.1.cmml" xref="S2.F2.31.m15.1.1">subscript</csymbol><ci id="S2.F2.31.m15.1.1.2.cmml" xref="S2.F2.31.m15.1.1.2">𝐴</ci><cn id="S2.F2.31.m15.1.1.3.cmml" type="integer" xref="S2.F2.31.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.31.m15.1d">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.31.m15.1e">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.12">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F2" title="Figure 2 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> tells one fictional story about how it may happen that a codon composed by <math alttext="\{N_{1},N_{2},N_{3}\}" class="ltx_Math" display="inline" id="S2.p8.1.m1.3"><semantics id="S2.p8.1.m1.3a"><mrow id="S2.p8.1.m1.3.3.3" xref="S2.p8.1.m1.3.3.4.cmml"><mo id="S2.p8.1.m1.3.3.3.4" stretchy="false" xref="S2.p8.1.m1.3.3.4.cmml">{</mo><msub id="S2.p8.1.m1.1.1.1.1" xref="S2.p8.1.m1.1.1.1.1.cmml"><mi id="S2.p8.1.m1.1.1.1.1.2" xref="S2.p8.1.m1.1.1.1.1.2.cmml">N</mi><mn id="S2.p8.1.m1.1.1.1.1.3" xref="S2.p8.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p8.1.m1.3.3.3.5" xref="S2.p8.1.m1.3.3.4.cmml">,</mo><msub id="S2.p8.1.m1.2.2.2.2" xref="S2.p8.1.m1.2.2.2.2.cmml"><mi id="S2.p8.1.m1.2.2.2.2.2" xref="S2.p8.1.m1.2.2.2.2.2.cmml">N</mi><mn id="S2.p8.1.m1.2.2.2.2.3" xref="S2.p8.1.m1.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p8.1.m1.3.3.3.6" xref="S2.p8.1.m1.3.3.4.cmml">,</mo><msub id="S2.p8.1.m1.3.3.3.3" xref="S2.p8.1.m1.3.3.3.3.cmml"><mi id="S2.p8.1.m1.3.3.3.3.2" xref="S2.p8.1.m1.3.3.3.3.2.cmml">N</mi><mn id="S2.p8.1.m1.3.3.3.3.3" xref="S2.p8.1.m1.3.3.3.3.3.cmml">3</mn></msub><mo id="S2.p8.1.m1.3.3.3.7" stretchy="false" xref="S2.p8.1.m1.3.3.4.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.1.m1.3b"><set id="S2.p8.1.m1.3.3.4.cmml" xref="S2.p8.1.m1.3.3.3"><apply id="S2.p8.1.m1.1.1.1.1.cmml" xref="S2.p8.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p8.1.m1.1.1.1.1.1.cmml" xref="S2.p8.1.m1.1.1.1.1">subscript</csymbol><ci id="S2.p8.1.m1.1.1.1.1.2.cmml" xref="S2.p8.1.m1.1.1.1.1.2">𝑁</ci><cn id="S2.p8.1.m1.1.1.1.1.3.cmml" type="integer" xref="S2.p8.1.m1.1.1.1.1.3">1</cn></apply><apply id="S2.p8.1.m1.2.2.2.2.cmml" xref="S2.p8.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S2.p8.1.m1.2.2.2.2.1.cmml" xref="S2.p8.1.m1.2.2.2.2">subscript</csymbol><ci id="S2.p8.1.m1.2.2.2.2.2.cmml" xref="S2.p8.1.m1.2.2.2.2.2">𝑁</ci><cn id="S2.p8.1.m1.2.2.2.2.3.cmml" type="integer" xref="S2.p8.1.m1.2.2.2.2.3">2</cn></apply><apply id="S2.p8.1.m1.3.3.3.3.cmml" xref="S2.p8.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S2.p8.1.m1.3.3.3.3.1.cmml" xref="S2.p8.1.m1.3.3.3.3">subscript</csymbol><ci id="S2.p8.1.m1.3.3.3.3.2.cmml" xref="S2.p8.1.m1.3.3.3.3.2">𝑁</ci><cn id="S2.p8.1.m1.3.3.3.3.3.cmml" type="integer" xref="S2.p8.1.m1.3.3.3.3.3">3</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.1.m1.3c">\{N_{1},N_{2},N_{3}\}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.1.m1.3d">{ italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }</annotation></semantics></math> is misunderstood as <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p8.2.m2.1"><semantics id="S2.p8.2.m2.1a"><msub id="S2.p8.2.m2.1.1" xref="S2.p8.2.m2.1.1.cmml"><mi id="S2.p8.2.m2.1.1.2" xref="S2.p8.2.m2.1.1.2.cmml">A</mi><mn id="S2.p8.2.m2.1.1.3" xref="S2.p8.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.2.m2.1b"><apply id="S2.p8.2.m2.1.1.cmml" xref="S2.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p8.2.m2.1.1.1.cmml" xref="S2.p8.2.m2.1.1">subscript</csymbol><ci id="S2.p8.2.m2.1.1.2.cmml" xref="S2.p8.2.m2.1.1.2">𝐴</ci><cn id="S2.p8.2.m2.1.1.3.cmml" type="integer" xref="S2.p8.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.2.m2.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.2.m2.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> instead of <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p8.3.m3.1"><semantics id="S2.p8.3.m3.1a"><msub id="S2.p8.3.m3.1.1" xref="S2.p8.3.m3.1.1.cmml"><mi id="S2.p8.3.m3.1.1.2" xref="S2.p8.3.m3.1.1.2.cmml">A</mi><mn id="S2.p8.3.m3.1.1.3" xref="S2.p8.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.3.m3.1b"><apply id="S2.p8.3.m3.1.1.cmml" xref="S2.p8.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p8.3.m3.1.1.1.cmml" xref="S2.p8.3.m3.1.1">subscript</csymbol><ci id="S2.p8.3.m3.1.1.2.cmml" xref="S2.p8.3.m3.1.1.2">𝐴</ci><cn id="S2.p8.3.m3.1.1.3.cmml" type="integer" xref="S2.p8.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.3.m3.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.3.m3.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. This story assumes that nucleotides are captured by transfer-RNAs in sequence, and that the first two nucleotides are common to both <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p8.4.m4.1"><semantics id="S2.p8.4.m4.1a"><msub id="S2.p8.4.m4.1.1" xref="S2.p8.4.m4.1.1.cmml"><mi id="S2.p8.4.m4.1.1.2" xref="S2.p8.4.m4.1.1.2.cmml">A</mi><mn id="S2.p8.4.m4.1.1.3" xref="S2.p8.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.4.m4.1b"><apply id="S2.p8.4.m4.1.1.cmml" xref="S2.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p8.4.m4.1.1.1.cmml" xref="S2.p8.4.m4.1.1">subscript</csymbol><ci id="S2.p8.4.m4.1.1.2.cmml" xref="S2.p8.4.m4.1.1.2">𝐴</ci><cn id="S2.p8.4.m4.1.1.3.cmml" type="integer" xref="S2.p8.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.4.m4.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.4.m4.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p8.5.m5.1"><semantics id="S2.p8.5.m5.1a"><msub id="S2.p8.5.m5.1.1" xref="S2.p8.5.m5.1.1.cmml"><mi id="S2.p8.5.m5.1.1.2" xref="S2.p8.5.m5.1.1.2.cmml">A</mi><mn id="S2.p8.5.m5.1.1.3" xref="S2.p8.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.5.m5.1b"><apply id="S2.p8.5.m5.1.1.cmml" xref="S2.p8.5.m5.1.1"><csymbol cd="ambiguous" id="S2.p8.5.m5.1.1.1.cmml" xref="S2.p8.5.m5.1.1">subscript</csymbol><ci id="S2.p8.5.m5.1.1.2.cmml" xref="S2.p8.5.m5.1.1.2">𝐴</ci><cn id="S2.p8.5.m5.1.1.3.cmml" type="integer" xref="S2.p8.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.5.m5.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.5.m5.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> whereas the third one is specific to <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p8.6.m6.1"><semantics id="S2.p8.6.m6.1a"><msub id="S2.p8.6.m6.1.1" xref="S2.p8.6.m6.1.1.cmml"><mi id="S2.p8.6.m6.1.1.2" xref="S2.p8.6.m6.1.1.2.cmml">A</mi><mn id="S2.p8.6.m6.1.1.3" xref="S2.p8.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.6.m6.1b"><apply id="S2.p8.6.m6.1.1.cmml" xref="S2.p8.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p8.6.m6.1.1.1.cmml" xref="S2.p8.6.m6.1.1">subscript</csymbol><ci id="S2.p8.6.m6.1.1.2.cmml" xref="S2.p8.6.m6.1.1.2">𝐴</ci><cn id="S2.p8.6.m6.1.1.3.cmml" type="integer" xref="S2.p8.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.6.m6.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.6.m6.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. If this codon is captured by a transfer-RNA that transports <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p8.7.m7.1"><semantics id="S2.p8.7.m7.1a"><msub id="S2.p8.7.m7.1.1" xref="S2.p8.7.m7.1.1.cmml"><mi id="S2.p8.7.m7.1.1.2" xref="S2.p8.7.m7.1.1.2.cmml">A</mi><mn id="S2.p8.7.m7.1.1.3" xref="S2.p8.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.7.m7.1b"><apply id="S2.p8.7.m7.1.1.cmml" xref="S2.p8.7.m7.1.1"><csymbol cd="ambiguous" id="S2.p8.7.m7.1.1.1.cmml" xref="S2.p8.7.m7.1.1">subscript</csymbol><ci id="S2.p8.7.m7.1.1.2.cmml" xref="S2.p8.7.m7.1.1.2">𝐴</ci><cn id="S2.p8.7.m7.1.1.3.cmml" type="integer" xref="S2.p8.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.7.m7.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.7.m7.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then it is correctly recognized. However, since the first two nucleotides are common to <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p8.8.m8.1"><semantics id="S2.p8.8.m8.1a"><msub id="S2.p8.8.m8.1.1" xref="S2.p8.8.m8.1.1.cmml"><mi id="S2.p8.8.m8.1.1.2" xref="S2.p8.8.m8.1.1.2.cmml">A</mi><mn id="S2.p8.8.m8.1.1.3" xref="S2.p8.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.8.m8.1b"><apply id="S2.p8.8.m8.1.1.cmml" xref="S2.p8.8.m8.1.1"><csymbol cd="ambiguous" id="S2.p8.8.m8.1.1.1.cmml" xref="S2.p8.8.m8.1.1">subscript</csymbol><ci id="S2.p8.8.m8.1.1.2.cmml" xref="S2.p8.8.m8.1.1.2">𝐴</ci><cn id="S2.p8.8.m8.1.1.3.cmml" type="integer" xref="S2.p8.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.8.m8.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.8.m8.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p8.9.m9.1"><semantics id="S2.p8.9.m9.1a"><msub id="S2.p8.9.m9.1.1" xref="S2.p8.9.m9.1.1.cmml"><mi id="S2.p8.9.m9.1.1.2" xref="S2.p8.9.m9.1.1.2.cmml">A</mi><mn id="S2.p8.9.m9.1.1.3" xref="S2.p8.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.9.m9.1b"><apply id="S2.p8.9.m9.1.1.cmml" xref="S2.p8.9.m9.1.1"><csymbol cd="ambiguous" id="S2.p8.9.m9.1.1.1.cmml" xref="S2.p8.9.m9.1.1">subscript</csymbol><ci id="S2.p8.9.m9.1.1.2.cmml" xref="S2.p8.9.m9.1.1.2">𝐴</ci><cn id="S2.p8.9.m9.1.1.3.cmml" type="integer" xref="S2.p8.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.9.m9.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.9.m9.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, this codon may be captured by a transfer-RNA that transports <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p8.10.m10.1"><semantics id="S2.p8.10.m10.1a"><msub id="S2.p8.10.m10.1.1" xref="S2.p8.10.m10.1.1.cmml"><mi id="S2.p8.10.m10.1.1.2" xref="S2.p8.10.m10.1.1.2.cmml">A</mi><mn id="S2.p8.10.m10.1.1.3" xref="S2.p8.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.10.m10.1b"><apply id="S2.p8.10.m10.1.1.cmml" xref="S2.p8.10.m10.1.1"><csymbol cd="ambiguous" id="S2.p8.10.m10.1.1.1.cmml" xref="S2.p8.10.m10.1.1">subscript</csymbol><ci id="S2.p8.10.m10.1.1.2.cmml" xref="S2.p8.10.m10.1.1.2">𝐴</ci><cn id="S2.p8.10.m10.1.1.3.cmml" type="integer" xref="S2.p8.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.10.m10.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.10.m10.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> which correctly recognizes the first two nucleotides and, if its anti-codon for <math alttext="N_{3}" class="ltx_Math" display="inline" id="S2.p8.11.m11.1"><semantics id="S2.p8.11.m11.1a"><msub id="S2.p8.11.m11.1.1" xref="S2.p8.11.m11.1.1.cmml"><mi id="S2.p8.11.m11.1.1.2" xref="S2.p8.11.m11.1.1.2.cmml">N</mi><mn id="S2.p8.11.m11.1.1.3" xref="S2.p8.11.m11.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.11.m11.1b"><apply id="S2.p8.11.m11.1.1.cmml" xref="S2.p8.11.m11.1.1"><csymbol cd="ambiguous" id="S2.p8.11.m11.1.1.1.cmml" xref="S2.p8.11.m11.1.1">subscript</csymbol><ci id="S2.p8.11.m11.1.1.2.cmml" xref="S2.p8.11.m11.1.1.2">𝑁</ci><cn id="S2.p8.11.m11.1.1.3.cmml" type="integer" xref="S2.p8.11.m11.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.11.m11.1c">N_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.11.m11.1d">italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> is sufficiently generic, ends up with recognizing the whole codon as <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p8.12.m12.1"><semantics id="S2.p8.12.m12.1a"><msub id="S2.p8.12.m12.1.1" xref="S2.p8.12.m12.1.1.cmml"><mi id="S2.p8.12.m12.1.1.2" xref="S2.p8.12.m12.1.1.2.cmml">A</mi><mn id="S2.p8.12.m12.1.1.3" xref="S2.p8.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p8.12.m12.1b"><apply id="S2.p8.12.m12.1.1.cmml" xref="S2.p8.12.m12.1.1"><csymbol cd="ambiguous" id="S2.p8.12.m12.1.1.1.cmml" xref="S2.p8.12.m12.1.1">subscript</csymbol><ci id="S2.p8.12.m12.1.1.2.cmml" xref="S2.p8.12.m12.1.1.2">𝐴</ci><cn id="S2.p8.12.m12.1.1.3.cmml" type="integer" xref="S2.p8.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.12.m12.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p8.12.m12.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, the statistical proteins.</p> </div> <div class="ltx_para" id="S2.p9"> <p class="ltx_p" id="S2.p9.1">Let us adapt ET to an organism so simple that bodies of evidence cannot be memorized and compared to one another. Testimonies simply arrive in time sequence and, as soon as three of them have arrived, a decision is made in the sense that a sequence of three nucleotides is finally interpreted as a specific amino acid.</p> </div> <div class="ltx_para" id="S2.p10"> <p class="ltx_p" id="S2.p10.8">Let <math alttext="\Theta=\{A_{1},A_{2}\}" class="ltx_Math" display="inline" id="S2.p10.1.m1.2"><semantics id="S2.p10.1.m1.2a"><mrow id="S2.p10.1.m1.2.2" xref="S2.p10.1.m1.2.2.cmml"><mi id="S2.p10.1.m1.2.2.4" mathvariant="normal" xref="S2.p10.1.m1.2.2.4.cmml">Θ</mi><mo id="S2.p10.1.m1.2.2.3" xref="S2.p10.1.m1.2.2.3.cmml">=</mo><mrow id="S2.p10.1.m1.2.2.2.2" xref="S2.p10.1.m1.2.2.2.3.cmml"><mo id="S2.p10.1.m1.2.2.2.2.3" stretchy="false" xref="S2.p10.1.m1.2.2.2.3.cmml">{</mo><msub id="S2.p10.1.m1.1.1.1.1.1" xref="S2.p10.1.m1.1.1.1.1.1.cmml"><mi id="S2.p10.1.m1.1.1.1.1.1.2" xref="S2.p10.1.m1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p10.1.m1.1.1.1.1.1.3" xref="S2.p10.1.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p10.1.m1.2.2.2.2.4" xref="S2.p10.1.m1.2.2.2.3.cmml">,</mo><msub id="S2.p10.1.m1.2.2.2.2.2" xref="S2.p10.1.m1.2.2.2.2.2.cmml"><mi id="S2.p10.1.m1.2.2.2.2.2.2" xref="S2.p10.1.m1.2.2.2.2.2.2.cmml">A</mi><mn id="S2.p10.1.m1.2.2.2.2.2.3" xref="S2.p10.1.m1.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p10.1.m1.2.2.2.2.5" stretchy="false" xref="S2.p10.1.m1.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.1.m1.2b"><apply id="S2.p10.1.m1.2.2.cmml" xref="S2.p10.1.m1.2.2"><eq id="S2.p10.1.m1.2.2.3.cmml" xref="S2.p10.1.m1.2.2.3"></eq><ci id="S2.p10.1.m1.2.2.4.cmml" xref="S2.p10.1.m1.2.2.4">Θ</ci><set id="S2.p10.1.m1.2.2.2.3.cmml" xref="S2.p10.1.m1.2.2.2.2"><apply id="S2.p10.1.m1.1.1.1.1.1.cmml" xref="S2.p10.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p10.1.m1.1.1.1.1.1.1.cmml" xref="S2.p10.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.p10.1.m1.1.1.1.1.1.2.cmml" xref="S2.p10.1.m1.1.1.1.1.1.2">𝐴</ci><cn id="S2.p10.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p10.1.m1.1.1.1.1.1.3">1</cn></apply><apply id="S2.p10.1.m1.2.2.2.2.2.cmml" xref="S2.p10.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.p10.1.m1.2.2.2.2.2.1.cmml" xref="S2.p10.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S2.p10.1.m1.2.2.2.2.2.2.cmml" xref="S2.p10.1.m1.2.2.2.2.2.2">𝐴</ci><cn id="S2.p10.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S2.p10.1.m1.2.2.2.2.2.3">2</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.1.m1.2c">\Theta=\{A_{1},A_{2}\}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.1.m1.2d">roman_Θ = { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT }</annotation></semantics></math> represent a FoD where either amino acid <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p10.2.m2.1"><semantics id="S2.p10.2.m2.1a"><msub id="S2.p10.2.m2.1.1" xref="S2.p10.2.m2.1.1.cmml"><mi id="S2.p10.2.m2.1.1.2" xref="S2.p10.2.m2.1.1.2.cmml">A</mi><mn id="S2.p10.2.m2.1.1.3" xref="S2.p10.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p10.2.m2.1b"><apply id="S2.p10.2.m2.1.1.cmml" xref="S2.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p10.2.m2.1.1.1.cmml" xref="S2.p10.2.m2.1.1">subscript</csymbol><ci id="S2.p10.2.m2.1.1.2.cmml" xref="S2.p10.2.m2.1.1.2">𝐴</ci><cn id="S2.p10.2.m2.1.1.3.cmml" type="integer" xref="S2.p10.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.2.m2.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.2.m2.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or amino acid <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p10.3.m3.1"><semantics id="S2.p10.3.m3.1a"><msub id="S2.p10.3.m3.1.1" xref="S2.p10.3.m3.1.1.cmml"><mi id="S2.p10.3.m3.1.1.2" xref="S2.p10.3.m3.1.1.2.cmml">A</mi><mn id="S2.p10.3.m3.1.1.3" xref="S2.p10.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p10.3.m3.1b"><apply id="S2.p10.3.m3.1.1.cmml" xref="S2.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p10.3.m3.1.1.1.cmml" xref="S2.p10.3.m3.1.1">subscript</csymbol><ci id="S2.p10.3.m3.1.1.2.cmml" xref="S2.p10.3.m3.1.1.2">𝐴</ci><cn id="S2.p10.3.m3.1.1.3.cmml" type="integer" xref="S2.p10.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.3.m3.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.3.m3.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> can be recognized. Let the testimonies be expressed by <em class="ltx_emph ltx_font_italic" id="S2.p10.8.1">bodies of evidence</em> represented by the nucleic acids that are being received. At <math alttext="t=t_{1}" class="ltx_Math" display="inline" id="S2.p10.4.m4.1"><semantics id="S2.p10.4.m4.1a"><mrow id="S2.p10.4.m4.1.1" xref="S2.p10.4.m4.1.1.cmml"><mi id="S2.p10.4.m4.1.1.2" xref="S2.p10.4.m4.1.1.2.cmml">t</mi><mo id="S2.p10.4.m4.1.1.1" xref="S2.p10.4.m4.1.1.1.cmml">=</mo><msub id="S2.p10.4.m4.1.1.3" xref="S2.p10.4.m4.1.1.3.cmml"><mi id="S2.p10.4.m4.1.1.3.2" xref="S2.p10.4.m4.1.1.3.2.cmml">t</mi><mn id="S2.p10.4.m4.1.1.3.3" xref="S2.p10.4.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.4.m4.1b"><apply id="S2.p10.4.m4.1.1.cmml" xref="S2.p10.4.m4.1.1"><eq id="S2.p10.4.m4.1.1.1.cmml" xref="S2.p10.4.m4.1.1.1"></eq><ci id="S2.p10.4.m4.1.1.2.cmml" xref="S2.p10.4.m4.1.1.2">𝑡</ci><apply id="S2.p10.4.m4.1.1.3.cmml" xref="S2.p10.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p10.4.m4.1.1.3.1.cmml" xref="S2.p10.4.m4.1.1.3">subscript</csymbol><ci id="S2.p10.4.m4.1.1.3.2.cmml" xref="S2.p10.4.m4.1.1.3.2">𝑡</ci><cn id="S2.p10.4.m4.1.1.3.3.cmml" type="integer" xref="S2.p10.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.4.m4.1c">t=t_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.4.m4.1d">italic_t = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> only the first nucleotide has been captured by the transfer RNA. Let <math alttext="m(A_{1})" class="ltx_Math" display="inline" id="S2.p10.5.m5.1"><semantics id="S2.p10.5.m5.1a"><mrow id="S2.p10.5.m5.1.1" xref="S2.p10.5.m5.1.1.cmml"><mi id="S2.p10.5.m5.1.1.3" xref="S2.p10.5.m5.1.1.3.cmml">m</mi><mo id="S2.p10.5.m5.1.1.2" xref="S2.p10.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.p10.5.m5.1.1.1.1" xref="S2.p10.5.m5.1.1.1.1.1.cmml"><mo id="S2.p10.5.m5.1.1.1.1.2" stretchy="false" xref="S2.p10.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S2.p10.5.m5.1.1.1.1.1" xref="S2.p10.5.m5.1.1.1.1.1.cmml"><mi id="S2.p10.5.m5.1.1.1.1.1.2" xref="S2.p10.5.m5.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p10.5.m5.1.1.1.1.1.3" xref="S2.p10.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p10.5.m5.1.1.1.1.3" stretchy="false" xref="S2.p10.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.5.m5.1b"><apply id="S2.p10.5.m5.1.1.cmml" xref="S2.p10.5.m5.1.1"><times id="S2.p10.5.m5.1.1.2.cmml" xref="S2.p10.5.m5.1.1.2"></times><ci id="S2.p10.5.m5.1.1.3.cmml" xref="S2.p10.5.m5.1.1.3">𝑚</ci><apply id="S2.p10.5.m5.1.1.1.1.1.cmml" xref="S2.p10.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.p10.5.m5.1.1.1.1.1.1.cmml" xref="S2.p10.5.m5.1.1.1.1">subscript</csymbol><ci id="S2.p10.5.m5.1.1.1.1.1.2.cmml" xref="S2.p10.5.m5.1.1.1.1.1.2">𝐴</ci><cn id="S2.p10.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S2.p10.5.m5.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.5.m5.1c">m(A_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.p10.5.m5.1d">italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="m(A_{2})" class="ltx_Math" display="inline" id="S2.p10.6.m6.1"><semantics id="S2.p10.6.m6.1a"><mrow id="S2.p10.6.m6.1.1" xref="S2.p10.6.m6.1.1.cmml"><mi id="S2.p10.6.m6.1.1.3" xref="S2.p10.6.m6.1.1.3.cmml">m</mi><mo id="S2.p10.6.m6.1.1.2" xref="S2.p10.6.m6.1.1.2.cmml">⁢</mo><mrow id="S2.p10.6.m6.1.1.1.1" xref="S2.p10.6.m6.1.1.1.1.1.cmml"><mo id="S2.p10.6.m6.1.1.1.1.2" stretchy="false" xref="S2.p10.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S2.p10.6.m6.1.1.1.1.1" xref="S2.p10.6.m6.1.1.1.1.1.cmml"><mi id="S2.p10.6.m6.1.1.1.1.1.2" xref="S2.p10.6.m6.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p10.6.m6.1.1.1.1.1.3" xref="S2.p10.6.m6.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.p10.6.m6.1.1.1.1.3" stretchy="false" xref="S2.p10.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.6.m6.1b"><apply id="S2.p10.6.m6.1.1.cmml" xref="S2.p10.6.m6.1.1"><times id="S2.p10.6.m6.1.1.2.cmml" xref="S2.p10.6.m6.1.1.2"></times><ci id="S2.p10.6.m6.1.1.3.cmml" xref="S2.p10.6.m6.1.1.3">𝑚</ci><apply id="S2.p10.6.m6.1.1.1.1.1.cmml" xref="S2.p10.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.p10.6.m6.1.1.1.1.1.1.cmml" xref="S2.p10.6.m6.1.1.1.1">subscript</csymbol><ci id="S2.p10.6.m6.1.1.1.1.1.2.cmml" xref="S2.p10.6.m6.1.1.1.1.1.2">𝐴</ci><cn id="S2.p10.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S2.p10.6.m6.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.6.m6.1c">m(A_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.p10.6.m6.1d">italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> denote the amount (mass) of evidence that the amino acid to be added to the protein is either <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p10.7.m7.1"><semantics id="S2.p10.7.m7.1a"><msub id="S2.p10.7.m7.1.1" xref="S2.p10.7.m7.1.1.cmml"><mi id="S2.p10.7.m7.1.1.2" xref="S2.p10.7.m7.1.1.2.cmml">A</mi><mn id="S2.p10.7.m7.1.1.3" xref="S2.p10.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p10.7.m7.1b"><apply id="S2.p10.7.m7.1.1.cmml" xref="S2.p10.7.m7.1.1"><csymbol cd="ambiguous" id="S2.p10.7.m7.1.1.1.cmml" xref="S2.p10.7.m7.1.1">subscript</csymbol><ci id="S2.p10.7.m7.1.1.2.cmml" xref="S2.p10.7.m7.1.1.2">𝐴</ci><cn id="S2.p10.7.m7.1.1.3.cmml" type="integer" xref="S2.p10.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.7.m7.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.7.m7.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p10.8.m8.1"><semantics id="S2.p10.8.m8.1a"><msub id="S2.p10.8.m8.1.1" xref="S2.p10.8.m8.1.1.cmml"><mi id="S2.p10.8.m8.1.1.2" xref="S2.p10.8.m8.1.1.2.cmml">A</mi><mn id="S2.p10.8.m8.1.1.3" xref="S2.p10.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p10.8.m8.1b"><apply id="S2.p10.8.m8.1.1.cmml" xref="S2.p10.8.m8.1.1"><csymbol cd="ambiguous" id="S2.p10.8.m8.1.1.1.cmml" xref="S2.p10.8.m8.1.1">subscript</csymbol><ci id="S2.p10.8.m8.1.1.2.cmml" xref="S2.p10.8.m8.1.1.2">𝐴</ci><cn id="S2.p10.8.m8.1.1.3.cmml" type="integer" xref="S2.p10.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.8.m8.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.8.m8.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively.</p> </div> <div class="ltx_para" id="S2.p11"> <p class="ltx_p" id="S2.p11.2">Let <math alttext="m(\Theta)" class="ltx_Math" display="inline" id="S2.p11.1.m1.1"><semantics id="S2.p11.1.m1.1a"><mrow id="S2.p11.1.m1.1.2" xref="S2.p11.1.m1.1.2.cmml"><mi id="S2.p11.1.m1.1.2.2" xref="S2.p11.1.m1.1.2.2.cmml">m</mi><mo id="S2.p11.1.m1.1.2.1" xref="S2.p11.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p11.1.m1.1.2.3.2" xref="S2.p11.1.m1.1.2.cmml"><mo id="S2.p11.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p11.1.m1.1.2.cmml">(</mo><mi id="S2.p11.1.m1.1.1" mathvariant="normal" xref="S2.p11.1.m1.1.1.cmml">Θ</mi><mo id="S2.p11.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p11.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p11.1.m1.1b"><apply id="S2.p11.1.m1.1.2.cmml" xref="S2.p11.1.m1.1.2"><times id="S2.p11.1.m1.1.2.1.cmml" xref="S2.p11.1.m1.1.2.1"></times><ci id="S2.p11.1.m1.1.2.2.cmml" xref="S2.p11.1.m1.1.2.2">𝑚</ci><ci id="S2.p11.1.m1.1.1.cmml" xref="S2.p11.1.m1.1.1">Θ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p11.1.m1.1c">m(\Theta)</annotation><annotation encoding="application/x-llamapun" id="S2.p11.1.m1.1d">italic_m ( roman_Θ )</annotation></semantics></math> denote lack of information. Since the transfer-RNA is unable to identify an amino acid out of one single nucleotide, at <math alttext="t=t_{1}" class="ltx_Math" display="inline" id="S2.p11.2.m2.1"><semantics id="S2.p11.2.m2.1a"><mrow id="S2.p11.2.m2.1.1" xref="S2.p11.2.m2.1.1.cmml"><mi id="S2.p11.2.m2.1.1.2" xref="S2.p11.2.m2.1.1.2.cmml">t</mi><mo id="S2.p11.2.m2.1.1.1" xref="S2.p11.2.m2.1.1.1.cmml">=</mo><msub id="S2.p11.2.m2.1.1.3" xref="S2.p11.2.m2.1.1.3.cmml"><mi id="S2.p11.2.m2.1.1.3.2" xref="S2.p11.2.m2.1.1.3.2.cmml">t</mi><mn id="S2.p11.2.m2.1.1.3.3" xref="S2.p11.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p11.2.m2.1b"><apply id="S2.p11.2.m2.1.1.cmml" xref="S2.p11.2.m2.1.1"><eq id="S2.p11.2.m2.1.1.1.cmml" xref="S2.p11.2.m2.1.1.1"></eq><ci id="S2.p11.2.m2.1.1.2.cmml" xref="S2.p11.2.m2.1.1.2">𝑡</ci><apply id="S2.p11.2.m2.1.1.3.cmml" xref="S2.p11.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.p11.2.m2.1.1.3.1.cmml" xref="S2.p11.2.m2.1.1.3">subscript</csymbol><ci id="S2.p11.2.m2.1.1.3.2.cmml" xref="S2.p11.2.m2.1.1.3.2">𝑡</ci><cn id="S2.p11.2.m2.1.1.3.3.cmml" type="integer" xref="S2.p11.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p11.2.m2.1c">t=t_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p11.2.m2.1d">italic_t = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> substantial information is still missing.</p> </div> <div class="ltx_para" id="S2.p12"> <p class="ltx_p" id="S2.p12.6">To fix ideas, let us assume that at <math alttext="t=t_{1}" class="ltx_Math" display="inline" id="S2.p12.1.m1.1"><semantics id="S2.p12.1.m1.1a"><mrow id="S2.p12.1.m1.1.1" xref="S2.p12.1.m1.1.1.cmml"><mi id="S2.p12.1.m1.1.1.2" xref="S2.p12.1.m1.1.1.2.cmml">t</mi><mo id="S2.p12.1.m1.1.1.1" xref="S2.p12.1.m1.1.1.1.cmml">=</mo><msub id="S2.p12.1.m1.1.1.3" xref="S2.p12.1.m1.1.1.3.cmml"><mi id="S2.p12.1.m1.1.1.3.2" xref="S2.p12.1.m1.1.1.3.2.cmml">t</mi><mn id="S2.p12.1.m1.1.1.3.3" xref="S2.p12.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p12.1.m1.1b"><apply id="S2.p12.1.m1.1.1.cmml" xref="S2.p12.1.m1.1.1"><eq id="S2.p12.1.m1.1.1.1.cmml" xref="S2.p12.1.m1.1.1.1"></eq><ci id="S2.p12.1.m1.1.1.2.cmml" xref="S2.p12.1.m1.1.1.2">𝑡</ci><apply id="S2.p12.1.m1.1.1.3.cmml" xref="S2.p12.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p12.1.m1.1.1.3.1.cmml" xref="S2.p12.1.m1.1.1.3">subscript</csymbol><ci id="S2.p12.1.m1.1.1.3.2.cmml" xref="S2.p12.1.m1.1.1.3.2">𝑡</ci><cn id="S2.p12.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p12.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p12.1.m1.1c">t=t_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p12.1.m1.1d">italic_t = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the first nucleotide brings the following body of evidence: <math alttext="A=\{m(A_{1})=1/3," class="ltx_math_unparsed" display="inline" id="S2.p12.2.m2.1"><semantics id="S2.p12.2.m2.1a"><mrow id="S2.p12.2.m2.1b"><mi id="S2.p12.2.m2.1.1">A</mi><mo id="S2.p12.2.m2.1.2">=</mo><mrow id="S2.p12.2.m2.1.3"><mo id="S2.p12.2.m2.1.3.1" stretchy="false">{</mo><mi id="S2.p12.2.m2.1.3.2">m</mi><mrow id="S2.p12.2.m2.1.3.3"><mo id="S2.p12.2.m2.1.3.3.1" stretchy="false">(</mo><msub id="S2.p12.2.m2.1.3.3.2"><mi id="S2.p12.2.m2.1.3.3.2.2">A</mi><mn id="S2.p12.2.m2.1.3.3.2.3">1</mn></msub><mo id="S2.p12.2.m2.1.3.3.3" stretchy="false">)</mo></mrow><mo id="S2.p12.2.m2.1.3.4">=</mo><mn id="S2.p12.2.m2.1.3.5">1</mn><mo id="S2.p12.2.m2.1.3.6">/</mo><mn id="S2.p12.2.m2.1.3.7">3</mn><mo id="S2.p12.2.m2.1.3.8">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.p12.2.m2.1c">A=\{m(A_{1})=1/3,</annotation><annotation encoding="application/x-llamapun" id="S2.p12.2.m2.1d">italic_A = { italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 1 / 3 ,</annotation></semantics></math> <math alttext="m(A_{2})=1/3," class="ltx_Math" display="inline" id="S2.p12.3.m3.1"><semantics id="S2.p12.3.m3.1a"><mrow id="S2.p12.3.m3.1.1.1" xref="S2.p12.3.m3.1.1.1.1.cmml"><mrow id="S2.p12.3.m3.1.1.1.1" xref="S2.p12.3.m3.1.1.1.1.cmml"><mrow id="S2.p12.3.m3.1.1.1.1.1" xref="S2.p12.3.m3.1.1.1.1.1.cmml"><mi id="S2.p12.3.m3.1.1.1.1.1.3" xref="S2.p12.3.m3.1.1.1.1.1.3.cmml">m</mi><mo id="S2.p12.3.m3.1.1.1.1.1.2" xref="S2.p12.3.m3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p12.3.m3.1.1.1.1.1.1.1" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.cmml"><mo id="S2.p12.3.m3.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p12.3.m3.1.1.1.1.1.1.1.1" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="S2.p12.3.m3.1.1.1.1.1.1.1.1.2" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p12.3.m3.1.1.1.1.1.1.1.1.3" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.p12.3.m3.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p12.3.m3.1.1.1.1.2" xref="S2.p12.3.m3.1.1.1.1.2.cmml">=</mo><mrow id="S2.p12.3.m3.1.1.1.1.3" xref="S2.p12.3.m3.1.1.1.1.3.cmml"><mn id="S2.p12.3.m3.1.1.1.1.3.2" xref="S2.p12.3.m3.1.1.1.1.3.2.cmml">1</mn><mo id="S2.p12.3.m3.1.1.1.1.3.1" xref="S2.p12.3.m3.1.1.1.1.3.1.cmml">/</mo><mn id="S2.p12.3.m3.1.1.1.1.3.3" xref="S2.p12.3.m3.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S2.p12.3.m3.1.1.1.2" xref="S2.p12.3.m3.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p12.3.m3.1b"><apply id="S2.p12.3.m3.1.1.1.1.cmml" xref="S2.p12.3.m3.1.1.1"><eq id="S2.p12.3.m3.1.1.1.1.2.cmml" xref="S2.p12.3.m3.1.1.1.1.2"></eq><apply id="S2.p12.3.m3.1.1.1.1.1.cmml" xref="S2.p12.3.m3.1.1.1.1.1"><times id="S2.p12.3.m3.1.1.1.1.1.2.cmml" xref="S2.p12.3.m3.1.1.1.1.1.2"></times><ci id="S2.p12.3.m3.1.1.1.1.1.3.cmml" xref="S2.p12.3.m3.1.1.1.1.1.3">𝑚</ci><apply id="S2.p12.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S2.p12.3.m3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p12.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="S2.p12.3.m3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p12.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.p12.3.m3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p12.3.m3.1.1.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S2.p12.3.m3.1.1.1.1.3.cmml" xref="S2.p12.3.m3.1.1.1.1.3"><divide id="S2.p12.3.m3.1.1.1.1.3.1.cmml" xref="S2.p12.3.m3.1.1.1.1.3.1"></divide><cn id="S2.p12.3.m3.1.1.1.1.3.2.cmml" type="integer" xref="S2.p12.3.m3.1.1.1.1.3.2">1</cn><cn id="S2.p12.3.m3.1.1.1.1.3.3.cmml" type="integer" xref="S2.p12.3.m3.1.1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p12.3.m3.1c">m(A_{2})=1/3,</annotation><annotation encoding="application/x-llamapun" id="S2.p12.3.m3.1d">italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 / 3 ,</annotation></semantics></math> <math alttext="m_{A}(\Theta)=1/3\}" class="ltx_math_unparsed" display="inline" id="S2.p12.4.m4.1"><semantics id="S2.p12.4.m4.1a"><mrow id="S2.p12.4.m4.1b"><msub id="S2.p12.4.m4.1.2"><mi id="S2.p12.4.m4.1.2.2">m</mi><mi id="S2.p12.4.m4.1.2.3">A</mi></msub><mrow id="S2.p12.4.m4.1.3"><mo id="S2.p12.4.m4.1.3.1" stretchy="false">(</mo><mi id="S2.p12.4.m4.1.1" mathvariant="normal">Θ</mi><mo id="S2.p12.4.m4.1.3.2" stretchy="false">)</mo></mrow><mo id="S2.p12.4.m4.1.4">=</mo><mn id="S2.p12.4.m4.1.5">1</mn><mo id="S2.p12.4.m4.1.6">/</mo><mn id="S2.p12.4.m4.1.7">3</mn><mo id="S2.p12.4.m4.1.8" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S2.p12.4.m4.1c">m_{A}(\Theta)=1/3\}</annotation><annotation encoding="application/x-llamapun" id="S2.p12.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) = 1 / 3 }</annotation></semantics></math>. Note that, since <math alttext="A_{1}\cap A_{2}\neq\emptyset" class="ltx_Math" display="inline" id="S2.p12.5.m5.1"><semantics id="S2.p12.5.m5.1a"><mrow id="S2.p12.5.m5.1.1" xref="S2.p12.5.m5.1.1.cmml"><mrow id="S2.p12.5.m5.1.1.2" xref="S2.p12.5.m5.1.1.2.cmml"><msub id="S2.p12.5.m5.1.1.2.2" xref="S2.p12.5.m5.1.1.2.2.cmml"><mi id="S2.p12.5.m5.1.1.2.2.2" xref="S2.p12.5.m5.1.1.2.2.2.cmml">A</mi><mn id="S2.p12.5.m5.1.1.2.2.3" xref="S2.p12.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.p12.5.m5.1.1.2.1" xref="S2.p12.5.m5.1.1.2.1.cmml">∩</mo><msub id="S2.p12.5.m5.1.1.2.3" xref="S2.p12.5.m5.1.1.2.3.cmml"><mi id="S2.p12.5.m5.1.1.2.3.2" xref="S2.p12.5.m5.1.1.2.3.2.cmml">A</mi><mn id="S2.p12.5.m5.1.1.2.3.3" xref="S2.p12.5.m5.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S2.p12.5.m5.1.1.1" xref="S2.p12.5.m5.1.1.1.cmml">≠</mo><mi id="S2.p12.5.m5.1.1.3" mathvariant="normal" xref="S2.p12.5.m5.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p12.5.m5.1b"><apply id="S2.p12.5.m5.1.1.cmml" xref="S2.p12.5.m5.1.1"><neq id="S2.p12.5.m5.1.1.1.cmml" xref="S2.p12.5.m5.1.1.1"></neq><apply id="S2.p12.5.m5.1.1.2.cmml" xref="S2.p12.5.m5.1.1.2"><intersect id="S2.p12.5.m5.1.1.2.1.cmml" xref="S2.p12.5.m5.1.1.2.1"></intersect><apply id="S2.p12.5.m5.1.1.2.2.cmml" xref="S2.p12.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S2.p12.5.m5.1.1.2.2.1.cmml" xref="S2.p12.5.m5.1.1.2.2">subscript</csymbol><ci id="S2.p12.5.m5.1.1.2.2.2.cmml" xref="S2.p12.5.m5.1.1.2.2.2">𝐴</ci><cn id="S2.p12.5.m5.1.1.2.2.3.cmml" type="integer" xref="S2.p12.5.m5.1.1.2.2.3">1</cn></apply><apply id="S2.p12.5.m5.1.1.2.3.cmml" xref="S2.p12.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S2.p12.5.m5.1.1.2.3.1.cmml" xref="S2.p12.5.m5.1.1.2.3">subscript</csymbol><ci id="S2.p12.5.m5.1.1.2.3.2.cmml" xref="S2.p12.5.m5.1.1.2.3.2">𝐴</ci><cn id="S2.p12.5.m5.1.1.2.3.3.cmml" type="integer" xref="S2.p12.5.m5.1.1.2.3.3">2</cn></apply></apply><emptyset id="S2.p12.5.m5.1.1.3.cmml" xref="S2.p12.5.m5.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p12.5.m5.1c">A_{1}\cap A_{2}\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.p12.5.m5.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math>, the equation <math alttext="m(A_{1})+m(A_{2})+m_{A}(\Theta)=1" class="ltx_Math" display="inline" id="S2.p12.6.m6.3"><semantics id="S2.p12.6.m6.3a"><mrow id="S2.p12.6.m6.3.3" xref="S2.p12.6.m6.3.3.cmml"><mrow id="S2.p12.6.m6.3.3.2" xref="S2.p12.6.m6.3.3.2.cmml"><mrow id="S2.p12.6.m6.2.2.1.1" xref="S2.p12.6.m6.2.2.1.1.cmml"><mi id="S2.p12.6.m6.2.2.1.1.3" xref="S2.p12.6.m6.2.2.1.1.3.cmml">m</mi><mo id="S2.p12.6.m6.2.2.1.1.2" xref="S2.p12.6.m6.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.p12.6.m6.2.2.1.1.1.1" xref="S2.p12.6.m6.2.2.1.1.1.1.1.cmml"><mo id="S2.p12.6.m6.2.2.1.1.1.1.2" stretchy="false" xref="S2.p12.6.m6.2.2.1.1.1.1.1.cmml">(</mo><msub id="S2.p12.6.m6.2.2.1.1.1.1.1" xref="S2.p12.6.m6.2.2.1.1.1.1.1.cmml"><mi id="S2.p12.6.m6.2.2.1.1.1.1.1.2" xref="S2.p12.6.m6.2.2.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p12.6.m6.2.2.1.1.1.1.1.3" xref="S2.p12.6.m6.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p12.6.m6.2.2.1.1.1.1.3" stretchy="false" xref="S2.p12.6.m6.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p12.6.m6.3.3.2.3" xref="S2.p12.6.m6.3.3.2.3.cmml">+</mo><mrow id="S2.p12.6.m6.3.3.2.2" xref="S2.p12.6.m6.3.3.2.2.cmml"><mi id="S2.p12.6.m6.3.3.2.2.3" xref="S2.p12.6.m6.3.3.2.2.3.cmml">m</mi><mo id="S2.p12.6.m6.3.3.2.2.2" xref="S2.p12.6.m6.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.p12.6.m6.3.3.2.2.1.1" xref="S2.p12.6.m6.3.3.2.2.1.1.1.cmml"><mo id="S2.p12.6.m6.3.3.2.2.1.1.2" stretchy="false" xref="S2.p12.6.m6.3.3.2.2.1.1.1.cmml">(</mo><msub id="S2.p12.6.m6.3.3.2.2.1.1.1" xref="S2.p12.6.m6.3.3.2.2.1.1.1.cmml"><mi id="S2.p12.6.m6.3.3.2.2.1.1.1.2" xref="S2.p12.6.m6.3.3.2.2.1.1.1.2.cmml">A</mi><mn id="S2.p12.6.m6.3.3.2.2.1.1.1.3" xref="S2.p12.6.m6.3.3.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S2.p12.6.m6.3.3.2.2.1.1.3" stretchy="false" xref="S2.p12.6.m6.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p12.6.m6.3.3.2.3a" xref="S2.p12.6.m6.3.3.2.3.cmml">+</mo><mrow id="S2.p12.6.m6.3.3.2.4" xref="S2.p12.6.m6.3.3.2.4.cmml"><msub id="S2.p12.6.m6.3.3.2.4.2" xref="S2.p12.6.m6.3.3.2.4.2.cmml"><mi id="S2.p12.6.m6.3.3.2.4.2.2" xref="S2.p12.6.m6.3.3.2.4.2.2.cmml">m</mi><mi id="S2.p12.6.m6.3.3.2.4.2.3" xref="S2.p12.6.m6.3.3.2.4.2.3.cmml">A</mi></msub><mo id="S2.p12.6.m6.3.3.2.4.1" xref="S2.p12.6.m6.3.3.2.4.1.cmml">⁢</mo><mrow id="S2.p12.6.m6.3.3.2.4.3.2" xref="S2.p12.6.m6.3.3.2.4.cmml"><mo id="S2.p12.6.m6.3.3.2.4.3.2.1" stretchy="false" xref="S2.p12.6.m6.3.3.2.4.cmml">(</mo><mi id="S2.p12.6.m6.1.1" mathvariant="normal" xref="S2.p12.6.m6.1.1.cmml">Θ</mi><mo id="S2.p12.6.m6.3.3.2.4.3.2.2" stretchy="false" xref="S2.p12.6.m6.3.3.2.4.cmml">)</mo></mrow></mrow></mrow><mo id="S2.p12.6.m6.3.3.3" xref="S2.p12.6.m6.3.3.3.cmml">=</mo><mn id="S2.p12.6.m6.3.3.4" xref="S2.p12.6.m6.3.3.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p12.6.m6.3b"><apply id="S2.p12.6.m6.3.3.cmml" xref="S2.p12.6.m6.3.3"><eq id="S2.p12.6.m6.3.3.3.cmml" xref="S2.p12.6.m6.3.3.3"></eq><apply id="S2.p12.6.m6.3.3.2.cmml" xref="S2.p12.6.m6.3.3.2"><plus id="S2.p12.6.m6.3.3.2.3.cmml" xref="S2.p12.6.m6.3.3.2.3"></plus><apply id="S2.p12.6.m6.2.2.1.1.cmml" xref="S2.p12.6.m6.2.2.1.1"><times id="S2.p12.6.m6.2.2.1.1.2.cmml" xref="S2.p12.6.m6.2.2.1.1.2"></times><ci id="S2.p12.6.m6.2.2.1.1.3.cmml" xref="S2.p12.6.m6.2.2.1.1.3">𝑚</ci><apply id="S2.p12.6.m6.2.2.1.1.1.1.1.cmml" xref="S2.p12.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.p12.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S2.p12.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S2.p12.6.m6.2.2.1.1.1.1.1.2.cmml" xref="S2.p12.6.m6.2.2.1.1.1.1.1.2">𝐴</ci><cn id="S2.p12.6.m6.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.p12.6.m6.2.2.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.p12.6.m6.3.3.2.2.cmml" xref="S2.p12.6.m6.3.3.2.2"><times id="S2.p12.6.m6.3.3.2.2.2.cmml" xref="S2.p12.6.m6.3.3.2.2.2"></times><ci id="S2.p12.6.m6.3.3.2.2.3.cmml" xref="S2.p12.6.m6.3.3.2.2.3">𝑚</ci><apply id="S2.p12.6.m6.3.3.2.2.1.1.1.cmml" xref="S2.p12.6.m6.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S2.p12.6.m6.3.3.2.2.1.1.1.1.cmml" xref="S2.p12.6.m6.3.3.2.2.1.1">subscript</csymbol><ci id="S2.p12.6.m6.3.3.2.2.1.1.1.2.cmml" xref="S2.p12.6.m6.3.3.2.2.1.1.1.2">𝐴</ci><cn id="S2.p12.6.m6.3.3.2.2.1.1.1.3.cmml" type="integer" xref="S2.p12.6.m6.3.3.2.2.1.1.1.3">2</cn></apply></apply><apply id="S2.p12.6.m6.3.3.2.4.cmml" xref="S2.p12.6.m6.3.3.2.4"><times id="S2.p12.6.m6.3.3.2.4.1.cmml" xref="S2.p12.6.m6.3.3.2.4.1"></times><apply id="S2.p12.6.m6.3.3.2.4.2.cmml" xref="S2.p12.6.m6.3.3.2.4.2"><csymbol cd="ambiguous" id="S2.p12.6.m6.3.3.2.4.2.1.cmml" xref="S2.p12.6.m6.3.3.2.4.2">subscript</csymbol><ci id="S2.p12.6.m6.3.3.2.4.2.2.cmml" xref="S2.p12.6.m6.3.3.2.4.2.2">𝑚</ci><ci id="S2.p12.6.m6.3.3.2.4.2.3.cmml" xref="S2.p12.6.m6.3.3.2.4.2.3">𝐴</ci></apply><ci id="S2.p12.6.m6.1.1.cmml" xref="S2.p12.6.m6.1.1">Θ</ci></apply></apply><cn id="S2.p12.6.m6.3.3.4.cmml" type="integer" xref="S2.p12.6.m6.3.3.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p12.6.m6.3c">m(A_{1})+m(A_{2})+m_{A}(\Theta)=1</annotation><annotation encoding="application/x-llamapun" id="S2.p12.6.m6.3d">italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) = 1</annotation></semantics></math> does not correspond to the additivity condition in PT.</p> </div> <div class="ltx_para" id="S2.p13"> <p class="ltx_p" id="S2.p13.3">Let us introduce two functions <math alttext="Bel(A_{i})" class="ltx_Math" display="inline" id="S2.p13.1.m1.1"><semantics id="S2.p13.1.m1.1a"><mrow id="S2.p13.1.m1.1.1" xref="S2.p13.1.m1.1.1.cmml"><mi id="S2.p13.1.m1.1.1.3" xref="S2.p13.1.m1.1.1.3.cmml">B</mi><mo id="S2.p13.1.m1.1.1.2" xref="S2.p13.1.m1.1.1.2.cmml">⁢</mo><mi id="S2.p13.1.m1.1.1.4" xref="S2.p13.1.m1.1.1.4.cmml">e</mi><mo id="S2.p13.1.m1.1.1.2a" xref="S2.p13.1.m1.1.1.2.cmml">⁢</mo><mi id="S2.p13.1.m1.1.1.5" xref="S2.p13.1.m1.1.1.5.cmml">l</mi><mo id="S2.p13.1.m1.1.1.2b" xref="S2.p13.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.p13.1.m1.1.1.1.1" xref="S2.p13.1.m1.1.1.1.1.1.cmml"><mo id="S2.p13.1.m1.1.1.1.1.2" stretchy="false" xref="S2.p13.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S2.p13.1.m1.1.1.1.1.1" xref="S2.p13.1.m1.1.1.1.1.1.cmml"><mi id="S2.p13.1.m1.1.1.1.1.1.2" xref="S2.p13.1.m1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.p13.1.m1.1.1.1.1.1.3" xref="S2.p13.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.p13.1.m1.1.1.1.1.3" stretchy="false" xref="S2.p13.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p13.1.m1.1b"><apply id="S2.p13.1.m1.1.1.cmml" xref="S2.p13.1.m1.1.1"><times id="S2.p13.1.m1.1.1.2.cmml" xref="S2.p13.1.m1.1.1.2"></times><ci id="S2.p13.1.m1.1.1.3.cmml" xref="S2.p13.1.m1.1.1.3">𝐵</ci><ci id="S2.p13.1.m1.1.1.4.cmml" xref="S2.p13.1.m1.1.1.4">𝑒</ci><ci id="S2.p13.1.m1.1.1.5.cmml" xref="S2.p13.1.m1.1.1.5">𝑙</ci><apply id="S2.p13.1.m1.1.1.1.1.1.cmml" xref="S2.p13.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p13.1.m1.1.1.1.1.1.1.cmml" xref="S2.p13.1.m1.1.1.1.1">subscript</csymbol><ci id="S2.p13.1.m1.1.1.1.1.1.2.cmml" xref="S2.p13.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="S2.p13.1.m1.1.1.1.1.1.3.cmml" xref="S2.p13.1.m1.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p13.1.m1.1c">Bel(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.p13.1.m1.1d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="Pl(A_{i})" class="ltx_Math" display="inline" id="S2.p13.2.m2.1"><semantics id="S2.p13.2.m2.1a"><mrow id="S2.p13.2.m2.1.1" xref="S2.p13.2.m2.1.1.cmml"><mi id="S2.p13.2.m2.1.1.3" xref="S2.p13.2.m2.1.1.3.cmml">P</mi><mo id="S2.p13.2.m2.1.1.2" xref="S2.p13.2.m2.1.1.2.cmml">⁢</mo><mi id="S2.p13.2.m2.1.1.4" xref="S2.p13.2.m2.1.1.4.cmml">l</mi><mo id="S2.p13.2.m2.1.1.2a" xref="S2.p13.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.p13.2.m2.1.1.1.1" xref="S2.p13.2.m2.1.1.1.1.1.cmml"><mo id="S2.p13.2.m2.1.1.1.1.2" stretchy="false" xref="S2.p13.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S2.p13.2.m2.1.1.1.1.1" xref="S2.p13.2.m2.1.1.1.1.1.cmml"><mi id="S2.p13.2.m2.1.1.1.1.1.2" xref="S2.p13.2.m2.1.1.1.1.1.2.cmml">A</mi><mi id="S2.p13.2.m2.1.1.1.1.1.3" xref="S2.p13.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.p13.2.m2.1.1.1.1.3" stretchy="false" xref="S2.p13.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p13.2.m2.1b"><apply id="S2.p13.2.m2.1.1.cmml" xref="S2.p13.2.m2.1.1"><times id="S2.p13.2.m2.1.1.2.cmml" xref="S2.p13.2.m2.1.1.2"></times><ci id="S2.p13.2.m2.1.1.3.cmml" xref="S2.p13.2.m2.1.1.3">𝑃</ci><ci id="S2.p13.2.m2.1.1.4.cmml" xref="S2.p13.2.m2.1.1.4">𝑙</ci><apply id="S2.p13.2.m2.1.1.1.1.1.cmml" xref="S2.p13.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.p13.2.m2.1.1.1.1.1.1.cmml" xref="S2.p13.2.m2.1.1.1.1">subscript</csymbol><ci id="S2.p13.2.m2.1.1.1.1.1.2.cmml" xref="S2.p13.2.m2.1.1.1.1.1.2">𝐴</ci><ci id="S2.p13.2.m2.1.1.1.1.1.3.cmml" xref="S2.p13.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p13.2.m2.1c">Pl(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.p13.2.m2.1d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> that sum all the evidence that is either strictly entailed or partly supports <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.p13.3.m3.1"><semantics id="S2.p13.3.m3.1a"><msub id="S2.p13.3.m3.1.1" xref="S2.p13.3.m3.1.1.cmml"><mi id="S2.p13.3.m3.1.1.2" xref="S2.p13.3.m3.1.1.2.cmml">A</mi><mi id="S2.p13.3.m3.1.1.3" xref="S2.p13.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p13.3.m3.1b"><apply id="S2.p13.3.m3.1.1.cmml" xref="S2.p13.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p13.3.m3.1.1.1.cmml" xref="S2.p13.3.m3.1.1">subscript</csymbol><ci id="S2.p13.3.m3.1.1.2.cmml" xref="S2.p13.3.m3.1.1.2">𝐴</ci><ci id="S2.p13.3.m3.1.1.3.cmml" xref="S2.p13.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p13.3.m3.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.p13.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, respectively:</p> </div> <div class="ltx_para" id="S2.p14"> <table class="ltx_equation ltx_eqn_table" id="S2.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="Bel(A_{i})\;=\;\sum_{A_{j}\subseteq A_{i}}\;m(A_{j})" class="ltx_Math" display="block" id="S2.E1.m1.2"><semantics id="S2.E1.m1.2a"><mrow id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.3" xref="S2.E1.m1.1.1.1.3.cmml">B</mi><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.4" xref="S2.E1.m1.1.1.1.4.cmml">e</mi><mo id="S2.E1.m1.1.1.1.2a" xref="S2.E1.m1.1.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.5" xref="S2.E1.m1.1.1.1.5.cmml">l</mi><mo id="S2.E1.m1.1.1.1.2b" xref="S2.E1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.E1.m1.1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.E1.m1.1.1.1.1.1.3" rspace="0.280em" stretchy="false" xref="S2.E1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.2.2.3" rspace="0.391em" xref="S2.E1.m1.2.2.3.cmml">=</mo><mrow id="S2.E1.m1.2.2.2" xref="S2.E1.m1.2.2.2.cmml"><munder id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml"><mo id="S2.E1.m1.2.2.2.2.2" movablelimits="false" xref="S2.E1.m1.2.2.2.2.2.cmml">∑</mo><mrow id="S2.E1.m1.2.2.2.2.3" xref="S2.E1.m1.2.2.2.2.3.cmml"><msub id="S2.E1.m1.2.2.2.2.3.2" xref="S2.E1.m1.2.2.2.2.3.2.cmml"><mi id="S2.E1.m1.2.2.2.2.3.2.2" xref="S2.E1.m1.2.2.2.2.3.2.2.cmml">A</mi><mi id="S2.E1.m1.2.2.2.2.3.2.3" xref="S2.E1.m1.2.2.2.2.3.2.3.cmml">j</mi></msub><mo id="S2.E1.m1.2.2.2.2.3.1" xref="S2.E1.m1.2.2.2.2.3.1.cmml">⊆</mo><msub id="S2.E1.m1.2.2.2.2.3.3" xref="S2.E1.m1.2.2.2.2.3.3.cmml"><mi id="S2.E1.m1.2.2.2.2.3.3.2" xref="S2.E1.m1.2.2.2.2.3.3.2.cmml">A</mi><mi id="S2.E1.m1.2.2.2.2.3.3.3" xref="S2.E1.m1.2.2.2.2.3.3.3.cmml">i</mi></msub></mrow></munder><mrow id="S2.E1.m1.2.2.2.1" xref="S2.E1.m1.2.2.2.1.cmml"><mi id="S2.E1.m1.2.2.2.1.3" xref="S2.E1.m1.2.2.2.1.3.cmml">m</mi><mo id="S2.E1.m1.2.2.2.1.2" xref="S2.E1.m1.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.2.1.1.1" xref="S2.E1.m1.2.2.2.1.1.1.1.cmml"><mo id="S2.E1.m1.2.2.2.1.1.1.2" stretchy="false" xref="S2.E1.m1.2.2.2.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.2.2.2.1.1.1.1" xref="S2.E1.m1.2.2.2.1.1.1.1.cmml"><mi id="S2.E1.m1.2.2.2.1.1.1.1.2" xref="S2.E1.m1.2.2.2.1.1.1.1.2.cmml">A</mi><mi id="S2.E1.m1.2.2.2.1.1.1.1.3" xref="S2.E1.m1.2.2.2.1.1.1.1.3.cmml">j</mi></msub><mo id="S2.E1.m1.2.2.2.1.1.1.3" stretchy="false" xref="S2.E1.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.2b"><apply id="S2.E1.m1.2.2.cmml" xref="S2.E1.m1.2.2"><eq id="S2.E1.m1.2.2.3.cmml" xref="S2.E1.m1.2.2.3"></eq><apply id="S2.E1.m1.1.1.1.cmml" xref="S2.E1.m1.1.1.1"><times 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id="S2.E1.m1.2.2.2.2.3.2.cmml" xref="S2.E1.m1.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.2.3.2.1.cmml" xref="S2.E1.m1.2.2.2.2.3.2">subscript</csymbol><ci id="S2.E1.m1.2.2.2.2.3.2.2.cmml" xref="S2.E1.m1.2.2.2.2.3.2.2">𝐴</ci><ci id="S2.E1.m1.2.2.2.2.3.2.3.cmml" xref="S2.E1.m1.2.2.2.2.3.2.3">𝑗</ci></apply><apply id="S2.E1.m1.2.2.2.2.3.3.cmml" xref="S2.E1.m1.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.2.3.3.1.cmml" xref="S2.E1.m1.2.2.2.2.3.3">subscript</csymbol><ci id="S2.E1.m1.2.2.2.2.3.3.2.cmml" xref="S2.E1.m1.2.2.2.2.3.3.2">𝐴</ci><ci id="S2.E1.m1.2.2.2.2.3.3.3.cmml" xref="S2.E1.m1.2.2.2.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="S2.E1.m1.2.2.2.1.cmml" xref="S2.E1.m1.2.2.2.1"><times id="S2.E1.m1.2.2.2.1.2.cmml" xref="S2.E1.m1.2.2.2.1.2"></times><ci id="S2.E1.m1.2.2.2.1.3.cmml" xref="S2.E1.m1.2.2.2.1.3">𝑚</ci><apply id="S2.E1.m1.2.2.2.1.1.1.1.cmml" xref="S2.E1.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.1.1.1.1.1.cmml" xref="S2.E1.m1.2.2.2.1.1.1">subscript</csymbol><ci id="S2.E1.m1.2.2.2.1.1.1.1.2.cmml" xref="S2.E1.m1.2.2.2.1.1.1.1.2">𝐴</ci><ci id="S2.E1.m1.2.2.2.1.1.1.1.3.cmml" xref="S2.E1.m1.2.2.2.1.1.1.1.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.2c">Bel(A_{i})\;=\;\sum_{A_{j}\subseteq A_{i}}\;m(A_{j})</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.2d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊆ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_m ( italic_A start_POSTSUBSCRIPT italic_j 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">(1)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p15"> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <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="Pl(A_{i})\;=\;\sum_{A_{j}\cap A_{i}\neq\emptyset}\;m(A_{j})" class="ltx_Math" display="block" id="S2.E2.m1.2"><semantics id="S2.E2.m1.2a"><mrow id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml">P</mi><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.4" xref="S2.E2.m1.1.1.1.4.cmml">l</mi><mo id="S2.E2.m1.1.1.1.2a" xref="S2.E2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.cmml"><mo id="S2.E2.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E2.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E2.m1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.E2.m1.1.1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.E2.m1.1.1.1.1.1.3" rspace="0.280em" stretchy="false" xref="S2.E2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.2.2.3" rspace="0.391em" xref="S2.E2.m1.2.2.3.cmml">=</mo><mrow id="S2.E2.m1.2.2.2" xref="S2.E2.m1.2.2.2.cmml"><munder id="S2.E2.m1.2.2.2.2" xref="S2.E2.m1.2.2.2.2.cmml"><mo id="S2.E2.m1.2.2.2.2.2" movablelimits="false" xref="S2.E2.m1.2.2.2.2.2.cmml">∑</mo><mrow id="S2.E2.m1.2.2.2.2.3" xref="S2.E2.m1.2.2.2.2.3.cmml"><mrow id="S2.E2.m1.2.2.2.2.3.2" xref="S2.E2.m1.2.2.2.2.3.2.cmml"><msub id="S2.E2.m1.2.2.2.2.3.2.2" xref="S2.E2.m1.2.2.2.2.3.2.2.cmml"><mi id="S2.E2.m1.2.2.2.2.3.2.2.2" xref="S2.E2.m1.2.2.2.2.3.2.2.2.cmml">A</mi><mi id="S2.E2.m1.2.2.2.2.3.2.2.3" xref="S2.E2.m1.2.2.2.2.3.2.2.3.cmml">j</mi></msub><mo id="S2.E2.m1.2.2.2.2.3.2.1" xref="S2.E2.m1.2.2.2.2.3.2.1.cmml">∩</mo><msub 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italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ ∅ end_POSTSUBSCRIPT italic_m ( italic_A start_POSTSUBSCRIPT italic_j 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">(2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p16"> <p class="ltx_p" id="S2.p16.9">In § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E1" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E2" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> will be generalized into <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E1" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E2" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively. In the current context, eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E1" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E2" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> represent ways in which the ribosomal RNA of ancestral code may sum the available evidence in order to interpret it either in favour of amino acid <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p16.1.m1.1"><semantics id="S2.p16.1.m1.1a"><msub id="S2.p16.1.m1.1.1" xref="S2.p16.1.m1.1.1.cmml"><mi id="S2.p16.1.m1.1.1.2" xref="S2.p16.1.m1.1.1.2.cmml">A</mi><mn id="S2.p16.1.m1.1.1.3" xref="S2.p16.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p16.1.m1.1b"><apply id="S2.p16.1.m1.1.1.cmml" xref="S2.p16.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p16.1.m1.1.1.1.cmml" xref="S2.p16.1.m1.1.1">subscript</csymbol><ci id="S2.p16.1.m1.1.1.2.cmml" xref="S2.p16.1.m1.1.1.2">𝐴</ci><cn id="S2.p16.1.m1.1.1.3.cmml" type="integer" xref="S2.p16.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p16.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p16.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p16.2.m2.1"><semantics id="S2.p16.2.m2.1a"><msub id="S2.p16.2.m2.1.1" xref="S2.p16.2.m2.1.1.cmml"><mi id="S2.p16.2.m2.1.1.2" xref="S2.p16.2.m2.1.1.2.cmml">A</mi><mn id="S2.p16.2.m2.1.1.3" xref="S2.p16.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p16.2.m2.1b"><apply id="S2.p16.2.m2.1.1.cmml" xref="S2.p16.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p16.2.m2.1.1.1.cmml" xref="S2.p16.2.m2.1.1">subscript</csymbol><ci id="S2.p16.2.m2.1.1.2.cmml" xref="S2.p16.2.m2.1.1.2">𝐴</ci><cn id="S2.p16.2.m2.1.1.3.cmml" type="integer" xref="S2.p16.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p16.2.m2.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p16.2.m2.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. 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xref="S2.p16.7.m7.1.1.1.3.cmml">P</mi><mo id="S2.p16.7.m7.1.1.1.2" xref="S2.p16.7.m7.1.1.1.2.cmml">⁢</mo><mi id="S2.p16.7.m7.1.1.1.4" xref="S2.p16.7.m7.1.1.1.4.cmml">l</mi><mo id="S2.p16.7.m7.1.1.1.2a" xref="S2.p16.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.p16.7.m7.1.1.1.1.1" xref="S2.p16.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.p16.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.p16.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p16.7.m7.1.1.1.1.1.1" xref="S2.p16.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.p16.7.m7.1.1.1.1.1.1.2" xref="S2.p16.7.m7.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p16.7.m7.1.1.1.1.1.1.3" xref="S2.p16.7.m7.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p16.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.p16.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p16.7.m7.1.1.2" xref="S2.p16.7.m7.1.1.2.cmml">=</mo><mrow id="S2.p16.7.m7.1.1.3" xref="S2.p16.7.m7.1.1.3.cmml"><mn id="S2.p16.7.m7.1.1.3.2" xref="S2.p16.7.m7.1.1.3.2.cmml">1</mn><mo id="S2.p16.7.m7.1.1.3.1" xref="S2.p16.7.m7.1.1.3.1.cmml">/</mo><mn id="S2.p16.7.m7.1.1.3.3" xref="S2.p16.7.m7.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p16.7.m7.1b"><apply id="S2.p16.7.m7.1.1.cmml" xref="S2.p16.7.m7.1.1"><eq id="S2.p16.7.m7.1.1.2.cmml" xref="S2.p16.7.m7.1.1.2"></eq><apply id="S2.p16.7.m7.1.1.1.cmml" xref="S2.p16.7.m7.1.1.1"><times id="S2.p16.7.m7.1.1.1.2.cmml" xref="S2.p16.7.m7.1.1.1.2"></times><ci id="S2.p16.7.m7.1.1.1.3.cmml" xref="S2.p16.7.m7.1.1.1.3">𝑃</ci><ci id="S2.p16.7.m7.1.1.1.4.cmml" xref="S2.p16.7.m7.1.1.1.4">𝑙</ci><apply id="S2.p16.7.m7.1.1.1.1.1.1.cmml" xref="S2.p16.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p16.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.p16.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S2.p16.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.p16.7.m7.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.p16.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p16.7.m7.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.p16.7.m7.1.1.3.cmml" xref="S2.p16.7.m7.1.1.3"><divide id="S2.p16.7.m7.1.1.3.1.cmml" xref="S2.p16.7.m7.1.1.3.1"></divide><cn id="S2.p16.7.m7.1.1.3.2.cmml" type="integer" xref="S2.p16.7.m7.1.1.3.2">1</cn><cn id="S2.p16.7.m7.1.1.3.3.cmml" type="integer" xref="S2.p16.7.m7.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p16.7.m7.1c">Pl(A_{1})=1/3</annotation><annotation encoding="application/x-llamapun" id="S2.p16.7.m7.1d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 1 / 3</annotation></semantics></math> and <math alttext="Bel(A_{2})=" class="ltx_Math" display="inline" id="S2.p16.8.m8.1"><semantics id="S2.p16.8.m8.1a"><mrow id="S2.p16.8.m8.1.1" xref="S2.p16.8.m8.1.1.cmml"><mrow id="S2.p16.8.m8.1.1.1" xref="S2.p16.8.m8.1.1.1.cmml"><mi id="S2.p16.8.m8.1.1.1.3" xref="S2.p16.8.m8.1.1.1.3.cmml">B</mi><mo id="S2.p16.8.m8.1.1.1.2" xref="S2.p16.8.m8.1.1.1.2.cmml">⁢</mo><mi id="S2.p16.8.m8.1.1.1.4" xref="S2.p16.8.m8.1.1.1.4.cmml">e</mi><mo id="S2.p16.8.m8.1.1.1.2a" xref="S2.p16.8.m8.1.1.1.2.cmml">⁢</mo><mi id="S2.p16.8.m8.1.1.1.5" xref="S2.p16.8.m8.1.1.1.5.cmml">l</mi><mo id="S2.p16.8.m8.1.1.1.2b" xref="S2.p16.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S2.p16.8.m8.1.1.1.1.1" xref="S2.p16.8.m8.1.1.1.1.1.1.cmml"><mo id="S2.p16.8.m8.1.1.1.1.1.2" stretchy="false" xref="S2.p16.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p16.8.m8.1.1.1.1.1.1" xref="S2.p16.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.p16.8.m8.1.1.1.1.1.1.2" xref="S2.p16.8.m8.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p16.8.m8.1.1.1.1.1.1.3" xref="S2.p16.8.m8.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.p16.8.m8.1.1.1.1.1.3" stretchy="false" xref="S2.p16.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p16.8.m8.1.1.2" xref="S2.p16.8.m8.1.1.2.cmml">=</mo><mi id="S2.p16.8.m8.1.1.3" xref="S2.p16.8.m8.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p16.8.m8.1b"><apply id="S2.p16.8.m8.1.1.cmml" xref="S2.p16.8.m8.1.1"><eq id="S2.p16.8.m8.1.1.2.cmml" xref="S2.p16.8.m8.1.1.2"></eq><apply id="S2.p16.8.m8.1.1.1.cmml" xref="S2.p16.8.m8.1.1.1"><times id="S2.p16.8.m8.1.1.1.2.cmml" xref="S2.p16.8.m8.1.1.1.2"></times><ci id="S2.p16.8.m8.1.1.1.3.cmml" xref="S2.p16.8.m8.1.1.1.3">𝐵</ci><ci id="S2.p16.8.m8.1.1.1.4.cmml" xref="S2.p16.8.m8.1.1.1.4">𝑒</ci><ci id="S2.p16.8.m8.1.1.1.5.cmml" xref="S2.p16.8.m8.1.1.1.5">𝑙</ci><apply id="S2.p16.8.m8.1.1.1.1.1.1.cmml" xref="S2.p16.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p16.8.m8.1.1.1.1.1.1.1.cmml" xref="S2.p16.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S2.p16.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.p16.8.m8.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.p16.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p16.8.m8.1.1.1.1.1.1.3">2</cn></apply></apply><csymbol cd="latexml" id="S2.p16.8.m8.1.1.3.cmml" xref="S2.p16.8.m8.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p16.8.m8.1c">Bel(A_{2})=</annotation><annotation encoding="application/x-llamapun" id="S2.p16.8.m8.1d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) =</annotation></semantics></math> <math alttext="Pl(A_{2})=1/3" class="ltx_Math" display="inline" id="S2.p16.9.m9.1"><semantics id="S2.p16.9.m9.1a"><mrow id="S2.p16.9.m9.1.1" xref="S2.p16.9.m9.1.1.cmml"><mrow id="S2.p16.9.m9.1.1.1" xref="S2.p16.9.m9.1.1.1.cmml"><mi id="S2.p16.9.m9.1.1.1.3" xref="S2.p16.9.m9.1.1.1.3.cmml">P</mi><mo id="S2.p16.9.m9.1.1.1.2" xref="S2.p16.9.m9.1.1.1.2.cmml">⁢</mo><mi id="S2.p16.9.m9.1.1.1.4" xref="S2.p16.9.m9.1.1.1.4.cmml">l</mi><mo id="S2.p16.9.m9.1.1.1.2a" xref="S2.p16.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S2.p16.9.m9.1.1.1.1.1" xref="S2.p16.9.m9.1.1.1.1.1.1.cmml"><mo id="S2.p16.9.m9.1.1.1.1.1.2" stretchy="false" xref="S2.p16.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p16.9.m9.1.1.1.1.1.1" xref="S2.p16.9.m9.1.1.1.1.1.1.cmml"><mi id="S2.p16.9.m9.1.1.1.1.1.1.2" xref="S2.p16.9.m9.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.p16.9.m9.1.1.1.1.1.1.3" xref="S2.p16.9.m9.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.p16.9.m9.1.1.1.1.1.3" stretchy="false" xref="S2.p16.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p16.9.m9.1.1.2" xref="S2.p16.9.m9.1.1.2.cmml">=</mo><mrow id="S2.p16.9.m9.1.1.3" xref="S2.p16.9.m9.1.1.3.cmml"><mn id="S2.p16.9.m9.1.1.3.2" xref="S2.p16.9.m9.1.1.3.2.cmml">1</mn><mo id="S2.p16.9.m9.1.1.3.1" xref="S2.p16.9.m9.1.1.3.1.cmml">/</mo><mn id="S2.p16.9.m9.1.1.3.3" xref="S2.p16.9.m9.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p16.9.m9.1b"><apply id="S2.p16.9.m9.1.1.cmml" xref="S2.p16.9.m9.1.1"><eq id="S2.p16.9.m9.1.1.2.cmml" xref="S2.p16.9.m9.1.1.2"></eq><apply id="S2.p16.9.m9.1.1.1.cmml" xref="S2.p16.9.m9.1.1.1"><times id="S2.p16.9.m9.1.1.1.2.cmml" xref="S2.p16.9.m9.1.1.1.2"></times><ci id="S2.p16.9.m9.1.1.1.3.cmml" xref="S2.p16.9.m9.1.1.1.3">𝑃</ci><ci id="S2.p16.9.m9.1.1.1.4.cmml" xref="S2.p16.9.m9.1.1.1.4">𝑙</ci><apply id="S2.p16.9.m9.1.1.1.1.1.1.cmml" xref="S2.p16.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p16.9.m9.1.1.1.1.1.1.1.cmml" xref="S2.p16.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S2.p16.9.m9.1.1.1.1.1.1.2.cmml" xref="S2.p16.9.m9.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.p16.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p16.9.m9.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S2.p16.9.m9.1.1.3.cmml" xref="S2.p16.9.m9.1.1.3"><divide id="S2.p16.9.m9.1.1.3.1.cmml" xref="S2.p16.9.m9.1.1.3.1"></divide><cn id="S2.p16.9.m9.1.1.3.2.cmml" type="integer" xref="S2.p16.9.m9.1.1.3.2">1</cn><cn id="S2.p16.9.m9.1.1.3.3.cmml" type="integer" xref="S2.p16.9.m9.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p16.9.m9.1c">Pl(A_{2})=1/3</annotation><annotation encoding="application/x-llamapun" id="S2.p16.9.m9.1d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 / 3</annotation></semantics></math>, respectively. This state of affairs appears in the first line of Table <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.T1" title="Table 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S2.p17"> <p class="ltx_p" id="S2.p17.7">At <math alttext="t=t_{2}" class="ltx_Math" display="inline" id="S2.p17.1.m1.1"><semantics id="S2.p17.1.m1.1a"><mrow id="S2.p17.1.m1.1.1" xref="S2.p17.1.m1.1.1.cmml"><mi id="S2.p17.1.m1.1.1.2" xref="S2.p17.1.m1.1.1.2.cmml">t</mi><mo id="S2.p17.1.m1.1.1.1" xref="S2.p17.1.m1.1.1.1.cmml">=</mo><msub id="S2.p17.1.m1.1.1.3" xref="S2.p17.1.m1.1.1.3.cmml"><mi id="S2.p17.1.m1.1.1.3.2" xref="S2.p17.1.m1.1.1.3.2.cmml">t</mi><mn id="S2.p17.1.m1.1.1.3.3" xref="S2.p17.1.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p17.1.m1.1b"><apply id="S2.p17.1.m1.1.1.cmml" xref="S2.p17.1.m1.1.1"><eq id="S2.p17.1.m1.1.1.1.cmml" xref="S2.p17.1.m1.1.1.1"></eq><ci id="S2.p17.1.m1.1.1.2.cmml" xref="S2.p17.1.m1.1.1.2">𝑡</ci><apply id="S2.p17.1.m1.1.1.3.cmml" xref="S2.p17.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p17.1.m1.1.1.3.1.cmml" xref="S2.p17.1.m1.1.1.3">subscript</csymbol><ci id="S2.p17.1.m1.1.1.3.2.cmml" xref="S2.p17.1.m1.1.1.3.2">𝑡</ci><cn id="S2.p17.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p17.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p17.1.m1.1c">t=t_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p17.1.m1.1d">italic_t = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> the second nucleotide is captured, namely <math alttext="N_{2}" class="ltx_Math" display="inline" id="S2.p17.2.m2.1"><semantics id="S2.p17.2.m2.1a"><msub id="S2.p17.2.m2.1.1" xref="S2.p17.2.m2.1.1.cmml"><mi id="S2.p17.2.m2.1.1.2" xref="S2.p17.2.m2.1.1.2.cmml">N</mi><mn id="S2.p17.2.m2.1.1.3" xref="S2.p17.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p17.2.m2.1b"><apply id="S2.p17.2.m2.1.1.cmml" xref="S2.p17.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p17.2.m2.1.1.1.cmml" xref="S2.p17.2.m2.1.1">subscript</csymbol><ci id="S2.p17.2.m2.1.1.2.cmml" xref="S2.p17.2.m2.1.1.2">𝑁</ci><cn id="S2.p17.2.m2.1.1.3.cmml" type="integer" xref="S2.p17.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p17.2.m2.1c">N_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p17.2.m2.1d">italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Since also the second nucleotide is compatible with either <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p17.3.m3.1"><semantics id="S2.p17.3.m3.1a"><msub id="S2.p17.3.m3.1.1" xref="S2.p17.3.m3.1.1.cmml"><mi id="S2.p17.3.m3.1.1.2" xref="S2.p17.3.m3.1.1.2.cmml">A</mi><mn id="S2.p17.3.m3.1.1.3" xref="S2.p17.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p17.3.m3.1b"><apply id="S2.p17.3.m3.1.1.cmml" xref="S2.p17.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p17.3.m3.1.1.1.cmml" xref="S2.p17.3.m3.1.1">subscript</csymbol><ci id="S2.p17.3.m3.1.1.2.cmml" xref="S2.p17.3.m3.1.1.2">𝐴</ci><cn id="S2.p17.3.m3.1.1.3.cmml" type="integer" xref="S2.p17.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p17.3.m3.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p17.3.m3.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p17.4.m4.1"><semantics id="S2.p17.4.m4.1a"><msub id="S2.p17.4.m4.1.1" xref="S2.p17.4.m4.1.1.cmml"><mi id="S2.p17.4.m4.1.1.2" xref="S2.p17.4.m4.1.1.2.cmml">A</mi><mn id="S2.p17.4.m4.1.1.3" xref="S2.p17.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p17.4.m4.1b"><apply id="S2.p17.4.m4.1.1.cmml" xref="S2.p17.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p17.4.m4.1.1.1.cmml" xref="S2.p17.4.m4.1.1">subscript</csymbol><ci id="S2.p17.4.m4.1.1.2.cmml" xref="S2.p17.4.m4.1.1.2">𝐴</ci><cn id="S2.p17.4.m4.1.1.3.cmml" type="integer" xref="S2.p17.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p17.4.m4.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p17.4.m4.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, it brings the same body of evidence, namely <math alttext="B=\{m(B_{1}\equiv A_{1})=1/3," class="ltx_math_unparsed" display="inline" id="S2.p17.5.m5.1"><semantics id="S2.p17.5.m5.1a"><mrow id="S2.p17.5.m5.1b"><mi id="S2.p17.5.m5.1.1">B</mi><mo id="S2.p17.5.m5.1.2">=</mo><mrow id="S2.p17.5.m5.1.3"><mo id="S2.p17.5.m5.1.3.1" stretchy="false">{</mo><mi id="S2.p17.5.m5.1.3.2">m</mi><mrow id="S2.p17.5.m5.1.3.3"><mo id="S2.p17.5.m5.1.3.3.1" stretchy="false">(</mo><msub id="S2.p17.5.m5.1.3.3.2"><mi id="S2.p17.5.m5.1.3.3.2.2">B</mi><mn id="S2.p17.5.m5.1.3.3.2.3">1</mn></msub><mo id="S2.p17.5.m5.1.3.3.3">≡</mo><msub id="S2.p17.5.m5.1.3.3.4"><mi id="S2.p17.5.m5.1.3.3.4.2">A</mi><mn id="S2.p17.5.m5.1.3.3.4.3">1</mn></msub><mo id="S2.p17.5.m5.1.3.3.5" stretchy="false">)</mo></mrow><mo id="S2.p17.5.m5.1.3.4">=</mo><mn id="S2.p17.5.m5.1.3.5">1</mn><mo id="S2.p17.5.m5.1.3.6">/</mo><mn id="S2.p17.5.m5.1.3.7">3</mn><mo id="S2.p17.5.m5.1.3.8">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.p17.5.m5.1c">B=\{m(B_{1}\equiv A_{1})=1/3,</annotation><annotation encoding="application/x-llamapun" id="S2.p17.5.m5.1d">italic_B = { italic_m ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 1 / 3 ,</annotation></semantics></math> <math alttext="m(B_{2}\equiv A_{2})=1/3," class="ltx_Math" display="inline" id="S2.p17.6.m6.1"><semantics id="S2.p17.6.m6.1a"><mrow id="S2.p17.6.m6.1.1.1" xref="S2.p17.6.m6.1.1.1.1.cmml"><mrow id="S2.p17.6.m6.1.1.1.1" xref="S2.p17.6.m6.1.1.1.1.cmml"><mrow id="S2.p17.6.m6.1.1.1.1.1" xref="S2.p17.6.m6.1.1.1.1.1.cmml"><mi id="S2.p17.6.m6.1.1.1.1.1.3" xref="S2.p17.6.m6.1.1.1.1.1.3.cmml">m</mi><mo id="S2.p17.6.m6.1.1.1.1.1.2" xref="S2.p17.6.m6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p17.6.m6.1.1.1.1.1.1.1" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.cmml"><mo id="S2.p17.6.m6.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p17.6.m6.1.1.1.1.1.1.1.1" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.cmml"><msub id="S2.p17.6.m6.1.1.1.1.1.1.1.1.2" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.cmml"><mi 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xref="S2.p17.6.m6.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S2.p17.6.m6.1.1.1.2" xref="S2.p17.6.m6.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p17.6.m6.1b"><apply id="S2.p17.6.m6.1.1.1.1.cmml" xref="S2.p17.6.m6.1.1.1"><eq id="S2.p17.6.m6.1.1.1.1.2.cmml" xref="S2.p17.6.m6.1.1.1.1.2"></eq><apply id="S2.p17.6.m6.1.1.1.1.1.cmml" xref="S2.p17.6.m6.1.1.1.1.1"><times id="S2.p17.6.m6.1.1.1.1.1.2.cmml" xref="S2.p17.6.m6.1.1.1.1.1.2"></times><ci id="S2.p17.6.m6.1.1.1.1.1.3.cmml" xref="S2.p17.6.m6.1.1.1.1.1.3">𝑚</ci><apply id="S2.p17.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1"><equivalent id="S2.p17.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.1"></equivalent><apply id="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.2">𝐵</ci><cn id="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.2.3">2</cn></apply><apply id="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.2">𝐴</ci><cn id="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.p17.6.m6.1.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><apply id="S2.p17.6.m6.1.1.1.1.3.cmml" xref="S2.p17.6.m6.1.1.1.1.3"><divide id="S2.p17.6.m6.1.1.1.1.3.1.cmml" xref="S2.p17.6.m6.1.1.1.1.3.1"></divide><cn id="S2.p17.6.m6.1.1.1.1.3.2.cmml" type="integer" xref="S2.p17.6.m6.1.1.1.1.3.2">1</cn><cn id="S2.p17.6.m6.1.1.1.1.3.3.cmml" type="integer" xref="S2.p17.6.m6.1.1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p17.6.m6.1c">m(B_{2}\equiv A_{2})=1/3,</annotation><annotation encoding="application/x-llamapun" id="S2.p17.6.m6.1d">italic_m ( italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≡ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 / 3 ,</annotation></semantics></math> <math alttext="m_{B}(\Theta)=1/3\}" class="ltx_math_unparsed" display="inline" id="S2.p17.7.m7.1"><semantics id="S2.p17.7.m7.1a"><mrow id="S2.p17.7.m7.1b"><msub id="S2.p17.7.m7.1.2"><mi id="S2.p17.7.m7.1.2.2">m</mi><mi id="S2.p17.7.m7.1.2.3">B</mi></msub><mrow id="S2.p17.7.m7.1.3"><mo id="S2.p17.7.m7.1.3.1" stretchy="false">(</mo><mi id="S2.p17.7.m7.1.1" mathvariant="normal">Θ</mi><mo id="S2.p17.7.m7.1.3.2" stretchy="false">)</mo></mrow><mo id="S2.p17.7.m7.1.4">=</mo><mn id="S2.p17.7.m7.1.5">1</mn><mo id="S2.p17.7.m7.1.6">/</mo><mn id="S2.p17.7.m7.1.7">3</mn><mo id="S2.p17.7.m7.1.8" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S2.p17.7.m7.1c">m_{B}(\Theta)=1/3\}</annotation><annotation encoding="application/x-llamapun" id="S2.p17.7.m7.1d">italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Θ ) = 1 / 3 }</annotation></semantics></math>. Since these two bodies of evidence originate from different sources, they support one another insofar they are coherent (they intersect) whereas they discredit one another insofar they are incoherent (they do not intersect).</p> </div> <div class="ltx_para" id="S2.p18"> <p class="ltx_p" id="S2.p18.2">In this case, all possibilities in evidence bodies <math alttext="A" class="ltx_Math" display="inline" id="S2.p18.1.m1.1"><semantics id="S2.p18.1.m1.1a"><mi id="S2.p18.1.m1.1.1" xref="S2.p18.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p18.1.m1.1b"><ci id="S2.p18.1.m1.1.1.cmml" xref="S2.p18.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p18.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p18.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S2.p18.2.m2.1"><semantics id="S2.p18.2.m2.1a"><mi id="S2.p18.2.m2.1.1" xref="S2.p18.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.p18.2.m2.1b"><ci id="S2.p18.2.m2.1.1.cmml" xref="S2.p18.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p18.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.p18.2.m2.1d">italic_B</annotation></semantics></math> intersect one another. Thus, the evidence supporting these intersections is:</p> </div> <div class="ltx_para" id="S2.p19"> <table class="ltx_equation ltx_eqn_table" id="S2.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="\begin{array}[]{rclcl}m(A_{1})&=&m(A_{1})\;m(A_{1})+m(A_{1})\;m_{B}(\Theta)+m_% {A}(\Theta)\;m(A_{1})&=&3/9\\ m(A_{2})&=&m(A_{2})\;m(A_{2})+m(A_{2})\;m_{B}(\Theta)+m_{A}(\Theta)\;m(A_{2})&% =&3/9\\ m(A_{1}\cap A_{2})&=&m(A_{1})\;m(A_{2})+m(A_{2})\;m(A_{1})&=&2/9\\ m(\Theta)&=&m_{A}(\Theta)\;m_{B}(\Theta)&=&1/9\end{array}" class="ltx_Math" display="block" id="S2.Ex1.m1.22"><semantics id="S2.Ex1.m1.22a"><mtable columnspacing="5pt" displaystyle="true" id="S2.Ex1.m1.22.22" rowspacing="0pt" xref="S2.Ex1.m1.22.22.cmml"><mtr id="S2.Ex1.m1.22.22a" xref="S2.Ex1.m1.22.22.cmml"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.22.22b" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.3.cmml">m</mi><mo id="S2.Ex1.m1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S2.Ex1.m1.22.22c" xref="S2.Ex1.m1.22.22.cmml"><mo id="S2.Ex1.m1.7.7.7.8.1" xref="S2.Ex1.m1.7.7.7.8.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.22.22d" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.7.7.7.7.6" xref="S2.Ex1.m1.7.7.7.7.6.cmml"><mrow id="S2.Ex1.m1.5.5.5.5.4.4" xref="S2.Ex1.m1.5.5.5.5.4.4.cmml"><mi id="S2.Ex1.m1.5.5.5.5.4.4.4" xref="S2.Ex1.m1.5.5.5.5.4.4.4.cmml">m</mi><mo id="S2.Ex1.m1.5.5.5.5.4.4.3" xref="S2.Ex1.m1.5.5.5.5.4.4.3.cmml">⁢</mo><mrow id="S2.Ex1.m1.4.4.4.4.3.3.1.1" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.cmml"><mo id="S2.Ex1.m1.4.4.4.4.3.3.1.1.2" stretchy="false" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.4.4.4.4.3.3.1.1.1" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.cmml"><mi id="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.2" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.3" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.4.4.4.4.3.3.1.1.3" stretchy="false" xref="S2.Ex1.m1.4.4.4.4.3.3.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.5.5.5.5.4.4.3a" lspace="0.280em" xref="S2.Ex1.m1.5.5.5.5.4.4.3.cmml">⁢</mo><mi id="S2.Ex1.m1.5.5.5.5.4.4.5" xref="S2.Ex1.m1.5.5.5.5.4.4.5.cmml">m</mi><mo id="S2.Ex1.m1.5.5.5.5.4.4.3b" xref="S2.Ex1.m1.5.5.5.5.4.4.3.cmml">⁢</mo><mrow id="S2.Ex1.m1.5.5.5.5.4.4.2.1" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.cmml"><mo id="S2.Ex1.m1.5.5.5.5.4.4.2.1.2" stretchy="false" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.cmml">(</mo><msub id="S2.Ex1.m1.5.5.5.5.4.4.2.1.1" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.cmml"><mi id="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.2" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.3" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.5.5.5.5.4.4.2.1.3" stretchy="false" xref="S2.Ex1.m1.5.5.5.5.4.4.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.7.7.7.7.6.7" xref="S2.Ex1.m1.7.7.7.7.6.7.cmml">+</mo><mrow id="S2.Ex1.m1.6.6.6.6.5.5" xref="S2.Ex1.m1.6.6.6.6.5.5.cmml"><mi id="S2.Ex1.m1.6.6.6.6.5.5.3" xref="S2.Ex1.m1.6.6.6.6.5.5.3.cmml">m</mi><mo id="S2.Ex1.m1.6.6.6.6.5.5.2" xref="S2.Ex1.m1.6.6.6.6.5.5.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.6.6.6.6.5.5.1.1" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.cmml"><mo id="S2.Ex1.m1.6.6.6.6.5.5.1.1.2" stretchy="false" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.6.6.6.6.5.5.1.1.1" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.cmml"><mi id="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.2" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.3" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.6.6.6.6.5.5.1.1.3" stretchy="false" xref="S2.Ex1.m1.6.6.6.6.5.5.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.6.6.6.6.5.5.2a" lspace="0.280em" xref="S2.Ex1.m1.6.6.6.6.5.5.2.cmml">⁢</mo><msub id="S2.Ex1.m1.6.6.6.6.5.5.4" xref="S2.Ex1.m1.6.6.6.6.5.5.4.cmml"><mi id="S2.Ex1.m1.6.6.6.6.5.5.4.2" xref="S2.Ex1.m1.6.6.6.6.5.5.4.2.cmml">m</mi><mi id="S2.Ex1.m1.6.6.6.6.5.5.4.3" xref="S2.Ex1.m1.6.6.6.6.5.5.4.3.cmml">B</mi></msub><mo id="S2.Ex1.m1.6.6.6.6.5.5.2b" xref="S2.Ex1.m1.6.6.6.6.5.5.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.6.6.6.6.5.5.5.2" xref="S2.Ex1.m1.6.6.6.6.5.5.cmml"><mo id="S2.Ex1.m1.6.6.6.6.5.5.5.2.1" stretchy="false" xref="S2.Ex1.m1.6.6.6.6.5.5.cmml">(</mo><mi id="S2.Ex1.m1.2.2.2.2.1.1" mathvariant="normal" xref="S2.Ex1.m1.2.2.2.2.1.1.cmml">Θ</mi><mo id="S2.Ex1.m1.6.6.6.6.5.5.5.2.2" stretchy="false" xref="S2.Ex1.m1.6.6.6.6.5.5.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.7.7.7.7.6.7a" xref="S2.Ex1.m1.7.7.7.7.6.7.cmml">+</mo><mrow id="S2.Ex1.m1.7.7.7.7.6.6" xref="S2.Ex1.m1.7.7.7.7.6.6.cmml"><msub id="S2.Ex1.m1.7.7.7.7.6.6.3" xref="S2.Ex1.m1.7.7.7.7.6.6.3.cmml"><mi id="S2.Ex1.m1.7.7.7.7.6.6.3.2" xref="S2.Ex1.m1.7.7.7.7.6.6.3.2.cmml">m</mi><mi id="S2.Ex1.m1.7.7.7.7.6.6.3.3" xref="S2.Ex1.m1.7.7.7.7.6.6.3.3.cmml">A</mi></msub><mo id="S2.Ex1.m1.7.7.7.7.6.6.2" xref="S2.Ex1.m1.7.7.7.7.6.6.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.7.7.7.7.6.6.4.2" xref="S2.Ex1.m1.7.7.7.7.6.6.cmml"><mo id="S2.Ex1.m1.7.7.7.7.6.6.4.2.1" stretchy="false" xref="S2.Ex1.m1.7.7.7.7.6.6.cmml">(</mo><mi id="S2.Ex1.m1.3.3.3.3.2.2" mathvariant="normal" xref="S2.Ex1.m1.3.3.3.3.2.2.cmml">Θ</mi><mo id="S2.Ex1.m1.7.7.7.7.6.6.4.2.2" stretchy="false" xref="S2.Ex1.m1.7.7.7.7.6.6.cmml">)</mo></mrow><mo id="S2.Ex1.m1.7.7.7.7.6.6.2a" lspace="0.280em" xref="S2.Ex1.m1.7.7.7.7.6.6.2.cmml">⁢</mo><mi id="S2.Ex1.m1.7.7.7.7.6.6.5" xref="S2.Ex1.m1.7.7.7.7.6.6.5.cmml">m</mi><mo id="S2.Ex1.m1.7.7.7.7.6.6.2b" xref="S2.Ex1.m1.7.7.7.7.6.6.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.7.7.7.7.6.6.1.1" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.cmml"><mo id="S2.Ex1.m1.7.7.7.7.6.6.1.1.2" stretchy="false" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.7.7.7.7.6.6.1.1.1" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.cmml"><mi id="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.2" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.3" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.7.7.7.7.6.6.1.1.3" stretchy="false" xref="S2.Ex1.m1.7.7.7.7.6.6.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S2.Ex1.m1.22.22e" xref="S2.Ex1.m1.22.22.cmml"><mo id="S2.Ex1.m1.7.7.7.9.1" xref="S2.Ex1.m1.7.7.7.9.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.22.22f" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.7.7.7.10.1" xref="S2.Ex1.m1.7.7.7.10.1.cmml"><mn id="S2.Ex1.m1.7.7.7.10.1.2" xref="S2.Ex1.m1.7.7.7.10.1.2.cmml">3</mn><mo id="S2.Ex1.m1.7.7.7.10.1.1" xref="S2.Ex1.m1.7.7.7.10.1.1.cmml">/</mo><mn id="S2.Ex1.m1.7.7.7.10.1.3" xref="S2.Ex1.m1.7.7.7.10.1.3.cmml">9</mn></mrow></mtd></mtr><mtr id="S2.Ex1.m1.22.22g" xref="S2.Ex1.m1.22.22.cmml"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.22.22h" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.8.8.8.1.1" xref="S2.Ex1.m1.8.8.8.1.1.cmml"><mi id="S2.Ex1.m1.8.8.8.1.1.3" xref="S2.Ex1.m1.8.8.8.1.1.3.cmml">m</mi><mo id="S2.Ex1.m1.8.8.8.1.1.2" xref="S2.Ex1.m1.8.8.8.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.8.8.8.1.1.1.1" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.8.8.8.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.8.8.8.1.1.1.1.1" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.8.8.8.1.1.1.1.1.2" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.8.8.8.1.1.1.1.1.3" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.8.8.8.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.8.8.8.1.1.1.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S2.Ex1.m1.22.22i" xref="S2.Ex1.m1.22.22.cmml"><mo id="S2.Ex1.m1.14.14.14.8.1" xref="S2.Ex1.m1.14.14.14.8.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.22.22j" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.14.14.14.7.6" xref="S2.Ex1.m1.14.14.14.7.6.cmml"><mrow id="S2.Ex1.m1.12.12.12.5.4.4" xref="S2.Ex1.m1.12.12.12.5.4.4.cmml"><mi id="S2.Ex1.m1.12.12.12.5.4.4.4" xref="S2.Ex1.m1.12.12.12.5.4.4.4.cmml">m</mi><mo id="S2.Ex1.m1.12.12.12.5.4.4.3" xref="S2.Ex1.m1.12.12.12.5.4.4.3.cmml">⁢</mo><mrow id="S2.Ex1.m1.11.11.11.4.3.3.1.1" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.cmml"><mo id="S2.Ex1.m1.11.11.11.4.3.3.1.1.2" stretchy="false" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.11.11.11.4.3.3.1.1.1" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.cmml"><mi id="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.2" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.3" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.11.11.11.4.3.3.1.1.3" stretchy="false" xref="S2.Ex1.m1.11.11.11.4.3.3.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.12.12.12.5.4.4.3a" lspace="0.280em" xref="S2.Ex1.m1.12.12.12.5.4.4.3.cmml">⁢</mo><mi id="S2.Ex1.m1.12.12.12.5.4.4.5" xref="S2.Ex1.m1.12.12.12.5.4.4.5.cmml">m</mi><mo id="S2.Ex1.m1.12.12.12.5.4.4.3b" xref="S2.Ex1.m1.12.12.12.5.4.4.3.cmml">⁢</mo><mrow id="S2.Ex1.m1.12.12.12.5.4.4.2.1" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.cmml"><mo id="S2.Ex1.m1.12.12.12.5.4.4.2.1.2" stretchy="false" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.cmml">(</mo><msub id="S2.Ex1.m1.12.12.12.5.4.4.2.1.1" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.cmml"><mi id="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.2" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.3" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.12.12.12.5.4.4.2.1.3" stretchy="false" xref="S2.Ex1.m1.12.12.12.5.4.4.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.14.14.14.7.6.7" xref="S2.Ex1.m1.14.14.14.7.6.7.cmml">+</mo><mrow id="S2.Ex1.m1.13.13.13.6.5.5" xref="S2.Ex1.m1.13.13.13.6.5.5.cmml"><mi id="S2.Ex1.m1.13.13.13.6.5.5.3" xref="S2.Ex1.m1.13.13.13.6.5.5.3.cmml">m</mi><mo id="S2.Ex1.m1.13.13.13.6.5.5.2" xref="S2.Ex1.m1.13.13.13.6.5.5.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.13.13.13.6.5.5.1.1" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.cmml"><mo id="S2.Ex1.m1.13.13.13.6.5.5.1.1.2" stretchy="false" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.13.13.13.6.5.5.1.1.1" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.cmml"><mi id="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.2" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.3" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.13.13.13.6.5.5.1.1.3" stretchy="false" xref="S2.Ex1.m1.13.13.13.6.5.5.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.13.13.13.6.5.5.2a" lspace="0.280em" xref="S2.Ex1.m1.13.13.13.6.5.5.2.cmml">⁢</mo><msub id="S2.Ex1.m1.13.13.13.6.5.5.4" xref="S2.Ex1.m1.13.13.13.6.5.5.4.cmml"><mi id="S2.Ex1.m1.13.13.13.6.5.5.4.2" xref="S2.Ex1.m1.13.13.13.6.5.5.4.2.cmml">m</mi><mi id="S2.Ex1.m1.13.13.13.6.5.5.4.3" xref="S2.Ex1.m1.13.13.13.6.5.5.4.3.cmml">B</mi></msub><mo id="S2.Ex1.m1.13.13.13.6.5.5.2b" xref="S2.Ex1.m1.13.13.13.6.5.5.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.13.13.13.6.5.5.5.2" xref="S2.Ex1.m1.13.13.13.6.5.5.cmml"><mo id="S2.Ex1.m1.13.13.13.6.5.5.5.2.1" stretchy="false" xref="S2.Ex1.m1.13.13.13.6.5.5.cmml">(</mo><mi id="S2.Ex1.m1.9.9.9.2.1.1" mathvariant="normal" xref="S2.Ex1.m1.9.9.9.2.1.1.cmml">Θ</mi><mo id="S2.Ex1.m1.13.13.13.6.5.5.5.2.2" stretchy="false" xref="S2.Ex1.m1.13.13.13.6.5.5.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.14.14.14.7.6.7a" xref="S2.Ex1.m1.14.14.14.7.6.7.cmml">+</mo><mrow id="S2.Ex1.m1.14.14.14.7.6.6" xref="S2.Ex1.m1.14.14.14.7.6.6.cmml"><msub id="S2.Ex1.m1.14.14.14.7.6.6.3" xref="S2.Ex1.m1.14.14.14.7.6.6.3.cmml"><mi id="S2.Ex1.m1.14.14.14.7.6.6.3.2" xref="S2.Ex1.m1.14.14.14.7.6.6.3.2.cmml">m</mi><mi id="S2.Ex1.m1.14.14.14.7.6.6.3.3" xref="S2.Ex1.m1.14.14.14.7.6.6.3.3.cmml">A</mi></msub><mo id="S2.Ex1.m1.14.14.14.7.6.6.2" xref="S2.Ex1.m1.14.14.14.7.6.6.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.14.14.14.7.6.6.4.2" xref="S2.Ex1.m1.14.14.14.7.6.6.cmml"><mo id="S2.Ex1.m1.14.14.14.7.6.6.4.2.1" stretchy="false" xref="S2.Ex1.m1.14.14.14.7.6.6.cmml">(</mo><mi id="S2.Ex1.m1.10.10.10.3.2.2" mathvariant="normal" xref="S2.Ex1.m1.10.10.10.3.2.2.cmml">Θ</mi><mo id="S2.Ex1.m1.14.14.14.7.6.6.4.2.2" stretchy="false" xref="S2.Ex1.m1.14.14.14.7.6.6.cmml">)</mo></mrow><mo id="S2.Ex1.m1.14.14.14.7.6.6.2a" lspace="0.280em" xref="S2.Ex1.m1.14.14.14.7.6.6.2.cmml">⁢</mo><mi id="S2.Ex1.m1.14.14.14.7.6.6.5" xref="S2.Ex1.m1.14.14.14.7.6.6.5.cmml">m</mi><mo id="S2.Ex1.m1.14.14.14.7.6.6.2b" xref="S2.Ex1.m1.14.14.14.7.6.6.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.14.14.14.7.6.6.1.1" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.cmml"><mo id="S2.Ex1.m1.14.14.14.7.6.6.1.1.2" stretchy="false" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.14.14.14.7.6.6.1.1.1" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.cmml"><mi id="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.2" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.3" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.14.14.14.7.6.6.1.1.3" stretchy="false" xref="S2.Ex1.m1.14.14.14.7.6.6.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S2.Ex1.m1.22.22k" xref="S2.Ex1.m1.22.22.cmml"><mo id="S2.Ex1.m1.14.14.14.9.1" xref="S2.Ex1.m1.14.14.14.9.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.22.22l" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.14.14.14.10.1" xref="S2.Ex1.m1.14.14.14.10.1.cmml"><mn id="S2.Ex1.m1.14.14.14.10.1.2" xref="S2.Ex1.m1.14.14.14.10.1.2.cmml">3</mn><mo id="S2.Ex1.m1.14.14.14.10.1.1" xref="S2.Ex1.m1.14.14.14.10.1.1.cmml">/</mo><mn id="S2.Ex1.m1.14.14.14.10.1.3" xref="S2.Ex1.m1.14.14.14.10.1.3.cmml">9</mn></mrow></mtd></mtr><mtr id="S2.Ex1.m1.22.22m" xref="S2.Ex1.m1.22.22.cmml"><mtd class="ltx_align_right" columnalign="right" id="S2.Ex1.m1.22.22n" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.15.15.15.1.1" xref="S2.Ex1.m1.15.15.15.1.1.cmml"><mi id="S2.Ex1.m1.15.15.15.1.1.3" xref="S2.Ex1.m1.15.15.15.1.1.3.cmml">m</mi><mo id="S2.Ex1.m1.15.15.15.1.1.2" xref="S2.Ex1.m1.15.15.15.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.15.15.15.1.1.1.1" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.15.15.15.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex1.m1.15.15.15.1.1.1.1.1" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.cmml"><msub id="S2.Ex1.m1.15.15.15.1.1.1.1.1.2" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.2.cmml"><mi id="S2.Ex1.m1.15.15.15.1.1.1.1.1.2.2" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.2.2.cmml">A</mi><mn id="S2.Ex1.m1.15.15.15.1.1.1.1.1.2.3" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.15.15.15.1.1.1.1.1.1" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.1.cmml">∩</mo><msub id="S2.Ex1.m1.15.15.15.1.1.1.1.1.3" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.3.cmml"><mi id="S2.Ex1.m1.15.15.15.1.1.1.1.1.3.2" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.3.2.cmml">A</mi><mn id="S2.Ex1.m1.15.15.15.1.1.1.1.1.3.3" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S2.Ex1.m1.15.15.15.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.15.15.15.1.1.1.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S2.Ex1.m1.22.22o" xref="S2.Ex1.m1.22.22.cmml"><mo id="S2.Ex1.m1.19.19.19.6.1" xref="S2.Ex1.m1.19.19.19.6.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex1.m1.22.22p" xref="S2.Ex1.m1.22.22.cmml"><mrow id="S2.Ex1.m1.19.19.19.5.4" xref="S2.Ex1.m1.19.19.19.5.4.cmml"><mrow id="S2.Ex1.m1.17.17.17.3.2.2" xref="S2.Ex1.m1.17.17.17.3.2.2.cmml"><mi id="S2.Ex1.m1.17.17.17.3.2.2.4" xref="S2.Ex1.m1.17.17.17.3.2.2.4.cmml">m</mi><mo id="S2.Ex1.m1.17.17.17.3.2.2.3" xref="S2.Ex1.m1.17.17.17.3.2.2.3.cmml">⁢</mo><mrow id="S2.Ex1.m1.16.16.16.2.1.1.1.1" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.16.16.16.2.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.16.16.16.2.1.1.1.1.1" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.2" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.2.cmml">A</mi><mn id="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.3" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.16.16.16.2.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.16.16.16.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.17.17.17.3.2.2.3a" lspace="0.280em" 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id="S2.Ex1.m1.22.22.22.6.1.2.cmml" type="integer" xref="S2.Ex1.m1.22.22.22.6.1.2">1</cn><cn id="S2.Ex1.m1.22.22.22.6.1.3.cmml" type="integer" xref="S2.Ex1.m1.22.22.22.6.1.3">9</cn></apply></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.22c">\begin{array}[]{rclcl}m(A_{1})&=&m(A_{1})\;m(A_{1})+m(A_{1})\;m_{B}(\Theta)+m_% {A}(\Theta)\;m(A_{1})&=&3/9\\ m(A_{2})&=&m(A_{2})\;m(A_{2})+m(A_{2})\;m_{B}(\Theta)+m_{A}(\Theta)\;m(A_{2})&% =&3/9\\ m(A_{1}\cap A_{2})&=&m(A_{1})\;m(A_{2})+m(A_{2})\;m(A_{1})&=&2/9\\ m(\Theta)&=&m_{A}(\Theta)\;m_{B}(\Theta)&=&1/9\end{array}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.22d">start_ARRAY start_ROW start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Θ ) + italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL 3 / 9 end_CELL end_ROW start_ROW start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Θ ) + italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL 3 / 9 end_CELL end_ROW start_ROW start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL 2 / 9 end_CELL end_ROW start_ROW start_CELL italic_m ( roman_Θ ) end_CELL start_CELL = end_CELL start_CELL italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Θ ) end_CELL start_CELL = end_CELL start_CELL 1 / 9 end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p20"> <p class="ltx_p" id="S2.p20.1">The results are displayed in the second line of Table <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.T1" title="Table 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a>. Since more information has become available, <math alttext="m(\Theta)" class="ltx_Math" display="inline" id="S2.p20.1.m1.1"><semantics id="S2.p20.1.m1.1a"><mrow id="S2.p20.1.m1.1.2" xref="S2.p20.1.m1.1.2.cmml"><mi id="S2.p20.1.m1.1.2.2" xref="S2.p20.1.m1.1.2.2.cmml">m</mi><mo id="S2.p20.1.m1.1.2.1" xref="S2.p20.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p20.1.m1.1.2.3.2" xref="S2.p20.1.m1.1.2.cmml"><mo id="S2.p20.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p20.1.m1.1.2.cmml">(</mo><mi id="S2.p20.1.m1.1.1" mathvariant="normal" xref="S2.p20.1.m1.1.1.cmml">Θ</mi><mo id="S2.p20.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p20.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p20.1.m1.1b"><apply id="S2.p20.1.m1.1.2.cmml" xref="S2.p20.1.m1.1.2"><times id="S2.p20.1.m1.1.2.1.cmml" xref="S2.p20.1.m1.1.2.1"></times><ci id="S2.p20.1.m1.1.2.2.cmml" xref="S2.p20.1.m1.1.2.2">𝑚</ci><ci id="S2.p20.1.m1.1.1.cmml" xref="S2.p20.1.m1.1.1">Θ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p20.1.m1.1c">m(\Theta)</annotation><annotation encoding="application/x-llamapun" id="S2.p20.1.m1.1d">italic_m ( roman_Θ )</annotation></semantics></math> has decreased. However, the two interpretations of the codon are still equally likely.</p> </div> <div class="ltx_para" id="S2.p21"> <p class="ltx_p" id="S2.p21.6">Suppose that this is an unambiguous code. At <math alttext="t=t_{3}" class="ltx_Math" display="inline" id="S2.p21.1.m1.1"><semantics id="S2.p21.1.m1.1a"><mrow id="S2.p21.1.m1.1.1" xref="S2.p21.1.m1.1.1.cmml"><mi id="S2.p21.1.m1.1.1.2" xref="S2.p21.1.m1.1.1.2.cmml">t</mi><mo id="S2.p21.1.m1.1.1.1" xref="S2.p21.1.m1.1.1.1.cmml">=</mo><msub id="S2.p21.1.m1.1.1.3" xref="S2.p21.1.m1.1.1.3.cmml"><mi id="S2.p21.1.m1.1.1.3.2" xref="S2.p21.1.m1.1.1.3.2.cmml">t</mi><mn id="S2.p21.1.m1.1.1.3.3" xref="S2.p21.1.m1.1.1.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p21.1.m1.1b"><apply id="S2.p21.1.m1.1.1.cmml" xref="S2.p21.1.m1.1.1"><eq id="S2.p21.1.m1.1.1.1.cmml" xref="S2.p21.1.m1.1.1.1"></eq><ci id="S2.p21.1.m1.1.1.2.cmml" xref="S2.p21.1.m1.1.1.2">𝑡</ci><apply id="S2.p21.1.m1.1.1.3.cmml" xref="S2.p21.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p21.1.m1.1.1.3.1.cmml" xref="S2.p21.1.m1.1.1.3">subscript</csymbol><ci id="S2.p21.1.m1.1.1.3.2.cmml" xref="S2.p21.1.m1.1.1.3.2">𝑡</ci><cn id="S2.p21.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p21.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.1.m1.1c">t=t_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.p21.1.m1.1d">italic_t = italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> nucleotide <math alttext="N_{3}" class="ltx_Math" display="inline" id="S2.p21.2.m2.1"><semantics id="S2.p21.2.m2.1a"><msub id="S2.p21.2.m2.1.1" xref="S2.p21.2.m2.1.1.cmml"><mi id="S2.p21.2.m2.1.1.2" xref="S2.p21.2.m2.1.1.2.cmml">N</mi><mn id="S2.p21.2.m2.1.1.3" xref="S2.p21.2.m2.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p21.2.m2.1b"><apply id="S2.p21.2.m2.1.1.cmml" xref="S2.p21.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p21.2.m2.1.1.1.cmml" xref="S2.p21.2.m2.1.1">subscript</csymbol><ci id="S2.p21.2.m2.1.1.2.cmml" xref="S2.p21.2.m2.1.1.2">𝑁</ci><cn id="S2.p21.2.m2.1.1.3.cmml" type="integer" xref="S2.p21.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.2.m2.1c">N_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.p21.2.m2.1d">italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> arrives, bearing evidence that this codom must be interpreted as amino acid is <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p21.3.m3.1"><semantics id="S2.p21.3.m3.1a"><msub id="S2.p21.3.m3.1.1" xref="S2.p21.3.m3.1.1.cmml"><mi id="S2.p21.3.m3.1.1.2" xref="S2.p21.3.m3.1.1.2.cmml">A</mi><mn id="S2.p21.3.m3.1.1.3" xref="S2.p21.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p21.3.m3.1b"><apply id="S2.p21.3.m3.1.1.cmml" xref="S2.p21.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p21.3.m3.1.1.1.cmml" xref="S2.p21.3.m3.1.1">subscript</csymbol><ci id="S2.p21.3.m3.1.1.2.cmml" xref="S2.p21.3.m3.1.1.2">𝐴</ci><cn id="S2.p21.3.m3.1.1.3.cmml" type="integer" xref="S2.p21.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.3.m3.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p21.3.m3.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. This corresponds to a body of evidence <math alttext="C=" class="ltx_Math" display="inline" id="S2.p21.4.m4.1"><semantics id="S2.p21.4.m4.1a"><mrow id="S2.p21.4.m4.1.1" xref="S2.p21.4.m4.1.1.cmml"><mi id="S2.p21.4.m4.1.1.2" xref="S2.p21.4.m4.1.1.2.cmml">C</mi><mo id="S2.p21.4.m4.1.1.1" xref="S2.p21.4.m4.1.1.1.cmml">=</mo><mi id="S2.p21.4.m4.1.1.3" xref="S2.p21.4.m4.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p21.4.m4.1b"><apply id="S2.p21.4.m4.1.1.cmml" xref="S2.p21.4.m4.1.1"><eq id="S2.p21.4.m4.1.1.1.cmml" xref="S2.p21.4.m4.1.1.1"></eq><ci id="S2.p21.4.m4.1.1.2.cmml" xref="S2.p21.4.m4.1.1.2">𝐶</ci><csymbol cd="latexml" id="S2.p21.4.m4.1.1.3.cmml" xref="S2.p21.4.m4.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.4.m4.1c">C=</annotation><annotation encoding="application/x-llamapun" id="S2.p21.4.m4.1d">italic_C =</annotation></semantics></math> <math alttext="\{m(C_{1}\equiv A_{2})=1\}" class="ltx_Math" display="inline" id="S2.p21.5.m5.1"><semantics id="S2.p21.5.m5.1a"><mrow id="S2.p21.5.m5.1.1.1" xref="S2.p21.5.m5.1.1.2.cmml"><mo id="S2.p21.5.m5.1.1.1.2" stretchy="false" xref="S2.p21.5.m5.1.1.2.cmml">{</mo><mrow id="S2.p21.5.m5.1.1.1.1" xref="S2.p21.5.m5.1.1.1.1.cmml"><mrow id="S2.p21.5.m5.1.1.1.1.1" xref="S2.p21.5.m5.1.1.1.1.1.cmml"><mi id="S2.p21.5.m5.1.1.1.1.1.3" xref="S2.p21.5.m5.1.1.1.1.1.3.cmml">m</mi><mo id="S2.p21.5.m5.1.1.1.1.1.2" xref="S2.p21.5.m5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p21.5.m5.1.1.1.1.1.1.1" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.cmml"><mo id="S2.p21.5.m5.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p21.5.m5.1.1.1.1.1.1.1.1" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.cmml"><msub id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.2" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.2.cmml">C</mi><mn id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.3" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p21.5.m5.1.1.1.1.1.1.1.1.1" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.1.cmml">≡</mo><msub id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.2" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.3" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S2.p21.5.m5.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p21.5.m5.1.1.1.1.2" xref="S2.p21.5.m5.1.1.1.1.2.cmml">=</mo><mn id="S2.p21.5.m5.1.1.1.1.3" xref="S2.p21.5.m5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.p21.5.m5.1.1.1.3" stretchy="false" xref="S2.p21.5.m5.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p21.5.m5.1b"><set id="S2.p21.5.m5.1.1.2.cmml" xref="S2.p21.5.m5.1.1.1"><apply id="S2.p21.5.m5.1.1.1.1.cmml" xref="S2.p21.5.m5.1.1.1.1"><eq id="S2.p21.5.m5.1.1.1.1.2.cmml" xref="S2.p21.5.m5.1.1.1.1.2"></eq><apply id="S2.p21.5.m5.1.1.1.1.1.cmml" xref="S2.p21.5.m5.1.1.1.1.1"><times id="S2.p21.5.m5.1.1.1.1.1.2.cmml" xref="S2.p21.5.m5.1.1.1.1.1.2"></times><ci id="S2.p21.5.m5.1.1.1.1.1.3.cmml" xref="S2.p21.5.m5.1.1.1.1.1.3">𝑚</ci><apply id="S2.p21.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1"><equivalent id="S2.p21.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.1"></equivalent><apply id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.2">𝐶</ci><cn id="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.2">𝐴</ci><cn id="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.p21.5.m5.1.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><cn id="S2.p21.5.m5.1.1.1.1.3.cmml" type="integer" xref="S2.p21.5.m5.1.1.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.5.m5.1c">\{m(C_{1}\equiv A_{2})=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.p21.5.m5.1d">{ italic_m ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 }</annotation></semantics></math>. The third line of Table <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.T1" title="Table 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> makes clear that in this case the available evidence strongly supports interpreting this codon as amino acid <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.p21.6.m6.1"><semantics id="S2.p21.6.m6.1a"><msub id="S2.p21.6.m6.1.1" xref="S2.p21.6.m6.1.1.cmml"><mi id="S2.p21.6.m6.1.1.2" xref="S2.p21.6.m6.1.1.2.cmml">A</mi><mn id="S2.p21.6.m6.1.1.3" xref="S2.p21.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p21.6.m6.1b"><apply id="S2.p21.6.m6.1.1.cmml" xref="S2.p21.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p21.6.m6.1.1.1.cmml" xref="S2.p21.6.m6.1.1">subscript</csymbol><ci id="S2.p21.6.m6.1.1.2.cmml" xref="S2.p21.6.m6.1.1.2">𝐴</ci><cn id="S2.p21.6.m6.1.1.3.cmml" type="integer" xref="S2.p21.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p21.6.m6.1c">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p21.6.m6.1d">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p22"> <p class="ltx_p" id="S2.p22.4">Conversely, let us suppose that this is an ambiguous code in the previously outlined sense that the third nucleotide can be misinterpreted as coding for <math alttext="A_{1}" class="ltx_Math" display="inline" id="S2.p22.1.m1.1"><semantics id="S2.p22.1.m1.1a"><msub id="S2.p22.1.m1.1.1" xref="S2.p22.1.m1.1.1.cmml"><mi id="S2.p22.1.m1.1.1.2" xref="S2.p22.1.m1.1.1.2.cmml">A</mi><mn id="S2.p22.1.m1.1.1.3" xref="S2.p22.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p22.1.m1.1b"><apply id="S2.p22.1.m1.1.1.cmml" xref="S2.p22.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p22.1.m1.1.1.1.cmml" xref="S2.p22.1.m1.1.1">subscript</csymbol><ci id="S2.p22.1.m1.1.1.2.cmml" xref="S2.p22.1.m1.1.1.2">𝐴</ci><cn id="S2.p22.1.m1.1.1.3.cmml" type="integer" xref="S2.p22.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p22.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p22.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. In this case, the third body of evidence is <math alttext="C=" class="ltx_Math" display="inline" id="S2.p22.2.m2.1"><semantics id="S2.p22.2.m2.1a"><mrow id="S2.p22.2.m2.1.1" xref="S2.p22.2.m2.1.1.cmml"><mi id="S2.p22.2.m2.1.1.2" xref="S2.p22.2.m2.1.1.2.cmml">C</mi><mo id="S2.p22.2.m2.1.1.1" xref="S2.p22.2.m2.1.1.1.cmml">=</mo><mi id="S2.p22.2.m2.1.1.3" xref="S2.p22.2.m2.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p22.2.m2.1b"><apply id="S2.p22.2.m2.1.1.cmml" xref="S2.p22.2.m2.1.1"><eq id="S2.p22.2.m2.1.1.1.cmml" xref="S2.p22.2.m2.1.1.1"></eq><ci id="S2.p22.2.m2.1.1.2.cmml" xref="S2.p22.2.m2.1.1.2">𝐶</ci><csymbol cd="latexml" id="S2.p22.2.m2.1.1.3.cmml" xref="S2.p22.2.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p22.2.m2.1c">C=</annotation><annotation encoding="application/x-llamapun" id="S2.p22.2.m2.1d">italic_C =</annotation></semantics></math> <math alttext="\{m_{C}(\Theta)=1\}" class="ltx_Math" display="inline" id="S2.p22.3.m3.2"><semantics id="S2.p22.3.m3.2a"><mrow id="S2.p22.3.m3.2.2.1" xref="S2.p22.3.m3.2.2.2.cmml"><mo id="S2.p22.3.m3.2.2.1.2" stretchy="false" xref="S2.p22.3.m3.2.2.2.cmml">{</mo><mrow id="S2.p22.3.m3.2.2.1.1" xref="S2.p22.3.m3.2.2.1.1.cmml"><mrow id="S2.p22.3.m3.2.2.1.1.2" xref="S2.p22.3.m3.2.2.1.1.2.cmml"><msub id="S2.p22.3.m3.2.2.1.1.2.2" xref="S2.p22.3.m3.2.2.1.1.2.2.cmml"><mi id="S2.p22.3.m3.2.2.1.1.2.2.2" xref="S2.p22.3.m3.2.2.1.1.2.2.2.cmml">m</mi><mi id="S2.p22.3.m3.2.2.1.1.2.2.3" xref="S2.p22.3.m3.2.2.1.1.2.2.3.cmml">C</mi></msub><mo id="S2.p22.3.m3.2.2.1.1.2.1" xref="S2.p22.3.m3.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.p22.3.m3.2.2.1.1.2.3.2" xref="S2.p22.3.m3.2.2.1.1.2.cmml"><mo id="S2.p22.3.m3.2.2.1.1.2.3.2.1" stretchy="false" xref="S2.p22.3.m3.2.2.1.1.2.cmml">(</mo><mi id="S2.p22.3.m3.1.1" mathvariant="normal" xref="S2.p22.3.m3.1.1.cmml">Θ</mi><mo id="S2.p22.3.m3.2.2.1.1.2.3.2.2" stretchy="false" xref="S2.p22.3.m3.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p22.3.m3.2.2.1.1.1" xref="S2.p22.3.m3.2.2.1.1.1.cmml">=</mo><mn id="S2.p22.3.m3.2.2.1.1.3" xref="S2.p22.3.m3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S2.p22.3.m3.2.2.1.3" stretchy="false" xref="S2.p22.3.m3.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p22.3.m3.2b"><set id="S2.p22.3.m3.2.2.2.cmml" xref="S2.p22.3.m3.2.2.1"><apply id="S2.p22.3.m3.2.2.1.1.cmml" xref="S2.p22.3.m3.2.2.1.1"><eq id="S2.p22.3.m3.2.2.1.1.1.cmml" xref="S2.p22.3.m3.2.2.1.1.1"></eq><apply id="S2.p22.3.m3.2.2.1.1.2.cmml" xref="S2.p22.3.m3.2.2.1.1.2"><times id="S2.p22.3.m3.2.2.1.1.2.1.cmml" xref="S2.p22.3.m3.2.2.1.1.2.1"></times><apply id="S2.p22.3.m3.2.2.1.1.2.2.cmml" xref="S2.p22.3.m3.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S2.p22.3.m3.2.2.1.1.2.2.1.cmml" xref="S2.p22.3.m3.2.2.1.1.2.2">subscript</csymbol><ci id="S2.p22.3.m3.2.2.1.1.2.2.2.cmml" xref="S2.p22.3.m3.2.2.1.1.2.2.2">𝑚</ci><ci id="S2.p22.3.m3.2.2.1.1.2.2.3.cmml" xref="S2.p22.3.m3.2.2.1.1.2.2.3">𝐶</ci></apply><ci id="S2.p22.3.m3.1.1.cmml" xref="S2.p22.3.m3.1.1">Θ</ci></apply><cn id="S2.p22.3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.p22.3.m3.2.2.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p22.3.m3.2c">\{m_{C}(\Theta)=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.p22.3.m3.2d">{ italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Θ ) = 1 }</annotation></semantics></math> and, as reported in the fourth line of Table <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.T1" title="Table 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a>, the available evidence is just the same as at <math alttext="t=t_{2}" class="ltx_Math" display="inline" id="S2.p22.4.m4.1"><semantics id="S2.p22.4.m4.1a"><mrow id="S2.p22.4.m4.1.1" xref="S2.p22.4.m4.1.1.cmml"><mi 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id="S2.p22.4.m4.1c">t=t_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p22.4.m4.1d">italic_t = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_table" id="S2.T1"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S2.T1.22.22"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.22.22.23.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.22.22.23.1.1">Time</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.22.22.23.1.2">Body of Evidence</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.22.22.23.1.3">Evaluation</td> </tr> <tr class="ltx_tr" id="S2.T1.3.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.1.1.1.1"><math alttext="t_{1}" class="ltx_Math" display="inline" id="S2.T1.1.1.1.1.m1.1"><semantics 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xref="S2.T1.2.2.2.2.m1.1.1">Θ</ci></apply></apply><apply id="S2.T1.2.2.2.2.m1.2.2.1.1e.cmml" xref="S2.T1.2.2.2.2.m1.2.2.1.1"><eq id="S2.T1.2.2.2.2.m1.2.2.1.1.7.cmml" xref="S2.T1.2.2.2.2.m1.2.2.1.1.7"></eq><share href="https://arxiv.org/html/2503.18984v1#S2.T1.2.2.2.2.m1.2.2.1.1.6.cmml" id="S2.T1.2.2.2.2.m1.2.2.1.1f.cmml" xref="S2.T1.2.2.2.2.m1.2.2.1.1"></share><apply id="S2.T1.2.2.2.2.m1.2.2.1.1.8.cmml" xref="S2.T1.2.2.2.2.m1.2.2.1.1.8"><divide id="S2.T1.2.2.2.2.m1.2.2.1.1.8.1.cmml" xref="S2.T1.2.2.2.2.m1.2.2.1.1.8.1"></divide><cn id="S2.T1.2.2.2.2.m1.2.2.1.1.8.2.cmml" type="integer" xref="S2.T1.2.2.2.2.m1.2.2.1.1.8.2">1</cn><cn id="S2.T1.2.2.2.2.m1.2.2.1.1.8.3.cmml" type="integer" xref="S2.T1.2.2.2.2.m1.2.2.1.1.8.3">3</cn></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.2.2.2.2.m1.2c">\{m(A_{1})=m(A_{2})=m(\Theta)=1/3\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.2.2.2.2.m1.2d">{ italic_m ( italic_A 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id="S2.T1.4.4.4.1.m1.2.2.2.2.3.2.cmml" type="integer" xref="S2.T1.4.4.4.1.m1.2.2.2.2.3.2">1</cn><cn id="S2.T1.4.4.4.1.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.T1.4.4.4.1.m1.2.2.2.2.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.4.4.4.1.m1.2c">Bel(A_{2})=1/3,\;Pl(A_{2})=1/3</annotation><annotation encoding="application/x-llamapun" id="S2.T1.4.4.4.1.m1.2d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 / 3 , italic_P italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 / 3</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.T1.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.5.5.5.1"><math alttext="t_{2}" class="ltx_Math" display="inline" id="S2.T1.5.5.5.1.m1.1"><semantics id="S2.T1.5.5.5.1.m1.1a"><msub id="S2.T1.5.5.5.1.m1.1.1" xref="S2.T1.5.5.5.1.m1.1.1.cmml"><mi id="S2.T1.5.5.5.1.m1.1.1.2" 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href="https://arxiv.org/html/2503.18984v1#S2.T1.6.6.6.2.m1.2.2.1.1.6.cmml" id="S2.T1.6.6.6.2.m1.2.2.1.1f.cmml" xref="S2.T1.6.6.6.2.m1.2.2.1.1"></share><apply id="S2.T1.6.6.6.2.m1.2.2.1.1.8.cmml" xref="S2.T1.6.6.6.2.m1.2.2.1.1.8"><divide id="S2.T1.6.6.6.2.m1.2.2.1.1.8.1.cmml" xref="S2.T1.6.6.6.2.m1.2.2.1.1.8.1"></divide><cn id="S2.T1.6.6.6.2.m1.2.2.1.1.8.2.cmml" type="integer" xref="S2.T1.6.6.6.2.m1.2.2.1.1.8.2">1</cn><cn id="S2.T1.6.6.6.2.m1.2.2.1.1.8.3.cmml" type="integer" xref="S2.T1.6.6.6.2.m1.2.2.1.1.8.3">3</cn></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.6.6.6.2.m1.2c">\{m(A_{1})=m(A_{2})=m(\Theta)=1/3\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.6.6.6.2.m1.2d">{ italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_m ( roman_Θ ) = 1 / 3 }</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S2.T1.6.6.6.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.8.8.8"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S2.T1.8.8.8.3"></th> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T1.7.7.7.1"><math alttext="\{m(A_{1})=m(A_{2})=3/9," class="ltx_math_unparsed" display="inline" id="S2.T1.7.7.7.1.m1.1"><semantics id="S2.T1.7.7.7.1.m1.1a"><mrow id="S2.T1.7.7.7.1.m1.1b"><mo id="S2.T1.7.7.7.1.m1.1.1" stretchy="false">{</mo><mi id="S2.T1.7.7.7.1.m1.1.2">m</mi><mrow id="S2.T1.7.7.7.1.m1.1.3"><mo id="S2.T1.7.7.7.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.T1.7.7.7.1.m1.1.3.2"><mi id="S2.T1.7.7.7.1.m1.1.3.2.2">A</mi><mn id="S2.T1.7.7.7.1.m1.1.3.2.3">1</mn></msub><mo id="S2.T1.7.7.7.1.m1.1.3.3" stretchy="false">)</mo></mrow><mo id="S2.T1.7.7.7.1.m1.1.4">=</mo><mi id="S2.T1.7.7.7.1.m1.1.5">m</mi><mrow id="S2.T1.7.7.7.1.m1.1.6"><mo id="S2.T1.7.7.7.1.m1.1.6.1" stretchy="false">(</mo><msub id="S2.T1.7.7.7.1.m1.1.6.2"><mi id="S2.T1.7.7.7.1.m1.1.6.2.2">A</mi><mn 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xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.T1.8.8.8.2.m1.2.2.2.2.2" xref="S2.T1.8.8.8.2.m1.2.2.2.2.2.cmml">=</mo><mrow id="S2.T1.8.8.8.2.m1.2.2.2.2.3" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.cmml"><mn id="S2.T1.8.8.8.2.m1.2.2.2.2.3.2" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.2.cmml">3</mn><mo id="S2.T1.8.8.8.2.m1.2.2.2.2.3.1" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.1.cmml">/</mo><mn id="S2.T1.8.8.8.2.m1.2.2.2.2.3.3" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.3.cmml">9</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.8.8.8.2.m1.2b"><apply id="S2.T1.8.8.8.2.m1.2.2.3.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2"><csymbol cd="ambiguous" id="S2.T1.8.8.8.2.m1.2.2.3a.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S2.T1.8.8.8.2.m1.1.1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.1.1.1.1"><eq id="S2.T1.8.8.8.2.m1.1.1.1.1.2.cmml" xref="S2.T1.8.8.8.2.m1.1.1.1.1.2"></eq><apply id="S2.T1.8.8.8.2.m1.1.1.1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.1.1.1.1.1"><times 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type="integer" xref="S2.T1.8.8.8.2.m1.1.1.1.1.3.2">3</cn><cn id="S2.T1.8.8.8.2.m1.1.1.1.1.3.3.cmml" type="integer" xref="S2.T1.8.8.8.2.m1.1.1.1.1.3.3">9</cn></apply></apply><apply id="S2.T1.8.8.8.2.m1.2.2.2.2.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2"><eq id="S2.T1.8.8.8.2.m1.2.2.2.2.2.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.2"></eq><apply id="S2.T1.8.8.8.2.m1.2.2.2.2.1.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1"><times id="S2.T1.8.8.8.2.m1.2.2.2.2.1.2.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.2"></times><ci id="S2.T1.8.8.8.2.m1.2.2.2.2.1.3.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.3">𝐵</ci><ci id="S2.T1.8.8.8.2.m1.2.2.2.2.1.4.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.4">𝑒</ci><ci id="S2.T1.8.8.8.2.m1.2.2.2.2.1.5.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.5">𝑙</ci><apply id="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.2">𝐴</ci><cn id="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.T1.8.8.8.2.m1.2.2.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="S2.T1.8.8.8.2.m1.2.2.2.2.3.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3"><divide id="S2.T1.8.8.8.2.m1.2.2.2.2.3.1.cmml" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.1"></divide><cn id="S2.T1.8.8.8.2.m1.2.2.2.2.3.2.cmml" type="integer" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.2">3</cn><cn id="S2.T1.8.8.8.2.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.T1.8.8.8.2.m1.2.2.2.2.3.3">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.8.8.8.2.m1.2c">Bel(A_{1})=3/9,\;Bel(A_{2})=3/9</annotation><annotation encoding="application/x-llamapun" id="S2.T1.8.8.8.2.m1.2d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 3 / 9 , italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 3 / 9</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.T1.10.10.10"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S2.T1.10.10.10.3"></th> <td class="ltx_td ltx_align_right ltx_border_r" id="S2.T1.9.9.9.1"><math alttext="m(A_{1}\cap A_{2})=2/9,\;m(\Theta)=1/9\}" class="ltx_math_unparsed" display="inline" id="S2.T1.9.9.9.1.m1.1"><semantics id="S2.T1.9.9.9.1.m1.1a"><mrow id="S2.T1.9.9.9.1.m1.1b"><mi id="S2.T1.9.9.9.1.m1.1.2">m</mi><mrow id="S2.T1.9.9.9.1.m1.1.3"><mo id="S2.T1.9.9.9.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.T1.9.9.9.1.m1.1.3.2"><mi id="S2.T1.9.9.9.1.m1.1.3.2.2">A</mi><mn id="S2.T1.9.9.9.1.m1.1.3.2.3">1</mn></msub><mo id="S2.T1.9.9.9.1.m1.1.3.3">∩</mo><msub id="S2.T1.9.9.9.1.m1.1.3.4"><mi id="S2.T1.9.9.9.1.m1.1.3.4.2">A</mi><mn id="S2.T1.9.9.9.1.m1.1.3.4.3">2</mn></msub><mo id="S2.T1.9.9.9.1.m1.1.3.5" stretchy="false">)</mo></mrow><mo id="S2.T1.9.9.9.1.m1.1.4">=</mo><mn id="S2.T1.9.9.9.1.m1.1.5">2</mn><mo 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cd="ambiguous" id="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1.1.2">𝐴</ci><cn id="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.T1.10.10.10.2.m1.2.2.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="S2.T1.10.10.10.2.m1.2.2.2.2.3.cmml" xref="S2.T1.10.10.10.2.m1.2.2.2.2.3"><divide id="S2.T1.10.10.10.2.m1.2.2.2.2.3.1.cmml" xref="S2.T1.10.10.10.2.m1.2.2.2.2.3.1"></divide><cn id="S2.T1.10.10.10.2.m1.2.2.2.2.3.2.cmml" type="integer" xref="S2.T1.10.10.10.2.m1.2.2.2.2.3.2">5</cn><cn id="S2.T1.10.10.10.2.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.T1.10.10.10.2.m1.2.2.2.2.3.3">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.10.10.10.2.m1.2c">Pl(A_{1})=5/9,\;Pl(A_{2})=5/9</annotation><annotation encoding="application/x-llamapun" id="S2.T1.10.10.10.2.m1.2d">italic_P italic_l ( 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xref="S2.T1.11.11.11.1.m1.1.1.2">𝑡</ci><cn id="S2.T1.11.11.11.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.11.11.11.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.11.11.11.1.m1.1c">t_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.11.11.11.1.m1.1d">italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T1.12.12.12.2"><math alttext="\{m(A_{2})=1\}" class="ltx_Math" display="inline" id="S2.T1.12.12.12.2.m1.1"><semantics id="S2.T1.12.12.12.2.m1.1a"><mrow id="S2.T1.12.12.12.2.m1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.2.cmml"><mo id="S2.T1.12.12.12.2.m1.1.1.1.2" stretchy="false" xref="S2.T1.12.12.12.2.m1.1.1.2.cmml">{</mo><mrow id="S2.T1.12.12.12.2.m1.1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.1.1.cmml"><mrow id="S2.T1.12.12.12.2.m1.1.1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.cmml"><mi id="S2.T1.12.12.12.2.m1.1.1.1.1.1.3" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.3.cmml">m</mi><mo id="S2.T1.12.12.12.2.m1.1.1.1.1.1.2" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.2" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.2.cmml">A</mi><mn id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.3" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.T1.12.12.12.2.m1.1.1.1.1.2" xref="S2.T1.12.12.12.2.m1.1.1.1.1.2.cmml">=</mo><mn id="S2.T1.12.12.12.2.m1.1.1.1.1.3" xref="S2.T1.12.12.12.2.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.T1.12.12.12.2.m1.1.1.1.3" stretchy="false" xref="S2.T1.12.12.12.2.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.12.12.12.2.m1.1b"><set id="S2.T1.12.12.12.2.m1.1.1.2.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1"><apply id="S2.T1.12.12.12.2.m1.1.1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1"><eq id="S2.T1.12.12.12.2.m1.1.1.1.1.2.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.2"></eq><apply id="S2.T1.12.12.12.2.m1.1.1.1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1"><times id="S2.T1.12.12.12.2.m1.1.1.1.1.1.2.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.2"></times><ci id="S2.T1.12.12.12.2.m1.1.1.1.1.1.3.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.3">𝑚</ci><apply id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.T1.12.12.12.2.m1.1.1.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="S2.T1.12.12.12.2.m1.1.1.1.1.3.cmml" type="integer" xref="S2.T1.12.12.12.2.m1.1.1.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.12.12.12.2.m1.1c">\{m(A_{2})=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.12.12.12.2.m1.1d">{ italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1 }</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S2.T1.12.12.12.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.14.14.14"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S2.T1.14.14.14.3">non</th> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T1.13.13.13.1"><math alttext="\{m(A_{1})=0,\;m(A_{2})=4/9," class="ltx_math_unparsed" display="inline" 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id="S2.T1.13.13.13.1.m1.1.10">4</mn><mo id="S2.T1.13.13.13.1.m1.1.11">/</mo><mn id="S2.T1.13.13.13.1.m1.1.12">9</mn><mo id="S2.T1.13.13.13.1.m1.1.13">,</mo></mrow><annotation encoding="application/x-tex" id="S2.T1.13.13.13.1.m1.1c">\{m(A_{1})=0,\;m(A_{2})=4/9,</annotation><annotation encoding="application/x-llamapun" id="S2.T1.13.13.13.1.m1.1d">{ italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0 , italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 4 / 9 ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T1.14.14.14.2"><math alttext="Bel(A_{1})=0,\;Bel(A_{2})=4/9" class="ltx_Math" display="inline" id="S2.T1.14.14.14.2.m1.2"><semantics id="S2.T1.14.14.14.2.m1.2a"><mrow id="S2.T1.14.14.14.2.m1.2.2.2" xref="S2.T1.14.14.14.2.m1.2.2.3.cmml"><mrow id="S2.T1.14.14.14.2.m1.1.1.1.1" xref="S2.T1.14.14.14.2.m1.1.1.1.1.cmml"><mrow id="S2.T1.14.14.14.2.m1.1.1.1.1.1" xref="S2.T1.14.14.14.2.m1.1.1.1.1.1.cmml"><mi 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xref="S2.T1.14.14.14.2.m1.2.2.2.2.2"></eq><apply id="S2.T1.14.14.14.2.m1.2.2.2.2.1.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1"><times id="S2.T1.14.14.14.2.m1.2.2.2.2.1.2.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.2"></times><ci id="S2.T1.14.14.14.2.m1.2.2.2.2.1.3.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.3">𝐵</ci><ci id="S2.T1.14.14.14.2.m1.2.2.2.2.1.4.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.4">𝑒</ci><ci id="S2.T1.14.14.14.2.m1.2.2.2.2.1.5.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.5">𝑙</ci><apply id="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.2">𝐴</ci><cn id="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.T1.14.14.14.2.m1.2.2.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="S2.T1.14.14.14.2.m1.2.2.2.2.3.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.3"><divide id="S2.T1.14.14.14.2.m1.2.2.2.2.3.1.cmml" xref="S2.T1.14.14.14.2.m1.2.2.2.2.3.1"></divide><cn id="S2.T1.14.14.14.2.m1.2.2.2.2.3.2.cmml" type="integer" xref="S2.T1.14.14.14.2.m1.2.2.2.2.3.2">4</cn><cn id="S2.T1.14.14.14.2.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.T1.14.14.14.2.m1.2.2.2.2.3.3">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.14.14.14.2.m1.2c">Bel(A_{1})=0,\;Bel(A_{2})=4/9</annotation><annotation encoding="application/x-llamapun" id="S2.T1.14.14.14.2.m1.2d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0 , italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 4 / 9</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.T1.16.16.16"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S2.T1.16.16.16.3">amb.</th> <td class="ltx_td ltx_align_right ltx_border_r" id="S2.T1.15.15.15.1"><math alttext="m(A_{1}\cap A_{2})=5/9,\;m(\Theta)=0\}" class="ltx_math_unparsed" display="inline" id="S2.T1.15.15.15.1.m1.1"><semantics id="S2.T1.15.15.15.1.m1.1a"><mrow id="S2.T1.15.15.15.1.m1.1b"><mi id="S2.T1.15.15.15.1.m1.1.2">m</mi><mrow id="S2.T1.15.15.15.1.m1.1.3"><mo id="S2.T1.15.15.15.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.T1.15.15.15.1.m1.1.3.2"><mi id="S2.T1.15.15.15.1.m1.1.3.2.2">A</mi><mn id="S2.T1.15.15.15.1.m1.1.3.2.3">1</mn></msub><mo id="S2.T1.15.15.15.1.m1.1.3.3">∩</mo><msub id="S2.T1.15.15.15.1.m1.1.3.4"><mi id="S2.T1.15.15.15.1.m1.1.3.4.2">A</mi><mn id="S2.T1.15.15.15.1.m1.1.3.4.3">2</mn></msub><mo id="S2.T1.15.15.15.1.m1.1.3.5" stretchy="false">)</mo></mrow><mo id="S2.T1.15.15.15.1.m1.1.4">=</mo><mn id="S2.T1.15.15.15.1.m1.1.5">5</mn><mo id="S2.T1.15.15.15.1.m1.1.6">/</mo><mn id="S2.T1.15.15.15.1.m1.1.7">9</mn><mo id="S2.T1.15.15.15.1.m1.1.8" rspace="0.447em">,</mo><mi id="S2.T1.15.15.15.1.m1.1.9">m</mi><mrow id="S2.T1.15.15.15.1.m1.1.10"><mo id="S2.T1.15.15.15.1.m1.1.10.1" stretchy="false">(</mo><mi id="S2.T1.15.15.15.1.m1.1.1" mathvariant="normal">Θ</mi><mo id="S2.T1.15.15.15.1.m1.1.10.2" stretchy="false">)</mo></mrow><mo id="S2.T1.15.15.15.1.m1.1.11">=</mo><mn id="S2.T1.15.15.15.1.m1.1.12">0</mn><mo id="S2.T1.15.15.15.1.m1.1.13" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S2.T1.15.15.15.1.m1.1c">m(A_{1}\cap A_{2})=5/9,\;m(\Theta)=0\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.15.15.15.1.m1.1d">italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 5 / 9 , italic_m ( roman_Θ ) = 0 }</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T1.16.16.16.2"><math alttext="Pl(A_{1})=5/9,\;Pl(A_{2})=1" class="ltx_Math" display="inline" id="S2.T1.16.16.16.2.m1.2"><semantics id="S2.T1.16.16.16.2.m1.2a"><mrow 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id="S2.T1.16.16.16.2.m1.1.1.1.1.1.4.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.1.4">𝑙</ci><apply id="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.2">𝐴</ci><cn id="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.T1.16.16.16.2.m1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.T1.16.16.16.2.m1.1.1.1.1.3.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.3"><divide id="S2.T1.16.16.16.2.m1.1.1.1.1.3.1.cmml" xref="S2.T1.16.16.16.2.m1.1.1.1.1.3.1"></divide><cn id="S2.T1.16.16.16.2.m1.1.1.1.1.3.2.cmml" type="integer" xref="S2.T1.16.16.16.2.m1.1.1.1.1.3.2">5</cn><cn id="S2.T1.16.16.16.2.m1.1.1.1.1.3.3.cmml" type="integer" xref="S2.T1.16.16.16.2.m1.1.1.1.1.3.3">9</cn></apply></apply><apply id="S2.T1.16.16.16.2.m1.2.2.2.2.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2"><eq id="S2.T1.16.16.16.2.m1.2.2.2.2.2.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.2"></eq><apply id="S2.T1.16.16.16.2.m1.2.2.2.2.1.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1"><times id="S2.T1.16.16.16.2.m1.2.2.2.2.1.2.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.2"></times><ci id="S2.T1.16.16.16.2.m1.2.2.2.2.1.3.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.3">𝑃</ci><ci id="S2.T1.16.16.16.2.m1.2.2.2.2.1.4.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.4">𝑙</ci><apply id="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.2">𝐴</ci><cn id="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.T1.16.16.16.2.m1.2.2.2.2.1.1.1.1.3">2</cn></apply></apply><cn id="S2.T1.16.16.16.2.m1.2.2.2.2.3.cmml" type="integer" xref="S2.T1.16.16.16.2.m1.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.16.16.16.2.m1.2c">Pl(A_{1})=5/9,\;Pl(A_{2})=1</annotation><annotation encoding="application/x-llamapun" id="S2.T1.16.16.16.2.m1.2d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 5 / 9 , italic_P italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.T1.18.18.18"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.17.17.17.1"><math alttext="t_{3}" class="ltx_Math" display="inline" id="S2.T1.17.17.17.1.m1.1"><semantics id="S2.T1.17.17.17.1.m1.1a"><msub id="S2.T1.17.17.17.1.m1.1.1" xref="S2.T1.17.17.17.1.m1.1.1.cmml"><mi id="S2.T1.17.17.17.1.m1.1.1.2" xref="S2.T1.17.17.17.1.m1.1.1.2.cmml">t</mi><mn id="S2.T1.17.17.17.1.m1.1.1.3" xref="S2.T1.17.17.17.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.T1.17.17.17.1.m1.1b"><apply id="S2.T1.17.17.17.1.m1.1.1.cmml" xref="S2.T1.17.17.17.1.m1.1.1"><csymbol cd="ambiguous" id="S2.T1.17.17.17.1.m1.1.1.1.cmml" xref="S2.T1.17.17.17.1.m1.1.1">subscript</csymbol><ci id="S2.T1.17.17.17.1.m1.1.1.2.cmml" xref="S2.T1.17.17.17.1.m1.1.1.2">𝑡</ci><cn id="S2.T1.17.17.17.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.17.17.17.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.17.17.17.1.m1.1c">t_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.17.17.17.1.m1.1d">italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S2.T1.18.18.18.2"><math alttext="\{m(\Theta)=1\}" class="ltx_Math" display="inline" id="S2.T1.18.18.18.2.m1.2"><semantics id="S2.T1.18.18.18.2.m1.2a"><mrow id="S2.T1.18.18.18.2.m1.2.2.1" 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xref="S2.T1.18.18.18.2.m1.2.2.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.18.18.18.2.m1.2c">\{m(\Theta)=1\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.18.18.18.2.m1.2d">{ italic_m ( roman_Θ ) = 1 }</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S2.T1.18.18.18.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.20.20.20"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S2.T1.20.20.20.3">ambi-</th> <td class="ltx_td ltx_align_left ltx_border_r" id="S2.T1.19.19.19.1"><math alttext="\{m(A_{1})=m(A_{2})=3/9," class="ltx_math_unparsed" display="inline" id="S2.T1.19.19.19.1.m1.1"><semantics id="S2.T1.19.19.19.1.m1.1a"><mrow id="S2.T1.19.19.19.1.m1.1b"><mo id="S2.T1.19.19.19.1.m1.1.1" stretchy="false">{</mo><mi id="S2.T1.19.19.19.1.m1.1.2">m</mi><mrow id="S2.T1.19.19.19.1.m1.1.3"><mo id="S2.T1.19.19.19.1.m1.1.3.1" stretchy="false">(</mo><msub 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id="S2.T1.20.20.20.2.m1.2.2.2.2.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2"><eq id="S2.T1.20.20.20.2.m1.2.2.2.2.2.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.2"></eq><apply id="S2.T1.20.20.20.2.m1.2.2.2.2.1.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1"><times id="S2.T1.20.20.20.2.m1.2.2.2.2.1.2.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.2"></times><ci id="S2.T1.20.20.20.2.m1.2.2.2.2.1.3.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.3">𝐵</ci><ci id="S2.T1.20.20.20.2.m1.2.2.2.2.1.4.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.4">𝑒</ci><ci id="S2.T1.20.20.20.2.m1.2.2.2.2.1.5.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.5">𝑙</ci><apply id="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.2">𝐴</ci><cn id="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.T1.20.20.20.2.m1.2.2.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="S2.T1.20.20.20.2.m1.2.2.2.2.3.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.3"><divide id="S2.T1.20.20.20.2.m1.2.2.2.2.3.1.cmml" xref="S2.T1.20.20.20.2.m1.2.2.2.2.3.1"></divide><cn id="S2.T1.20.20.20.2.m1.2.2.2.2.3.2.cmml" type="integer" xref="S2.T1.20.20.20.2.m1.2.2.2.2.3.2">3</cn><cn id="S2.T1.20.20.20.2.m1.2.2.2.2.3.3.cmml" type="integer" xref="S2.T1.20.20.20.2.m1.2.2.2.2.3.3">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.20.20.20.2.m1.2c">Bel(A_{1})=3/9,\;Bel(A_{2})=3/9</annotation><annotation encoding="application/x-llamapun" id="S2.T1.20.20.20.2.m1.2d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 3 / 9 , italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 3 / 9</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.T1.22.22.22"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r" id="S2.T1.22.22.22.3">guous</th> <td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" id="S2.T1.21.21.21.1"><math alttext="m(A_{1}\cap A_{2})=2/9,\;m(\Theta)=1/9\}" class="ltx_math_unparsed" display="inline" id="S2.T1.21.21.21.1.m1.1"><semantics id="S2.T1.21.21.21.1.m1.1a"><mrow id="S2.T1.21.21.21.1.m1.1b"><mi id="S2.T1.21.21.21.1.m1.1.2">m</mi><mrow id="S2.T1.21.21.21.1.m1.1.3"><mo id="S2.T1.21.21.21.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.T1.21.21.21.1.m1.1.3.2"><mi id="S2.T1.21.21.21.1.m1.1.3.2.2">A</mi><mn id="S2.T1.21.21.21.1.m1.1.3.2.3">1</mn></msub><mo id="S2.T1.21.21.21.1.m1.1.3.3">∩</mo><msub id="S2.T1.21.21.21.1.m1.1.3.4"><mi id="S2.T1.21.21.21.1.m1.1.3.4.2">A</mi><mn id="S2.T1.21.21.21.1.m1.1.3.4.3">2</mn></msub><mo id="S2.T1.21.21.21.1.m1.1.3.5" stretchy="false">)</mo></mrow><mo id="S2.T1.21.21.21.1.m1.1.4">=</mo><mn id="S2.T1.21.21.21.1.m1.1.5">2</mn><mo id="S2.T1.21.21.21.1.m1.1.6">/</mo><mn id="S2.T1.21.21.21.1.m1.1.7">9</mn><mo id="S2.T1.21.21.21.1.m1.1.8" rspace="0.447em">,</mo><mi id="S2.T1.21.21.21.1.m1.1.9">m</mi><mrow id="S2.T1.21.21.21.1.m1.1.10"><mo id="S2.T1.21.21.21.1.m1.1.10.1" stretchy="false">(</mo><mi id="S2.T1.21.21.21.1.m1.1.1" mathvariant="normal">Θ</mi><mo id="S2.T1.21.21.21.1.m1.1.10.2" stretchy="false">)</mo></mrow><mo id="S2.T1.21.21.21.1.m1.1.11">=</mo><mn id="S2.T1.21.21.21.1.m1.1.12">1</mn><mo id="S2.T1.21.21.21.1.m1.1.13">/</mo><mn id="S2.T1.21.21.21.1.m1.1.14">9</mn><mo id="S2.T1.21.21.21.1.m1.1.15" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S2.T1.21.21.21.1.m1.1c">m(A_{1}\cap A_{2})=2/9,\;m(\Theta)=1/9\}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.21.21.21.1.m1.1d">italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 2 / 9 , italic_m ( roman_Θ ) = 1 / 9 }</annotation></semantics></math></td> <td class="ltx_td ltx_align_left 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encoding="application/x-llamapun" id="S2.T1.22.22.22.2.m1.2d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 5 / 9 , italic_P italic_l ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 5 / 9</annotation></semantics></math></td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>The bodies of evidence that become available to the transfer-DNA at subsequent points in time. At <math alttext="t_{1}" class="ltx_Math" display="inline" id="S2.T1.27.m1.1"><semantics id="S2.T1.27.m1.1b"><msub id="S2.T1.27.m1.1.1" xref="S2.T1.27.m1.1.1.cmml"><mi id="S2.T1.27.m1.1.1.2" xref="S2.T1.27.m1.1.1.2.cmml">t</mi><mn id="S2.T1.27.m1.1.1.3" xref="S2.T1.27.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.T1.27.m1.1c"><apply id="S2.T1.27.m1.1.1.cmml" xref="S2.T1.27.m1.1.1"><csymbol cd="ambiguous" id="S2.T1.27.m1.1.1.1.cmml" xref="S2.T1.27.m1.1.1">subscript</csymbol><ci id="S2.T1.27.m1.1.1.2.cmml" xref="S2.T1.27.m1.1.1.2">𝑡</ci><cn id="S2.T1.27.m1.1.1.3.cmml" type="integer" xref="S2.T1.27.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.27.m1.1d">t_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.27.m1.1e">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> only one nucleotide has been captured and the available evidence does not favour one over the other interpretation. At <math alttext="t=t_{2}" class="ltx_Math" display="inline" id="S2.T1.28.m2.1"><semantics id="S2.T1.28.m2.1b"><mrow id="S2.T1.28.m2.1.1" xref="S2.T1.28.m2.1.1.cmml"><mi id="S2.T1.28.m2.1.1.2" xref="S2.T1.28.m2.1.1.2.cmml">t</mi><mo id="S2.T1.28.m2.1.1.1" xref="S2.T1.28.m2.1.1.1.cmml">=</mo><msub id="S2.T1.28.m2.1.1.3" xref="S2.T1.28.m2.1.1.3.cmml"><mi id="S2.T1.28.m2.1.1.3.2" xref="S2.T1.28.m2.1.1.3.2.cmml">t</mi><mn id="S2.T1.28.m2.1.1.3.3" xref="S2.T1.28.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.28.m2.1c"><apply id="S2.T1.28.m2.1.1.cmml" xref="S2.T1.28.m2.1.1"><eq id="S2.T1.28.m2.1.1.1.cmml" xref="S2.T1.28.m2.1.1.1"></eq><ci id="S2.T1.28.m2.1.1.2.cmml" xref="S2.T1.28.m2.1.1.2">𝑡</ci><apply id="S2.T1.28.m2.1.1.3.cmml" xref="S2.T1.28.m2.1.1.3"><csymbol cd="ambiguous" id="S2.T1.28.m2.1.1.3.1.cmml" xref="S2.T1.28.m2.1.1.3">subscript</csymbol><ci id="S2.T1.28.m2.1.1.3.2.cmml" xref="S2.T1.28.m2.1.1.3.2">𝑡</ci><cn id="S2.T1.28.m2.1.1.3.3.cmml" type="integer" xref="S2.T1.28.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.28.m2.1d">t=t_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.28.m2.1e">italic_t = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> two nucleotides have been captured but the evidence is still inconclusive. At <math alttext="t=t_{3}" class="ltx_Math" display="inline" id="S2.T1.29.m3.1"><semantics id="S2.T1.29.m3.1b"><mrow id="S2.T1.29.m3.1.1" xref="S2.T1.29.m3.1.1.cmml"><mi id="S2.T1.29.m3.1.1.2" xref="S2.T1.29.m3.1.1.2.cmml">t</mi><mo id="S2.T1.29.m3.1.1.1" xref="S2.T1.29.m3.1.1.1.cmml">=</mo><msub id="S2.T1.29.m3.1.1.3" xref="S2.T1.29.m3.1.1.3.cmml"><mi id="S2.T1.29.m3.1.1.3.2" xref="S2.T1.29.m3.1.1.3.2.cmml">t</mi><mn id="S2.T1.29.m3.1.1.3.3" xref="S2.T1.29.m3.1.1.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.29.m3.1c"><apply id="S2.T1.29.m3.1.1.cmml" xref="S2.T1.29.m3.1.1"><eq id="S2.T1.29.m3.1.1.1.cmml" xref="S2.T1.29.m3.1.1.1"></eq><ci id="S2.T1.29.m3.1.1.2.cmml" xref="S2.T1.29.m3.1.1.2">𝑡</ci><apply id="S2.T1.29.m3.1.1.3.cmml" xref="S2.T1.29.m3.1.1.3"><csymbol cd="ambiguous" id="S2.T1.29.m3.1.1.3.1.cmml" xref="S2.T1.29.m3.1.1.3">subscript</csymbol><ci id="S2.T1.29.m3.1.1.3.2.cmml" xref="S2.T1.29.m3.1.1.3.2">𝑡</ci><cn id="S2.T1.29.m3.1.1.3.3.cmml" type="integer" xref="S2.T1.29.m3.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.29.m3.1d">t=t_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.29.m3.1e">italic_t = italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>, an unambiguous code shifts the balance towards <math alttext="A_{2}" class="ltx_Math" display="inline" id="S2.T1.30.m4.1"><semantics id="S2.T1.30.m4.1b"><msub id="S2.T1.30.m4.1.1" xref="S2.T1.30.m4.1.1.cmml"><mi id="S2.T1.30.m4.1.1.2" xref="S2.T1.30.m4.1.1.2.cmml">A</mi><mn id="S2.T1.30.m4.1.1.3" xref="S2.T1.30.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.T1.30.m4.1c"><apply id="S2.T1.30.m4.1.1.cmml" xref="S2.T1.30.m4.1.1"><csymbol cd="ambiguous" id="S2.T1.30.m4.1.1.1.cmml" xref="S2.T1.30.m4.1.1">subscript</csymbol><ci id="S2.T1.30.m4.1.1.2.cmml" xref="S2.T1.30.m4.1.1.2">𝐴</ci><cn id="S2.T1.30.m4.1.1.3.cmml" type="integer" xref="S2.T1.30.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.30.m4.1d">A_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.T1.30.m4.1e">italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, whereas an ambiguous code keeps the issue undecided.</figcaption> </figure> <div class="ltx_para" id="S2.p23"> <p class="ltx_p" id="S2.p23.1">The computational procedure carried out hitherto is an instance of Dempster-Shafer combination rule <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> that will be discussed in § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a>. Specifically, it was based on the numerator of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> because the evidence carried by the nucleotides was assumed never to be contradictory.</p> </div> <div class="ltx_para" id="S2.p24"> <p class="ltx_p" id="S2.p24.2">This exercise has explorative value insofar it highlighted the benefits of representing possibilities as sets that may partially overlap, as well as the usefulness of restraining from assigning all the evidence to the possibilities that are currently being envisaged by assigning a portion to <math alttext="\Theta" class="ltx_Math" display="inline" id="S2.p24.1.m1.1"><semantics id="S2.p24.1.m1.1a"><mi id="S2.p24.1.m1.1.1" mathvariant="normal" xref="S2.p24.1.m1.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S2.p24.1.m1.1b"><ci id="S2.p24.1.m1.1.1.cmml" xref="S2.p24.1.m1.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p24.1.m1.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S2.p24.1.m1.1d">roman_Θ</annotation></semantics></math>. 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Clearly, more appropriate rules should be devised for living organisms so simple as those considered in this section. Suitable candidates may be derived from combining threshold-based rules, such as those employed to model neuron firing, with some geometrical encoding of the time sequence. Careful observation of the way bacteria and similarly simple organisms make their “decisions” could yield rules for the combination of evidence that are appropriate for them, as well as for more complex organisms in situations where they lack the time and resources to carry out a judge’s work.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Anticipatory Brains</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">A snake stops chasing its prey as soon as the prey hides behind a tree, whereas a wolf goes around the tree to seek its prey. This anecdote highlights that the wolf has the capability to figure out states that are not communicated by the sensory organs, which the snake has not. In other words, the wolf has the capability to anticipate where the prey is.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">It is not important for my discourse whether the divide between animals who are capable of anticipation and those who are not coincides with any specific taxonomy of animal species, be they mammals (wolves) and reptilia (snakes) or other classes. All what matters is that a sufficiently sharp divide exists, at some point in the evolutionary tree. There can be intermediate levels of anticipatory capabilities, all I need to assume is that at some point in animal space a sufficiently sharp transition exists.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">ET displays much wider potentialities once the capability of anticipating events is assumed. Hypotheses can be formulated, that go beyond the currently available evidence. 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id="S3.p4.6.m6.2b"><list id="S3.p4.6.m6.2.2.2.cmml" xref="S3.p4.6.m6.2.2.1"><apply id="S3.p4.6.m6.2.2.1.1.cmml" xref="S3.p4.6.m6.2.2.1.1"><times id="S3.p4.6.m6.2.2.1.1.2.cmml" xref="S3.p4.6.m6.2.2.1.1.2"></times><ci id="S3.p4.6.m6.2.2.1.1.3.cmml" xref="S3.p4.6.m6.2.2.1.1.3">𝑚</ci><apply id="S3.p4.6.m6.2.2.1.1.1.1.1.cmml" xref="S3.p4.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p4.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S3.p4.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S3.p4.6.m6.2.2.1.1.1.1.1.2.cmml" xref="S3.p4.6.m6.2.2.1.1.1.1.1.2">𝐴</ci><cn id="S3.p4.6.m6.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.6.m6.2.2.1.1.1.1.1.3">2</cn></apply></apply><ci id="S3.p4.6.m6.1.1.cmml" xref="S3.p4.6.m6.1.1">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m6.2c">m(A_{2}),\ldots</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m6.2d">italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , …</annotation></semantics></math> <math alttext="m(A_{N})" class="ltx_Math" display="inline" id="S3.p4.7.m7.1"><semantics id="S3.p4.7.m7.1a"><mrow id="S3.p4.7.m7.1.1" xref="S3.p4.7.m7.1.1.cmml"><mi id="S3.p4.7.m7.1.1.3" xref="S3.p4.7.m7.1.1.3.cmml">m</mi><mo id="S3.p4.7.m7.1.1.2" xref="S3.p4.7.m7.1.1.2.cmml">⁢</mo><mrow id="S3.p4.7.m7.1.1.1.1" xref="S3.p4.7.m7.1.1.1.1.1.cmml"><mo id="S3.p4.7.m7.1.1.1.1.2" stretchy="false" xref="S3.p4.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S3.p4.7.m7.1.1.1.1.1" xref="S3.p4.7.m7.1.1.1.1.1.cmml"><mi id="S3.p4.7.m7.1.1.1.1.1.2" xref="S3.p4.7.m7.1.1.1.1.1.2.cmml">A</mi><mi id="S3.p4.7.m7.1.1.1.1.1.3" xref="S3.p4.7.m7.1.1.1.1.1.3.cmml">N</mi></msub><mo id="S3.p4.7.m7.1.1.1.1.3" stretchy="false" xref="S3.p4.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.7.m7.1b"><apply id="S3.p4.7.m7.1.1.cmml" xref="S3.p4.7.m7.1.1"><times id="S3.p4.7.m7.1.1.2.cmml" xref="S3.p4.7.m7.1.1.2"></times><ci id="S3.p4.7.m7.1.1.3.cmml" xref="S3.p4.7.m7.1.1.3">𝑚</ci><apply id="S3.p4.7.m7.1.1.1.1.1.cmml" xref="S3.p4.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S3.p4.7.m7.1.1.1.1.1.1.cmml" xref="S3.p4.7.m7.1.1.1.1">subscript</csymbol><ci id="S3.p4.7.m7.1.1.1.1.1.2.cmml" xref="S3.p4.7.m7.1.1.1.1.1.2">𝐴</ci><ci id="S3.p4.7.m7.1.1.1.1.1.3.cmml" xref="S3.p4.7.m7.1.1.1.1.1.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m7.1c">m(A_{N})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m7.1d">italic_m ( italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT )</annotation></semantics></math>. ET allows to assign a positive mass to the frame of discernment as a whole. An <math alttext="m(\Theta)>0" class="ltx_Math" display="inline" id="S3.p4.8.m8.1"><semantics id="S3.p4.8.m8.1a"><mrow id="S3.p4.8.m8.1.2" xref="S3.p4.8.m8.1.2.cmml"><mrow id="S3.p4.8.m8.1.2.2" xref="S3.p4.8.m8.1.2.2.cmml"><mi id="S3.p4.8.m8.1.2.2.2" xref="S3.p4.8.m8.1.2.2.2.cmml">m</mi><mo id="S3.p4.8.m8.1.2.2.1" xref="S3.p4.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.p4.8.m8.1.2.2.3.2" xref="S3.p4.8.m8.1.2.2.cmml"><mo id="S3.p4.8.m8.1.2.2.3.2.1" stretchy="false" xref="S3.p4.8.m8.1.2.2.cmml">(</mo><mi id="S3.p4.8.m8.1.1" mathvariant="normal" xref="S3.p4.8.m8.1.1.cmml">Θ</mi><mo id="S3.p4.8.m8.1.2.2.3.2.2" stretchy="false" xref="S3.p4.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.8.m8.1.2.1" xref="S3.p4.8.m8.1.2.1.cmml">></mo><mn id="S3.p4.8.m8.1.2.3" xref="S3.p4.8.m8.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.8.m8.1b"><apply id="S3.p4.8.m8.1.2.cmml" xref="S3.p4.8.m8.1.2"><gt id="S3.p4.8.m8.1.2.1.cmml" xref="S3.p4.8.m8.1.2.1"></gt><apply id="S3.p4.8.m8.1.2.2.cmml" xref="S3.p4.8.m8.1.2.2"><times id="S3.p4.8.m8.1.2.2.1.cmml" xref="S3.p4.8.m8.1.2.2.1"></times><ci id="S3.p4.8.m8.1.2.2.2.cmml" xref="S3.p4.8.m8.1.2.2.2">𝑚</ci><ci id="S3.p4.8.m8.1.1.cmml" xref="S3.p4.8.m8.1.1">Θ</ci></apply><cn id="S3.p4.8.m8.1.2.3.cmml" type="integer" xref="S3.p4.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m8.1c">m(\Theta)>0</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m8.1d">italic_m ( roman_Θ ) > 0</annotation></semantics></math> represents suspension of judgement, non-assigned belief in the conviction that new information will become available at a later point in time.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.1">Though not essential to the theory, masses <math alttext="m(.)" class="ltx_math_unparsed" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mrow id="S3.p5.1.m1.1b"><mi id="S3.p5.1.m1.1.1">m</mi><mrow id="S3.p5.1.m1.1.2"><mo id="S3.p5.1.m1.1.2.1" stretchy="false">(</mo><mo id="S3.p5.1.m1.1.2.2" lspace="0em" rspace="0.167em">.</mo><mo id="S3.p5.1.m1.1.2.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">m(.)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_m ( . )</annotation></semantics></math> can be normalized in order to obtain that:</p> </div> <div class="ltx_para" id="S3.p6"> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <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="\sum_{i=1}^{N}\>m_{i}(A_{i})\;+\;m_{A}(\Theta)\;=\;1" class="ltx_Math" display="block" id="S3.E3.m1.2"><semantics id="S3.E3.m1.2a"><mrow id="S3.E3.m1.2.2" xref="S3.E3.m1.2.2.cmml"><mrow id="S3.E3.m1.2.2.1" xref="S3.E3.m1.2.2.1.cmml"><mrow id="S3.E3.m1.2.2.1.1" xref="S3.E3.m1.2.2.1.1.cmml"><munderover id="S3.E3.m1.2.2.1.1.2" xref="S3.E3.m1.2.2.1.1.2.cmml"><mo id="S3.E3.m1.2.2.1.1.2.2.2" movablelimits="false" xref="S3.E3.m1.2.2.1.1.2.2.2.cmml">∑</mo><mrow id="S3.E3.m1.2.2.1.1.2.2.3" xref="S3.E3.m1.2.2.1.1.2.2.3.cmml"><mi id="S3.E3.m1.2.2.1.1.2.2.3.2" xref="S3.E3.m1.2.2.1.1.2.2.3.2.cmml">i</mi><mo id="S3.E3.m1.2.2.1.1.2.2.3.1" xref="S3.E3.m1.2.2.1.1.2.2.3.1.cmml">=</mo><mn id="S3.E3.m1.2.2.1.1.2.2.3.3" xref="S3.E3.m1.2.2.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="S3.E3.m1.2.2.1.1.2.3" xref="S3.E3.m1.2.2.1.1.2.3.cmml">N</mi></munderover><mrow id="S3.E3.m1.2.2.1.1.1" xref="S3.E3.m1.2.2.1.1.1.cmml"><msub id="S3.E3.m1.2.2.1.1.1.3" xref="S3.E3.m1.2.2.1.1.1.3.cmml"><mi id="S3.E3.m1.2.2.1.1.1.3.2" xref="S3.E3.m1.2.2.1.1.1.3.2.cmml">m</mi><mi id="S3.E3.m1.2.2.1.1.1.3.3" xref="S3.E3.m1.2.2.1.1.1.3.3.cmml">i</mi></msub><mo id="S3.E3.m1.2.2.1.1.1.2" xref="S3.E3.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.E3.m1.2.2.1.1.1.1.1" 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id="S3.E3.m1.2.2.1.3.3.2.1" stretchy="false" xref="S3.E3.m1.2.2.1.3.cmml">(</mo><mi id="S3.E3.m1.1.1" mathvariant="normal" xref="S3.E3.m1.1.1.cmml">Θ</mi><mo id="S3.E3.m1.2.2.1.3.3.2.2" rspace="0.280em" stretchy="false" xref="S3.E3.m1.2.2.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E3.m1.2.2.2" xref="S3.E3.m1.2.2.2.cmml">=</mo><mn id="S3.E3.m1.2.2.3" xref="S3.E3.m1.2.2.3.cmml"> 1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.2b"><apply id="S3.E3.m1.2.2.cmml" xref="S3.E3.m1.2.2"><eq id="S3.E3.m1.2.2.2.cmml" xref="S3.E3.m1.2.2.2"></eq><apply id="S3.E3.m1.2.2.1.cmml" xref="S3.E3.m1.2.2.1"><plus id="S3.E3.m1.2.2.1.2.cmml" xref="S3.E3.m1.2.2.1.2"></plus><apply id="S3.E3.m1.2.2.1.1.cmml" xref="S3.E3.m1.2.2.1.1"><apply id="S3.E3.m1.2.2.1.1.2.cmml" xref="S3.E3.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.2.2.1.1.2.1.cmml" xref="S3.E3.m1.2.2.1.1.2">superscript</csymbol><apply id="S3.E3.m1.2.2.1.1.2.2.cmml" xref="S3.E3.m1.2.2.1.1.2"><csymbol cd="ambiguous" 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xref="S3.E3.m1.2.2.1.1.1.3.3">𝑖</ci></apply><apply id="S3.E3.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E3.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E3.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E3.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.E3.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.E3.m1.2.2.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.E3.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.E3.m1.2.2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S3.E3.m1.2.2.1.3.cmml" xref="S3.E3.m1.2.2.1.3"><times id="S3.E3.m1.2.2.1.3.1.cmml" xref="S3.E3.m1.2.2.1.3.1"></times><apply id="S3.E3.m1.2.2.1.3.2.cmml" xref="S3.E3.m1.2.2.1.3.2"><csymbol cd="ambiguous" id="S3.E3.m1.2.2.1.3.2.1.cmml" xref="S3.E3.m1.2.2.1.3.2">subscript</csymbol><ci id="S3.E3.m1.2.2.1.3.2.2.cmml" xref="S3.E3.m1.2.2.1.3.2.2">𝑚</ci><ci id="S3.E3.m1.2.2.1.3.2.3.cmml" xref="S3.E3.m1.2.2.1.3.2.3">𝐴</ci></apply><ci id="S3.E3.m1.1.1.cmml" xref="S3.E3.m1.1.1">Θ</ci></apply></apply><cn id="S3.E3.m1.2.2.3.cmml" type="integer" xref="S3.E3.m1.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.2c">\sum_{i=1}^{N}\>m_{i}(A_{i})\;+\;m_{A}(\Theta)\;=\;1</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.2d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) = 1</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 class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.3">In general, possibilities <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><msub id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml"><mi id="S3.p7.1.m1.1.1.2" xref="S3.p7.1.m1.1.1.2.cmml">A</mi><mi id="S3.p7.1.m1.1.1.3" xref="S3.p7.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><apply id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.1.cmml" xref="S3.p7.1.m1.1.1">subscript</csymbol><ci id="S3.p7.1.m1.1.1.2.cmml" xref="S3.p7.1.m1.1.1.2">𝐴</ci><ci id="S3.p7.1.m1.1.1.3.cmml" xref="S3.p7.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are not disjoint sets. Since <math alttext="m(A_{i})+m(A_{j})" class="ltx_Math" display="inline" id="S3.p7.2.m2.2"><semantics id="S3.p7.2.m2.2a"><mrow id="S3.p7.2.m2.2.2" xref="S3.p7.2.m2.2.2.cmml"><mrow id="S3.p7.2.m2.1.1.1" xref="S3.p7.2.m2.1.1.1.cmml"><mi id="S3.p7.2.m2.1.1.1.3" xref="S3.p7.2.m2.1.1.1.3.cmml">m</mi><mo id="S3.p7.2.m2.1.1.1.2" xref="S3.p7.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.p7.2.m2.1.1.1.1.1" xref="S3.p7.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.p7.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.p7.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p7.2.m2.1.1.1.1.1.1" xref="S3.p7.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.p7.2.m2.1.1.1.1.1.1.2" xref="S3.p7.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mi id="S3.p7.2.m2.1.1.1.1.1.1.3" xref="S3.p7.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.p7.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.p7.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p7.2.m2.2.2.3" xref="S3.p7.2.m2.2.2.3.cmml">+</mo><mrow id="S3.p7.2.m2.2.2.2" xref="S3.p7.2.m2.2.2.2.cmml"><mi id="S3.p7.2.m2.2.2.2.3" xref="S3.p7.2.m2.2.2.2.3.cmml">m</mi><mo id="S3.p7.2.m2.2.2.2.2" xref="S3.p7.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S3.p7.2.m2.2.2.2.1.1" xref="S3.p7.2.m2.2.2.2.1.1.1.cmml"><mo id="S3.p7.2.m2.2.2.2.1.1.2" stretchy="false" xref="S3.p7.2.m2.2.2.2.1.1.1.cmml">(</mo><msub id="S3.p7.2.m2.2.2.2.1.1.1" xref="S3.p7.2.m2.2.2.2.1.1.1.cmml"><mi id="S3.p7.2.m2.2.2.2.1.1.1.2" xref="S3.p7.2.m2.2.2.2.1.1.1.2.cmml">A</mi><mi id="S3.p7.2.m2.2.2.2.1.1.1.3" xref="S3.p7.2.m2.2.2.2.1.1.1.3.cmml">j</mi></msub><mo id="S3.p7.2.m2.2.2.2.1.1.3" stretchy="false" xref="S3.p7.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.2b"><apply id="S3.p7.2.m2.2.2.cmml" xref="S3.p7.2.m2.2.2"><plus id="S3.p7.2.m2.2.2.3.cmml" xref="S3.p7.2.m2.2.2.3"></plus><apply id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1.1"><times id="S3.p7.2.m2.1.1.1.2.cmml" xref="S3.p7.2.m2.1.1.1.2"></times><ci id="S3.p7.2.m2.1.1.1.3.cmml" xref="S3.p7.2.m2.1.1.1.3">𝑚</ci><apply id="S3.p7.2.m2.1.1.1.1.1.1.cmml" xref="S3.p7.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.p7.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.p7.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.p7.2.m2.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.p7.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.p7.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.p7.2.m2.2.2.2.cmml" xref="S3.p7.2.m2.2.2.2"><times id="S3.p7.2.m2.2.2.2.2.cmml" xref="S3.p7.2.m2.2.2.2.2"></times><ci id="S3.p7.2.m2.2.2.2.3.cmml" xref="S3.p7.2.m2.2.2.2.3">𝑚</ci><apply id="S3.p7.2.m2.2.2.2.1.1.1.cmml" xref="S3.p7.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.2.2.2.1.1.1.1.cmml" xref="S3.p7.2.m2.2.2.2.1.1">subscript</csymbol><ci id="S3.p7.2.m2.2.2.2.1.1.1.2.cmml" xref="S3.p7.2.m2.2.2.2.1.1.1.2">𝐴</ci><ci id="S3.p7.2.m2.2.2.2.1.1.1.3.cmml" xref="S3.p7.2.m2.2.2.2.1.1.1.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.2c">m(A_{i})+m(A_{j})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.2d">italic_m ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_m ( italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> is not equivalent to <math alttext="m(A_{i}\cup A_{j})" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><mrow id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml"><mi id="S3.p7.3.m3.1.1.3" xref="S3.p7.3.m3.1.1.3.cmml">m</mi><mo id="S3.p7.3.m3.1.1.2" xref="S3.p7.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.p7.3.m3.1.1.1.1" xref="S3.p7.3.m3.1.1.1.1.1.cmml"><mo id="S3.p7.3.m3.1.1.1.1.2" stretchy="false" xref="S3.p7.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.p7.3.m3.1.1.1.1.1" xref="S3.p7.3.m3.1.1.1.1.1.cmml"><msub id="S3.p7.3.m3.1.1.1.1.1.2" xref="S3.p7.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.p7.3.m3.1.1.1.1.1.2.2" xref="S3.p7.3.m3.1.1.1.1.1.2.2.cmml">A</mi><mi id="S3.p7.3.m3.1.1.1.1.1.2.3" xref="S3.p7.3.m3.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.p7.3.m3.1.1.1.1.1.1" xref="S3.p7.3.m3.1.1.1.1.1.1.cmml">∪</mo><msub id="S3.p7.3.m3.1.1.1.1.1.3" xref="S3.p7.3.m3.1.1.1.1.1.3.cmml"><mi id="S3.p7.3.m3.1.1.1.1.1.3.2" xref="S3.p7.3.m3.1.1.1.1.1.3.2.cmml">A</mi><mi id="S3.p7.3.m3.1.1.1.1.1.3.3" xref="S3.p7.3.m3.1.1.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="S3.p7.3.m3.1.1.1.1.3" stretchy="false" xref="S3.p7.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><apply id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1"><times id="S3.p7.3.m3.1.1.2.cmml" xref="S3.p7.3.m3.1.1.2"></times><ci id="S3.p7.3.m3.1.1.3.cmml" xref="S3.p7.3.m3.1.1.3">𝑚</ci><apply id="S3.p7.3.m3.1.1.1.1.1.cmml" xref="S3.p7.3.m3.1.1.1.1"><union id="S3.p7.3.m3.1.1.1.1.1.1.cmml" xref="S3.p7.3.m3.1.1.1.1.1.1"></union><apply id="S3.p7.3.m3.1.1.1.1.1.2.cmml" xref="S3.p7.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p7.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.p7.3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p7.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.p7.3.m3.1.1.1.1.1.2.2">𝐴</ci><ci id="S3.p7.3.m3.1.1.1.1.1.2.3.cmml" xref="S3.p7.3.m3.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S3.p7.3.m3.1.1.1.1.1.3.cmml" xref="S3.p7.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p7.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.p7.3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p7.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.p7.3.m3.1.1.1.1.1.3.2">𝐴</ci><ci id="S3.p7.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.p7.3.m3.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">m(A_{i}\cup A_{j})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">italic_m ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∪ italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>, eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E3" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> does not amount to distributing a given mass among distinct possibilities.</p> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.13">Let us assume that evidence <math alttext="A=\{m(A_{1})," class="ltx_math_unparsed" display="inline" id="S3.p8.1.m1.1"><semantics id="S3.p8.1.m1.1a"><mrow id="S3.p8.1.m1.1b"><mi id="S3.p8.1.m1.1.1">A</mi><mo id="S3.p8.1.m1.1.2">=</mo><mrow id="S3.p8.1.m1.1.3"><mo id="S3.p8.1.m1.1.3.1" stretchy="false">{</mo><mi id="S3.p8.1.m1.1.3.2">m</mi><mrow id="S3.p8.1.m1.1.3.3"><mo id="S3.p8.1.m1.1.3.3.1" stretchy="false">(</mo><msub id="S3.p8.1.m1.1.3.3.2"><mi id="S3.p8.1.m1.1.3.3.2.2">A</mi><mn id="S3.p8.1.m1.1.3.3.2.3">1</mn></msub><mo id="S3.p8.1.m1.1.3.3.3" stretchy="false">)</mo></mrow><mo id="S3.p8.1.m1.1.3.4">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.p8.1.m1.1c">A=\{m(A_{1}),</annotation><annotation encoding="application/x-llamapun" id="S3.p8.1.m1.1d">italic_A = { italic_m ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ,</annotation></semantics></math> <math alttext="m(A_{2}),\ldots" class="ltx_Math" display="inline" id="S3.p8.2.m2.2"><semantics id="S3.p8.2.m2.2a"><mrow id="S3.p8.2.m2.2.2.1" xref="S3.p8.2.m2.2.2.2.cmml"><mrow id="S3.p8.2.m2.2.2.1.1" xref="S3.p8.2.m2.2.2.1.1.cmml"><mi id="S3.p8.2.m2.2.2.1.1.3" xref="S3.p8.2.m2.2.2.1.1.3.cmml">m</mi><mo id="S3.p8.2.m2.2.2.1.1.2" xref="S3.p8.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S3.p8.2.m2.2.2.1.1.1.1" xref="S3.p8.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S3.p8.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S3.p8.2.m2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S3.p8.2.m2.2.2.1.1.1.1.1" xref="S3.p8.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S3.p8.2.m2.2.2.1.1.1.1.1.2" xref="S3.p8.2.m2.2.2.1.1.1.1.1.2.cmml">A</mi><mn id="S3.p8.2.m2.2.2.1.1.1.1.1.3" xref="S3.p8.2.m2.2.2.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S3.p8.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="S3.p8.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p8.2.m2.2.2.1.2" xref="S3.p8.2.m2.2.2.2.cmml">,</mo><mi id="S3.p8.2.m2.1.1" mathvariant="normal" xref="S3.p8.2.m2.1.1.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.2.m2.2b"><list id="S3.p8.2.m2.2.2.2.cmml" xref="S3.p8.2.m2.2.2.1"><apply id="S3.p8.2.m2.2.2.1.1.cmml" xref="S3.p8.2.m2.2.2.1.1"><times id="S3.p8.2.m2.2.2.1.1.2.cmml" xref="S3.p8.2.m2.2.2.1.1.2"></times><ci id="S3.p8.2.m2.2.2.1.1.3.cmml" xref="S3.p8.2.m2.2.2.1.1.3">𝑚</ci><apply id="S3.p8.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.p8.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p8.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S3.p8.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S3.p8.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S3.p8.2.m2.2.2.1.1.1.1.1.2">𝐴</ci><cn id="S3.p8.2.m2.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S3.p8.2.m2.2.2.1.1.1.1.1.3">2</cn></apply></apply><ci id="S3.p8.2.m2.1.1.cmml" xref="S3.p8.2.m2.1.1">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.2.m2.2c">m(A_{2}),\ldots</annotation><annotation encoding="application/x-llamapun" id="S3.p8.2.m2.2d">italic_m ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , …</annotation></semantics></math> <math alttext="m(A_{N_{A}})," class="ltx_Math" display="inline" id="S3.p8.3.m3.1"><semantics id="S3.p8.3.m3.1a"><mrow id="S3.p8.3.m3.1.1.1" xref="S3.p8.3.m3.1.1.1.1.cmml"><mrow id="S3.p8.3.m3.1.1.1.1" xref="S3.p8.3.m3.1.1.1.1.cmml"><mi id="S3.p8.3.m3.1.1.1.1.3" xref="S3.p8.3.m3.1.1.1.1.3.cmml">m</mi><mo id="S3.p8.3.m3.1.1.1.1.2" xref="S3.p8.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.p8.3.m3.1.1.1.1.1.1" xref="S3.p8.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.p8.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.p8.3.m3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p8.3.m3.1.1.1.1.1.1.1" xref="S3.p8.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S3.p8.3.m3.1.1.1.1.1.1.1.2" xref="S3.p8.3.m3.1.1.1.1.1.1.1.2.cmml">A</mi><msub id="S3.p8.3.m3.1.1.1.1.1.1.1.3" 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xref="S3.p8.3.m3.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p8.3.m3.1.1.1.1.1.1.1.3.1.cmml" xref="S3.p8.3.m3.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p8.3.m3.1.1.1.1.1.1.1.3.2.cmml" xref="S3.p8.3.m3.1.1.1.1.1.1.1.3.2">𝑁</ci><ci id="S3.p8.3.m3.1.1.1.1.1.1.1.3.3.cmml" xref="S3.p8.3.m3.1.1.1.1.1.1.1.3.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.3.m3.1c">m(A_{N_{A}}),</annotation><annotation encoding="application/x-llamapun" id="S3.p8.3.m3.1d">italic_m ( italic_A start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math> <math alttext="m_{A}(\Theta)\}" class="ltx_math_unparsed" display="inline" id="S3.p8.4.m4.1"><semantics id="S3.p8.4.m4.1a"><mrow id="S3.p8.4.m4.1b"><msub id="S3.p8.4.m4.1.2"><mi id="S3.p8.4.m4.1.2.2">m</mi><mi id="S3.p8.4.m4.1.2.3">A</mi></msub><mrow id="S3.p8.4.m4.1.3"><mo id="S3.p8.4.m4.1.3.1" stretchy="false">(</mo><mi id="S3.p8.4.m4.1.1" mathvariant="normal">Θ</mi><mo id="S3.p8.4.m4.1.3.2" stretchy="false">)</mo></mrow><mo id="S3.p8.4.m4.1.4" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S3.p8.4.m4.1c">m_{A}(\Theta)\}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Θ ) }</annotation></semantics></math> is available when a new body of evidence arrives — for instance, the wolf is receiving visual information about a prey that suddenly makes a noise. 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xref="S3.p8.7.m7.1.1.1.1.1.1.1.3.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.7.m7.1c">m(B_{N_{B}}),</annotation><annotation encoding="application/x-llamapun" id="S3.p8.7.m7.1d">italic_m ( italic_B start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math> <math alttext="m_{B}(\Theta)\}" class="ltx_math_unparsed" display="inline" id="S3.p8.8.m8.1"><semantics id="S3.p8.8.m8.1a"><mrow id="S3.p8.8.m8.1b"><msub id="S3.p8.8.m8.1.2"><mi id="S3.p8.8.m8.1.2.2">m</mi><mi id="S3.p8.8.m8.1.2.3">B</mi></msub><mrow id="S3.p8.8.m8.1.3"><mo id="S3.p8.8.m8.1.3.1" stretchy="false">(</mo><mi id="S3.p8.8.m8.1.1" mathvariant="normal">Θ</mi><mo id="S3.p8.8.m8.1.3.2" stretchy="false">)</mo></mrow><mo id="S3.p8.8.m8.1.4" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S3.p8.8.m8.1c">m_{B}(\Theta)\}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.8.m8.1d">italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Θ ) }</annotation></semantics></math> be the new body of evidence carried by auditory signals. Just like the sets entailed in one single body of evidence are not necessarily disjoint, <math alttext="\forall i,j" class="ltx_Math" display="inline" id="S3.p8.9.m9.2"><semantics id="S3.p8.9.m9.2a"><mrow id="S3.p8.9.m9.2.2.1" xref="S3.p8.9.m9.2.2.2.cmml"><mrow id="S3.p8.9.m9.2.2.1.1" xref="S3.p8.9.m9.2.2.1.1.cmml"><mo id="S3.p8.9.m9.2.2.1.1.1" rspace="0.167em" xref="S3.p8.9.m9.2.2.1.1.1.cmml">∀</mo><mi id="S3.p8.9.m9.2.2.1.1.2" xref="S3.p8.9.m9.2.2.1.1.2.cmml">i</mi></mrow><mo id="S3.p8.9.m9.2.2.1.2" xref="S3.p8.9.m9.2.2.2.cmml">,</mo><mi id="S3.p8.9.m9.1.1" xref="S3.p8.9.m9.1.1.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.9.m9.2b"><list id="S3.p8.9.m9.2.2.2.cmml" xref="S3.p8.9.m9.2.2.1"><apply id="S3.p8.9.m9.2.2.1.1.cmml" xref="S3.p8.9.m9.2.2.1.1"><csymbol cd="latexml" id="S3.p8.9.m9.2.2.1.1.1.cmml" xref="S3.p8.9.m9.2.2.1.1.1">for-all</csymbol><ci id="S3.p8.9.m9.2.2.1.1.2.cmml" xref="S3.p8.9.m9.2.2.1.1.2">𝑖</ci></apply><ci id="S3.p8.9.m9.1.1.cmml" xref="S3.p8.9.m9.1.1">𝑗</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.9.m9.2c">\forall i,j</annotation><annotation encoding="application/x-llamapun" id="S3.p8.9.m9.2d">∀ italic_i , italic_j</annotation></semantics></math> it may either be <math alttext="A_{i}\subseteq B_{j}" class="ltx_Math" display="inline" id="S3.p8.10.m10.1"><semantics id="S3.p8.10.m10.1a"><mrow id="S3.p8.10.m10.1.1" xref="S3.p8.10.m10.1.1.cmml"><msub id="S3.p8.10.m10.1.1.2" xref="S3.p8.10.m10.1.1.2.cmml"><mi id="S3.p8.10.m10.1.1.2.2" xref="S3.p8.10.m10.1.1.2.2.cmml">A</mi><mi id="S3.p8.10.m10.1.1.2.3" xref="S3.p8.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S3.p8.10.m10.1.1.1" xref="S3.p8.10.m10.1.1.1.cmml">⊆</mo><msub id="S3.p8.10.m10.1.1.3" xref="S3.p8.10.m10.1.1.3.cmml"><mi id="S3.p8.10.m10.1.1.3.2" xref="S3.p8.10.m10.1.1.3.2.cmml">B</mi><mi id="S3.p8.10.m10.1.1.3.3" xref="S3.p8.10.m10.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" 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end_POSTSUBSCRIPT ⊆ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, or <math alttext="A_{i}\supseteq B_{j}" class="ltx_Math" display="inline" id="S3.p8.11.m11.1"><semantics id="S3.p8.11.m11.1a"><mrow id="S3.p8.11.m11.1.1" xref="S3.p8.11.m11.1.1.cmml"><msub id="S3.p8.11.m11.1.1.2" xref="S3.p8.11.m11.1.1.2.cmml"><mi id="S3.p8.11.m11.1.1.2.2" xref="S3.p8.11.m11.1.1.2.2.cmml">A</mi><mi id="S3.p8.11.m11.1.1.2.3" xref="S3.p8.11.m11.1.1.2.3.cmml">i</mi></msub><mo id="S3.p8.11.m11.1.1.1" xref="S3.p8.11.m11.1.1.cmml">⊇</mo><msub id="S3.p8.11.m11.1.1.3" xref="S3.p8.11.m11.1.1.3.cmml"><mi id="S3.p8.11.m11.1.1.3.2" xref="S3.p8.11.m11.1.1.3.2.cmml">B</mi><mi id="S3.p8.11.m11.1.1.3.3" xref="S3.p8.11.m11.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.11.m11.1b"><apply id="S3.p8.11.m11.1.1.cmml" xref="S3.p8.11.m11.1.1"><subset id="S3.p8.11.m11.1.1a.cmml" xref="S3.p8.11.m11.1.1"></subset><apply id="S3.p8.11.m11.1.1.3.cmml" xref="S3.p8.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.p8.11.m11.1.1.3.1.cmml" xref="S3.p8.11.m11.1.1.3">subscript</csymbol><ci id="S3.p8.11.m11.1.1.3.2.cmml" xref="S3.p8.11.m11.1.1.3.2">𝐵</ci><ci id="S3.p8.11.m11.1.1.3.3.cmml" xref="S3.p8.11.m11.1.1.3.3">𝑗</ci></apply><apply id="S3.p8.11.m11.1.1.2.cmml" xref="S3.p8.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.p8.11.m11.1.1.2.1.cmml" xref="S3.p8.11.m11.1.1.2">subscript</csymbol><ci id="S3.p8.11.m11.1.1.2.2.cmml" xref="S3.p8.11.m11.1.1.2.2">𝐴</ci><ci id="S3.p8.11.m11.1.1.2.3.cmml" xref="S3.p8.11.m11.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.11.m11.1c">A_{i}\supseteq B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.11.m11.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊇ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, or <math alttext="Ai\cap B_{j}\neq\emptyset" class="ltx_Math" display="inline" id="S3.p8.12.m12.1"><semantics id="S3.p8.12.m12.1a"><mrow id="S3.p8.12.m12.1.1" xref="S3.p8.12.m12.1.1.cmml"><mrow id="S3.p8.12.m12.1.1.2" xref="S3.p8.12.m12.1.1.2.cmml"><mrow id="S3.p8.12.m12.1.1.2.2" xref="S3.p8.12.m12.1.1.2.2.cmml"><mi id="S3.p8.12.m12.1.1.2.2.2" xref="S3.p8.12.m12.1.1.2.2.2.cmml">A</mi><mo id="S3.p8.12.m12.1.1.2.2.1" xref="S3.p8.12.m12.1.1.2.2.1.cmml">⁢</mo><mi id="S3.p8.12.m12.1.1.2.2.3" xref="S3.p8.12.m12.1.1.2.2.3.cmml">i</mi></mrow><mo id="S3.p8.12.m12.1.1.2.1" xref="S3.p8.12.m12.1.1.2.1.cmml">∩</mo><msub id="S3.p8.12.m12.1.1.2.3" xref="S3.p8.12.m12.1.1.2.3.cmml"><mi id="S3.p8.12.m12.1.1.2.3.2" xref="S3.p8.12.m12.1.1.2.3.2.cmml">B</mi><mi id="S3.p8.12.m12.1.1.2.3.3" xref="S3.p8.12.m12.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S3.p8.12.m12.1.1.1" xref="S3.p8.12.m12.1.1.1.cmml">≠</mo><mi id="S3.p8.12.m12.1.1.3" mathvariant="normal" xref="S3.p8.12.m12.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.12.m12.1b"><apply 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xref="S3.p8.12.m12.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.12.m12.1c">Ai\cap B_{j}\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p8.12.m12.1d">italic_A italic_i ∩ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math>, or <math alttext="A_{i}\cap B_{j}=\emptyset" class="ltx_Math" display="inline" id="S3.p8.13.m13.1"><semantics id="S3.p8.13.m13.1a"><mrow id="S3.p8.13.m13.1.1" xref="S3.p8.13.m13.1.1.cmml"><mrow id="S3.p8.13.m13.1.1.2" xref="S3.p8.13.m13.1.1.2.cmml"><msub id="S3.p8.13.m13.1.1.2.2" xref="S3.p8.13.m13.1.1.2.2.cmml"><mi id="S3.p8.13.m13.1.1.2.2.2" xref="S3.p8.13.m13.1.1.2.2.2.cmml">A</mi><mi id="S3.p8.13.m13.1.1.2.2.3" xref="S3.p8.13.m13.1.1.2.2.3.cmml">i</mi></msub><mo id="S3.p8.13.m13.1.1.2.1" xref="S3.p8.13.m13.1.1.2.1.cmml">∩</mo><msub id="S3.p8.13.m13.1.1.2.3" xref="S3.p8.13.m13.1.1.2.3.cmml"><mi id="S3.p8.13.m13.1.1.2.3.2" xref="S3.p8.13.m13.1.1.2.3.2.cmml">B</mi><mi id="S3.p8.13.m13.1.1.2.3.3" xref="S3.p8.13.m13.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S3.p8.13.m13.1.1.1" xref="S3.p8.13.m13.1.1.1.cmml">=</mo><mi id="S3.p8.13.m13.1.1.3" mathvariant="normal" xref="S3.p8.13.m13.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.13.m13.1b"><apply id="S3.p8.13.m13.1.1.cmml" xref="S3.p8.13.m13.1.1"><eq id="S3.p8.13.m13.1.1.1.cmml" xref="S3.p8.13.m13.1.1.1"></eq><apply id="S3.p8.13.m13.1.1.2.cmml" xref="S3.p8.13.m13.1.1.2"><intersect id="S3.p8.13.m13.1.1.2.1.cmml" xref="S3.p8.13.m13.1.1.2.1"></intersect><apply id="S3.p8.13.m13.1.1.2.2.cmml" xref="S3.p8.13.m13.1.1.2.2"><csymbol cd="ambiguous" id="S3.p8.13.m13.1.1.2.2.1.cmml" xref="S3.p8.13.m13.1.1.2.2">subscript</csymbol><ci id="S3.p8.13.m13.1.1.2.2.2.cmml" xref="S3.p8.13.m13.1.1.2.2.2">𝐴</ci><ci id="S3.p8.13.m13.1.1.2.2.3.cmml" xref="S3.p8.13.m13.1.1.2.2.3">𝑖</ci></apply><apply id="S3.p8.13.m13.1.1.2.3.cmml" xref="S3.p8.13.m13.1.1.2.3"><csymbol cd="ambiguous" id="S3.p8.13.m13.1.1.2.3.1.cmml" xref="S3.p8.13.m13.1.1.2.3">subscript</csymbol><ci id="S3.p8.13.m13.1.1.2.3.2.cmml" xref="S3.p8.13.m13.1.1.2.3.2">𝐵</ci><ci id="S3.p8.13.m13.1.1.2.3.3.cmml" xref="S3.p8.13.m13.1.1.2.3.3">𝑗</ci></apply></apply><emptyset id="S3.p8.13.m13.1.1.3.cmml" xref="S3.p8.13.m13.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.13.m13.1c">A_{i}\cap B_{j}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p8.13.m13.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∅</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p9"> <p class="ltx_p" id="S3.p9.4">ET is concerned with combining coherent pieces of evidence while weighing them against contradictory items. In particular, Dempster-Shafer’s combination rule <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib14" title="">14</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib51" title="">51</a>]</cite> yields the components of a new body of evidence <math alttext="m_{C}" class="ltx_Math" display="inline" id="S3.p9.1.m1.1"><semantics id="S3.p9.1.m1.1a"><msub id="S3.p9.1.m1.1.1" xref="S3.p9.1.m1.1.1.cmml"><mi id="S3.p9.1.m1.1.1.2" xref="S3.p9.1.m1.1.1.2.cmml">m</mi><mi id="S3.p9.1.m1.1.1.3" xref="S3.p9.1.m1.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p9.1.m1.1b"><apply id="S3.p9.1.m1.1.1.cmml" xref="S3.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p9.1.m1.1.1.1.cmml" xref="S3.p9.1.m1.1.1">subscript</csymbol><ci id="S3.p9.1.m1.1.1.2.cmml" xref="S3.p9.1.m1.1.1.2">𝑚</ci><ci id="S3.p9.1.m1.1.1.3.cmml" xref="S3.p9.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.1.m1.1c">m_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> that unites two bodies <math alttext="m_{A}" class="ltx_Math" display="inline" id="S3.p9.2.m2.1"><semantics id="S3.p9.2.m2.1a"><msub id="S3.p9.2.m2.1.1" xref="S3.p9.2.m2.1.1.cmml"><mi id="S3.p9.2.m2.1.1.2" xref="S3.p9.2.m2.1.1.2.cmml">m</mi><mi id="S3.p9.2.m2.1.1.3" xref="S3.p9.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p9.2.m2.1b"><apply id="S3.p9.2.m2.1.1.cmml" xref="S3.p9.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p9.2.m2.1.1.1.cmml" xref="S3.p9.2.m2.1.1">subscript</csymbol><ci id="S3.p9.2.m2.1.1.2.cmml" xref="S3.p9.2.m2.1.1.2">𝑚</ci><ci id="S3.p9.2.m2.1.1.3.cmml" xref="S3.p9.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.2.m2.1c">m_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.2.m2.1d">italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="m_{B}" class="ltx_Math" display="inline" id="S3.p9.3.m3.1"><semantics id="S3.p9.3.m3.1a"><msub id="S3.p9.3.m3.1.1" xref="S3.p9.3.m3.1.1.cmml"><mi id="S3.p9.3.m3.1.1.2" xref="S3.p9.3.m3.1.1.2.cmml">m</mi><mi id="S3.p9.3.m3.1.1.3" xref="S3.p9.3.m3.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p9.3.m3.1b"><apply id="S3.p9.3.m3.1.1.cmml" xref="S3.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p9.3.m3.1.1.1.cmml" xref="S3.p9.3.m3.1.1">subscript</csymbol><ci id="S3.p9.3.m3.1.1.2.cmml" xref="S3.p9.3.m3.1.1.2">𝑚</ci><ci id="S3.p9.3.m3.1.1.3.cmml" xref="S3.p9.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.3.m3.1c">m_{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.3.m3.1d">italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>. Note that intersections with <math alttext="\Theta" class="ltx_Math" display="inline" id="S3.p9.4.m4.1"><semantics id="S3.p9.4.m4.1a"><mi id="S3.p9.4.m4.1.1" mathvariant="normal" xref="S3.p9.4.m4.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S3.p9.4.m4.1b"><ci id="S3.p9.4.m4.1.1.cmml" xref="S3.p9.4.m4.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.4.m4.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S3.p9.4.m4.1d">roman_Θ</annotation></semantics></math> enter the computation.</p> </div> <div class="ltx_para" id="S3.p10"> <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="m(C_{k})\;=\;\frac{\sum_{X_{i}\cap Y_{j}=C_{k}}\;m_{A}(X_{i})\,m_{B}(Y_{j})}{1% -\>\sum_{X_{i}\cap Y_{j}=\emptyset}\;m_{A}(X_{i})\,m_{B}(Y_{j})}" class="ltx_Math" 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id="S3.E4.m1.5c">m(C_{k})\;=\;\frac{\sum_{X_{i}\cap Y_{j}=C_{k}}\;m_{A}(X_{i})\,m_{B}(Y_{j})}{1% -\>\sum_{X_{i}\cap Y_{j}=\emptyset}\;m_{A}(X_{i})\,m_{B}(Y_{j})}</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.5d">italic_m ( italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = divide start_ARG ∑ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG 1 - ∑ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∅ end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG</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.p10.5">where <math alttext="X_{i}\in\{A_{i}\,\forall i,\;\Theta\}" class="ltx_Math" display="inline" id="S3.p10.1.m1.2"><semantics id="S3.p10.1.m1.2a"><mrow id="S3.p10.1.m1.2.2" xref="S3.p10.1.m1.2.2.cmml"><msub id="S3.p10.1.m1.2.2.3" xref="S3.p10.1.m1.2.2.3.cmml"><mi id="S3.p10.1.m1.2.2.3.2" xref="S3.p10.1.m1.2.2.3.2.cmml">X</mi><mi id="S3.p10.1.m1.2.2.3.3" xref="S3.p10.1.m1.2.2.3.3.cmml">i</mi></msub><mo id="S3.p10.1.m1.2.2.2" xref="S3.p10.1.m1.2.2.2.cmml">∈</mo><mrow id="S3.p10.1.m1.2.2.1.1" 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xref="S3.p10.1.m1.2.2.1.1.1.2.2">𝐴</ci><ci id="S3.p10.1.m1.2.2.1.1.1.2.3.cmml" xref="S3.p10.1.m1.2.2.1.1.1.2.3">𝑖</ci></apply><apply id="S3.p10.1.m1.2.2.1.1.1.3.cmml" xref="S3.p10.1.m1.2.2.1.1.1.3"><csymbol cd="latexml" id="S3.p10.1.m1.2.2.1.1.1.3.1.cmml" xref="S3.p10.1.m1.2.2.1.1.1.3.1">for-all</csymbol><ci id="S3.p10.1.m1.2.2.1.1.1.3.2.cmml" xref="S3.p10.1.m1.2.2.1.1.1.3.2">𝑖</ci></apply></apply><ci id="S3.p10.1.m1.1.1.cmml" xref="S3.p10.1.m1.1.1">Θ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.1.m1.2c">X_{i}\in\{A_{i}\,\forall i,\;\Theta\}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.1.m1.2d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∀ italic_i , roman_Θ }</annotation></semantics></math>, <math alttext="Y_{j}\in\{B_{j}\,\forall j,\;\Theta\}" class="ltx_Math" display="inline" id="S3.p10.2.m2.2"><semantics id="S3.p10.2.m2.2a"><mrow id="S3.p10.2.m2.2.2" xref="S3.p10.2.m2.2.2.cmml"><msub id="S3.p10.2.m2.2.2.3" xref="S3.p10.2.m2.2.2.3.cmml"><mi id="S3.p10.2.m2.2.2.3.2" xref="S3.p10.2.m2.2.2.3.2.cmml">Y</mi><mi id="S3.p10.2.m2.2.2.3.3" xref="S3.p10.2.m2.2.2.3.3.cmml">j</mi></msub><mo id="S3.p10.2.m2.2.2.2" xref="S3.p10.2.m2.2.2.2.cmml">∈</mo><mrow id="S3.p10.2.m2.2.2.1.1" xref="S3.p10.2.m2.2.2.1.2.cmml"><mo id="S3.p10.2.m2.2.2.1.1.2" stretchy="false" xref="S3.p10.2.m2.2.2.1.2.cmml">{</mo><mrow id="S3.p10.2.m2.2.2.1.1.1" xref="S3.p10.2.m2.2.2.1.1.1.cmml"><msub id="S3.p10.2.m2.2.2.1.1.1.2" xref="S3.p10.2.m2.2.2.1.1.1.2.cmml"><mi id="S3.p10.2.m2.2.2.1.1.1.2.2" xref="S3.p10.2.m2.2.2.1.1.1.2.2.cmml">B</mi><mi id="S3.p10.2.m2.2.2.1.1.1.2.3" xref="S3.p10.2.m2.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="S3.p10.2.m2.2.2.1.1.1.1" lspace="0.167em" xref="S3.p10.2.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.p10.2.m2.2.2.1.1.1.3" xref="S3.p10.2.m2.2.2.1.1.1.3.cmml"><mo id="S3.p10.2.m2.2.2.1.1.1.3.1" rspace="0.167em" xref="S3.p10.2.m2.2.2.1.1.1.3.1.cmml">∀</mo><mi id="S3.p10.2.m2.2.2.1.1.1.3.2" xref="S3.p10.2.m2.2.2.1.1.1.3.2.cmml">j</mi></mrow></mrow><mo id="S3.p10.2.m2.2.2.1.1.3" rspace="0.447em" xref="S3.p10.2.m2.2.2.1.2.cmml">,</mo><mi id="S3.p10.2.m2.1.1" mathvariant="normal" xref="S3.p10.2.m2.1.1.cmml">Θ</mi><mo id="S3.p10.2.m2.2.2.1.1.4" stretchy="false" xref="S3.p10.2.m2.2.2.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p10.2.m2.2b"><apply id="S3.p10.2.m2.2.2.cmml" xref="S3.p10.2.m2.2.2"><in id="S3.p10.2.m2.2.2.2.cmml" xref="S3.p10.2.m2.2.2.2"></in><apply id="S3.p10.2.m2.2.2.3.cmml" xref="S3.p10.2.m2.2.2.3"><csymbol cd="ambiguous" id="S3.p10.2.m2.2.2.3.1.cmml" xref="S3.p10.2.m2.2.2.3">subscript</csymbol><ci id="S3.p10.2.m2.2.2.3.2.cmml" xref="S3.p10.2.m2.2.2.3.2">𝑌</ci><ci id="S3.p10.2.m2.2.2.3.3.cmml" xref="S3.p10.2.m2.2.2.3.3">𝑗</ci></apply><set id="S3.p10.2.m2.2.2.1.2.cmml" xref="S3.p10.2.m2.2.2.1.1"><apply id="S3.p10.2.m2.2.2.1.1.1.cmml" xref="S3.p10.2.m2.2.2.1.1.1"><times id="S3.p10.2.m2.2.2.1.1.1.1.cmml" xref="S3.p10.2.m2.2.2.1.1.1.1"></times><apply id="S3.p10.2.m2.2.2.1.1.1.2.cmml" xref="S3.p10.2.m2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3.p10.2.m2.2.2.1.1.1.2.1.cmml" xref="S3.p10.2.m2.2.2.1.1.1.2">subscript</csymbol><ci id="S3.p10.2.m2.2.2.1.1.1.2.2.cmml" xref="S3.p10.2.m2.2.2.1.1.1.2.2">𝐵</ci><ci id="S3.p10.2.m2.2.2.1.1.1.2.3.cmml" xref="S3.p10.2.m2.2.2.1.1.1.2.3">𝑗</ci></apply><apply id="S3.p10.2.m2.2.2.1.1.1.3.cmml" xref="S3.p10.2.m2.2.2.1.1.1.3"><csymbol cd="latexml" id="S3.p10.2.m2.2.2.1.1.1.3.1.cmml" xref="S3.p10.2.m2.2.2.1.1.1.3.1">for-all</csymbol><ci id="S3.p10.2.m2.2.2.1.1.1.3.2.cmml" xref="S3.p10.2.m2.2.2.1.1.1.3.2">𝑗</ci></apply></apply><ci id="S3.p10.2.m2.1.1.cmml" xref="S3.p10.2.m2.1.1">Θ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.2.m2.2c">Y_{j}\in\{B_{j}\,\forall j,\;\Theta\}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.2.m2.2d">italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ { italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∀ italic_j , roman_Θ }</annotation></semantics></math>, and where the <math alttext="C_{k}" class="ltx_Math" display="inline" id="S3.p10.3.m3.1"><semantics id="S3.p10.3.m3.1a"><msub id="S3.p10.3.m3.1.1" xref="S3.p10.3.m3.1.1.cmml"><mi id="S3.p10.3.m3.1.1.2" xref="S3.p10.3.m3.1.1.2.cmml">C</mi><mi id="S3.p10.3.m3.1.1.3" xref="S3.p10.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p10.3.m3.1b"><apply id="S3.p10.3.m3.1.1.cmml" xref="S3.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p10.3.m3.1.1.1.cmml" xref="S3.p10.3.m3.1.1">subscript</csymbol><ci id="S3.p10.3.m3.1.1.2.cmml" xref="S3.p10.3.m3.1.1.2">𝐶</ci><ci id="S3.p10.3.m3.1.1.3.cmml" xref="S3.p10.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.3.m3.1c">C_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>s are defined by all possible intersections of the <math alttext="X_{i}" class="ltx_Math" display="inline" id="S3.p10.4.m4.1"><semantics id="S3.p10.4.m4.1a"><msub id="S3.p10.4.m4.1.1" xref="S3.p10.4.m4.1.1.cmml"><mi id="S3.p10.4.m4.1.1.2" xref="S3.p10.4.m4.1.1.2.cmml">X</mi><mi id="S3.p10.4.m4.1.1.3" xref="S3.p10.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p10.4.m4.1b"><apply id="S3.p10.4.m4.1.1.cmml" xref="S3.p10.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p10.4.m4.1.1.1.cmml" xref="S3.p10.4.m4.1.1">subscript</csymbol><ci id="S3.p10.4.m4.1.1.2.cmml" xref="S3.p10.4.m4.1.1.2">𝑋</ci><ci id="S3.p10.4.m4.1.1.3.cmml" xref="S3.p10.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.4.m4.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.4.m4.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>s with the <math alttext="Y_{j}" class="ltx_Math" display="inline" id="S3.p10.5.m5.1"><semantics id="S3.p10.5.m5.1a"><msub id="S3.p10.5.m5.1.1" xref="S3.p10.5.m5.1.1.cmml"><mi id="S3.p10.5.m5.1.1.2" xref="S3.p10.5.m5.1.1.2.cmml">Y</mi><mi id="S3.p10.5.m5.1.1.3" xref="S3.p10.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p10.5.m5.1b"><apply id="S3.p10.5.m5.1.1.cmml" xref="S3.p10.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p10.5.m5.1.1.1.cmml" xref="S3.p10.5.m5.1.1">subscript</csymbol><ci id="S3.p10.5.m5.1.1.2.cmml" xref="S3.p10.5.m5.1.1.2">𝑌</ci><ci id="S3.p10.5.m5.1.1.3.cmml" xref="S3.p10.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.5.m5.1c">Y_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.5.m5.1d">italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>s.</p> </div> <div class="ltx_para" id="S3.p11"> <p class="ltx_p" id="S3.p11.3">The numerator of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> measures the extent to which the two bodies of evidence coherently support <math alttext="C_{k}" class="ltx_Math" display="inline" id="S3.p11.1.m1.1"><semantics id="S3.p11.1.m1.1a"><msub id="S3.p11.1.m1.1.1" xref="S3.p11.1.m1.1.1.cmml"><mi id="S3.p11.1.m1.1.1.2" xref="S3.p11.1.m1.1.1.2.cmml">C</mi><mi id="S3.p11.1.m1.1.1.3" xref="S3.p11.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p11.1.m1.1b"><apply id="S3.p11.1.m1.1.1.cmml" xref="S3.p11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p11.1.m1.1.1.1.cmml" xref="S3.p11.1.m1.1.1">subscript</csymbol><ci id="S3.p11.1.m1.1.1.2.cmml" xref="S3.p11.1.m1.1.1.2">𝐶</ci><ci id="S3.p11.1.m1.1.1.3.cmml" xref="S3.p11.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p11.1.m1.1c">C_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p11.1.m1.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, whereas the denominator measures the extent to which they are not contradictory with one another. Equivalently, one can say that the numerator expresses the logic of serial testimonies whereas the denominator expresses the logic of parallel testimonies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib49" title="">49</a>]</cite>. In our example where <math alttext="A" class="ltx_Math" display="inline" id="S3.p11.2.m2.1"><semantics id="S3.p11.2.m2.1a"><mi id="S3.p11.2.m2.1.1" xref="S3.p11.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.p11.2.m2.1b"><ci id="S3.p11.2.m2.1.1.cmml" xref="S3.p11.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p11.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.p11.2.m2.1d">italic_A</annotation></semantics></math> represents the visual evidence whereas <math alttext="B" class="ltx_Math" display="inline" id="S3.p11.3.m3.1"><semantics id="S3.p11.3.m3.1a"><mi id="S3.p11.3.m3.1.1" xref="S3.p11.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.p11.3.m3.1b"><ci id="S3.p11.3.m3.1.1.cmml" xref="S3.p11.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p11.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.p11.3.m3.1d">italic_B</annotation></semantics></math> represents the auditory evidence received by the wolf about its prey, evidence about the actual position of the prey is enhanced if the noise made by the prey comes from the area where the prey has been seen the last time, whereas the available evidence becomes scant if the noise contradicts visual information.</p> </div> <div class="ltx_para" id="S3.p12"> <p class="ltx_p" id="S3.p12.1">Dempster-Shafer’s combination rule <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> can be iterated to combine any number of evidence bodies. Its outcome is independent of the order in which evidence bodies are combined. In other words, the ability to memorize past states and anticipate future states allows to ignore the sequence of arrival of bodies of evidence in order to focus on their content.</p> </div> <div class="ltx_para" id="S3.p13"> <p class="ltx_p" id="S3.p13.1">Masses <math alttext="m(.)" class="ltx_math_unparsed" display="inline" id="S3.p13.1.m1.1"><semantics id="S3.p13.1.m1.1a"><mrow id="S3.p13.1.m1.1b"><mi id="S3.p13.1.m1.1.1">m</mi><mrow id="S3.p13.1.m1.1.2"><mo id="S3.p13.1.m1.1.2.1" stretchy="false">(</mo><mo id="S3.p13.1.m1.1.2.2" lspace="0em" rspace="0.167em">.</mo><mo id="S3.p13.1.m1.1.2.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.p13.1.m1.1c">m(.)</annotation><annotation encoding="application/x-llamapun" id="S3.p13.1.m1.1d">italic_m ( . )</annotation></semantics></math> can be discounted in order to account for the reliability of bodies of evidence, or the existence of causal relations between them. However, these are features of each specific setting, rather than ET.</p> </div> <div class="ltx_para" id="S3.p14"> <p class="ltx_p" id="S3.p14.3">Let us now suppose that the decision-maker formulates a hypothesis <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S3.p14.1.m1.1"><semantics id="S3.p14.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p14.1.m1.1.1" xref="S3.p14.1.m1.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="S3.p14.1.m1.1b"><ci id="S3.p14.1.m1.1.1.cmml" xref="S3.p14.1.m1.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p14.1.m1.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S3.p14.1.m1.1d">caligraphic_H</annotation></semantics></math>. For instance, a wolf may hypothesize that the prey is hiding behind the tree, or a detective may hypothesize that the butler did it. This hypothesis is a possibility and therefore it is subset of <math alttext="\Theta" class="ltx_Math" display="inline" id="S3.p14.2.m2.1"><semantics id="S3.p14.2.m2.1a"><mi id="S3.p14.2.m2.1.1" mathvariant="normal" xref="S3.p14.2.m2.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S3.p14.2.m2.1b"><ci id="S3.p14.2.m2.1.1.cmml" xref="S3.p14.2.m2.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p14.2.m2.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S3.p14.2.m2.1d">roman_Θ</annotation></semantics></math> but, unlike the <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.p14.3.m3.1"><semantics id="S3.p14.3.m3.1a"><msub id="S3.p14.3.m3.1.1" xref="S3.p14.3.m3.1.1.cmml"><mi id="S3.p14.3.m3.1.1.2" xref="S3.p14.3.m3.1.1.2.cmml">A</mi><mi id="S3.p14.3.m3.1.1.3" xref="S3.p14.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p14.3.m3.1b"><apply id="S3.p14.3.m3.1.1.cmml" xref="S3.p14.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p14.3.m3.1.1.1.cmml" xref="S3.p14.3.m3.1.1">subscript</csymbol><ci id="S3.p14.3.m3.1.1.2.cmml" xref="S3.p14.3.m3.1.1.2">𝐴</ci><ci id="S3.p14.3.m3.1.1.3.cmml" xref="S3.p14.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p14.3.m3.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p14.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>s, it does not represent empirical evidence but rather a mental construct.</p> </div> <div class="ltx_para" id="S3.p15"> <p class="ltx_p" id="S3.p15.6">The belief that can reasonably attached to <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S3.p15.1.m1.1"><semantics id="S3.p15.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p15.1.m1.1.1" xref="S3.p15.1.m1.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="S3.p15.1.m1.1b"><ci id="S3.p15.1.m1.1.1.cmml" xref="S3.p15.1.m1.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p15.1.m1.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S3.p15.1.m1.1d">caligraphic_H</annotation></semantics></math> is given by the amount of evidence supporting it. Assuming a body of evidence <math alttext="\{m(C_{1})," class="ltx_math_unparsed" display="inline" id="S3.p15.2.m2.1"><semantics id="S3.p15.2.m2.1a"><mrow id="S3.p15.2.m2.1b"><mo id="S3.p15.2.m2.1.1" stretchy="false">{</mo><mi id="S3.p15.2.m2.1.2">m</mi><mrow id="S3.p15.2.m2.1.3"><mo id="S3.p15.2.m2.1.3.1" stretchy="false">(</mo><msub id="S3.p15.2.m2.1.3.2"><mi id="S3.p15.2.m2.1.3.2.2">C</mi><mn id="S3.p15.2.m2.1.3.2.3">1</mn></msub><mo id="S3.p15.2.m2.1.3.3" stretchy="false">)</mo></mrow><mo id="S3.p15.2.m2.1.4">,</mo></mrow><annotation encoding="application/x-tex" id="S3.p15.2.m2.1c">\{m(C_{1}),</annotation><annotation encoding="application/x-llamapun" id="S3.p15.2.m2.1d">{ italic_m ( italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ,</annotation></semantics></math> <math alttext="m(C_{2}),\ldots" class="ltx_Math" display="inline" id="S3.p15.3.m3.2"><semantics id="S3.p15.3.m3.2a"><mrow id="S3.p15.3.m3.2.2.1" xref="S3.p15.3.m3.2.2.2.cmml"><mrow id="S3.p15.3.m3.2.2.1.1" 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id="S3.p15.4.m4.1"><semantics id="S3.p15.4.m4.1a"><mrow id="S3.p15.4.m4.1.1.1" xref="S3.p15.4.m4.1.1.1.1.cmml"><mrow id="S3.p15.4.m4.1.1.1.1" xref="S3.p15.4.m4.1.1.1.1.cmml"><mi id="S3.p15.4.m4.1.1.1.1.3" xref="S3.p15.4.m4.1.1.1.1.3.cmml">m</mi><mo id="S3.p15.4.m4.1.1.1.1.2" xref="S3.p15.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.p15.4.m4.1.1.1.1.1.1" xref="S3.p15.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S3.p15.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S3.p15.4.m4.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p15.4.m4.1.1.1.1.1.1.1" xref="S3.p15.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S3.p15.4.m4.1.1.1.1.1.1.1.2" xref="S3.p15.4.m4.1.1.1.1.1.1.1.2.cmml">C</mi><msub id="S3.p15.4.m4.1.1.1.1.1.1.1.3" xref="S3.p15.4.m4.1.1.1.1.1.1.1.3.cmml"><mi id="S3.p15.4.m4.1.1.1.1.1.1.1.3.2" xref="S3.p15.4.m4.1.1.1.1.1.1.1.3.2.cmml">N</mi><mi id="S3.p15.4.m4.1.1.1.1.1.1.1.3.3" xref="S3.p15.4.m4.1.1.1.1.1.1.1.3.3.cmml">C</mi></msub></msub><mo id="S3.p15.4.m4.1.1.1.1.1.1.3" stretchy="false" 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xref="S3.p15.4.m4.1.1.1.1.1.1.1.3.3">𝐶</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p15.4.m4.1c">m(C_{N_{C}}),</annotation><annotation encoding="application/x-llamapun" id="S3.p15.4.m4.1d">italic_m ( italic_C start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math> <math alttext="m_{C}(\Theta)\}" class="ltx_math_unparsed" display="inline" id="S3.p15.5.m5.1"><semantics id="S3.p15.5.m5.1a"><mrow id="S3.p15.5.m5.1b"><msub id="S3.p15.5.m5.1.2"><mi id="S3.p15.5.m5.1.2.2">m</mi><mi id="S3.p15.5.m5.1.2.3">C</mi></msub><mrow id="S3.p15.5.m5.1.3"><mo id="S3.p15.5.m5.1.3.1" stretchy="false">(</mo><mi id="S3.p15.5.m5.1.1" mathvariant="normal">Θ</mi><mo id="S3.p15.5.m5.1.3.2" stretchy="false">)</mo></mrow><mo id="S3.p15.5.m5.1.4" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S3.p15.5.m5.1c">m_{C}(\Theta)\}</annotation><annotation encoding="application/x-llamapun" id="S3.p15.5.m5.1d">italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Θ ) }</annotation></semantics></math>, the following <em class="ltx_emph ltx_font_italic" id="S3.p15.6.1">Belief Function</em> expresses the belief in <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S3.p15.6.m6.1"><semantics id="S3.p15.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p15.6.m6.1.1" xref="S3.p15.6.m6.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="S3.p15.6.m6.1b"><ci id="S3.p15.6.m6.1.1.cmml" xref="S3.p15.6.m6.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p15.6.m6.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S3.p15.6.m6.1d">caligraphic_H</annotation></semantics></math> supported by the available evidence:</p> </div> <div class="ltx_para" id="S3.p16"> <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="Bel(\mathcal{H})\;=\;\sum_{C_{k}\subset\mathcal{H}}\;m(C_{k})" class="ltx_Math" display="block" id="S3.E5.m1.2"><semantics id="S3.E5.m1.2a"><mrow id="S3.E5.m1.2.2" xref="S3.E5.m1.2.2.cmml"><mrow id="S3.E5.m1.2.2.3" xref="S3.E5.m1.2.2.3.cmml"><mi id="S3.E5.m1.2.2.3.2" xref="S3.E5.m1.2.2.3.2.cmml">B</mi><mo id="S3.E5.m1.2.2.3.1" xref="S3.E5.m1.2.2.3.1.cmml">⁢</mo><mi id="S3.E5.m1.2.2.3.3" xref="S3.E5.m1.2.2.3.3.cmml">e</mi><mo id="S3.E5.m1.2.2.3.1a" xref="S3.E5.m1.2.2.3.1.cmml">⁢</mo><mi id="S3.E5.m1.2.2.3.4" xref="S3.E5.m1.2.2.3.4.cmml">l</mi><mo id="S3.E5.m1.2.2.3.1b" xref="S3.E5.m1.2.2.3.1.cmml">⁢</mo><mrow id="S3.E5.m1.2.2.3.5.2" xref="S3.E5.m1.2.2.3.cmml"><mo id="S3.E5.m1.2.2.3.5.2.1" stretchy="false" xref="S3.E5.m1.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.E5.m1.1.1" xref="S3.E5.m1.1.1.cmml">ℋ</mi><mo id="S3.E5.m1.2.2.3.5.2.2" rspace="0.280em" stretchy="false" xref="S3.E5.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E5.m1.2.2.2" rspace="0.391em" xref="S3.E5.m1.2.2.2.cmml">=</mo><mrow id="S3.E5.m1.2.2.1" xref="S3.E5.m1.2.2.1.cmml"><munder id="S3.E5.m1.2.2.1.2" xref="S3.E5.m1.2.2.1.2.cmml"><mo id="S3.E5.m1.2.2.1.2.2" movablelimits="false" xref="S3.E5.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S3.E5.m1.2.2.1.2.3" xref="S3.E5.m1.2.2.1.2.3.cmml"><msub id="S3.E5.m1.2.2.1.2.3.2" xref="S3.E5.m1.2.2.1.2.3.2.cmml"><mi id="S3.E5.m1.2.2.1.2.3.2.2" xref="S3.E5.m1.2.2.1.2.3.2.2.cmml">C</mi><mi id="S3.E5.m1.2.2.1.2.3.2.3" xref="S3.E5.m1.2.2.1.2.3.2.3.cmml">k</mi></msub><mo id="S3.E5.m1.2.2.1.2.3.1" xref="S3.E5.m1.2.2.1.2.3.1.cmml">⊂</mo><mi class="ltx_font_mathcaligraphic" id="S3.E5.m1.2.2.1.2.3.3" xref="S3.E5.m1.2.2.1.2.3.3.cmml">ℋ</mi></mrow></munder><mrow id="S3.E5.m1.2.2.1.1" xref="S3.E5.m1.2.2.1.1.cmml"><mi id="S3.E5.m1.2.2.1.1.3" xref="S3.E5.m1.2.2.1.1.3.cmml">m</mi><mo id="S3.E5.m1.2.2.1.1.2" xref="S3.E5.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S3.E5.m1.2.2.1.1.1.1" xref="S3.E5.m1.2.2.1.1.1.1.1.cmml"><mo id="S3.E5.m1.2.2.1.1.1.1.2" stretchy="false" xref="S3.E5.m1.2.2.1.1.1.1.1.cmml">(</mo><msub id="S3.E5.m1.2.2.1.1.1.1.1" xref="S3.E5.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.E5.m1.2.2.1.1.1.1.1.2" xref="S3.E5.m1.2.2.1.1.1.1.1.2.cmml">C</mi><mi id="S3.E5.m1.2.2.1.1.1.1.1.3" xref="S3.E5.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.E5.m1.2.2.1.1.1.1.3" stretchy="false" xref="S3.E5.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E5.m1.2b"><apply id="S3.E5.m1.2.2.cmml" xref="S3.E5.m1.2.2"><eq id="S3.E5.m1.2.2.2.cmml" xref="S3.E5.m1.2.2.2"></eq><apply id="S3.E5.m1.2.2.3.cmml" xref="S3.E5.m1.2.2.3"><times id="S3.E5.m1.2.2.3.1.cmml" xref="S3.E5.m1.2.2.3.1"></times><ci id="S3.E5.m1.2.2.3.2.cmml" xref="S3.E5.m1.2.2.3.2">𝐵</ci><ci id="S3.E5.m1.2.2.3.3.cmml" xref="S3.E5.m1.2.2.3.3">𝑒</ci><ci id="S3.E5.m1.2.2.3.4.cmml" xref="S3.E5.m1.2.2.3.4">𝑙</ci><ci id="S3.E5.m1.1.1.cmml" xref="S3.E5.m1.1.1">ℋ</ci></apply><apply id="S3.E5.m1.2.2.1.cmml" xref="S3.E5.m1.2.2.1"><apply id="S3.E5.m1.2.2.1.2.cmml" xref="S3.E5.m1.2.2.1.2"><csymbol cd="ambiguous" id="S3.E5.m1.2.2.1.2.1.cmml" xref="S3.E5.m1.2.2.1.2">subscript</csymbol><sum id="S3.E5.m1.2.2.1.2.2.cmml" xref="S3.E5.m1.2.2.1.2.2"></sum><apply id="S3.E5.m1.2.2.1.2.3.cmml" xref="S3.E5.m1.2.2.1.2.3"><subset id="S3.E5.m1.2.2.1.2.3.1.cmml" xref="S3.E5.m1.2.2.1.2.3.1"></subset><apply id="S3.E5.m1.2.2.1.2.3.2.cmml" xref="S3.E5.m1.2.2.1.2.3.2"><csymbol cd="ambiguous" id="S3.E5.m1.2.2.1.2.3.2.1.cmml" xref="S3.E5.m1.2.2.1.2.3.2">subscript</csymbol><ci id="S3.E5.m1.2.2.1.2.3.2.2.cmml" xref="S3.E5.m1.2.2.1.2.3.2.2">𝐶</ci><ci id="S3.E5.m1.2.2.1.2.3.2.3.cmml" xref="S3.E5.m1.2.2.1.2.3.2.3">𝑘</ci></apply><ci id="S3.E5.m1.2.2.1.2.3.3.cmml" xref="S3.E5.m1.2.2.1.2.3.3">ℋ</ci></apply></apply><apply id="S3.E5.m1.2.2.1.1.cmml" xref="S3.E5.m1.2.2.1.1"><times id="S3.E5.m1.2.2.1.1.2.cmml" xref="S3.E5.m1.2.2.1.1.2"></times><ci id="S3.E5.m1.2.2.1.1.3.cmml" xref="S3.E5.m1.2.2.1.1.3">𝑚</ci><apply id="S3.E5.m1.2.2.1.1.1.1.1.cmml" xref="S3.E5.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.E5.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E5.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S3.E5.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.E5.m1.2.2.1.1.1.1.1.2">𝐶</ci><ci id="S3.E5.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.E5.m1.2.2.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.2c">Bel(\mathcal{H})\;=\;\sum_{C_{k}\subset\mathcal{H}}\;m(C_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.2d">italic_B italic_e italic_l ( caligraphic_H ) = ∑ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊂ caligraphic_H end_POSTSUBSCRIPT italic_m ( italic_C start_POSTSUBSCRIPT italic_k 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.p16.2">where by definition <math alttext="Bel(\Theta)=1" class="ltx_Math" display="inline" id="S3.p16.1.m1.1"><semantics id="S3.p16.1.m1.1a"><mrow id="S3.p16.1.m1.1.2" xref="S3.p16.1.m1.1.2.cmml"><mrow id="S3.p16.1.m1.1.2.2" xref="S3.p16.1.m1.1.2.2.cmml"><mi id="S3.p16.1.m1.1.2.2.2" xref="S3.p16.1.m1.1.2.2.2.cmml">B</mi><mo id="S3.p16.1.m1.1.2.2.1" xref="S3.p16.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S3.p16.1.m1.1.2.2.3" xref="S3.p16.1.m1.1.2.2.3.cmml">e</mi><mo id="S3.p16.1.m1.1.2.2.1a" xref="S3.p16.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S3.p16.1.m1.1.2.2.4" xref="S3.p16.1.m1.1.2.2.4.cmml">l</mi><mo id="S3.p16.1.m1.1.2.2.1b" xref="S3.p16.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.p16.1.m1.1.2.2.5.2" xref="S3.p16.1.m1.1.2.2.cmml"><mo id="S3.p16.1.m1.1.2.2.5.2.1" stretchy="false" xref="S3.p16.1.m1.1.2.2.cmml">(</mo><mi id="S3.p16.1.m1.1.1" mathvariant="normal" xref="S3.p16.1.m1.1.1.cmml">Θ</mi><mo id="S3.p16.1.m1.1.2.2.5.2.2" stretchy="false" xref="S3.p16.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p16.1.m1.1.2.1" xref="S3.p16.1.m1.1.2.1.cmml">=</mo><mn id="S3.p16.1.m1.1.2.3" xref="S3.p16.1.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p16.1.m1.1b"><apply id="S3.p16.1.m1.1.2.cmml" xref="S3.p16.1.m1.1.2"><eq id="S3.p16.1.m1.1.2.1.cmml" xref="S3.p16.1.m1.1.2.1"></eq><apply id="S3.p16.1.m1.1.2.2.cmml" xref="S3.p16.1.m1.1.2.2"><times id="S3.p16.1.m1.1.2.2.1.cmml" xref="S3.p16.1.m1.1.2.2.1"></times><ci id="S3.p16.1.m1.1.2.2.2.cmml" xref="S3.p16.1.m1.1.2.2.2">𝐵</ci><ci id="S3.p16.1.m1.1.2.2.3.cmml" xref="S3.p16.1.m1.1.2.2.3">𝑒</ci><ci id="S3.p16.1.m1.1.2.2.4.cmml" xref="S3.p16.1.m1.1.2.2.4">𝑙</ci><ci id="S3.p16.1.m1.1.1.cmml" xref="S3.p16.1.m1.1.1">Θ</ci></apply><cn id="S3.p16.1.m1.1.2.3.cmml" type="integer" xref="S3.p16.1.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p16.1.m1.1c">Bel(\Theta)=1</annotation><annotation encoding="application/x-llamapun" id="S3.p16.1.m1.1d">italic_B italic_e italic_l ( roman_Θ ) = 1</annotation></semantics></math> and <math alttext="Bel(\emptyset)=0" class="ltx_Math" display="inline" id="S3.p16.2.m2.1"><semantics id="S3.p16.2.m2.1a"><mrow id="S3.p16.2.m2.1.2" xref="S3.p16.2.m2.1.2.cmml"><mrow id="S3.p16.2.m2.1.2.2" xref="S3.p16.2.m2.1.2.2.cmml"><mi id="S3.p16.2.m2.1.2.2.2" xref="S3.p16.2.m2.1.2.2.2.cmml">B</mi><mo id="S3.p16.2.m2.1.2.2.1" xref="S3.p16.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S3.p16.2.m2.1.2.2.3" xref="S3.p16.2.m2.1.2.2.3.cmml">e</mi><mo id="S3.p16.2.m2.1.2.2.1a" xref="S3.p16.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S3.p16.2.m2.1.2.2.4" xref="S3.p16.2.m2.1.2.2.4.cmml">l</mi><mo id="S3.p16.2.m2.1.2.2.1b" xref="S3.p16.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.p16.2.m2.1.2.2.5.2" xref="S3.p16.2.m2.1.2.2.cmml"><mo id="S3.p16.2.m2.1.2.2.5.2.1" stretchy="false" xref="S3.p16.2.m2.1.2.2.cmml">(</mo><mi id="S3.p16.2.m2.1.1" mathvariant="normal" xref="S3.p16.2.m2.1.1.cmml">∅</mi><mo id="S3.p16.2.m2.1.2.2.5.2.2" stretchy="false" xref="S3.p16.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p16.2.m2.1.2.1" xref="S3.p16.2.m2.1.2.1.cmml">=</mo><mn id="S3.p16.2.m2.1.2.3" xref="S3.p16.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p16.2.m2.1b"><apply id="S3.p16.2.m2.1.2.cmml" xref="S3.p16.2.m2.1.2"><eq id="S3.p16.2.m2.1.2.1.cmml" xref="S3.p16.2.m2.1.2.1"></eq><apply id="S3.p16.2.m2.1.2.2.cmml" xref="S3.p16.2.m2.1.2.2"><times id="S3.p16.2.m2.1.2.2.1.cmml" xref="S3.p16.2.m2.1.2.2.1"></times><ci id="S3.p16.2.m2.1.2.2.2.cmml" xref="S3.p16.2.m2.1.2.2.2">𝐵</ci><ci id="S3.p16.2.m2.1.2.2.3.cmml" xref="S3.p16.2.m2.1.2.2.3">𝑒</ci><ci id="S3.p16.2.m2.1.2.2.4.cmml" xref="S3.p16.2.m2.1.2.2.4">𝑙</ci><emptyset id="S3.p16.2.m2.1.1.cmml" xref="S3.p16.2.m2.1.1"></emptyset></apply><cn id="S3.p16.2.m2.1.2.3.cmml" type="integer" xref="S3.p16.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p16.2.m2.1c">Bel(\emptyset)=0</annotation><annotation encoding="application/x-llamapun" id="S3.p16.2.m2.1d">italic_B italic_e italic_l ( ∅ ) = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p17"> <p class="ltx_p" id="S3.p17.2">While belief in <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S3.p17.1.m1.1"><semantics id="S3.p17.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p17.1.m1.1.1" xref="S3.p17.1.m1.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="S3.p17.1.m1.1b"><ci id="S3.p17.1.m1.1.1.cmml" xref="S3.p17.1.m1.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p17.1.m1.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S3.p17.1.m1.1d">caligraphic_H</annotation></semantics></math> is only supported by the evidence bearing specifically on <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S3.p17.2.m2.1"><semantics id="S3.p17.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p17.2.m2.1.1" xref="S3.p17.2.m2.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="S3.p17.2.m2.1b"><ci id="S3.p17.2.m2.1.1.cmml" xref="S3.p17.2.m2.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p17.2.m2.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S3.p17.2.m2.1d">caligraphic_H</annotation></semantics></math>, it may be desirable to include also the evidence that partially supports it. This is achieved by the <em class="ltx_emph ltx_font_italic" id="S3.p17.2.1">Plausibility Function</em>:</p> </div> <div class="ltx_para" id="S3.p18"> <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="Pl(\mathcal{H})\;=\;\sum_{C_{k}\cap\mathcal{H}\neq\emptyset}\;m(C_{k})" class="ltx_Math" display="block" id="S3.E6.m1.2"><semantics id="S3.E6.m1.2a"><mrow id="S3.E6.m1.2.2" xref="S3.E6.m1.2.2.cmml"><mrow id="S3.E6.m1.2.2.3" xref="S3.E6.m1.2.2.3.cmml"><mi id="S3.E6.m1.2.2.3.2" xref="S3.E6.m1.2.2.3.2.cmml">P</mi><mo id="S3.E6.m1.2.2.3.1" xref="S3.E6.m1.2.2.3.1.cmml">⁢</mo><mi id="S3.E6.m1.2.2.3.3" xref="S3.E6.m1.2.2.3.3.cmml">l</mi><mo id="S3.E6.m1.2.2.3.1a" xref="S3.E6.m1.2.2.3.1.cmml">⁢</mo><mrow id="S3.E6.m1.2.2.3.4.2" xref="S3.E6.m1.2.2.3.cmml"><mo id="S3.E6.m1.2.2.3.4.2.1" stretchy="false" xref="S3.E6.m1.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.E6.m1.1.1" xref="S3.E6.m1.1.1.cmml">ℋ</mi><mo id="S3.E6.m1.2.2.3.4.2.2" rspace="0.280em" stretchy="false" xref="S3.E6.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E6.m1.2.2.2" rspace="0.391em" xref="S3.E6.m1.2.2.2.cmml">=</mo><mrow id="S3.E6.m1.2.2.1" xref="S3.E6.m1.2.2.1.cmml"><munder id="S3.E6.m1.2.2.1.2" xref="S3.E6.m1.2.2.1.2.cmml"><mo id="S3.E6.m1.2.2.1.2.2" movablelimits="false" xref="S3.E6.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S3.E6.m1.2.2.1.2.3" xref="S3.E6.m1.2.2.1.2.3.cmml"><mrow id="S3.E6.m1.2.2.1.2.3.2" xref="S3.E6.m1.2.2.1.2.3.2.cmml"><msub id="S3.E6.m1.2.2.1.2.3.2.2" xref="S3.E6.m1.2.2.1.2.3.2.2.cmml"><mi id="S3.E6.m1.2.2.1.2.3.2.2.2" xref="S3.E6.m1.2.2.1.2.3.2.2.2.cmml">C</mi><mi id="S3.E6.m1.2.2.1.2.3.2.2.3" xref="S3.E6.m1.2.2.1.2.3.2.2.3.cmml">k</mi></msub><mo id="S3.E6.m1.2.2.1.2.3.2.1" xref="S3.E6.m1.2.2.1.2.3.2.1.cmml">∩</mo><mi class="ltx_font_mathcaligraphic" id="S3.E6.m1.2.2.1.2.3.2.3" xref="S3.E6.m1.2.2.1.2.3.2.3.cmml">ℋ</mi></mrow><mo id="S3.E6.m1.2.2.1.2.3.1" xref="S3.E6.m1.2.2.1.2.3.1.cmml">≠</mo><mi id="S3.E6.m1.2.2.1.2.3.3" mathvariant="normal" xref="S3.E6.m1.2.2.1.2.3.3.cmml">∅</mi></mrow></munder><mrow id="S3.E6.m1.2.2.1.1" xref="S3.E6.m1.2.2.1.1.cmml"><mi id="S3.E6.m1.2.2.1.1.3" xref="S3.E6.m1.2.2.1.1.3.cmml">m</mi><mo id="S3.E6.m1.2.2.1.1.2" xref="S3.E6.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S3.E6.m1.2.2.1.1.1.1" xref="S3.E6.m1.2.2.1.1.1.1.1.cmml"><mo id="S3.E6.m1.2.2.1.1.1.1.2" stretchy="false" xref="S3.E6.m1.2.2.1.1.1.1.1.cmml">(</mo><msub id="S3.E6.m1.2.2.1.1.1.1.1" xref="S3.E6.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.E6.m1.2.2.1.1.1.1.1.2" xref="S3.E6.m1.2.2.1.1.1.1.1.2.cmml">C</mi><mi id="S3.E6.m1.2.2.1.1.1.1.1.3" xref="S3.E6.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.E6.m1.2.2.1.1.1.1.3" stretchy="false" xref="S3.E6.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E6.m1.2b"><apply id="S3.E6.m1.2.2.cmml" xref="S3.E6.m1.2.2"><eq id="S3.E6.m1.2.2.2.cmml" xref="S3.E6.m1.2.2.2"></eq><apply id="S3.E6.m1.2.2.3.cmml" xref="S3.E6.m1.2.2.3"><times id="S3.E6.m1.2.2.3.1.cmml" xref="S3.E6.m1.2.2.3.1"></times><ci id="S3.E6.m1.2.2.3.2.cmml" xref="S3.E6.m1.2.2.3.2">𝑃</ci><ci id="S3.E6.m1.2.2.3.3.cmml" xref="S3.E6.m1.2.2.3.3">𝑙</ci><ci id="S3.E6.m1.1.1.cmml" xref="S3.E6.m1.1.1">ℋ</ci></apply><apply id="S3.E6.m1.2.2.1.cmml" xref="S3.E6.m1.2.2.1"><apply id="S3.E6.m1.2.2.1.2.cmml" xref="S3.E6.m1.2.2.1.2"><csymbol cd="ambiguous" id="S3.E6.m1.2.2.1.2.1.cmml" xref="S3.E6.m1.2.2.1.2">subscript</csymbol><sum id="S3.E6.m1.2.2.1.2.2.cmml" xref="S3.E6.m1.2.2.1.2.2"></sum><apply id="S3.E6.m1.2.2.1.2.3.cmml" xref="S3.E6.m1.2.2.1.2.3"><neq id="S3.E6.m1.2.2.1.2.3.1.cmml" xref="S3.E6.m1.2.2.1.2.3.1"></neq><apply id="S3.E6.m1.2.2.1.2.3.2.cmml" xref="S3.E6.m1.2.2.1.2.3.2"><intersect id="S3.E6.m1.2.2.1.2.3.2.1.cmml" xref="S3.E6.m1.2.2.1.2.3.2.1"></intersect><apply id="S3.E6.m1.2.2.1.2.3.2.2.cmml" xref="S3.E6.m1.2.2.1.2.3.2.2"><csymbol cd="ambiguous" id="S3.E6.m1.2.2.1.2.3.2.2.1.cmml" xref="S3.E6.m1.2.2.1.2.3.2.2">subscript</csymbol><ci id="S3.E6.m1.2.2.1.2.3.2.2.2.cmml" xref="S3.E6.m1.2.2.1.2.3.2.2.2">𝐶</ci><ci id="S3.E6.m1.2.2.1.2.3.2.2.3.cmml" xref="S3.E6.m1.2.2.1.2.3.2.2.3">𝑘</ci></apply><ci id="S3.E6.m1.2.2.1.2.3.2.3.cmml" xref="S3.E6.m1.2.2.1.2.3.2.3">ℋ</ci></apply><emptyset id="S3.E6.m1.2.2.1.2.3.3.cmml" xref="S3.E6.m1.2.2.1.2.3.3"></emptyset></apply></apply><apply id="S3.E6.m1.2.2.1.1.cmml" xref="S3.E6.m1.2.2.1.1"><times id="S3.E6.m1.2.2.1.1.2.cmml" xref="S3.E6.m1.2.2.1.1.2"></times><ci id="S3.E6.m1.2.2.1.1.3.cmml" xref="S3.E6.m1.2.2.1.1.3">𝑚</ci><apply id="S3.E6.m1.2.2.1.1.1.1.1.cmml" xref="S3.E6.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.E6.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E6.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S3.E6.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.E6.m1.2.2.1.1.1.1.1.2">𝐶</ci><ci id="S3.E6.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.E6.m1.2.2.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.2c">Pl(\mathcal{H})\;=\;\sum_{C_{k}\cap\mathcal{H}\neq\emptyset}\;m(C_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.2d">italic_P italic_l ( caligraphic_H ) = ∑ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ caligraphic_H ≠ ∅ end_POSTSUBSCRIPT italic_m ( italic_C start_POSTSUBSCRIPT italic_k 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.p18.2">where by definition <math alttext="Pl(\Theta)=1" class="ltx_Math" display="inline" id="S3.p18.1.m1.1"><semantics id="S3.p18.1.m1.1a"><mrow id="S3.p18.1.m1.1.2" xref="S3.p18.1.m1.1.2.cmml"><mrow id="S3.p18.1.m1.1.2.2" xref="S3.p18.1.m1.1.2.2.cmml"><mi id="S3.p18.1.m1.1.2.2.2" xref="S3.p18.1.m1.1.2.2.2.cmml">P</mi><mo id="S3.p18.1.m1.1.2.2.1" xref="S3.p18.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S3.p18.1.m1.1.2.2.3" xref="S3.p18.1.m1.1.2.2.3.cmml">l</mi><mo id="S3.p18.1.m1.1.2.2.1a" xref="S3.p18.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.p18.1.m1.1.2.2.4.2" xref="S3.p18.1.m1.1.2.2.cmml"><mo id="S3.p18.1.m1.1.2.2.4.2.1" stretchy="false" xref="S3.p18.1.m1.1.2.2.cmml">(</mo><mi id="S3.p18.1.m1.1.1" mathvariant="normal" xref="S3.p18.1.m1.1.1.cmml">Θ</mi><mo id="S3.p18.1.m1.1.2.2.4.2.2" stretchy="false" xref="S3.p18.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p18.1.m1.1.2.1" xref="S3.p18.1.m1.1.2.1.cmml">=</mo><mn id="S3.p18.1.m1.1.2.3" xref="S3.p18.1.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p18.1.m1.1b"><apply id="S3.p18.1.m1.1.2.cmml" xref="S3.p18.1.m1.1.2"><eq id="S3.p18.1.m1.1.2.1.cmml" xref="S3.p18.1.m1.1.2.1"></eq><apply id="S3.p18.1.m1.1.2.2.cmml" xref="S3.p18.1.m1.1.2.2"><times id="S3.p18.1.m1.1.2.2.1.cmml" xref="S3.p18.1.m1.1.2.2.1"></times><ci id="S3.p18.1.m1.1.2.2.2.cmml" xref="S3.p18.1.m1.1.2.2.2">𝑃</ci><ci id="S3.p18.1.m1.1.2.2.3.cmml" xref="S3.p18.1.m1.1.2.2.3">𝑙</ci><ci id="S3.p18.1.m1.1.1.cmml" xref="S3.p18.1.m1.1.1">Θ</ci></apply><cn id="S3.p18.1.m1.1.2.3.cmml" type="integer" xref="S3.p18.1.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p18.1.m1.1c">Pl(\Theta)=1</annotation><annotation encoding="application/x-llamapun" id="S3.p18.1.m1.1d">italic_P italic_l ( roman_Θ ) = 1</annotation></semantics></math> and <math alttext="Pl(\emptyset)=0" class="ltx_Math" display="inline" id="S3.p18.2.m2.1"><semantics id="S3.p18.2.m2.1a"><mrow id="S3.p18.2.m2.1.2" xref="S3.p18.2.m2.1.2.cmml"><mrow id="S3.p18.2.m2.1.2.2" xref="S3.p18.2.m2.1.2.2.cmml"><mi id="S3.p18.2.m2.1.2.2.2" xref="S3.p18.2.m2.1.2.2.2.cmml">P</mi><mo id="S3.p18.2.m2.1.2.2.1" xref="S3.p18.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S3.p18.2.m2.1.2.2.3" xref="S3.p18.2.m2.1.2.2.3.cmml">l</mi><mo id="S3.p18.2.m2.1.2.2.1a" xref="S3.p18.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.p18.2.m2.1.2.2.4.2" xref="S3.p18.2.m2.1.2.2.cmml"><mo id="S3.p18.2.m2.1.2.2.4.2.1" stretchy="false" xref="S3.p18.2.m2.1.2.2.cmml">(</mo><mi id="S3.p18.2.m2.1.1" mathvariant="normal" xref="S3.p18.2.m2.1.1.cmml">∅</mi><mo id="S3.p18.2.m2.1.2.2.4.2.2" stretchy="false" xref="S3.p18.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.p18.2.m2.1.2.1" xref="S3.p18.2.m2.1.2.1.cmml">=</mo><mn id="S3.p18.2.m2.1.2.3" xref="S3.p18.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p18.2.m2.1b"><apply id="S3.p18.2.m2.1.2.cmml" xref="S3.p18.2.m2.1.2"><eq id="S3.p18.2.m2.1.2.1.cmml" xref="S3.p18.2.m2.1.2.1"></eq><apply id="S3.p18.2.m2.1.2.2.cmml" xref="S3.p18.2.m2.1.2.2"><times id="S3.p18.2.m2.1.2.2.1.cmml" xref="S3.p18.2.m2.1.2.2.1"></times><ci id="S3.p18.2.m2.1.2.2.2.cmml" xref="S3.p18.2.m2.1.2.2.2">𝑃</ci><ci id="S3.p18.2.m2.1.2.2.3.cmml" xref="S3.p18.2.m2.1.2.2.3">𝑙</ci><emptyset id="S3.p18.2.m2.1.1.cmml" xref="S3.p18.2.m2.1.1"></emptyset></apply><cn id="S3.p18.2.m2.1.2.3.cmml" type="integer" xref="S3.p18.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p18.2.m2.1c">Pl(\emptyset)=0</annotation><annotation encoding="application/x-llamapun" id="S3.p18.2.m2.1d">italic_P italic_l ( ∅ ) = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p19"> <p class="ltx_p" id="S3.p19.1">In general, <math alttext="Bel(\mathcal{H})\leq Pl(\mathcal{H})" class="ltx_Math" display="inline" id="S3.p19.1.m1.2"><semantics id="S3.p19.1.m1.2a"><mrow id="S3.p19.1.m1.2.3" xref="S3.p19.1.m1.2.3.cmml"><mrow id="S3.p19.1.m1.2.3.2" xref="S3.p19.1.m1.2.3.2.cmml"><mi id="S3.p19.1.m1.2.3.2.2" xref="S3.p19.1.m1.2.3.2.2.cmml">B</mi><mo id="S3.p19.1.m1.2.3.2.1" xref="S3.p19.1.m1.2.3.2.1.cmml">⁢</mo><mi id="S3.p19.1.m1.2.3.2.3" xref="S3.p19.1.m1.2.3.2.3.cmml">e</mi><mo id="S3.p19.1.m1.2.3.2.1a" xref="S3.p19.1.m1.2.3.2.1.cmml">⁢</mo><mi id="S3.p19.1.m1.2.3.2.4" xref="S3.p19.1.m1.2.3.2.4.cmml">l</mi><mo id="S3.p19.1.m1.2.3.2.1b" xref="S3.p19.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.p19.1.m1.2.3.2.5.2" xref="S3.p19.1.m1.2.3.2.cmml"><mo id="S3.p19.1.m1.2.3.2.5.2.1" stretchy="false" xref="S3.p19.1.m1.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.p19.1.m1.1.1" xref="S3.p19.1.m1.1.1.cmml">ℋ</mi><mo id="S3.p19.1.m1.2.3.2.5.2.2" stretchy="false" xref="S3.p19.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p19.1.m1.2.3.1" xref="S3.p19.1.m1.2.3.1.cmml">≤</mo><mrow id="S3.p19.1.m1.2.3.3" xref="S3.p19.1.m1.2.3.3.cmml"><mi id="S3.p19.1.m1.2.3.3.2" xref="S3.p19.1.m1.2.3.3.2.cmml">P</mi><mo id="S3.p19.1.m1.2.3.3.1" xref="S3.p19.1.m1.2.3.3.1.cmml">⁢</mo><mi id="S3.p19.1.m1.2.3.3.3" xref="S3.p19.1.m1.2.3.3.3.cmml">l</mi><mo id="S3.p19.1.m1.2.3.3.1a" xref="S3.p19.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.p19.1.m1.2.3.3.4.2" xref="S3.p19.1.m1.2.3.3.cmml"><mo id="S3.p19.1.m1.2.3.3.4.2.1" stretchy="false" xref="S3.p19.1.m1.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.p19.1.m1.2.2" xref="S3.p19.1.m1.2.2.cmml">ℋ</mi><mo id="S3.p19.1.m1.2.3.3.4.2.2" stretchy="false" xref="S3.p19.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p19.1.m1.2b"><apply id="S3.p19.1.m1.2.3.cmml" xref="S3.p19.1.m1.2.3"><leq id="S3.p19.1.m1.2.3.1.cmml" xref="S3.p19.1.m1.2.3.1"></leq><apply id="S3.p19.1.m1.2.3.2.cmml" xref="S3.p19.1.m1.2.3.2"><times id="S3.p19.1.m1.2.3.2.1.cmml" xref="S3.p19.1.m1.2.3.2.1"></times><ci id="S3.p19.1.m1.2.3.2.2.cmml" xref="S3.p19.1.m1.2.3.2.2">𝐵</ci><ci id="S3.p19.1.m1.2.3.2.3.cmml" xref="S3.p19.1.m1.2.3.2.3">𝑒</ci><ci id="S3.p19.1.m1.2.3.2.4.cmml" xref="S3.p19.1.m1.2.3.2.4">𝑙</ci><ci id="S3.p19.1.m1.1.1.cmml" xref="S3.p19.1.m1.1.1">ℋ</ci></apply><apply id="S3.p19.1.m1.2.3.3.cmml" xref="S3.p19.1.m1.2.3.3"><times id="S3.p19.1.m1.2.3.3.1.cmml" xref="S3.p19.1.m1.2.3.3.1"></times><ci id="S3.p19.1.m1.2.3.3.2.cmml" xref="S3.p19.1.m1.2.3.3.2">𝑃</ci><ci id="S3.p19.1.m1.2.3.3.3.cmml" xref="S3.p19.1.m1.2.3.3.3">𝑙</ci><ci id="S3.p19.1.m1.2.2.cmml" xref="S3.p19.1.m1.2.2">ℋ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p19.1.m1.2c">Bel(\mathcal{H})\leq Pl(\mathcal{H})</annotation><annotation encoding="application/x-llamapun" id="S3.p19.1.m1.2d">italic_B italic_e italic_l ( caligraphic_H ) ≤ italic_P italic_l ( caligraphic_H )</annotation></semantics></math>. In many applications, belief is more important than plausibility.</p> </div> <div class="ltx_para" id="S3.p20"> <p class="ltx_p" id="S3.p20.1">It is obvious that eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E5" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E6" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a> are generalizations of eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E1" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E2" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively. What has been added is the ability to formulate hypotheses that go beyond assessing incoming information.</p> </div> <div class="ltx_para" id="S3.p21"> <p class="ltx_p" id="S3.p21.6">In general, decision-makers may formulate several alternative hypotheses, which they may wish to compare to one another given the available evidence. For instance, the wolf may either make the hypothesis that the prey is hiding behind the tree, or that it has climbed the tree. Alternative hypotheses <math alttext="\mathcal{H}_{1}" class="ltx_Math" display="inline" id="S3.p21.1.m1.1"><semantics id="S3.p21.1.m1.1a"><msub id="S3.p21.1.m1.1.1" xref="S3.p21.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p21.1.m1.1.1.2" xref="S3.p21.1.m1.1.1.2.cmml">ℋ</mi><mn id="S3.p21.1.m1.1.1.3" xref="S3.p21.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p21.1.m1.1b"><apply id="S3.p21.1.m1.1.1.cmml" xref="S3.p21.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p21.1.m1.1.1.1.cmml" xref="S3.p21.1.m1.1.1">subscript</csymbol><ci id="S3.p21.1.m1.1.1.2.cmml" xref="S3.p21.1.m1.1.1.2">ℋ</ci><cn id="S3.p21.1.m1.1.1.3.cmml" type="integer" xref="S3.p21.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p21.1.m1.1c">\mathcal{H}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p21.1.m1.1d">caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{H}_{2}" class="ltx_Math" display="inline" id="S3.p21.2.m2.1"><semantics id="S3.p21.2.m2.1a"><msub id="S3.p21.2.m2.1.1" xref="S3.p21.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p21.2.m2.1.1.2" xref="S3.p21.2.m2.1.1.2.cmml">ℋ</mi><mn id="S3.p21.2.m2.1.1.3" xref="S3.p21.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p21.2.m2.1b"><apply id="S3.p21.2.m2.1.1.cmml" xref="S3.p21.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p21.2.m2.1.1.1.cmml" xref="S3.p21.2.m2.1.1">subscript</csymbol><ci id="S3.p21.2.m2.1.1.2.cmml" xref="S3.p21.2.m2.1.1.2">ℋ</ci><cn id="S3.p21.2.m2.1.1.3.cmml" type="integer" xref="S3.p21.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p21.2.m2.1c">\mathcal{H}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p21.2.m2.1d">caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> might be compared by evaluating either <math alttext="Bel(\mathcal{H}_{1})\lessgtr" class="ltx_Math" display="inline" id="S3.p21.3.m3.1"><semantics id="S3.p21.3.m3.1a"><mrow id="S3.p21.3.m3.1.1" xref="S3.p21.3.m3.1.1.cmml"><mrow id="S3.p21.3.m3.1.1.1" xref="S3.p21.3.m3.1.1.1.cmml"><mi id="S3.p21.3.m3.1.1.1.3" xref="S3.p21.3.m3.1.1.1.3.cmml">B</mi><mo id="S3.p21.3.m3.1.1.1.2" xref="S3.p21.3.m3.1.1.1.2.cmml">⁢</mo><mi id="S3.p21.3.m3.1.1.1.4" xref="S3.p21.3.m3.1.1.1.4.cmml">e</mi><mo id="S3.p21.3.m3.1.1.1.2a" xref="S3.p21.3.m3.1.1.1.2.cmml">⁢</mo><mi id="S3.p21.3.m3.1.1.1.5" xref="S3.p21.3.m3.1.1.1.5.cmml">l</mi><mo id="S3.p21.3.m3.1.1.1.2b" xref="S3.p21.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.p21.3.m3.1.1.1.1.1" xref="S3.p21.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.p21.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.p21.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p21.3.m3.1.1.1.1.1.1" xref="S3.p21.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p21.3.m3.1.1.1.1.1.1.2" xref="S3.p21.3.m3.1.1.1.1.1.1.2.cmml">ℋ</mi><mn id="S3.p21.3.m3.1.1.1.1.1.1.3" xref="S3.p21.3.m3.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.p21.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.p21.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p21.3.m3.1.1.2" xref="S3.p21.3.m3.1.1.2.cmml">≶</mo><mi id="S3.p21.3.m3.1.1.3" xref="S3.p21.3.m3.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p21.3.m3.1b"><apply id="S3.p21.3.m3.1.1.cmml" xref="S3.p21.3.m3.1.1"><csymbol cd="latexml" id="S3.p21.3.m3.1.1.2.cmml" xref="S3.p21.3.m3.1.1.2">less-than-or-greater-than</csymbol><apply id="S3.p21.3.m3.1.1.1.cmml" xref="S3.p21.3.m3.1.1.1"><times id="S3.p21.3.m3.1.1.1.2.cmml" xref="S3.p21.3.m3.1.1.1.2"></times><ci id="S3.p21.3.m3.1.1.1.3.cmml" xref="S3.p21.3.m3.1.1.1.3">𝐵</ci><ci id="S3.p21.3.m3.1.1.1.4.cmml" xref="S3.p21.3.m3.1.1.1.4">𝑒</ci><ci id="S3.p21.3.m3.1.1.1.5.cmml" xref="S3.p21.3.m3.1.1.1.5">𝑙</ci><apply id="S3.p21.3.m3.1.1.1.1.1.1.cmml" xref="S3.p21.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" 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id="S3.p21.4.m4.1.1.4" xref="S3.p21.4.m4.1.1.4.cmml">e</mi><mo id="S3.p21.4.m4.1.1.2a" xref="S3.p21.4.m4.1.1.2.cmml">⁢</mo><mi id="S3.p21.4.m4.1.1.5" xref="S3.p21.4.m4.1.1.5.cmml">l</mi><mo id="S3.p21.4.m4.1.1.2b" xref="S3.p21.4.m4.1.1.2.cmml">⁢</mo><mrow id="S3.p21.4.m4.1.1.1.1" xref="S3.p21.4.m4.1.1.1.1.1.cmml"><mo id="S3.p21.4.m4.1.1.1.1.2" stretchy="false" xref="S3.p21.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S3.p21.4.m4.1.1.1.1.1" xref="S3.p21.4.m4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p21.4.m4.1.1.1.1.1.2" xref="S3.p21.4.m4.1.1.1.1.1.2.cmml">ℋ</mi><mn id="S3.p21.4.m4.1.1.1.1.1.3" xref="S3.p21.4.m4.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S3.p21.4.m4.1.1.1.1.3" stretchy="false" xref="S3.p21.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p21.4.m4.1b"><apply id="S3.p21.4.m4.1.1.cmml" xref="S3.p21.4.m4.1.1"><times id="S3.p21.4.m4.1.1.2.cmml" xref="S3.p21.4.m4.1.1.2"></times><ci id="S3.p21.4.m4.1.1.3.cmml" xref="S3.p21.4.m4.1.1.3">𝐵</ci><ci id="S3.p21.4.m4.1.1.4.cmml" xref="S3.p21.4.m4.1.1.4">𝑒</ci><ci id="S3.p21.4.m4.1.1.5.cmml" xref="S3.p21.4.m4.1.1.5">𝑙</ci><apply id="S3.p21.4.m4.1.1.1.1.1.cmml" xref="S3.p21.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S3.p21.4.m4.1.1.1.1.1.1.cmml" xref="S3.p21.4.m4.1.1.1.1">subscript</csymbol><ci id="S3.p21.4.m4.1.1.1.1.1.2.cmml" xref="S3.p21.4.m4.1.1.1.1.1.2">ℋ</ci><cn id="S3.p21.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.p21.4.m4.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p21.4.m4.1c">Bel(\mathcal{H}_{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p21.4.m4.1d">italic_B italic_e italic_l ( caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, or <math alttext="Pl(\mathcal{H}_{1})\lessgtr" class="ltx_Math" display="inline" id="S3.p21.5.m5.1"><semantics id="S3.p21.5.m5.1a"><mrow id="S3.p21.5.m5.1.1" xref="S3.p21.5.m5.1.1.cmml"><mrow 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xref="S3.p21.5.m5.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p21.5.m5.1c">Pl(\mathcal{H}_{1})\lessgtr</annotation><annotation encoding="application/x-llamapun" id="S3.p21.5.m5.1d">italic_P italic_l ( caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≶</annotation></semantics></math> <math alttext="Pl(\mathcal{H}_{2})" class="ltx_Math" display="inline" id="S3.p21.6.m6.1"><semantics id="S3.p21.6.m6.1a"><mrow id="S3.p21.6.m6.1.1" xref="S3.p21.6.m6.1.1.cmml"><mi id="S3.p21.6.m6.1.1.3" xref="S3.p21.6.m6.1.1.3.cmml">P</mi><mo id="S3.p21.6.m6.1.1.2" xref="S3.p21.6.m6.1.1.2.cmml">⁢</mo><mi id="S3.p21.6.m6.1.1.4" xref="S3.p21.6.m6.1.1.4.cmml">l</mi><mo id="S3.p21.6.m6.1.1.2a" xref="S3.p21.6.m6.1.1.2.cmml">⁢</mo><mrow id="S3.p21.6.m6.1.1.1.1" xref="S3.p21.6.m6.1.1.1.1.1.cmml"><mo id="S3.p21.6.m6.1.1.1.1.2" stretchy="false" xref="S3.p21.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S3.p21.6.m6.1.1.1.1.1" xref="S3.p21.6.m6.1.1.1.1.1.cmml"><mi 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encoding="application/x-tex" id="S3.p21.6.m6.1c">Pl(\mathcal{H}_{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p21.6.m6.1d">italic_P italic_l ( caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p22"> <p class="ltx_p" id="S3.p22.1">ET does not prescribe any specific means to formulate hypotheses. Wolves are likely to formulate wolves quite differently from judges, and even individual wolves or individual judges may differ. Equations <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E5" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E6" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a> simply say that, if you want to check the hypothesis that the prey has climbed the tree rather than simply hiding behind it, then different pieces of evidence may be relevant — for instance, whether they prey was a chicken or a cat.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>What do They Think about Me?</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">Having a <em class="ltx_emph ltx_font_italic" id="S4.p1.1.1">Theory of Mind</em> (ToM), also known as <em class="ltx_emph ltx_font_italic" id="S4.p1.1.2">mentalizing</em>, <em class="ltx_emph ltx_font_italic" id="S4.p1.1.3">meta-representation</em>, <em class="ltx_emph ltx_font_italic" id="S4.p1.1.4">second-order intentionality</em> or <em class="ltx_emph ltx_font_italic" id="S4.p1.1.5">mind-reading</em>, indicates the ability to figure out what others think about oneself. Developing ToM is a clear transition in child development, which takes place in parallel with language acquisition <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib62" title="">62</a>]</cite>. Among adults, it correlates with recursive language constructs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib41" title="">41</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">Mentalizing is a sophisticated ability that marks a sharp divide between humans and most other animals, albeit some animals other than humans appear to have it to some extent <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib12" title="">12</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib11" title="">11</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib17" title="">17</a>]</cite>. For instance, chimpanzees are organized in a hierarchy where all females are for the dominant male but, unlike most animals with similar social organizations, female chimps and non-dominant males are capable of arranging secret intercourses. Such arrangements, as well as those enacted in order to escape from the dominant male’s wrath, point to the existence of a substantial degree of mind-reading <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib11" title="">11</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">Just like the transition from simply processing exogenous information to being able to anticipate the future marks a sharp divide in spite of coming in degrees, so does ToM. In particular, by coupling ToM with language humans achieve a degree of complexity of their social relations that sets them apart from all other animals <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib17" title="">17</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">One important consequence of ToM is that it generates indetermination of the possibilities that can be conceived. ToM can generate infinite regressions of the sort “What is this person thinking about me?”, “What is this person thinking that I am thinking about her?”, and so on. Even when thinking is restricted to one specific issue, ToM can slip into regressions of the sort “I think that you think that I think that…”</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">In practice, most of the times humans avoid infinite regressions by limiting mind-reading to 2-3 levels <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib9" title="">9</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib3" title="">3</a>]</cite>, and in any case they appear to be incapable of more than 5 levels <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib41" title="">41</a>]</cite>. However, even limited levels of mind-reading can easily trigger the generation of a large number of possibilities <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib63" title="">63</a>]</cite>, marking a sharp transition of the number of hypotheses that humans can entertain.</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">Humans live in a social reality where novel possibilities continuously appear, and they are aware that they do. In contrast to probabilistic uncertainty on given possibilities, radical uncertainty concerns what possibilities may appear in the FoD <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib46" title="">46</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib13" title="">13</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib21" title="">21</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib16" title="">16</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib37" title="">37</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib33" title="">33</a>]</cite>. Since radical uncertainty undermines confidence in current possibilities and current causal relations, it can have dramatic consequences in terms of postponing or avoiding decision altogether <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib59" title="">59</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib19" title="">19</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib20" title="">20</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib26" title="">26</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.1">However, radical uncertainty originates from novel evidence that contradicts established causal relations, simply because once novel and unthinkable things have been observed, one expects others to appear in the future. Precisely this sort of conflicting evidence can trigger the abductive logic that eventually enables humans to conceive novel possibilities and novel causal relations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib38" title="">38</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib2" title="">2</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib58" title="">58</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.p8"> <p class="ltx_p" id="S4.p8.3">In ET, two possibilities <math alttext="A_{i}" class="ltx_Math" display="inline" id="S4.p8.1.m1.1"><semantics id="S4.p8.1.m1.1a"><msub id="S4.p8.1.m1.1.1" xref="S4.p8.1.m1.1.1.cmml"><mi id="S4.p8.1.m1.1.1.2" xref="S4.p8.1.m1.1.1.2.cmml">A</mi><mi id="S4.p8.1.m1.1.1.3" xref="S4.p8.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p8.1.m1.1b"><apply id="S4.p8.1.m1.1.1.cmml" xref="S4.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p8.1.m1.1.1.1.cmml" xref="S4.p8.1.m1.1.1">subscript</csymbol><ci id="S4.p8.1.m1.1.1.2.cmml" xref="S4.p8.1.m1.1.1.2">𝐴</ci><ci id="S4.p8.1.m1.1.1.3.cmml" xref="S4.p8.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p8.1.m1.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p8.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{j}" class="ltx_Math" display="inline" id="S4.p8.2.m2.1"><semantics id="S4.p8.2.m2.1a"><msub id="S4.p8.2.m2.1.1" xref="S4.p8.2.m2.1.1.cmml"><mi id="S4.p8.2.m2.1.1.2" xref="S4.p8.2.m2.1.1.2.cmml">A</mi><mi id="S4.p8.2.m2.1.1.3" xref="S4.p8.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p8.2.m2.1b"><apply id="S4.p8.2.m2.1.1.cmml" xref="S4.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p8.2.m2.1.1.1.cmml" xref="S4.p8.2.m2.1.1">subscript</csymbol><ci id="S4.p8.2.m2.1.1.2.cmml" xref="S4.p8.2.m2.1.1.2">𝐴</ci><ci id="S4.p8.2.m2.1.1.3.cmml" xref="S4.p8.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p8.2.m2.1c">A_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.p8.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> conflict with one another if <math alttext="A_{i}\cap A_{j}=\emptyset" class="ltx_Math" display="inline" id="S4.p8.3.m3.1"><semantics id="S4.p8.3.m3.1a"><mrow id="S4.p8.3.m3.1.1" xref="S4.p8.3.m3.1.1.cmml"><mrow id="S4.p8.3.m3.1.1.2" xref="S4.p8.3.m3.1.1.2.cmml"><msub id="S4.p8.3.m3.1.1.2.2" xref="S4.p8.3.m3.1.1.2.2.cmml"><mi id="S4.p8.3.m3.1.1.2.2.2" xref="S4.p8.3.m3.1.1.2.2.2.cmml">A</mi><mi id="S4.p8.3.m3.1.1.2.2.3" xref="S4.p8.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S4.p8.3.m3.1.1.2.1" xref="S4.p8.3.m3.1.1.2.1.cmml">∩</mo><msub id="S4.p8.3.m3.1.1.2.3" xref="S4.p8.3.m3.1.1.2.3.cmml"><mi id="S4.p8.3.m3.1.1.2.3.2" xref="S4.p8.3.m3.1.1.2.3.2.cmml">A</mi><mi id="S4.p8.3.m3.1.1.2.3.3" xref="S4.p8.3.m3.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S4.p8.3.m3.1.1.1" xref="S4.p8.3.m3.1.1.1.cmml">=</mo><mi id="S4.p8.3.m3.1.1.3" mathvariant="normal" xref="S4.p8.3.m3.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p8.3.m3.1b"><apply id="S4.p8.3.m3.1.1.cmml" xref="S4.p8.3.m3.1.1"><eq id="S4.p8.3.m3.1.1.1.cmml" xref="S4.p8.3.m3.1.1.1"></eq><apply id="S4.p8.3.m3.1.1.2.cmml" xref="S4.p8.3.m3.1.1.2"><intersect id="S4.p8.3.m3.1.1.2.1.cmml" xref="S4.p8.3.m3.1.1.2.1"></intersect><apply id="S4.p8.3.m3.1.1.2.2.cmml" xref="S4.p8.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S4.p8.3.m3.1.1.2.2.1.cmml" xref="S4.p8.3.m3.1.1.2.2">subscript</csymbol><ci id="S4.p8.3.m3.1.1.2.2.2.cmml" xref="S4.p8.3.m3.1.1.2.2.2">𝐴</ci><ci id="S4.p8.3.m3.1.1.2.2.3.cmml" xref="S4.p8.3.m3.1.1.2.2.3">𝑖</ci></apply><apply id="S4.p8.3.m3.1.1.2.3.cmml" xref="S4.p8.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S4.p8.3.m3.1.1.2.3.1.cmml" xref="S4.p8.3.m3.1.1.2.3">subscript</csymbol><ci id="S4.p8.3.m3.1.1.2.3.2.cmml" xref="S4.p8.3.m3.1.1.2.3.2">𝐴</ci><ci id="S4.p8.3.m3.1.1.2.3.3.cmml" xref="S4.p8.3.m3.1.1.2.3.3">𝑗</ci></apply></apply><emptyset id="S4.p8.3.m3.1.1.3.cmml" xref="S4.p8.3.m3.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p8.3.m3.1c">A_{i}\cap A_{j}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S4.p8.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∅</annotation></semantics></math>. In standard ET, conflicting evidence is redistributed among available possibilities through the denominator of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <div class="ltx_para" id="S4.p9"> <p class="ltx_p" id="S4.p9.2">By contrast, the Transferable Belief Model (TBM) assumes that conflicting evidence translates into <math alttext="m(\emptyset)>0" class="ltx_Math" display="inline" id="S4.p9.1.m1.1"><semantics id="S4.p9.1.m1.1a"><mrow id="S4.p9.1.m1.1.2" xref="S4.p9.1.m1.1.2.cmml"><mrow id="S4.p9.1.m1.1.2.2" xref="S4.p9.1.m1.1.2.2.cmml"><mi id="S4.p9.1.m1.1.2.2.2" xref="S4.p9.1.m1.1.2.2.2.cmml">m</mi><mo id="S4.p9.1.m1.1.2.2.1" xref="S4.p9.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.p9.1.m1.1.2.2.3.2" xref="S4.p9.1.m1.1.2.2.cmml"><mo id="S4.p9.1.m1.1.2.2.3.2.1" stretchy="false" xref="S4.p9.1.m1.1.2.2.cmml">(</mo><mi id="S4.p9.1.m1.1.1" mathvariant="normal" xref="S4.p9.1.m1.1.1.cmml">∅</mi><mo id="S4.p9.1.m1.1.2.2.3.2.2" stretchy="false" xref="S4.p9.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p9.1.m1.1.2.1" xref="S4.p9.1.m1.1.2.1.cmml">></mo><mn id="S4.p9.1.m1.1.2.3" xref="S4.p9.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p9.1.m1.1b"><apply id="S4.p9.1.m1.1.2.cmml" xref="S4.p9.1.m1.1.2"><gt id="S4.p9.1.m1.1.2.1.cmml" xref="S4.p9.1.m1.1.2.1"></gt><apply id="S4.p9.1.m1.1.2.2.cmml" xref="S4.p9.1.m1.1.2.2"><times id="S4.p9.1.m1.1.2.2.1.cmml" xref="S4.p9.1.m1.1.2.2.1"></times><ci id="S4.p9.1.m1.1.2.2.2.cmml" xref="S4.p9.1.m1.1.2.2.2">𝑚</ci><emptyset id="S4.p9.1.m1.1.1.cmml" xref="S4.p9.1.m1.1.1"></emptyset></apply><cn id="S4.p9.1.m1.1.2.3.cmml" type="integer" xref="S4.p9.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.1.m1.1c">m(\emptyset)>0</annotation><annotation encoding="application/x-llamapun" id="S4.p9.1.m1.1d">italic_m ( ∅ ) > 0</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib54" title="">54</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib56" title="">56</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib55" title="">55</a>]</cite>. The rationale of this assumption is that conflicting evidence, by suggesting that something may happen, that is currently not imaginable, moves some mass <math alttext="m" class="ltx_Math" display="inline" id="S4.p9.2.m2.1"><semantics id="S4.p9.2.m2.1a"><mi id="S4.p9.2.m2.1.1" xref="S4.p9.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.p9.2.m2.1b"><ci id="S4.p9.2.m2.1.1.cmml" xref="S4.p9.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.p9.2.m2.1d">italic_m</annotation></semantics></math> towards the void set.</p> </div> <div class="ltx_para" id="S4.p10"> <p class="ltx_p" id="S4.p10.1">Correspondingly, the TBM substitutes Dempster-Shafer’s with Smets’ combination rule, which is essentially the numerator of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E4" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a>:</p> </div> <div class="ltx_para" id="S4.p11"> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <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="m(C_{k})\;=\;\sum_{X_{i}\cap Y_{j}=C_{k}}\;m_{A}(X_{i})\,m_{B}(Y_{j})" class="ltx_Math" display="block" id="S4.E7.m1.3"><semantics id="S4.E7.m1.3a"><mrow id="S4.E7.m1.3.3" xref="S4.E7.m1.3.3.cmml"><mrow id="S4.E7.m1.1.1.1" xref="S4.E7.m1.1.1.1.cmml"><mi id="S4.E7.m1.1.1.1.3" xref="S4.E7.m1.1.1.1.3.cmml">m</mi><mo id="S4.E7.m1.1.1.1.2" xref="S4.E7.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E7.m1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.cmml"><mo id="S4.E7.m1.1.1.1.1.1.2" stretchy="false" xref="S4.E7.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E7.m1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.cmml"><mi id="S4.E7.m1.1.1.1.1.1.1.2" xref="S4.E7.m1.1.1.1.1.1.1.2.cmml">C</mi><mi id="S4.E7.m1.1.1.1.1.1.1.3" xref="S4.E7.m1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.E7.m1.1.1.1.1.1.3" rspace="0.280em" stretchy="false" xref="S4.E7.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.3.3.4" rspace="0.391em" xref="S4.E7.m1.3.3.4.cmml">=</mo><mrow id="S4.E7.m1.3.3.3" xref="S4.E7.m1.3.3.3.cmml"><munder id="S4.E7.m1.3.3.3.3" xref="S4.E7.m1.3.3.3.3.cmml"><mo id="S4.E7.m1.3.3.3.3.2" movablelimits="false" xref="S4.E7.m1.3.3.3.3.2.cmml">∑</mo><mrow id="S4.E7.m1.3.3.3.3.3" xref="S4.E7.m1.3.3.3.3.3.cmml"><mrow id="S4.E7.m1.3.3.3.3.3.2" xref="S4.E7.m1.3.3.3.3.3.2.cmml"><msub id="S4.E7.m1.3.3.3.3.3.2.2" xref="S4.E7.m1.3.3.3.3.3.2.2.cmml"><mi id="S4.E7.m1.3.3.3.3.3.2.2.2" xref="S4.E7.m1.3.3.3.3.3.2.2.2.cmml">X</mi><mi id="S4.E7.m1.3.3.3.3.3.2.2.3" xref="S4.E7.m1.3.3.3.3.3.2.2.3.cmml">i</mi></msub><mo id="S4.E7.m1.3.3.3.3.3.2.1" xref="S4.E7.m1.3.3.3.3.3.2.1.cmml">∩</mo><msub id="S4.E7.m1.3.3.3.3.3.2.3" 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xref="S4.p11.2.m2.1.1.cmml">∅</mi><mo id="S4.p11.2.m2.3.3.1.1.4" rspace="0.387em" xref="S4.p11.2.m2.3.3.1.2.cmml">,</mo><mi id="S4.p11.2.m2.2.2" mathvariant="normal" xref="S4.p11.2.m2.2.2.cmml">Θ</mi><mo id="S4.p11.2.m2.3.3.1.1.5" stretchy="false" xref="S4.p11.2.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p11.2.m2.3b"><apply id="S4.p11.2.m2.3.3.cmml" xref="S4.p11.2.m2.3.3"><in id="S4.p11.2.m2.3.3.2.cmml" xref="S4.p11.2.m2.3.3.2"></in><apply id="S4.p11.2.m2.3.3.3.cmml" xref="S4.p11.2.m2.3.3.3"><csymbol cd="ambiguous" id="S4.p11.2.m2.3.3.3.1.cmml" xref="S4.p11.2.m2.3.3.3">subscript</csymbol><ci id="S4.p11.2.m2.3.3.3.2.cmml" xref="S4.p11.2.m2.3.3.3.2">𝑌</ci><ci id="S4.p11.2.m2.3.3.3.3.cmml" xref="S4.p11.2.m2.3.3.3.3">𝑗</ci></apply><set id="S4.p11.2.m2.3.3.1.2.cmml" xref="S4.p11.2.m2.3.3.1.1"><apply id="S4.p11.2.m2.3.3.1.1.1.cmml" xref="S4.p11.2.m2.3.3.1.1.1"><times id="S4.p11.2.m2.3.3.1.1.1.1.cmml" xref="S4.p11.2.m2.3.3.1.1.1.1"></times><apply 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italic_j end_POSTSUBSCRIPT ∈ { italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∀ italic_j , ∅ , roman_Θ }</annotation></semantics></math> and the <math alttext="C_{k}" class="ltx_Math" display="inline" id="S4.p11.3.m3.1"><semantics id="S4.p11.3.m3.1a"><msub id="S4.p11.3.m3.1.1" xref="S4.p11.3.m3.1.1.cmml"><mi id="S4.p11.3.m3.1.1.2" xref="S4.p11.3.m3.1.1.2.cmml">C</mi><mi id="S4.p11.3.m3.1.1.3" xref="S4.p11.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p11.3.m3.1b"><apply id="S4.p11.3.m3.1.1.cmml" xref="S4.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p11.3.m3.1.1.1.cmml" xref="S4.p11.3.m3.1.1">subscript</csymbol><ci id="S4.p11.3.m3.1.1.2.cmml" xref="S4.p11.3.m3.1.1.2">𝐶</ci><ci id="S4.p11.3.m3.1.1.3.cmml" xref="S4.p11.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p11.3.m3.1c">C_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.p11.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>s are defined by all possible intersections of the <math alttext="X_{i}" class="ltx_Math" display="inline" id="S4.p11.4.m4.1"><semantics id="S4.p11.4.m4.1a"><msub id="S4.p11.4.m4.1.1" xref="S4.p11.4.m4.1.1.cmml"><mi id="S4.p11.4.m4.1.1.2" xref="S4.p11.4.m4.1.1.2.cmml">X</mi><mi id="S4.p11.4.m4.1.1.3" xref="S4.p11.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p11.4.m4.1b"><apply id="S4.p11.4.m4.1.1.cmml" xref="S4.p11.4.m4.1.1"><csymbol cd="ambiguous" id="S4.p11.4.m4.1.1.1.cmml" xref="S4.p11.4.m4.1.1">subscript</csymbol><ci id="S4.p11.4.m4.1.1.2.cmml" xref="S4.p11.4.m4.1.1.2">𝑋</ci><ci id="S4.p11.4.m4.1.1.3.cmml" xref="S4.p11.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p11.4.m4.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p11.4.m4.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>s with the <math alttext="Y_{j}" class="ltx_Math" display="inline" id="S4.p11.5.m5.1"><semantics id="S4.p11.5.m5.1a"><msub id="S4.p11.5.m5.1.1" xref="S4.p11.5.m5.1.1.cmml"><mi id="S4.p11.5.m5.1.1.2" xref="S4.p11.5.m5.1.1.2.cmml">Y</mi><mi id="S4.p11.5.m5.1.1.3" xref="S4.p11.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p11.5.m5.1b"><apply id="S4.p11.5.m5.1.1.cmml" xref="S4.p11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.p11.5.m5.1.1.1.cmml" xref="S4.p11.5.m5.1.1">subscript</csymbol><ci id="S4.p11.5.m5.1.1.2.cmml" xref="S4.p11.5.m5.1.1.2">𝑌</ci><ci id="S4.p11.5.m5.1.1.3.cmml" xref="S4.p11.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p11.5.m5.1c">Y_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.p11.5.m5.1d">italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>s.</p> </div> <div class="ltx_para" id="S4.p12"> <p class="ltx_p" id="S4.p12.1">Since eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4.E7" title="In 4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">7</span></a> does not redistribute conflicting evidence among available possibilities, renormalization is in order. In place of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E3" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a>, the following normalization entails <math alttext="m(\emptyset)" class="ltx_Math" display="inline" id="S4.p12.1.m1.1"><semantics id="S4.p12.1.m1.1a"><mrow id="S4.p12.1.m1.1.2" xref="S4.p12.1.m1.1.2.cmml"><mi id="S4.p12.1.m1.1.2.2" xref="S4.p12.1.m1.1.2.2.cmml">m</mi><mo id="S4.p12.1.m1.1.2.1" xref="S4.p12.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p12.1.m1.1.2.3.2" xref="S4.p12.1.m1.1.2.cmml"><mo id="S4.p12.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p12.1.m1.1.2.cmml">(</mo><mi id="S4.p12.1.m1.1.1" mathvariant="normal" xref="S4.p12.1.m1.1.1.cmml">∅</mi><mo id="S4.p12.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p12.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p12.1.m1.1b"><apply id="S4.p12.1.m1.1.2.cmml" xref="S4.p12.1.m1.1.2"><times id="S4.p12.1.m1.1.2.1.cmml" xref="S4.p12.1.m1.1.2.1"></times><ci id="S4.p12.1.m1.1.2.2.cmml" xref="S4.p12.1.m1.1.2.2">𝑚</ci><emptyset id="S4.p12.1.m1.1.1.cmml" xref="S4.p12.1.m1.1.1"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p12.1.m1.1c">m(\emptyset)</annotation><annotation encoding="application/x-llamapun" id="S4.p12.1.m1.1d">italic_m ( ∅ )</annotation></semantics></math>:</p> </div> <div class="ltx_para" id="S4.p13"> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <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="\sum_{i=1}^{N}\>m_{i}(C_{i})\;+\;m_{C}(\emptyset)\;+\;m_{C}(\Theta)\;=\;1" class="ltx_Math" display="block" id="S4.E8.m1.3"><semantics id="S4.E8.m1.3a"><mrow id="S4.E8.m1.3.3" xref="S4.E8.m1.3.3.cmml"><mrow id="S4.E8.m1.3.3.1" xref="S4.E8.m1.3.3.1.cmml"><mrow id="S4.E8.m1.3.3.1.1" xref="S4.E8.m1.3.3.1.1.cmml"><munderover id="S4.E8.m1.3.3.1.1.2" xref="S4.E8.m1.3.3.1.1.2.cmml"><mo id="S4.E8.m1.3.3.1.1.2.2.2" movablelimits="false" xref="S4.E8.m1.3.3.1.1.2.2.2.cmml">∑</mo><mrow id="S4.E8.m1.3.3.1.1.2.2.3" xref="S4.E8.m1.3.3.1.1.2.2.3.cmml"><mi id="S4.E8.m1.3.3.1.1.2.2.3.2" xref="S4.E8.m1.3.3.1.1.2.2.3.2.cmml">i</mi><mo id="S4.E8.m1.3.3.1.1.2.2.3.1" xref="S4.E8.m1.3.3.1.1.2.2.3.1.cmml">=</mo><mn id="S4.E8.m1.3.3.1.1.2.2.3.3" xref="S4.E8.m1.3.3.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="S4.E8.m1.3.3.1.1.2.3" xref="S4.E8.m1.3.3.1.1.2.3.cmml">N</mi></munderover><mrow id="S4.E8.m1.3.3.1.1.1" xref="S4.E8.m1.3.3.1.1.1.cmml"><msub id="S4.E8.m1.3.3.1.1.1.3" xref="S4.E8.m1.3.3.1.1.1.3.cmml"><mi id="S4.E8.m1.3.3.1.1.1.3.2" xref="S4.E8.m1.3.3.1.1.1.3.2.cmml">m</mi><mi id="S4.E8.m1.3.3.1.1.1.3.3" xref="S4.E8.m1.3.3.1.1.1.3.3.cmml">i</mi></msub><mo id="S4.E8.m1.3.3.1.1.1.2" xref="S4.E8.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E8.m1.3.3.1.1.1.1.1" 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id="S4.E8.m1.3.3.1.4.2.2.cmml" xref="S4.E8.m1.3.3.1.4.2.2">𝑚</ci><ci id="S4.E8.m1.3.3.1.4.2.3.cmml" xref="S4.E8.m1.3.3.1.4.2.3">𝐶</ci></apply><ci id="S4.E8.m1.2.2.cmml" xref="S4.E8.m1.2.2">Θ</ci></apply></apply><cn id="S4.E8.m1.3.3.3.cmml" type="integer" xref="S4.E8.m1.3.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.3c">\sum_{i=1}^{N}\>m_{i}(C_{i})\;+\;m_{C}(\emptyset)\;+\;m_{C}(\Theta)\;=\;1</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.3d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( ∅ ) + italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Θ ) = 1</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> </div> <div class="ltx_para" id="S4.p14"> <p class="ltx_p" id="S4.p14.2">The Belief and Plausibility functions expressed by eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E5" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E6" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a> must be amended on <math alttext="\emptyset" class="ltx_Math" display="inline" id="S4.p14.1.m1.1"><semantics id="S4.p14.1.m1.1a"><mi id="S4.p14.1.m1.1.1" mathvariant="normal" xref="S4.p14.1.m1.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="S4.p14.1.m1.1b"><emptyset id="S4.p14.1.m1.1.1.cmml" xref="S4.p14.1.m1.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="S4.p14.1.m1.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="S4.p14.1.m1.1d">∅</annotation></semantics></math> and <math alttext="\Theta" class="ltx_Math" display="inline" id="S4.p14.2.m2.1"><semantics id="S4.p14.2.m2.1a"><mi id="S4.p14.2.m2.1.1" mathvariant="normal" xref="S4.p14.2.m2.1.1.cmml">Θ</mi><annotation-xml encoding="MathML-Content" id="S4.p14.2.m2.1b"><ci id="S4.p14.2.m2.1.1.cmml" xref="S4.p14.2.m2.1.1">Θ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p14.2.m2.1c">\Theta</annotation><annotation encoding="application/x-llamapun" id="S4.p14.2.m2.1d">roman_Θ</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib57" title="">57</a>]</cite>:</p> </div> <div class="ltx_para" id="S4.p15"> <table class="ltx_equation 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id="S4.E9.m1.6c">Bel(\mathcal{H})\;=\left\{\begin{array}[]{lll}\sum_{C_{k}\subset\mathcal{H}}\;% m(C_{k})&\mathrm{if}&\mathcal{H}\subset\Theta,\;\mathcal{H}\neq\emptyset\\ m_{C}(\emptyset)&\mathrm{if}&\mathcal{H}\equiv\emptyset\\ m_{C}(\Theta)&\mathrm{if}&\mathcal{H}\equiv\Theta\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.6d">italic_B italic_e italic_l ( caligraphic_H ) = { start_ARRAY start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊂ caligraphic_H end_POSTSUBSCRIPT italic_m ( italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ⊂ roman_Θ , caligraphic_H ≠ ∅ end_CELL end_ROW start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( ∅ ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ≡ ∅ end_CELL end_ROW start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Θ ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ≡ roman_Θ end_CELL end_ROW end_ARRAY</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">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p15.2">where the second and third lines are different from eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E5" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a>, which by definition assumed <math alttext="Bel(\emptyset)=0" class="ltx_Math" display="inline" id="S4.p15.1.m1.1"><semantics id="S4.p15.1.m1.1a"><mrow id="S4.p15.1.m1.1.2" xref="S4.p15.1.m1.1.2.cmml"><mrow id="S4.p15.1.m1.1.2.2" xref="S4.p15.1.m1.1.2.2.cmml"><mi id="S4.p15.1.m1.1.2.2.2" xref="S4.p15.1.m1.1.2.2.2.cmml">B</mi><mo id="S4.p15.1.m1.1.2.2.1" xref="S4.p15.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S4.p15.1.m1.1.2.2.3" xref="S4.p15.1.m1.1.2.2.3.cmml">e</mi><mo id="S4.p15.1.m1.1.2.2.1a" xref="S4.p15.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S4.p15.1.m1.1.2.2.4" xref="S4.p15.1.m1.1.2.2.4.cmml">l</mi><mo id="S4.p15.1.m1.1.2.2.1b" xref="S4.p15.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.p15.1.m1.1.2.2.5.2" xref="S4.p15.1.m1.1.2.2.cmml"><mo id="S4.p15.1.m1.1.2.2.5.2.1" stretchy="false" xref="S4.p15.1.m1.1.2.2.cmml">(</mo><mi id="S4.p15.1.m1.1.1" mathvariant="normal" xref="S4.p15.1.m1.1.1.cmml">∅</mi><mo id="S4.p15.1.m1.1.2.2.5.2.2" stretchy="false" xref="S4.p15.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p15.1.m1.1.2.1" xref="S4.p15.1.m1.1.2.1.cmml">=</mo><mn id="S4.p15.1.m1.1.2.3" xref="S4.p15.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p15.1.m1.1b"><apply id="S4.p15.1.m1.1.2.cmml" xref="S4.p15.1.m1.1.2"><eq id="S4.p15.1.m1.1.2.1.cmml" xref="S4.p15.1.m1.1.2.1"></eq><apply id="S4.p15.1.m1.1.2.2.cmml" xref="S4.p15.1.m1.1.2.2"><times id="S4.p15.1.m1.1.2.2.1.cmml" xref="S4.p15.1.m1.1.2.2.1"></times><ci id="S4.p15.1.m1.1.2.2.2.cmml" xref="S4.p15.1.m1.1.2.2.2">𝐵</ci><ci id="S4.p15.1.m1.1.2.2.3.cmml" xref="S4.p15.1.m1.1.2.2.3">𝑒</ci><ci id="S4.p15.1.m1.1.2.2.4.cmml" xref="S4.p15.1.m1.1.2.2.4">𝑙</ci><emptyset id="S4.p15.1.m1.1.1.cmml" xref="S4.p15.1.m1.1.1"></emptyset></apply><cn id="S4.p15.1.m1.1.2.3.cmml" type="integer" xref="S4.p15.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p15.1.m1.1c">Bel(\emptyset)=0</annotation><annotation encoding="application/x-llamapun" id="S4.p15.1.m1.1d">italic_B italic_e italic_l ( ∅ ) = 0</annotation></semantics></math> and <math alttext="Bel(\Theta)=1" class="ltx_Math" display="inline" id="S4.p15.2.m2.1"><semantics id="S4.p15.2.m2.1a"><mrow id="S4.p15.2.m2.1.2" xref="S4.p15.2.m2.1.2.cmml"><mrow id="S4.p15.2.m2.1.2.2" xref="S4.p15.2.m2.1.2.2.cmml"><mi id="S4.p15.2.m2.1.2.2.2" xref="S4.p15.2.m2.1.2.2.2.cmml">B</mi><mo id="S4.p15.2.m2.1.2.2.1" xref="S4.p15.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S4.p15.2.m2.1.2.2.3" xref="S4.p15.2.m2.1.2.2.3.cmml">e</mi><mo id="S4.p15.2.m2.1.2.2.1a" xref="S4.p15.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S4.p15.2.m2.1.2.2.4" xref="S4.p15.2.m2.1.2.2.4.cmml">l</mi><mo id="S4.p15.2.m2.1.2.2.1b" xref="S4.p15.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S4.p15.2.m2.1.2.2.5.2" xref="S4.p15.2.m2.1.2.2.cmml"><mo id="S4.p15.2.m2.1.2.2.5.2.1" stretchy="false" xref="S4.p15.2.m2.1.2.2.cmml">(</mo><mi id="S4.p15.2.m2.1.1" mathvariant="normal" xref="S4.p15.2.m2.1.1.cmml">Θ</mi><mo id="S4.p15.2.m2.1.2.2.5.2.2" stretchy="false" xref="S4.p15.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p15.2.m2.1.2.1" xref="S4.p15.2.m2.1.2.1.cmml">=</mo><mn id="S4.p15.2.m2.1.2.3" xref="S4.p15.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p15.2.m2.1b"><apply id="S4.p15.2.m2.1.2.cmml" xref="S4.p15.2.m2.1.2"><eq id="S4.p15.2.m2.1.2.1.cmml" xref="S4.p15.2.m2.1.2.1"></eq><apply id="S4.p15.2.m2.1.2.2.cmml" xref="S4.p15.2.m2.1.2.2"><times id="S4.p15.2.m2.1.2.2.1.cmml" xref="S4.p15.2.m2.1.2.2.1"></times><ci id="S4.p15.2.m2.1.2.2.2.cmml" xref="S4.p15.2.m2.1.2.2.2">𝐵</ci><ci id="S4.p15.2.m2.1.2.2.3.cmml" xref="S4.p15.2.m2.1.2.2.3">𝑒</ci><ci id="S4.p15.2.m2.1.2.2.4.cmml" xref="S4.p15.2.m2.1.2.2.4">𝑙</ci><ci id="S4.p15.2.m2.1.1.cmml" xref="S4.p15.2.m2.1.1">Θ</ci></apply><cn id="S4.p15.2.m2.1.2.3.cmml" type="integer" xref="S4.p15.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p15.2.m2.1c">Bel(\Theta)=1</annotation><annotation encoding="application/x-llamapun" id="S4.p15.2.m2.1d">italic_B italic_e italic_l ( roman_Θ ) = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p16"> <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <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="Pl(\mathcal{H})\;=\left\{\begin{array}[]{lll}\sum_{C_{k}\cap\mathcal{H}\neq% \emptyset}\;m(C_{k})&\mathrm{if}&\mathcal{H}\subset\Theta,\;\mathcal{H}\neq% \emptyset\\ m_{C}(\emptyset)&\mathrm{if}&\mathcal{H}\equiv\emptyset\\ m_{C}(\Theta)&\mathrm{if}&\mathcal{H}\equiv\Theta\end{array}\right." class="ltx_Math" display="block" id="S4.E10.m1.6"><semantics id="S4.E10.m1.6a"><mrow id="S4.E10.m1.6.7" xref="S4.E10.m1.6.7.cmml"><mrow id="S4.E10.m1.6.7.2" xref="S4.E10.m1.6.7.2.cmml"><mi id="S4.E10.m1.6.7.2.2" xref="S4.E10.m1.6.7.2.2.cmml">P</mi><mo id="S4.E10.m1.6.7.2.1" xref="S4.E10.m1.6.7.2.1.cmml">⁢</mo><mi id="S4.E10.m1.6.7.2.3" xref="S4.E10.m1.6.7.2.3.cmml">l</mi><mo id="S4.E10.m1.6.7.2.1a" xref="S4.E10.m1.6.7.2.1.cmml">⁢</mo><mrow id="S4.E10.m1.6.7.2.4.2" xref="S4.E10.m1.6.7.2.cmml"><mo id="S4.E10.m1.6.7.2.4.2.1" stretchy="false" xref="S4.E10.m1.6.7.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.6.6" xref="S4.E10.m1.6.6.cmml">ℋ</mi><mo id="S4.E10.m1.6.7.2.4.2.2" rspace="0.280em" stretchy="false" xref="S4.E10.m1.6.7.2.cmml">)</mo></mrow></mrow><mo id="S4.E10.m1.6.7.1" xref="S4.E10.m1.6.7.1.cmml">=</mo><mrow id="S4.E10.m1.6.7.3.2" xref="S4.E10.m1.6.7.3.1.cmml"><mo id="S4.E10.m1.6.7.3.2.1" xref="S4.E10.m1.6.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.E10.m1.5.5" rowspacing="0pt" xref="S4.E10.m1.5.5.cmml"><mtr id="S4.E10.m1.5.5a" xref="S4.E10.m1.5.5.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E10.m1.5.5b" xref="S4.E10.m1.5.5.cmml"><mrow id="S4.E10.m1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.cmml"><mstyle displaystyle="false" id="S4.E10.m1.1.1.1.1.1.2" xref="S4.E10.m1.1.1.1.1.1.2.cmml"><msub id="S4.E10.m1.1.1.1.1.1.2a" xref="S4.E10.m1.1.1.1.1.1.2.cmml"><mo id="S4.E10.m1.1.1.1.1.1.2.2" xref="S4.E10.m1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S4.E10.m1.1.1.1.1.1.2.3" xref="S4.E10.m1.1.1.1.1.1.2.3.cmml"><mrow id="S4.E10.m1.1.1.1.1.1.2.3.2" xref="S4.E10.m1.1.1.1.1.1.2.3.2.cmml"><msub id="S4.E10.m1.1.1.1.1.1.2.3.2.2" xref="S4.E10.m1.1.1.1.1.1.2.3.2.2.cmml"><mi id="S4.E10.m1.1.1.1.1.1.2.3.2.2.2" xref="S4.E10.m1.1.1.1.1.1.2.3.2.2.2.cmml">C</mi><mi id="S4.E10.m1.1.1.1.1.1.2.3.2.2.3" xref="S4.E10.m1.1.1.1.1.1.2.3.2.2.3.cmml">k</mi></msub><mo id="S4.E10.m1.1.1.1.1.1.2.3.2.1" xref="S4.E10.m1.1.1.1.1.1.2.3.2.1.cmml">∩</mo><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.1.1.1.1.1.2.3.2.3" xref="S4.E10.m1.1.1.1.1.1.2.3.2.3.cmml">ℋ</mi></mrow><mo id="S4.E10.m1.1.1.1.1.1.2.3.1" xref="S4.E10.m1.1.1.1.1.1.2.3.1.cmml">≠</mo><mi id="S4.E10.m1.1.1.1.1.1.2.3.3" mathvariant="normal" xref="S4.E10.m1.1.1.1.1.1.2.3.3.cmml">∅</mi></mrow></msub></mstyle><mrow id="S4.E10.m1.1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.1.cmml"><mi id="S4.E10.m1.1.1.1.1.1.1.3" xref="S4.E10.m1.1.1.1.1.1.1.3.cmml">m</mi><mo id="S4.E10.m1.1.1.1.1.1.1.2" xref="S4.E10.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E10.m1.1.1.1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E10.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E10.m1.1.1.1.1.1.1.1.1.1" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E10.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.2.cmml">C</mi><mi id="S4.E10.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.E10.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E10.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E10.m1.5.5c" xref="S4.E10.m1.5.5.cmml"><mi id="S4.E10.m1.3.3.3.4.1" xref="S4.E10.m1.3.3.3.4.1.cmml">if</mi></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E10.m1.5.5d" xref="S4.E10.m1.5.5.cmml"><mrow id="S4.E10.m1.3.3.3.3.2.2" xref="S4.E10.m1.3.3.3.3.2.3.cmml"><mrow id="S4.E10.m1.2.2.2.2.1.1.1" xref="S4.E10.m1.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E10.m1.2.2.2.2.1.1.1.2" 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id="S4.E10.m1.5.5.5.1.1.1.cmml" xref="S4.E10.m1.5.5.5.1.1.1">Θ</ci></apply><ci id="S4.E10.m1.5.5.5.2.1.cmml" xref="S4.E10.m1.5.5.5.2.1">if</ci><apply id="S4.E10.m1.5.5.5.3.1.cmml" xref="S4.E10.m1.5.5.5.3.1"><equivalent id="S4.E10.m1.5.5.5.3.1.1.cmml" xref="S4.E10.m1.5.5.5.3.1.1"></equivalent><ci id="S4.E10.m1.5.5.5.3.1.2.cmml" xref="S4.E10.m1.5.5.5.3.1.2">ℋ</ci><ci id="S4.E10.m1.5.5.5.3.1.3.cmml" xref="S4.E10.m1.5.5.5.3.1.3">Θ</ci></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.6c">Pl(\mathcal{H})\;=\left\{\begin{array}[]{lll}\sum_{C_{k}\cap\mathcal{H}\neq% \emptyset}\;m(C_{k})&\mathrm{if}&\mathcal{H}\subset\Theta,\;\mathcal{H}\neq% \emptyset\\ m_{C}(\emptyset)&\mathrm{if}&\mathcal{H}\equiv\emptyset\\ m_{C}(\Theta)&\mathrm{if}&\mathcal{H}\equiv\Theta\end{array}\right.</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.6d">italic_P italic_l ( caligraphic_H ) = { start_ARRAY start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ caligraphic_H ≠ ∅ end_POSTSUBSCRIPT italic_m ( italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ⊂ roman_Θ , caligraphic_H ≠ ∅ end_CELL end_ROW start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( ∅ ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ≡ ∅ end_CELL end_ROW start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Θ ) end_CELL start_CELL roman_if end_CELL start_CELL caligraphic_H ≡ roman_Θ end_CELL end_ROW end_ARRAY</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">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p16.2">where the second and third lines are different from eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E6" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a>, which by definition assumed <math alttext="Pl(\emptyset)=0" class="ltx_Math" display="inline" id="S4.p16.1.m1.1"><semantics id="S4.p16.1.m1.1a"><mrow id="S4.p16.1.m1.1.2" xref="S4.p16.1.m1.1.2.cmml"><mrow id="S4.p16.1.m1.1.2.2" xref="S4.p16.1.m1.1.2.2.cmml"><mi id="S4.p16.1.m1.1.2.2.2" xref="S4.p16.1.m1.1.2.2.2.cmml">P</mi><mo id="S4.p16.1.m1.1.2.2.1" xref="S4.p16.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S4.p16.1.m1.1.2.2.3" xref="S4.p16.1.m1.1.2.2.3.cmml">l</mi><mo id="S4.p16.1.m1.1.2.2.1a" xref="S4.p16.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.p16.1.m1.1.2.2.4.2" xref="S4.p16.1.m1.1.2.2.cmml"><mo id="S4.p16.1.m1.1.2.2.4.2.1" stretchy="false" xref="S4.p16.1.m1.1.2.2.cmml">(</mo><mi id="S4.p16.1.m1.1.1" mathvariant="normal" xref="S4.p16.1.m1.1.1.cmml">∅</mi><mo 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id="S4.p16.1.m1.1d">italic_P italic_l ( ∅ ) = 0</annotation></semantics></math> and <math alttext="Pl(\Theta)=1" class="ltx_Math" display="inline" id="S4.p16.2.m2.1"><semantics id="S4.p16.2.m2.1a"><mrow id="S4.p16.2.m2.1.2" xref="S4.p16.2.m2.1.2.cmml"><mrow id="S4.p16.2.m2.1.2.2" xref="S4.p16.2.m2.1.2.2.cmml"><mi id="S4.p16.2.m2.1.2.2.2" xref="S4.p16.2.m2.1.2.2.2.cmml">P</mi><mo id="S4.p16.2.m2.1.2.2.1" xref="S4.p16.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S4.p16.2.m2.1.2.2.3" xref="S4.p16.2.m2.1.2.2.3.cmml">l</mi><mo id="S4.p16.2.m2.1.2.2.1a" xref="S4.p16.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S4.p16.2.m2.1.2.2.4.2" xref="S4.p16.2.m2.1.2.2.cmml"><mo id="S4.p16.2.m2.1.2.2.4.2.1" stretchy="false" xref="S4.p16.2.m2.1.2.2.cmml">(</mo><mi id="S4.p16.2.m2.1.1" mathvariant="normal" xref="S4.p16.2.m2.1.1.cmml">Θ</mi><mo id="S4.p16.2.m2.1.2.2.4.2.2" stretchy="false" xref="S4.p16.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p16.2.m2.1.2.1" xref="S4.p16.2.m2.1.2.1.cmml">=</mo><mn id="S4.p16.2.m2.1.2.3" xref="S4.p16.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p16.2.m2.1b"><apply id="S4.p16.2.m2.1.2.cmml" xref="S4.p16.2.m2.1.2"><eq id="S4.p16.2.m2.1.2.1.cmml" xref="S4.p16.2.m2.1.2.1"></eq><apply id="S4.p16.2.m2.1.2.2.cmml" xref="S4.p16.2.m2.1.2.2"><times id="S4.p16.2.m2.1.2.2.1.cmml" xref="S4.p16.2.m2.1.2.2.1"></times><ci id="S4.p16.2.m2.1.2.2.2.cmml" xref="S4.p16.2.m2.1.2.2.2">𝑃</ci><ci id="S4.p16.2.m2.1.2.2.3.cmml" xref="S4.p16.2.m2.1.2.2.3">𝑙</ci><ci id="S4.p16.2.m2.1.1.cmml" xref="S4.p16.2.m2.1.1">Θ</ci></apply><cn id="S4.p16.2.m2.1.2.3.cmml" type="integer" xref="S4.p16.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p16.2.m2.1c">Pl(\Theta)=1</annotation><annotation encoding="application/x-llamapun" id="S4.p16.2.m2.1d">italic_P italic_l ( roman_Θ ) = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p17"> <p class="ltx_p" id="S4.p17.1">The above equations express what evidence decision-makers see, but humans can react very differently to one and the same information. In particular, conflicting evidence expressed by <math alttext="m(\emptyset)>0" class="ltx_Math" display="inline" id="S4.p17.1.m1.1"><semantics id="S4.p17.1.m1.1a"><mrow id="S4.p17.1.m1.1.2" xref="S4.p17.1.m1.1.2.cmml"><mrow id="S4.p17.1.m1.1.2.2" xref="S4.p17.1.m1.1.2.2.cmml"><mi id="S4.p17.1.m1.1.2.2.2" xref="S4.p17.1.m1.1.2.2.2.cmml">m</mi><mo id="S4.p17.1.m1.1.2.2.1" xref="S4.p17.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.p17.1.m1.1.2.2.3.2" xref="S4.p17.1.m1.1.2.2.cmml"><mo id="S4.p17.1.m1.1.2.2.3.2.1" stretchy="false" xref="S4.p17.1.m1.1.2.2.cmml">(</mo><mi id="S4.p17.1.m1.1.1" mathvariant="normal" xref="S4.p17.1.m1.1.1.cmml">∅</mi><mo id="S4.p17.1.m1.1.2.2.3.2.2" stretchy="false" xref="S4.p17.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p17.1.m1.1.2.1" xref="S4.p17.1.m1.1.2.1.cmml">></mo><mn id="S4.p17.1.m1.1.2.3" xref="S4.p17.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p17.1.m1.1b"><apply id="S4.p17.1.m1.1.2.cmml" xref="S4.p17.1.m1.1.2"><gt id="S4.p17.1.m1.1.2.1.cmml" xref="S4.p17.1.m1.1.2.1"></gt><apply id="S4.p17.1.m1.1.2.2.cmml" xref="S4.p17.1.m1.1.2.2"><times id="S4.p17.1.m1.1.2.2.1.cmml" xref="S4.p17.1.m1.1.2.2.1"></times><ci id="S4.p17.1.m1.1.2.2.2.cmml" xref="S4.p17.1.m1.1.2.2.2">𝑚</ci><emptyset id="S4.p17.1.m1.1.1.cmml" xref="S4.p17.1.m1.1.1"></emptyset></apply><cn id="S4.p17.1.m1.1.2.3.cmml" type="integer" xref="S4.p17.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p17.1.m1.1c">m(\emptyset)>0</annotation><annotation encoding="application/x-llamapun" id="S4.p17.1.m1.1d">italic_m ( ∅ ) > 0</annotation></semantics></math> can trigger different reactions from different individuals, or from one and the same individuals at different points in time.</p> </div> <div class="ltx_para" id="S4.p18"> <p class="ltx_p" id="S4.p18.1">To be sure, Dr. Watson knows from the very beginning who’s guilty. All clues point to one and only one direction so if the case had been in his hands, it had been closed immediately. However, Sherlock Holmes is profoundly disturbed by a tiny detail that contradicts Dr. Watson’s interpretation. Thus, he interrogates other testimonies, finds other cues that do not fit with the rest of the picture, ascertains that certain testimonies are unreliable and, in the end, the denouement finally comes. Sherlock Holmes comes out with an entirely different interpretation, where certain details have a prominent place in causal explanations whereas others have been discarded.</p> </div> <div class="ltx_para" id="S4.p19"> <p class="ltx_p" id="S4.p19.1">ET understands the process of formulating novel hypotheses and looking for novel evidence, again and again until a coherent interpretation is reached, as refining and coarsening the FoD <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib27" title="">27</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib29" title="">29</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib57" title="">57</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib50" title="">50</a>]</cite>. This process is neither irrational nor obscure, but rather follows its own rationale:</p> </div> <div class="ltx_para" id="S4.p20"> <br class="ltx_break"/> </div> <div class="ltx_para" id="S4.p21"> <blockquote class="ltx_quote" id="S4.p21.1"> <p class="ltx_p" id="S4.p21.1.1">Like any creative act, the act of constructing a frame of discernment does not lend itself to thorough analysis. But we can pick out two considerations that influence it: (1) we want our evidence to interact in an interesting way, and (2) we do not want it to exhibit too much internal conflict.</p> <p class="ltx_p" id="S4.p21.1.2">Two items of evidence can always be said to interact, but they interact in an interesting way only if they jointly support a proposition more interesting than the propositions supported by either alone. (…) Since it depends on what we are interested in, any judgment as to whether our frame is successful in making our evidence interact in an interesting way is a subjective one. But since interesting interactions can always be destroyed by loosening relevant assumptions and thus enlarging our frame, it is clear that our desire for interesting interaction will incline us towards abridging or tightening our frame.</p> <p class="ltx_p" id="S4.p21.1.3">Our desire to avoid excessive internal conflict in our evidence will have precisely the opposite effect: it will incline us towards enlarging or loosening our frame. For internal conflict is itself a form of interaction — the most extreme form of it. And it too tends to increase as the frame is tightened, decrease as it is loosened.</p> </blockquote> <p class="ltx_p" id="S4.p21.2">Glenn Shafer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib51" title="">51</a>]</cite>, Ch. XII.</p> </div> <div class="ltx_para" id="S4.p22"> <br class="ltx_break"/> </div> <div class="ltx_para" id="S4.p23"> <p class="ltx_p" id="S4.p23.1">Interestingly, this quote has a rationale for what Sherlock Holmes does — tightening the FoD in order to highlight contradictions — as well as what Dr. Watson does — coarsening the FoD in order to arrive at a decision. Detective stories present us with contrived cases where the Sherlock Holmes are the heroes, but it is easy to think of simple everyday-problems where reasoning like Sherlock Holmes would lead to irrealistic plot theories. Moreover, even Sherlock Holmes resorts to coarsening the FoD as soon as he determines that certain details are irrelevant.</p> </div> <div class="ltx_para" id="S4.p24"> <p class="ltx_p" id="S4.p24.1">In the end, it is evident that albeit ET can offer a partial formalization of the interpretation of cultural codes, their inherent indeterminacy cannot be eliminated. Even social conventions are not a definitive solution, because although they often operate in the sense of stabilizing cultural codes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib15" title="">15</a>]</cite>, they can at times work in just the opposite direction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib10" title="">10</a>]</cite>. Because of the explosion of possibilities generated by recursive mind-reading and the ambiguity of natural languages, cultural codes are likely to be the most unstable of all <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib42" title="">42</a>]</cite>.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Evolutionary Pressures on Communication Codes</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this section I discuss the direction of evolutionary pressures on biological codes. Since testimonies reporting to a judge can be seen as a communication channel <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib52" title="">52</a>]</cite>, entropic considerations can be adapted from Shannon-Weaver’s Information Theory (IT) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib53" title="">53</a>]</cite> onto ET.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">IT assumes that a source emits characters drawn from a given alphabet, which travel through a noisy channel until they reach a receiver. Noise is able to alter characters, swithing them into other characters from the given alphabet. Thus, in order to minimize errors each single character is coded into a sequence of characters. In this way, even if noise alters one character in the sequence, the damaged sequence is still sufficiently similar to the original one to enable the receiver to reconstruct the original character. Notably, in order for this mechanism to work it is essential that the receiver knows the alphabet of the source, i.e., the set of all possible characters.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">ET generalizes the framework of IT with multiple sources emitting partially overlapping character sets (the evidence) whose overlap is further enhanced by coding and subsequently by transmission through a noisy channel. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.F3" title="Figure 3 ‣ 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> illustrates this generalization.</p> </div> <figure class="ltx_figure" id="S5.F3"> <p class="ltx_p ltx_align_center" id="S5.F3.1"> <span class="ltx_inline-block ltx_framed ltx_framed_rectangle ltx_transformed_outer" id="S5.F3.1.1.1" style="width:346.9pt;height:336.3pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-4.7pt,4.6pt) scale(0.97363448416511,0.97363448416511) ;"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="664" id="S5.F3.1.1.1.g1" src="x3.png" width="685"/> </span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>On top (a), the classical framework of Information Theory where single characters <math alttext="\{A_{i}\}" class="ltx_Math" display="inline" id="S5.F3.4.m1.1"><semantics id="S5.F3.4.m1.1b"><mrow id="S5.F3.4.m1.1.1.1" xref="S5.F3.4.m1.1.1.2.cmml"><mo id="S5.F3.4.m1.1.1.1.2" stretchy="false" xref="S5.F3.4.m1.1.1.2.cmml">{</mo><msub id="S5.F3.4.m1.1.1.1.1" xref="S5.F3.4.m1.1.1.1.1.cmml"><mi id="S5.F3.4.m1.1.1.1.1.2" xref="S5.F3.4.m1.1.1.1.1.2.cmml">A</mi><mi id="S5.F3.4.m1.1.1.1.1.3" xref="S5.F3.4.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.F3.4.m1.1.1.1.3" stretchy="false" xref="S5.F3.4.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.F3.4.m1.1c"><set id="S5.F3.4.m1.1.1.2.cmml" xref="S5.F3.4.m1.1.1.1"><apply id="S5.F3.4.m1.1.1.1.1.cmml" xref="S5.F3.4.m1.1.1.1.1"><csymbol cd="ambiguous" id="S5.F3.4.m1.1.1.1.1.1.cmml" xref="S5.F3.4.m1.1.1.1.1">subscript</csymbol><ci id="S5.F3.4.m1.1.1.1.1.2.cmml" xref="S5.F3.4.m1.1.1.1.1.2">𝐴</ci><ci id="S5.F3.4.m1.1.1.1.1.3.cmml" xref="S5.F3.4.m1.1.1.1.1.3">𝑖</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.F3.4.m1.1d">\{A_{i}\}</annotation><annotation encoding="application/x-llamapun" id="S5.F3.4.m1.1e">{ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math> are first coded into sets <math alttext="A_{i}" class="ltx_Math" display="inline" id="S5.F3.5.m2.1"><semantics id="S5.F3.5.m2.1b"><msub id="S5.F3.5.m2.1.1" xref="S5.F3.5.m2.1.1.cmml"><mi id="S5.F3.5.m2.1.1.2" xref="S5.F3.5.m2.1.1.2.cmml">A</mi><mi id="S5.F3.5.m2.1.1.3" xref="S5.F3.5.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.F3.5.m2.1c"><apply id="S5.F3.5.m2.1.1.cmml" xref="S5.F3.5.m2.1.1"><csymbol cd="ambiguous" id="S5.F3.5.m2.1.1.1.cmml" xref="S5.F3.5.m2.1.1">subscript</csymbol><ci id="S5.F3.5.m2.1.1.2.cmml" xref="S5.F3.5.m2.1.1.2">𝐴</ci><ci id="S5.F3.5.m2.1.1.3.cmml" xref="S5.F3.5.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F3.5.m2.1d">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.F3.5.m2.1e">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, then transmitted through a noisy channel which may generate intersections between these sets, and finally decoded. Bottom (b), the fusion of partially overlapping information originating from different sources. The original overlap may be enhanced by coding and further enhanced by transmission through a noisy channel.</figcaption> </figure> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">IT defines <em class="ltx_emph ltx_font_italic" id="S5.p4.1.1">information</em> as the reduction of uncertainty upon receiving a character. Thus, rare characters that have a low probability to be emitted convey more information than common characters do.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">The <em class="ltx_emph ltx_font_italic" id="S5.p5.1.1">information entropy</em> of a source is an an average of the information conveyed by the single characters. It is maximum when all characters are emitted with equal probability.</p> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.1">Life does not escape the general trend towards greater thermodynamic entropy, but it can macroscopically decrease entropy — the structures of living organisms — by compensating it with higher entropy at more microscopic levels (e.g., heat dissipation) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib31" title="">31</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib32" title="">32</a>]</cite>. Information entropy can be used as a measure of the macroscopic order generated by the communication codes living organisms. This measure does not need to capture exact dynamics, for all we need is a suitable Lyapunov function to describe the trend towards greater structure in the information conveyed by communication codes (see § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#A1" title="Appendix A Lyapunov Functions ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">A</span></a>).</p> </div> <div class="ltx_para" id="S5.p7"> <p class="ltx_p" id="S5.p7.1">Since ET generalizes IT to multiple sources of partially overlapping sets, Shannon’s information entropy requires some adaptation. The quest for a suitable entropy function is a subject of debates that did not yet reach a universally accepted conclusion <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib35" title="">35</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib1" title="">1</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib40" title="">40</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib18" title="">18</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib47" title="">47</a>]</cite>, but the following recent proposal <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib43" title="">43</a>]</cite> is indicative of the sort of functionals that are being scrutinized:</p> </div> <div class="ltx_para" id="S5.p8"> <table class="ltx_equation ltx_eqn_table" id="S5.E11"> <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(A)\;=\;-\sum_{A_{i}\in\Theta}\;\frac{Pl(A_{i})\;\;\lg\>Pl(A_{i})}{e^{Pl(A_{i% })-Bel(A_{i})}}\;\;+\sum_{A_{i}\in\Theta}\;\big{[}Pl(A_{i})-Bel(A_{i})\big{]}" class="ltx_Math" display="block" id="S5.E11.m1.6"><semantics id="S5.E11.m1.6a"><mrow id="S5.E11.m1.6.6" xref="S5.E11.m1.6.6.cmml"><mrow id="S5.E11.m1.6.6.3" xref="S5.E11.m1.6.6.3.cmml"><mi id="S5.E11.m1.6.6.3.2" xref="S5.E11.m1.6.6.3.2.cmml">H</mi><mo id="S5.E11.m1.6.6.3.1" xref="S5.E11.m1.6.6.3.1.cmml">⁢</mo><mrow id="S5.E11.m1.6.6.3.3.2" xref="S5.E11.m1.6.6.3.cmml"><mo id="S5.E11.m1.6.6.3.3.2.1" stretchy="false" xref="S5.E11.m1.6.6.3.cmml">(</mo><mi id="S5.E11.m1.5.5" xref="S5.E11.m1.5.5.cmml">A</mi><mo id="S5.E11.m1.6.6.3.3.2.2" rspace="0.280em" stretchy="false" xref="S5.E11.m1.6.6.3.cmml">)</mo></mrow></mrow><mo id="S5.E11.m1.6.6.2" rspace="0.558em" xref="S5.E11.m1.6.6.2.cmml">=</mo><mrow id="S5.E11.m1.6.6.1" xref="S5.E11.m1.6.6.1.cmml"><mrow id="S5.E11.m1.6.6.1.3" xref="S5.E11.m1.6.6.1.3.cmml"><mo id="S5.E11.m1.6.6.1.3a" xref="S5.E11.m1.6.6.1.3.cmml">−</mo><mrow id="S5.E11.m1.6.6.1.3.2" xref="S5.E11.m1.6.6.1.3.2.cmml"><munder id="S5.E11.m1.6.6.1.3.2.1" xref="S5.E11.m1.6.6.1.3.2.1.cmml"><mo id="S5.E11.m1.6.6.1.3.2.1.2" movablelimits="false" xref="S5.E11.m1.6.6.1.3.2.1.2.cmml">∑</mo><mrow id="S5.E11.m1.6.6.1.3.2.1.3" xref="S5.E11.m1.6.6.1.3.2.1.3.cmml"><msub id="S5.E11.m1.6.6.1.3.2.1.3.2" xref="S5.E11.m1.6.6.1.3.2.1.3.2.cmml"><mi id="S5.E11.m1.6.6.1.3.2.1.3.2.2" xref="S5.E11.m1.6.6.1.3.2.1.3.2.2.cmml">A</mi><mi id="S5.E11.m1.6.6.1.3.2.1.3.2.3" xref="S5.E11.m1.6.6.1.3.2.1.3.2.3.cmml">i</mi></msub><mo id="S5.E11.m1.6.6.1.3.2.1.3.1" xref="S5.E11.m1.6.6.1.3.2.1.3.1.cmml">∈</mo><mi id="S5.E11.m1.6.6.1.3.2.1.3.3" mathvariant="normal" xref="S5.E11.m1.6.6.1.3.2.1.3.3.cmml">Θ</mi></mrow></munder><mfrac id="S5.E11.m1.4.4" xref="S5.E11.m1.4.4.cmml"><mrow id="S5.E11.m1.2.2.2" xref="S5.E11.m1.2.2.2.cmml"><mi id="S5.E11.m1.2.2.2.4" xref="S5.E11.m1.2.2.2.4.cmml">P</mi><mo id="S5.E11.m1.2.2.2.3" xref="S5.E11.m1.2.2.2.3.cmml">⁢</mo><mi id="S5.E11.m1.2.2.2.5" xref="S5.E11.m1.2.2.2.5.cmml">l</mi><mo id="S5.E11.m1.2.2.2.3a" xref="S5.E11.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.E11.m1.1.1.1.1.1" xref="S5.E11.m1.1.1.1.1.1.1.cmml"><mo id="S5.E11.m1.1.1.1.1.1.2" stretchy="false" xref="S5.E11.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.E11.m1.1.1.1.1.1.1" xref="S5.E11.m1.1.1.1.1.1.1.cmml"><mi id="S5.E11.m1.1.1.1.1.1.1.2" xref="S5.E11.m1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S5.E11.m1.1.1.1.1.1.1.3" xref="S5.E11.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.E11.m1.1.1.1.1.1.3" stretchy="false" xref="S5.E11.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.E11.m1.2.2.2.3b" lspace="0.727em" xref="S5.E11.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.E11.m1.2.2.2.6" xref="S5.E11.m1.2.2.2.6.cmml"><mi id="S5.E11.m1.2.2.2.6.1" xref="S5.E11.m1.2.2.2.6.1.cmml">lg</mi><mo id="S5.E11.m1.2.2.2.6a" lspace="0.387em" xref="S5.E11.m1.2.2.2.6.cmml">⁡</mo><mrow id="S5.E11.m1.2.2.2.6.2" xref="S5.E11.m1.2.2.2.6.2.cmml"><mi id="S5.E11.m1.2.2.2.6.2.2" xref="S5.E11.m1.2.2.2.6.2.2.cmml">P</mi><mo id="S5.E11.m1.2.2.2.6.2.1" xref="S5.E11.m1.2.2.2.6.2.1.cmml">⁢</mo><mi id="S5.E11.m1.2.2.2.6.2.3" xref="S5.E11.m1.2.2.2.6.2.3.cmml">l</mi></mrow></mrow><mo id="S5.E11.m1.2.2.2.3c" xref="S5.E11.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.E11.m1.2.2.2.2.1" xref="S5.E11.m1.2.2.2.2.1.1.cmml"><mo id="S5.E11.m1.2.2.2.2.1.2" stretchy="false" xref="S5.E11.m1.2.2.2.2.1.1.cmml">(</mo><msub id="S5.E11.m1.2.2.2.2.1.1" xref="S5.E11.m1.2.2.2.2.1.1.cmml"><mi id="S5.E11.m1.2.2.2.2.1.1.2" xref="S5.E11.m1.2.2.2.2.1.1.2.cmml">A</mi><mi id="S5.E11.m1.2.2.2.2.1.1.3" xref="S5.E11.m1.2.2.2.2.1.1.3.cmml">i</mi></msub><mo id="S5.E11.m1.2.2.2.2.1.3" stretchy="false" xref="S5.E11.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><msup id="S5.E11.m1.4.4.4" xref="S5.E11.m1.4.4.4.cmml"><mi id="S5.E11.m1.4.4.4.4" xref="S5.E11.m1.4.4.4.4.cmml">e</mi><mrow id="S5.E11.m1.4.4.4.2.2" xref="S5.E11.m1.4.4.4.2.2.cmml"><mrow id="S5.E11.m1.3.3.3.1.1.1" xref="S5.E11.m1.3.3.3.1.1.1.cmml"><mi id="S5.E11.m1.3.3.3.1.1.1.3" xref="S5.E11.m1.3.3.3.1.1.1.3.cmml">P</mi><mo id="S5.E11.m1.3.3.3.1.1.1.2" xref="S5.E11.m1.3.3.3.1.1.1.2.cmml">⁢</mo><mi id="S5.E11.m1.3.3.3.1.1.1.4" xref="S5.E11.m1.3.3.3.1.1.1.4.cmml">l</mi><mo id="S5.E11.m1.3.3.3.1.1.1.2a" xref="S5.E11.m1.3.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S5.E11.m1.3.3.3.1.1.1.1.1" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.cmml"><mo id="S5.E11.m1.3.3.3.1.1.1.1.1.2" stretchy="false" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S5.E11.m1.3.3.3.1.1.1.1.1.1" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.cmml"><mi id="S5.E11.m1.3.3.3.1.1.1.1.1.1.2" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.2.cmml">A</mi><mi id="S5.E11.m1.3.3.3.1.1.1.1.1.1.3" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.E11.m1.3.3.3.1.1.1.1.1.3" stretchy="false" xref="S5.E11.m1.3.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.E11.m1.4.4.4.2.2.3" xref="S5.E11.m1.4.4.4.2.2.3.cmml">−</mo><mrow id="S5.E11.m1.4.4.4.2.2.2" xref="S5.E11.m1.4.4.4.2.2.2.cmml"><mi id="S5.E11.m1.4.4.4.2.2.2.3" xref="S5.E11.m1.4.4.4.2.2.2.3.cmml">B</mi><mo id="S5.E11.m1.4.4.4.2.2.2.2" xref="S5.E11.m1.4.4.4.2.2.2.2.cmml">⁢</mo><mi id="S5.E11.m1.4.4.4.2.2.2.4" xref="S5.E11.m1.4.4.4.2.2.2.4.cmml">e</mi><mo id="S5.E11.m1.4.4.4.2.2.2.2a" xref="S5.E11.m1.4.4.4.2.2.2.2.cmml">⁢</mo><mi id="S5.E11.m1.4.4.4.2.2.2.5" xref="S5.E11.m1.4.4.4.2.2.2.5.cmml">l</mi><mo id="S5.E11.m1.4.4.4.2.2.2.2b" xref="S5.E11.m1.4.4.4.2.2.2.2.cmml">⁢</mo><mrow id="S5.E11.m1.4.4.4.2.2.2.1.1" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.cmml"><mo id="S5.E11.m1.4.4.4.2.2.2.1.1.2" stretchy="false" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.cmml">(</mo><msub id="S5.E11.m1.4.4.4.2.2.2.1.1.1" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.cmml"><mi id="S5.E11.m1.4.4.4.2.2.2.1.1.1.2" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.2.cmml">A</mi><mi id="S5.E11.m1.4.4.4.2.2.2.1.1.1.3" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S5.E11.m1.4.4.4.2.2.2.1.1.3" stretchy="false" xref="S5.E11.m1.4.4.4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mfrac></mrow></mrow><mo id="S5.E11.m1.6.6.1.2" lspace="0.782em" rspace="0.055em" xref="S5.E11.m1.6.6.1.2.cmml">+</mo><mrow id="S5.E11.m1.6.6.1.1" xref="S5.E11.m1.6.6.1.1.cmml"><munder id="S5.E11.m1.6.6.1.1.2" xref="S5.E11.m1.6.6.1.1.2.cmml"><mo id="S5.E11.m1.6.6.1.1.2.2" movablelimits="false" rspace="0em" xref="S5.E11.m1.6.6.1.1.2.2.cmml">∑</mo><mrow id="S5.E11.m1.6.6.1.1.2.3" xref="S5.E11.m1.6.6.1.1.2.3.cmml"><msub id="S5.E11.m1.6.6.1.1.2.3.2" xref="S5.E11.m1.6.6.1.1.2.3.2.cmml"><mi id="S5.E11.m1.6.6.1.1.2.3.2.2" 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xref="S5.E11.m1.6.6.1.1.1.1.1.2.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E11.m1.6c">H(A)\;=\;-\sum_{A_{i}\in\Theta}\;\frac{Pl(A_{i})\;\;\lg\>Pl(A_{i})}{e^{Pl(A_{i% })-Bel(A_{i})}}\;\;+\sum_{A_{i}\in\Theta}\;\big{[}Pl(A_{i})-Bel(A_{i})\big{]}</annotation><annotation encoding="application/x-llamapun" id="S5.E11.m1.6d">italic_H ( italic_A ) = - ∑ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Θ end_POSTSUBSCRIPT divide start_ARG italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) roman_lg italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_e start_POSTSUPERSCRIPT italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Θ end_POSTSUBSCRIPT [ italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT italic_i 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">(11)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.p9"> <p class="ltx_p" id="S5.p9.2">In eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.E11" title="In 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">11</span></a>, belief and plausibility appear in the most basic version of eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E1" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.E2" title="In 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a>. For living organisms who are capable of anticipation, eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E5" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3.E6" title="In 3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">6</span></a> apply with <math alttext="\mathcal{H}=A_{i}" class="ltx_Math" display="inline" id="S5.p9.1.m1.1"><semantics id="S5.p9.1.m1.1a"><mrow id="S5.p9.1.m1.1.1" xref="S5.p9.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p9.1.m1.1.1.2" xref="S5.p9.1.m1.1.1.2.cmml">ℋ</mi><mo id="S5.p9.1.m1.1.1.1" xref="S5.p9.1.m1.1.1.1.cmml">=</mo><msub id="S5.p9.1.m1.1.1.3" xref="S5.p9.1.m1.1.1.3.cmml"><mi id="S5.p9.1.m1.1.1.3.2" xref="S5.p9.1.m1.1.1.3.2.cmml">A</mi><mi id="S5.p9.1.m1.1.1.3.3" xref="S5.p9.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p9.1.m1.1b"><apply id="S5.p9.1.m1.1.1.cmml" xref="S5.p9.1.m1.1.1"><eq id="S5.p9.1.m1.1.1.1.cmml" xref="S5.p9.1.m1.1.1.1"></eq><ci id="S5.p9.1.m1.1.1.2.cmml" xref="S5.p9.1.m1.1.1.2">ℋ</ci><apply id="S5.p9.1.m1.1.1.3.cmml" xref="S5.p9.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.p9.1.m1.1.1.3.1.cmml" xref="S5.p9.1.m1.1.1.3">subscript</csymbol><ci id="S5.p9.1.m1.1.1.3.2.cmml" xref="S5.p9.1.m1.1.1.3.2">𝐴</ci><ci id="S5.p9.1.m1.1.1.3.3.cmml" xref="S5.p9.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.1.m1.1c">\mathcal{H}=A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p9.1.m1.1d">caligraphic_H = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. For living organisms who are capable of abduction, eqs. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4.E9" title="In 4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">9</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4.E10" title="In 4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">10</span></a> apply with <math alttext="\mathcal{H}=A_{i}" class="ltx_Math" display="inline" id="S5.p9.2.m2.1"><semantics id="S5.p9.2.m2.1a"><mrow id="S5.p9.2.m2.1.1" xref="S5.p9.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p9.2.m2.1.1.2" xref="S5.p9.2.m2.1.1.2.cmml">ℋ</mi><mo id="S5.p9.2.m2.1.1.1" xref="S5.p9.2.m2.1.1.1.cmml">=</mo><msub id="S5.p9.2.m2.1.1.3" xref="S5.p9.2.m2.1.1.3.cmml"><mi id="S5.p9.2.m2.1.1.3.2" xref="S5.p9.2.m2.1.1.3.2.cmml">A</mi><mi id="S5.p9.2.m2.1.1.3.3" xref="S5.p9.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p9.2.m2.1b"><apply id="S5.p9.2.m2.1.1.cmml" xref="S5.p9.2.m2.1.1"><eq id="S5.p9.2.m2.1.1.1.cmml" xref="S5.p9.2.m2.1.1.1"></eq><ci id="S5.p9.2.m2.1.1.2.cmml" xref="S5.p9.2.m2.1.1.2">ℋ</ci><apply id="S5.p9.2.m2.1.1.3.cmml" xref="S5.p9.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.p9.2.m2.1.1.3.1.cmml" xref="S5.p9.2.m2.1.1.3">subscript</csymbol><ci id="S5.p9.2.m2.1.1.3.2.cmml" xref="S5.p9.2.m2.1.1.3.2">𝐴</ci><ci id="S5.p9.2.m2.1.1.3.3.cmml" xref="S5.p9.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.2.m2.1c">\mathcal{H}=A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p9.2.m2.1d">caligraphic_H = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p10"> <p class="ltx_p" id="S5.p10.6">The first term of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.E11" title="In 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">11</span></a> reduces to Shannon’s entropy if the <math alttext="A_{i}" class="ltx_Math" display="inline" id="S5.p10.1.m1.1"><semantics id="S5.p10.1.m1.1a"><msub id="S5.p10.1.m1.1.1" xref="S5.p10.1.m1.1.1.cmml"><mi id="S5.p10.1.m1.1.1.2" xref="S5.p10.1.m1.1.1.2.cmml">A</mi><mi id="S5.p10.1.m1.1.1.3" xref="S5.p10.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p10.1.m1.1b"><apply id="S5.p10.1.m1.1.1.cmml" xref="S5.p10.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p10.1.m1.1.1.1.cmml" xref="S5.p10.1.m1.1.1">subscript</csymbol><ci id="S5.p10.1.m1.1.1.2.cmml" xref="S5.p10.1.m1.1.1.2">𝐴</ci><ci id="S5.p10.1.m1.1.1.3.cmml" xref="S5.p10.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.1.m1.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p10.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>s are singletons <math alttext="\{A_{i}\}" class="ltx_Math" display="inline" id="S5.p10.2.m2.1"><semantics id="S5.p10.2.m2.1a"><mrow id="S5.p10.2.m2.1.1.1" xref="S5.p10.2.m2.1.1.2.cmml"><mo id="S5.p10.2.m2.1.1.1.2" stretchy="false" xref="S5.p10.2.m2.1.1.2.cmml">{</mo><msub id="S5.p10.2.m2.1.1.1.1" xref="S5.p10.2.m2.1.1.1.1.cmml"><mi id="S5.p10.2.m2.1.1.1.1.2" xref="S5.p10.2.m2.1.1.1.1.2.cmml">A</mi><mi id="S5.p10.2.m2.1.1.1.1.3" xref="S5.p10.2.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.p10.2.m2.1.1.1.3" stretchy="false" xref="S5.p10.2.m2.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p10.2.m2.1b"><set id="S5.p10.2.m2.1.1.2.cmml" xref="S5.p10.2.m2.1.1.1"><apply id="S5.p10.2.m2.1.1.1.1.cmml" xref="S5.p10.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S5.p10.2.m2.1.1.1.1.1.cmml" xref="S5.p10.2.m2.1.1.1.1">subscript</csymbol><ci id="S5.p10.2.m2.1.1.1.1.2.cmml" xref="S5.p10.2.m2.1.1.1.1.2">𝐴</ci><ci id="S5.p10.2.m2.1.1.1.1.3.cmml" xref="S5.p10.2.m2.1.1.1.1.3">𝑖</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.2.m2.1c">\{A_{i}\}</annotation><annotation encoding="application/x-llamapun" id="S5.p10.2.m2.1d">{ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>s and, consequently, <math alttext="Bel(A_{i})=" class="ltx_Math" display="inline" id="S5.p10.3.m3.1"><semantics id="S5.p10.3.m3.1a"><mrow id="S5.p10.3.m3.1.1" xref="S5.p10.3.m3.1.1.cmml"><mrow id="S5.p10.3.m3.1.1.1" xref="S5.p10.3.m3.1.1.1.cmml"><mi id="S5.p10.3.m3.1.1.1.3" xref="S5.p10.3.m3.1.1.1.3.cmml">B</mi><mo id="S5.p10.3.m3.1.1.1.2" xref="S5.p10.3.m3.1.1.1.2.cmml">⁢</mo><mi id="S5.p10.3.m3.1.1.1.4" xref="S5.p10.3.m3.1.1.1.4.cmml">e</mi><mo id="S5.p10.3.m3.1.1.1.2a" xref="S5.p10.3.m3.1.1.1.2.cmml">⁢</mo><mi 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end_POSTSUBSCRIPT ) =</annotation></semantics></math> <math alttext="Pl(A_{i})=" class="ltx_Math" display="inline" id="S5.p10.4.m4.1"><semantics id="S5.p10.4.m4.1a"><mrow id="S5.p10.4.m4.1.1" xref="S5.p10.4.m4.1.1.cmml"><mrow id="S5.p10.4.m4.1.1.1" xref="S5.p10.4.m4.1.1.1.cmml"><mi id="S5.p10.4.m4.1.1.1.3" xref="S5.p10.4.m4.1.1.1.3.cmml">P</mi><mo id="S5.p10.4.m4.1.1.1.2" xref="S5.p10.4.m4.1.1.1.2.cmml">⁢</mo><mi id="S5.p10.4.m4.1.1.1.4" xref="S5.p10.4.m4.1.1.1.4.cmml">l</mi><mo id="S5.p10.4.m4.1.1.1.2a" xref="S5.p10.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.p10.4.m4.1.1.1.1.1" xref="S5.p10.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.p10.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.p10.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p10.4.m4.1.1.1.1.1.1" xref="S5.p10.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.p10.4.m4.1.1.1.1.1.1.2" xref="S5.p10.4.m4.1.1.1.1.1.1.2.cmml">A</mi><mi id="S5.p10.4.m4.1.1.1.1.1.1.3" xref="S5.p10.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.p10.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.p10.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p10.4.m4.1.1.2" xref="S5.p10.4.m4.1.1.2.cmml">=</mo><mi id="S5.p10.4.m4.1.1.3" xref="S5.p10.4.m4.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p10.4.m4.1b"><apply id="S5.p10.4.m4.1.1.cmml" xref="S5.p10.4.m4.1.1"><eq id="S5.p10.4.m4.1.1.2.cmml" xref="S5.p10.4.m4.1.1.2"></eq><apply id="S5.p10.4.m4.1.1.1.cmml" xref="S5.p10.4.m4.1.1.1"><times id="S5.p10.4.m4.1.1.1.2.cmml" xref="S5.p10.4.m4.1.1.1.2"></times><ci id="S5.p10.4.m4.1.1.1.3.cmml" xref="S5.p10.4.m4.1.1.1.3">𝑃</ci><ci id="S5.p10.4.m4.1.1.1.4.cmml" xref="S5.p10.4.m4.1.1.1.4">𝑙</ci><apply id="S5.p10.4.m4.1.1.1.1.1.1.cmml" xref="S5.p10.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p10.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.p10.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S5.p10.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.p10.4.m4.1.1.1.1.1.1.2">𝐴</ci><ci id="S5.p10.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.p10.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><csymbol cd="latexml" id="S5.p10.4.m4.1.1.3.cmml" xref="S5.p10.4.m4.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.4.m4.1c">Pl(A_{i})=</annotation><annotation encoding="application/x-llamapun" id="S5.p10.4.m4.1d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) =</annotation></semantics></math> <math alttext="p(\{A_{i}\})" class="ltx_Math" display="inline" id="S5.p10.5.m5.1"><semantics id="S5.p10.5.m5.1a"><mrow id="S5.p10.5.m5.1.1" xref="S5.p10.5.m5.1.1.cmml"><mi id="S5.p10.5.m5.1.1.3" xref="S5.p10.5.m5.1.1.3.cmml">p</mi><mo id="S5.p10.5.m5.1.1.2" xref="S5.p10.5.m5.1.1.2.cmml">⁢</mo><mrow id="S5.p10.5.m5.1.1.1.1" xref="S5.p10.5.m5.1.1.cmml"><mo id="S5.p10.5.m5.1.1.1.1.2" stretchy="false" xref="S5.p10.5.m5.1.1.cmml">(</mo><mrow id="S5.p10.5.m5.1.1.1.1.1.1" xref="S5.p10.5.m5.1.1.1.1.1.2.cmml"><mo id="S5.p10.5.m5.1.1.1.1.1.1.2" stretchy="false" xref="S5.p10.5.m5.1.1.1.1.1.2.cmml">{</mo><msub id="S5.p10.5.m5.1.1.1.1.1.1.1" xref="S5.p10.5.m5.1.1.1.1.1.1.1.cmml"><mi id="S5.p10.5.m5.1.1.1.1.1.1.1.2" xref="S5.p10.5.m5.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S5.p10.5.m5.1.1.1.1.1.1.1.3" xref="S5.p10.5.m5.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.p10.5.m5.1.1.1.1.1.1.3" stretchy="false" xref="S5.p10.5.m5.1.1.1.1.1.2.cmml">}</mo></mrow><mo id="S5.p10.5.m5.1.1.1.1.3" stretchy="false" xref="S5.p10.5.m5.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p10.5.m5.1b"><apply id="S5.p10.5.m5.1.1.cmml" xref="S5.p10.5.m5.1.1"><times id="S5.p10.5.m5.1.1.2.cmml" xref="S5.p10.5.m5.1.1.2"></times><ci id="S5.p10.5.m5.1.1.3.cmml" xref="S5.p10.5.m5.1.1.3">𝑝</ci><set id="S5.p10.5.m5.1.1.1.1.1.2.cmml" xref="S5.p10.5.m5.1.1.1.1.1.1"><apply id="S5.p10.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.p10.5.m5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p10.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S5.p10.5.m5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p10.5.m5.1.1.1.1.1.1.1.2.cmml" xref="S5.p10.5.m5.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S5.p10.5.m5.1.1.1.1.1.1.1.3.cmml" xref="S5.p10.5.m5.1.1.1.1.1.1.1.3">𝑖</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.5.m5.1c">p(\{A_{i}\})</annotation><annotation encoding="application/x-llamapun" id="S5.p10.5.m5.1d">italic_p ( { italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } )</annotation></semantics></math> where <math alttext="p" class="ltx_Math" display="inline" id="S5.p10.6.m6.1"><semantics id="S5.p10.6.m6.1a"><mi id="S5.p10.6.m6.1.1" xref="S5.p10.6.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.p10.6.m6.1b"><ci id="S5.p10.6.m6.1.1.cmml" xref="S5.p10.6.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.6.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.p10.6.m6.1d">italic_p</annotation></semantics></math> denotes probability. This term expresses contradiction of competing evidence. The higher this term, the more difficult an interpretation.</p> </div> <div class="ltx_para" id="S5.p11"> <p class="ltx_p" id="S5.p11.1">In the context of IT, this term can be minimized by adopting redundant codes that allow receivers to (partially) correct the mistakes introduced by the noisy channel (see the central portion of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2.F1" title="Figure 1 ‣ 2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a>). Living organisms do exploit this option; for instance, the genetic code is redundant (or <em class="ltx_emph ltx_font_italic" id="S5.p11.1.1">degenerate</em>) and, while errors are most often made on the third nucleotide, this is precisely the one nucleotide on which most multiple codifications of one single amino acid differ from one another. However, one other option is available to living organisms in order to minimize the first term of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.E11" title="In 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">11</span></a>.</p> </div> <div class="ltx_para" id="S5.p12"> <p class="ltx_p" id="S5.p12.1">In IT, the receiver knows the alphabet of the source. Therefore, any character that has been received must belong to one of those in the alphabet. In IT, the set of possibilities is given once and for all.</p> </div> <div class="ltx_para" id="S5.p13"> <p class="ltx_p" id="S5.p13.1">By contrast, living beings can give novel meanings to novel possibilities generated by either random mutations, or random codings, or both. Whenever this happens, novel possibilities are added to the FoD, and by increasing the number of possibilities, information entropy can decrease <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib4" title="">4</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib5" title="">5</a>]</cite>. This may have happened, for instance, each time the ancestral genetic code increased the number of amino acids from a likely initial number of 10 to the current 20 amino acids.</p> </div> <div class="ltx_para" id="S5.p14"> <p class="ltx_p" id="S5.p14.3">The second term of eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.E11" title="In 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">11</span></a> has no counterpart in Shannon’s entropy. The difference between <math alttext="Pl(A_{i})" class="ltx_Math" display="inline" id="S5.p14.1.m1.1"><semantics id="S5.p14.1.m1.1a"><mrow id="S5.p14.1.m1.1.1" xref="S5.p14.1.m1.1.1.cmml"><mi id="S5.p14.1.m1.1.1.3" xref="S5.p14.1.m1.1.1.3.cmml">P</mi><mo id="S5.p14.1.m1.1.1.2" xref="S5.p14.1.m1.1.1.2.cmml">⁢</mo><mi id="S5.p14.1.m1.1.1.4" xref="S5.p14.1.m1.1.1.4.cmml">l</mi><mo id="S5.p14.1.m1.1.1.2a" xref="S5.p14.1.m1.1.1.2.cmml">⁢</mo><mrow id="S5.p14.1.m1.1.1.1.1" xref="S5.p14.1.m1.1.1.1.1.1.cmml"><mo id="S5.p14.1.m1.1.1.1.1.2" stretchy="false" xref="S5.p14.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S5.p14.1.m1.1.1.1.1.1" xref="S5.p14.1.m1.1.1.1.1.1.cmml"><mi id="S5.p14.1.m1.1.1.1.1.1.2" xref="S5.p14.1.m1.1.1.1.1.1.2.cmml">A</mi><mi id="S5.p14.1.m1.1.1.1.1.1.3" xref="S5.p14.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.p14.1.m1.1.1.1.1.3" stretchy="false" xref="S5.p14.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p14.1.m1.1b"><apply id="S5.p14.1.m1.1.1.cmml" xref="S5.p14.1.m1.1.1"><times id="S5.p14.1.m1.1.1.2.cmml" xref="S5.p14.1.m1.1.1.2"></times><ci id="S5.p14.1.m1.1.1.3.cmml" xref="S5.p14.1.m1.1.1.3">𝑃</ci><ci id="S5.p14.1.m1.1.1.4.cmml" xref="S5.p14.1.m1.1.1.4">𝑙</ci><apply id="S5.p14.1.m1.1.1.1.1.1.cmml" xref="S5.p14.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p14.1.m1.1.1.1.1.1.1.cmml" xref="S5.p14.1.m1.1.1.1.1">subscript</csymbol><ci id="S5.p14.1.m1.1.1.1.1.1.2.cmml" xref="S5.p14.1.m1.1.1.1.1.1.2">𝐴</ci><ci id="S5.p14.1.m1.1.1.1.1.1.3.cmml" xref="S5.p14.1.m1.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p14.1.m1.1c">Pl(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.p14.1.m1.1d">italic_P italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="Bel(A_{i})" class="ltx_Math" display="inline" id="S5.p14.2.m2.1"><semantics id="S5.p14.2.m2.1a"><mrow id="S5.p14.2.m2.1.1" xref="S5.p14.2.m2.1.1.cmml"><mi id="S5.p14.2.m2.1.1.3" xref="S5.p14.2.m2.1.1.3.cmml">B</mi><mo id="S5.p14.2.m2.1.1.2" xref="S5.p14.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.p14.2.m2.1.1.4" xref="S5.p14.2.m2.1.1.4.cmml">e</mi><mo id="S5.p14.2.m2.1.1.2a" xref="S5.p14.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.p14.2.m2.1.1.5" xref="S5.p14.2.m2.1.1.5.cmml">l</mi><mo id="S5.p14.2.m2.1.1.2b" xref="S5.p14.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.p14.2.m2.1.1.1.1" xref="S5.p14.2.m2.1.1.1.1.1.cmml"><mo id="S5.p14.2.m2.1.1.1.1.2" stretchy="false" xref="S5.p14.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S5.p14.2.m2.1.1.1.1.1" xref="S5.p14.2.m2.1.1.1.1.1.cmml"><mi id="S5.p14.2.m2.1.1.1.1.1.2" xref="S5.p14.2.m2.1.1.1.1.1.2.cmml">A</mi><mi id="S5.p14.2.m2.1.1.1.1.1.3" xref="S5.p14.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.p14.2.m2.1.1.1.1.3" stretchy="false" xref="S5.p14.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p14.2.m2.1b"><apply id="S5.p14.2.m2.1.1.cmml" xref="S5.p14.2.m2.1.1"><times id="S5.p14.2.m2.1.1.2.cmml" xref="S5.p14.2.m2.1.1.2"></times><ci id="S5.p14.2.m2.1.1.3.cmml" xref="S5.p14.2.m2.1.1.3">𝐵</ci><ci id="S5.p14.2.m2.1.1.4.cmml" xref="S5.p14.2.m2.1.1.4">𝑒</ci><ci id="S5.p14.2.m2.1.1.5.cmml" xref="S5.p14.2.m2.1.1.5">𝑙</ci><apply id="S5.p14.2.m2.1.1.1.1.1.cmml" xref="S5.p14.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S5.p14.2.m2.1.1.1.1.1.1.cmml" xref="S5.p14.2.m2.1.1.1.1">subscript</csymbol><ci id="S5.p14.2.m2.1.1.1.1.1.2.cmml" xref="S5.p14.2.m2.1.1.1.1.1.2">𝐴</ci><ci id="S5.p14.2.m2.1.1.1.1.1.3.cmml" xref="S5.p14.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p14.2.m2.1c">Bel(A_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.p14.2.m2.1d">italic_B italic_e italic_l ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> measures to what extent the available evidence goes beyond <math alttext="A_{i}" class="ltx_Math" display="inline" id="S5.p14.3.m3.1"><semantics id="S5.p14.3.m3.1a"><msub id="S5.p14.3.m3.1.1" xref="S5.p14.3.m3.1.1.cmml"><mi id="S5.p14.3.m3.1.1.2" xref="S5.p14.3.m3.1.1.2.cmml">A</mi><mi id="S5.p14.3.m3.1.1.3" xref="S5.p14.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p14.3.m3.1b"><apply id="S5.p14.3.m3.1.1.cmml" xref="S5.p14.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p14.3.m3.1.1.1.cmml" xref="S5.p14.3.m3.1.1">subscript</csymbol><ci id="S5.p14.3.m3.1.1.2.cmml" xref="S5.p14.3.m3.1.1.2">𝐴</ci><ci id="S5.p14.3.m3.1.1.3.cmml" xref="S5.p14.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p14.3.m3.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p14.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to support other possibilities as well. Thus, it measures code ambiguity. Its minimization expresses the evolutiuonary trend towards less ambiguous codes; for instance, the ambiguous ancestral genetic code has been substituted by the current non-ambiguous code.</p> </div> <div class="ltx_para" id="S5.p15"> <p class="ltx_p" id="S5.p15.1">To summarize, eq. <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S5.E11" title="In 5 Evolutionary Pressures on Communication Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">11</span></a> is a Lyapunov function whose minimization describes the evolutionary trends of the communication codes employed by living organisms in terms of: (i) Reduction of communication errors; (ii) Appearance of novel meanings, and (iii) Reduction of ambiguity.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusions</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">Adapting ET to the transmission of information through arbitrary codes between and within living organisms required taking sides in the unsettled debate on what constitutes an “interpretation.” While semiotics understands interpretation as necessarily dialogical and inxtricably bound to abduction, many applied disciplines ascribe interpretation to induction and deduction as well.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">In particular, Law studies mention the interpretation of laws through jurisprudence as an instance of induction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib30" title="">30</a>]</cite> whereas the interpretation of laws from higher principles — expressed, e.g., in a Constitution — is rather seen as an instance of deduction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib39" title="">39</a>]</cite>. In its turn, abduction is eventually recognized to be fundamental to resolve legal disputes characterized by conflicting evidence <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib28" title="">28</a>]</cite>. Likewise, medical doctors stress the importance of induction for interpreting clinical tests <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib44" title="">44</a>]</cite>, abduction for interpreting symptoms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib60" title="">60</a>]</cite>, and deduction in order to ensure that the consequences of critical information have been explored <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib25" title="">25</a>]</cite>. Notably, even deduction can generate multiple interpretations because of degrees of freedom, random disturbances, or Gödel’s indecidability theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib24" title="">24</a>]</cite>.</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">In § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S1" title="1 Introduction ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a> I defined “interpretation” as the process of attaching meaning to information. It is now appropriate to specify that this can either happen by means of deduction, or induction, or deduction.</p> </div> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.1">In particular, coarsening and refining a FoD following the observation of conflicting evidence described in § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4" title="4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> are abductions, whereas the interpretations based on extrapolation of § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> are instances of induction. I assumed that even the simple organisms of § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2" title="2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a> are capable of interpretation, but their interpretations are only based on deduction. Such a low status for deduction may strike many readers, but decades of psychological research have shown that humans capabilities in deductive logic are very limited, certainly inferior to those of chess-playing computers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib22" title="">22</a>]</cite>. Deduction, once regarded as the quintessence of human intelligence, is nowadays the province of artificial intelligence.</p> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">Deduction, induction and abduction are not meant to be the exclusive domain of the living beings of § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S2" title="2 Ambiguous Ancestral Genetic Codes ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">2</span></a>, § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S3" title="3 Anticipatory Brains ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">3</span></a> and § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S4" title="4 What do They Think about Me? ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a>, respectively. Humans are clearly capable of all reasoning logics, whereas animals capable of anticipation display both deduction and induction.</p> </div> <div class="ltx_para" id="S6.p6"> <p class="ltx_p" id="S6.p6.1">As outlined in § <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#S1" title="1 Introduction ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">1</span></a>, I defined meaning as deriving from feed-backs that have an impact on living organisms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib45" title="">45</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib36" title="">36</a>]</cite>. But since interpretation is the process of attaching meaning to information, it follows that interpretation implies receiving a fitness value on which natural selection will act. And since the novel correspondences of arbitrary codes ultimately arise from random fluctuations, in the end we are back to the classical understanding of evolution as arising from mutation + selection.</p> </div> <div class="ltx_para" id="S6.p7"> <p class="ltx_p" id="S6.p7.1">At this level of generality, this is true. However, the Modern Synthesis (MS) is based on applying to the Life Sciences the very same reductionism that had proven so successful in the physics of inanimate objects, whereas the Semiotic Theory of Evolution (STE) is based on relations rather than elements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib6" title="">6</a>]</cite>. MS explores the combinations of given ©<em class="ltx_emph ltx_font_italic" id="S6.p7.1.1">Lego</em> bricks, whereas STE moves from communication codes to explore how those ©<em class="ltx_emph ltx_font_italic" id="S6.p7.1.2">Lego</em> bricks might change with time. Tracing a parallel to human affairs, MS understands Nature as a football coach who hires the perfect combination of individual players in the conviction that a team will spontaneously emerge. By contrast, STE understands Nature as a football coach who cares that each player makes the effort of reading the minds of all other players, in the conviction that all of them are able to learn and coordinate <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib34" title="">34</a>]</cite> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#bib.bib61" title="">61</a>]</cite>.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Legal Disclaimers</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">This research was not funded. The author has no conflict of interests. No copyrighted material was used without permission. Neither humans nor other animals were involved in experiments.</p> </div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Lyapunov Functions</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.4">A Lyapunov function can be used to prove the stability of an equilibrium point. A Lyapunov function is continuous, has continuous first derivatives, is strictly positive except for the equilibrium point, and its time derivative is non-positive. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#A1.F4" title="Figure 4 ‣ Appendix A Lyapunov Functions ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> illustrates a Lyapunov function for a system described by two state variables <math alttext="x_{1}" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><msub id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mi id="A1.p1.1.m1.1.1.2" xref="A1.p1.1.m1.1.1.2.cmml">x</mi><mn id="A1.p1.1.m1.1.1.3" xref="A1.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><apply id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p1.1.m1.1.1.1.cmml" xref="A1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.p1.1.m1.1.1.2.cmml" xref="A1.p1.1.m1.1.1.2">𝑥</ci><cn id="A1.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{2}" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><msub id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml"><mi id="A1.p1.2.m2.1.1.2" xref="A1.p1.2.m2.1.1.2.cmml">x</mi><mn id="A1.p1.2.m2.1.1.3" xref="A1.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><apply id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p1.2.m2.1.1.1.cmml" xref="A1.p1.2.m2.1.1">subscript</csymbol><ci id="A1.p1.2.m2.1.1.2.cmml" xref="A1.p1.2.m2.1.1.2">𝑥</ci><cn id="A1.p1.2.m2.1.1.3.cmml" type="integer" xref="A1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> with a stable equilibrium at <math alttext="(0,0)" class="ltx_Math" display="inline" id="A1.p1.3.m3.2"><semantics id="A1.p1.3.m3.2a"><mrow id="A1.p1.3.m3.2.3.2" xref="A1.p1.3.m3.2.3.1.cmml"><mo id="A1.p1.3.m3.2.3.2.1" stretchy="false" xref="A1.p1.3.m3.2.3.1.cmml">(</mo><mn id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml">0</mn><mo id="A1.p1.3.m3.2.3.2.2" xref="A1.p1.3.m3.2.3.1.cmml">,</mo><mn id="A1.p1.3.m3.2.2" xref="A1.p1.3.m3.2.2.cmml">0</mn><mo id="A1.p1.3.m3.2.3.2.3" stretchy="false" xref="A1.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.2b"><interval closure="open" id="A1.p1.3.m3.2.3.1.cmml" xref="A1.p1.3.m3.2.3.2"><cn id="A1.p1.3.m3.1.1.cmml" type="integer" xref="A1.p1.3.m3.1.1">0</cn><cn id="A1.p1.3.m3.2.2.cmml" type="integer" xref="A1.p1.3.m3.2.2">0</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.2c">(0,0)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.2d">( 0 , 0 )</annotation></semantics></math>. The equilibrium is reached by minimizing <math alttext="V" class="ltx_Math" display="inline" id="A1.p1.4.m4.1"><semantics id="A1.p1.4.m4.1a"><mi id="A1.p1.4.m4.1.1" xref="A1.p1.4.m4.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A1.p1.4.m4.1b"><ci id="A1.p1.4.m4.1.1.cmml" xref="A1.p1.4.m4.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m4.1c">V</annotation><annotation encoding="application/x-llamapun" id="A1.p1.4.m4.1d">italic_V</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.1">Several Lyapunov functions can exist for one and the same equilibrium point, all what is required is that the Lyapunov function has the required shape. For instance, the Lyapunov function of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.18984v1#A1.F4" title="Figure 4 ‣ Appendix A Lyapunov Functions ‣ The Misinterpretable Evidence Conveyed by Arbitrary Codes"><span class="ltx_text ltx_ref_tag">4</span></a> would identify <math alttext="(0,0)" class="ltx_Math" display="inline" id="A1.p2.1.m1.2"><semantics id="A1.p2.1.m1.2a"><mrow id="A1.p2.1.m1.2.3.2" xref="A1.p2.1.m1.2.3.1.cmml"><mo id="A1.p2.1.m1.2.3.2.1" stretchy="false" xref="A1.p2.1.m1.2.3.1.cmml">(</mo><mn id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml">0</mn><mo id="A1.p2.1.m1.2.3.2.2" xref="A1.p2.1.m1.2.3.1.cmml">,</mo><mn id="A1.p2.1.m1.2.2" xref="A1.p2.1.m1.2.2.cmml">0</mn><mo id="A1.p2.1.m1.2.3.2.3" stretchy="false" xref="A1.p2.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.2b"><interval closure="open" id="A1.p2.1.m1.2.3.1.cmml" xref="A1.p2.1.m1.2.3.2"><cn id="A1.p2.1.m1.1.1.cmml" type="integer" xref="A1.p2.1.m1.1.1">0</cn><cn id="A1.p2.1.m1.2.2.cmml" type="integer" xref="A1.p2.1.m1.2.2">0</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.2c">(0,0)</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.2d">( 0 , 0 )</annotation></semantics></math> as a stable equilibrium point even if the surrounding basing would be narrower, or wider than it is.</p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.1">Lyapunov functions shaped like a Mexican hat can represent the trend towards a limit cycle between the edge of the hat and the height in the centre. More complex Lyapunov functions can entail several locally stable equilibria, in which case the Lyapunov function illustrates the capability to switch between different equilibria by jumping through saddles. Lyapunov functions cannot represent strange attractors.</p> </div> <div class="ltx_para" id="A1.p4"> <p class="ltx_p" id="A1.p4.1">The construction of a Lyapunov function is more an art than a science, though it is known that in simple cases with one equilibrium quadratic functions work. Construction is eased by the awareness that in general several Lyapunov functions can exist, and that any of them works.</p> </div> <figure class="ltx_figure" id="A1.F4"> <p class="ltx_p ltx_align_center" id="A1.F4.1"> <span class="ltx_inline-block ltx_framed ltx_framed_rectangle ltx_transformed_outer" id="A1.F4.1.1.1" style="width:346.9pt;height:321.7pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(113.8pt,-105.5pt) scale(2.90910191374831,2.90910191374831) ;"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="212" id="A1.F4.1.1.1.g1" src="x4.png" width="229"/> </span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>A Lyapunov function <math alttext="V" class="ltx_Math" display="inline" id="A1.F4.8.m1.1"><semantics id="A1.F4.8.m1.1b"><mi id="A1.F4.8.m1.1.1" xref="A1.F4.8.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A1.F4.8.m1.1c"><ci id="A1.F4.8.m1.1.1.cmml" xref="A1.F4.8.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.8.m1.1d">V</annotation><annotation encoding="application/x-llamapun" id="A1.F4.8.m1.1e">italic_V</annotation></semantics></math> of two state variables <math alttext="x_{1}" class="ltx_Math" display="inline" id="A1.F4.9.m2.1"><semantics id="A1.F4.9.m2.1b"><msub id="A1.F4.9.m2.1.1" xref="A1.F4.9.m2.1.1.cmml"><mi id="A1.F4.9.m2.1.1.2" xref="A1.F4.9.m2.1.1.2.cmml">x</mi><mn id="A1.F4.9.m2.1.1.3" xref="A1.F4.9.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F4.9.m2.1c"><apply id="A1.F4.9.m2.1.1.cmml" xref="A1.F4.9.m2.1.1"><csymbol cd="ambiguous" id="A1.F4.9.m2.1.1.1.cmml" xref="A1.F4.9.m2.1.1">subscript</csymbol><ci id="A1.F4.9.m2.1.1.2.cmml" xref="A1.F4.9.m2.1.1.2">𝑥</ci><cn id="A1.F4.9.m2.1.1.3.cmml" type="integer" xref="A1.F4.9.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.9.m2.1d">x_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.F4.9.m2.1e">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{2}" class="ltx_Math" display="inline" id="A1.F4.10.m3.1"><semantics id="A1.F4.10.m3.1b"><msub id="A1.F4.10.m3.1.1" xref="A1.F4.10.m3.1.1.cmml"><mi id="A1.F4.10.m3.1.1.2" xref="A1.F4.10.m3.1.1.2.cmml">x</mi><mn id="A1.F4.10.m3.1.1.3" xref="A1.F4.10.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F4.10.m3.1c"><apply id="A1.F4.10.m3.1.1.cmml" xref="A1.F4.10.m3.1.1"><csymbol cd="ambiguous" id="A1.F4.10.m3.1.1.1.cmml" xref="A1.F4.10.m3.1.1">subscript</csymbol><ci id="A1.F4.10.m3.1.1.2.cmml" xref="A1.F4.10.m3.1.1.2">𝑥</ci><cn id="A1.F4.10.m3.1.1.3.cmml" type="integer" xref="A1.F4.10.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.10.m3.1d">x_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.F4.10.m3.1e">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> describing movement towards equilibrium at <math alttext="(0,0)" class="ltx_Math" display="inline" id="A1.F4.11.m4.2"><semantics id="A1.F4.11.m4.2b"><mrow id="A1.F4.11.m4.2.3.2" xref="A1.F4.11.m4.2.3.1.cmml"><mo id="A1.F4.11.m4.2.3.2.1" stretchy="false" xref="A1.F4.11.m4.2.3.1.cmml">(</mo><mn id="A1.F4.11.m4.1.1" xref="A1.F4.11.m4.1.1.cmml">0</mn><mo id="A1.F4.11.m4.2.3.2.2" xref="A1.F4.11.m4.2.3.1.cmml">,</mo><mn id="A1.F4.11.m4.2.2" xref="A1.F4.11.m4.2.2.cmml">0</mn><mo id="A1.F4.11.m4.2.3.2.3" stretchy="false" xref="A1.F4.11.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.F4.11.m4.2c"><interval closure="open" id="A1.F4.11.m4.2.3.1.cmml" xref="A1.F4.11.m4.2.3.2"><cn id="A1.F4.11.m4.1.1.cmml" type="integer" xref="A1.F4.11.m4.1.1">0</cn><cn id="A1.F4.11.m4.2.2.cmml" type="integer" xref="A1.F4.11.m4.2.2">0</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.F4.11.m4.2d">(0,0)</annotation><annotation encoding="application/x-llamapun" id="A1.F4.11.m4.2e">( 0 , 0 )</annotation></semantics></math> along the path projected on the <math alttext="x_{1},x_{2}" class="ltx_Math" display="inline" id="A1.F4.12.m5.2"><semantics id="A1.F4.12.m5.2b"><mrow id="A1.F4.12.m5.2.2.2" xref="A1.F4.12.m5.2.2.3.cmml"><msub id="A1.F4.12.m5.1.1.1.1" xref="A1.F4.12.m5.1.1.1.1.cmml"><mi id="A1.F4.12.m5.1.1.1.1.2" xref="A1.F4.12.m5.1.1.1.1.2.cmml">x</mi><mn id="A1.F4.12.m5.1.1.1.1.3" xref="A1.F4.12.m5.1.1.1.1.3.cmml">1</mn></msub><mo id="A1.F4.12.m5.2.2.2.3" xref="A1.F4.12.m5.2.2.3.cmml">,</mo><msub id="A1.F4.12.m5.2.2.2.2" xref="A1.F4.12.m5.2.2.2.2.cmml"><mi id="A1.F4.12.m5.2.2.2.2.2" xref="A1.F4.12.m5.2.2.2.2.2.cmml">x</mi><mn id="A1.F4.12.m5.2.2.2.2.3" xref="A1.F4.12.m5.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.F4.12.m5.2c"><list id="A1.F4.12.m5.2.2.3.cmml" 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A different function with a similar shape had worked equally well. 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Electric potential is a Lyapunov function for electrons moving towards the positive pole. For ecosystems, fitness is a Lyapunov function with a minus sign. 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