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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation</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/2408.06786v1/"/></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/2408.06786v1#S1" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">I Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S2" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">II The Majorana Demonstrator</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">III A unified approach to combine assay results</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">IV Background index</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V Promoting single values to distributions</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">VI Monte Carlo uncertainty propagation</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S7" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">VII Projected background and conclusions</a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#A1" title="In An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">A Efficiency distribution prior derivation</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> Majorana Collaboration </div> <h1 class="ltx_title ltx_title_document">An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation</h1> <div class="ltx_authors"> I.J. Arnquist Pacific Northwest National Laboratory, Richland, WA 99354, USA F.T. Avignone III Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA A.S. Barabash \orcidlink0000-0002-5130-0922 National Research Center “Kurchatov Institute”, Kurchatov Complex of Theoretical and Experimental Physics, Moscow, 117218 Russia C.J. Barton Department of Physics, University of South Dakota, Vermillion, SD 57069, USA K.H. Bhimani Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA E. Blalock Department of Physics, North Carolina State University, Raleigh, NC 27695, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA B. Bos Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA M. Busch Department of Physics, Duke University, Durham, NC 27708, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA T.S. Caldwell Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA Y.-D. Chan Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA C.D. Christofferson South Dakota Mines, Rapid City, SD 57701, USA P.-H. Chu \orcidlink0000-0003-1372-2910 Los Alamos National Laboratory, Los Alamos, NM 87545, USA M.L. Clark Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA C. Cuesta \orcidlink0000-0003-1190-7233 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040, Madrid, Spain J.A. Detwiler \orcidlink0000-0002-9050-4610 Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA Yu. Efremenko Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37916, USA Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA H. Ejiri Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan S.R. Elliott \orcidlink0000-0001-9361-9870 Los Alamos National Laboratory, Los Alamos, NM 87545, USA N. Fuad \orcidlink0000-0002-5445-2534 IU Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA G.K. Giovanetti Physics Department, Williams College, Williamstown, MA 01267, USA M.P. Green \orcidlink0000-0002-1958-8030 Department of Physics, North Carolina State University, Raleigh, NC 27695, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA J. Gruszko \orcidlink0000-0002-3777-2237 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA I.S. Guinn \orcidlink0000-0002-2424-3272 Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA V.E. Guiseppe \orcidlink0000-0002-0078-7101 Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA C.R. Haufe Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA R. Henning Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA D. Hervas Aguilar \orcidlink0000-0002-9686-0659 Present address: Technical University of Munich, 85748 Garching, Germany Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA E.W. Hoppe Pacific Northwest National Laboratory, Richland, WA 99354, USA A. Hostiuc Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA M.F. Kidd Tennessee Tech University, Cookeville, TN 38505, USA I. Kim Present address: Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Los Alamos National Laboratory, Los Alamos, NM 87545, USA R.T. Kouzes Pacific Northwest National Laboratory, Richland, WA 99354, USA T.E. Lannen V Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA A. Li \orcidlink0000-0002-4844-9339 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA J.M. López-Castaño Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA R.D. Martin,\orcidlink0000-0001-8648-1658 Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6, Canada R. Massarczyk Los Alamos National Laboratory, Los Alamos, NM 87545, USA S.J. Meijer \orcidlink0000-0002-1366-0361 Los Alamos National Laboratory, Los Alamos, NM 87545, USA T.K. Oli \orcidlink0000-0001-8857-3716 Present address: Argonne National Laboratory, Lemont, IL 60439, USA Department of Physics, University of South Dakota, Vermillion, SD 57069, USA L.S. Paudel \orcidlink0000-0003-3100-4074 Department of Physics, University of South Dakota, Vermillion, SD 57069, USA W. Pettus \orcidlink0000-0003-4947-7400 IU Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA A.W.P. Poon \orcidlink0000-0003-2684-6402 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA D.C. Radford Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA A.L. Reine \orcidlink0000-0002-5900-8299 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA K. Rielage \orcidlink0000-0002-7392-7152 Los Alamos National Laboratory, Los Alamos, NM 87545, USA N.W. Ruof \orcidlink0000-0002-0477-7488 Present address: Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA D.C. Schaper Los Alamos National Laboratory, Los Alamos, NM 87545, USA S.J. Schleich \orcidlink0000-0003-1878-9102 IU Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA D. Tedeschi Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA R.L. Varner \orcidlink0000-0002-0477-7488 Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA S. Vasilyev Joint Institute for Nuclear Research, Dubna, 141980 Russia S.L. Watkins \orcidlink0000-0003-0649-1923 Los Alamos National Laboratory, Los Alamos, NM 87545, USA J.F. Wilkerson \orcidlink0000-0002-0342-0217 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA C. Wiseman \orcidlink0000-0002-4232-1326 Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA W. Xu Department of Physics, University of South Dakota, Vermillion, SD 57069, USA C.-H. Yu \orcidlink0000-0002-9849-842X Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA </div> <div class="ltx_dates">(August 13, 2024)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> The background index is an important quantity which is used in projecting and calculating the half-life sensitivity of neutrinoless double-beta decay (<math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mrow id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml"><mn id="id1.1.m1.1.1.2" xref="id1.1.m1.1.1.2.cmml">0</mn><mo id="id1.1.m1.1.1.1" xref="id1.1.m1.1.1.1.cmml">⁢</mo><mi id="id1.1.m1.1.1.3" xref="id1.1.m1.1.1.3.cmml">ν</mi><mo id="id1.1.m1.1.1.1a" xref="id1.1.m1.1.1.1.cmml">⁢</mo><mi id="id1.1.m1.1.1.4" xref="id1.1.m1.1.1.4.cmml">β</mi><mo id="id1.1.m1.1.1.1b" xref="id1.1.m1.1.1.1.cmml">⁢</mo><mi id="id1.1.m1.1.1.5" xref="id1.1.m1.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><apply id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1"><times id="id1.1.m1.1.1.1.cmml" xref="id1.1.m1.1.1.1"></times><cn id="id1.1.m1.1.1.2.cmml" type="integer" xref="id1.1.m1.1.1.2">0</cn><ci id="id1.1.m1.1.1.3.cmml" xref="id1.1.m1.1.1.3">𝜈</ci><ci id="id1.1.m1.1.1.4.cmml" xref="id1.1.m1.1.1.4">𝛽</ci><ci id="id1.1.m1.1.1.5.cmml" xref="id1.1.m1.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">0 italic_ν italic_β italic_β</annotation></semantics></math>) experiments. A novel analysis framework is presented to calculate the background index using the specific activities, masses and simulated efficiencies of an experiment’s components as distributions. This Bayesian framework includes a unified approach to combine specific activities from assay. Monte Carlo uncertainty propagation is used to build a background index distribution from the specific activity, mass and efficiency distributions. This analysis method is applied to the Majorana Demonstrator, which deployed arrays of high-purity Ge detectors enriched in 76Ge to search for <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml"><mn id="id3.3.m3.1.1.2" xref="id3.3.m3.1.1.2.cmml">0</mn><mo id="id3.3.m3.1.1.1" xref="id3.3.m3.1.1.1.cmml">⁢</mo><mi id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml">ν</mi><mo id="id3.3.m3.1.1.1a" xref="id3.3.m3.1.1.1.cmml">⁢</mo><mi id="id3.3.m3.1.1.4" xref="id3.3.m3.1.1.4.cmml">β</mi><mo id="id3.3.m3.1.1.1b" xref="id3.3.m3.1.1.1.cmml">⁢</mo><mi id="id3.3.m3.1.1.5" xref="id3.3.m3.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1"><times id="id3.3.m3.1.1.1.cmml" xref="id3.3.m3.1.1.1"></times><cn id="id3.3.m3.1.1.2.cmml" type="integer" xref="id3.3.m3.1.1.2">0</cn><ci id="id3.3.m3.1.1.3.cmml" xref="id3.3.m3.1.1.3">𝜈</ci><ci id="id3.3.m3.1.1.4.cmml" xref="id3.3.m3.1.1.4">𝛽</ci><ci id="id3.3.m3.1.1.5.cmml" xref="id3.3.m3.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">0 italic_ν italic_β italic_β</annotation></semantics></math>. The framework projects a mean background index of <math alttext="\left[8.95\pm 0.36\right]\times 10^{-4}~{}\text{cts/(keV\,kg\,yr)}" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mrow id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml"><mrow id="id4.4.m4.1.1.1" xref="id4.4.m4.1.1.1.cmml"><mrow id="id4.4.m4.1.1.1.1.1" xref="id4.4.m4.1.1.1.1.2.cmml"><mo id="id4.4.m4.1.1.1.1.1.2" xref="id4.4.m4.1.1.1.1.2.1.cmml">[</mo><mrow id="id4.4.m4.1.1.1.1.1.1" xref="id4.4.m4.1.1.1.1.1.1.cmml"><mn id="id4.4.m4.1.1.1.1.1.1.2" xref="id4.4.m4.1.1.1.1.1.1.2.cmml">8.95</mn><mo id="id4.4.m4.1.1.1.1.1.1.1" xref="id4.4.m4.1.1.1.1.1.1.1.cmml">±</mo><mn id="id4.4.m4.1.1.1.1.1.1.3" xref="id4.4.m4.1.1.1.1.1.1.3.cmml">0.36</mn></mrow><mo id="id4.4.m4.1.1.1.1.1.3" rspace="0.055em" xref="id4.4.m4.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="id4.4.m4.1.1.1.2" rspace="0.222em" xref="id4.4.m4.1.1.1.2.cmml">×</mo><msup id="id4.4.m4.1.1.1.3" xref="id4.4.m4.1.1.1.3.cmml"><mn id="id4.4.m4.1.1.1.3.2" xref="id4.4.m4.1.1.1.3.2.cmml">10</mn><mrow id="id4.4.m4.1.1.1.3.3" xref="id4.4.m4.1.1.1.3.3.cmml"><mo id="id4.4.m4.1.1.1.3.3a" xref="id4.4.m4.1.1.1.3.3.cmml">−</mo><mn id="id4.4.m4.1.1.1.3.3.2" xref="id4.4.m4.1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><mo id="id4.4.m4.1.1.2" xref="id4.4.m4.1.1.2.cmml">⁢</mo><mtext id="id4.4.m4.1.1.3" xref="id4.4.m4.1.1.3a.cmml">cts/(keV kg yr)</mtext></mrow><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><apply id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1"><times id="id4.4.m4.1.1.2.cmml" xref="id4.4.m4.1.1.2"></times><apply id="id4.4.m4.1.1.1.cmml" xref="id4.4.m4.1.1.1"><times id="id4.4.m4.1.1.1.2.cmml" xref="id4.4.m4.1.1.1.2"></times><apply id="id4.4.m4.1.1.1.1.2.cmml" xref="id4.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="id4.4.m4.1.1.1.1.2.1.cmml" xref="id4.4.m4.1.1.1.1.1.2">delimited-[]</csymbol><apply id="id4.4.m4.1.1.1.1.1.1.cmml" xref="id4.4.m4.1.1.1.1.1.1"><csymbol cd="latexml" id="id4.4.m4.1.1.1.1.1.1.1.cmml" xref="id4.4.m4.1.1.1.1.1.1.1">plus-or-minus</csymbol><cn id="id4.4.m4.1.1.1.1.1.1.2.cmml" type="float" xref="id4.4.m4.1.1.1.1.1.1.2">8.95</cn><cn id="id4.4.m4.1.1.1.1.1.1.3.cmml" type="float" xref="id4.4.m4.1.1.1.1.1.1.3">0.36</cn></apply></apply><apply id="id4.4.m4.1.1.1.3.cmml" xref="id4.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="id4.4.m4.1.1.1.3.1.cmml" xref="id4.4.m4.1.1.1.3">superscript</csymbol><cn id="id4.4.m4.1.1.1.3.2.cmml" type="integer" xref="id4.4.m4.1.1.1.3.2">10</cn><apply id="id4.4.m4.1.1.1.3.3.cmml" xref="id4.4.m4.1.1.1.3.3"><minus id="id4.4.m4.1.1.1.3.3.1.cmml" xref="id4.4.m4.1.1.1.3.3"></minus><cn id="id4.4.m4.1.1.1.3.3.2.cmml" type="integer" xref="id4.4.m4.1.1.1.3.3.2">4</cn></apply></apply></apply><ci id="id4.4.m4.1.1.3a.cmml" xref="id4.4.m4.1.1.3"><mtext id="id4.4.m4.1.1.3.cmml" xref="id4.4.m4.1.1.3">cts/(keV kg yr)</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">\left[8.95\pm 0.36\right]\times 10^{-4}~{}\text{cts/(keV\,kg\,yr)}</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">[ 8.95 ± 0.36 ] × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT cts/(keV kg yr)</annotation></semantics></math> from 232Th and 238U in the Demonstrator’s components. </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">pacs: </h6>23.40-s, 23.40.Bw, 14.60.Pq, 27.50.+j </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> I Introduction</h2> <div class="ltx_para" id="S1.p1"> A variety of low background experiments form the rich experimental program searching for neutrinoless double-beta decay (<math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mn id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">0</mn><mo id="S1.p1.1.m1.1.1.1" xref="S1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml">ν</mi><mo id="S1.p1.1.m1.1.1.1a" xref="S1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p1.1.m1.1.1.4" xref="S1.p1.1.m1.1.1.4.cmml">β</mi><mo id="S1.p1.1.m1.1.1.1b" xref="S1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p1.1.m1.1.1.5" xref="S1.p1.1.m1.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><apply id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1"><times id="S1.p1.1.m1.1.1.1.cmml" xref="S1.p1.1.m1.1.1.1"></times><cn id="S1.p1.1.m1.1.1.2.cmml" type="integer" xref="S1.p1.1.m1.1.1.2">0</cn><ci id="S1.p1.1.m1.1.1.3.cmml" xref="S1.p1.1.m1.1.1.3">𝜈</ci><ci id="S1.p1.1.m1.1.1.4.cmml" xref="S1.p1.1.m1.1.1.4">𝛽</ci><ci id="S1.p1.1.m1.1.1.5.cmml" xref="S1.p1.1.m1.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">0 italic_ν italic_β italic_β</annotation></semantics></math>). While the decay remains unobserved in all candidate isotopes, the half-life has been constrained to be above <math alttext="10^{25}-10^{26}" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mrow id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><msup id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml"><mn id="S1.p1.2.m2.1.1.2.2" xref="S1.p1.2.m2.1.1.2.2.cmml">10</mn><mn id="S1.p1.2.m2.1.1.2.3" xref="S1.p1.2.m2.1.1.2.3.cmml">25</mn></msup><mo id="S1.p1.2.m2.1.1.1" xref="S1.p1.2.m2.1.1.1.cmml">−</mo><msup id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml"><mn id="S1.p1.2.m2.1.1.3.2" xref="S1.p1.2.m2.1.1.3.2.cmml">10</mn><mn id="S1.p1.2.m2.1.1.3.3" xref="S1.p1.2.m2.1.1.3.3.cmml">26</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1"><minus id="S1.p1.2.m2.1.1.1.cmml" xref="S1.p1.2.m2.1.1.1"></minus><apply id="S1.p1.2.m2.1.1.2.cmml" xref="S1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p1.2.m2.1.1.2.1.cmml" xref="S1.p1.2.m2.1.1.2">superscript</csymbol><cn id="S1.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S1.p1.2.m2.1.1.2.2">10</cn><cn id="S1.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S1.p1.2.m2.1.1.2.3">25</cn></apply><apply id="S1.p1.2.m2.1.1.3.cmml" xref="S1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p1.2.m2.1.1.3.1.cmml" xref="S1.p1.2.m2.1.1.3">superscript</csymbol><cn id="S1.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S1.p1.2.m2.1.1.3.2">10</cn><cn id="S1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S1.p1.2.m2.1.1.3.3">26</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">10^{25}-10^{26}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">10 start_POSTSUPERSCRIPT 25 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 26 end_POSTSUPERSCRIPT</annotation></semantics></math> years by recent experiments <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib5" title="">5</a>]</cite>. The next generation of proposed experiments target a half-life sensitivity, <math alttext="T_{1/2}" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><msub id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">T</mi><mrow id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml"><mn id="S1.p1.3.m3.1.1.3.2" xref="S1.p1.3.m3.1.1.3.2.cmml">1</mn><mo id="S1.p1.3.m3.1.1.3.1" xref="S1.p1.3.m3.1.1.3.1.cmml">/</mo><mn id="S1.p1.3.m3.1.1.3.3" xref="S1.p1.3.m3.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1">subscript</csymbol><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">𝑇</ci><apply id="S1.p1.3.m3.1.1.3.cmml" xref="S1.p1.3.m3.1.1.3"><divide id="S1.p1.3.m3.1.1.3.1.cmml" xref="S1.p1.3.m3.1.1.3.1"></divide><cn id="S1.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S1.p1.3.m3.1.1.3.2">1</cn><cn id="S1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S1.p1.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">T_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>, of <math alttext="10^{27}-10^{28}" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mrow id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml"><msup id="S1.p1.4.m4.1.1.2" xref="S1.p1.4.m4.1.1.2.cmml"><mn id="S1.p1.4.m4.1.1.2.2" xref="S1.p1.4.m4.1.1.2.2.cmml">10</mn><mn id="S1.p1.4.m4.1.1.2.3" xref="S1.p1.4.m4.1.1.2.3.cmml">27</mn></msup><mo id="S1.p1.4.m4.1.1.1" xref="S1.p1.4.m4.1.1.1.cmml">−</mo><msup id="S1.p1.4.m4.1.1.3" xref="S1.p1.4.m4.1.1.3.cmml"><mn id="S1.p1.4.m4.1.1.3.2" xref="S1.p1.4.m4.1.1.3.2.cmml">10</mn><mn id="S1.p1.4.m4.1.1.3.3" xref="S1.p1.4.m4.1.1.3.3.cmml">28</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><apply id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1"><minus id="S1.p1.4.m4.1.1.1.cmml" xref="S1.p1.4.m4.1.1.1"></minus><apply id="S1.p1.4.m4.1.1.2.cmml" xref="S1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.p1.4.m4.1.1.2.1.cmml" xref="S1.p1.4.m4.1.1.2">superscript</csymbol><cn id="S1.p1.4.m4.1.1.2.2.cmml" type="integer" xref="S1.p1.4.m4.1.1.2.2">10</cn><cn id="S1.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S1.p1.4.m4.1.1.2.3">27</cn></apply><apply id="S1.p1.4.m4.1.1.3.cmml" xref="S1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.p1.4.m4.1.1.3.1.cmml" xref="S1.p1.4.m4.1.1.3">superscript</csymbol><cn id="S1.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S1.p1.4.m4.1.1.3.2">10</cn><cn id="S1.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S1.p1.4.m4.1.1.3.3">28</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">10^{27}-10^{28}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">10 start_POSTSUPERSCRIPT 27 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 28 end_POSTSUPERSCRIPT</annotation></semantics></math> years <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib8" title="">8</a>]</cite>. To achieve these sensitivities, <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mrow id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml"><mn id="S1.p1.5.m5.1.1.2" xref="S1.p1.5.m5.1.1.2.cmml">0</mn><mo id="S1.p1.5.m5.1.1.1" xref="S1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.p1.5.m5.1.1.3" xref="S1.p1.5.m5.1.1.3.cmml">ν</mi><mo id="S1.p1.5.m5.1.1.1a" xref="S1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.p1.5.m5.1.1.4" xref="S1.p1.5.m5.1.1.4.cmml">β</mi><mo id="S1.p1.5.m5.1.1.1b" xref="S1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.p1.5.m5.1.1.5" xref="S1.p1.5.m5.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><apply id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1"><times id="S1.p1.5.m5.1.1.1.cmml" xref="S1.p1.5.m5.1.1.1"></times><cn id="S1.p1.5.m5.1.1.2.cmml" type="integer" xref="S1.p1.5.m5.1.1.2">0</cn><ci id="S1.p1.5.m5.1.1.3.cmml" xref="S1.p1.5.m5.1.1.3">𝜈</ci><ci id="S1.p1.5.m5.1.1.4.cmml" xref="S1.p1.5.m5.1.1.4">𝛽</ci><ci id="S1.p1.5.m5.1.1.5.cmml" xref="S1.p1.5.m5.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments require underground locations, large isotopic mass and low-background construction materials. Extensive assay screenings are conducted to determine if the experiments structural components meet the targeted background levels. </div> <div class="ltx_para" id="S1.p2"> In terms of the total electron kinetic energy, the experimental signature of <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><mn id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml">0</mn><mo id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">ν</mi><mo id="S1.p2.1.m1.1.1.1a" xref="S1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p2.1.m1.1.1.4" xref="S1.p2.1.m1.1.1.4.cmml">β</mi><mo id="S1.p2.1.m1.1.1.1b" xref="S1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p2.1.m1.1.1.5" xref="S1.p2.1.m1.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><times id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"></times><cn id="S1.p2.1.m1.1.1.2.cmml" type="integer" xref="S1.p2.1.m1.1.1.2">0</cn><ci id="S1.p2.1.m1.1.1.3.cmml" xref="S1.p2.1.m1.1.1.3">𝜈</ci><ci id="S1.p2.1.m1.1.1.4.cmml" xref="S1.p2.1.m1.1.1.4">𝛽</ci><ci id="S1.p2.1.m1.1.1.5.cmml" xref="S1.p2.1.m1.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> is a peak at the Q-value of the decay, <math alttext="Q_{\beta\beta}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><msub id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">Q</mi><mrow id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml"><mi id="S1.p2.2.m2.1.1.3.2" xref="S1.p2.2.m2.1.1.3.2.cmml">β</mi><mo id="S1.p2.2.m2.1.1.3.1" xref="S1.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S1.p2.2.m2.1.1.3.3" xref="S1.p2.2.m2.1.1.3.3.cmml">β</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1">subscript</csymbol><ci id="S1.p2.2.m2.1.1.2.cmml" xref="S1.p2.2.m2.1.1.2">𝑄</ci><apply id="S1.p2.2.m2.1.1.3.cmml" xref="S1.p2.2.m2.1.1.3"><times id="S1.p2.2.m2.1.1.3.1.cmml" xref="S1.p2.2.m2.1.1.3.1"></times><ci id="S1.p2.2.m2.1.1.3.2.cmml" xref="S1.p2.2.m2.1.1.3.2">𝛽</ci><ci id="S1.p2.2.m2.1.1.3.3.cmml" xref="S1.p2.2.m2.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">Q_{\beta\beta}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT</annotation></semantics></math>. If no background is present in this region of interest (ROI), the sensitivity of a <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mn id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">0</mn><mo id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">ν</mi><mo id="S1.p2.3.m3.1.1.1a" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S1.p2.3.m3.1.1.4" xref="S1.p2.3.m3.1.1.4.cmml">β</mi><mo id="S1.p2.3.m3.1.1.1b" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S1.p2.3.m3.1.1.5" xref="S1.p2.3.m3.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><times id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1.1"></times><cn id="S1.p2.3.m3.1.1.2.cmml" type="integer" xref="S1.p2.3.m3.1.1.2">0</cn><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">𝜈</ci><ci id="S1.p2.3.m3.1.1.4.cmml" xref="S1.p2.3.m3.1.1.4">𝛽</ci><ci id="S1.p2.3.m3.1.1.5.cmml" xref="S1.p2.3.m3.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiment scales linearly with the product of its isotopic mass, <math alttext="M" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">M</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_M</annotation></semantics></math>, and exposure time, <math alttext="t" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_t</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib9" title="">9</a>]</cite>. However, if the specific background rate, <math alttext="b" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">b</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_b</annotation></semantics></math>, is large enough (such that the uncertainty on the background level is proportional to <math alttext="\sqrt{b\Delta E}" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><msqrt id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml"><mrow id="S1.p2.7.m7.1.1.2" xref="S1.p2.7.m7.1.1.2.cmml"><mi id="S1.p2.7.m7.1.1.2.2" xref="S1.p2.7.m7.1.1.2.2.cmml">b</mi><mo id="S1.p2.7.m7.1.1.2.1" xref="S1.p2.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S1.p2.7.m7.1.1.2.3" mathvariant="normal" xref="S1.p2.7.m7.1.1.2.3.cmml">Δ</mi><mo id="S1.p2.7.m7.1.1.2.1a" xref="S1.p2.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S1.p2.7.m7.1.1.2.4" xref="S1.p2.7.m7.1.1.2.4.cmml">E</mi></mrow></msqrt><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1"><root id="S1.p2.7.m7.1.1a.cmml" xref="S1.p2.7.m7.1.1"></root><apply id="S1.p2.7.m7.1.1.2.cmml" xref="S1.p2.7.m7.1.1.2"><times id="S1.p2.7.m7.1.1.2.1.cmml" xref="S1.p2.7.m7.1.1.2.1"></times><ci id="S1.p2.7.m7.1.1.2.2.cmml" xref="S1.p2.7.m7.1.1.2.2">𝑏</ci><ci id="S1.p2.7.m7.1.1.2.3.cmml" xref="S1.p2.7.m7.1.1.2.3">Δ</ci><ci id="S1.p2.7.m7.1.1.2.4.cmml" xref="S1.p2.7.m7.1.1.2.4">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.1c">\sqrt{b\Delta E}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">square-root start_ARG italic_b roman_Δ italic_E end_ARG</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib10" title="">10</a>]</cite>) the half-life sensitivity (<math alttext="T_{1/2}" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><msub id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml"><mi id="S1.p2.8.m8.1.1.2" xref="S1.p2.8.m8.1.1.2.cmml">T</mi><mrow id="S1.p2.8.m8.1.1.3" xref="S1.p2.8.m8.1.1.3.cmml"><mn id="S1.p2.8.m8.1.1.3.2" xref="S1.p2.8.m8.1.1.3.2.cmml">1</mn><mo id="S1.p2.8.m8.1.1.3.1" xref="S1.p2.8.m8.1.1.3.1.cmml">/</mo><mn id="S1.p2.8.m8.1.1.3.3" xref="S1.p2.8.m8.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><apply id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S1.p2.8.m8.1.1.1.cmml" xref="S1.p2.8.m8.1.1">subscript</csymbol><ci id="S1.p2.8.m8.1.1.2.cmml" xref="S1.p2.8.m8.1.1.2">𝑇</ci><apply id="S1.p2.8.m8.1.1.3.cmml" xref="S1.p2.8.m8.1.1.3"><divide id="S1.p2.8.m8.1.1.3.1.cmml" 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xref="S1.E1.m1.1.1.1.1.3.2.3.1"></times><ci id="S1.E1.m1.1.1.1.1.3.2.3.2.cmml" xref="S1.E1.m1.1.1.1.1.3.2.3.2">𝑏</ci><ci id="S1.E1.m1.1.1.1.1.3.2.3.3.cmml" xref="S1.E1.m1.1.1.1.1.3.2.3.3">Δ</ci><ci id="S1.E1.m1.1.1.1.1.3.2.3.4.cmml" xref="S1.E1.m1.1.1.1.1.3.2.3.4">𝐸</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.1c">T_{1/2}\propto\sqrt{\frac{Mt}{b\Delta E}}.</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.1d">italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT ∝ square-root start_ARG divide start_ARG italic_M italic_t end_ARG start_ARG italic_b roman_Δ italic_E end_ARG 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">(1)</td> </tr></tbody> </table> The width of the ROI, <math alttext="\Delta E" class="ltx_Math" display="inline" id="S1.p2.9.m1.1"><semantics id="S1.p2.9.m1.1a"><mrow id="S1.p2.9.m1.1.1" xref="S1.p2.9.m1.1.1.cmml"><mi id="S1.p2.9.m1.1.1.2" mathvariant="normal" xref="S1.p2.9.m1.1.1.2.cmml">Δ</mi><mo id="S1.p2.9.m1.1.1.1" xref="S1.p2.9.m1.1.1.1.cmml">⁢</mo><mi id="S1.p2.9.m1.1.1.3" xref="S1.p2.9.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.9.m1.1b"><apply id="S1.p2.9.m1.1.1.cmml" xref="S1.p2.9.m1.1.1"><times id="S1.p2.9.m1.1.1.1.cmml" xref="S1.p2.9.m1.1.1.1"></times><ci id="S1.p2.9.m1.1.1.2.cmml" xref="S1.p2.9.m1.1.1.2">Δ</ci><ci id="S1.p2.9.m1.1.1.3.cmml" xref="S1.p2.9.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m1.1c">\Delta E</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m1.1d">roman_Δ italic_E</annotation></semantics></math>, is related to <math alttext="b" class="ltx_Math" display="inline" id="S1.p2.10.m2.1"><semantics id="S1.p2.10.m2.1a"><mi id="S1.p2.10.m2.1.1" xref="S1.p2.10.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S1.p2.10.m2.1b"><ci id="S1.p2.10.m2.1.1.cmml" xref="S1.p2.10.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m2.1d">italic_b</annotation></semantics></math> and the energy resolution at <math alttext="Q_{\beta\beta}" class="ltx_Math" display="inline" id="S1.p2.11.m3.1"><semantics id="S1.p2.11.m3.1a"><msub id="S1.p2.11.m3.1.1" xref="S1.p2.11.m3.1.1.cmml"><mi id="S1.p2.11.m3.1.1.2" xref="S1.p2.11.m3.1.1.2.cmml">Q</mi><mrow id="S1.p2.11.m3.1.1.3" xref="S1.p2.11.m3.1.1.3.cmml"><mi id="S1.p2.11.m3.1.1.3.2" xref="S1.p2.11.m3.1.1.3.2.cmml">β</mi><mo id="S1.p2.11.m3.1.1.3.1" xref="S1.p2.11.m3.1.1.3.1.cmml">⁢</mo><mi id="S1.p2.11.m3.1.1.3.3" xref="S1.p2.11.m3.1.1.3.3.cmml">β</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p2.11.m3.1b"><apply id="S1.p2.11.m3.1.1.cmml" xref="S1.p2.11.m3.1.1"><csymbol cd="ambiguous" id="S1.p2.11.m3.1.1.1.cmml" xref="S1.p2.11.m3.1.1">subscript</csymbol><ci id="S1.p2.11.m3.1.1.2.cmml" xref="S1.p2.11.m3.1.1.2">𝑄</ci><apply id="S1.p2.11.m3.1.1.3.cmml" xref="S1.p2.11.m3.1.1.3"><times id="S1.p2.11.m3.1.1.3.1.cmml" xref="S1.p2.11.m3.1.1.3.1"></times><ci id="S1.p2.11.m3.1.1.3.2.cmml" xref="S1.p2.11.m3.1.1.3.2">𝛽</ci><ci id="S1.p2.11.m3.1.1.3.3.cmml" xref="S1.p2.11.m3.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.11.m3.1c">Q_{\beta\beta}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.11.m3.1d">italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT</annotation></semantics></math>. The specific background rate is measured in counts per keV in the ROI, per kg of detector mass, per year (cts/(keV kg yr)). This observable is also referred to as the background index (BI). Given the low-background nature of <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p2.12.m4.1"><semantics id="S1.p2.12.m4.1a"><mrow id="S1.p2.12.m4.1.1" xref="S1.p2.12.m4.1.1.cmml"><mn id="S1.p2.12.m4.1.1.2" xref="S1.p2.12.m4.1.1.2.cmml">0</mn><mo id="S1.p2.12.m4.1.1.1" xref="S1.p2.12.m4.1.1.1.cmml">⁢</mo><mi id="S1.p2.12.m4.1.1.3" xref="S1.p2.12.m4.1.1.3.cmml">ν</mi><mo id="S1.p2.12.m4.1.1.1a" xref="S1.p2.12.m4.1.1.1.cmml">⁢</mo><mi id="S1.p2.12.m4.1.1.4" xref="S1.p2.12.m4.1.1.4.cmml">β</mi><mo id="S1.p2.12.m4.1.1.1b" xref="S1.p2.12.m4.1.1.1.cmml">⁢</mo><mi id="S1.p2.12.m4.1.1.5" xref="S1.p2.12.m4.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.12.m4.1b"><apply id="S1.p2.12.m4.1.1.cmml" xref="S1.p2.12.m4.1.1"><times id="S1.p2.12.m4.1.1.1.cmml" xref="S1.p2.12.m4.1.1.1"></times><cn id="S1.p2.12.m4.1.1.2.cmml" type="integer" xref="S1.p2.12.m4.1.1.2">0</cn><ci id="S1.p2.12.m4.1.1.3.cmml" xref="S1.p2.12.m4.1.1.3">𝜈</ci><ci id="S1.p2.12.m4.1.1.4.cmml" xref="S1.p2.12.m4.1.1.4">𝛽</ci><ci id="S1.p2.12.m4.1.1.5.cmml" xref="S1.p2.12.m4.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m4.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m4.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments, the number of background counts in the ROI is typically too low to estimate the BI (note that no signal would have to be assumed). In such cases, a wider proxy region is needed to increase statistics. This background estimation window (BEW) is often asymmetric around <math alttext="Q_{\beta\beta}" class="ltx_Math" display="inline" id="S1.p2.13.m5.1"><semantics id="S1.p2.13.m5.1a"><msub id="S1.p2.13.m5.1.1" xref="S1.p2.13.m5.1.1.cmml"><mi id="S1.p2.13.m5.1.1.2" xref="S1.p2.13.m5.1.1.2.cmml">Q</mi><mrow id="S1.p2.13.m5.1.1.3" xref="S1.p2.13.m5.1.1.3.cmml"><mi id="S1.p2.13.m5.1.1.3.2" xref="S1.p2.13.m5.1.1.3.2.cmml">β</mi><mo id="S1.p2.13.m5.1.1.3.1" xref="S1.p2.13.m5.1.1.3.1.cmml">⁢</mo><mi id="S1.p2.13.m5.1.1.3.3" xref="S1.p2.13.m5.1.1.3.3.cmml">β</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p2.13.m5.1b"><apply id="S1.p2.13.m5.1.1.cmml" xref="S1.p2.13.m5.1.1"><csymbol cd="ambiguous" id="S1.p2.13.m5.1.1.1.cmml" xref="S1.p2.13.m5.1.1">subscript</csymbol><ci id="S1.p2.13.m5.1.1.2.cmml" xref="S1.p2.13.m5.1.1.2">𝑄</ci><apply id="S1.p2.13.m5.1.1.3.cmml" xref="S1.p2.13.m5.1.1.3"><times id="S1.p2.13.m5.1.1.3.1.cmml" xref="S1.p2.13.m5.1.1.3.1"></times><ci id="S1.p2.13.m5.1.1.3.2.cmml" xref="S1.p2.13.m5.1.1.3.2">𝛽</ci><ci id="S1.p2.13.m5.1.1.3.3.cmml" xref="S1.p2.13.m5.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.13.m5.1c">Q_{\beta\beta}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.13.m5.1d">italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT</annotation></semantics></math> to avoid running into the <math alttext="2\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p2.14.m6.1"><semantics id="S1.p2.14.m6.1a"><mrow id="S1.p2.14.m6.1.1" xref="S1.p2.14.m6.1.1.cmml"><mn id="S1.p2.14.m6.1.1.2" xref="S1.p2.14.m6.1.1.2.cmml">2</mn><mo id="S1.p2.14.m6.1.1.1" xref="S1.p2.14.m6.1.1.1.cmml">⁢</mo><mi id="S1.p2.14.m6.1.1.3" xref="S1.p2.14.m6.1.1.3.cmml">ν</mi><mo id="S1.p2.14.m6.1.1.1a" xref="S1.p2.14.m6.1.1.1.cmml">⁢</mo><mi id="S1.p2.14.m6.1.1.4" xref="S1.p2.14.m6.1.1.4.cmml">β</mi><mo id="S1.p2.14.m6.1.1.1b" xref="S1.p2.14.m6.1.1.1.cmml">⁢</mo><mi id="S1.p2.14.m6.1.1.5" xref="S1.p2.14.m6.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.14.m6.1b"><apply id="S1.p2.14.m6.1.1.cmml" xref="S1.p2.14.m6.1.1"><times id="S1.p2.14.m6.1.1.1.cmml" xref="S1.p2.14.m6.1.1.1"></times><cn id="S1.p2.14.m6.1.1.2.cmml" type="integer" xref="S1.p2.14.m6.1.1.2">2</cn><ci id="S1.p2.14.m6.1.1.3.cmml" xref="S1.p2.14.m6.1.1.3">𝜈</ci><ci id="S1.p2.14.m6.1.1.4.cmml" xref="S1.p2.14.m6.1.1.4">𝛽</ci><ci id="S1.p2.14.m6.1.1.5.cmml" xref="S1.p2.14.m6.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.14.m6.1c">2\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p2.14.m6.1d">2 italic_ν italic_β italic_β</annotation></semantics></math> spectrum and known gamma lines. </div> <div class="ltx_para" id="S1.p3"> Germanium detector technology has been developed for decades, finding applications in radiometric assays and <math alttext="\gamma" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_γ</annotation></semantics></math>-ray spectroscopy. Ge detectors offer superb energy resolution – resulting in a narrower ROI – and can be readily enriched to <math alttext="\sim 90" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mi id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml"></mi><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">∼</mo><mn id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">90</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><csymbol cd="latexml" id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2">absent</csymbol><cn id="S1.p3.2.m2.1.1.3.cmml" type="integer" xref="S1.p3.2.m2.1.1.3">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\sim 90</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">∼ 90</annotation></semantics></math>% 76Ge <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib11" title="">11</a>]</cite>. The Majorana Demonstrator and GERDA experiments exploited this technology to search for <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mrow id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mn id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml">0</mn><mo id="S1.p3.4.m4.1.1.1" xref="S1.p3.4.m4.1.1.1.cmml">⁢</mo><mi id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml">ν</mi><mo id="S1.p3.4.m4.1.1.1a" xref="S1.p3.4.m4.1.1.1.cmml">⁢</mo><mi id="S1.p3.4.m4.1.1.4" xref="S1.p3.4.m4.1.1.4.cmml">β</mi><mo id="S1.p3.4.m4.1.1.1b" xref="S1.p3.4.m4.1.1.1.cmml">⁢</mo><mi id="S1.p3.4.m4.1.1.5" xref="S1.p3.4.m4.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><times id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1.1"></times><cn id="S1.p3.4.m4.1.1.2.cmml" type="integer" xref="S1.p3.4.m4.1.1.2">0</cn><ci id="S1.p3.4.m4.1.1.3.cmml" xref="S1.p3.4.m4.1.1.3">𝜈</ci><ci id="S1.p3.4.m4.1.1.4.cmml" xref="S1.p3.4.m4.1.1.4">𝛽</ci><ci id="S1.p3.4.m4.1.1.5.cmml" xref="S1.p3.4.m4.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> in 76Ge-enriched high purity Ge (HPGe) detectors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib12" title="">12</a>]</cite>. Both experiments have the lowest backgrounds of any experiment in the present <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mrow id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml"><mn id="S1.p3.6.m6.1.1.2" xref="S1.p3.6.m6.1.1.2.cmml">0</mn><mo id="S1.p3.6.m6.1.1.1" xref="S1.p3.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p3.6.m6.1.1.3" xref="S1.p3.6.m6.1.1.3.cmml">ν</mi><mo id="S1.p3.6.m6.1.1.1a" xref="S1.p3.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p3.6.m6.1.1.4" xref="S1.p3.6.m6.1.1.4.cmml">β</mi><mo id="S1.p3.6.m6.1.1.1b" xref="S1.p3.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p3.6.m6.1.1.5" xref="S1.p3.6.m6.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><apply id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1"><times id="S1.p3.6.m6.1.1.1.cmml" xref="S1.p3.6.m6.1.1.1"></times><cn id="S1.p3.6.m6.1.1.2.cmml" type="integer" xref="S1.p3.6.m6.1.1.2">0</cn><ci id="S1.p3.6.m6.1.1.3.cmml" xref="S1.p3.6.m6.1.1.3">𝜈</ci><ci id="S1.p3.6.m6.1.1.4.cmml" xref="S1.p3.6.m6.1.1.4">𝛽</ci><ci id="S1.p3.6.m6.1.1.5.cmml" xref="S1.p3.6.m6.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experimental landscape <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib1" title="">1</a>]</cite>. </div> <div class="ltx_para" id="S1.p4"> At the 2039 keV Q-value of 76Ge <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib14" title="">14</a>]</cite>, the Demonstrator achieved a BI of <math alttext="6.2^{+0.6}_{-0.5}\times 10^{-3}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mrow id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><msubsup id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml"><mn id="S1.p4.2.m2.1.1.2.2.2" xref="S1.p4.2.m2.1.1.2.2.2.cmml">6.2</mn><mrow id="S1.p4.2.m2.1.1.2.3" xref="S1.p4.2.m2.1.1.2.3.cmml"><mo id="S1.p4.2.m2.1.1.2.3a" xref="S1.p4.2.m2.1.1.2.3.cmml">−</mo><mn id="S1.p4.2.m2.1.1.2.3.2" xref="S1.p4.2.m2.1.1.2.3.2.cmml">0.5</mn></mrow><mrow id="S1.p4.2.m2.1.1.2.2.3" xref="S1.p4.2.m2.1.1.2.2.3.cmml"><mo id="S1.p4.2.m2.1.1.2.2.3a" xref="S1.p4.2.m2.1.1.2.2.3.cmml">+</mo><mn id="S1.p4.2.m2.1.1.2.2.3.2" xref="S1.p4.2.m2.1.1.2.2.3.2.cmml">0.6</mn></mrow></msubsup><mo id="S1.p4.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.p4.2.m2.1.1.1.cmml">×</mo><msup id="S1.p4.2.m2.1.1.3" xref="S1.p4.2.m2.1.1.3.cmml"><mn id="S1.p4.2.m2.1.1.3.2" xref="S1.p4.2.m2.1.1.3.2.cmml">10</mn><mrow id="S1.p4.2.m2.1.1.3.3" xref="S1.p4.2.m2.1.1.3.3.cmml"><mo id="S1.p4.2.m2.1.1.3.3a" xref="S1.p4.2.m2.1.1.3.3.cmml">−</mo><mn id="S1.p4.2.m2.1.1.3.3.2" xref="S1.p4.2.m2.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><times id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1.1"></times><apply id="S1.p4.2.m2.1.1.2.cmml" xref="S1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.2.1.cmml" xref="S1.p4.2.m2.1.1.2">subscript</csymbol><apply id="S1.p4.2.m2.1.1.2.2.cmml" xref="S1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.2.2.1.cmml" xref="S1.p4.2.m2.1.1.2">superscript</csymbol><cn id="S1.p4.2.m2.1.1.2.2.2.cmml" type="float" xref="S1.p4.2.m2.1.1.2.2.2">6.2</cn><apply id="S1.p4.2.m2.1.1.2.2.3.cmml" xref="S1.p4.2.m2.1.1.2.2.3"><plus id="S1.p4.2.m2.1.1.2.2.3.1.cmml" xref="S1.p4.2.m2.1.1.2.2.3"></plus><cn id="S1.p4.2.m2.1.1.2.2.3.2.cmml" type="float" xref="S1.p4.2.m2.1.1.2.2.3.2">0.6</cn></apply></apply><apply id="S1.p4.2.m2.1.1.2.3.cmml" xref="S1.p4.2.m2.1.1.2.3"><minus id="S1.p4.2.m2.1.1.2.3.1.cmml" xref="S1.p4.2.m2.1.1.2.3"></minus><cn id="S1.p4.2.m2.1.1.2.3.2.cmml" type="float" xref="S1.p4.2.m2.1.1.2.3.2">0.5</cn></apply></apply><apply id="S1.p4.2.m2.1.1.3.cmml" xref="S1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.3.1.cmml" xref="S1.p4.2.m2.1.1.3">superscript</csymbol><cn id="S1.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.3.2">10</cn><apply id="S1.p4.2.m2.1.1.3.3.cmml" xref="S1.p4.2.m2.1.1.3.3"><minus id="S1.p4.2.m2.1.1.3.3.1.cmml" xref="S1.p4.2.m2.1.1.3.3"></minus><cn id="S1.p4.2.m2.1.1.3.3.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.3.3.2">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">6.2^{+0.6}_{-0.5}\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">6.2 start_POSTSUPERSCRIPT + 0.6 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.5 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> cts/(keV kg yr) in its low background configuration <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib1" title="">1</a>]</cite>. This is not in agreement with the originally projected BI of <math alttext="<8.75\times 10^{-4}" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mrow id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml"><mi id="S1.p4.3.m3.1.1.2" xref="S1.p4.3.m3.1.1.2.cmml"></mi><mo id="S1.p4.3.m3.1.1.1" xref="S1.p4.3.m3.1.1.1.cmml"><</mo><mrow id="S1.p4.3.m3.1.1.3" xref="S1.p4.3.m3.1.1.3.cmml"><mn id="S1.p4.3.m3.1.1.3.2" xref="S1.p4.3.m3.1.1.3.2.cmml">8.75</mn><mo id="S1.p4.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p4.3.m3.1.1.3.1.cmml">×</mo><msup id="S1.p4.3.m3.1.1.3.3" xref="S1.p4.3.m3.1.1.3.3.cmml"><mn id="S1.p4.3.m3.1.1.3.3.2" xref="S1.p4.3.m3.1.1.3.3.2.cmml">10</mn><mrow id="S1.p4.3.m3.1.1.3.3.3" xref="S1.p4.3.m3.1.1.3.3.3.cmml"><mo id="S1.p4.3.m3.1.1.3.3.3a" xref="S1.p4.3.m3.1.1.3.3.3.cmml">−</mo><mn id="S1.p4.3.m3.1.1.3.3.3.2" xref="S1.p4.3.m3.1.1.3.3.3.2.cmml">4</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><apply id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1"><lt id="S1.p4.3.m3.1.1.1.cmml" xref="S1.p4.3.m3.1.1.1"></lt><csymbol cd="latexml" id="S1.p4.3.m3.1.1.2.cmml" xref="S1.p4.3.m3.1.1.2">absent</csymbol><apply id="S1.p4.3.m3.1.1.3.cmml" xref="S1.p4.3.m3.1.1.3"><times id="S1.p4.3.m3.1.1.3.1.cmml" xref="S1.p4.3.m3.1.1.3.1"></times><cn id="S1.p4.3.m3.1.1.3.2.cmml" type="float" xref="S1.p4.3.m3.1.1.3.2">8.75</cn><apply id="S1.p4.3.m3.1.1.3.3.cmml" xref="S1.p4.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S1.p4.3.m3.1.1.3.3.1.cmml" xref="S1.p4.3.m3.1.1.3.3">superscript</csymbol><cn id="S1.p4.3.m3.1.1.3.3.2.cmml" type="integer" xref="S1.p4.3.m3.1.1.3.3.2">10</cn><apply id="S1.p4.3.m3.1.1.3.3.3.cmml" xref="S1.p4.3.m3.1.1.3.3.3"><minus id="S1.p4.3.m3.1.1.3.3.3.1.cmml" xref="S1.p4.3.m3.1.1.3.3.3"></minus><cn id="S1.p4.3.m3.1.1.3.3.3.2.cmml" type="integer" xref="S1.p4.3.m3.1.1.3.3.3.2">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c"><8.75\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">< 8.75 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> cts/(keV kg yr) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite>. The projection was based on simulations of the design geometry and the results of an extensive radioassay program. To account for any discrepancies between the design and as-built geometries of the Demonstrator, a new set of simulations was performed with the updated geometries. This as-built model did not find a significant deviation from the originally predicted BI. Nevertheless, neither calculation captured the often significant uncertainties from assay in the contribution of natural radiation (<math alttext="{}^{\text{assay}}" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><msup id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml"><mi id="S1.p4.4.m4.1.1a" xref="S1.p4.4.m4.1.1.cmml"></mi><mtext id="S1.p4.4.m4.1.1.1" xref="S1.p4.4.m4.1.1.1a.cmml">assay</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><apply id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1"><ci id="S1.p4.4.m4.1.1.1a.cmml" xref="S1.p4.4.m4.1.1.1"><mtext id="S1.p4.4.m4.1.1.1.cmml" mathsize="70%" xref="S1.p4.4.m4.1.1.1">assay</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">{}^{\text{assay}}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">start_FLOATSUPERSCRIPT assay end_FLOATSUPERSCRIPT</annotation></semantics></math>BI) to the total BI. Additionally, a systematic review of assay results highlighted the prevalence of measured values that did not agree within error, motivating the development of a technique which properly accounts for this spread and propagates it into the BI. </div> <div class="ltx_para" id="S1.p5"> The total BI includes subdominant contributions from, amongst others, external and cosmogenically induced backgrounds. For this work, only backgrounds quantified by assay – 232Th, 238U – are considered. While the components in the Demonstrator are also screened for 40K, it is excluded from the analysis since the <math alttext="\gamma" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mi id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><ci id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">italic_γ</annotation></semantics></math>-rays produced in the 40K decay chain have energies well below the BEW. </div> <div class="ltx_para" id="S1.p6"> The <math alttext="{}^{\text{assay}}" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><msup id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml"><mi id="S1.p6.1.m1.1.1a" xref="S1.p6.1.m1.1.1.cmml"></mi><mtext id="S1.p6.1.m1.1.1.1" xref="S1.p6.1.m1.1.1.1a.cmml">assay</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1"><ci id="S1.p6.1.m1.1.1.1a.cmml" xref="S1.p6.1.m1.1.1.1"><mtext id="S1.p6.1.m1.1.1.1.cmml" mathsize="70%" xref="S1.p6.1.m1.1.1.1">assay</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">{}^{\text{assay}}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">start_FLOATSUPERSCRIPT assay end_FLOATSUPERSCRIPT</annotation></semantics></math>BI is calculated by weighting the activity of a component with the Demonstrator’s simulated array-detection efficiency of decays originating from it. An upper limit on the activity of a component translates into an upper limit on its BI. In the previous projections, such upper limits were directly summed to BIs calculated from measured activities when summing over all modeled components of the Demonstrator. Thus, the total projected BI was reported as an upper limit itself. Null efficiencies were computed for some components far away from the detector array and failed to contribute to the BI. However, given the high activity of some of these components, their true contribution could still be significant. Therefore, simulation statistics must be properly taken into account. </div> <div class="ltx_para" id="S1.p7"> Other <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><mrow id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mn id="S1.p7.1.m1.1.1.2" xref="S1.p7.1.m1.1.1.2.cmml">0</mn><mo id="S1.p7.1.m1.1.1.1" xref="S1.p7.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">ν</mi><mo id="S1.p7.1.m1.1.1.1a" xref="S1.p7.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p7.1.m1.1.1.4" xref="S1.p7.1.m1.1.1.4.cmml">β</mi><mo id="S1.p7.1.m1.1.1.1b" xref="S1.p7.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p7.1.m1.1.1.5" xref="S1.p7.1.m1.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><times id="S1.p7.1.m1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1"></times><cn id="S1.p7.1.m1.1.1.2.cmml" type="integer" xref="S1.p7.1.m1.1.1.2">0</cn><ci id="S1.p7.1.m1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.3">𝜈</ci><ci id="S1.p7.1.m1.1.1.4.cmml" xref="S1.p7.1.m1.1.1.4">𝛽</ci><ci id="S1.p7.1.m1.1.1.5.cmml" xref="S1.p7.1.m1.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments use techniques which address these issues in part. By repeatedly drawing samples from an activity probability density function (PDF) and weighting them by the simulated efficiency, a BI distribution is generated. The CUORE Collaboration generates activity PDFs from fits to preliminary data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib16" title="">16</a>]</cite>. On the other hand, the nEXO Collaboration promotes assay-based activities to a truncated-at-zero Gaussian PDF <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib17" title="">17</a>]</cite>. The latter technique is adopted by this work and expanded on by promoting the single-valued efficiency to a distribution as well. Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib18" title="">18</a>]</cite> provides a comprehensive summary of the methods used to estimate the BI of various <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S1.p7.2.m2.1"><semantics id="S1.p7.2.m2.1a"><mrow id="S1.p7.2.m2.1.1" xref="S1.p7.2.m2.1.1.cmml"><mn id="S1.p7.2.m2.1.1.2" xref="S1.p7.2.m2.1.1.2.cmml">0</mn><mo id="S1.p7.2.m2.1.1.1" xref="S1.p7.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p7.2.m2.1.1.3" xref="S1.p7.2.m2.1.1.3.cmml">ν</mi><mo id="S1.p7.2.m2.1.1.1a" xref="S1.p7.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p7.2.m2.1.1.4" xref="S1.p7.2.m2.1.1.4.cmml">β</mi><mo id="S1.p7.2.m2.1.1.1b" xref="S1.p7.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p7.2.m2.1.1.5" xref="S1.p7.2.m2.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.2.m2.1b"><apply id="S1.p7.2.m2.1.1.cmml" xref="S1.p7.2.m2.1.1"><times id="S1.p7.2.m2.1.1.1.cmml" xref="S1.p7.2.m2.1.1.1"></times><cn id="S1.p7.2.m2.1.1.2.cmml" type="integer" xref="S1.p7.2.m2.1.1.2">0</cn><ci id="S1.p7.2.m2.1.1.3.cmml" xref="S1.p7.2.m2.1.1.3">𝜈</ci><ci id="S1.p7.2.m2.1.1.4.cmml" xref="S1.p7.2.m2.1.1.4">𝛽</ci><ci id="S1.p7.2.m2.1.1.5.cmml" xref="S1.p7.2.m2.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.2.m2.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S1.p7.2.m2.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments. </div> <div class="ltx_para" id="S1.p8"> In this article, an assay-based Bayesian framework to project the BI of low-background experiments is presented and applied to the Majorana Demonstrator. The framework takes as input all the assay and simulation efficiency data with the goal of: <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> 1. <div class="ltx_para" id="S1.I1.i1.p1"> Combining multiple assay measurements, including upper limits, using a unified averaging method which properly accounts for the spread in data. </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> 2. <div class="ltx_para" id="S1.I1.i2.p1"> Calculating uncertainties in non-Gaussian regimes such as those posed by null simulation efficiencies. </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> 3. <div class="ltx_para" id="S1.I1.i3.p1"> Combining uncertainties in assay results, component masses and simulation statistics. </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> 4. <div class="ltx_para" id="S1.I1.i4.p1"> Preserving the generality of this method, allowing for its adoption by the low-background community. </div> </li> </ol> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> II The Majorana Demonstrator </h2> <div class="ltx_para" id="S2.p1"> The Majorana Demonstrator consisted of two modules (M1, M2) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib19" title="">19</a>]</cite> where a total of 40.4 kg of HPGe detectors (27.2 kg enriched to 88% in 76Ge) operated in vacuum at the 4850-foot-level (4300 meter water equivalent) of the Sanford Underground Research Facility (SURF) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib20" title="">20</a>]</cite> in Lead, South Dakota. The ultra-low background and world-leading energy resolution achieved by the Demonstrator enabled a sensitive <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mrow id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mn id="S2.p1.2.m2.1.1.2" xref="S2.p1.2.m2.1.1.2.cmml">0</mn><mo id="S2.p1.2.m2.1.1.1" xref="S2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S2.p1.2.m2.1.1.3" xref="S2.p1.2.m2.1.1.3.cmml">ν</mi><mo id="S2.p1.2.m2.1.1.1a" xref="S2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S2.p1.2.m2.1.1.4" xref="S2.p1.2.m2.1.1.4.cmml">β</mi><mo id="S2.p1.2.m2.1.1.1b" xref="S2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S2.p1.2.m2.1.1.5" xref="S2.p1.2.m2.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><apply id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1"><times id="S2.p1.2.m2.1.1.1.cmml" xref="S2.p1.2.m2.1.1.1"></times><cn id="S2.p1.2.m2.1.1.2.cmml" type="integer" xref="S2.p1.2.m2.1.1.2">0</cn><ci id="S2.p1.2.m2.1.1.3.cmml" xref="S2.p1.2.m2.1.1.3">𝜈</ci><ci id="S2.p1.2.m2.1.1.4.cmml" xref="S2.p1.2.m2.1.1.4">𝛽</ci><ci id="S2.p1.2.m2.1.1.5.cmml" xref="S2.p1.2.m2.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> decay search, as well as additional searches for physics beyond the Standard Model. </div> <div class="ltx_para" id="S2.p2"> Both vacuum cryostats and all structural components of the detector arrays were machined from ultra-low-background underground electroformed copper (UGEFCu) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite> and low-background plastics, such as DuPontTM Vespel® and polytetrafluoroethylene (PTFE). Low-radioactivity parylene was used to coat UGEFCu threads to prevent galling and for the cryostat seal <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib19" title="">19</a>]</cite>. A layered shield enclosed both modules. The innermost layer consisted of 5 cm of UGEFCu. Five cm of commercial oxygen-free high conductivity copper (OFHCCu) and 45 cm of high-purity lead followed. The shield and module volume were constantly purged with low-radon liquid nitrogen boil-off gas. The aluminum enclosure that isolated this Rn-excluded region was covered with a plastic active muon veto which provided near-<math alttext="4\pi" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mn id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml">4</mn><mo id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><times id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1.1"></times><cn id="S2.p2.1.m1.1.1.2.cmml" type="integer" xref="S2.p2.1.m1.1.1.2">4</cn><ci id="S2.p2.1.m1.1.1.3.cmml" xref="S2.p2.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">4\pi</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">4 italic_π</annotation></semantics></math> coverage <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib22" title="">22</a>]</cite>. The near-detector readout system, which was designed for the Demonstrator, included low-mass front end (LMFE) electronics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib23" title="">23</a>]</cite> and low-mass cables and connectors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib24" title="">24</a>]</cite>. Cables were guided out of each module following a UGEFCu cross-arm which penetrated the layered shield. The cross-arm connected the cryostat with vacuum and cryogenic hardware. Control and readout electronics were just outside the Rn-excluded region. The entire assembly was surrounded by 5 cm of borated polyethylene and 25 cm of pure polyethylene to shield against neutrons. </div> <div class="ltx_para" id="S2.p3"> All components inside the Rn-excluded region populate the as-built model. These were modeled in MaGe, the Geant4-based simulation software jointly developed by the GERDA and Majorana collaborations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib25" title="">25</a>]</cite>. Due to possible radioactive shine through the cross-arm, the vacuum hardware was included as well. The as-built model is based on the Aug. 2016 to Nov. 2019 configuration of the Demonstrator, where up to 32.1 kg of detectors were operational. In Nov. 2019, M2 was upgraded with an improved set of cables and connectors and additional cross-arm shielding. The upgrade, in combination with a reconfiguration of M2 detectors, resulted in the final configuration of up to 40.4 kg of operational HPGe detectors. The final active enriched exposure of the Demonstrator was 64.5 kg yr. A low-background dataset – with an active exposure of 63.3 kg yr – was obtained by excluding data taken previous to the installation of the inner UGEFCu shield. From this dataset, a BI of <math alttext="6.2^{+0.6}_{-0.5}\times 10^{-3}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><msubsup id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml"><mn id="S2.p3.1.m1.1.1.2.2.2" xref="S2.p3.1.m1.1.1.2.2.2.cmml">6.2</mn><mrow id="S2.p3.1.m1.1.1.2.3" xref="S2.p3.1.m1.1.1.2.3.cmml"><mo id="S2.p3.1.m1.1.1.2.3a" xref="S2.p3.1.m1.1.1.2.3.cmml">−</mo><mn id="S2.p3.1.m1.1.1.2.3.2" xref="S2.p3.1.m1.1.1.2.3.2.cmml">0.5</mn></mrow><mrow id="S2.p3.1.m1.1.1.2.2.3" xref="S2.p3.1.m1.1.1.2.2.3.cmml"><mo id="S2.p3.1.m1.1.1.2.2.3a" xref="S2.p3.1.m1.1.1.2.2.3.cmml">+</mo><mn id="S2.p3.1.m1.1.1.2.2.3.2" xref="S2.p3.1.m1.1.1.2.2.3.2.cmml">0.6</mn></mrow></msubsup><mo id="S2.p3.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.p3.1.m1.1.1.1.cmml">×</mo><msup id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml"><mn id="S2.p3.1.m1.1.1.3.2" xref="S2.p3.1.m1.1.1.3.2.cmml">10</mn><mrow id="S2.p3.1.m1.1.1.3.3" xref="S2.p3.1.m1.1.1.3.3.cmml"><mo id="S2.p3.1.m1.1.1.3.3a" xref="S2.p3.1.m1.1.1.3.3.cmml">−</mo><mn id="S2.p3.1.m1.1.1.3.3.2" xref="S2.p3.1.m1.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><times id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1"></times><apply id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.2.1.cmml" xref="S2.p3.1.m1.1.1.2">subscript</csymbol><apply id="S2.p3.1.m1.1.1.2.2.cmml" xref="S2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.2.2.1.cmml" xref="S2.p3.1.m1.1.1.2">superscript</csymbol><cn id="S2.p3.1.m1.1.1.2.2.2.cmml" type="float" xref="S2.p3.1.m1.1.1.2.2.2">6.2</cn><apply id="S2.p3.1.m1.1.1.2.2.3.cmml" xref="S2.p3.1.m1.1.1.2.2.3"><plus id="S2.p3.1.m1.1.1.2.2.3.1.cmml" xref="S2.p3.1.m1.1.1.2.2.3"></plus><cn id="S2.p3.1.m1.1.1.2.2.3.2.cmml" type="float" xref="S2.p3.1.m1.1.1.2.2.3.2">0.6</cn></apply></apply><apply id="S2.p3.1.m1.1.1.2.3.cmml" xref="S2.p3.1.m1.1.1.2.3"><minus id="S2.p3.1.m1.1.1.2.3.1.cmml" xref="S2.p3.1.m1.1.1.2.3"></minus><cn id="S2.p3.1.m1.1.1.2.3.2.cmml" type="float" xref="S2.p3.1.m1.1.1.2.3.2">0.5</cn></apply></apply><apply id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.3.1.cmml" xref="S2.p3.1.m1.1.1.3">superscript</csymbol><cn id="S2.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S2.p3.1.m1.1.1.3.2">10</cn><apply id="S2.p3.1.m1.1.1.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3"><minus id="S2.p3.1.m1.1.1.3.3.1.cmml" xref="S2.p3.1.m1.1.1.3.3"></minus><cn id="S2.p3.1.m1.1.1.3.3.2.cmml" type="integer" xref="S2.p3.1.m1.1.1.3.3.2">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">6.2^{+0.6}_{-0.5}\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">6.2 start_POSTSUPERSCRIPT + 0.6 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.5 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> cts/(keV kg yr) was calculated <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib1" title="">1</a>]</cite>. The previous data release of the Demonstrator does not include post-M2-upgrade data. The low-background dataset BI in this release is <math alttext="4.7\pm 0.8\times 10^{-3}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mrow id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mn id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">4.7</mn><mo id="S2.p3.2.m2.1.1.1" xref="S2.p3.2.m2.1.1.1.cmml">±</mo><mrow id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml"><mn id="S2.p3.2.m2.1.1.3.2" xref="S2.p3.2.m2.1.1.3.2.cmml">0.8</mn><mo id="S2.p3.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.p3.2.m2.1.1.3.1.cmml">×</mo><msup id="S2.p3.2.m2.1.1.3.3" xref="S2.p3.2.m2.1.1.3.3.cmml"><mn id="S2.p3.2.m2.1.1.3.3.2" xref="S2.p3.2.m2.1.1.3.3.2.cmml">10</mn><mrow id="S2.p3.2.m2.1.1.3.3.3" xref="S2.p3.2.m2.1.1.3.3.3.cmml"><mo id="S2.p3.2.m2.1.1.3.3.3a" xref="S2.p3.2.m2.1.1.3.3.3.cmml">−</mo><mn id="S2.p3.2.m2.1.1.3.3.3.2" xref="S2.p3.2.m2.1.1.3.3.3.2.cmml">3</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="latexml" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1.1">plus-or-minus</csymbol><cn id="S2.p3.2.m2.1.1.2.cmml" type="float" xref="S2.p3.2.m2.1.1.2">4.7</cn><apply id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3"><times id="S2.p3.2.m2.1.1.3.1.cmml" xref="S2.p3.2.m2.1.1.3.1"></times><cn id="S2.p3.2.m2.1.1.3.2.cmml" type="float" xref="S2.p3.2.m2.1.1.3.2">0.8</cn><apply id="S2.p3.2.m2.1.1.3.3.cmml" xref="S2.p3.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.3.3.1.cmml" xref="S2.p3.2.m2.1.1.3.3">superscript</csymbol><cn id="S2.p3.2.m2.1.1.3.3.2.cmml" type="integer" xref="S2.p3.2.m2.1.1.3.3.2">10</cn><apply id="S2.p3.2.m2.1.1.3.3.3.cmml" xref="S2.p3.2.m2.1.1.3.3.3"><minus id="S2.p3.2.m2.1.1.3.3.3.1.cmml" xref="S2.p3.2.m2.1.1.3.3.3"></minus><cn id="S2.p3.2.m2.1.1.3.3.3.2.cmml" type="integer" xref="S2.p3.2.m2.1.1.3.3.3.2">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">4.7\pm 0.8\times 10^{-3}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">4.7 ± 0.8 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> cts/(keV kg yr) with 21.3 kg yr of active exposure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib26" title="">26</a>]</cite>. </div> <div class="ltx_para" id="S2.p4"> To simulate 232Th and 238U decays originating from the hardware components of the Demonstrator, their respective decay chains were divided into 10 and 4 segments respectively, following the prescription in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib27" title="">27</a>]</cite>. Within each segment secular equilibrium was assumed. The breakpoints in the chains – which generally correspond to isotopes with half-lives longer than 3 days – allow a break in secular equilibrium. However, given that the concentration of these isotopes is unknown, secular equilibrium of the chain as a whole was assumed as well. For a particular component and decay chain, the same number of decays were simulated for each segment and the energy depositions in the detectors were recorded. These were later combined with the appropriate branching ratios to produce a spectrum. Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4" title="IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">IV</a> describes how the component efficiency is extracted from this simulated spectrum. </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> III A unified approach to combine assay results</h2> <div class="ltx_para" id="S3.p1"> The Demonstrator’s radioassay program delivered an extensive specific activity database (232Th, 238U, 40K) of the materials used to build the experiment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite>. During the commissioning of the Demonstrator, additional samples were collected and assayed, thus continually growing this database. Amongst others, inductively coupled plasma mass spectrometry (ICPMS), <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_γ</annotation></semantics></math>-count and neutron activation analysis (NAA) measurements were performed. The most sensitive technique, ICPMS, measures specific elements within a decay chain and therefore secular equilibrium is assumed to project a specific activity. The concentration of 232Th and 238U in HPGe detectors is far too low to be detected by ICPMS measurements. Thus, the 232Th and 238U contaminations are deduced by searching for time-correlated <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.p1.9.m9.1"><semantics id="S3.p1.9.m9.1a"><mi id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.1b"><ci id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.1d">italic_α</annotation></semantics></math> decays from their respective decay chains in the Demonstrator’s low-background dataset. These data-driven results (as opposed to the assay-driven projections published here) will be reported in a future publication. Despite the high detection efficiency of decays originating within Ge, the bulk 232Th and 238U contamination is anticipated to be so low <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib28" title="">28</a>]</cite> that the BI contribution from these sources is expected to be sub-dominant. </div> <div class="ltx_para" id="S3.p2"> In many cases, measurements of duplicate parts returned specific activities that did not agree within error. A method is thus needed to properly combine these results. The method should take into account the possible sample-to-sample variation of contaminants and the different detection limits of assay methods. It should also allow for the inclusion of assays which result in upper limits. </div> <div class="ltx_para" id="S3.p3"> Following the methodology of the Particle Data Group (PDG) for unconstrained averaging, a standard weighted least-squares approach is employed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib29" title="">29</a>]</cite>. The average specific activity and its uncertainty are calculated as <table class="ltx_equation ltx_eqn_table" id="S3.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="\overline{a}\pm\delta\overline{a}=\frac{\sum_{k}w_{k}a_{k}}{\sum_{k}w_{k}}\pm% \left(\sum_{k}w_{k}\right)^{-1/2}," class="ltx_Math" display="block" id="S3.E2.m1.1"><semantics id="S3.E2.m1.1a"><mrow id="S3.E2.m1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><mrow id="S3.E2.m1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><mrow id="S3.E2.m1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.3.cmml"><mover accent="true" id="S3.E2.m1.1.1.1.1.3.2" xref="S3.E2.m1.1.1.1.1.3.2.cmml"><mi id="S3.E2.m1.1.1.1.1.3.2.2" xref="S3.E2.m1.1.1.1.1.3.2.2.cmml">a</mi><mo id="S3.E2.m1.1.1.1.1.3.2.1" xref="S3.E2.m1.1.1.1.1.3.2.1.cmml">¯</mo></mover><mo id="S3.E2.m1.1.1.1.1.3.1" xref="S3.E2.m1.1.1.1.1.3.1.cmml">±</mo><mrow id="S3.E2.m1.1.1.1.1.3.3" xref="S3.E2.m1.1.1.1.1.3.3.cmml"><mi id="S3.E2.m1.1.1.1.1.3.3.2" xref="S3.E2.m1.1.1.1.1.3.3.2.cmml">δ</mi><mo id="S3.E2.m1.1.1.1.1.3.3.1" xref="S3.E2.m1.1.1.1.1.3.3.1.cmml">⁢</mo><mover accent="true" id="S3.E2.m1.1.1.1.1.3.3.3" xref="S3.E2.m1.1.1.1.1.3.3.3.cmml"><mi id="S3.E2.m1.1.1.1.1.3.3.3.2" xref="S3.E2.m1.1.1.1.1.3.3.3.2.cmml">a</mi><mo id="S3.E2.m1.1.1.1.1.3.3.3.1" xref="S3.E2.m1.1.1.1.1.3.3.3.1.cmml">¯</mo></mover></mrow></mrow><mo id="S3.E2.m1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.2.cmml">=</mo><mrow id="S3.E2.m1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.cmml"><mfrac id="S3.E2.m1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.3.cmml"><mrow id="S3.E2.m1.1.1.1.1.1.3.2" xref="S3.E2.m1.1.1.1.1.1.3.2.cmml"><msub id="S3.E2.m1.1.1.1.1.1.3.2.1" xref="S3.E2.m1.1.1.1.1.1.3.2.1.cmml"><mo id="S3.E2.m1.1.1.1.1.1.3.2.1.2" xref="S3.E2.m1.1.1.1.1.1.3.2.1.2.cmml">∑</mo><mi id="S3.E2.m1.1.1.1.1.1.3.2.1.3" xref="S3.E2.m1.1.1.1.1.1.3.2.1.3.cmml">k</mi></msub><mrow id="S3.E2.m1.1.1.1.1.1.3.2.2" 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xref="S3.E2.m1.1.1.1.1.1.1.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.1c">\overline{a}\pm\delta\overline{a}=\frac{\sum_{k}w_{k}a_{k}}{\sum_{k}w_{k}}\pm% \left(\sum_{k}w_{k}\right)^{-1/2},</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.1d">over¯ start_ARG italic_a end_ARG ± italic_δ over¯ start_ARG italic_a end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ± ( ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> </table> where the <math alttext="k" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mi id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_k</annotation></semantics></math>-th specific activity from assay, <math alttext="a_{k}" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><msub id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">a</mi><mi id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply 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The PDG states “We do not average or combine upper limits except in a very few cases where they may be re-expressed as measured numbers with Gaussian errors.” <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib29" title="">29</a>]</cite> Exactly the latter is used to treat specific activity upper limits. 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xref="S3.p3.6.m4.2.2.1.1.1.1.3.2">erf</mtext></ci><apply id="S3.p3.6.m4.2.2.1.1.1.1.3.3.cmml" xref="S3.p3.6.m4.2.2.1.1.1.1.3.3"><minus id="S3.p3.6.m4.2.2.1.1.1.1.3.3.1.cmml" xref="S3.p3.6.m4.2.2.1.1.1.1.3.3"></minus><cn id="S3.p3.6.m4.2.2.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.p3.6.m4.2.2.1.1.1.1.3.3.2">1</cn></apply></apply><cn id="S3.p3.6.m4.1.1.cmml" type="float" xref="S3.p3.6.m4.1.1">0.9</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.6.m4.2c">\delta a_{k}=l/[\sqrt{2}\text{erf}^{-1}(0.9)]</annotation><annotation encoding="application/x-llamapun" id="S3.p3.6.m4.2d">italic_δ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_l / [ square-root start_ARG 2 end_ARG erf start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 0.9 ) ]</annotation></semantics></math>. Note that an issue would arise if multiple upper limits are combined. Combining two identical <math alttext="a_{k}\pm\delta a_{k}" class="ltx_Math" display="inline" id="S3.p3.7.m5.1"><semantics id="S3.p3.7.m5.1a"><mrow id="S3.p3.7.m5.1.1" xref="S3.p3.7.m5.1.1.cmml"><msub id="S3.p3.7.m5.1.1.2" xref="S3.p3.7.m5.1.1.2.cmml"><mi id="S3.p3.7.m5.1.1.2.2" xref="S3.p3.7.m5.1.1.2.2.cmml">a</mi><mi id="S3.p3.7.m5.1.1.2.3" xref="S3.p3.7.m5.1.1.2.3.cmml">k</mi></msub><mo id="S3.p3.7.m5.1.1.1" xref="S3.p3.7.m5.1.1.1.cmml">±</mo><mrow id="S3.p3.7.m5.1.1.3" xref="S3.p3.7.m5.1.1.3.cmml"><mi id="S3.p3.7.m5.1.1.3.2" xref="S3.p3.7.m5.1.1.3.2.cmml">δ</mi><mo id="S3.p3.7.m5.1.1.3.1" xref="S3.p3.7.m5.1.1.3.1.cmml">⁢</mo><msub id="S3.p3.7.m5.1.1.3.3" xref="S3.p3.7.m5.1.1.3.3.cmml"><mi id="S3.p3.7.m5.1.1.3.3.2" xref="S3.p3.7.m5.1.1.3.3.2.cmml">a</mi><mi id="S3.p3.7.m5.1.1.3.3.3" xref="S3.p3.7.m5.1.1.3.3.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.7.m5.1b"><apply id="S3.p3.7.m5.1.1.cmml" xref="S3.p3.7.m5.1.1"><csymbol cd="latexml" id="S3.p3.7.m5.1.1.1.cmml" xref="S3.p3.7.m5.1.1.1">plus-or-minus</csymbol><apply id="S3.p3.7.m5.1.1.2.cmml" xref="S3.p3.7.m5.1.1.2"><csymbol cd="ambiguous" id="S3.p3.7.m5.1.1.2.1.cmml" xref="S3.p3.7.m5.1.1.2">subscript</csymbol><ci id="S3.p3.7.m5.1.1.2.2.cmml" xref="S3.p3.7.m5.1.1.2.2">𝑎</ci><ci id="S3.p3.7.m5.1.1.2.3.cmml" xref="S3.p3.7.m5.1.1.2.3">𝑘</ci></apply><apply id="S3.p3.7.m5.1.1.3.cmml" xref="S3.p3.7.m5.1.1.3"><times id="S3.p3.7.m5.1.1.3.1.cmml" xref="S3.p3.7.m5.1.1.3.1"></times><ci id="S3.p3.7.m5.1.1.3.2.cmml" xref="S3.p3.7.m5.1.1.3.2">𝛿</ci><apply id="S3.p3.7.m5.1.1.3.3.cmml" xref="S3.p3.7.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.p3.7.m5.1.1.3.3.1.cmml" xref="S3.p3.7.m5.1.1.3.3">subscript</csymbol><ci id="S3.p3.7.m5.1.1.3.3.2.cmml" xref="S3.p3.7.m5.1.1.3.3.2">𝑎</ci><ci id="S3.p3.7.m5.1.1.3.3.3.cmml" xref="S3.p3.7.m5.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.7.m5.1c">a_{k}\pm\delta a_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.7.m5.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ± italic_δ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> results in a smaller combined uncertainty <math alttext="\delta\overline{a}=\delta a_{k}/\sqrt{2}" class="ltx_Math" display="inline" id="S3.p3.8.m6.1"><semantics id="S3.p3.8.m6.1a"><mrow id="S3.p3.8.m6.1.1" xref="S3.p3.8.m6.1.1.cmml"><mrow id="S3.p3.8.m6.1.1.2" xref="S3.p3.8.m6.1.1.2.cmml"><mi id="S3.p3.8.m6.1.1.2.2" xref="S3.p3.8.m6.1.1.2.2.cmml">δ</mi><mo id="S3.p3.8.m6.1.1.2.1" xref="S3.p3.8.m6.1.1.2.1.cmml">⁢</mo><mover accent="true" id="S3.p3.8.m6.1.1.2.3" xref="S3.p3.8.m6.1.1.2.3.cmml"><mi id="S3.p3.8.m6.1.1.2.3.2" xref="S3.p3.8.m6.1.1.2.3.2.cmml">a</mi><mo id="S3.p3.8.m6.1.1.2.3.1" xref="S3.p3.8.m6.1.1.2.3.1.cmml">¯</mo></mover></mrow><mo id="S3.p3.8.m6.1.1.1" xref="S3.p3.8.m6.1.1.1.cmml">=</mo><mrow id="S3.p3.8.m6.1.1.3" xref="S3.p3.8.m6.1.1.3.cmml"><mrow id="S3.p3.8.m6.1.1.3.2" xref="S3.p3.8.m6.1.1.3.2.cmml"><mi id="S3.p3.8.m6.1.1.3.2.2" xref="S3.p3.8.m6.1.1.3.2.2.cmml">δ</mi><mo id="S3.p3.8.m6.1.1.3.2.1" xref="S3.p3.8.m6.1.1.3.2.1.cmml">⁢</mo><msub id="S3.p3.8.m6.1.1.3.2.3" xref="S3.p3.8.m6.1.1.3.2.3.cmml"><mi id="S3.p3.8.m6.1.1.3.2.3.2" xref="S3.p3.8.m6.1.1.3.2.3.2.cmml">a</mi><mi id="S3.p3.8.m6.1.1.3.2.3.3" xref="S3.p3.8.m6.1.1.3.2.3.3.cmml">k</mi></msub></mrow><mo id="S3.p3.8.m6.1.1.3.1" xref="S3.p3.8.m6.1.1.3.1.cmml">/</mo><msqrt id="S3.p3.8.m6.1.1.3.3" xref="S3.p3.8.m6.1.1.3.3.cmml"><mn id="S3.p3.8.m6.1.1.3.3.2" xref="S3.p3.8.m6.1.1.3.3.2.cmml">2</mn></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.8.m6.1b"><apply id="S3.p3.8.m6.1.1.cmml" xref="S3.p3.8.m6.1.1"><eq id="S3.p3.8.m6.1.1.1.cmml" xref="S3.p3.8.m6.1.1.1"></eq><apply id="S3.p3.8.m6.1.1.2.cmml" xref="S3.p3.8.m6.1.1.2"><times id="S3.p3.8.m6.1.1.2.1.cmml" xref="S3.p3.8.m6.1.1.2.1"></times><ci id="S3.p3.8.m6.1.1.2.2.cmml" xref="S3.p3.8.m6.1.1.2.2">𝛿</ci><apply id="S3.p3.8.m6.1.1.2.3.cmml" xref="S3.p3.8.m6.1.1.2.3"><ci id="S3.p3.8.m6.1.1.2.3.1.cmml" xref="S3.p3.8.m6.1.1.2.3.1">¯</ci><ci id="S3.p3.8.m6.1.1.2.3.2.cmml" xref="S3.p3.8.m6.1.1.2.3.2">𝑎</ci></apply></apply><apply id="S3.p3.8.m6.1.1.3.cmml" xref="S3.p3.8.m6.1.1.3"><divide id="S3.p3.8.m6.1.1.3.1.cmml" xref="S3.p3.8.m6.1.1.3.1"></divide><apply id="S3.p3.8.m6.1.1.3.2.cmml" xref="S3.p3.8.m6.1.1.3.2"><times id="S3.p3.8.m6.1.1.3.2.1.cmml" xref="S3.p3.8.m6.1.1.3.2.1"></times><ci id="S3.p3.8.m6.1.1.3.2.2.cmml" xref="S3.p3.8.m6.1.1.3.2.2">𝛿</ci><apply id="S3.p3.8.m6.1.1.3.2.3.cmml" xref="S3.p3.8.m6.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.p3.8.m6.1.1.3.2.3.1.cmml" xref="S3.p3.8.m6.1.1.3.2.3">subscript</csymbol><ci id="S3.p3.8.m6.1.1.3.2.3.2.cmml" xref="S3.p3.8.m6.1.1.3.2.3.2">𝑎</ci><ci id="S3.p3.8.m6.1.1.3.2.3.3.cmml" xref="S3.p3.8.m6.1.1.3.2.3.3">𝑘</ci></apply></apply><apply id="S3.p3.8.m6.1.1.3.3.cmml" xref="S3.p3.8.m6.1.1.3.3"><root id="S3.p3.8.m6.1.1.3.3a.cmml" xref="S3.p3.8.m6.1.1.3.3"></root><cn id="S3.p3.8.m6.1.1.3.3.2.cmml" type="integer" xref="S3.p3.8.m6.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m6.1c">\delta\overline{a}=\delta a_{k}/\sqrt{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m6.1d">italic_δ over¯ start_ARG italic_a end_ARG = italic_δ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT / square-root start_ARG 2 end_ARG</annotation></semantics></math>. This is a desirable property for measured values, but not for upper limits. Repeated results are likely if the activity of a source is significantly below the detection limit of the assay apparatus. Combining such measurements would result in an unrealistically lowered upper limit. Therefore only the most stringent upper limit is chosen. A maximum of one upper limit is thus included in the sum of Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.E2" title="In III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a>. </div> <div class="ltx_para" id="S3.p4"> Once the average is computed, <math alttext="\chi^{2}=\sum_{k}w_{k}(\overline{a}-a_{k})^{2}" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mrow id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><msup id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml"><mi id="S3.p4.1.m1.1.1.3.2" xref="S3.p4.1.m1.1.1.3.2.cmml">χ</mi><mn id="S3.p4.1.m1.1.1.3.3" xref="S3.p4.1.m1.1.1.3.3.cmml">2</mn></msup><mo id="S3.p4.1.m1.1.1.2" rspace="0.111em" xref="S3.p4.1.m1.1.1.2.cmml">=</mo><mrow id="S3.p4.1.m1.1.1.1" xref="S3.p4.1.m1.1.1.1.cmml"><msub id="S3.p4.1.m1.1.1.1.2" xref="S3.p4.1.m1.1.1.1.2.cmml"><mo id="S3.p4.1.m1.1.1.1.2.2" xref="S3.p4.1.m1.1.1.1.2.2.cmml">∑</mo><mi id="S3.p4.1.m1.1.1.1.2.3" xref="S3.p4.1.m1.1.1.1.2.3.cmml">k</mi></msub><mrow id="S3.p4.1.m1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.cmml"><msub id="S3.p4.1.m1.1.1.1.1.3" xref="S3.p4.1.m1.1.1.1.1.3.cmml"><mi id="S3.p4.1.m1.1.1.1.1.3.2" xref="S3.p4.1.m1.1.1.1.1.3.2.cmml">w</mi><mi id="S3.p4.1.m1.1.1.1.1.3.3" xref="S3.p4.1.m1.1.1.1.1.3.3.cmml">k</mi></msub><mo id="S3.p4.1.m1.1.1.1.1.2" xref="S3.p4.1.m1.1.1.1.1.2.cmml">⁢</mo><msup id="S3.p4.1.m1.1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.1.cmml"><mrow id="S3.p4.1.m1.1.1.1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.p4.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.1.m1.1.1.1.1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.2" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.2.cmml">a</mi><mo id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.1" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S3.p4.1.m1.1.1.1.1.1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.2.cmml">a</mi><mi id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.3" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S3.p4.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.p4.1.m1.1.1.1.1.1.3" xref="S3.p4.1.m1.1.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><eq id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2"></eq><apply id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.3.1.cmml" xref="S3.p4.1.m1.1.1.3">superscript</csymbol><ci id="S3.p4.1.m1.1.1.3.2.cmml" xref="S3.p4.1.m1.1.1.3.2">𝜒</ci><cn id="S3.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p4.1.m1.1.1.3.3">2</cn></apply><apply id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1"><apply id="S3.p4.1.m1.1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.2.1.cmml" xref="S3.p4.1.m1.1.1.1.2">subscript</csymbol><sum id="S3.p4.1.m1.1.1.1.2.2.cmml" xref="S3.p4.1.m1.1.1.1.2.2"></sum><ci id="S3.p4.1.m1.1.1.1.2.3.cmml" xref="S3.p4.1.m1.1.1.1.2.3">𝑘</ci></apply><apply id="S3.p4.1.m1.1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1.1"><times id="S3.p4.1.m1.1.1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.1.1.2"></times><apply id="S3.p4.1.m1.1.1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.1.3.1.cmml" xref="S3.p4.1.m1.1.1.1.1.3">subscript</csymbol><ci id="S3.p4.1.m1.1.1.1.1.3.2.cmml" xref="S3.p4.1.m1.1.1.1.1.3.2">𝑤</ci><ci id="S3.p4.1.m1.1.1.1.1.3.3.cmml" xref="S3.p4.1.m1.1.1.1.1.3.3">𝑘</ci></apply><apply id="S3.p4.1.m1.1.1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.1.1.1">superscript</csymbol><apply id="S3.p4.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1"><minus id="S3.p4.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2"><ci id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.1">¯</ci><ci id="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.2.2">𝑎</ci></apply><apply id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.2">𝑎</ci><ci id="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.p4.1.m1.1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply><cn id="S3.p4.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.1.m1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\chi^{2}=\sum_{k}w_{k}(\overline{a}-a_{k})^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( over¯ start_ARG italic_a end_ARG - italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is calculated. <math alttext="\chi^{2}/(N_{a}-1)" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mrow id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><msup id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml"><mi id="S3.p4.2.m2.1.1.3.2" xref="S3.p4.2.m2.1.1.3.2.cmml">χ</mi><mn id="S3.p4.2.m2.1.1.3.3" xref="S3.p4.2.m2.1.1.3.3.cmml">2</mn></msup><mo id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">/</mo><mrow id="S3.p4.2.m2.1.1.1.1" xref="S3.p4.2.m2.1.1.1.1.1.cmml"><mo id="S3.p4.2.m2.1.1.1.1.2" stretchy="false" xref="S3.p4.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.2.m2.1.1.1.1.1" xref="S3.p4.2.m2.1.1.1.1.1.cmml"><msub id="S3.p4.2.m2.1.1.1.1.1.2" xref="S3.p4.2.m2.1.1.1.1.1.2.cmml"><mi id="S3.p4.2.m2.1.1.1.1.1.2.2" xref="S3.p4.2.m2.1.1.1.1.1.2.2.cmml">N</mi><mi id="S3.p4.2.m2.1.1.1.1.1.2.3" xref="S3.p4.2.m2.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S3.p4.2.m2.1.1.1.1.1.1" xref="S3.p4.2.m2.1.1.1.1.1.1.cmml">−</mo><mn id="S3.p4.2.m2.1.1.1.1.1.3" xref="S3.p4.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p4.2.m2.1.1.1.1.3" stretchy="false" xref="S3.p4.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><divide id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2"></divide><apply id="S3.p4.2.m2.1.1.3.cmml" xref="S3.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.1.3.1.cmml" xref="S3.p4.2.m2.1.1.3">superscript</csymbol><ci id="S3.p4.2.m2.1.1.3.2.cmml" xref="S3.p4.2.m2.1.1.3.2">𝜒</ci><cn id="S3.p4.2.m2.1.1.3.3.cmml" type="integer" xref="S3.p4.2.m2.1.1.3.3">2</cn></apply><apply id="S3.p4.2.m2.1.1.1.1.1.cmml" xref="S3.p4.2.m2.1.1.1.1"><minus id="S3.p4.2.m2.1.1.1.1.1.1.cmml" xref="S3.p4.2.m2.1.1.1.1.1.1"></minus><apply id="S3.p4.2.m2.1.1.1.1.1.2.cmml" xref="S3.p4.2.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.1.1.1.1.2.1.cmml" xref="S3.p4.2.m2.1.1.1.1.1.2">subscript</csymbol><ci id="S3.p4.2.m2.1.1.1.1.1.2.2.cmml" xref="S3.p4.2.m2.1.1.1.1.1.2.2">𝑁</ci><ci id="S3.p4.2.m2.1.1.1.1.1.2.3.cmml" xref="S3.p4.2.m2.1.1.1.1.1.2.3">𝑎</ci></apply><cn id="S3.p4.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.p4.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">\chi^{2}/(N_{a}-1)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT - 1 )</annotation></semantics></math> is used as a discriminant as follows: <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> 1. <div class="ltx_para" id="S3.I1.i1.p1"> If <math alttext="\chi^{2}/(N_{a}-1)\leq 1" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mrow id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml"><msup id="S3.I1.i1.p1.1.m1.1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.1.3.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.1.3.2" xref="S3.I1.i1.p1.1.m1.1.1.1.3.2.cmml">χ</mi><mn id="S3.I1.i1.p1.1.m1.1.1.1.3.3" xref="S3.I1.i1.p1.1.m1.1.1.1.3.3.cmml">2</mn></msup><mo id="S3.I1.i1.p1.1.m1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.1.2.cmml">/</mo><mrow id="S3.I1.i1.p1.1.m1.1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.2" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.2.cmml">N</mi><mi id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.3" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">≤</mo><mn id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><leq id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"></leq><apply id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"><divide id="S3.I1.i1.p1.1.m1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.2"></divide><apply id="S3.I1.i1.p1.1.m1.1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.1.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.3">superscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.1.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.3.2">𝜒</ci><cn id="S3.I1.i1.p1.1.m1.1.1.1.3.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.1.3.3">2</cn></apply><apply id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1"><minus id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.1"></minus><apply id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.2">𝑁</ci><ci id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.2.3">𝑎</ci></apply><cn id="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="S3.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">\chi^{2}/(N_{a}-1)\leq 1</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT - 1 ) ≤ 1</annotation></semantics></math> the average, <math alttext="\overline{a}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mover accent="true" id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.2" xref="S3.I1.i1.p1.2.m2.1.1.2.cmml">a</mi><mo id="S3.I1.i1.p1.2.m2.1.1.1" xref="S3.I1.i1.p1.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><apply id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><ci id="S3.I1.i1.p1.2.m2.1.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1.1">¯</ci><ci id="S3.I1.i1.p1.2.m2.1.1.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math>, and uncertainty, <math alttext="\delta\overline{a}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><mrow id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.2.cmml">δ</mi><mo id="S3.I1.i1.p1.3.m3.1.1.1" xref="S3.I1.i1.p1.3.m3.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.3.2" xref="S3.I1.i1.p1.3.m3.1.1.3.2.cmml">a</mi><mo id="S3.I1.i1.p1.3.m3.1.1.3.1" xref="S3.I1.i1.p1.3.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><times id="S3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1"></times><ci id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2">𝛿</ci><apply id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3"><ci id="S3.I1.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.1">¯</ci><ci id="S3.I1.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math>, are accepted. </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> 2. <div class="ltx_para" id="S3.I1.i2.p1"> If <math alttext="\chi^{2}/(N_{a}-1)>1" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mrow id="S3.I1.i2.p1.1.m1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml"><msup id="S3.I1.i2.p1.1.m1.1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.1.3.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.1.3.2" xref="S3.I1.i2.p1.1.m1.1.1.1.3.2.cmml">χ</mi><mn id="S3.I1.i2.p1.1.m1.1.1.1.3.3" xref="S3.I1.i2.p1.1.m1.1.1.1.3.3.cmml">2</mn></msup><mo id="S3.I1.i2.p1.1.m1.1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.1.2.cmml">/</mo><mrow id="S3.I1.i2.p1.1.m1.1.1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.I1.i2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.2" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.2.cmml">N</mi><mi id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.3" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i2.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">></mo><mn id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><gt id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2"></gt><apply id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1"><divide id="S3.I1.i2.p1.1.m1.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.2"></divide><apply id="S3.I1.i2.p1.1.m1.1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.3">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.3.2">𝜒</ci><cn id="S3.I1.i2.p1.1.m1.1.1.1.3.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1.1.3.3">2</cn></apply><apply id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1"><minus id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.1"></minus><apply id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.2">𝑁</ci><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.2.3">𝑎</ci></apply><cn id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="S3.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">\chi^{2}/(N_{a}-1)>1</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT - 1 ) > 1</annotation></semantics></math> the uncertainty, <math alttext="\delta\overline{a}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mrow id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml">δ</mi><mo id="S3.I1.i2.p1.2.m2.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.3.2" xref="S3.I1.i2.p1.2.m2.1.1.3.2.cmml">a</mi><mo id="S3.I1.i2.p1.2.m2.1.1.3.1" xref="S3.I1.i2.p1.2.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><times id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1"></times><ci id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">𝛿</ci><apply id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3"><ci id="S3.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.1">¯</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math>, is scaled by a factor <math alttext="\Sigma=\left[\chi^{2}/(N_{a}-1)\right]^{1/2}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><mrow id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.3" mathvariant="normal" xref="S3.I1.i2.p1.3.m3.1.1.3.cmml">Σ</mi><mo id="S3.I1.i2.p1.3.m3.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.2.cmml">=</mo><msup id="S3.I1.i2.p1.3.m3.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.cmml"><mrow id="S3.I1.i2.p1.3.m3.1.1.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.1.2.cmml"><mo id="S3.I1.i2.p1.3.m3.1.1.1.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><msup id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3.2" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3.2.cmml">χ</mi><mn id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3.3" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">/</mo><mrow id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml"><msub id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.2.2" xref="S3.I1.i2.p1.3.m3.1.1.1.1.1.1.1.1.1.2.2.cmml">N</mi><mi 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ltx_centering">Figure 1: The combination of an upper limit with a measured value (a,b) and the combination of two measured values (c,d) are shown as truncated-at-zero Gaussian distributions in solid red. Toy data is used for the raw upper limits and measured values, which are also depicted as truncated-at-zero Gaussian distributions but in dashed grey.</figcaption> </figure> <div class="ltx_para" id="S3.p5"> Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a> exemplifies the technique for combining assay results. Following the nEXO collaboration’s treatment of activities, these are visualized as truncated-at-zero Gaussian distributions. In the figure two example assay results, <math alttext="a_{k}\,\pm\,\delta a_{k}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mrow id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><msub id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml"><mi id="S3.p5.1.m1.1.1.2.2" xref="S3.p5.1.m1.1.1.2.2.cmml">a</mi><mi id="S3.p5.1.m1.1.1.2.3" xref="S3.p5.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S3.p5.1.m1.1.1.1" lspace="0.392em" rspace="0.392em" xref="S3.p5.1.m1.1.1.1.cmml">±</mo><mrow id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml"><mi id="S3.p5.1.m1.1.1.3.2" xref="S3.p5.1.m1.1.1.3.2.cmml">δ</mi><mo id="S3.p5.1.m1.1.1.3.1" xref="S3.p5.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S3.p5.1.m1.1.1.3.3" xref="S3.p5.1.m1.1.1.3.3.cmml"><mi id="S3.p5.1.m1.1.1.3.3.2" xref="S3.p5.1.m1.1.1.3.3.2.cmml">a</mi><mi id="S3.p5.1.m1.1.1.3.3.3" xref="S3.p5.1.m1.1.1.3.3.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><csymbol cd="latexml" id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1.1">plus-or-minus</csymbol><apply id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.1.2.1.cmml" xref="S3.p5.1.m1.1.1.2">subscript</csymbol><ci id="S3.p5.1.m1.1.1.2.2.cmml" xref="S3.p5.1.m1.1.1.2.2">𝑎</ci><ci id="S3.p5.1.m1.1.1.2.3.cmml" xref="S3.p5.1.m1.1.1.2.3">𝑘</ci></apply><apply id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3"><times id="S3.p5.1.m1.1.1.3.1.cmml" xref="S3.p5.1.m1.1.1.3.1"></times><ci id="S3.p5.1.m1.1.1.3.2.cmml" xref="S3.p5.1.m1.1.1.3.2">𝛿</ci><apply id="S3.p5.1.m1.1.1.3.3.cmml" xref="S3.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.1.3.3.1.cmml" xref="S3.p5.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.p5.1.m1.1.1.3.3.2.cmml" xref="S3.p5.1.m1.1.1.3.3.2">𝑎</ci><ci id="S3.p5.1.m1.1.1.3.3.3.cmml" xref="S3.p5.1.m1.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">a_{k}\,\pm\,\delta a_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ± italic_δ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, are plotted as such. These are averaged using the technique described above and the result, <math alttext="\overline{a}\,\pm\,\delta\overline{a}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mrow id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml"><mover accent="true" id="S3.p5.2.m2.1.1.2" xref="S3.p5.2.m2.1.1.2.cmml"><mi id="S3.p5.2.m2.1.1.2.2" xref="S3.p5.2.m2.1.1.2.2.cmml">a</mi><mo id="S3.p5.2.m2.1.1.2.1" xref="S3.p5.2.m2.1.1.2.1.cmml">¯</mo></mover><mo id="S3.p5.2.m2.1.1.1" lspace="0.392em" rspace="0.392em" xref="S3.p5.2.m2.1.1.1.cmml">±</mo><mrow id="S3.p5.2.m2.1.1.3" xref="S3.p5.2.m2.1.1.3.cmml"><mi id="S3.p5.2.m2.1.1.3.2" xref="S3.p5.2.m2.1.1.3.2.cmml">δ</mi><mo id="S3.p5.2.m2.1.1.3.1" xref="S3.p5.2.m2.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S3.p5.2.m2.1.1.3.3" xref="S3.p5.2.m2.1.1.3.3.cmml"><mi id="S3.p5.2.m2.1.1.3.3.2" xref="S3.p5.2.m2.1.1.3.3.2.cmml">a</mi><mo id="S3.p5.2.m2.1.1.3.3.1" xref="S3.p5.2.m2.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><apply id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1"><csymbol cd="latexml" id="S3.p5.2.m2.1.1.1.cmml" xref="S3.p5.2.m2.1.1.1">plus-or-minus</csymbol><apply id="S3.p5.2.m2.1.1.2.cmml" xref="S3.p5.2.m2.1.1.2"><ci id="S3.p5.2.m2.1.1.2.1.cmml" xref="S3.p5.2.m2.1.1.2.1">¯</ci><ci id="S3.p5.2.m2.1.1.2.2.cmml" xref="S3.p5.2.m2.1.1.2.2">𝑎</ci></apply><apply id="S3.p5.2.m2.1.1.3.cmml" xref="S3.p5.2.m2.1.1.3"><times id="S3.p5.2.m2.1.1.3.1.cmml" xref="S3.p5.2.m2.1.1.3.1"></times><ci id="S3.p5.2.m2.1.1.3.2.cmml" xref="S3.p5.2.m2.1.1.3.2">𝛿</ci><apply id="S3.p5.2.m2.1.1.3.3.cmml" xref="S3.p5.2.m2.1.1.3.3"><ci id="S3.p5.2.m2.1.1.3.3.1.cmml" xref="S3.p5.2.m2.1.1.3.3.1">¯</ci><ci id="S3.p5.2.m2.1.1.3.3.2.cmml" xref="S3.p5.2.m2.1.1.3.3.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\overline{a}\,\pm\,\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">over¯ start_ARG italic_a end_ARG ± italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math>, is shown. An upper limit lower than a measured value lowers the measured value Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>(a), whereas an upper limit higher than a measured value has almost no effect on the same Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>(b). Note that in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>(a) a scaling factor of <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mi id="S3.p5.3.m3.1.1" mathvariant="normal" xref="S3.p5.3.m3.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">roman_Σ</annotation></semantics></math> is applied since <math alttext="\chi^{2}>1" class="ltx_Math" display="inline" id="S3.p5.4.m4.1"><semantics id="S3.p5.4.m4.1a"><mrow id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml"><msup id="S3.p5.4.m4.1.1.2" xref="S3.p5.4.m4.1.1.2.cmml"><mi id="S3.p5.4.m4.1.1.2.2" xref="S3.p5.4.m4.1.1.2.2.cmml">χ</mi><mn id="S3.p5.4.m4.1.1.2.3" xref="S3.p5.4.m4.1.1.2.3.cmml">2</mn></msup><mo id="S3.p5.4.m4.1.1.1" xref="S3.p5.4.m4.1.1.1.cmml">></mo><mn id="S3.p5.4.m4.1.1.3" xref="S3.p5.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.1b"><apply id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1"><gt id="S3.p5.4.m4.1.1.1.cmml" xref="S3.p5.4.m4.1.1.1"></gt><apply id="S3.p5.4.m4.1.1.2.cmml" xref="S3.p5.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p5.4.m4.1.1.2.1.cmml" xref="S3.p5.4.m4.1.1.2">superscript</csymbol><ci id="S3.p5.4.m4.1.1.2.2.cmml" xref="S3.p5.4.m4.1.1.2.2">𝜒</ci><cn id="S3.p5.4.m4.1.1.2.3.cmml" type="integer" xref="S3.p5.4.m4.1.1.2.3">2</cn></apply><cn id="S3.p5.4.m4.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.1c">\chi^{2}>1</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT > 1</annotation></semantics></math>. If two measured values agree within error, the combined uncertainty decreases Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>(c). If there is no agreement, the combined uncertainty increases since a scaling factor of <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.p5.5.m5.1"><semantics id="S3.p5.5.m5.1a"><mi id="S3.p5.5.m5.1.1" mathvariant="normal" xref="S3.p5.5.m5.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.1b"><ci id="S3.p5.5.m5.1.1.cmml" xref="S3.p5.5.m5.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.1d">roman_Σ</annotation></semantics></math> is applied Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F1" title="Figure 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>(d). </div> <div class="ltx_para" id="S3.p6"> The levels of 232Th and 238U in the Demonstrator’s components often lie below the detection limits of the most sensitive assays, leading to a high prevalence of upper limits. The motivation to include upper limits, not only in the sum of Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.E2" title="In III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a> but also in the calculation of the scaling factor <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.p6.3.m3.1"><semantics id="S3.p6.3.m3.1a"><mi id="S3.p6.3.m3.1.1" mathvariant="normal" xref="S3.p6.3.m3.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.1b"><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.1d">roman_Σ</annotation></semantics></math>, stems from this prevalence. To justify their use, the effect of combining an upper limit, <math alttext="l" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><mi id="S3.p6.4.m4.1.1" xref="S3.p6.4.m4.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><ci id="S3.p6.4.m4.1.1.cmml" xref="S3.p6.4.m4.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">l</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">italic_l</annotation></semantics></math>, with a fixed measured value, <math alttext="a\,\pm\,\delta a" class="ltx_Math" display="inline" id="S3.p6.5.m5.1"><semantics id="S3.p6.5.m5.1a"><mrow id="S3.p6.5.m5.1.1" xref="S3.p6.5.m5.1.1.cmml"><mi id="S3.p6.5.m5.1.1.2" xref="S3.p6.5.m5.1.1.2.cmml">a</mi><mo id="S3.p6.5.m5.1.1.1" lspace="0.392em" rspace="0.392em" xref="S3.p6.5.m5.1.1.1.cmml">±</mo><mrow id="S3.p6.5.m5.1.1.3" xref="S3.p6.5.m5.1.1.3.cmml"><mi id="S3.p6.5.m5.1.1.3.2" xref="S3.p6.5.m5.1.1.3.2.cmml">δ</mi><mo id="S3.p6.5.m5.1.1.3.1" xref="S3.p6.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S3.p6.5.m5.1.1.3.3" xref="S3.p6.5.m5.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.1b"><apply id="S3.p6.5.m5.1.1.cmml" xref="S3.p6.5.m5.1.1"><csymbol cd="latexml" id="S3.p6.5.m5.1.1.1.cmml" xref="S3.p6.5.m5.1.1.1">plus-or-minus</csymbol><ci id="S3.p6.5.m5.1.1.2.cmml" xref="S3.p6.5.m5.1.1.2">𝑎</ci><apply id="S3.p6.5.m5.1.1.3.cmml" xref="S3.p6.5.m5.1.1.3"><times id="S3.p6.5.m5.1.1.3.1.cmml" xref="S3.p6.5.m5.1.1.3.1"></times><ci id="S3.p6.5.m5.1.1.3.2.cmml" xref="S3.p6.5.m5.1.1.3.2">𝛿</ci><ci id="S3.p6.5.m5.1.1.3.3.cmml" xref="S3.p6.5.m5.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.1c">a\,\pm\,\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.1d">italic_a ± italic_δ italic_a</annotation></semantics></math>, with small error (<math alttext="\delta a<a" class="ltx_Math" display="inline" id="S3.p6.6.m6.1"><semantics id="S3.p6.6.m6.1a"><mrow id="S3.p6.6.m6.1.1" xref="S3.p6.6.m6.1.1.cmml"><mrow id="S3.p6.6.m6.1.1.2" xref="S3.p6.6.m6.1.1.2.cmml"><mi id="S3.p6.6.m6.1.1.2.2" xref="S3.p6.6.m6.1.1.2.2.cmml">δ</mi><mo id="S3.p6.6.m6.1.1.2.1" xref="S3.p6.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S3.p6.6.m6.1.1.2.3" xref="S3.p6.6.m6.1.1.2.3.cmml">a</mi></mrow><mo id="S3.p6.6.m6.1.1.1" xref="S3.p6.6.m6.1.1.1.cmml"><</mo><mi id="S3.p6.6.m6.1.1.3" xref="S3.p6.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.1b"><apply id="S3.p6.6.m6.1.1.cmml" xref="S3.p6.6.m6.1.1"><lt id="S3.p6.6.m6.1.1.1.cmml" xref="S3.p6.6.m6.1.1.1"></lt><apply id="S3.p6.6.m6.1.1.2.cmml" xref="S3.p6.6.m6.1.1.2"><times id="S3.p6.6.m6.1.1.2.1.cmml" xref="S3.p6.6.m6.1.1.2.1"></times><ci id="S3.p6.6.m6.1.1.2.2.cmml" xref="S3.p6.6.m6.1.1.2.2">𝛿</ci><ci id="S3.p6.6.m6.1.1.2.3.cmml" xref="S3.p6.6.m6.1.1.2.3">𝑎</ci></apply><ci id="S3.p6.6.m6.1.1.3.cmml" xref="S3.p6.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.1c">\delta a<a</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.1d">italic_δ italic_a < italic_a</annotation></semantics></math>), was evaluated. This case is representative of many assay results. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.F2" title="Figure 2 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a> shows the <math alttext="\chi^{2}" class="ltx_Math" display="inline" id="S3.p6.7.m7.1"><semantics id="S3.p6.7.m7.1a"><msup id="S3.p6.7.m7.1.1" xref="S3.p6.7.m7.1.1.cmml"><mi id="S3.p6.7.m7.1.1.2" xref="S3.p6.7.m7.1.1.2.cmml">χ</mi><mn id="S3.p6.7.m7.1.1.3" xref="S3.p6.7.m7.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.p6.7.m7.1b"><apply id="S3.p6.7.m7.1.1.cmml" xref="S3.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p6.7.m7.1.1.1.cmml" xref="S3.p6.7.m7.1.1">superscript</csymbol><ci id="S3.p6.7.m7.1.1.2.cmml" xref="S3.p6.7.m7.1.1.2">𝜒</ci><cn id="S3.p6.7.m7.1.1.3.cmml" type="integer" xref="S3.p6.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m7.1c">\chi^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m7.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, average and uncertainty with respect to a Gaussian upper limit, <math alttext="\delta a_{l}" class="ltx_Math" display="inline" id="S3.p6.8.m8.1"><semantics id="S3.p6.8.m8.1a"><mrow id="S3.p6.8.m8.1.1" xref="S3.p6.8.m8.1.1.cmml"><mi id="S3.p6.8.m8.1.1.2" xref="S3.p6.8.m8.1.1.2.cmml">δ</mi><mo id="S3.p6.8.m8.1.1.1" xref="S3.p6.8.m8.1.1.1.cmml">⁢</mo><msub id="S3.p6.8.m8.1.1.3" xref="S3.p6.8.m8.1.1.3.cmml"><mi id="S3.p6.8.m8.1.1.3.2" xref="S3.p6.8.m8.1.1.3.2.cmml">a</mi><mi id="S3.p6.8.m8.1.1.3.3" xref="S3.p6.8.m8.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.8.m8.1b"><apply id="S3.p6.8.m8.1.1.cmml" xref="S3.p6.8.m8.1.1"><times id="S3.p6.8.m8.1.1.1.cmml" xref="S3.p6.8.m8.1.1.1"></times><ci id="S3.p6.8.m8.1.1.2.cmml" xref="S3.p6.8.m8.1.1.2">𝛿</ci><apply id="S3.p6.8.m8.1.1.3.cmml" xref="S3.p6.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.p6.8.m8.1.1.3.1.cmml" xref="S3.p6.8.m8.1.1.3">subscript</csymbol><ci id="S3.p6.8.m8.1.1.3.2.cmml" xref="S3.p6.8.m8.1.1.3.2">𝑎</ci><ci id="S3.p6.8.m8.1.1.3.3.cmml" xref="S3.p6.8.m8.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.8.m8.1c">\delta a_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.8.m8.1d">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>, between 0 and <math alttext="1.1a" class="ltx_Math" display="inline" id="S3.p6.9.m9.1"><semantics id="S3.p6.9.m9.1a"><mrow id="S3.p6.9.m9.1.1" xref="S3.p6.9.m9.1.1.cmml"><mn id="S3.p6.9.m9.1.1.2" xref="S3.p6.9.m9.1.1.2.cmml">1.1</mn><mo id="S3.p6.9.m9.1.1.1" xref="S3.p6.9.m9.1.1.1.cmml">⁢</mo><mi id="S3.p6.9.m9.1.1.3" xref="S3.p6.9.m9.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.9.m9.1b"><apply id="S3.p6.9.m9.1.1.cmml" xref="S3.p6.9.m9.1.1"><times id="S3.p6.9.m9.1.1.1.cmml" xref="S3.p6.9.m9.1.1.1"></times><cn id="S3.p6.9.m9.1.1.2.cmml" type="float" xref="S3.p6.9.m9.1.1.2">1.1</cn><ci id="S3.p6.9.m9.1.1.3.cmml" xref="S3.p6.9.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.9.m9.1c">1.1a</annotation><annotation encoding="application/x-llamapun" id="S3.p6.9.m9.1d">1.1 italic_a</annotation></semantics></math>. It demonstrates that averaging with an upper limit leads to the following desirable properties: <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> 1. <div class="ltx_para" id="S3.I2.i1.p1"> For <math alttext="\delta a_{l}>a+\delta a" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mrow id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml"><mrow id="S3.I2.i1.p1.1.m1.1.1.2" xref="S3.I2.i1.p1.1.m1.1.1.2.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.2.2" xref="S3.I2.i1.p1.1.m1.1.1.2.2.cmml">δ</mi><mo id="S3.I2.i1.p1.1.m1.1.1.2.1" xref="S3.I2.i1.p1.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S3.I2.i1.p1.1.m1.1.1.2.3" xref="S3.I2.i1.p1.1.m1.1.1.2.3.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.2.3.2" xref="S3.I2.i1.p1.1.m1.1.1.2.3.2.cmml">a</mi><mi id="S3.I2.i1.p1.1.m1.1.1.2.3.3" xref="S3.I2.i1.p1.1.m1.1.1.2.3.3.cmml">l</mi></msub></mrow><mo id="S3.I2.i1.p1.1.m1.1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.1.cmml">></mo><mrow id="S3.I2.i1.p1.1.m1.1.1.3" xref="S3.I2.i1.p1.1.m1.1.1.3.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.3.2" xref="S3.I2.i1.p1.1.m1.1.1.3.2.cmml">a</mi><mo id="S3.I2.i1.p1.1.m1.1.1.3.1" xref="S3.I2.i1.p1.1.m1.1.1.3.1.cmml">+</mo><mrow id="S3.I2.i1.p1.1.m1.1.1.3.3" xref="S3.I2.i1.p1.1.m1.1.1.3.3.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.3.3.2" xref="S3.I2.i1.p1.1.m1.1.1.3.3.2.cmml">δ</mi><mo id="S3.I2.i1.p1.1.m1.1.1.3.3.1" xref="S3.I2.i1.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.I2.i1.p1.1.m1.1.1.3.3.3" xref="S3.I2.i1.p1.1.m1.1.1.3.3.3.cmml">a</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><apply id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1"><gt id="S3.I2.i1.p1.1.m1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1"></gt><apply id="S3.I2.i1.p1.1.m1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2"><times id="S3.I2.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.1"></times><ci id="S3.I2.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.2">𝛿</ci><apply id="S3.I2.i1.p1.1.m1.1.1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.1.1.2.3.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.3">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.1.1.2.3.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.3.2">𝑎</ci><ci id="S3.I2.i1.p1.1.m1.1.1.2.3.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.3.3">𝑙</ci></apply></apply><apply id="S3.I2.i1.p1.1.m1.1.1.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3"><plus id="S3.I2.i1.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.1"></plus><ci id="S3.I2.i1.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.2">𝑎</ci><apply id="S3.I2.i1.p1.1.m1.1.1.3.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.3"><times id="S3.I2.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.3.1"></times><ci id="S3.I2.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.3.2">𝛿</ci><ci id="S3.I2.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">\delta a_{l}>a+\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT > italic_a + italic_δ italic_a</annotation></semantics></math>, <math alttext="\overline{a}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mover accent="true" id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.2.cmml">a</mi><mo id="S3.I2.i1.p1.2.m2.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1"><ci id="S3.I2.i1.p1.2.m2.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1">¯</ci><ci id="S3.I2.i1.p1.2.m2.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math> and <math alttext="\delta\overline{a}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><mrow id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml">δ</mi><mo id="S3.I2.i1.p1.3.m3.1.1.1" xref="S3.I2.i1.p1.3.m3.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.3.2" xref="S3.I2.i1.p1.3.m3.1.1.3.2.cmml">a</mi><mo id="S3.I2.i1.p1.3.m3.1.1.3.1" xref="S3.I2.i1.p1.3.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><times id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.1"></times><ci id="S3.I2.i1.p1.3.m3.1.1.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2">𝛿</ci><apply id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3"><ci id="S3.I2.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3.1">¯</ci><ci id="S3.I2.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math> rapidly converge to <math alttext="a" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.1"><semantics id="S3.I2.i1.p1.4.m4.1a"><mi id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.1b"><ci id="S3.I2.i1.p1.4.m4.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.1d">italic_a</annotation></semantics></math> and <math alttext="\delta a" class="ltx_Math" display="inline" id="S3.I2.i1.p1.5.m5.1"><semantics id="S3.I2.i1.p1.5.m5.1a"><mrow id="S3.I2.i1.p1.5.m5.1.1" xref="S3.I2.i1.p1.5.m5.1.1.cmml"><mi id="S3.I2.i1.p1.5.m5.1.1.2" xref="S3.I2.i1.p1.5.m5.1.1.2.cmml">δ</mi><mo id="S3.I2.i1.p1.5.m5.1.1.1" xref="S3.I2.i1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.I2.i1.p1.5.m5.1.1.3" xref="S3.I2.i1.p1.5.m5.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.5.m5.1b"><apply id="S3.I2.i1.p1.5.m5.1.1.cmml" xref="S3.I2.i1.p1.5.m5.1.1"><times id="S3.I2.i1.p1.5.m5.1.1.1.cmml" xref="S3.I2.i1.p1.5.m5.1.1.1"></times><ci id="S3.I2.i1.p1.5.m5.1.1.2.cmml" xref="S3.I2.i1.p1.5.m5.1.1.2">𝛿</ci><ci id="S3.I2.i1.p1.5.m5.1.1.3.cmml" xref="S3.I2.i1.p1.5.m5.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.5.m5.1c">\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.5.m5.1d">italic_δ italic_a</annotation></semantics></math> respectively. Smaller <math alttext="\delta a" class="ltx_Math" display="inline" id="S3.I2.i1.p1.6.m6.1"><semantics id="S3.I2.i1.p1.6.m6.1a"><mrow id="S3.I2.i1.p1.6.m6.1.1" xref="S3.I2.i1.p1.6.m6.1.1.cmml"><mi id="S3.I2.i1.p1.6.m6.1.1.2" xref="S3.I2.i1.p1.6.m6.1.1.2.cmml">δ</mi><mo id="S3.I2.i1.p1.6.m6.1.1.1" xref="S3.I2.i1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.I2.i1.p1.6.m6.1.1.3" xref="S3.I2.i1.p1.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.6.m6.1b"><apply id="S3.I2.i1.p1.6.m6.1.1.cmml" xref="S3.I2.i1.p1.6.m6.1.1"><times id="S3.I2.i1.p1.6.m6.1.1.1.cmml" xref="S3.I2.i1.p1.6.m6.1.1.1"></times><ci id="S3.I2.i1.p1.6.m6.1.1.2.cmml" xref="S3.I2.i1.p1.6.m6.1.1.2">𝛿</ci><ci id="S3.I2.i1.p1.6.m6.1.1.3.cmml" xref="S3.I2.i1.p1.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.6.m6.1c">\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.6.m6.1d">italic_δ italic_a</annotation></semantics></math> lead to faster convergence. In other words, upper limits higher than measured values have little effect on the same. The lower the measured value’s error, the lower the impact. In this regime <math alttext="\chi^{2}<1" class="ltx_Math" display="inline" id="S3.I2.i1.p1.7.m7.1"><semantics id="S3.I2.i1.p1.7.m7.1a"><mrow id="S3.I2.i1.p1.7.m7.1.1" xref="S3.I2.i1.p1.7.m7.1.1.cmml"><msup id="S3.I2.i1.p1.7.m7.1.1.2" xref="S3.I2.i1.p1.7.m7.1.1.2.cmml"><mi id="S3.I2.i1.p1.7.m7.1.1.2.2" xref="S3.I2.i1.p1.7.m7.1.1.2.2.cmml">χ</mi><mn id="S3.I2.i1.p1.7.m7.1.1.2.3" xref="S3.I2.i1.p1.7.m7.1.1.2.3.cmml">2</mn></msup><mo id="S3.I2.i1.p1.7.m7.1.1.1" xref="S3.I2.i1.p1.7.m7.1.1.1.cmml"><</mo><mn id="S3.I2.i1.p1.7.m7.1.1.3" xref="S3.I2.i1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.7.m7.1b"><apply id="S3.I2.i1.p1.7.m7.1.1.cmml" xref="S3.I2.i1.p1.7.m7.1.1"><lt id="S3.I2.i1.p1.7.m7.1.1.1.cmml" xref="S3.I2.i1.p1.7.m7.1.1.1"></lt><apply id="S3.I2.i1.p1.7.m7.1.1.2.cmml" xref="S3.I2.i1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.7.m7.1.1.2.1.cmml" xref="S3.I2.i1.p1.7.m7.1.1.2">superscript</csymbol><ci id="S3.I2.i1.p1.7.m7.1.1.2.2.cmml" xref="S3.I2.i1.p1.7.m7.1.1.2.2">𝜒</ci><cn id="S3.I2.i1.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S3.I2.i1.p1.7.m7.1.1.2.3">2</cn></apply><cn id="S3.I2.i1.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.I2.i1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.7.m7.1c">\chi^{2}<1</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.7.m7.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < 1</annotation></semantics></math>. </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> 2. <div class="ltx_para" id="S3.I2.i2.p1"> For <math alttext="\delta a_{l}<a+\delta a" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><mrow id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml"><mrow id="S3.I2.i2.p1.1.m1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.2.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.2.2" xref="S3.I2.i2.p1.1.m1.1.1.2.2.cmml">δ</mi><mo id="S3.I2.i2.p1.1.m1.1.1.2.1" xref="S3.I2.i2.p1.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S3.I2.i2.p1.1.m1.1.1.2.3" xref="S3.I2.i2.p1.1.m1.1.1.2.3.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.2.3.2" xref="S3.I2.i2.p1.1.m1.1.1.2.3.2.cmml">a</mi><mi id="S3.I2.i2.p1.1.m1.1.1.2.3.3" xref="S3.I2.i2.p1.1.m1.1.1.2.3.3.cmml">l</mi></msub></mrow><mo id="S3.I2.i2.p1.1.m1.1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.1.cmml"><</mo><mrow id="S3.I2.i2.p1.1.m1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.3.2" xref="S3.I2.i2.p1.1.m1.1.1.3.2.cmml">a</mi><mo id="S3.I2.i2.p1.1.m1.1.1.3.1" xref="S3.I2.i2.p1.1.m1.1.1.3.1.cmml">+</mo><mrow id="S3.I2.i2.p1.1.m1.1.1.3.3" xref="S3.I2.i2.p1.1.m1.1.1.3.3.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.3.3.2" xref="S3.I2.i2.p1.1.m1.1.1.3.3.2.cmml">δ</mi><mo id="S3.I2.i2.p1.1.m1.1.1.3.3.1" xref="S3.I2.i2.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.I2.i2.p1.1.m1.1.1.3.3.3" xref="S3.I2.i2.p1.1.m1.1.1.3.3.3.cmml">a</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><lt id="S3.I2.i2.p1.1.m1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1"></lt><apply id="S3.I2.i2.p1.1.m1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2"><times id="S3.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.1"></times><ci id="S3.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.2">𝛿</ci><apply id="S3.I2.i2.p1.1.m1.1.1.2.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.2.3.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.3">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.2.3.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.3.2">𝑎</ci><ci id="S3.I2.i2.p1.1.m1.1.1.2.3.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.3.3">𝑙</ci></apply></apply><apply id="S3.I2.i2.p1.1.m1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3"><plus id="S3.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.1"></plus><ci id="S3.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.2">𝑎</ci><apply id="S3.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.3"><times id="S3.I2.i2.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.3.1"></times><ci id="S3.I2.i2.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.3.2">𝛿</ci><ci id="S3.I2.i2.p1.1.m1.1.1.3.3.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">\delta a_{l}<a+\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < italic_a + italic_δ italic_a</annotation></semantics></math>, <math alttext="\overline{a}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><mover accent="true" id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">a</mi><mo id="S3.I2.i2.p1.2.m2.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><ci id="S3.I2.i2.p1.2.m2.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1">¯</ci><ci id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math> decreases monotonically with <math alttext="\delta a_{l}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><mrow id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml">δ</mi><mo id="S3.I2.i2.p1.3.m3.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.cmml">⁢</mo><msub id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.3.2" xref="S3.I2.i2.p1.3.m3.1.1.3.2.cmml">a</mi><mi id="S3.I2.i2.p1.3.m3.1.1.3.3" xref="S3.I2.i2.p1.3.m3.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><times id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1"></times><ci id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">𝛿</ci><apply id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3.2">𝑎</ci><ci id="S3.I2.i2.p1.3.m3.1.1.3.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">\delta a_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>. </div> </li> <li class="ltx_item" id="S3.I2.i3" style="list-style-type:none;"> 3. <div class="ltx_para" id="S3.I2.i3.p1"> The uncertainty, <math alttext="\delta\overline{a}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.1.m1.1"><semantics id="S3.I2.i3.p1.1.m1.1a"><mrow id="S3.I2.i3.p1.1.m1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.cmml"><mi id="S3.I2.i3.p1.1.m1.1.1.2" xref="S3.I2.i3.p1.1.m1.1.1.2.cmml">δ</mi><mo id="S3.I2.i3.p1.1.m1.1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.I2.i3.p1.1.m1.1.1.3" xref="S3.I2.i3.p1.1.m1.1.1.3.cmml"><mi id="S3.I2.i3.p1.1.m1.1.1.3.2" xref="S3.I2.i3.p1.1.m1.1.1.3.2.cmml">a</mi><mo id="S3.I2.i3.p1.1.m1.1.1.3.1" xref="S3.I2.i3.p1.1.m1.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.1.m1.1b"><apply id="S3.I2.i3.p1.1.m1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1"><times id="S3.I2.i3.p1.1.m1.1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.1"></times><ci id="S3.I2.i3.p1.1.m1.1.1.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.2">𝛿</ci><apply id="S3.I2.i3.p1.1.m1.1.1.3.cmml" xref="S3.I2.i3.p1.1.m1.1.1.3"><ci id="S3.I2.i3.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1.3.1">¯</ci><ci id="S3.I2.i3.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.i3.p1.1.m1.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.1c">\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.1.m1.1d">italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math> is maximized when <math alttext="\delta a_{l}=\delta a" class="ltx_Math" display="inline" id="S3.I2.i3.p1.2.m2.1"><semantics id="S3.I2.i3.p1.2.m2.1a"><mrow id="S3.I2.i3.p1.2.m2.1.1" xref="S3.I2.i3.p1.2.m2.1.1.cmml"><mrow id="S3.I2.i3.p1.2.m2.1.1.2" xref="S3.I2.i3.p1.2.m2.1.1.2.cmml"><mi id="S3.I2.i3.p1.2.m2.1.1.2.2" xref="S3.I2.i3.p1.2.m2.1.1.2.2.cmml">δ</mi><mo id="S3.I2.i3.p1.2.m2.1.1.2.1" xref="S3.I2.i3.p1.2.m2.1.1.2.1.cmml">⁢</mo><msub id="S3.I2.i3.p1.2.m2.1.1.2.3" xref="S3.I2.i3.p1.2.m2.1.1.2.3.cmml"><mi id="S3.I2.i3.p1.2.m2.1.1.2.3.2" xref="S3.I2.i3.p1.2.m2.1.1.2.3.2.cmml">a</mi><mi id="S3.I2.i3.p1.2.m2.1.1.2.3.3" xref="S3.I2.i3.p1.2.m2.1.1.2.3.3.cmml">l</mi></msub></mrow><mo id="S3.I2.i3.p1.2.m2.1.1.1" xref="S3.I2.i3.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.I2.i3.p1.2.m2.1.1.3" xref="S3.I2.i3.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.i3.p1.2.m2.1.1.3.2" xref="S3.I2.i3.p1.2.m2.1.1.3.2.cmml">δ</mi><mo id="S3.I2.i3.p1.2.m2.1.1.3.1" xref="S3.I2.i3.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.I2.i3.p1.2.m2.1.1.3.3" xref="S3.I2.i3.p1.2.m2.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.2.m2.1b"><apply id="S3.I2.i3.p1.2.m2.1.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1"><eq id="S3.I2.i3.p1.2.m2.1.1.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1.1"></eq><apply id="S3.I2.i3.p1.2.m2.1.1.2.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2"><times id="S3.I2.i3.p1.2.m2.1.1.2.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.1"></times><ci id="S3.I2.i3.p1.2.m2.1.1.2.2.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.2">𝛿</ci><apply id="S3.I2.i3.p1.2.m2.1.1.2.3.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S3.I2.i3.p1.2.m2.1.1.2.3.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.3">subscript</csymbol><ci id="S3.I2.i3.p1.2.m2.1.1.2.3.2.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.3.2">𝑎</ci><ci id="S3.I2.i3.p1.2.m2.1.1.2.3.3.cmml" xref="S3.I2.i3.p1.2.m2.1.1.2.3.3">𝑙</ci></apply></apply><apply id="S3.I2.i3.p1.2.m2.1.1.3.cmml" xref="S3.I2.i3.p1.2.m2.1.1.3"><times id="S3.I2.i3.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1.3.1"></times><ci id="S3.I2.i3.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.i3.p1.2.m2.1.1.3.2">𝛿</ci><ci id="S3.I2.i3.p1.2.m2.1.1.3.3.cmml" xref="S3.I2.i3.p1.2.m2.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.2.m2.1c">\delta a_{l}=\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.2.m2.1d">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT = italic_δ italic_a</annotation></semantics></math>. For this value, <math alttext="\overline{a}=a/2" class="ltx_Math" display="inline" id="S3.I2.i3.p1.3.m3.1"><semantics id="S3.I2.i3.p1.3.m3.1a"><mrow id="S3.I2.i3.p1.3.m3.1.1" xref="S3.I2.i3.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.I2.i3.p1.3.m3.1.1.2" xref="S3.I2.i3.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i3.p1.3.m3.1.1.2.2" xref="S3.I2.i3.p1.3.m3.1.1.2.2.cmml">a</mi><mo id="S3.I2.i3.p1.3.m3.1.1.2.1" xref="S3.I2.i3.p1.3.m3.1.1.2.1.cmml">¯</mo></mover><mo id="S3.I2.i3.p1.3.m3.1.1.1" xref="S3.I2.i3.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S3.I2.i3.p1.3.m3.1.1.3" xref="S3.I2.i3.p1.3.m3.1.1.3.cmml"><mi id="S3.I2.i3.p1.3.m3.1.1.3.2" xref="S3.I2.i3.p1.3.m3.1.1.3.2.cmml">a</mi><mo id="S3.I2.i3.p1.3.m3.1.1.3.1" xref="S3.I2.i3.p1.3.m3.1.1.3.1.cmml">/</mo><mn id="S3.I2.i3.p1.3.m3.1.1.3.3" xref="S3.I2.i3.p1.3.m3.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.3.m3.1b"><apply id="S3.I2.i3.p1.3.m3.1.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1"><eq id="S3.I2.i3.p1.3.m3.1.1.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1.1"></eq><apply id="S3.I2.i3.p1.3.m3.1.1.2.cmml" xref="S3.I2.i3.p1.3.m3.1.1.2"><ci id="S3.I2.i3.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1.2.1">¯</ci><ci id="S3.I2.i3.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i3.p1.3.m3.1.1.2.2">𝑎</ci></apply><apply id="S3.I2.i3.p1.3.m3.1.1.3.cmml" xref="S3.I2.i3.p1.3.m3.1.1.3"><divide id="S3.I2.i3.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1.3.1"></divide><ci id="S3.I2.i3.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.i3.p1.3.m3.1.1.3.2">𝑎</ci><cn id="S3.I2.i3.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S3.I2.i3.p1.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.3.m3.1c">\overline{a}=a/2</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.3.m3.1d">over¯ start_ARG italic_a end_ARG = italic_a / 2</annotation></semantics></math>. Note the importance of scaling the uncertainty and thus the need to include the upper limit in the calculation of <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.I2.i3.p1.4.m4.1"><semantics id="S3.I2.i3.p1.4.m4.1a"><mi id="S3.I2.i3.p1.4.m4.1.1" mathvariant="normal" xref="S3.I2.i3.p1.4.m4.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.4.m4.1b"><ci id="S3.I2.i3.p1.4.m4.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.4.m4.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.4.m4.1d">roman_Σ</annotation></semantics></math>. </div> </li> </ol> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="830" id="S3.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 2: The <math alttext="\chi^{2}" class="ltx_Math" display="inline" id="S3.F2.11.m1.1"><semantics id="S3.F2.11.m1.1b"><msup id="S3.F2.11.m1.1.1" xref="S3.F2.11.m1.1.1.cmml"><mi id="S3.F2.11.m1.1.1.2" xref="S3.F2.11.m1.1.1.2.cmml">χ</mi><mn id="S3.F2.11.m1.1.1.3" xref="S3.F2.11.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.F2.11.m1.1c"><apply id="S3.F2.11.m1.1.1.cmml" xref="S3.F2.11.m1.1.1"><csymbol cd="ambiguous" id="S3.F2.11.m1.1.1.1.cmml" xref="S3.F2.11.m1.1.1">superscript</csymbol><ci id="S3.F2.11.m1.1.1.2.cmml" xref="S3.F2.11.m1.1.1.2">𝜒</ci><cn id="S3.F2.11.m1.1.1.3.cmml" type="integer" xref="S3.F2.11.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.11.m1.1d">\chi^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.11.m1.1e">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, average (<math alttext="\overline{a}" class="ltx_Math" display="inline" id="S3.F2.12.m2.1"><semantics id="S3.F2.12.m2.1b"><mover accent="true" id="S3.F2.12.m2.1.1" xref="S3.F2.12.m2.1.1.cmml"><mi id="S3.F2.12.m2.1.1.2" xref="S3.F2.12.m2.1.1.2.cmml">a</mi><mo id="S3.F2.12.m2.1.1.1" xref="S3.F2.12.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.F2.12.m2.1c"><apply id="S3.F2.12.m2.1.1.cmml" xref="S3.F2.12.m2.1.1"><ci id="S3.F2.12.m2.1.1.1.cmml" xref="S3.F2.12.m2.1.1.1">¯</ci><ci id="S3.F2.12.m2.1.1.2.cmml" xref="S3.F2.12.m2.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.12.m2.1d">\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.12.m2.1e">over¯ start_ARG italic_a end_ARG</annotation></semantics></math>) and uncertainty (<math alttext="\delta\overline{a}" class="ltx_Math" display="inline" id="S3.F2.13.m3.1"><semantics id="S3.F2.13.m3.1b"><mrow id="S3.F2.13.m3.1.1" xref="S3.F2.13.m3.1.1.cmml"><mi id="S3.F2.13.m3.1.1.2" xref="S3.F2.13.m3.1.1.2.cmml">δ</mi><mo id="S3.F2.13.m3.1.1.1" xref="S3.F2.13.m3.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.F2.13.m3.1.1.3" xref="S3.F2.13.m3.1.1.3.cmml"><mi id="S3.F2.13.m3.1.1.3.2" xref="S3.F2.13.m3.1.1.3.2.cmml">a</mi><mo id="S3.F2.13.m3.1.1.3.1" xref="S3.F2.13.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.13.m3.1c"><apply id="S3.F2.13.m3.1.1.cmml" xref="S3.F2.13.m3.1.1"><times id="S3.F2.13.m3.1.1.1.cmml" xref="S3.F2.13.m3.1.1.1"></times><ci id="S3.F2.13.m3.1.1.2.cmml" xref="S3.F2.13.m3.1.1.2">𝛿</ci><apply id="S3.F2.13.m3.1.1.3.cmml" xref="S3.F2.13.m3.1.1.3"><ci id="S3.F2.13.m3.1.1.3.1.cmml" xref="S3.F2.13.m3.1.1.3.1">¯</ci><ci id="S3.F2.13.m3.1.1.3.2.cmml" xref="S3.F2.13.m3.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.13.m3.1d">\delta\overline{a}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.13.m3.1e">italic_δ over¯ start_ARG italic_a end_ARG</annotation></semantics></math>), of a measured value, <math alttext="a\pm\delta a" class="ltx_Math" display="inline" id="S3.F2.14.m4.1"><semantics id="S3.F2.14.m4.1b"><mrow id="S3.F2.14.m4.1.1" xref="S3.F2.14.m4.1.1.cmml"><mi id="S3.F2.14.m4.1.1.2" xref="S3.F2.14.m4.1.1.2.cmml">a</mi><mo id="S3.F2.14.m4.1.1.1" xref="S3.F2.14.m4.1.1.1.cmml">±</mo><mrow id="S3.F2.14.m4.1.1.3" xref="S3.F2.14.m4.1.1.3.cmml"><mi id="S3.F2.14.m4.1.1.3.2" xref="S3.F2.14.m4.1.1.3.2.cmml">δ</mi><mo id="S3.F2.14.m4.1.1.3.1" xref="S3.F2.14.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.F2.14.m4.1.1.3.3" xref="S3.F2.14.m4.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.14.m4.1c"><apply id="S3.F2.14.m4.1.1.cmml" xref="S3.F2.14.m4.1.1"><csymbol cd="latexml" id="S3.F2.14.m4.1.1.1.cmml" xref="S3.F2.14.m4.1.1.1">plus-or-minus</csymbol><ci id="S3.F2.14.m4.1.1.2.cmml" xref="S3.F2.14.m4.1.1.2">𝑎</ci><apply id="S3.F2.14.m4.1.1.3.cmml" xref="S3.F2.14.m4.1.1.3"><times id="S3.F2.14.m4.1.1.3.1.cmml" xref="S3.F2.14.m4.1.1.3.1"></times><ci id="S3.F2.14.m4.1.1.3.2.cmml" xref="S3.F2.14.m4.1.1.3.2">𝛿</ci><ci id="S3.F2.14.m4.1.1.3.3.cmml" xref="S3.F2.14.m4.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.14.m4.1d">a\pm\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.F2.14.m4.1e">italic_a ± italic_δ italic_a</annotation></semantics></math>, and the Gaussian upper limit, <math alttext="\delta a_{l}" class="ltx_Math" display="inline" id="S3.F2.15.m5.1"><semantics id="S3.F2.15.m5.1b"><mrow id="S3.F2.15.m5.1.1" xref="S3.F2.15.m5.1.1.cmml"><mi id="S3.F2.15.m5.1.1.2" xref="S3.F2.15.m5.1.1.2.cmml">δ</mi><mo id="S3.F2.15.m5.1.1.1" xref="S3.F2.15.m5.1.1.1.cmml">⁢</mo><msub id="S3.F2.15.m5.1.1.3" xref="S3.F2.15.m5.1.1.3.cmml"><mi id="S3.F2.15.m5.1.1.3.2" xref="S3.F2.15.m5.1.1.3.2.cmml">a</mi><mi id="S3.F2.15.m5.1.1.3.3" xref="S3.F2.15.m5.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.15.m5.1c"><apply id="S3.F2.15.m5.1.1.cmml" xref="S3.F2.15.m5.1.1"><times id="S3.F2.15.m5.1.1.1.cmml" xref="S3.F2.15.m5.1.1.1"></times><ci id="S3.F2.15.m5.1.1.2.cmml" xref="S3.F2.15.m5.1.1.2">𝛿</ci><apply id="S3.F2.15.m5.1.1.3.cmml" xref="S3.F2.15.m5.1.1.3"><csymbol cd="ambiguous" id="S3.F2.15.m5.1.1.3.1.cmml" xref="S3.F2.15.m5.1.1.3">subscript</csymbol><ci id="S3.F2.15.m5.1.1.3.2.cmml" xref="S3.F2.15.m5.1.1.3.2">𝑎</ci><ci id="S3.F2.15.m5.1.1.3.3.cmml" xref="S3.F2.15.m5.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.15.m5.1d">\delta a_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.15.m5.1e">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>, defined in the text. <math alttext="\delta a" class="ltx_Math" display="inline" id="S3.F2.16.m6.1"><semantics id="S3.F2.16.m6.1b"><mrow id="S3.F2.16.m6.1.1" xref="S3.F2.16.m6.1.1.cmml"><mi id="S3.F2.16.m6.1.1.2" xref="S3.F2.16.m6.1.1.2.cmml">δ</mi><mo id="S3.F2.16.m6.1.1.1" xref="S3.F2.16.m6.1.1.1.cmml">⁢</mo><mi id="S3.F2.16.m6.1.1.3" xref="S3.F2.16.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.16.m6.1c"><apply id="S3.F2.16.m6.1.1.cmml" xref="S3.F2.16.m6.1.1"><times id="S3.F2.16.m6.1.1.1.cmml" xref="S3.F2.16.m6.1.1.1"></times><ci id="S3.F2.16.m6.1.1.2.cmml" xref="S3.F2.16.m6.1.1.2">𝛿</ci><ci id="S3.F2.16.m6.1.1.3.cmml" xref="S3.F2.16.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.16.m6.1d">\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.F2.16.m6.1e">italic_δ italic_a</annotation></semantics></math> is set to <math alttext="0.2a" class="ltx_Math" display="inline" id="S3.F2.17.m7.1"><semantics id="S3.F2.17.m7.1b"><mrow id="S3.F2.17.m7.1.1" xref="S3.F2.17.m7.1.1.cmml"><mn id="S3.F2.17.m7.1.1.2" xref="S3.F2.17.m7.1.1.2.cmml">0.2</mn><mo id="S3.F2.17.m7.1.1.1" xref="S3.F2.17.m7.1.1.1.cmml">⁢</mo><mi id="S3.F2.17.m7.1.1.3" xref="S3.F2.17.m7.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.17.m7.1c"><apply id="S3.F2.17.m7.1.1.cmml" xref="S3.F2.17.m7.1.1"><times id="S3.F2.17.m7.1.1.1.cmml" xref="S3.F2.17.m7.1.1.1"></times><cn id="S3.F2.17.m7.1.1.2.cmml" type="float" xref="S3.F2.17.m7.1.1.2">0.2</cn><ci id="S3.F2.17.m7.1.1.3.cmml" xref="S3.F2.17.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.17.m7.1d">0.2a</annotation><annotation encoding="application/x-llamapun" id="S3.F2.17.m7.1e">0.2 italic_a</annotation></semantics></math>. A dashed and dash-dotted vertical line is shown at <math alttext="\delta a_{l}=\delta a" class="ltx_Math" display="inline" id="S3.F2.18.m8.1"><semantics id="S3.F2.18.m8.1b"><mrow id="S3.F2.18.m8.1.1" xref="S3.F2.18.m8.1.1.cmml"><mrow id="S3.F2.18.m8.1.1.2" xref="S3.F2.18.m8.1.1.2.cmml"><mi id="S3.F2.18.m8.1.1.2.2" xref="S3.F2.18.m8.1.1.2.2.cmml">δ</mi><mo id="S3.F2.18.m8.1.1.2.1" xref="S3.F2.18.m8.1.1.2.1.cmml">⁢</mo><msub id="S3.F2.18.m8.1.1.2.3" xref="S3.F2.18.m8.1.1.2.3.cmml"><mi id="S3.F2.18.m8.1.1.2.3.2" xref="S3.F2.18.m8.1.1.2.3.2.cmml">a</mi><mi id="S3.F2.18.m8.1.1.2.3.3" xref="S3.F2.18.m8.1.1.2.3.3.cmml">l</mi></msub></mrow><mo id="S3.F2.18.m8.1.1.1" xref="S3.F2.18.m8.1.1.1.cmml">=</mo><mrow id="S3.F2.18.m8.1.1.3" xref="S3.F2.18.m8.1.1.3.cmml"><mi id="S3.F2.18.m8.1.1.3.2" xref="S3.F2.18.m8.1.1.3.2.cmml">δ</mi><mo id="S3.F2.18.m8.1.1.3.1" xref="S3.F2.18.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.F2.18.m8.1.1.3.3" xref="S3.F2.18.m8.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml 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encoding="application/x-tex" id="S3.F2.18.m8.1d">\delta a_{l}=\delta a</annotation><annotation encoding="application/x-llamapun" id="S3.F2.18.m8.1e">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT = italic_δ italic_a</annotation></semantics></math> and at the <math alttext="\delta a_{l}" class="ltx_Math" display="inline" id="S3.F2.19.m9.1"><semantics id="S3.F2.19.m9.1b"><mrow id="S3.F2.19.m9.1.1" xref="S3.F2.19.m9.1.1.cmml"><mi id="S3.F2.19.m9.1.1.2" xref="S3.F2.19.m9.1.1.2.cmml">δ</mi><mo id="S3.F2.19.m9.1.1.1" xref="S3.F2.19.m9.1.1.1.cmml">⁢</mo><msub id="S3.F2.19.m9.1.1.3" xref="S3.F2.19.m9.1.1.3.cmml"><mi id="S3.F2.19.m9.1.1.3.2" xref="S3.F2.19.m9.1.1.3.2.cmml">a</mi><mi id="S3.F2.19.m9.1.1.3.3" xref="S3.F2.19.m9.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.19.m9.1c"><apply id="S3.F2.19.m9.1.1.cmml" xref="S3.F2.19.m9.1.1"><times id="S3.F2.19.m9.1.1.1.cmml" xref="S3.F2.19.m9.1.1.1"></times><ci id="S3.F2.19.m9.1.1.2.cmml" xref="S3.F2.19.m9.1.1.2">𝛿</ci><apply id="S3.F2.19.m9.1.1.3.cmml" xref="S3.F2.19.m9.1.1.3"><csymbol cd="ambiguous" id="S3.F2.19.m9.1.1.3.1.cmml" xref="S3.F2.19.m9.1.1.3">subscript</csymbol><ci id="S3.F2.19.m9.1.1.3.2.cmml" xref="S3.F2.19.m9.1.1.3.2">𝑎</ci><ci id="S3.F2.19.m9.1.1.3.3.cmml" xref="S3.F2.19.m9.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.19.m9.1d">\delta a_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.19.m9.1e">italic_δ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> which results in <math alttext="\chi^{2}=1" class="ltx_Math" display="inline" id="S3.F2.20.m10.1"><semantics id="S3.F2.20.m10.1b"><mrow id="S3.F2.20.m10.1.1" xref="S3.F2.20.m10.1.1.cmml"><msup id="S3.F2.20.m10.1.1.2" xref="S3.F2.20.m10.1.1.2.cmml"><mi id="S3.F2.20.m10.1.1.2.2" xref="S3.F2.20.m10.1.1.2.2.cmml">χ</mi><mn id="S3.F2.20.m10.1.1.2.3" xref="S3.F2.20.m10.1.1.2.3.cmml">2</mn></msup><mo id="S3.F2.20.m10.1.1.1" xref="S3.F2.20.m10.1.1.1.cmml">=</mo><mn id="S3.F2.20.m10.1.1.3" xref="S3.F2.20.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.20.m10.1c"><apply id="S3.F2.20.m10.1.1.cmml" xref="S3.F2.20.m10.1.1"><eq id="S3.F2.20.m10.1.1.1.cmml" xref="S3.F2.20.m10.1.1.1"></eq><apply id="S3.F2.20.m10.1.1.2.cmml" xref="S3.F2.20.m10.1.1.2"><csymbol cd="ambiguous" id="S3.F2.20.m10.1.1.2.1.cmml" xref="S3.F2.20.m10.1.1.2">superscript</csymbol><ci id="S3.F2.20.m10.1.1.2.2.cmml" xref="S3.F2.20.m10.1.1.2.2">𝜒</ci><cn id="S3.F2.20.m10.1.1.2.3.cmml" type="integer" xref="S3.F2.20.m10.1.1.2.3">2</cn></apply><cn id="S3.F2.20.m10.1.1.3.cmml" type="integer" xref="S3.F2.20.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.20.m10.1d">\chi^{2}=1</annotation><annotation encoding="application/x-llamapun" id="S3.F2.20.m10.1e">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S3.p7"> If assays have to be averaged more than once, multiple upper limits may contribute to the result. For example, 40 of the Demonstrator’s lead bricks were assayed with the following procedure. One or more samples were taken from each brick, and each sample was used to prepare one or more dilutions to conduct ICPMS measurements. In some cases each dilution was separated into multiple vials, which were separately measured. Thus averaging is performed at the dilution, sample, brick and global levels, with upper limits contributing at each stage. </div> <div class="ltx_para" id="S3.p8"> The specific activity averages and uncertainties of the materials used in the Majorana Demonstrator are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3.T1" title="Table 1 ‣ III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">1</a>. These values are used to calculate the BI contributions of all 232Th and 238U sources used to model the experiment. In this table, <math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.p8.3.m3.1"><semantics id="S3.p8.3.m3.1a"><msub id="S3.p8.3.m3.1.1" xref="S3.p8.3.m3.1.1.cmml"><mi id="S3.p8.3.m3.1.1.2" xref="S3.p8.3.m3.1.1.2.cmml">N</mi><mi id="S3.p8.3.m3.1.1.3" xref="S3.p8.3.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.3.m3.1b"><apply id="S3.p8.3.m3.1.1.cmml" xref="S3.p8.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p8.3.m3.1.1.1.cmml" xref="S3.p8.3.m3.1.1">subscript</csymbol><ci id="S3.p8.3.m3.1.1.2.cmml" xref="S3.p8.3.m3.1.1.2">𝑁</ci><ci id="S3.p8.3.m3.1.1.3.cmml" xref="S3.p8.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.3.m3.1c">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.3.m3.1d">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> represents the number of assays contributing at the global averaging level. While the set of assayed components is limited, it is assumed to be representative of all those used in the Demonstrator. </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption">Table 1: Measured (<math alttext="N_{a}=1" class="ltx_Math" display="inline" id="S3.T1.10.m1.1"><semantics id="S3.T1.10.m1.1b"><mrow id="S3.T1.10.m1.1.1" xref="S3.T1.10.m1.1.1.cmml"><msub id="S3.T1.10.m1.1.1.2" xref="S3.T1.10.m1.1.1.2.cmml"><mi id="S3.T1.10.m1.1.1.2.2" xref="S3.T1.10.m1.1.1.2.2.cmml">N</mi><mi id="S3.T1.10.m1.1.1.2.3" xref="S3.T1.10.m1.1.1.2.3.cmml">a</mi></msub><mo id="S3.T1.10.m1.1.1.1" xref="S3.T1.10.m1.1.1.1.cmml">=</mo><mn id="S3.T1.10.m1.1.1.3" xref="S3.T1.10.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.10.m1.1c"><apply id="S3.T1.10.m1.1.1.cmml" xref="S3.T1.10.m1.1.1"><eq id="S3.T1.10.m1.1.1.1.cmml" xref="S3.T1.10.m1.1.1.1"></eq><apply id="S3.T1.10.m1.1.1.2.cmml" xref="S3.T1.10.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T1.10.m1.1.1.2.1.cmml" xref="S3.T1.10.m1.1.1.2">subscript</csymbol><ci id="S3.T1.10.m1.1.1.2.2.cmml" xref="S3.T1.10.m1.1.1.2.2">𝑁</ci><ci id="S3.T1.10.m1.1.1.2.3.cmml" xref="S3.T1.10.m1.1.1.2.3">𝑎</ci></apply><cn id="S3.T1.10.m1.1.1.3.cmml" type="integer" xref="S3.T1.10.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.10.m1.1d">N_{a}=1</annotation><annotation encoding="application/x-llamapun" id="S3.T1.10.m1.1e">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = 1</annotation></semantics></math>) or combined (<math alttext="N_{a}>1" class="ltx_Math" display="inline" id="S3.T1.11.m2.1"><semantics id="S3.T1.11.m2.1b"><mrow id="S3.T1.11.m2.1.1" xref="S3.T1.11.m2.1.1.cmml"><msub id="S3.T1.11.m2.1.1.2" xref="S3.T1.11.m2.1.1.2.cmml"><mi id="S3.T1.11.m2.1.1.2.2" xref="S3.T1.11.m2.1.1.2.2.cmml">N</mi><mi id="S3.T1.11.m2.1.1.2.3" xref="S3.T1.11.m2.1.1.2.3.cmml">a</mi></msub><mo id="S3.T1.11.m2.1.1.1" xref="S3.T1.11.m2.1.1.1.cmml">></mo><mn id="S3.T1.11.m2.1.1.3" xref="S3.T1.11.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.11.m2.1c"><apply id="S3.T1.11.m2.1.1.cmml" xref="S3.T1.11.m2.1.1"><gt id="S3.T1.11.m2.1.1.1.cmml" xref="S3.T1.11.m2.1.1.1"></gt><apply id="S3.T1.11.m2.1.1.2.cmml" xref="S3.T1.11.m2.1.1.2"><csymbol cd="ambiguous" id="S3.T1.11.m2.1.1.2.1.cmml" xref="S3.T1.11.m2.1.1.2">subscript</csymbol><ci id="S3.T1.11.m2.1.1.2.2.cmml" xref="S3.T1.11.m2.1.1.2.2">𝑁</ci><ci id="S3.T1.11.m2.1.1.2.3.cmml" xref="S3.T1.11.m2.1.1.2.3">𝑎</ci></apply><cn id="S3.T1.11.m2.1.1.3.cmml" type="integer" xref="S3.T1.11.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.11.m2.1d">N_{a}>1</annotation><annotation encoding="application/x-llamapun" id="S3.T1.11.m2.1e">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT > 1</annotation></semantics></math>, using the technique described in Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3" title="III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">III</a>) bulk 232Th and 238U specific activities of materials used in the Majorana Demonstrator. Where applicable, the most stringent 90% C.L. upper limit is used. The assay methodology is the same for all <math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.T1.14.m5.1"><semantics id="S3.T1.14.m5.1b"><msub id="S3.T1.14.m5.1.1" xref="S3.T1.14.m5.1.1.cmml"><mi id="S3.T1.14.m5.1.1.2" xref="S3.T1.14.m5.1.1.2.cmml">N</mi><mi id="S3.T1.14.m5.1.1.3" xref="S3.T1.14.m5.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.14.m5.1c"><apply id="S3.T1.14.m5.1.1.cmml" xref="S3.T1.14.m5.1.1"><csymbol cd="ambiguous" id="S3.T1.14.m5.1.1.1.cmml" xref="S3.T1.14.m5.1.1">subscript</csymbol><ci id="S3.T1.14.m5.1.1.2.cmml" xref="S3.T1.14.m5.1.1.2">𝑁</ci><ci id="S3.T1.14.m5.1.1.3.cmml" xref="S3.T1.14.m5.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.14.m5.1d">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.14.m5.1e">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> assays in a given row. If <math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.T1.15.m6.1"><semantics id="S3.T1.15.m6.1b"><msub id="S3.T1.15.m6.1.1" xref="S3.T1.15.m6.1.1.cmml"><mi id="S3.T1.15.m6.1.1.2" xref="S3.T1.15.m6.1.1.2.cmml">N</mi><mi id="S3.T1.15.m6.1.1.3" xref="S3.T1.15.m6.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.15.m6.1c"><apply id="S3.T1.15.m6.1.1.cmml" xref="S3.T1.15.m6.1.1"><csymbol cd="ambiguous" id="S3.T1.15.m6.1.1.1.cmml" xref="S3.T1.15.m6.1.1">subscript</csymbol><ci id="S3.T1.15.m6.1.1.2.cmml" xref="S3.T1.15.m6.1.1.2">𝑁</ci><ci id="S3.T1.15.m6.1.1.3.cmml" xref="S3.T1.15.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.15.m6.1d">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.15.m6.1e">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is different for 232Th and 238U, then comma separated values are given. Machined UGEFCu samples were fabricated from stock UGEFCu. The latter is assumed for the specific activity of the inner copper shield of the Demonstrator. Signal cables were separated into two sections. Custom connectors were designed and built to connect these sections, with the female end wired to an LMFE and the male end wired to pre-amplification and readout electronics. When calculating the BI, all components made from a given material are collected in the group shown in the second column. A total of <math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.T1.18.m9.1"><semantics id="S3.T1.18.m9.1b"><msub id="S3.T1.18.m9.1.1" xref="S3.T1.18.m9.1.1.cmml"><mi id="S3.T1.18.m9.1.1.2" xref="S3.T1.18.m9.1.1.2.cmml">N</mi><mi id="S3.T1.18.m9.1.1.3" xref="S3.T1.18.m9.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.18.m9.1c"><apply id="S3.T1.18.m9.1.1.cmml" xref="S3.T1.18.m9.1.1"><csymbol cd="ambiguous" id="S3.T1.18.m9.1.1.1.cmml" xref="S3.T1.18.m9.1.1">subscript</csymbol><ci id="S3.T1.18.m9.1.1.2.cmml" xref="S3.T1.18.m9.1.1.2">𝑁</ci><ci id="S3.T1.18.m9.1.1.3.cmml" xref="S3.T1.18.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.18.m9.1d">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.18.m9.1e">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> assays – sourced from Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite> and additional measurement campaigns – are combined. </figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S3.T1.59"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.23.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column" id="S3.T1.23.5.6">Material</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.23.5.7">Group</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.20.2.2"> 232Th [<math alttext="\upmu" class="ltx_Math" display="inline" id="S3.T1.20.2.2.m2.1"><semantics id="S3.T1.20.2.2.m2.1a"><mi id="S3.T1.20.2.2.m2.1.1" mathvariant="normal" xref="S3.T1.20.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.T1.20.2.2.m2.1b"><ci id="S3.T1.20.2.2.m2.1.1.cmml" xref="S3.T1.20.2.2.m2.1.1">μ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.20.2.2.m2.1c">\upmu</annotation><annotation encoding="application/x-llamapun" id="S3.T1.20.2.2.m2.1d">roman_μ</annotation></semantics></math>Bq/kg]</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.22.4.4"> 238U [<math alttext="\upmu" class="ltx_Math" display="inline" id="S3.T1.22.4.4.m2.1"><semantics id="S3.T1.22.4.4.m2.1a"><mi id="S3.T1.22.4.4.m2.1.1" mathvariant="normal" xref="S3.T1.22.4.4.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.T1.22.4.4.m2.1b"><ci id="S3.T1.22.4.4.m2.1.1.cmml" xref="S3.T1.22.4.4.m2.1.1">μ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.22.4.4.m2.1c">\upmu</annotation><annotation encoding="application/x-llamapun" id="S3.T1.22.4.4.m2.1d">roman_μ</annotation></semantics></math>Bq/kg]</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.23.5.5"><math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.T1.23.5.5.m1.1"><semantics id="S3.T1.23.5.5.m1.1a"><msub id="S3.T1.23.5.5.m1.1.1" xref="S3.T1.23.5.5.m1.1.1.cmml"><mi id="S3.T1.23.5.5.m1.1.1.2" xref="S3.T1.23.5.5.m1.1.1.2.cmml">N</mi><mi id="S3.T1.23.5.5.m1.1.1.3" xref="S3.T1.23.5.5.m1.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.23.5.5.m1.1b"><apply id="S3.T1.23.5.5.m1.1.1.cmml" xref="S3.T1.23.5.5.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.23.5.5.m1.1.1.1.cmml" xref="S3.T1.23.5.5.m1.1.1">subscript</csymbol><ci id="S3.T1.23.5.5.m1.1.1.2.cmml" xref="S3.T1.23.5.5.m1.1.1.2">𝑁</ci><ci id="S3.T1.23.5.5.m1.1.1.3.cmml" xref="S3.T1.23.5.5.m1.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.23.5.5.m1.1c">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.23.5.5.m1.1d">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.23.5.8">Method</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.25.7"> <td class="ltx_td ltx_align_left ltx_border_t" id="S3.T1.25.7.3">LMFE</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.25.7.4">Front Ends</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.24.6.1">6950 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.24.6.1.m1.1"><semantics id="S3.T1.24.6.1.m1.1a"><mo id="S3.T1.24.6.1.m1.1.1" xref="S3.T1.24.6.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.24.6.1.m1.1b"><csymbol cd="latexml" id="S3.T1.24.6.1.m1.1.1.cmml" xref="S3.T1.24.6.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.24.6.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.24.6.1.m1.1d">±</annotation></semantics></math> 830</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.25.7.2">10600 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.25.7.2.m1.1"><semantics id="S3.T1.25.7.2.m1.1a"><mo id="S3.T1.25.7.2.m1.1.1" xref="S3.T1.25.7.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.25.7.2.m1.1b"><csymbol cd="latexml" id="S3.T1.25.7.2.m1.1.1.cmml" xref="S3.T1.25.7.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.25.7.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.25.7.2.m1.1d">±</annotation></semantics></math> 300</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.25.7.5">2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.25.7.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.27.9"> <td class="ltx_td ltx_align_left" id="S3.T1.27.9.3">HV Cable</td> <td class="ltx_td ltx_align_center" id="S3.T1.27.9.4">Cables</td> <td class="ltx_td ltx_align_center" id="S3.T1.26.8.1">87.9 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.26.8.1.m1.1"><semantics id="S3.T1.26.8.1.m1.1a"><mo id="S3.T1.26.8.1.m1.1.1" xref="S3.T1.26.8.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.26.8.1.m1.1b"><csymbol cd="latexml" id="S3.T1.26.8.1.m1.1.1.cmml" xref="S3.T1.26.8.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.26.8.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.26.8.1.m1.1d">±</annotation></semantics></math> 52.4</td> <td class="ltx_td ltx_align_center" id="S3.T1.27.9.2">231 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.27.9.2.m1.1"><semantics id="S3.T1.27.9.2.m1.1a"><mo id="S3.T1.27.9.2.m1.1.1" xref="S3.T1.27.9.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.27.9.2.m1.1b"><csymbol cd="latexml" id="S3.T1.27.9.2.m1.1.1.cmml" xref="S3.T1.27.9.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.27.9.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.27.9.2.m1.1d">±</annotation></semantics></math> 34</td> <td class="ltx_td ltx_align_center" id="S3.T1.27.9.5">3</td> <td class="ltx_td ltx_align_center" id="S3.T1.27.9.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.29.11"> <td class="ltx_td ltx_align_left" id="S3.T1.29.11.3">Signal Cable</td> <td class="ltx_td ltx_align_center" id="S3.T1.29.11.4">Cables</td> <td class="ltx_td ltx_align_center" id="S3.T1.28.10.1">546 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.28.10.1.m1.1"><semantics id="S3.T1.28.10.1.m1.1a"><mo id="S3.T1.28.10.1.m1.1.1" xref="S3.T1.28.10.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.28.10.1.m1.1b"><csymbol cd="latexml" id="S3.T1.28.10.1.m1.1.1.cmml" xref="S3.T1.28.10.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.28.10.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.28.10.1.m1.1d">±</annotation></semantics></math> 112</td> <td class="ltx_td ltx_align_center" id="S3.T1.29.11.2">530 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.29.11.2.m1.1"><semantics id="S3.T1.29.11.2.m1.1a"><mo id="S3.T1.29.11.2.m1.1.1" xref="S3.T1.29.11.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.29.11.2.m1.1b"><csymbol cd="latexml" id="S3.T1.29.11.2.m1.1.1.cmml" xref="S3.T1.29.11.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.29.11.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.29.11.2.m1.1d">±</annotation></semantics></math> 58</td> <td class="ltx_td ltx_align_center" id="S3.T1.29.11.5">3</td> <td class="ltx_td ltx_align_center" id="S3.T1.29.11.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.31.13"> <td class="ltx_td ltx_align_left" id="S3.T1.31.13.3">Stock UGEFCu</td> <td class="ltx_td ltx_align_center" id="S3.T1.31.13.4">Electroformed Cu</td> <td class="ltx_td ltx_align_center" id="S3.T1.30.12.1">0.188 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.30.12.1.m1.1"><semantics id="S3.T1.30.12.1.m1.1a"><mo id="S3.T1.30.12.1.m1.1.1" xref="S3.T1.30.12.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.30.12.1.m1.1b"><csymbol cd="latexml" id="S3.T1.30.12.1.m1.1.1.cmml" xref="S3.T1.30.12.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.30.12.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.30.12.1.m1.1d">±</annotation></semantics></math> 0.029</td> <td class="ltx_td ltx_align_center" id="S3.T1.31.13.2">0.137 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.31.13.2.m1.1"><semantics id="S3.T1.31.13.2.m1.1a"><mo id="S3.T1.31.13.2.m1.1.1" xref="S3.T1.31.13.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.31.13.2.m1.1b"><csymbol cd="latexml" id="S3.T1.31.13.2.m1.1.1.cmml" xref="S3.T1.31.13.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.31.13.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.31.13.2.m1.1d">±</annotation></semantics></math> 0.044</td> <td class="ltx_td ltx_align_center" id="S3.T1.31.13.5">5</td> <td class="ltx_td ltx_align_center" id="S3.T1.31.13.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.33.15"> <td class="ltx_td ltx_align_left" id="S3.T1.33.15.3">Machined UGEFCu</td> <td class="ltx_td ltx_align_center" id="S3.T1.33.15.4">Electroformed Cu</td> <td class="ltx_td ltx_align_center" id="S3.T1.32.14.1">0.575 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.32.14.1.m1.1"><semantics id="S3.T1.32.14.1.m1.1a"><mo id="S3.T1.32.14.1.m1.1.1" xref="S3.T1.32.14.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.32.14.1.m1.1b"><csymbol cd="latexml" id="S3.T1.32.14.1.m1.1.1.cmml" xref="S3.T1.32.14.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.32.14.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.32.14.1.m1.1d">±</annotation></semantics></math> 0.088</td> <td class="ltx_td ltx_align_center" id="S3.T1.33.15.2">0.752 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.33.15.2.m1.1"><semantics id="S3.T1.33.15.2.m1.1a"><mo id="S3.T1.33.15.2.m1.1.1" xref="S3.T1.33.15.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.33.15.2.m1.1b"><csymbol cd="latexml" id="S3.T1.33.15.2.m1.1.1.cmml" xref="S3.T1.33.15.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.33.15.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.33.15.2.m1.1d">±</annotation></semantics></math> 0.083</td> <td class="ltx_td ltx_align_center" id="S3.T1.33.15.5">12</td> <td class="ltx_td ltx_align_center" id="S3.T1.33.15.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.35.17"> <td class="ltx_td ltx_align_left" id="S3.T1.35.17.3">OFHCCu</td> <td class="ltx_td ltx_align_center" id="S3.T1.35.17.4">OFHC Cu Shielding</td> <td class="ltx_td ltx_align_center" id="S3.T1.34.16.1">1.10 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.34.16.1.m1.1"><semantics id="S3.T1.34.16.1.m1.1a"><mo id="S3.T1.34.16.1.m1.1.1" xref="S3.T1.34.16.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.34.16.1.m1.1b"><csymbol cd="latexml" id="S3.T1.34.16.1.m1.1.1.cmml" xref="S3.T1.34.16.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.34.16.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.34.16.1.m1.1d">±</annotation></semantics></math> 0.14</td> <td class="ltx_td ltx_align_center" id="S3.T1.35.17.2">1.37 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.35.17.2.m1.1"><semantics id="S3.T1.35.17.2.m1.1a"><mo id="S3.T1.35.17.2.m1.1.1" xref="S3.T1.35.17.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.35.17.2.m1.1b"><csymbol cd="latexml" id="S3.T1.35.17.2.m1.1.1.cmml" xref="S3.T1.35.17.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.35.17.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.35.17.2.m1.1d">±</annotation></semantics></math> 0.18</td> <td class="ltx_td ltx_align_center" id="S3.T1.35.17.5">2</td> <td class="ltx_td ltx_align_center" id="S3.T1.35.17.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.37.19"> <td class="ltx_td ltx_align_left" id="S3.T1.37.19.3">Pb Bricks</td> <td class="ltx_td ltx_align_center" id="S3.T1.37.19.4">Pb Shielding</td> <td class="ltx_td ltx_align_center" id="S3.T1.36.18.1">9.53 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.36.18.1.m1.1"><semantics id="S3.T1.36.18.1.m1.1a"><mo id="S3.T1.36.18.1.m1.1.1" xref="S3.T1.36.18.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.36.18.1.m1.1b"><csymbol cd="latexml" id="S3.T1.36.18.1.m1.1.1.cmml" xref="S3.T1.36.18.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.36.18.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.36.18.1.m1.1d">±</annotation></semantics></math> 1.01</td> <td class="ltx_td ltx_align_center" id="S3.T1.37.19.2">25.6 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.37.19.2.m1.1"><semantics id="S3.T1.37.19.2.m1.1a"><mo id="S3.T1.37.19.2.m1.1.1" xref="S3.T1.37.19.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.37.19.2.m1.1b"><csymbol cd="latexml" id="S3.T1.37.19.2.m1.1.1.cmml" xref="S3.T1.37.19.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.37.19.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.37.19.2.m1.1d">±</annotation></semantics></math> 1.5</td> <td class="ltx_td ltx_align_center" id="S3.T1.37.19.5">29, 19</td> <td class="ltx_td ltx_align_center" id="S3.T1.37.19.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.39.21"> <td class="ltx_td ltx_align_left" id="S3.T1.39.21.3">Female Connector</td> <td class="ltx_td ltx_align_center" id="S3.T1.39.21.4">Connectors</td> <td class="ltx_td ltx_align_center" id="S3.T1.38.20.1">390 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.38.20.1.m1.1"><semantics id="S3.T1.38.20.1.m1.1a"><mo id="S3.T1.38.20.1.m1.1.1" xref="S3.T1.38.20.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.38.20.1.m1.1b"><csymbol cd="latexml" id="S3.T1.38.20.1.m1.1.1.cmml" xref="S3.T1.38.20.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.38.20.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.38.20.1.m1.1d">±</annotation></semantics></math> 7</td> <td class="ltx_td ltx_align_center" id="S3.T1.39.21.2">540 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.39.21.2.m1.1"><semantics id="S3.T1.39.21.2.m1.1a"><mo id="S3.T1.39.21.2.m1.1.1" xref="S3.T1.39.21.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.39.21.2.m1.1b"><csymbol cd="latexml" id="S3.T1.39.21.2.m1.1.1.cmml" xref="S3.T1.39.21.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.39.21.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.39.21.2.m1.1d">±</annotation></semantics></math> 9</td> <td class="ltx_td ltx_align_center" id="S3.T1.39.21.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.39.21.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.41.23"> <td class="ltx_td ltx_align_left" id="S3.T1.41.23.3">Male Connector</td> <td class="ltx_td ltx_align_center" id="S3.T1.41.23.4">Connectors</td> <td class="ltx_td ltx_align_center" id="S3.T1.40.22.1">28.8 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.40.22.1.m1.1"><semantics id="S3.T1.40.22.1.m1.1a"><mo id="S3.T1.40.22.1.m1.1.1" xref="S3.T1.40.22.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.40.22.1.m1.1b"><csymbol cd="latexml" id="S3.T1.40.22.1.m1.1.1.cmml" xref="S3.T1.40.22.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.40.22.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.40.22.1.m1.1d">±</annotation></semantics></math> 2.0</td> <td class="ltx_td ltx_align_center" id="S3.T1.41.23.2">130 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.41.23.2.m1.1"><semantics id="S3.T1.41.23.2.m1.1a"><mo id="S3.T1.41.23.2.m1.1.1" xref="S3.T1.41.23.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.41.23.2.m1.1b"><csymbol cd="latexml" id="S3.T1.41.23.2.m1.1.1.cmml" xref="S3.T1.41.23.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.41.23.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.41.23.2.m1.1d">±</annotation></semantics></math> 11</td> <td class="ltx_td ltx_align_center" id="S3.T1.41.23.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.41.23.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.43.25"> <td class="ltx_td ltx_align_left" id="S3.T1.43.25.3">PTFE O-ring</td> <td class="ltx_td ltx_align_center" id="S3.T1.43.25.4">Other Plastics</td> <td class="ltx_td ltx_align_center" id="S3.T1.42.24.1">39.7 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.42.24.1.m1.1"><semantics id="S3.T1.42.24.1.m1.1a"><mo id="S3.T1.42.24.1.m1.1.1" xref="S3.T1.42.24.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.42.24.1.m1.1b"><csymbol cd="latexml" id="S3.T1.42.24.1.m1.1.1.cmml" xref="S3.T1.42.24.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.42.24.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.42.24.1.m1.1d">±</annotation></semantics></math> 31.9</td> <td class="ltx_td ltx_align_center" id="S3.T1.43.25.2"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.43.25.2.m1.1"><semantics id="S3.T1.43.25.2.m1.1a"><mo id="S3.T1.43.25.2.m1.1.1" xref="S3.T1.43.25.2.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.43.25.2.m1.1b"><lt id="S3.T1.43.25.2.m1.1.1.cmml" xref="S3.T1.43.25.2.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.43.25.2.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.43.25.2.m1.1d"><</annotation></semantics></math> 105</td> <td class="ltx_td ltx_align_center" id="S3.T1.43.25.5">2, 1</td> <td class="ltx_td ltx_align_center" id="S3.T1.43.25.6">NAA</td> </tr> <tr class="ltx_tr" id="S3.T1.45.27"> <td class="ltx_td ltx_align_left" id="S3.T1.45.27.3">PTFE</td> <td class="ltx_td ltx_align_center" id="S3.T1.45.27.4">Detector Unit PTFE</td> <td class="ltx_td ltx_align_center" id="S3.T1.44.26.1">0.101 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.44.26.1.m1.1"><semantics id="S3.T1.44.26.1.m1.1a"><mo id="S3.T1.44.26.1.m1.1.1" xref="S3.T1.44.26.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.44.26.1.m1.1b"><csymbol cd="latexml" id="S3.T1.44.26.1.m1.1.1.cmml" xref="S3.T1.44.26.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.44.26.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.44.26.1.m1.1d">±</annotation></semantics></math> 0.008</td> <td class="ltx_td ltx_align_center" id="S3.T1.45.27.2"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.45.27.2.m1.1"><semantics id="S3.T1.45.27.2.m1.1a"><mo id="S3.T1.45.27.2.m1.1.1" xref="S3.T1.45.27.2.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.45.27.2.m1.1b"><lt id="S3.T1.45.27.2.m1.1.1.cmml" xref="S3.T1.45.27.2.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.45.27.2.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.45.27.2.m1.1d"><</annotation></semantics></math>4.97</td> <td class="ltx_td ltx_align_center" id="S3.T1.45.27.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.45.27.6">NAA</td> </tr> <tr class="ltx_tr" id="S3.T1.47.29"> <td class="ltx_td ltx_align_left" id="S3.T1.47.29.3">DuPontTM Vespel® </td> <td class="ltx_td ltx_align_center" id="S3.T1.47.29.4">Other Plastics</td> <td class="ltx_td ltx_align_center" id="S3.T1.46.28.1">360 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.46.28.1.m1.1"><semantics id="S3.T1.46.28.1.m1.1a"><mo id="S3.T1.46.28.1.m1.1.1" xref="S3.T1.46.28.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.46.28.1.m1.1b"><csymbol cd="latexml" id="S3.T1.46.28.1.m1.1.1.cmml" xref="S3.T1.46.28.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.46.28.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.46.28.1.m1.1d">±</annotation></semantics></math> 234</td> <td class="ltx_td ltx_align_center" id="S3.T1.47.29.2">403 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.47.29.2.m1.1"><semantics id="S3.T1.47.29.2.m1.1a"><mo id="S3.T1.47.29.2.m1.1.1" xref="S3.T1.47.29.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.47.29.2.m1.1b"><csymbol cd="latexml" id="S3.T1.47.29.2.m1.1.1.cmml" xref="S3.T1.47.29.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.47.29.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.47.29.2.m1.1d">±</annotation></semantics></math> 179</td> <td class="ltx_td ltx_align_center" id="S3.T1.47.29.5">3</td> <td class="ltx_td ltx_align_center" id="S3.T1.47.29.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.49.31"> <td class="ltx_td ltx_align_left" id="S3.T1.49.31.3">PTFE Gasket</td> <td class="ltx_td ltx_align_center" id="S3.T1.49.31.4">Other Plastics</td> <td class="ltx_td ltx_align_center" id="S3.T1.48.30.1"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.48.30.1.m1.1"><semantics id="S3.T1.48.30.1.m1.1a"><mo id="S3.T1.48.30.1.m1.1.1" xref="S3.T1.48.30.1.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.48.30.1.m1.1b"><lt id="S3.T1.48.30.1.m1.1.1.cmml" xref="S3.T1.48.30.1.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.48.30.1.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.48.30.1.m1.1d"><</annotation></semantics></math>20.7</td> <td class="ltx_td ltx_align_center" id="S3.T1.49.31.2"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.49.31.2.m1.1"><semantics id="S3.T1.49.31.2.m1.1a"><mo id="S3.T1.49.31.2.m1.1.1" xref="S3.T1.49.31.2.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.49.31.2.m1.1b"><lt id="S3.T1.49.31.2.m1.1.1.cmml" xref="S3.T1.49.31.2.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.49.31.2.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.49.31.2.m1.1d"><</annotation></semantics></math>94.5</td> <td class="ltx_td ltx_align_center" id="S3.T1.49.31.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.49.31.6">NAA</td> </tr> <tr class="ltx_tr" id="S3.T1.52.34"> <td class="ltx_td ltx_align_left" id="S3.T1.52.34.4">Stainless Steel</td> <td class="ltx_td ltx_align_center" id="S3.T1.52.34.5">Vacuum Hardware</td> <td class="ltx_td ltx_align_center" id="S3.T1.50.32.1">13000 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.50.32.1.m1.1"><semantics id="S3.T1.50.32.1.m1.1a"><mo id="S3.T1.50.32.1.m1.1.1" xref="S3.T1.50.32.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.50.32.1.m1.1b"><csymbol cd="latexml" id="S3.T1.50.32.1.m1.1.1.cmml" xref="S3.T1.50.32.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.50.32.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.50.32.1.m1.1d">±</annotation></semantics></math> 4000</td> <td class="ltx_td ltx_align_center" id="S3.T1.51.33.2"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.51.33.2.m1.1"><semantics id="S3.T1.51.33.2.m1.1a"><mo id="S3.T1.51.33.2.m1.1.1" xref="S3.T1.51.33.2.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.51.33.2.m1.1b"><lt id="S3.T1.51.33.2.m1.1.1.cmml" xref="S3.T1.51.33.2.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.51.33.2.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.51.33.2.m1.1d"><</annotation></semantics></math>5000</td> <td class="ltx_td ltx_align_center" id="S3.T1.52.34.6">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.52.34.3"> <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.T1.52.34.3.m1.1"><semantics id="S3.T1.52.34.3.m1.1a"><mi id="S3.T1.52.34.3.m1.1.1" xref="S3.T1.52.34.3.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.T1.52.34.3.m1.1b"><ci id="S3.T1.52.34.3.m1.1.1.cmml" xref="S3.T1.52.34.3.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.52.34.3.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.T1.52.34.3.m1.1d">italic_γ</annotation></semantics></math>-count</td> </tr> <tr class="ltx_tr" id="S3.T1.55.37"> <td class="ltx_td ltx_align_left" id="S3.T1.55.37.4">Glass Break</td> <td class="ltx_td ltx_align_center" id="S3.T1.55.37.5">Vacuum Hardware</td> <td class="ltx_td ltx_align_center" id="S3.T1.53.35.1">49000 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.53.35.1.m1.1"><semantics id="S3.T1.53.35.1.m1.1a"><mo id="S3.T1.53.35.1.m1.1.1" xref="S3.T1.53.35.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.53.35.1.m1.1b"><csymbol cd="latexml" id="S3.T1.53.35.1.m1.1.1.cmml" xref="S3.T1.53.35.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.53.35.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.53.35.1.m1.1d">±</annotation></semantics></math> 8000</td> <td class="ltx_td ltx_align_center" id="S3.T1.54.36.2">160000 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.54.36.2.m1.1"><semantics id="S3.T1.54.36.2.m1.1a"><mo id="S3.T1.54.36.2.m1.1.1" xref="S3.T1.54.36.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.54.36.2.m1.1b"><csymbol cd="latexml" id="S3.T1.54.36.2.m1.1.1.cmml" xref="S3.T1.54.36.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.54.36.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.54.36.2.m1.1d">±</annotation></semantics></math> 10000</td> <td class="ltx_td ltx_align_center" id="S3.T1.55.37.6">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.55.37.3"> <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.T1.55.37.3.m1.1"><semantics id="S3.T1.55.37.3.m1.1a"><mi id="S3.T1.55.37.3.m1.1.1" xref="S3.T1.55.37.3.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.T1.55.37.3.m1.1b"><ci id="S3.T1.55.37.3.m1.1.1.cmml" xref="S3.T1.55.37.3.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.55.37.3.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.T1.55.37.3.m1.1d">italic_γ</annotation></semantics></math>-count</td> </tr> <tr class="ltx_tr" id="S3.T1.57.39"> <td class="ltx_td ltx_align_left" id="S3.T1.57.39.3">PTFE Tubing</td> <td class="ltx_td ltx_align_center" id="S3.T1.57.39.4">Other Plastics</td> <td class="ltx_td ltx_align_center" id="S3.T1.56.38.1">6.09 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.56.38.1.m1.1"><semantics id="S3.T1.56.38.1.m1.1a"><mo id="S3.T1.56.38.1.m1.1.1" xref="S3.T1.56.38.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.56.38.1.m1.1b"><csymbol cd="latexml" id="S3.T1.56.38.1.m1.1.1.cmml" xref="S3.T1.56.38.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.56.38.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.56.38.1.m1.1d">±</annotation></semantics></math> 7.30</td> <td class="ltx_td ltx_align_center" id="S3.T1.57.39.2"> <math alttext="<" class="ltx_Math" display="inline" id="S3.T1.57.39.2.m1.1"><semantics id="S3.T1.57.39.2.m1.1a"><mo id="S3.T1.57.39.2.m1.1.1" xref="S3.T1.57.39.2.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S3.T1.57.39.2.m1.1b"><lt id="S3.T1.57.39.2.m1.1.1.cmml" xref="S3.T1.57.39.2.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.57.39.2.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="S3.T1.57.39.2.m1.1d"><</annotation></semantics></math>38.6</td> <td class="ltx_td ltx_align_center" id="S3.T1.57.39.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.57.39.6">ICPMS</td> </tr> <tr class="ltx_tr" id="S3.T1.59.41"> <td class="ltx_td ltx_align_left" id="S3.T1.59.41.3">Parylene</td> <td class="ltx_td ltx_align_center" id="S3.T1.59.41.4">Parylene</td> <td class="ltx_td ltx_align_center" id="S3.T1.58.40.1">2150 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.58.40.1.m1.1"><semantics id="S3.T1.58.40.1.m1.1a"><mo id="S3.T1.58.40.1.m1.1.1" xref="S3.T1.58.40.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.58.40.1.m1.1b"><csymbol cd="latexml" id="S3.T1.58.40.1.m1.1.1.cmml" xref="S3.T1.58.40.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.58.40.1.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.58.40.1.m1.1d">±</annotation></semantics></math> 120</td> <td class="ltx_td ltx_align_center" id="S3.T1.59.41.2">3110 <math alttext="\pm" class="ltx_Math" display="inline" id="S3.T1.59.41.2.m1.1"><semantics id="S3.T1.59.41.2.m1.1a"><mo id="S3.T1.59.41.2.m1.1.1" xref="S3.T1.59.41.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S3.T1.59.41.2.m1.1b"><csymbol cd="latexml" id="S3.T1.59.41.2.m1.1.1.cmml" xref="S3.T1.59.41.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.59.41.2.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S3.T1.59.41.2.m1.1d">±</annotation></semantics></math> 750</td> <td class="ltx_td ltx_align_center" id="S3.T1.59.41.5">1</td> <td class="ltx_td ltx_align_center" id="S3.T1.59.41.6">ICPMS</td> </tr> </tbody> </table> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> IV Background index</h2> <div class="ltx_para" id="S4.p1"> Simulations predict an approximately flat background in the 370 keV split BEW covering 1950-2350 keV. The BEW has three 10 keV regions removed, centered at 2103 keV, the 208Tl (232Th) single escape peak, and 2118 keV and 2204 keV, the 214Bi (238U) peaks. This BEW is used to estimate the BI at <math alttext="Q_{\beta\beta}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><msub id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml">Q</mi><mrow id="S4.p1.5.m5.1.1.3" xref="S4.p1.5.m5.1.1.3.cmml"><mi id="S4.p1.5.m5.1.1.3.2" xref="S4.p1.5.m5.1.1.3.2.cmml">β</mi><mo id="S4.p1.5.m5.1.1.3.1" xref="S4.p1.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S4.p1.5.m5.1.1.3.3" xref="S4.p1.5.m5.1.1.3.3.cmml">β</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1">subscript</csymbol><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">𝑄</ci><apply id="S4.p1.5.m5.1.1.3.cmml" xref="S4.p1.5.m5.1.1.3"><times id="S4.p1.5.m5.1.1.3.1.cmml" xref="S4.p1.5.m5.1.1.3.1"></times><ci id="S4.p1.5.m5.1.1.3.2.cmml" xref="S4.p1.5.m5.1.1.3.2">𝛽</ci><ci id="S4.p1.5.m5.1.1.3.3.cmml" xref="S4.p1.5.m5.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">Q_{\beta\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT</annotation></semantics></math>, for both simulations and data. When estimating the BI from data an additional 10 keV region, centered at <math alttext="Q_{\beta\beta}" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><msub id="S4.p1.6.m6.1.1" xref="S4.p1.6.m6.1.1.cmml"><mi id="S4.p1.6.m6.1.1.2" xref="S4.p1.6.m6.1.1.2.cmml">Q</mi><mrow id="S4.p1.6.m6.1.1.3" xref="S4.p1.6.m6.1.1.3.cmml"><mi id="S4.p1.6.m6.1.1.3.2" xref="S4.p1.6.m6.1.1.3.2.cmml">β</mi><mo id="S4.p1.6.m6.1.1.3.1" xref="S4.p1.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S4.p1.6.m6.1.1.3.3" xref="S4.p1.6.m6.1.1.3.3.cmml">β</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><apply id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.p1.6.m6.1.1.1.cmml" xref="S4.p1.6.m6.1.1">subscript</csymbol><ci id="S4.p1.6.m6.1.1.2.cmml" xref="S4.p1.6.m6.1.1.2">𝑄</ci><apply id="S4.p1.6.m6.1.1.3.cmml" xref="S4.p1.6.m6.1.1.3"><times id="S4.p1.6.m6.1.1.3.1.cmml" xref="S4.p1.6.m6.1.1.3.1"></times><ci id="S4.p1.6.m6.1.1.3.2.cmml" xref="S4.p1.6.m6.1.1.3.2">𝛽</ci><ci id="S4.p1.6.m6.1.1.3.3.cmml" xref="S4.p1.6.m6.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">Q_{\beta\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT</annotation></semantics></math>, is removed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib1" title="">1</a>]</cite>. </div> <div class="ltx_para" id="S4.p2"> The contribution to the Demonstrator’s BI from a natural radiation source <math alttext="i" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_i</annotation></semantics></math> – the 232Th or 238U contamination in a component of mass <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><msub id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml"><mi id="S4.p2.4.m4.1.1.2" xref="S4.p2.4.m4.1.1.2.cmml">m</mi><mi id="S4.p2.4.m4.1.1.3" xref="S4.p2.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.1b"><apply id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.p2.4.m4.1.1.1.cmml" xref="S4.p2.4.m4.1.1">subscript</csymbol><ci id="S4.p2.4.m4.1.1.2.cmml" xref="S4.p2.4.m4.1.1.2">𝑚</ci><ci id="S4.p2.4.m4.1.1.3.cmml" xref="S4.p2.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> – is calculated as follows. For each segment, <math alttext="j" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mi id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><ci id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">italic_j</annotation></semantics></math>, of the decay chain of the source’s contaminant, MaGe simulates <math alttext="N_{ij}" class="ltx_Math" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><msub id="S4.p2.6.m6.1.1" xref="S4.p2.6.m6.1.1.cmml"><mi id="S4.p2.6.m6.1.1.2" xref="S4.p2.6.m6.1.1.2.cmml">N</mi><mrow id="S4.p2.6.m6.1.1.3" xref="S4.p2.6.m6.1.1.3.cmml"><mi id="S4.p2.6.m6.1.1.3.2" xref="S4.p2.6.m6.1.1.3.2.cmml">i</mi><mo id="S4.p2.6.m6.1.1.3.1" xref="S4.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.6.m6.1.1.3.3" xref="S4.p2.6.m6.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.6.m6.1b"><apply id="S4.p2.6.m6.1.1.cmml" xref="S4.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.p2.6.m6.1.1.1.cmml" xref="S4.p2.6.m6.1.1">subscript</csymbol><ci id="S4.p2.6.m6.1.1.2.cmml" xref="S4.p2.6.m6.1.1.2">𝑁</ci><apply id="S4.p2.6.m6.1.1.3.cmml" xref="S4.p2.6.m6.1.1.3"><times id="S4.p2.6.m6.1.1.3.1.cmml" xref="S4.p2.6.m6.1.1.3.1"></times><ci id="S4.p2.6.m6.1.1.3.2.cmml" xref="S4.p2.6.m6.1.1.3.2">𝑖</ci><ci id="S4.p2.6.m6.1.1.3.3.cmml" xref="S4.p2.6.m6.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">N_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">italic_N start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> decays. For a given source <math alttext="i" class="ltx_Math" display="inline" id="S4.p2.7.m7.1"><semantics id="S4.p2.7.m7.1a"><mi id="S4.p2.7.m7.1.1" xref="S4.p2.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.p2.7.m7.1b"><ci id="S4.p2.7.m7.1.1.cmml" xref="S4.p2.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m7.1d">italic_i</annotation></semantics></math>, all <math alttext="N_{ij}" class="ltx_Math" display="inline" id="S4.p2.8.m8.1"><semantics id="S4.p2.8.m8.1a"><msub id="S4.p2.8.m8.1.1" xref="S4.p2.8.m8.1.1.cmml"><mi id="S4.p2.8.m8.1.1.2" xref="S4.p2.8.m8.1.1.2.cmml">N</mi><mrow id="S4.p2.8.m8.1.1.3" xref="S4.p2.8.m8.1.1.3.cmml"><mi id="S4.p2.8.m8.1.1.3.2" xref="S4.p2.8.m8.1.1.3.2.cmml">i</mi><mo id="S4.p2.8.m8.1.1.3.1" xref="S4.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.8.m8.1.1.3.3" xref="S4.p2.8.m8.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.8.m8.1b"><apply id="S4.p2.8.m8.1.1.cmml" xref="S4.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.p2.8.m8.1.1.1.cmml" xref="S4.p2.8.m8.1.1">subscript</csymbol><ci id="S4.p2.8.m8.1.1.2.cmml" xref="S4.p2.8.m8.1.1.2">𝑁</ci><apply id="S4.p2.8.m8.1.1.3.cmml" xref="S4.p2.8.m8.1.1.3"><times id="S4.p2.8.m8.1.1.3.1.cmml" xref="S4.p2.8.m8.1.1.3.1"></times><ci id="S4.p2.8.m8.1.1.3.2.cmml" xref="S4.p2.8.m8.1.1.3.2">𝑖</ci><ci id="S4.p2.8.m8.1.1.3.3.cmml" xref="S4.p2.8.m8.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m8.1c">N_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m8.1d">italic_N start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are equal up to computational errors. Simulated events leading to energy depositions in an operational detector are subject to the same anti-coincidence and pulse shape analysis cuts applied to data (designed to select <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S4.p2.9.m9.1"><semantics id="S4.p2.9.m9.1a"><mrow id="S4.p2.9.m9.1.1" xref="S4.p2.9.m9.1.1.cmml"><mn id="S4.p2.9.m9.1.1.2" xref="S4.p2.9.m9.1.1.2.cmml">0</mn><mo id="S4.p2.9.m9.1.1.1" xref="S4.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S4.p2.9.m9.1.1.3" xref="S4.p2.9.m9.1.1.3.cmml">ν</mi><mo id="S4.p2.9.m9.1.1.1a" xref="S4.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S4.p2.9.m9.1.1.4" xref="S4.p2.9.m9.1.1.4.cmml">β</mi><mo id="S4.p2.9.m9.1.1.1b" xref="S4.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S4.p2.9.m9.1.1.5" xref="S4.p2.9.m9.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.9.m9.1b"><apply id="S4.p2.9.m9.1.1.cmml" xref="S4.p2.9.m9.1.1"><times id="S4.p2.9.m9.1.1.1.cmml" xref="S4.p2.9.m9.1.1.1"></times><cn id="S4.p2.9.m9.1.1.2.cmml" type="integer" xref="S4.p2.9.m9.1.1.2">0</cn><ci id="S4.p2.9.m9.1.1.3.cmml" xref="S4.p2.9.m9.1.1.3">𝜈</ci><ci id="S4.p2.9.m9.1.1.4.cmml" xref="S4.p2.9.m9.1.1.4">𝛽</ci><ci id="S4.p2.9.m9.1.1.5.cmml" xref="S4.p2.9.m9.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.9.m9.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S4.p2.9.m9.1d">0 italic_ν italic_β italic_β</annotation></semantics></math>-like single-site events). The number of counts passing all cuts, <math alttext="n_{ij}" class="ltx_Math" display="inline" id="S4.p2.10.m10.1"><semantics id="S4.p2.10.m10.1a"><msub id="S4.p2.10.m10.1.1" xref="S4.p2.10.m10.1.1.cmml"><mi id="S4.p2.10.m10.1.1.2" xref="S4.p2.10.m10.1.1.2.cmml">n</mi><mrow id="S4.p2.10.m10.1.1.3" xref="S4.p2.10.m10.1.1.3.cmml"><mi id="S4.p2.10.m10.1.1.3.2" xref="S4.p2.10.m10.1.1.3.2.cmml">i</mi><mo id="S4.p2.10.m10.1.1.3.1" xref="S4.p2.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.10.m10.1.1.3.3" xref="S4.p2.10.m10.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.10.m10.1b"><apply id="S4.p2.10.m10.1.1.cmml" xref="S4.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.p2.10.m10.1.1.1.cmml" xref="S4.p2.10.m10.1.1">subscript</csymbol><ci id="S4.p2.10.m10.1.1.2.cmml" xref="S4.p2.10.m10.1.1.2">𝑛</ci><apply id="S4.p2.10.m10.1.1.3.cmml" xref="S4.p2.10.m10.1.1.3"><times id="S4.p2.10.m10.1.1.3.1.cmml" xref="S4.p2.10.m10.1.1.3.1"></times><ci id="S4.p2.10.m10.1.1.3.2.cmml" xref="S4.p2.10.m10.1.1.3.2">𝑖</ci><ci id="S4.p2.10.m10.1.1.3.3.cmml" xref="S4.p2.10.m10.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m10.1c">n_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m10.1d">italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, is calculated by integrating the resulting combined detector spectra over the BEW. The corresponding segment efficiency is thus: <table class="ltx_equation ltx_eqn_table" id="S4.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="\epsilon_{ij}^{\textsc{Mjd}}=n_{ij}/N_{ij}" class="ltx_Math" display="block" id="S4.E4.m1.1"><semantics id="S4.E4.m1.1a"><mrow id="S4.E4.m1.1.1" xref="S4.E4.m1.1.1.cmml"><msubsup id="S4.E4.m1.1.1.2" xref="S4.E4.m1.1.1.2.cmml"><mi id="S4.E4.m1.1.1.2.2.2" xref="S4.E4.m1.1.1.2.2.2.cmml">ϵ</mi><mrow id="S4.E4.m1.1.1.2.2.3" xref="S4.E4.m1.1.1.2.2.3.cmml"><mi id="S4.E4.m1.1.1.2.2.3.2" xref="S4.E4.m1.1.1.2.2.3.2.cmml">i</mi><mo id="S4.E4.m1.1.1.2.2.3.1" xref="S4.E4.m1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E4.m1.1.1.2.2.3.3" xref="S4.E4.m1.1.1.2.2.3.3.cmml">j</mi></mrow><mtext class="ltx_font_smallcaps" id="S4.E4.m1.1.1.2.3" xref="S4.E4.m1.1.1.2.3a.cmml">Mjd</mtext></msubsup><mo id="S4.E4.m1.1.1.1" xref="S4.E4.m1.1.1.1.cmml">=</mo><mrow id="S4.E4.m1.1.1.3" xref="S4.E4.m1.1.1.3.cmml"><msub id="S4.E4.m1.1.1.3.2" xref="S4.E4.m1.1.1.3.2.cmml"><mi id="S4.E4.m1.1.1.3.2.2" xref="S4.E4.m1.1.1.3.2.2.cmml">n</mi><mrow id="S4.E4.m1.1.1.3.2.3" xref="S4.E4.m1.1.1.3.2.3.cmml"><mi id="S4.E4.m1.1.1.3.2.3.2" xref="S4.E4.m1.1.1.3.2.3.2.cmml">i</mi><mo id="S4.E4.m1.1.1.3.2.3.1" xref="S4.E4.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S4.E4.m1.1.1.3.2.3.3" xref="S4.E4.m1.1.1.3.2.3.3.cmml">j</mi></mrow></msub><mo id="S4.E4.m1.1.1.3.1" xref="S4.E4.m1.1.1.3.1.cmml">/</mo><msub id="S4.E4.m1.1.1.3.3" xref="S4.E4.m1.1.1.3.3.cmml"><mi id="S4.E4.m1.1.1.3.3.2" xref="S4.E4.m1.1.1.3.3.2.cmml">N</mi><mrow id="S4.E4.m1.1.1.3.3.3" xref="S4.E4.m1.1.1.3.3.3.cmml"><mi id="S4.E4.m1.1.1.3.3.3.2" xref="S4.E4.m1.1.1.3.3.3.2.cmml">i</mi><mo id="S4.E4.m1.1.1.3.3.3.1" xref="S4.E4.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S4.E4.m1.1.1.3.3.3.3" xref="S4.E4.m1.1.1.3.3.3.3.cmml">j</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.1b"><apply id="S4.E4.m1.1.1.cmml" xref="S4.E4.m1.1.1"><eq id="S4.E4.m1.1.1.1.cmml" xref="S4.E4.m1.1.1.1"></eq><apply id="S4.E4.m1.1.1.2.cmml" xref="S4.E4.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.2.1.cmml" xref="S4.E4.m1.1.1.2">superscript</csymbol><apply id="S4.E4.m1.1.1.2.2.cmml" xref="S4.E4.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.2.2.1.cmml" xref="S4.E4.m1.1.1.2">subscript</csymbol><ci id="S4.E4.m1.1.1.2.2.2.cmml" xref="S4.E4.m1.1.1.2.2.2">italic-ϵ</ci><apply id="S4.E4.m1.1.1.2.2.3.cmml" xref="S4.E4.m1.1.1.2.2.3"><times id="S4.E4.m1.1.1.2.2.3.1.cmml" xref="S4.E4.m1.1.1.2.2.3.1"></times><ci id="S4.E4.m1.1.1.2.2.3.2.cmml" xref="S4.E4.m1.1.1.2.2.3.2">𝑖</ci><ci id="S4.E4.m1.1.1.2.2.3.3.cmml" xref="S4.E4.m1.1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.E4.m1.1.1.2.3a.cmml" xref="S4.E4.m1.1.1.2.3"><mtext class="ltx_font_smallcaps" id="S4.E4.m1.1.1.2.3.cmml" mathsize="70%" xref="S4.E4.m1.1.1.2.3">Mjd</mtext></ci></apply><apply 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id="S4.E4.m1.1.1.3.3.3.2.cmml" xref="S4.E4.m1.1.1.3.3.3.2">𝑖</ci><ci id="S4.E4.m1.1.1.3.3.3.3.cmml" xref="S4.E4.m1.1.1.3.3.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.1c">\epsilon_{ij}^{\textsc{Mjd}}=n_{ij}/N_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.1d">italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT / italic_N start_POSTSUBSCRIPT italic_i 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">(4)</td> </tr></tbody> </table> The segment efficiencies are weighted by the branching ratio of each segment, <math alttext="\mathcal{B}_{j}" class="ltx_Math" display="inline" id="S4.p2.11.m1.1"><semantics id="S4.p2.11.m1.1a"><msub id="S4.p2.11.m1.1.1" xref="S4.p2.11.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.11.m1.1.1.2" xref="S4.p2.11.m1.1.1.2.cmml">ℬ</mi><mi id="S4.p2.11.m1.1.1.3" xref="S4.p2.11.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.11.m1.1b"><apply id="S4.p2.11.m1.1.1.cmml" xref="S4.p2.11.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.11.m1.1.1.1.cmml" xref="S4.p2.11.m1.1.1">subscript</csymbol><ci id="S4.p2.11.m1.1.1.2.cmml" xref="S4.p2.11.m1.1.1.2">ℬ</ci><ci id="S4.p2.11.m1.1.1.3.cmml" xref="S4.p2.11.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m1.1c">\mathcal{B}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m1.1d">caligraphic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, and summed over the total number of segments, <math alttext="S_{i}" class="ltx_Math" display="inline" id="S4.p2.12.m2.1"><semantics id="S4.p2.12.m2.1a"><msub id="S4.p2.12.m2.1.1" xref="S4.p2.12.m2.1.1.cmml"><mi id="S4.p2.12.m2.1.1.2" xref="S4.p2.12.m2.1.1.2.cmml">S</mi><mi id="S4.p2.12.m2.1.1.3" xref="S4.p2.12.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.12.m2.1b"><apply id="S4.p2.12.m2.1.1.cmml" xref="S4.p2.12.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.12.m2.1.1.1.cmml" xref="S4.p2.12.m2.1.1">subscript</csymbol><ci id="S4.p2.12.m2.1.1.2.cmml" xref="S4.p2.12.m2.1.1.2">𝑆</ci><ci id="S4.p2.12.m2.1.1.3.cmml" xref="S4.p2.12.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m2.1c">S_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m2.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, to produce the source efficiency, <math alttext="\epsilon_{i}^{\textsc{Mjd}}" class="ltx_Math" display="inline" id="S4.p2.13.m3.1"><semantics id="S4.p2.13.m3.1a"><msubsup id="S4.p2.13.m3.1.1" xref="S4.p2.13.m3.1.1.cmml"><mi id="S4.p2.13.m3.1.1.2.2" xref="S4.p2.13.m3.1.1.2.2.cmml">ϵ</mi><mi id="S4.p2.13.m3.1.1.2.3" xref="S4.p2.13.m3.1.1.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S4.p2.13.m3.1.1.3" xref="S4.p2.13.m3.1.1.3a.cmml">Mjd</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S4.p2.13.m3.1b"><apply id="S4.p2.13.m3.1.1.cmml" xref="S4.p2.13.m3.1.1"><csymbol cd="ambiguous" id="S4.p2.13.m3.1.1.1.cmml" xref="S4.p2.13.m3.1.1">superscript</csymbol><apply id="S4.p2.13.m3.1.1.2.cmml" xref="S4.p2.13.m3.1.1"><csymbol cd="ambiguous" id="S4.p2.13.m3.1.1.2.1.cmml" xref="S4.p2.13.m3.1.1">subscript</csymbol><ci id="S4.p2.13.m3.1.1.2.2.cmml" xref="S4.p2.13.m3.1.1.2.2">italic-ϵ</ci><ci id="S4.p2.13.m3.1.1.2.3.cmml" xref="S4.p2.13.m3.1.1.2.3">𝑖</ci></apply><ci id="S4.p2.13.m3.1.1.3a.cmml" xref="S4.p2.13.m3.1.1.3"><mtext class="ltx_font_smallcaps" id="S4.p2.13.m3.1.1.3.cmml" mathsize="70%" xref="S4.p2.13.m3.1.1.3">Mjd</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.13.m3.1c">\epsilon_{i}^{\textsc{Mjd}}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.13.m3.1d">italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT</annotation></semantics></math>: <table class="ltx_equation ltx_eqn_table" id="S4.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="\epsilon_{i}^{\textsc{Mjd}}=\sum_{j}^{S_{i}}~{}\mathcal{B}_{j}\epsilon_{ij}^{% \textsc{Mjd}}" class="ltx_Math" display="block" id="S4.E5.m1.1"><semantics id="S4.E5.m1.1a"><mrow id="S4.E5.m1.1.1" xref="S4.E5.m1.1.1.cmml"><msubsup id="S4.E5.m1.1.1.2" xref="S4.E5.m1.1.1.2.cmml"><mi id="S4.E5.m1.1.1.2.2.2" xref="S4.E5.m1.1.1.2.2.2.cmml">ϵ</mi><mi id="S4.E5.m1.1.1.2.2.3" xref="S4.E5.m1.1.1.2.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S4.E5.m1.1.1.2.3" xref="S4.E5.m1.1.1.2.3a.cmml">Mjd</mtext></msubsup><mo id="S4.E5.m1.1.1.1" rspace="0.111em" xref="S4.E5.m1.1.1.1.cmml">=</mo><mrow id="S4.E5.m1.1.1.3" xref="S4.E5.m1.1.1.3.cmml"><munderover id="S4.E5.m1.1.1.3.1" xref="S4.E5.m1.1.1.3.1.cmml"><mo id="S4.E5.m1.1.1.3.1.2.2" movablelimits="false" xref="S4.E5.m1.1.1.3.1.2.2.cmml">∑</mo><mi id="S4.E5.m1.1.1.3.1.2.3" xref="S4.E5.m1.1.1.3.1.2.3.cmml">j</mi><msub id="S4.E5.m1.1.1.3.1.3" xref="S4.E5.m1.1.1.3.1.3.cmml"><mi id="S4.E5.m1.1.1.3.1.3.2" xref="S4.E5.m1.1.1.3.1.3.2.cmml">S</mi><mi id="S4.E5.m1.1.1.3.1.3.3" xref="S4.E5.m1.1.1.3.1.3.3.cmml">i</mi></msub></munderover><mrow id="S4.E5.m1.1.1.3.2" xref="S4.E5.m1.1.1.3.2.cmml"><msub id="S4.E5.m1.1.1.3.2.2" xref="S4.E5.m1.1.1.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.1.1.3.2.2.2" xref="S4.E5.m1.1.1.3.2.2.2.cmml">ℬ</mi><mi id="S4.E5.m1.1.1.3.2.2.3" xref="S4.E5.m1.1.1.3.2.2.3.cmml">j</mi></msub><mo id="S4.E5.m1.1.1.3.2.1" xref="S4.E5.m1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S4.E5.m1.1.1.3.2.3" xref="S4.E5.m1.1.1.3.2.3.cmml"><mi id="S4.E5.m1.1.1.3.2.3.2.2" xref="S4.E5.m1.1.1.3.2.3.2.2.cmml">ϵ</mi><mrow id="S4.E5.m1.1.1.3.2.3.2.3" xref="S4.E5.m1.1.1.3.2.3.2.3.cmml"><mi id="S4.E5.m1.1.1.3.2.3.2.3.2" xref="S4.E5.m1.1.1.3.2.3.2.3.2.cmml">i</mi><mo id="S4.E5.m1.1.1.3.2.3.2.3.1" xref="S4.E5.m1.1.1.3.2.3.2.3.1.cmml">⁢</mo><mi id="S4.E5.m1.1.1.3.2.3.2.3.3" xref="S4.E5.m1.1.1.3.2.3.2.3.3.cmml">j</mi></mrow><mtext class="ltx_font_smallcaps" id="S4.E5.m1.1.1.3.2.3.3" xref="S4.E5.m1.1.1.3.2.3.3a.cmml">Mjd</mtext></msubsup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E5.m1.1b"><apply id="S4.E5.m1.1.1.cmml" xref="S4.E5.m1.1.1"><eq id="S4.E5.m1.1.1.1.cmml" xref="S4.E5.m1.1.1.1"></eq><apply id="S4.E5.m1.1.1.2.cmml" xref="S4.E5.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.2.1.cmml" xref="S4.E5.m1.1.1.2">superscript</csymbol><apply id="S4.E5.m1.1.1.2.2.cmml" xref="S4.E5.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.2.2.1.cmml" xref="S4.E5.m1.1.1.2">subscript</csymbol><ci id="S4.E5.m1.1.1.2.2.2.cmml" xref="S4.E5.m1.1.1.2.2.2">italic-ϵ</ci><ci id="S4.E5.m1.1.1.2.2.3.cmml" xref="S4.E5.m1.1.1.2.2.3">𝑖</ci></apply><ci id="S4.E5.m1.1.1.2.3a.cmml" xref="S4.E5.m1.1.1.2.3"><mtext class="ltx_font_smallcaps" id="S4.E5.m1.1.1.2.3.cmml" mathsize="70%" xref="S4.E5.m1.1.1.2.3">Mjd</mtext></ci></apply><apply id="S4.E5.m1.1.1.3.cmml" xref="S4.E5.m1.1.1.3"><apply id="S4.E5.m1.1.1.3.1.cmml" xref="S4.E5.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.1.1.cmml" xref="S4.E5.m1.1.1.3.1">superscript</csymbol><apply id="S4.E5.m1.1.1.3.1.2.cmml" xref="S4.E5.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.1.2.1.cmml" xref="S4.E5.m1.1.1.3.1">subscript</csymbol><sum id="S4.E5.m1.1.1.3.1.2.2.cmml" xref="S4.E5.m1.1.1.3.1.2.2"></sum><ci id="S4.E5.m1.1.1.3.1.2.3.cmml" xref="S4.E5.m1.1.1.3.1.2.3">𝑗</ci></apply><apply id="S4.E5.m1.1.1.3.1.3.cmml" xref="S4.E5.m1.1.1.3.1.3"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.1.3.1.cmml" xref="S4.E5.m1.1.1.3.1.3">subscript</csymbol><ci id="S4.E5.m1.1.1.3.1.3.2.cmml" xref="S4.E5.m1.1.1.3.1.3.2">𝑆</ci><ci id="S4.E5.m1.1.1.3.1.3.3.cmml" xref="S4.E5.m1.1.1.3.1.3.3">𝑖</ci></apply></apply><apply id="S4.E5.m1.1.1.3.2.cmml" xref="S4.E5.m1.1.1.3.2"><times id="S4.E5.m1.1.1.3.2.1.cmml" xref="S4.E5.m1.1.1.3.2.1"></times><apply id="S4.E5.m1.1.1.3.2.2.cmml" xref="S4.E5.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.2.2.1.cmml" xref="S4.E5.m1.1.1.3.2.2">subscript</csymbol><ci id="S4.E5.m1.1.1.3.2.2.2.cmml" xref="S4.E5.m1.1.1.3.2.2.2">ℬ</ci><ci id="S4.E5.m1.1.1.3.2.2.3.cmml" xref="S4.E5.m1.1.1.3.2.2.3">𝑗</ci></apply><apply id="S4.E5.m1.1.1.3.2.3.cmml" xref="S4.E5.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.2.3.1.cmml" xref="S4.E5.m1.1.1.3.2.3">superscript</csymbol><apply id="S4.E5.m1.1.1.3.2.3.2.cmml" xref="S4.E5.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.3.2.3.2.1.cmml" xref="S4.E5.m1.1.1.3.2.3">subscript</csymbol><ci id="S4.E5.m1.1.1.3.2.3.2.2.cmml" xref="S4.E5.m1.1.1.3.2.3.2.2">italic-ϵ</ci><apply id="S4.E5.m1.1.1.3.2.3.2.3.cmml" xref="S4.E5.m1.1.1.3.2.3.2.3"><times id="S4.E5.m1.1.1.3.2.3.2.3.1.cmml" xref="S4.E5.m1.1.1.3.2.3.2.3.1"></times><ci id="S4.E5.m1.1.1.3.2.3.2.3.2.cmml" xref="S4.E5.m1.1.1.3.2.3.2.3.2">𝑖</ci><ci id="S4.E5.m1.1.1.3.2.3.2.3.3.cmml" xref="S4.E5.m1.1.1.3.2.3.2.3.3">𝑗</ci></apply></apply><ci id="S4.E5.m1.1.1.3.2.3.3a.cmml" xref="S4.E5.m1.1.1.3.2.3.3"><mtext class="ltx_font_smallcaps" id="S4.E5.m1.1.1.3.2.3.3.cmml" mathsize="70%" xref="S4.E5.m1.1.1.3.2.3.3">Mjd</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.1c">\epsilon_{i}^{\textsc{Mjd}}=\sum_{j}^{S_{i}}~{}\mathcal{B}_{j}\epsilon_{ij}^{% \textsc{Mjd}}</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.1d">italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(5)</td> </tr></tbody> </table> The source efficiency, mass and averaged specific activity, <math alttext="\overline{a}_{i}" class="ltx_Math" display="inline" id="S4.p2.14.m1.1"><semantics id="S4.p2.14.m1.1a"><msub id="S4.p2.14.m1.1.1" xref="S4.p2.14.m1.1.1.cmml"><mover accent="true" id="S4.p2.14.m1.1.1.2" xref="S4.p2.14.m1.1.1.2.cmml"><mi id="S4.p2.14.m1.1.1.2.2" xref="S4.p2.14.m1.1.1.2.2.cmml">a</mi><mo id="S4.p2.14.m1.1.1.2.1" xref="S4.p2.14.m1.1.1.2.1.cmml">¯</mo></mover><mi id="S4.p2.14.m1.1.1.3" xref="S4.p2.14.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.14.m1.1b"><apply id="S4.p2.14.m1.1.1.cmml" xref="S4.p2.14.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.14.m1.1.1.1.cmml" xref="S4.p2.14.m1.1.1">subscript</csymbol><apply id="S4.p2.14.m1.1.1.2.cmml" xref="S4.p2.14.m1.1.1.2"><ci id="S4.p2.14.m1.1.1.2.1.cmml" xref="S4.p2.14.m1.1.1.2.1">¯</ci><ci id="S4.p2.14.m1.1.1.2.2.cmml" xref="S4.p2.14.m1.1.1.2.2">𝑎</ci></apply><ci id="S4.p2.14.m1.1.1.3.cmml" xref="S4.p2.14.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.14.m1.1c">\overline{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.14.m1.1d">over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, are used to calculate the BI of the source: <table class="ltx_equation ltx_eqn_table" id="S4.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="\text{BI}_{i}=\frac{1}{M\times\Delta}\epsilon_{i}^{\textsc{Mjd}}\times% \overline{a}_{i}\times m_{i}" class="ltx_Math" display="block" id="S4.E6.m1.1"><semantics id="S4.E6.m1.1a"><mrow id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml"><msub id="S4.E6.m1.1.1.2" xref="S4.E6.m1.1.1.2.cmml"><mtext id="S4.E6.m1.1.1.2.2" xref="S4.E6.m1.1.1.2.2a.cmml">BI</mtext><mi id="S4.E6.m1.1.1.2.3" xref="S4.E6.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.E6.m1.1.1.1" xref="S4.E6.m1.1.1.1.cmml">=</mo><mrow id="S4.E6.m1.1.1.3" xref="S4.E6.m1.1.1.3.cmml"><mrow id="S4.E6.m1.1.1.3.2" xref="S4.E6.m1.1.1.3.2.cmml"><mfrac id="S4.E6.m1.1.1.3.2.2" xref="S4.E6.m1.1.1.3.2.2.cmml"><mn id="S4.E6.m1.1.1.3.2.2.2" xref="S4.E6.m1.1.1.3.2.2.2.cmml">1</mn><mrow id="S4.E6.m1.1.1.3.2.2.3" xref="S4.E6.m1.1.1.3.2.2.3.cmml"><mi id="S4.E6.m1.1.1.3.2.2.3.2" xref="S4.E6.m1.1.1.3.2.2.3.2.cmml">M</mi><mo id="S4.E6.m1.1.1.3.2.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.E6.m1.1.1.3.2.2.3.1.cmml">×</mo><mi id="S4.E6.m1.1.1.3.2.2.3.3" mathvariant="normal" xref="S4.E6.m1.1.1.3.2.2.3.3.cmml">Δ</mi></mrow></mfrac><mo id="S4.E6.m1.1.1.3.2.1" xref="S4.E6.m1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S4.E6.m1.1.1.3.2.3" xref="S4.E6.m1.1.1.3.2.3.cmml"><mi id="S4.E6.m1.1.1.3.2.3.2.2" xref="S4.E6.m1.1.1.3.2.3.2.2.cmml">ϵ</mi><mi id="S4.E6.m1.1.1.3.2.3.2.3" xref="S4.E6.m1.1.1.3.2.3.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S4.E6.m1.1.1.3.2.3.3" xref="S4.E6.m1.1.1.3.2.3.3a.cmml">Mjd</mtext></msubsup></mrow><mo id="S4.E6.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.E6.m1.1.1.3.1.cmml">×</mo><msub id="S4.E6.m1.1.1.3.3" xref="S4.E6.m1.1.1.3.3.cmml"><mover accent="true" id="S4.E6.m1.1.1.3.3.2" xref="S4.E6.m1.1.1.3.3.2.cmml"><mi id="S4.E6.m1.1.1.3.3.2.2" xref="S4.E6.m1.1.1.3.3.2.2.cmml">a</mi><mo id="S4.E6.m1.1.1.3.3.2.1" xref="S4.E6.m1.1.1.3.3.2.1.cmml">¯</mo></mover><mi id="S4.E6.m1.1.1.3.3.3" xref="S4.E6.m1.1.1.3.3.3.cmml">i</mi></msub><mo id="S4.E6.m1.1.1.3.1a" lspace="0.222em" rspace="0.222em" xref="S4.E6.m1.1.1.3.1.cmml">×</mo><msub id="S4.E6.m1.1.1.3.4" xref="S4.E6.m1.1.1.3.4.cmml"><mi id="S4.E6.m1.1.1.3.4.2" xref="S4.E6.m1.1.1.3.4.2.cmml">m</mi><mi id="S4.E6.m1.1.1.3.4.3" xref="S4.E6.m1.1.1.3.4.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.1b"><apply id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1"><eq id="S4.E6.m1.1.1.1.cmml" xref="S4.E6.m1.1.1.1"></eq><apply id="S4.E6.m1.1.1.2.cmml" xref="S4.E6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.2.1.cmml" xref="S4.E6.m1.1.1.2">subscript</csymbol><ci id="S4.E6.m1.1.1.2.2a.cmml" xref="S4.E6.m1.1.1.2.2"><mtext id="S4.E6.m1.1.1.2.2.cmml" xref="S4.E6.m1.1.1.2.2">BI</mtext></ci><ci id="S4.E6.m1.1.1.2.3.cmml" xref="S4.E6.m1.1.1.2.3">𝑖</ci></apply><apply id="S4.E6.m1.1.1.3.cmml" xref="S4.E6.m1.1.1.3"><times id="S4.E6.m1.1.1.3.1.cmml" xref="S4.E6.m1.1.1.3.1"></times><apply 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id="S4.E6.m1.1.1.3.2.3.2.2.cmml" xref="S4.E6.m1.1.1.3.2.3.2.2">italic-ϵ</ci><ci id="S4.E6.m1.1.1.3.2.3.2.3.cmml" xref="S4.E6.m1.1.1.3.2.3.2.3">𝑖</ci></apply><ci id="S4.E6.m1.1.1.3.2.3.3a.cmml" xref="S4.E6.m1.1.1.3.2.3.3"><mtext class="ltx_font_smallcaps" id="S4.E6.m1.1.1.3.2.3.3.cmml" mathsize="70%" xref="S4.E6.m1.1.1.3.2.3.3">Mjd</mtext></ci></apply></apply><apply id="S4.E6.m1.1.1.3.3.cmml" xref="S4.E6.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.3.1.cmml" xref="S4.E6.m1.1.1.3.3">subscript</csymbol><apply id="S4.E6.m1.1.1.3.3.2.cmml" xref="S4.E6.m1.1.1.3.3.2"><ci id="S4.E6.m1.1.1.3.3.2.1.cmml" xref="S4.E6.m1.1.1.3.3.2.1">¯</ci><ci id="S4.E6.m1.1.1.3.3.2.2.cmml" xref="S4.E6.m1.1.1.3.3.2.2">𝑎</ci></apply><ci id="S4.E6.m1.1.1.3.3.3.cmml" xref="S4.E6.m1.1.1.3.3.3">𝑖</ci></apply><apply id="S4.E6.m1.1.1.3.4.cmml" xref="S4.E6.m1.1.1.3.4"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.3.4.1.cmml" xref="S4.E6.m1.1.1.3.4">subscript</csymbol><ci id="S4.E6.m1.1.1.3.4.2.cmml" xref="S4.E6.m1.1.1.3.4.2">𝑚</ci><ci id="S4.E6.m1.1.1.3.4.3.cmml" xref="S4.E6.m1.1.1.3.4.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.1c">\text{BI}_{i}=\frac{1}{M\times\Delta}\epsilon_{i}^{\textsc{Mjd}}\times% \overline{a}_{i}\times m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.1d">BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_M × roman_Δ end_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT × over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT × italic_m 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">(6)</td> </tr></tbody> </table> Where <math alttext="\Delta=370" class="ltx_Math" display="inline" id="S4.p2.15.m1.1"><semantics id="S4.p2.15.m1.1a"><mrow id="S4.p2.15.m1.1.1" xref="S4.p2.15.m1.1.1.cmml"><mi id="S4.p2.15.m1.1.1.2" mathvariant="normal" xref="S4.p2.15.m1.1.1.2.cmml">Δ</mi><mo id="S4.p2.15.m1.1.1.1" xref="S4.p2.15.m1.1.1.1.cmml">=</mo><mn id="S4.p2.15.m1.1.1.3" xref="S4.p2.15.m1.1.1.3.cmml">370</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.15.m1.1b"><apply id="S4.p2.15.m1.1.1.cmml" xref="S4.p2.15.m1.1.1"><eq id="S4.p2.15.m1.1.1.1.cmml" xref="S4.p2.15.m1.1.1.1"></eq><ci id="S4.p2.15.m1.1.1.2.cmml" xref="S4.p2.15.m1.1.1.2">Δ</ci><cn id="S4.p2.15.m1.1.1.3.cmml" type="integer" xref="S4.p2.15.m1.1.1.3">370</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.15.m1.1c">\Delta=370</annotation><annotation encoding="application/x-llamapun" id="S4.p2.15.m1.1d">roman_Δ = 370</annotation></semantics></math> keV is the width of the BEW and <math alttext="M" class="ltx_Math" display="inline" id="S4.p2.16.m2.1"><semantics id="S4.p2.16.m2.1a"><mi id="S4.p2.16.m2.1.1" xref="S4.p2.16.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.p2.16.m2.1b"><ci id="S4.p2.16.m2.1.1.cmml" xref="S4.p2.16.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.16.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.p2.16.m2.1d">italic_M</annotation></semantics></math> is the mass of operational detectors in the array. Note that the total number of sources is equal to twice the number of components used to model the Demonstrator, since 232Th and 238U are accounted for in all components. The source contributions to the total <math alttext="{}^{\text{assay}}\text{BI}=\sum_{i}\text{BI}_{i}" class="ltx_Math" display="inline" id="S4.p2.19.m5.1"><semantics id="S4.p2.19.m5.1a"><mrow id="S4.p2.19.m5.1.1" xref="S4.p2.19.m5.1.1.cmml"><mmultiscripts id="S4.p2.19.m5.1.1.2" xref="S4.p2.19.m5.1.1.2.cmml"><mtext id="S4.p2.19.m5.1.1.2.2" xref="S4.p2.19.m5.1.1.2.2a.cmml">BI</mtext><mprescripts id="S4.p2.19.m5.1.1.2a" xref="S4.p2.19.m5.1.1.2.cmml"></mprescripts><mrow id="S4.p2.19.m5.1.1.2b" xref="S4.p2.19.m5.1.1.2.cmml"></mrow><mtext id="S4.p2.19.m5.1.1.2.3" xref="S4.p2.19.m5.1.1.2.3a.cmml">assay</mtext></mmultiscripts><mo id="S4.p2.19.m5.1.1.1" rspace="0.111em" xref="S4.p2.19.m5.1.1.1.cmml">=</mo><mrow id="S4.p2.19.m5.1.1.3" xref="S4.p2.19.m5.1.1.3.cmml"><msub id="S4.p2.19.m5.1.1.3.1" xref="S4.p2.19.m5.1.1.3.1.cmml"><mo id="S4.p2.19.m5.1.1.3.1.2" xref="S4.p2.19.m5.1.1.3.1.2.cmml">∑</mo><mi id="S4.p2.19.m5.1.1.3.1.3" xref="S4.p2.19.m5.1.1.3.1.3.cmml">i</mi></msub><msub id="S4.p2.19.m5.1.1.3.2" xref="S4.p2.19.m5.1.1.3.2.cmml"><mtext id="S4.p2.19.m5.1.1.3.2.2" xref="S4.p2.19.m5.1.1.3.2.2a.cmml">BI</mtext><mi id="S4.p2.19.m5.1.1.3.2.3" xref="S4.p2.19.m5.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.19.m5.1b"><apply id="S4.p2.19.m5.1.1.cmml" xref="S4.p2.19.m5.1.1"><eq id="S4.p2.19.m5.1.1.1.cmml" xref="S4.p2.19.m5.1.1.1"></eq><apply id="S4.p2.19.m5.1.1.2.cmml" xref="S4.p2.19.m5.1.1.2"><csymbol cd="ambiguous" id="S4.p2.19.m5.1.1.2.1.cmml" xref="S4.p2.19.m5.1.1.2">superscript</csymbol><ci id="S4.p2.19.m5.1.1.2.2a.cmml" xref="S4.p2.19.m5.1.1.2.2"><mtext id="S4.p2.19.m5.1.1.2.2.cmml" xref="S4.p2.19.m5.1.1.2.2">BI</mtext></ci><ci id="S4.p2.19.m5.1.1.2.3a.cmml" xref="S4.p2.19.m5.1.1.2.3"><mtext id="S4.p2.19.m5.1.1.2.3.cmml" mathsize="70%" xref="S4.p2.19.m5.1.1.2.3">assay</mtext></ci></apply><apply id="S4.p2.19.m5.1.1.3.cmml" xref="S4.p2.19.m5.1.1.3"><apply id="S4.p2.19.m5.1.1.3.1.cmml" xref="S4.p2.19.m5.1.1.3.1"><csymbol cd="ambiguous" id="S4.p2.19.m5.1.1.3.1.1.cmml" xref="S4.p2.19.m5.1.1.3.1">subscript</csymbol><sum id="S4.p2.19.m5.1.1.3.1.2.cmml" xref="S4.p2.19.m5.1.1.3.1.2"></sum><ci id="S4.p2.19.m5.1.1.3.1.3.cmml" xref="S4.p2.19.m5.1.1.3.1.3">𝑖</ci></apply><apply id="S4.p2.19.m5.1.1.3.2.cmml" xref="S4.p2.19.m5.1.1.3.2"><csymbol cd="ambiguous" id="S4.p2.19.m5.1.1.3.2.1.cmml" xref="S4.p2.19.m5.1.1.3.2">subscript</csymbol><ci id="S4.p2.19.m5.1.1.3.2.2a.cmml" xref="S4.p2.19.m5.1.1.3.2.2"><mtext id="S4.p2.19.m5.1.1.3.2.2.cmml" xref="S4.p2.19.m5.1.1.3.2.2">BI</mtext></ci><ci id="S4.p2.19.m5.1.1.3.2.3.cmml" xref="S4.p2.19.m5.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.19.m5.1c">{}^{\text{assay}}\text{BI}=\sum_{i}\text{BI}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.19.m5.1d">start_FLOATSUPERSCRIPT assay end_FLOATSUPERSCRIPT BI = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, can be summed as desired, either by source material, contaminant, or component group. If only an upper limit, <math alttext="l_{i}" class="ltx_Math" display="inline" id="S4.p2.20.m6.1"><semantics id="S4.p2.20.m6.1a"><msub id="S4.p2.20.m6.1.1" xref="S4.p2.20.m6.1.1.cmml"><mi id="S4.p2.20.m6.1.1.2" xref="S4.p2.20.m6.1.1.2.cmml">l</mi><mi id="S4.p2.20.m6.1.1.3" xref="S4.p2.20.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.20.m6.1b"><apply id="S4.p2.20.m6.1.1.cmml" xref="S4.p2.20.m6.1.1"><csymbol cd="ambiguous" id="S4.p2.20.m6.1.1.1.cmml" xref="S4.p2.20.m6.1.1">subscript</csymbol><ci id="S4.p2.20.m6.1.1.2.cmml" xref="S4.p2.20.m6.1.1.2">𝑙</ci><ci id="S4.p2.20.m6.1.1.3.cmml" xref="S4.p2.20.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.20.m6.1c">l_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.20.m6.1d">italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, is available, it can be used in place of <math alttext="\overline{a}_{i}" class="ltx_Math" display="inline" id="S4.p2.21.m7.1"><semantics id="S4.p2.21.m7.1a"><msub id="S4.p2.21.m7.1.1" xref="S4.p2.21.m7.1.1.cmml"><mover accent="true" id="S4.p2.21.m7.1.1.2" xref="S4.p2.21.m7.1.1.2.cmml"><mi id="S4.p2.21.m7.1.1.2.2" xref="S4.p2.21.m7.1.1.2.2.cmml">a</mi><mo id="S4.p2.21.m7.1.1.2.1" xref="S4.p2.21.m7.1.1.2.1.cmml">¯</mo></mover><mi id="S4.p2.21.m7.1.1.3" xref="S4.p2.21.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.21.m7.1b"><apply id="S4.p2.21.m7.1.1.cmml" xref="S4.p2.21.m7.1.1"><csymbol cd="ambiguous" id="S4.p2.21.m7.1.1.1.cmml" xref="S4.p2.21.m7.1.1">subscript</csymbol><apply id="S4.p2.21.m7.1.1.2.cmml" xref="S4.p2.21.m7.1.1.2"><ci id="S4.p2.21.m7.1.1.2.1.cmml" xref="S4.p2.21.m7.1.1.2.1">¯</ci><ci id="S4.p2.21.m7.1.1.2.2.cmml" xref="S4.p2.21.m7.1.1.2.2">𝑎</ci></apply><ci id="S4.p2.21.m7.1.1.3.cmml" xref="S4.p2.21.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.21.m7.1c">\overline{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.21.m7.1d">over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. This calculates a BI which is the direct sum of upper limits and central values and is referred to as the direct BI in the text. The component-group-combined 232Th and 238U direct BIs are calculated by summing over the appropriate sources and are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.T2" title="Table 2 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a>. Note that the design geometry projection of Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite> used this method. It is not possible to assign an uncertainty to the direct BI because of the inclusion of upper limits. However, a proper treatment of uncertainties can be obtained by promoting the single-valued <math alttext="\overline{a}_{i}" class="ltx_Math" display="inline" id="S4.p2.24.m10.1"><semantics id="S4.p2.24.m10.1a"><msub id="S4.p2.24.m10.1.1" xref="S4.p2.24.m10.1.1.cmml"><mover accent="true" id="S4.p2.24.m10.1.1.2" xref="S4.p2.24.m10.1.1.2.cmml"><mi id="S4.p2.24.m10.1.1.2.2" xref="S4.p2.24.m10.1.1.2.2.cmml">a</mi><mo id="S4.p2.24.m10.1.1.2.1" xref="S4.p2.24.m10.1.1.2.1.cmml">¯</mo></mover><mi id="S4.p2.24.m10.1.1.3" xref="S4.p2.24.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.24.m10.1b"><apply id="S4.p2.24.m10.1.1.cmml" xref="S4.p2.24.m10.1.1"><csymbol cd="ambiguous" id="S4.p2.24.m10.1.1.1.cmml" xref="S4.p2.24.m10.1.1">subscript</csymbol><apply id="S4.p2.24.m10.1.1.2.cmml" xref="S4.p2.24.m10.1.1.2"><ci id="S4.p2.24.m10.1.1.2.1.cmml" xref="S4.p2.24.m10.1.1.2.1">¯</ci><ci id="S4.p2.24.m10.1.1.2.2.cmml" xref="S4.p2.24.m10.1.1.2.2">𝑎</ci></apply><ci id="S4.p2.24.m10.1.1.3.cmml" xref="S4.p2.24.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.24.m10.1c">\overline{a}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.24.m10.1d">over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (or <math alttext="l_{i}" class="ltx_Math" display="inline" id="S4.p2.25.m11.1"><semantics id="S4.p2.25.m11.1a"><msub id="S4.p2.25.m11.1.1" xref="S4.p2.25.m11.1.1.cmml"><mi id="S4.p2.25.m11.1.1.2" xref="S4.p2.25.m11.1.1.2.cmml">l</mi><mi id="S4.p2.25.m11.1.1.3" xref="S4.p2.25.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.25.m11.1b"><apply id="S4.p2.25.m11.1.1.cmml" xref="S4.p2.25.m11.1.1"><csymbol cd="ambiguous" id="S4.p2.25.m11.1.1.1.cmml" xref="S4.p2.25.m11.1.1">subscript</csymbol><ci id="S4.p2.25.m11.1.1.2.cmml" xref="S4.p2.25.m11.1.1.2">𝑙</ci><ci id="S4.p2.25.m11.1.1.3.cmml" xref="S4.p2.25.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.25.m11.1c">l_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.25.m11.1d">italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>), <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.p2.26.m12.1"><semantics id="S4.p2.26.m12.1a"><msub id="S4.p2.26.m12.1.1" xref="S4.p2.26.m12.1.1.cmml"><mi id="S4.p2.26.m12.1.1.2" xref="S4.p2.26.m12.1.1.2.cmml">m</mi><mi id="S4.p2.26.m12.1.1.3" xref="S4.p2.26.m12.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.26.m12.1b"><apply id="S4.p2.26.m12.1.1.cmml" xref="S4.p2.26.m12.1.1"><csymbol cd="ambiguous" id="S4.p2.26.m12.1.1.1.cmml" xref="S4.p2.26.m12.1.1">subscript</csymbol><ci id="S4.p2.26.m12.1.1.2.cmml" xref="S4.p2.26.m12.1.1.2">𝑚</ci><ci id="S4.p2.26.m12.1.1.3.cmml" xref="S4.p2.26.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.26.m12.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.26.m12.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\epsilon^{\textsc{Mjd}}_{i}" class="ltx_Math" display="inline" id="S4.p2.27.m13.1"><semantics id="S4.p2.27.m13.1a"><msubsup id="S4.p2.27.m13.1.1" xref="S4.p2.27.m13.1.1.cmml"><mi id="S4.p2.27.m13.1.1.2.2" xref="S4.p2.27.m13.1.1.2.2.cmml">ϵ</mi><mi id="S4.p2.27.m13.1.1.3" xref="S4.p2.27.m13.1.1.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S4.p2.27.m13.1.1.2.3" xref="S4.p2.27.m13.1.1.2.3a.cmml">Mjd</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S4.p2.27.m13.1b"><apply id="S4.p2.27.m13.1.1.cmml" xref="S4.p2.27.m13.1.1"><csymbol cd="ambiguous" id="S4.p2.27.m13.1.1.1.cmml" xref="S4.p2.27.m13.1.1">subscript</csymbol><apply id="S4.p2.27.m13.1.1.2.cmml" xref="S4.p2.27.m13.1.1"><csymbol cd="ambiguous" id="S4.p2.27.m13.1.1.2.1.cmml" xref="S4.p2.27.m13.1.1">superscript</csymbol><ci id="S4.p2.27.m13.1.1.2.2.cmml" xref="S4.p2.27.m13.1.1.2.2">italic-ϵ</ci><ci id="S4.p2.27.m13.1.1.2.3a.cmml" xref="S4.p2.27.m13.1.1.2.3"><mtext class="ltx_font_smallcaps" id="S4.p2.27.m13.1.1.2.3.cmml" mathsize="70%" xref="S4.p2.27.m13.1.1.2.3">Mjd</mtext></ci></apply><ci id="S4.p2.27.m13.1.1.3.cmml" xref="S4.p2.27.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.27.m13.1c">\epsilon^{\textsc{Mjd}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.27.m13.1d">italic_ϵ start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to distributions and using Monte Carlo uncertainty propagation to combine these in a final BI distribution. </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> V Promoting single values to distributions</h2> <div class="ltx_para" id="S5.p1"> As described in Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S3" title="III A unified approach to combine assay results ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">III</a> the specific activity can be expressed as a truncated-at-zero Gaussian. The mass, <math alttext="m_{i}" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><msub id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">m</mi><mi id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1">subscript</csymbol><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝑚</ci><ci id="S5.p1.1.m1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, is also promoted to a distribution of this form. Depending on the method used to measure or estimate the mass, a Gaussian uncertainty is assigned. This uncertainty ranges from 1%, to account for sample to sample variation in direct mass measurements, to 10% for masses that were estimated from geometry. In practice the mass PDFs are indistinguishable from a true Gaussian given the assigned level of uncertainty. </div> <div class="ltx_para" id="S5.p2"> The derivation of the efficiency, <math alttext="\epsilon^{\textsc{Mjd}}_{i}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><msubsup id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi id="S5.p2.1.m1.1.1.2.2" xref="S5.p2.1.m1.1.1.2.2.cmml">ϵ</mi><mi id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S5.p2.1.m1.1.1.2.3" xref="S5.p2.1.m1.1.1.2.3a.cmml">Mjd</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1">subscript</csymbol><apply id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.2.1.cmml" xref="S5.p2.1.m1.1.1">superscript</csymbol><ci id="S5.p2.1.m1.1.1.2.2.cmml" xref="S5.p2.1.m1.1.1.2.2">italic-ϵ</ci><ci id="S5.p2.1.m1.1.1.2.3a.cmml" xref="S5.p2.1.m1.1.1.2.3"><mtext class="ltx_font_smallcaps" id="S5.p2.1.m1.1.1.2.3.cmml" mathsize="70%" xref="S5.p2.1.m1.1.1.2.3">Mjd</mtext></ci></apply><ci id="S5.p2.1.m1.1.1.3.cmml" xref="S5.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\epsilon^{\textsc{Mjd}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_ϵ start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, PDF follows. Starting from Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4.E4" title="In IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">4</a>, the probability of finding <math alttext="n" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mi id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><ci id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_n</annotation></semantics></math> counts in the BEW – with expectation value <math alttext="\lambda" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mi id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><ci id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_λ</annotation></semantics></math> – is given by the Poisson distribution, <math alttext="P(n|\lambda)=\lambda^{n}e^{-\lambda}/n!" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mrow id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml"><mrow id="S5.p2.4.m4.1.1.1" xref="S5.p2.4.m4.1.1.1.cmml"><mi id="S5.p2.4.m4.1.1.1.3" xref="S5.p2.4.m4.1.1.1.3.cmml">P</mi><mo id="S5.p2.4.m4.1.1.1.2" xref="S5.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.p2.4.m4.1.1.1.1.1" xref="S5.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.4.m4.1.1.1.1.1.1" xref="S5.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.p2.4.m4.1.1.1.1.1.1.2" 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id="S5.p2.4.m4.1.1.3.2.3.3a" xref="S5.p2.4.m4.1.1.3.2.3.3.cmml">−</mo><mi id="S5.p2.4.m4.1.1.3.2.3.3.2" xref="S5.p2.4.m4.1.1.3.2.3.3.2.cmml">λ</mi></mrow></msup></mrow><mo id="S5.p2.4.m4.1.1.3.1" xref="S5.p2.4.m4.1.1.3.1.cmml">/</mo><mrow id="S5.p2.4.m4.1.1.3.3" xref="S5.p2.4.m4.1.1.3.3.cmml"><mi id="S5.p2.4.m4.1.1.3.3.2" xref="S5.p2.4.m4.1.1.3.3.2.cmml">n</mi><mo id="S5.p2.4.m4.1.1.3.3.1" xref="S5.p2.4.m4.1.1.3.3.1.cmml">!</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><apply id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1"><eq id="S5.p2.4.m4.1.1.2.cmml" xref="S5.p2.4.m4.1.1.2"></eq><apply id="S5.p2.4.m4.1.1.1.cmml" xref="S5.p2.4.m4.1.1.1"><times id="S5.p2.4.m4.1.1.1.2.cmml" xref="S5.p2.4.m4.1.1.1.2"></times><ci id="S5.p2.4.m4.1.1.1.3.cmml" xref="S5.p2.4.m4.1.1.1.3">𝑃</ci><apply id="S5.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.p2.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="S5.p2.4.m4.1.1.1.1.1.1.1.cmml" 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id="S5.p2.4.m4.1.1.3.2.3.2.cmml" xref="S5.p2.4.m4.1.1.3.2.3.2">𝑒</ci><apply id="S5.p2.4.m4.1.1.3.2.3.3.cmml" xref="S5.p2.4.m4.1.1.3.2.3.3"><minus id="S5.p2.4.m4.1.1.3.2.3.3.1.cmml" xref="S5.p2.4.m4.1.1.3.2.3.3"></minus><ci id="S5.p2.4.m4.1.1.3.2.3.3.2.cmml" xref="S5.p2.4.m4.1.1.3.2.3.3.2">𝜆</ci></apply></apply></apply><apply id="S5.p2.4.m4.1.1.3.3.cmml" xref="S5.p2.4.m4.1.1.3.3"><factorial id="S5.p2.4.m4.1.1.3.3.1.cmml" xref="S5.p2.4.m4.1.1.3.3.1"></factorial><ci id="S5.p2.4.m4.1.1.3.3.2.cmml" xref="S5.p2.4.m4.1.1.3.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">P(n|\lambda)=\lambda^{n}e^{-\lambda}/n!</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">italic_P ( italic_n | italic_λ ) = italic_λ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ end_POSTSUPERSCRIPT / italic_n !</annotation></semantics></math>. Conversely, <math alttext="P(\lambda|n)" class="ltx_Math" display="inline" id="S5.p2.5.m5.1"><semantics id="S5.p2.5.m5.1a"><mrow id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml"><mi id="S5.p2.5.m5.1.1.3" xref="S5.p2.5.m5.1.1.3.cmml">P</mi><mo id="S5.p2.5.m5.1.1.2" xref="S5.p2.5.m5.1.1.2.cmml">⁢</mo><mrow id="S5.p2.5.m5.1.1.1.1" xref="S5.p2.5.m5.1.1.1.1.1.cmml"><mo id="S5.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S5.p2.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.5.m5.1.1.1.1.1" xref="S5.p2.5.m5.1.1.1.1.1.cmml"><mi id="S5.p2.5.m5.1.1.1.1.1.2" xref="S5.p2.5.m5.1.1.1.1.1.2.cmml">λ</mi><mo fence="false" id="S5.p2.5.m5.1.1.1.1.1.1" xref="S5.p2.5.m5.1.1.1.1.1.1.cmml">|</mo><mi id="S5.p2.5.m5.1.1.1.1.1.3" xref="S5.p2.5.m5.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="S5.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S5.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.1b"><apply id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1"><times id="S5.p2.5.m5.1.1.2.cmml" xref="S5.p2.5.m5.1.1.2"></times><ci id="S5.p2.5.m5.1.1.3.cmml" xref="S5.p2.5.m5.1.1.3">𝑃</ci><apply id="S5.p2.5.m5.1.1.1.1.1.cmml" xref="S5.p2.5.m5.1.1.1.1"><csymbol cd="latexml" id="S5.p2.5.m5.1.1.1.1.1.1.cmml" xref="S5.p2.5.m5.1.1.1.1.1.1">conditional</csymbol><ci id="S5.p2.5.m5.1.1.1.1.1.2.cmml" xref="S5.p2.5.m5.1.1.1.1.1.2">𝜆</ci><ci id="S5.p2.5.m5.1.1.1.1.1.3.cmml" xref="S5.p2.5.m5.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.1c">P(\lambda|n)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.1d">italic_P ( italic_λ | italic_n )</annotation></semantics></math> is deduced via Bayes’ theorem, <math alttext="P(\lambda|n)\propto P(\lambda)P(n|\lambda)" class="ltx_Math" display="inline" id="S5.p2.6.m6.3"><semantics id="S5.p2.6.m6.3a"><mrow id="S5.p2.6.m6.3.3" xref="S5.p2.6.m6.3.3.cmml"><mrow id="S5.p2.6.m6.2.2.1" xref="S5.p2.6.m6.2.2.1.cmml"><mi id="S5.p2.6.m6.2.2.1.3" xref="S5.p2.6.m6.2.2.1.3.cmml">P</mi><mo id="S5.p2.6.m6.2.2.1.2" xref="S5.p2.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S5.p2.6.m6.2.2.1.1.1" xref="S5.p2.6.m6.2.2.1.1.1.1.cmml"><mo id="S5.p2.6.m6.2.2.1.1.1.2" stretchy="false" xref="S5.p2.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.p2.6.m6.2.2.1.1.1.1" xref="S5.p2.6.m6.2.2.1.1.1.1.cmml"><mi id="S5.p2.6.m6.2.2.1.1.1.1.2" xref="S5.p2.6.m6.2.2.1.1.1.1.2.cmml">λ</mi><mo fence="false" id="S5.p2.6.m6.2.2.1.1.1.1.1" xref="S5.p2.6.m6.2.2.1.1.1.1.1.cmml">|</mo><mi id="S5.p2.6.m6.2.2.1.1.1.1.3" xref="S5.p2.6.m6.2.2.1.1.1.1.3.cmml">n</mi></mrow><mo id="S5.p2.6.m6.2.2.1.1.1.3" stretchy="false" xref="S5.p2.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p2.6.m6.3.3.3" xref="S5.p2.6.m6.3.3.3.cmml">∝</mo><mrow id="S5.p2.6.m6.3.3.2" xref="S5.p2.6.m6.3.3.2.cmml"><mi id="S5.p2.6.m6.3.3.2.3" xref="S5.p2.6.m6.3.3.2.3.cmml">P</mi><mo id="S5.p2.6.m6.3.3.2.2" xref="S5.p2.6.m6.3.3.2.2.cmml">⁢</mo><mrow id="S5.p2.6.m6.3.3.2.4.2" xref="S5.p2.6.m6.3.3.2.cmml"><mo id="S5.p2.6.m6.3.3.2.4.2.1" stretchy="false" xref="S5.p2.6.m6.3.3.2.cmml">(</mo><mi id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml">λ</mi><mo id="S5.p2.6.m6.3.3.2.4.2.2" stretchy="false" xref="S5.p2.6.m6.3.3.2.cmml">)</mo></mrow><mo id="S5.p2.6.m6.3.3.2.2a" xref="S5.p2.6.m6.3.3.2.2.cmml">⁢</mo><mi id="S5.p2.6.m6.3.3.2.5" xref="S5.p2.6.m6.3.3.2.5.cmml">P</mi><mo id="S5.p2.6.m6.3.3.2.2b" xref="S5.p2.6.m6.3.3.2.2.cmml">⁢</mo><mrow id="S5.p2.6.m6.3.3.2.1.1" xref="S5.p2.6.m6.3.3.2.1.1.1.cmml"><mo id="S5.p2.6.m6.3.3.2.1.1.2" stretchy="false" xref="S5.p2.6.m6.3.3.2.1.1.1.cmml">(</mo><mrow id="S5.p2.6.m6.3.3.2.1.1.1" xref="S5.p2.6.m6.3.3.2.1.1.1.cmml"><mi id="S5.p2.6.m6.3.3.2.1.1.1.2" xref="S5.p2.6.m6.3.3.2.1.1.1.2.cmml">n</mi><mo fence="false" id="S5.p2.6.m6.3.3.2.1.1.1.1" xref="S5.p2.6.m6.3.3.2.1.1.1.1.cmml">|</mo><mi id="S5.p2.6.m6.3.3.2.1.1.1.3" xref="S5.p2.6.m6.3.3.2.1.1.1.3.cmml">λ</mi></mrow><mo id="S5.p2.6.m6.3.3.2.1.1.3" stretchy="false" xref="S5.p2.6.m6.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.3b"><apply id="S5.p2.6.m6.3.3.cmml" xref="S5.p2.6.m6.3.3"><csymbol cd="latexml" id="S5.p2.6.m6.3.3.3.cmml" xref="S5.p2.6.m6.3.3.3">proportional-to</csymbol><apply id="S5.p2.6.m6.2.2.1.cmml" xref="S5.p2.6.m6.2.2.1"><times id="S5.p2.6.m6.2.2.1.2.cmml" xref="S5.p2.6.m6.2.2.1.2"></times><ci id="S5.p2.6.m6.2.2.1.3.cmml" xref="S5.p2.6.m6.2.2.1.3">𝑃</ci><apply id="S5.p2.6.m6.2.2.1.1.1.1.cmml" xref="S5.p2.6.m6.2.2.1.1.1"><csymbol cd="latexml" id="S5.p2.6.m6.2.2.1.1.1.1.1.cmml" xref="S5.p2.6.m6.2.2.1.1.1.1.1">conditional</csymbol><ci id="S5.p2.6.m6.2.2.1.1.1.1.2.cmml" xref="S5.p2.6.m6.2.2.1.1.1.1.2">𝜆</ci><ci id="S5.p2.6.m6.2.2.1.1.1.1.3.cmml" xref="S5.p2.6.m6.2.2.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S5.p2.6.m6.3.3.2.cmml" xref="S5.p2.6.m6.3.3.2"><times id="S5.p2.6.m6.3.3.2.2.cmml" xref="S5.p2.6.m6.3.3.2.2"></times><ci id="S5.p2.6.m6.3.3.2.3.cmml" xref="S5.p2.6.m6.3.3.2.3">𝑃</ci><ci id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1">𝜆</ci><ci id="S5.p2.6.m6.3.3.2.5.cmml" xref="S5.p2.6.m6.3.3.2.5">𝑃</ci><apply id="S5.p2.6.m6.3.3.2.1.1.1.cmml" xref="S5.p2.6.m6.3.3.2.1.1"><csymbol cd="latexml" id="S5.p2.6.m6.3.3.2.1.1.1.1.cmml" xref="S5.p2.6.m6.3.3.2.1.1.1.1">conditional</csymbol><ci id="S5.p2.6.m6.3.3.2.1.1.1.2.cmml" xref="S5.p2.6.m6.3.3.2.1.1.1.2">𝑛</ci><ci id="S5.p2.6.m6.3.3.2.1.1.1.3.cmml" xref="S5.p2.6.m6.3.3.2.1.1.1.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.3c">P(\lambda|n)\propto P(\lambda)P(n|\lambda)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.3d">italic_P ( italic_λ | italic_n ) ∝ italic_P ( italic_λ ) italic_P ( italic_n | italic_λ )</annotation></semantics></math>, using the prior, <table class="ltx_equation ltx_eqn_table" id="S5.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="P(\lambda)=\frac{1}{\lambda^{1-1/S}}." class="ltx_Math" display="block" id="S5.E7.m1.2"><semantics id="S5.E7.m1.2a"><mrow id="S5.E7.m1.2.2.1" xref="S5.E7.m1.2.2.1.1.cmml"><mrow id="S5.E7.m1.2.2.1.1" xref="S5.E7.m1.2.2.1.1.cmml"><mrow id="S5.E7.m1.2.2.1.1.2" xref="S5.E7.m1.2.2.1.1.2.cmml"><mi id="S5.E7.m1.2.2.1.1.2.2" xref="S5.E7.m1.2.2.1.1.2.2.cmml">P</mi><mo id="S5.E7.m1.2.2.1.1.2.1" xref="S5.E7.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S5.E7.m1.2.2.1.1.2.3.2" xref="S5.E7.m1.2.2.1.1.2.cmml"><mo id="S5.E7.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="S5.E7.m1.2.2.1.1.2.cmml">(</mo><mi id="S5.E7.m1.1.1" xref="S5.E7.m1.1.1.cmml">λ</mi><mo id="S5.E7.m1.2.2.1.1.2.3.2.2" stretchy="false" xref="S5.E7.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.E7.m1.2.2.1.1.1" xref="S5.E7.m1.2.2.1.1.1.cmml">=</mo><mfrac id="S5.E7.m1.2.2.1.1.3" xref="S5.E7.m1.2.2.1.1.3.cmml"><mn id="S5.E7.m1.2.2.1.1.3.2" xref="S5.E7.m1.2.2.1.1.3.2.cmml">1</mn><msup id="S5.E7.m1.2.2.1.1.3.3" xref="S5.E7.m1.2.2.1.1.3.3.cmml"><mi id="S5.E7.m1.2.2.1.1.3.3.2" xref="S5.E7.m1.2.2.1.1.3.3.2.cmml">λ</mi><mrow id="S5.E7.m1.2.2.1.1.3.3.3" xref="S5.E7.m1.2.2.1.1.3.3.3.cmml"><mn id="S5.E7.m1.2.2.1.1.3.3.3.2" xref="S5.E7.m1.2.2.1.1.3.3.3.2.cmml">1</mn><mo id="S5.E7.m1.2.2.1.1.3.3.3.1" xref="S5.E7.m1.2.2.1.1.3.3.3.1.cmml">−</mo><mrow id="S5.E7.m1.2.2.1.1.3.3.3.3" xref="S5.E7.m1.2.2.1.1.3.3.3.3.cmml"><mn id="S5.E7.m1.2.2.1.1.3.3.3.3.2" xref="S5.E7.m1.2.2.1.1.3.3.3.3.2.cmml">1</mn><mo id="S5.E7.m1.2.2.1.1.3.3.3.3.1" xref="S5.E7.m1.2.2.1.1.3.3.3.3.1.cmml">/</mo><mi id="S5.E7.m1.2.2.1.1.3.3.3.3.3" xref="S5.E7.m1.2.2.1.1.3.3.3.3.3.cmml">S</mi></mrow></mrow></msup></mfrac></mrow><mo id="S5.E7.m1.2.2.1.2" lspace="0em" xref="S5.E7.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E7.m1.2b"><apply id="S5.E7.m1.2.2.1.1.cmml" xref="S5.E7.m1.2.2.1"><eq id="S5.E7.m1.2.2.1.1.1.cmml" xref="S5.E7.m1.2.2.1.1.1"></eq><apply id="S5.E7.m1.2.2.1.1.2.cmml" xref="S5.E7.m1.2.2.1.1.2"><times id="S5.E7.m1.2.2.1.1.2.1.cmml" xref="S5.E7.m1.2.2.1.1.2.1"></times><ci id="S5.E7.m1.2.2.1.1.2.2.cmml" xref="S5.E7.m1.2.2.1.1.2.2">𝑃</ci><ci id="S5.E7.m1.1.1.cmml" xref="S5.E7.m1.1.1">𝜆</ci></apply><apply id="S5.E7.m1.2.2.1.1.3.cmml" xref="S5.E7.m1.2.2.1.1.3"><divide id="S5.E7.m1.2.2.1.1.3.1.cmml" xref="S5.E7.m1.2.2.1.1.3"></divide><cn id="S5.E7.m1.2.2.1.1.3.2.cmml" type="integer" xref="S5.E7.m1.2.2.1.1.3.2">1</cn><apply id="S5.E7.m1.2.2.1.1.3.3.cmml" xref="S5.E7.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S5.E7.m1.2.2.1.1.3.3.1.cmml" xref="S5.E7.m1.2.2.1.1.3.3">superscript</csymbol><ci id="S5.E7.m1.2.2.1.1.3.3.2.cmml" xref="S5.E7.m1.2.2.1.1.3.3.2">𝜆</ci><apply id="S5.E7.m1.2.2.1.1.3.3.3.cmml" xref="S5.E7.m1.2.2.1.1.3.3.3"><minus id="S5.E7.m1.2.2.1.1.3.3.3.1.cmml" xref="S5.E7.m1.2.2.1.1.3.3.3.1"></minus><cn id="S5.E7.m1.2.2.1.1.3.3.3.2.cmml" type="integer" xref="S5.E7.m1.2.2.1.1.3.3.3.2">1</cn><apply id="S5.E7.m1.2.2.1.1.3.3.3.3.cmml" xref="S5.E7.m1.2.2.1.1.3.3.3.3"><divide id="S5.E7.m1.2.2.1.1.3.3.3.3.1.cmml" xref="S5.E7.m1.2.2.1.1.3.3.3.3.1"></divide><cn id="S5.E7.m1.2.2.1.1.3.3.3.3.2.cmml" type="integer" xref="S5.E7.m1.2.2.1.1.3.3.3.3.2">1</cn><ci id="S5.E7.m1.2.2.1.1.3.3.3.3.3.cmml" xref="S5.E7.m1.2.2.1.1.3.3.3.3.3">𝑆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7.m1.2c">P(\lambda)=\frac{1}{\lambda^{1-1/S}}.</annotation><annotation encoding="application/x-llamapun" id="S5.E7.m1.2d">italic_P ( italic_λ ) = divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 1 - 1 / italic_S end_POSTSUPERSCRIPT 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">(7)</td> </tr></tbody> </table> The choice of uninformative prior corresponds to a flat prior on the sum of the <math alttext="S" class="ltx_Math" display="inline" id="S5.p2.7.m1.1"><semantics id="S5.p2.7.m1.1a"><mi id="S5.p2.7.m1.1.1" xref="S5.p2.7.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p2.7.m1.1b"><ci id="S5.p2.7.m1.1.1.cmml" xref="S5.p2.7.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m1.1d">italic_S</annotation></semantics></math> decay chain segments. The functional form of the prior is derived in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#A1" title="Appendix A Efficiency distribution prior derivation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">A</a>. <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx1"> <tbody id="S5.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P(\lambda|n)" class="ltx_Math" display="inline" id="S5.Ex1.m1.1"><semantics id="S5.Ex1.m1.1a"><mrow id="S5.Ex1.m1.1.1" xref="S5.Ex1.m1.1.1.cmml"><mi id="S5.Ex1.m1.1.1.3" xref="S5.Ex1.m1.1.1.3.cmml">P</mi><mo id="S5.Ex1.m1.1.1.2" xref="S5.Ex1.m1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex1.m1.1.1.1.1" xref="S5.Ex1.m1.1.1.1.1.1.cmml"><mo id="S5.Ex1.m1.1.1.1.1.2" stretchy="false" xref="S5.Ex1.m1.1.1.1.1.1.cmml">(</mo><mrow 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encoding="application/x-tex" id="S5.Ex1.m1.1c">\displaystyle P(\lambda|n)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex1.m1.1d">italic_P ( italic_λ | italic_n )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\propto\frac{\lambda^{n-1+1/S}e^{-\lambda}}{n!}" class="ltx_Math" display="inline" id="S5.Ex1.m2.1"><semantics id="S5.Ex1.m2.1a"><mrow id="S5.Ex1.m2.1.1" xref="S5.Ex1.m2.1.1.cmml"><mi id="S5.Ex1.m2.1.1.2" xref="S5.Ex1.m2.1.1.2.cmml"></mi><mo id="S5.Ex1.m2.1.1.1" xref="S5.Ex1.m2.1.1.1.cmml">∝</mo><mstyle displaystyle="true" id="S5.Ex1.m2.1.1.3" xref="S5.Ex1.m2.1.1.3.cmml"><mfrac id="S5.Ex1.m2.1.1.3a" xref="S5.Ex1.m2.1.1.3.cmml"><mrow id="S5.Ex1.m2.1.1.3.2" xref="S5.Ex1.m2.1.1.3.2.cmml"><msup id="S5.Ex1.m2.1.1.3.2.2" xref="S5.Ex1.m2.1.1.3.2.2.cmml"><mi id="S5.Ex1.m2.1.1.3.2.2.2" xref="S5.Ex1.m2.1.1.3.2.2.2.cmml">λ</mi><mrow id="S5.Ex1.m2.1.1.3.2.2.3" xref="S5.Ex1.m2.1.1.3.2.2.3.cmml"><mrow 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xref="S5.E8.m1.1.1.1.1.1.1.2.2.2">𝛽</ci><cn id="S5.E8.m1.1.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S5.E8.m1.1.1.1.1.1.1.2.2.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E8.m1.1c">\displaystyle\propto\Gamma(\lambda|\alpha=n+1/S,~{}\beta=1)</annotation><annotation encoding="application/x-llamapun" id="S5.E8.m1.1d">∝ roman_Γ ( italic_λ | italic_α = italic_n + 1 / italic_S , italic_β = 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">(8)</td> </tr></tbody> </table> The resulting posterior is a Gamma distribution with parameters <math alttext="\alpha=n+1/S" class="ltx_Math" display="inline" id="S5.p2.8.m1.1"><semantics id="S5.p2.8.m1.1a"><mrow id="S5.p2.8.m1.1.1" xref="S5.p2.8.m1.1.1.cmml"><mi id="S5.p2.8.m1.1.1.2" xref="S5.p2.8.m1.1.1.2.cmml">α</mi><mo id="S5.p2.8.m1.1.1.1" xref="S5.p2.8.m1.1.1.1.cmml">=</mo><mrow id="S5.p2.8.m1.1.1.3" xref="S5.p2.8.m1.1.1.3.cmml"><mi id="S5.p2.8.m1.1.1.3.2" xref="S5.p2.8.m1.1.1.3.2.cmml">n</mi><mo id="S5.p2.8.m1.1.1.3.1" xref="S5.p2.8.m1.1.1.3.1.cmml">+</mo><mrow id="S5.p2.8.m1.1.1.3.3" xref="S5.p2.8.m1.1.1.3.3.cmml"><mn id="S5.p2.8.m1.1.1.3.3.2" xref="S5.p2.8.m1.1.1.3.3.2.cmml">1</mn><mo id="S5.p2.8.m1.1.1.3.3.1" xref="S5.p2.8.m1.1.1.3.3.1.cmml">/</mo><mi id="S5.p2.8.m1.1.1.3.3.3" xref="S5.p2.8.m1.1.1.3.3.3.cmml">S</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.8.m1.1b"><apply id="S5.p2.8.m1.1.1.cmml" xref="S5.p2.8.m1.1.1"><eq id="S5.p2.8.m1.1.1.1.cmml" xref="S5.p2.8.m1.1.1.1"></eq><ci id="S5.p2.8.m1.1.1.2.cmml" xref="S5.p2.8.m1.1.1.2">𝛼</ci><apply id="S5.p2.8.m1.1.1.3.cmml" xref="S5.p2.8.m1.1.1.3"><plus id="S5.p2.8.m1.1.1.3.1.cmml" xref="S5.p2.8.m1.1.1.3.1"></plus><ci id="S5.p2.8.m1.1.1.3.2.cmml" xref="S5.p2.8.m1.1.1.3.2">𝑛</ci><apply id="S5.p2.8.m1.1.1.3.3.cmml" xref="S5.p2.8.m1.1.1.3.3"><divide id="S5.p2.8.m1.1.1.3.3.1.cmml" xref="S5.p2.8.m1.1.1.3.3.1"></divide><cn id="S5.p2.8.m1.1.1.3.3.2.cmml" type="integer" xref="S5.p2.8.m1.1.1.3.3.2">1</cn><ci id="S5.p2.8.m1.1.1.3.3.3.cmml" xref="S5.p2.8.m1.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m1.1c">\alpha=n+1/S</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m1.1d">italic_α = italic_n + 1 / italic_S</annotation></semantics></math> and <math alttext="\beta=1" class="ltx_Math" display="inline" id="S5.p2.9.m2.1"><semantics id="S5.p2.9.m2.1a"><mrow id="S5.p2.9.m2.1.1" xref="S5.p2.9.m2.1.1.cmml"><mi id="S5.p2.9.m2.1.1.2" xref="S5.p2.9.m2.1.1.2.cmml">β</mi><mo id="S5.p2.9.m2.1.1.1" xref="S5.p2.9.m2.1.1.1.cmml">=</mo><mn id="S5.p2.9.m2.1.1.3" xref="S5.p2.9.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.9.m2.1b"><apply id="S5.p2.9.m2.1.1.cmml" xref="S5.p2.9.m2.1.1"><eq id="S5.p2.9.m2.1.1.1.cmml" xref="S5.p2.9.m2.1.1.1"></eq><ci id="S5.p2.9.m2.1.1.2.cmml" xref="S5.p2.9.m2.1.1.2">𝛽</ci><cn id="S5.p2.9.m2.1.1.3.cmml" type="integer" xref="S5.p2.9.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.9.m2.1c">\beta=1</annotation><annotation encoding="application/x-llamapun" id="S5.p2.9.m2.1d">italic_β = 1</annotation></semantics></math>. The probability of obtaining an efficiency <math alttext="\epsilon_{i}^{\textsc{Mjd}}" class="ltx_Math" display="inline" id="S5.p2.10.m3.1"><semantics id="S5.p2.10.m3.1a"><msubsup id="S5.p2.10.m3.1.1" xref="S5.p2.10.m3.1.1.cmml"><mi id="S5.p2.10.m3.1.1.2.2" xref="S5.p2.10.m3.1.1.2.2.cmml">ϵ</mi><mi id="S5.p2.10.m3.1.1.2.3" xref="S5.p2.10.m3.1.1.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S5.p2.10.m3.1.1.3" xref="S5.p2.10.m3.1.1.3a.cmml">Mjd</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.p2.10.m3.1b"><apply id="S5.p2.10.m3.1.1.cmml" xref="S5.p2.10.m3.1.1"><csymbol cd="ambiguous" id="S5.p2.10.m3.1.1.1.cmml" xref="S5.p2.10.m3.1.1">superscript</csymbol><apply id="S5.p2.10.m3.1.1.2.cmml" xref="S5.p2.10.m3.1.1"><csymbol cd="ambiguous" id="S5.p2.10.m3.1.1.2.1.cmml" xref="S5.p2.10.m3.1.1">subscript</csymbol><ci id="S5.p2.10.m3.1.1.2.2.cmml" xref="S5.p2.10.m3.1.1.2.2">italic-ϵ</ci><ci id="S5.p2.10.m3.1.1.2.3.cmml" xref="S5.p2.10.m3.1.1.2.3">𝑖</ci></apply><ci id="S5.p2.10.m3.1.1.3a.cmml" xref="S5.p2.10.m3.1.1.3"><mtext class="ltx_font_smallcaps" id="S5.p2.10.m3.1.1.3.cmml" mathsize="70%" xref="S5.p2.10.m3.1.1.3">Mjd</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.10.m3.1c">\epsilon_{i}^{\textsc{Mjd}}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.10.m3.1d">italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT</annotation></semantics></math>, given an ensemble of counts <math alttext="\{n_{ij}\}" class="ltx_Math" display="inline" id="S5.p2.11.m4.1"><semantics id="S5.p2.11.m4.1a"><mrow id="S5.p2.11.m4.1.1.1" xref="S5.p2.11.m4.1.1.2.cmml"><mo id="S5.p2.11.m4.1.1.1.2" stretchy="false" xref="S5.p2.11.m4.1.1.2.cmml">{</mo><msub id="S5.p2.11.m4.1.1.1.1" xref="S5.p2.11.m4.1.1.1.1.cmml"><mi id="S5.p2.11.m4.1.1.1.1.2" xref="S5.p2.11.m4.1.1.1.1.2.cmml">n</mi><mrow id="S5.p2.11.m4.1.1.1.1.3" xref="S5.p2.11.m4.1.1.1.1.3.cmml"><mi id="S5.p2.11.m4.1.1.1.1.3.2" xref="S5.p2.11.m4.1.1.1.1.3.2.cmml">i</mi><mo id="S5.p2.11.m4.1.1.1.1.3.1" xref="S5.p2.11.m4.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.p2.11.m4.1.1.1.1.3.3" xref="S5.p2.11.m4.1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S5.p2.11.m4.1.1.1.3" stretchy="false" xref="S5.p2.11.m4.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.11.m4.1b"><set id="S5.p2.11.m4.1.1.2.cmml" xref="S5.p2.11.m4.1.1.1"><apply id="S5.p2.11.m4.1.1.1.1.cmml" xref="S5.p2.11.m4.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.11.m4.1.1.1.1.1.cmml" xref="S5.p2.11.m4.1.1.1.1">subscript</csymbol><ci id="S5.p2.11.m4.1.1.1.1.2.cmml" xref="S5.p2.11.m4.1.1.1.1.2">𝑛</ci><apply id="S5.p2.11.m4.1.1.1.1.3.cmml" xref="S5.p2.11.m4.1.1.1.1.3"><times id="S5.p2.11.m4.1.1.1.1.3.1.cmml" xref="S5.p2.11.m4.1.1.1.1.3.1"></times><ci id="S5.p2.11.m4.1.1.1.1.3.2.cmml" xref="S5.p2.11.m4.1.1.1.1.3.2">𝑖</ci><ci id="S5.p2.11.m4.1.1.1.1.3.3.cmml" xref="S5.p2.11.m4.1.1.1.1.3.3">𝑗</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.11.m4.1c">\{n_{ij}\}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.11.m4.1d">{ italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT }</annotation></semantics></math> is thus, <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx2"> <tbody id="S5.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P(\epsilon_{i}^{\textsc{Mjd}}|\{n_{ij}\})" class="ltx_Math" display="inline" id="S5.Ex2.m1.1"><semantics id="S5.Ex2.m1.1a"><mrow id="S5.Ex2.m1.1.1" xref="S5.Ex2.m1.1.1.cmml"><mi id="S5.Ex2.m1.1.1.3" xref="S5.Ex2.m1.1.1.3.cmml">P</mi><mo id="S5.Ex2.m1.1.1.2" xref="S5.Ex2.m1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex2.m1.1.1.1.1" xref="S5.Ex2.m1.1.1.1.1.1.cmml"><mo id="S5.Ex2.m1.1.1.1.1.2" stretchy="false" xref="S5.Ex2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex2.m1.1.1.1.1.1" xref="S5.Ex2.m1.1.1.1.1.1.cmml"><msubsup id="S5.Ex2.m1.1.1.1.1.1.3" xref="S5.Ex2.m1.1.1.1.1.1.3.cmml"><mi id="S5.Ex2.m1.1.1.1.1.1.3.2.2" xref="S5.Ex2.m1.1.1.1.1.1.3.2.2.cmml">ϵ</mi><mi id="S5.Ex2.m1.1.1.1.1.1.3.2.3" xref="S5.Ex2.m1.1.1.1.1.1.3.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S5.Ex2.m1.1.1.1.1.1.3.3" xref="S5.Ex2.m1.1.1.1.1.1.3.3a.cmml">Mjd</mtext></msubsup><mo fence="false" id="S5.Ex2.m1.1.1.1.1.1.2" xref="S5.Ex2.m1.1.1.1.1.1.2.cmml">|</mo><mrow id="S5.Ex2.m1.1.1.1.1.1.1.1" xref="S5.Ex2.m1.1.1.1.1.1.1.2.cmml"><mo id="S5.Ex2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex2.m1.1.1.1.1.1.1.2.cmml">{</mo><msub id="S5.Ex2.m1.1.1.1.1.1.1.1.1" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex2.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.2.cmml">n</mi><mrow id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.2" 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xref="S5.Ex2.m1.1.1.1.1.1.3">superscript</csymbol><apply id="S5.Ex2.m1.1.1.1.1.1.3.2.cmml" xref="S5.Ex2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex2.m1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex2.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex2.m1.1.1.1.1.1.3.2.2.cmml" xref="S5.Ex2.m1.1.1.1.1.1.3.2.2">italic-ϵ</ci><ci id="S5.Ex2.m1.1.1.1.1.1.3.2.3.cmml" xref="S5.Ex2.m1.1.1.1.1.1.3.2.3">𝑖</ci></apply><ci id="S5.Ex2.m1.1.1.1.1.1.3.3a.cmml" xref="S5.Ex2.m1.1.1.1.1.1.3.3"><mtext class="ltx_font_smallcaps" id="S5.Ex2.m1.1.1.1.1.1.3.3.cmml" mathsize="70%" xref="S5.Ex2.m1.1.1.1.1.1.3.3">Mjd</mtext></ci></apply><set id="S5.Ex2.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1"><apply id="S5.Ex2.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex2.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Ex2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.2">𝑛</ci><apply id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.3"><times id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.1"></times><ci id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.2">𝑖</ci><ci id="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S5.Ex2.m1.1.1.1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex2.m1.1c">\displaystyle P(\epsilon_{i}^{\textsc{Mjd}}|\{n_{ij}\})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex2.m1.1d">italic_P ( italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT | { italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT } )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j}^{S_{i}}~{}\mathcal{B}_{j}P(\epsilon_{ij}^{\textsc{Mjd}}% |n_{ij})" class="ltx_Math" display="inline" id="S5.Ex2.m2.1"><semantics 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xref="S5.E9.m1.1.1.3.3.2">𝑖</ci><ci id="S5.E9.m1.1.1.3.3.3.cmml" xref="S5.E9.m1.1.1.3.3.3">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E9.m1.2c">\displaystyle=\sum_{j}^{S_{i}}~{}\mathcal{B}_{j}\frac{P(\lambda_{ij}|n_{ij})}{% N_{ij}}~{}.</annotation><annotation encoding="application/x-llamapun" id="S5.E9.m1.2d">= ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG italic_P ( italic_λ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_N start_POSTSUBSCRIPT italic_i 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">(9)</td> </tr></tbody> </table> This result is obtained by replacing <math alttext="n_{ij}" class="ltx_Math" display="inline" id="S5.p2.12.m1.1"><semantics id="S5.p2.12.m1.1a"><msub id="S5.p2.12.m1.1.1" xref="S5.p2.12.m1.1.1.cmml"><mi id="S5.p2.12.m1.1.1.2" xref="S5.p2.12.m1.1.1.2.cmml">n</mi><mrow id="S5.p2.12.m1.1.1.3" xref="S5.p2.12.m1.1.1.3.cmml"><mi id="S5.p2.12.m1.1.1.3.2" xref="S5.p2.12.m1.1.1.3.2.cmml">i</mi><mo id="S5.p2.12.m1.1.1.3.1" xref="S5.p2.12.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.p2.12.m1.1.1.3.3" xref="S5.p2.12.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.p2.12.m1.1b"><apply id="S5.p2.12.m1.1.1.cmml" xref="S5.p2.12.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.12.m1.1.1.1.cmml" xref="S5.p2.12.m1.1.1">subscript</csymbol><ci id="S5.p2.12.m1.1.1.2.cmml" xref="S5.p2.12.m1.1.1.2">𝑛</ci><apply id="S5.p2.12.m1.1.1.3.cmml" xref="S5.p2.12.m1.1.1.3"><times id="S5.p2.12.m1.1.1.3.1.cmml" xref="S5.p2.12.m1.1.1.3.1"></times><ci id="S5.p2.12.m1.1.1.3.2.cmml" xref="S5.p2.12.m1.1.1.3.2">𝑖</ci><ci id="S5.p2.12.m1.1.1.3.3.cmml" xref="S5.p2.12.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.12.m1.1c">n_{ij}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.12.m1.1d">italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> by <math alttext="P(\lambda_{ij}|n_{ij})" class="ltx_Math" display="inline" id="S5.p2.13.m2.1"><semantics id="S5.p2.13.m2.1a"><mrow id="S5.p2.13.m2.1.1" xref="S5.p2.13.m2.1.1.cmml"><mi id="S5.p2.13.m2.1.1.3" xref="S5.p2.13.m2.1.1.3.cmml">P</mi><mo id="S5.p2.13.m2.1.1.2" xref="S5.p2.13.m2.1.1.2.cmml">⁢</mo><mrow id="S5.p2.13.m2.1.1.1.1" xref="S5.p2.13.m2.1.1.1.1.1.cmml"><mo id="S5.p2.13.m2.1.1.1.1.2" stretchy="false" xref="S5.p2.13.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.13.m2.1.1.1.1.1" xref="S5.p2.13.m2.1.1.1.1.1.cmml"><msub id="S5.p2.13.m2.1.1.1.1.1.2" xref="S5.p2.13.m2.1.1.1.1.1.2.cmml"><mi id="S5.p2.13.m2.1.1.1.1.1.2.2" xref="S5.p2.13.m2.1.1.1.1.1.2.2.cmml">λ</mi><mrow id="S5.p2.13.m2.1.1.1.1.1.2.3" xref="S5.p2.13.m2.1.1.1.1.1.2.3.cmml"><mi id="S5.p2.13.m2.1.1.1.1.1.2.3.2" xref="S5.p2.13.m2.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S5.p2.13.m2.1.1.1.1.1.2.3.1" xref="S5.p2.13.m2.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p2.13.m2.1.1.1.1.1.2.3.3" xref="S5.p2.13.m2.1.1.1.1.1.2.3.3.cmml">j</mi></mrow></msub><mo fence="false" id="S5.p2.13.m2.1.1.1.1.1.1" xref="S5.p2.13.m2.1.1.1.1.1.1.cmml">|</mo><msub id="S5.p2.13.m2.1.1.1.1.1.3" xref="S5.p2.13.m2.1.1.1.1.1.3.cmml"><mi id="S5.p2.13.m2.1.1.1.1.1.3.2" xref="S5.p2.13.m2.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S5.p2.13.m2.1.1.1.1.1.3.3" xref="S5.p2.13.m2.1.1.1.1.1.3.3.cmml"><mi id="S5.p2.13.m2.1.1.1.1.1.3.3.2" xref="S5.p2.13.m2.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="S5.p2.13.m2.1.1.1.1.1.3.3.1" xref="S5.p2.13.m2.1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S5.p2.13.m2.1.1.1.1.1.3.3.3" xref="S5.p2.13.m2.1.1.1.1.1.3.3.3.cmml">j</mi></mrow></msub></mrow><mo id="S5.p2.13.m2.1.1.1.1.3" stretchy="false" xref="S5.p2.13.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.13.m2.1b"><apply id="S5.p2.13.m2.1.1.cmml" xref="S5.p2.13.m2.1.1"><times id="S5.p2.13.m2.1.1.2.cmml" xref="S5.p2.13.m2.1.1.2"></times><ci id="S5.p2.13.m2.1.1.3.cmml" xref="S5.p2.13.m2.1.1.3">𝑃</ci><apply id="S5.p2.13.m2.1.1.1.1.1.cmml" xref="S5.p2.13.m2.1.1.1.1"><csymbol cd="latexml" id="S5.p2.13.m2.1.1.1.1.1.1.cmml" xref="S5.p2.13.m2.1.1.1.1.1.1">conditional</csymbol><apply id="S5.p2.13.m2.1.1.1.1.1.2.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.p2.13.m2.1.1.1.1.1.2.1.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2">subscript</csymbol><ci id="S5.p2.13.m2.1.1.1.1.1.2.2.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2.2">𝜆</ci><apply id="S5.p2.13.m2.1.1.1.1.1.2.3.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2.3"><times id="S5.p2.13.m2.1.1.1.1.1.2.3.1.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2.3.1"></times><ci id="S5.p2.13.m2.1.1.1.1.1.2.3.2.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2.3.2">𝑖</ci><ci id="S5.p2.13.m2.1.1.1.1.1.2.3.3.cmml" xref="S5.p2.13.m2.1.1.1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S5.p2.13.m2.1.1.1.1.1.3.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.13.m2.1.1.1.1.1.3.1.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3">subscript</csymbol><ci id="S5.p2.13.m2.1.1.1.1.1.3.2.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3.2">𝑛</ci><apply id="S5.p2.13.m2.1.1.1.1.1.3.3.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3.3"><times id="S5.p2.13.m2.1.1.1.1.1.3.3.1.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3.3.1"></times><ci id="S5.p2.13.m2.1.1.1.1.1.3.3.2.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3.3.2">𝑖</ci><ci id="S5.p2.13.m2.1.1.1.1.1.3.3.3.cmml" xref="S5.p2.13.m2.1.1.1.1.1.3.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.13.m2.1c">P(\lambda_{ij}|n_{ij})</annotation><annotation encoding="application/x-llamapun" id="S5.p2.13.m2.1d">italic_P ( italic_λ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4.E4" title="In IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">4</a> and carrying it into Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4.E5" title="In IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">5</a>. The PDF of <math alttext="\epsilon_{i}^{\textsc{Mjd}}" class="ltx_Math" display="inline" id="S5.p2.14.m3.1"><semantics id="S5.p2.14.m3.1a"><msubsup id="S5.p2.14.m3.1.1" xref="S5.p2.14.m3.1.1.cmml"><mi id="S5.p2.14.m3.1.1.2.2" xref="S5.p2.14.m3.1.1.2.2.cmml">ϵ</mi><mi id="S5.p2.14.m3.1.1.2.3" xref="S5.p2.14.m3.1.1.2.3.cmml">i</mi><mtext class="ltx_font_smallcaps" id="S5.p2.14.m3.1.1.3" xref="S5.p2.14.m3.1.1.3a.cmml">Mjd</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.p2.14.m3.1b"><apply id="S5.p2.14.m3.1.1.cmml" xref="S5.p2.14.m3.1.1"><csymbol cd="ambiguous" id="S5.p2.14.m3.1.1.1.cmml" xref="S5.p2.14.m3.1.1">superscript</csymbol><apply id="S5.p2.14.m3.1.1.2.cmml" xref="S5.p2.14.m3.1.1"><csymbol cd="ambiguous" id="S5.p2.14.m3.1.1.2.1.cmml" xref="S5.p2.14.m3.1.1">subscript</csymbol><ci id="S5.p2.14.m3.1.1.2.2.cmml" xref="S5.p2.14.m3.1.1.2.2">italic-ϵ</ci><ci id="S5.p2.14.m3.1.1.2.3.cmml" xref="S5.p2.14.m3.1.1.2.3">𝑖</ci></apply><ci id="S5.p2.14.m3.1.1.3a.cmml" xref="S5.p2.14.m3.1.1.3"><mtext class="ltx_font_smallcaps" id="S5.p2.14.m3.1.1.3.cmml" mathsize="70%" xref="S5.p2.14.m3.1.1.3">Mjd</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.14.m3.1c">\epsilon_{i}^{\textsc{Mjd}}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.14.m3.1d">italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Mjd end_POSTSUPERSCRIPT</annotation></semantics></math> is computed numerically. Taking a random draw from the PDF in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5.Ex1" title="V Promoting single values to distributions ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V</a> for each segment, weighing by the appropriate factors and summing them as in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5.Ex2" title="V Promoting single values to distributions ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V</a>. This process is repeated <math alttext=">10^{5}" class="ltx_Math" display="inline" id="S5.p2.15.m4.1"><semantics id="S5.p2.15.m4.1a"><mrow id="S5.p2.15.m4.1.1" xref="S5.p2.15.m4.1.1.cmml"><mi id="S5.p2.15.m4.1.1.2" xref="S5.p2.15.m4.1.1.2.cmml"></mi><mo id="S5.p2.15.m4.1.1.1" xref="S5.p2.15.m4.1.1.1.cmml">></mo><msup id="S5.p2.15.m4.1.1.3" xref="S5.p2.15.m4.1.1.3.cmml"><mn id="S5.p2.15.m4.1.1.3.2" xref="S5.p2.15.m4.1.1.3.2.cmml">10</mn><mn id="S5.p2.15.m4.1.1.3.3" xref="S5.p2.15.m4.1.1.3.3.cmml">5</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.15.m4.1b"><apply id="S5.p2.15.m4.1.1.cmml" xref="S5.p2.15.m4.1.1"><gt id="S5.p2.15.m4.1.1.1.cmml" xref="S5.p2.15.m4.1.1.1"></gt><csymbol cd="latexml" id="S5.p2.15.m4.1.1.2.cmml" xref="S5.p2.15.m4.1.1.2">absent</csymbol><apply id="S5.p2.15.m4.1.1.3.cmml" xref="S5.p2.15.m4.1.1.3"><csymbol cd="ambiguous" id="S5.p2.15.m4.1.1.3.1.cmml" xref="S5.p2.15.m4.1.1.3">superscript</csymbol><cn id="S5.p2.15.m4.1.1.3.2.cmml" type="integer" xref="S5.p2.15.m4.1.1.3.2">10</cn><cn id="S5.p2.15.m4.1.1.3.3.cmml" type="integer" xref="S5.p2.15.m4.1.1.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.15.m4.1c">>10^{5}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.15.m4.1d">> 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math> times to generate the PDF or only once if one efficiency sample is needed. </div> <div class="ltx_para" id="S5.p3"> Not all segments are included in the sum for far away sources, only segments which produce <math alttext="\gamma" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mi id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_γ</annotation></semantics></math>’s with sufficiently high energies to lead to energy depositions in the BEW. The excluded segments have a true zero efficiency. </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> VI Monte Carlo uncertainty propagation</h2> <div class="ltx_para" id="S6.p1"> The assay-based background index can be promoted to a distribution by sampling the PDFs described in Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5" title="V Promoting single values to distributions ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V</a>. The <math alttext="\text{BI}_{i}" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><msub id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mtext id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2a.cmml">BI</mtext><mi id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1">subscript</csymbol><ci id="S6.p1.1.m1.1.1.2a.cmml" xref="S6.p1.1.m1.1.1.2"><mtext id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">BI</mtext></ci><ci id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">\text{BI}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> distributions are thus generated in a similar manner as that of the efficiency. A random draw is taken from the specific activity, mass, and efficiency PDFs. These are multiplied and scaled as in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4.E6" title="In IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">6</a>. The process is repeated <math alttext="10^{6}" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><msup id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml"><mn id="S6.p1.2.m2.1.1.2" xref="S6.p1.2.m2.1.1.2.cmml">10</mn><mn id="S6.p1.2.m2.1.1.3" xref="S6.p1.2.m2.1.1.3.cmml">6</mn></msup><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><apply id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.p1.2.m2.1.1.1.cmml" xref="S6.p1.2.m2.1.1">superscript</csymbol><cn id="S6.p1.2.m2.1.1.2.cmml" type="integer" xref="S6.p1.2.m2.1.1.2">10</cn><cn id="S6.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.p1.2.m2.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">10^{6}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT</annotation></semantics></math> times to produce a PDF for the BI of the source. Each set of draws constitutes a toy experiment resulting in a different BI. The uncertainty of the efficiency, specific activity and mass – embedded in their corresponding PDFs – is propagated into the <math alttext="\text{BI}_{i}" class="ltx_Math" display="inline" id="S6.p1.3.m3.1"><semantics id="S6.p1.3.m3.1a"><msub id="S6.p1.3.m3.1.1" xref="S6.p1.3.m3.1.1.cmml"><mtext id="S6.p1.3.m3.1.1.2" xref="S6.p1.3.m3.1.1.2a.cmml">BI</mtext><mi id="S6.p1.3.m3.1.1.3" xref="S6.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p1.3.m3.1b"><apply id="S6.p1.3.m3.1.1.cmml" xref="S6.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.p1.3.m3.1.1.1.cmml" xref="S6.p1.3.m3.1.1">subscript</csymbol><ci id="S6.p1.3.m3.1.1.2a.cmml" xref="S6.p1.3.m3.1.1.2"><mtext id="S6.p1.3.m3.1.1.2.cmml" xref="S6.p1.3.m3.1.1.2">BI</mtext></ci><ci id="S6.p1.3.m3.1.1.3.cmml" xref="S6.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.m3.1c">\text{BI}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.m3.1d">BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> distribution through this process, referred to as Monte Carlo uncertainty propagation. </div> <div class="ltx_para" id="S6.p2"> The <math alttext="\text{BI}_{i}" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><msub id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml"><mtext id="S6.p2.1.m1.1.1.2" xref="S6.p2.1.m1.1.1.2a.cmml">BI</mtext><mi id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.p2.1.m1.1.1.1.cmml" xref="S6.p2.1.m1.1.1">subscript</csymbol><ci id="S6.p2.1.m1.1.1.2a.cmml" xref="S6.p2.1.m1.1.1.2"><mtext id="S6.p2.1.m1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.2">BI</mtext></ci><ci id="S6.p2.1.m1.1.1.3.cmml" xref="S6.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">\text{BI}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> PDFs are combined by taking the direct sum of the samples that were used to generate them. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.F3" title="Figure 3 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">3</a> shows the result of summing all the <math alttext="\text{BI}_{i}" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><msub id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml"><mtext id="S6.p2.2.m2.1.1.2" xref="S6.p2.2.m2.1.1.2a.cmml">BI</mtext><mi id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.p2.2.m2.1.1.1.cmml" xref="S6.p2.2.m2.1.1">subscript</csymbol><ci id="S6.p2.2.m2.1.1.2a.cmml" xref="S6.p2.2.m2.1.1.2"><mtext id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.2">BI</mtext></ci><ci id="S6.p2.2.m2.1.1.3.cmml" xref="S6.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\text{BI}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">BI start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> PDFs belonging to a component group in the Demonstrator. The mean and uncertainties that are extracted from these distributions are reported in Table <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.T2" title="Table 2 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a> and plotted as error bars in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.F3" title="Figure 3 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">3</a>. The total BI distribution of 232Th and 238U sources in the Demonstrator is computed in a similar manner and displayed at the bottom of the figure. </div> <figure class="ltx_figure" id="S6.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="1572" id="S6.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 3: The mean BI and 1<math alttext="\sigma" class="ltx_Math" display="inline" id="S6.F3.4.m1.1"><semantics id="S6.F3.4.m1.1b"><mi id="S6.F3.4.m1.1.1" xref="S6.F3.4.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.F3.4.m1.1c"><ci id="S6.F3.4.m1.1.1.cmml" xref="S6.F3.4.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.4.m1.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.F3.4.m1.1e">italic_σ</annotation></semantics></math> uncertainties of the Demonstrator’s component groups and their sum are plotted as a vertical black lines through dark grey error bars. Statistical uncertainty contributions are shown in red. These are overlaid on their corresponding PDFs in light grey (with the central 2<math alttext="\sigma" class="ltx_Math" display="inline" id="S6.F3.5.m2.1"><semantics id="S6.F3.5.m2.1b"><mi id="S6.F3.5.m2.1.1" xref="S6.F3.5.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.F3.5.m2.1c"><ci id="S6.F3.5.m2.1.1.cmml" xref="S6.F3.5.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.5.m2.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.F3.5.m2.1e">italic_σ</annotation></semantics></math> region shaded). All PDFs contain <math alttext="10^{6}" class="ltx_Math" display="inline" id="S6.F3.6.m3.1"><semantics id="S6.F3.6.m3.1b"><msup id="S6.F3.6.m3.1.1" xref="S6.F3.6.m3.1.1.cmml"><mn id="S6.F3.6.m3.1.1.2" xref="S6.F3.6.m3.1.1.2.cmml">10</mn><mn id="S6.F3.6.m3.1.1.3" xref="S6.F3.6.m3.1.1.3.cmml">6</mn></msup><annotation-xml encoding="MathML-Content" id="S6.F3.6.m3.1c"><apply id="S6.F3.6.m3.1.1.cmml" xref="S6.F3.6.m3.1.1"><csymbol cd="ambiguous" id="S6.F3.6.m3.1.1.1.cmml" xref="S6.F3.6.m3.1.1">superscript</csymbol><cn id="S6.F3.6.m3.1.1.2.cmml" type="integer" xref="S6.F3.6.m3.1.1.2">10</cn><cn id="S6.F3.6.m3.1.1.3.cmml" type="integer" xref="S6.F3.6.m3.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F3.6.m3.1d">10^{6}</annotation><annotation encoding="application/x-llamapun" id="S6.F3.6.m3.1e">10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT</annotation></semantics></math> samples and are scaled vertically such that they have the same height.</figcaption> </figure> <div class="ltx_para" id="S6.p3"> Simulation statistical uncertainties can be isolated by collapsing the specific activity and mass distributions to their respective means and using them to weigh the samples that were drawn from the efficiency distribution. This information can be used by the analyst as a guide to set the number of decays in future simulations. </div> <div class="ltx_para" id="S6.p4"> Monte Carlo uncertainty propagation allows for all potential sources of natural radiation to be incorporated in the model, not just those which produced counts in the BEW. With no counts in the BEW, the single-valued efficiency is zero. However, the efficiency distribution will take an exponential form, effectively setting an upper limit on the BI (once weighted properly) for the source in question. This upper limit is fully dependent on the number of simulated decays and given enough computational resources should be driven down to the point where it is negligible compared to other contributions to the BI. This is the case for the Vacuum Hardware and Pb Shielding 238U contamination. </div> <figure class="ltx_table" id="S6.T2"> <figcaption class="ltx_caption">Table 2: Background indices and <math alttext="1\sigma" class="ltx_Math" display="inline" id="S6.T2.4.m1.1"><semantics id="S6.T2.4.m1.1b"><mrow id="S6.T2.4.m1.1.1" xref="S6.T2.4.m1.1.1.cmml"><mn id="S6.T2.4.m1.1.1.2" xref="S6.T2.4.m1.1.1.2.cmml">1</mn><mo id="S6.T2.4.m1.1.1.1" xref="S6.T2.4.m1.1.1.1.cmml">⁢</mo><mi id="S6.T2.4.m1.1.1.3" xref="S6.T2.4.m1.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.4.m1.1c"><apply id="S6.T2.4.m1.1.1.cmml" xref="S6.T2.4.m1.1.1"><times id="S6.T2.4.m1.1.1.1.cmml" xref="S6.T2.4.m1.1.1.1"></times><cn id="S6.T2.4.m1.1.1.2.cmml" type="integer" xref="S6.T2.4.m1.1.1.2">1</cn><ci id="S6.T2.4.m1.1.1.3.cmml" xref="S6.T2.4.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.4.m1.1d">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.T2.4.m1.1e">1 italic_σ</annotation></semantics></math> uncertainties of the component groups of the Demonstrator as derived from their distributions. A 90% C.L. upper limit is given if the width of the central <math alttext="2\sigma" class="ltx_Math" display="inline" id="S6.T2.5.m2.1"><semantics id="S6.T2.5.m2.1b"><mrow id="S6.T2.5.m2.1.1" xref="S6.T2.5.m2.1.1.cmml"><mn id="S6.T2.5.m2.1.1.2" xref="S6.T2.5.m2.1.1.2.cmml">2</mn><mo id="S6.T2.5.m2.1.1.1" xref="S6.T2.5.m2.1.1.1.cmml">⁢</mo><mi id="S6.T2.5.m2.1.1.3" xref="S6.T2.5.m2.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.5.m2.1c"><apply id="S6.T2.5.m2.1.1.cmml" xref="S6.T2.5.m2.1.1"><times id="S6.T2.5.m2.1.1.1.cmml" xref="S6.T2.5.m2.1.1.1"></times><cn id="S6.T2.5.m2.1.1.2.cmml" type="integer" xref="S6.T2.5.m2.1.1.2">2</cn><ci id="S6.T2.5.m2.1.1.3.cmml" xref="S6.T2.5.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.5.m2.1d">2\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.T2.5.m2.1e">2 italic_σ</annotation></semantics></math> region is higher than the width of the lower <math alttext="2\sigma" class="ltx_Math" display="inline" id="S6.T2.6.m3.1"><semantics id="S6.T2.6.m3.1b"><mrow id="S6.T2.6.m3.1.1" xref="S6.T2.6.m3.1.1.cmml"><mn id="S6.T2.6.m3.1.1.2" xref="S6.T2.6.m3.1.1.2.cmml">2</mn><mo id="S6.T2.6.m3.1.1.1" xref="S6.T2.6.m3.1.1.1.cmml">⁢</mo><mi id="S6.T2.6.m3.1.1.3" xref="S6.T2.6.m3.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.6.m3.1c"><apply id="S6.T2.6.m3.1.1.cmml" xref="S6.T2.6.m3.1.1"><times id="S6.T2.6.m3.1.1.1.cmml" xref="S6.T2.6.m3.1.1.1"></times><cn id="S6.T2.6.m3.1.1.2.cmml" type="integer" xref="S6.T2.6.m3.1.1.2">2</cn><ci id="S6.T2.6.m3.1.1.3.cmml" xref="S6.T2.6.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.6.m3.1d">2\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.T2.6.m3.1e">2 italic_σ</annotation></semantics></math> region. The direct BIs are included for reference.</figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S6.T2.53"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S6.T2.8.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S6.T2.8.2.3">Group</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.7.1.1">Mean 232Th BI</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.8.2.2">Mean 238U BI</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.8.2.4">Mean BI</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.8.2.5">Direct BI</th> </tr> <tr class="ltx_tr" id="S6.T2.53.48.1"> <th class="ltx_td ltx_th ltx_th_row" id="S6.T2.53.48.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.53.48.1.2">[cts/(keV kg yr)]</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.53.48.1.3">[cts/(keV kg yr)]</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.53.48.1.4">[cts/(keV kg yr)]</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S6.T2.53.48.1.5">[cts/(keV kg yr)]</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S6.T2.12.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S6.T2.12.6.5"> Electroformed Cu</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S6.T2.9.3.1"><math alttext="2.44^{+0.19}_{-0.19}\times 10^{-4}" class="ltx_Math" display="inline" id="S6.T2.9.3.1.m1.1"><semantics id="S6.T2.9.3.1.m1.1a"><mrow id="S6.T2.9.3.1.m1.1.1" xref="S6.T2.9.3.1.m1.1.1.cmml"><msubsup id="S6.T2.9.3.1.m1.1.1.2" xref="S6.T2.9.3.1.m1.1.1.2.cmml"><mn id="S6.T2.9.3.1.m1.1.1.2.2.2" xref="S6.T2.9.3.1.m1.1.1.2.2.2.cmml">2.44</mn><mrow id="S6.T2.9.3.1.m1.1.1.2.3" xref="S6.T2.9.3.1.m1.1.1.2.3.cmml"><mo id="S6.T2.9.3.1.m1.1.1.2.3a" xref="S6.T2.9.3.1.m1.1.1.2.3.cmml">−</mo><mn id="S6.T2.9.3.1.m1.1.1.2.3.2" xref="S6.T2.9.3.1.m1.1.1.2.3.2.cmml">0.19</mn></mrow><mrow id="S6.T2.9.3.1.m1.1.1.2.2.3" xref="S6.T2.9.3.1.m1.1.1.2.2.3.cmml"><mo id="S6.T2.9.3.1.m1.1.1.2.2.3a" xref="S6.T2.9.3.1.m1.1.1.2.2.3.cmml">+</mo><mn id="S6.T2.9.3.1.m1.1.1.2.2.3.2" xref="S6.T2.9.3.1.m1.1.1.2.2.3.2.cmml">0.19</mn></mrow></msubsup><mo id="S6.T2.9.3.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.T2.9.3.1.m1.1.1.1.cmml">×</mo><msup id="S6.T2.9.3.1.m1.1.1.3" xref="S6.T2.9.3.1.m1.1.1.3.cmml"><mn id="S6.T2.9.3.1.m1.1.1.3.2" xref="S6.T2.9.3.1.m1.1.1.3.2.cmml">10</mn><mrow id="S6.T2.9.3.1.m1.1.1.3.3" xref="S6.T2.9.3.1.m1.1.1.3.3.cmml"><mo id="S6.T2.9.3.1.m1.1.1.3.3a" xref="S6.T2.9.3.1.m1.1.1.3.3.cmml">−</mo><mn id="S6.T2.9.3.1.m1.1.1.3.3.2" xref="S6.T2.9.3.1.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.9.3.1.m1.1b"><apply id="S6.T2.9.3.1.m1.1.1.cmml" xref="S6.T2.9.3.1.m1.1.1"><times id="S6.T2.9.3.1.m1.1.1.1.cmml" xref="S6.T2.9.3.1.m1.1.1.1"></times><apply id="S6.T2.9.3.1.m1.1.1.2.cmml" xref="S6.T2.9.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.T2.9.3.1.m1.1.1.2.1.cmml" xref="S6.T2.9.3.1.m1.1.1.2">subscript</csymbol><apply id="S6.T2.9.3.1.m1.1.1.2.2.cmml" xref="S6.T2.9.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.T2.9.3.1.m1.1.1.2.2.1.cmml" xref="S6.T2.9.3.1.m1.1.1.2">superscript</csymbol><cn id="S6.T2.9.3.1.m1.1.1.2.2.2.cmml" type="float" xref="S6.T2.9.3.1.m1.1.1.2.2.2">2.44</cn><apply id="S6.T2.9.3.1.m1.1.1.2.2.3.cmml" xref="S6.T2.9.3.1.m1.1.1.2.2.3"><plus id="S6.T2.9.3.1.m1.1.1.2.2.3.1.cmml" xref="S6.T2.9.3.1.m1.1.1.2.2.3"></plus><cn id="S6.T2.9.3.1.m1.1.1.2.2.3.2.cmml" type="float" xref="S6.T2.9.3.1.m1.1.1.2.2.3.2">0.19</cn></apply></apply><apply id="S6.T2.9.3.1.m1.1.1.2.3.cmml" xref="S6.T2.9.3.1.m1.1.1.2.3"><minus id="S6.T2.9.3.1.m1.1.1.2.3.1.cmml" xref="S6.T2.9.3.1.m1.1.1.2.3"></minus><cn id="S6.T2.9.3.1.m1.1.1.2.3.2.cmml" type="float" xref="S6.T2.9.3.1.m1.1.1.2.3.2">0.19</cn></apply></apply><apply id="S6.T2.9.3.1.m1.1.1.3.cmml" xref="S6.T2.9.3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.T2.9.3.1.m1.1.1.3.1.cmml" xref="S6.T2.9.3.1.m1.1.1.3">superscript</csymbol><cn id="S6.T2.9.3.1.m1.1.1.3.2.cmml" type="integer" xref="S6.T2.9.3.1.m1.1.1.3.2">10</cn><apply id="S6.T2.9.3.1.m1.1.1.3.3.cmml" xref="S6.T2.9.3.1.m1.1.1.3.3"><minus id="S6.T2.9.3.1.m1.1.1.3.3.1.cmml" xref="S6.T2.9.3.1.m1.1.1.3.3"></minus><cn id="S6.T2.9.3.1.m1.1.1.3.3.2.cmml" type="integer" xref="S6.T2.9.3.1.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.9.3.1.m1.1c">2.44^{+0.19}_{-0.19}\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S6.T2.9.3.1.m1.1d">2.44 start_POSTSUPERSCRIPT + 0.19 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 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0.57 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S6.T2.36.30.4"><math alttext="2.36\times 10^{-6}" class="ltx_Math" display="inline" id="S6.T2.36.30.4.m1.1"><semantics id="S6.T2.36.30.4.m1.1a"><mrow id="S6.T2.36.30.4.m1.1.1" xref="S6.T2.36.30.4.m1.1.1.cmml"><mn id="S6.T2.36.30.4.m1.1.1.2" xref="S6.T2.36.30.4.m1.1.1.2.cmml">2.36</mn><mo id="S6.T2.36.30.4.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.T2.36.30.4.m1.1.1.1.cmml">×</mo><msup id="S6.T2.36.30.4.m1.1.1.3" xref="S6.T2.36.30.4.m1.1.1.3.cmml"><mn id="S6.T2.36.30.4.m1.1.1.3.2" xref="S6.T2.36.30.4.m1.1.1.3.2.cmml">10</mn><mrow id="S6.T2.36.30.4.m1.1.1.3.3" xref="S6.T2.36.30.4.m1.1.1.3.3.cmml"><mo id="S6.T2.36.30.4.m1.1.1.3.3a" xref="S6.T2.36.30.4.m1.1.1.3.3.cmml">−</mo><mn id="S6.T2.36.30.4.m1.1.1.3.3.2" xref="S6.T2.36.30.4.m1.1.1.3.3.2.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.36.30.4.m1.1b"><apply id="S6.T2.36.30.4.m1.1.1.cmml" xref="S6.T2.36.30.4.m1.1.1"><times id="S6.T2.36.30.4.m1.1.1.1.cmml" xref="S6.T2.36.30.4.m1.1.1.1"></times><cn id="S6.T2.36.30.4.m1.1.1.2.cmml" type="float" xref="S6.T2.36.30.4.m1.1.1.2">2.36</cn><apply id="S6.T2.36.30.4.m1.1.1.3.cmml" xref="S6.T2.36.30.4.m1.1.1.3"><csymbol cd="ambiguous" id="S6.T2.36.30.4.m1.1.1.3.1.cmml" xref="S6.T2.36.30.4.m1.1.1.3">superscript</csymbol><cn id="S6.T2.36.30.4.m1.1.1.3.2.cmml" type="integer" xref="S6.T2.36.30.4.m1.1.1.3.2">10</cn><apply id="S6.T2.36.30.4.m1.1.1.3.3.cmml" xref="S6.T2.36.30.4.m1.1.1.3.3"><minus id="S6.T2.36.30.4.m1.1.1.3.3.1.cmml" xref="S6.T2.36.30.4.m1.1.1.3.3"></minus><cn id="S6.T2.36.30.4.m1.1.1.3.3.2.cmml" type="integer" xref="S6.T2.36.30.4.m1.1.1.3.3.2">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.36.30.4.m1.1c">2.36\times 10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S6.T2.36.30.4.m1.1d">2.36 × 10 start_POSTSUPERSCRIPT - 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0.36 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S6.T2.53.47.4"><math alttext="9.23\times 10^{-4}" class="ltx_Math" display="inline" id="S6.T2.53.47.4.m1.1"><semantics id="S6.T2.53.47.4.m1.1a"><mrow id="S6.T2.53.47.4.m1.1.1" xref="S6.T2.53.47.4.m1.1.1.cmml"><mn id="S6.T2.53.47.4.m1.1.1.2" xref="S6.T2.53.47.4.m1.1.1.2.cmml">9.23</mn><mo id="S6.T2.53.47.4.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.T2.53.47.4.m1.1.1.1.cmml">×</mo><msup id="S6.T2.53.47.4.m1.1.1.3" xref="S6.T2.53.47.4.m1.1.1.3.cmml"><mn id="S6.T2.53.47.4.m1.1.1.3.2" xref="S6.T2.53.47.4.m1.1.1.3.2.cmml">10</mn><mrow id="S6.T2.53.47.4.m1.1.1.3.3" xref="S6.T2.53.47.4.m1.1.1.3.3.cmml"><mo id="S6.T2.53.47.4.m1.1.1.3.3a" xref="S6.T2.53.47.4.m1.1.1.3.3.cmml">−</mo><mn id="S6.T2.53.47.4.m1.1.1.3.3.2" xref="S6.T2.53.47.4.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.T2.53.47.4.m1.1b"><apply id="S6.T2.53.47.4.m1.1.1.cmml" xref="S6.T2.53.47.4.m1.1.1"><times id="S6.T2.53.47.4.m1.1.1.1.cmml" xref="S6.T2.53.47.4.m1.1.1.1"></times><cn id="S6.T2.53.47.4.m1.1.1.2.cmml" type="float" xref="S6.T2.53.47.4.m1.1.1.2">9.23</cn><apply id="S6.T2.53.47.4.m1.1.1.3.cmml" xref="S6.T2.53.47.4.m1.1.1.3"><csymbol cd="ambiguous" id="S6.T2.53.47.4.m1.1.1.3.1.cmml" xref="S6.T2.53.47.4.m1.1.1.3">superscript</csymbol><cn id="S6.T2.53.47.4.m1.1.1.3.2.cmml" type="integer" xref="S6.T2.53.47.4.m1.1.1.3.2">10</cn><apply id="S6.T2.53.47.4.m1.1.1.3.3.cmml" xref="S6.T2.53.47.4.m1.1.1.3.3"><minus id="S6.T2.53.47.4.m1.1.1.3.3.1.cmml" xref="S6.T2.53.47.4.m1.1.1.3.3"></minus><cn id="S6.T2.53.47.4.m1.1.1.3.3.2.cmml" type="integer" xref="S6.T2.53.47.4.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.53.47.4.m1.1c">9.23\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S6.T2.53.47.4.m1.1d">9.23 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> VII Projected background and conclusions</h2> <div class="ltx_para" id="S7.p1"> As Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6" title="VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">VI</a> describes, the mean BI and the corresponding uncertainties of the Majorana Demonstrator’s component groups are extracted from their distributions in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.F3" title="Figure 3 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">3</a> and reported in Table <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6.T2" title="Table 2 ‣ VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">2</a>. The mean total natural radiation BI, determined from its distribution at the bottom of the figure, is <table class="ltx_equation ltx_eqn_table" id="S7.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="^{\text{assay}}\text{BI}=\left[8.95\pm 0.16(\text{stat.})\pm 0.20(\text{act.})% \right]\times 10^{-4}" class="ltx_math_unparsed" display="block" id="S7.Ex3.m1.1"><semantics id="S7.Ex3.m1.1a"><mrow id="S7.Ex3.m1.1b"><msup id="S7.Ex3.m1.1.1"><mi id="S7.Ex3.m1.1.1a"></mi><mtext id="S7.Ex3.m1.1.1.1">assay</mtext></msup><mtext id="S7.Ex3.m1.1.2">BI</mtext><mo id="S7.Ex3.m1.1.3">=</mo><mrow id="S7.Ex3.m1.1.4"><mo id="S7.Ex3.m1.1.4.1">[</mo><mn id="S7.Ex3.m1.1.4.2">8.95</mn><mo id="S7.Ex3.m1.1.4.3">±</mo><mn id="S7.Ex3.m1.1.4.4">0.16</mn><mrow id="S7.Ex3.m1.1.4.5"><mo id="S7.Ex3.m1.1.4.5.1" stretchy="false">(</mo><mtext id="S7.Ex3.m1.1.4.5.2">stat.</mtext><mo id="S7.Ex3.m1.1.4.5.3" stretchy="false">)</mo></mrow><mo id="S7.Ex3.m1.1.4.6">±</mo><mn id="S7.Ex3.m1.1.4.7">0.20</mn><mrow id="S7.Ex3.m1.1.4.8"><mo id="S7.Ex3.m1.1.4.8.1" stretchy="false">(</mo><mtext id="S7.Ex3.m1.1.4.8.2">act.</mtext><mo id="S7.Ex3.m1.1.4.8.3" stretchy="false">)</mo></mrow><mo id="S7.Ex3.m1.1.4.9" rspace="0.055em">]</mo></mrow><mo id="S7.Ex3.m1.1.5" rspace="0.222em">×</mo><msup id="S7.Ex3.m1.1.6"><mn id="S7.Ex3.m1.1.6.2">10</mn><mrow id="S7.Ex3.m1.1.6.3"><mo id="S7.Ex3.m1.1.6.3a">−</mo><mn id="S7.Ex3.m1.1.6.3.2">4</mn></mrow></msup></mrow><annotation encoding="application/x-tex" id="S7.Ex3.m1.1c">^{\text{assay}}\text{BI}=\left[8.95\pm 0.16(\text{stat.})\pm 0.20(\text{act.})% \right]\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S7.Ex3.m1.1d">start_POSTSUPERSCRIPT assay end_POSTSUPERSCRIPT BI = [ 8.95 ± 0.16 ( stat. ) ± 0.20 ( act. ) ] × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> in units of cts/(keV kg yr). The uncertainty from activity (act.) dominates <math alttext="{}^{\text{assay}}\text{BI}" class="ltx_Math" display="inline" id="S7.p1.1.m1.1"><semantics id="S7.p1.1.m1.1a"><mmultiscripts id="S7.p1.1.m1.1.1" xref="S7.p1.1.m1.1.1.cmml"><mtext id="S7.p1.1.m1.1.1.2" xref="S7.p1.1.m1.1.1.2a.cmml">BI</mtext><mprescripts id="S7.p1.1.m1.1.1a" xref="S7.p1.1.m1.1.1.cmml"></mprescripts><mrow id="S7.p1.1.m1.1.1b" xref="S7.p1.1.m1.1.1.cmml"></mrow><mtext id="S7.p1.1.m1.1.1.3" xref="S7.p1.1.m1.1.1.3a.cmml">assay</mtext></mmultiscripts><annotation-xml encoding="MathML-Content" id="S7.p1.1.m1.1b"><apply id="S7.p1.1.m1.1.1.cmml" xref="S7.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.p1.1.m1.1.1.1.cmml" xref="S7.p1.1.m1.1.1">superscript</csymbol><ci id="S7.p1.1.m1.1.1.2a.cmml" xref="S7.p1.1.m1.1.1.2"><mtext id="S7.p1.1.m1.1.1.2.cmml" xref="S7.p1.1.m1.1.1.2">BI</mtext></ci><ci id="S7.p1.1.m1.1.1.3a.cmml" xref="S7.p1.1.m1.1.1.3"><mtext id="S7.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S7.p1.1.m1.1.1.3">assay</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.1.m1.1c">{}^{\text{assay}}\text{BI}</annotation><annotation encoding="application/x-llamapun" id="S7.p1.1.m1.1d">start_FLOATSUPERSCRIPT assay end_FLOATSUPERSCRIPT BI</annotation></semantics></math> and most component group BIs. The statistical uncertainty from simulation (stat.) has been calculated with the method described in Section <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S6" title="VI Monte Carlo uncertainty propagation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">VI</a>. Note that the symmetric uncertainty around the mean does not capture the asymmetry of the distribution. The contributions from the 232Th and 238U decay chains, given in units of cts/(keV kg yr), follow: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx3"> <tbody id="S7.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="{}^{\text{232}}\text{BI}" class="ltx_Math" display="inline" id="S7.Ex4.m1.1"><semantics id="S7.Ex4.m1.1a"><mmultiscripts id="S7.Ex4.m1.1.1" xref="S7.Ex4.m1.1.1.cmml"><mtext id="S7.Ex4.m1.1.1.2" xref="S7.Ex4.m1.1.1.2a.cmml">BI</mtext><mprescripts id="S7.Ex4.m1.1.1a" xref="S7.Ex4.m1.1.1.cmml"></mprescripts><mrow id="S7.Ex4.m1.1.1b" xref="S7.Ex4.m1.1.1.cmml"></mrow><mtext id="S7.Ex4.m1.1.1.3" xref="S7.Ex4.m1.1.1.3a.cmml">232</mtext></mmultiscripts><annotation-xml encoding="MathML-Content" id="S7.Ex4.m1.1b"><apply id="S7.Ex4.m1.1.1.cmml" xref="S7.Ex4.m1.1.1"><csymbol cd="ambiguous" id="S7.Ex4.m1.1.1.1.cmml" xref="S7.Ex4.m1.1.1">superscript</csymbol><ci id="S7.Ex4.m1.1.1.2a.cmml" xref="S7.Ex4.m1.1.1.2"><mtext id="S7.Ex4.m1.1.1.2.cmml" xref="S7.Ex4.m1.1.1.2">BI</mtext></ci><ci id="S7.Ex4.m1.1.1.3a.cmml" xref="S7.Ex4.m1.1.1.3"><mtext id="S7.Ex4.m1.1.1.3.cmml" mathsize="70%" xref="S7.Ex4.m1.1.1.3">232</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Ex4.m1.1c">{}^{\text{232}}\text{BI}</annotation><annotation encoding="application/x-llamapun" id="S7.Ex4.m1.1d">start_FLOATSUPERSCRIPT 232 end_FLOATSUPERSCRIPT BI</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\left[7.37\pm 0.12(\text{stat.})\pm 0.22(\text{act.})\right]% \times 10^{-4}~{}," class="ltx_Math" display="inline" id="S7.Ex4.m2.3"><semantics id="S7.Ex4.m2.3a"><mrow id="S7.Ex4.m2.3.3.1" xref="S7.Ex4.m2.3.3.1.1.cmml"><mrow id="S7.Ex4.m2.3.3.1.1" xref="S7.Ex4.m2.3.3.1.1.cmml"><mi id="S7.Ex4.m2.3.3.1.1.3" xref="S7.Ex4.m2.3.3.1.1.3.cmml"></mi><mo id="S7.Ex4.m2.3.3.1.1.2" xref="S7.Ex4.m2.3.3.1.1.2.cmml">=</mo><mrow id="S7.Ex4.m2.3.3.1.1.1" xref="S7.Ex4.m2.3.3.1.1.1.cmml"><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1" xref="S7.Ex4.m2.3.3.1.1.1.1.2.cmml"><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.2" xref="S7.Ex4.m2.3.3.1.1.1.1.2.1.cmml">[</mo><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1.1" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.cmml"><mn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.2" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.2.cmml">7.37</mn><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.1" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.1.cmml">±</mo><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.cmml"><mn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.2" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.2.cmml">0.12</mn><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.1" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.3.2" xref="S7.Ex4.m2.1.1a.cmml"><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S7.Ex4.m2.1.1a.cmml">(</mo><mtext id="S7.Ex4.m2.1.1" xref="S7.Ex4.m2.1.1.cmml">stat.</mtext><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S7.Ex4.m2.1.1a.cmml">)</mo></mrow></mrow><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.1a" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.1.cmml">±</mo><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.cmml"><mn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.2" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.2.cmml">0.22</mn><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.1" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.3.2" xref="S7.Ex4.m2.2.2a.cmml"><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S7.Ex4.m2.2.2a.cmml">(</mo><mtext id="S7.Ex4.m2.2.2" xref="S7.Ex4.m2.2.2.cmml">act.</mtext><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.3.2.2" stretchy="false" xref="S7.Ex4.m2.2.2a.cmml">)</mo></mrow></mrow></mrow><mo id="S7.Ex4.m2.3.3.1.1.1.1.1.3" rspace="0.055em" xref="S7.Ex4.m2.3.3.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S7.Ex4.m2.3.3.1.1.1.2" rspace="0.222em" xref="S7.Ex4.m2.3.3.1.1.1.2.cmml">×</mo><msup id="S7.Ex4.m2.3.3.1.1.1.3" xref="S7.Ex4.m2.3.3.1.1.1.3.cmml"><mn id="S7.Ex4.m2.3.3.1.1.1.3.2" xref="S7.Ex4.m2.3.3.1.1.1.3.2.cmml">10</mn><mrow id="S7.Ex4.m2.3.3.1.1.1.3.3" xref="S7.Ex4.m2.3.3.1.1.1.3.3.cmml"><mo id="S7.Ex4.m2.3.3.1.1.1.3.3a" xref="S7.Ex4.m2.3.3.1.1.1.3.3.cmml">−</mo><mn id="S7.Ex4.m2.3.3.1.1.1.3.3.2" xref="S7.Ex4.m2.3.3.1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow></mrow><mo id="S7.Ex4.m2.3.3.1.2" xref="S7.Ex4.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Ex4.m2.3b"><apply id="S7.Ex4.m2.3.3.1.1.cmml" xref="S7.Ex4.m2.3.3.1"><eq id="S7.Ex4.m2.3.3.1.1.2.cmml" xref="S7.Ex4.m2.3.3.1.1.2"></eq><csymbol cd="latexml" id="S7.Ex4.m2.3.3.1.1.3.cmml" xref="S7.Ex4.m2.3.3.1.1.3">absent</csymbol><apply id="S7.Ex4.m2.3.3.1.1.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1"><times id="S7.Ex4.m2.3.3.1.1.1.2.cmml" xref="S7.Ex4.m2.3.3.1.1.1.2"></times><apply id="S7.Ex4.m2.3.3.1.1.1.1.2.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S7.Ex4.m2.3.3.1.1.1.1.2.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S7.Ex4.m2.3.3.1.1.1.1.1.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1"><csymbol cd="latexml" id="S7.Ex4.m2.3.3.1.1.1.1.1.1.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.1">plus-or-minus</csymbol><cn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.2.cmml" type="float" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.2">7.37</cn><apply id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3"><times id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.1"></times><cn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.2.cmml" type="float" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.2">0.12</cn><ci id="S7.Ex4.m2.1.1a.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.3.3.2"><mtext id="S7.Ex4.m2.1.1.cmml" xref="S7.Ex4.m2.1.1">stat.</mtext></ci></apply><apply id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4"><times id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.1"></times><cn id="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.2.cmml" type="float" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.2">0.22</cn><ci id="S7.Ex4.m2.2.2a.cmml" xref="S7.Ex4.m2.3.3.1.1.1.1.1.1.4.3.2"><mtext id="S7.Ex4.m2.2.2.cmml" xref="S7.Ex4.m2.2.2">act.</mtext></ci></apply></apply></apply><apply id="S7.Ex4.m2.3.3.1.1.1.3.cmml" xref="S7.Ex4.m2.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S7.Ex4.m2.3.3.1.1.1.3.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.3">superscript</csymbol><cn id="S7.Ex4.m2.3.3.1.1.1.3.2.cmml" type="integer" xref="S7.Ex4.m2.3.3.1.1.1.3.2">10</cn><apply id="S7.Ex4.m2.3.3.1.1.1.3.3.cmml" xref="S7.Ex4.m2.3.3.1.1.1.3.3"><minus id="S7.Ex4.m2.3.3.1.1.1.3.3.1.cmml" xref="S7.Ex4.m2.3.3.1.1.1.3.3"></minus><cn id="S7.Ex4.m2.3.3.1.1.1.3.3.2.cmml" type="integer" xref="S7.Ex4.m2.3.3.1.1.1.3.3.2">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Ex4.m2.3c">\displaystyle=\left[7.37\pm 0.12(\text{stat.})\pm 0.22(\text{act.})\right]% \times 10^{-4}~{},</annotation><annotation encoding="application/x-llamapun" id="S7.Ex4.m2.3d">= [ 7.37 ± 0.12 ( stat. ) ± 0.22 ( act. ) ] × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S7.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="{}^{\text{238}}\text{BI}" class="ltx_Math" display="inline" id="S7.Ex5.m1.1"><semantics id="S7.Ex5.m1.1a"><mmultiscripts id="S7.Ex5.m1.1.1" xref="S7.Ex5.m1.1.1.cmml"><mtext id="S7.Ex5.m1.1.1.2" xref="S7.Ex5.m1.1.1.2a.cmml">BI</mtext><mprescripts id="S7.Ex5.m1.1.1a" xref="S7.Ex5.m1.1.1.cmml"></mprescripts><mrow id="S7.Ex5.m1.1.1b" xref="S7.Ex5.m1.1.1.cmml"></mrow><mtext id="S7.Ex5.m1.1.1.3" xref="S7.Ex5.m1.1.1.3a.cmml">238</mtext></mmultiscripts><annotation-xml 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href="https://arxiv.org/html/2408.06786v1#bib.bib15" title="">15</a>]</cite> for the same components, a 44% increase is found. Uncertainties were not calculated for the design geometry projection. On the other hand, Monte Carlo uncertainty propagation captured the uncertainties from specific activity, component mass and simulation efficiencies in the BI. Multiple contaminants, which have null single-valued efficiencies, contribute to the BI under this new framework. </div> <div class="ltx_para" id="S7.p3"> The projected <math alttext="{}^{\text{assay}}\text{BI}" class="ltx_Math" display="inline" id="S7.p3.1.m1.1"><semantics id="S7.p3.1.m1.1a"><mmultiscripts id="S7.p3.1.m1.1.1" xref="S7.p3.1.m1.1.1.cmml"><mtext id="S7.p3.1.m1.1.1.2" xref="S7.p3.1.m1.1.1.2a.cmml">BI</mtext><mprescripts id="S7.p3.1.m1.1.1a" xref="S7.p3.1.m1.1.1.cmml"></mprescripts><mrow id="S7.p3.1.m1.1.1b" xref="S7.p3.1.m1.1.1.cmml"></mrow><mtext id="S7.p3.1.m1.1.1.3" xref="S7.p3.1.m1.1.1.3a.cmml">assay</mtext></mmultiscripts><annotation-xml encoding="MathML-Content" id="S7.p3.1.m1.1b"><apply id="S7.p3.1.m1.1.1.cmml" xref="S7.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S7.p3.1.m1.1.1.1.cmml" xref="S7.p3.1.m1.1.1">superscript</csymbol><ci id="S7.p3.1.m1.1.1.2a.cmml" xref="S7.p3.1.m1.1.1.2"><mtext id="S7.p3.1.m1.1.1.2.cmml" xref="S7.p3.1.m1.1.1.2">BI</mtext></ci><ci id="S7.p3.1.m1.1.1.3a.cmml" xref="S7.p3.1.m1.1.1.3"><mtext id="S7.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="S7.p3.1.m1.1.1.3">assay</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.1.m1.1c">{}^{\text{assay}}\text{BI}</annotation><annotation encoding="application/x-llamapun" id="S7.p3.1.m1.1d">start_FLOATSUPERSCRIPT assay end_FLOATSUPERSCRIPT BI</annotation></semantics></math> does not capture the uncertainty associated with the assumption of secular equilibrium or account for the possible introduction of backgrounds during the construction of the Demonstrator. Additionally, given the limited number of assayed components, the variation in contaminants may be larger than captured by the averaged assay uncertainty. Furthermore, possible systematic uncertainties in simulated component geometry were not taken into account. Nevertheless, the Monte Carlo uncertainty propagation framework that has been developed can be extended to account for these effects. </div> <div class="ltx_para" id="S7.p4"> The techniques outlined in this article can be exploited to project the BI of future <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S7.p4.1.m1.1"><semantics id="S7.p4.1.m1.1a"><mrow id="S7.p4.1.m1.1.1" xref="S7.p4.1.m1.1.1.cmml"><mn id="S7.p4.1.m1.1.1.2" xref="S7.p4.1.m1.1.1.2.cmml">0</mn><mo id="S7.p4.1.m1.1.1.1" xref="S7.p4.1.m1.1.1.1.cmml">⁢</mo><mi id="S7.p4.1.m1.1.1.3" xref="S7.p4.1.m1.1.1.3.cmml">ν</mi><mo id="S7.p4.1.m1.1.1.1a" xref="S7.p4.1.m1.1.1.1.cmml">⁢</mo><mi id="S7.p4.1.m1.1.1.4" xref="S7.p4.1.m1.1.1.4.cmml">β</mi><mo id="S7.p4.1.m1.1.1.1b" xref="S7.p4.1.m1.1.1.1.cmml">⁢</mo><mi id="S7.p4.1.m1.1.1.5" xref="S7.p4.1.m1.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.p4.1.m1.1b"><apply id="S7.p4.1.m1.1.1.cmml" xref="S7.p4.1.m1.1.1"><times id="S7.p4.1.m1.1.1.1.cmml" xref="S7.p4.1.m1.1.1.1"></times><cn id="S7.p4.1.m1.1.1.2.cmml" type="integer" xref="S7.p4.1.m1.1.1.2">0</cn><ci id="S7.p4.1.m1.1.1.3.cmml" xref="S7.p4.1.m1.1.1.3">𝜈</ci><ci id="S7.p4.1.m1.1.1.4.cmml" xref="S7.p4.1.m1.1.1.4">𝛽</ci><ci id="S7.p4.1.m1.1.1.5.cmml" xref="S7.p4.1.m1.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p4.1.m1.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S7.p4.1.m1.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments. Particularly, the design of such experiments can benefit from the ability to include component mass uncertainties, since the designed and as-built geometries often differ. Additionally, the framework informs the analyst on the number of decays to simulate, thus optimizing computational resources. The unified approach to average assay results can facilitate the standardization of assay reporting. This is of importance in the field, given that the design of new experiments often draws from assay data collected by others. The uncertainty extracted from the BI can be propagated into the projected sensitivity, thus further illuminating the physics reach of the next generation of <math alttext="0\nu\beta\beta" class="ltx_Math" display="inline" id="S7.p4.2.m2.1"><semantics id="S7.p4.2.m2.1a"><mrow id="S7.p4.2.m2.1.1" xref="S7.p4.2.m2.1.1.cmml"><mn id="S7.p4.2.m2.1.1.2" xref="S7.p4.2.m2.1.1.2.cmml">0</mn><mo id="S7.p4.2.m2.1.1.1" xref="S7.p4.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.p4.2.m2.1.1.3" xref="S7.p4.2.m2.1.1.3.cmml">ν</mi><mo id="S7.p4.2.m2.1.1.1a" xref="S7.p4.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.p4.2.m2.1.1.4" xref="S7.p4.2.m2.1.1.4.cmml">β</mi><mo id="S7.p4.2.m2.1.1.1b" xref="S7.p4.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.p4.2.m2.1.1.5" xref="S7.p4.2.m2.1.1.5.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.p4.2.m2.1b"><apply id="S7.p4.2.m2.1.1.cmml" xref="S7.p4.2.m2.1.1"><times id="S7.p4.2.m2.1.1.1.cmml" xref="S7.p4.2.m2.1.1.1"></times><cn id="S7.p4.2.m2.1.1.2.cmml" type="integer" xref="S7.p4.2.m2.1.1.2">0</cn><ci id="S7.p4.2.m2.1.1.3.cmml" xref="S7.p4.2.m2.1.1.3">𝜈</ci><ci id="S7.p4.2.m2.1.1.4.cmml" xref="S7.p4.2.m2.1.1.4">𝛽</ci><ci id="S7.p4.2.m2.1.1.5.cmml" xref="S7.p4.2.m2.1.1.5">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p4.2.m2.1c">0\nu\beta\beta</annotation><annotation encoding="application/x-llamapun" id="S7.p4.2.m2.1d">0 italic_ν italic_β italic_β</annotation></semantics></math> experiments. </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract / award numbers DE-AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0017594, DE-SC0018060, DE-SC0022339, and LANLEM77/LANLEM78. We acknowledge support from the Particle Astrophysics Program and Nuclear Physics Program of the National Science Foundation through grant numbers MRI-0923142, PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611, PHY-1812409, PHY-1812356, PHY-2111140, and PHY-2209530. We gratefully acknowledge the support of the Laboratory Directed Research & Development (LDRD) program at Lawrence Berkeley National Laboratory for this work. We gratefully acknowledge the support of the U.S. Department of Energy through the Los Alamos National Laboratory LDRD Program, the Oak Ridge National Laboratory LDRD Program, and the Pacific Northwest National Laboratory LDRD Program for this work. We gratefully acknowledge the support of the South Dakota Board of Regents Competitive Research Grant. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, funding reference number SAPIN-2017-00023, and from the Canada Foundation for Innovation John R. Evans Leaders Fund. We acknowledge support from the 2020/2021 L’Oréal-UNESCO for Women in Science Programme. This research used resources provided by the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory and by the National Energy Research Scientific Computing Center, a U.S. Department of Energy Office of Science User Facility. We thank our hosts and colleagues at the Sanford Underground Research Facility for their support. </div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> Appendix A Efficiency distribution prior derivation</h2> <div class="ltx_para" id="A1.p1"> The choice of prior, <math alttext="P(\lambda_{ij})=1/\lambda_{i}j^{1-1/S_{i}}" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><mrow id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mrow id="A1.p1.1.m1.1.1.1" xref="A1.p1.1.m1.1.1.1.cmml"><mi id="A1.p1.1.m1.1.1.1.3" xref="A1.p1.1.m1.1.1.1.3.cmml">P</mi><mo id="A1.p1.1.m1.1.1.1.2" xref="A1.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="A1.p1.1.m1.1.1.1.1.1" xref="A1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="A1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="A1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.p1.1.m1.1.1.1.1.1.1" xref="A1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="A1.p1.1.m1.1.1.1.1.1.1.2" xref="A1.p1.1.m1.1.1.1.1.1.1.2.cmml">λ</mi><mrow 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end_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT 1 - 1 / italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, becomes apparent here when all <math alttext="n_{ij}" 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">n</mi><mrow id="A1.p1.2.m2.1.1.3" xref="A1.p1.2.m2.1.1.3.cmml"><mi id="A1.p1.2.m2.1.1.3.2" xref="A1.p1.2.m2.1.1.3.2.cmml">i</mi><mo id="A1.p1.2.m2.1.1.3.1" xref="A1.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A1.p1.2.m2.1.1.3.3" xref="A1.p1.2.m2.1.1.3.3.cmml">j</mi></mrow></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">𝑛</ci><apply id="A1.p1.2.m2.1.1.3.cmml" xref="A1.p1.2.m2.1.1.3"><times id="A1.p1.2.m2.1.1.3.1.cmml" xref="A1.p1.2.m2.1.1.3.1"></times><ci id="A1.p1.2.m2.1.1.3.2.cmml" xref="A1.p1.2.m2.1.1.3.2">𝑖</ci><ci id="A1.p1.2.m2.1.1.3.3.cmml" xref="A1.p1.2.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">n_{ij}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="B_{j}" class="ltx_Math" display="inline" id="A1.p1.3.m3.1"><semantics id="A1.p1.3.m3.1a"><msub id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml"><mi id="A1.p1.3.m3.1.1.2" xref="A1.p1.3.m3.1.1.2.cmml">B</mi><mi id="A1.p1.3.m3.1.1.3" xref="A1.p1.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.1b"><apply id="A1.p1.3.m3.1.1.cmml" xref="A1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.p1.3.m3.1.1.1.cmml" xref="A1.p1.3.m3.1.1">subscript</csymbol><ci id="A1.p1.3.m3.1.1.2.cmml" xref="A1.p1.3.m3.1.1.2">𝐵</ci><ci id="A1.p1.3.m3.1.1.3.cmml" xref="A1.p1.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="N_{ij}" class="ltx_Math" display="inline" id="A1.p1.4.m4.1"><semantics id="A1.p1.4.m4.1a"><msub id="A1.p1.4.m4.1.1" xref="A1.p1.4.m4.1.1.cmml"><mi id="A1.p1.4.m4.1.1.2" xref="A1.p1.4.m4.1.1.2.cmml">N</mi><mrow id="A1.p1.4.m4.1.1.3" xref="A1.p1.4.m4.1.1.3.cmml"><mi id="A1.p1.4.m4.1.1.3.2" xref="A1.p1.4.m4.1.1.3.2.cmml">i</mi><mo id="A1.p1.4.m4.1.1.3.1" xref="A1.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="A1.p1.4.m4.1.1.3.3" xref="A1.p1.4.m4.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.p1.4.m4.1b"><apply id="A1.p1.4.m4.1.1.cmml" xref="A1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A1.p1.4.m4.1.1.1.cmml" xref="A1.p1.4.m4.1.1">subscript</csymbol><ci id="A1.p1.4.m4.1.1.2.cmml" xref="A1.p1.4.m4.1.1.2">𝑁</ci><apply id="A1.p1.4.m4.1.1.3.cmml" xref="A1.p1.4.m4.1.1.3"><times id="A1.p1.4.m4.1.1.3.1.cmml" xref="A1.p1.4.m4.1.1.3.1"></times><ci id="A1.p1.4.m4.1.1.3.2.cmml" xref="A1.p1.4.m4.1.1.3.2">𝑖</ci><ci id="A1.p1.4.m4.1.1.3.3.cmml" xref="A1.p1.4.m4.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m4.1c">N_{ij}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.4.m4.1d">italic_N start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are equal for a given <math alttext="i" class="ltx_Math" display="inline" id="A1.p1.5.m5.1"><semantics id="A1.p1.5.m5.1a"><mi id="A1.p1.5.m5.1.1" xref="A1.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.p1.5.m5.1b"><ci id="A1.p1.5.m5.1.1.cmml" xref="A1.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.p1.5.m5.1d">italic_i</annotation></semantics></math> in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S4.E5" title="In IV Background index ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">5</a>. Setting <math alttext="n_{ij}=n_{j}" class="ltx_Math" display="inline" id="A1.p1.6.m6.1"><semantics id="A1.p1.6.m6.1a"><mrow id="A1.p1.6.m6.1.1" xref="A1.p1.6.m6.1.1.cmml"><msub id="A1.p1.6.m6.1.1.2" xref="A1.p1.6.m6.1.1.2.cmml"><mi id="A1.p1.6.m6.1.1.2.2" xref="A1.p1.6.m6.1.1.2.2.cmml">n</mi><mrow id="A1.p1.6.m6.1.1.2.3" xref="A1.p1.6.m6.1.1.2.3.cmml"><mi id="A1.p1.6.m6.1.1.2.3.2" xref="A1.p1.6.m6.1.1.2.3.2.cmml">i</mi><mo id="A1.p1.6.m6.1.1.2.3.1" xref="A1.p1.6.m6.1.1.2.3.1.cmml">⁢</mo><mi id="A1.p1.6.m6.1.1.2.3.3" xref="A1.p1.6.m6.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="A1.p1.6.m6.1.1.1" xref="A1.p1.6.m6.1.1.1.cmml">=</mo><msub id="A1.p1.6.m6.1.1.3" xref="A1.p1.6.m6.1.1.3.cmml"><mi id="A1.p1.6.m6.1.1.3.2" xref="A1.p1.6.m6.1.1.3.2.cmml">n</mi><mi id="A1.p1.6.m6.1.1.3.3" xref="A1.p1.6.m6.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.6.m6.1b"><apply id="A1.p1.6.m6.1.1.cmml" xref="A1.p1.6.m6.1.1"><eq id="A1.p1.6.m6.1.1.1.cmml" xref="A1.p1.6.m6.1.1.1"></eq><apply id="A1.p1.6.m6.1.1.2.cmml" xref="A1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="A1.p1.6.m6.1.1.2.1.cmml" xref="A1.p1.6.m6.1.1.2">subscript</csymbol><ci id="A1.p1.6.m6.1.1.2.2.cmml" xref="A1.p1.6.m6.1.1.2.2">𝑛</ci><apply id="A1.p1.6.m6.1.1.2.3.cmml" xref="A1.p1.6.m6.1.1.2.3"><times id="A1.p1.6.m6.1.1.2.3.1.cmml" xref="A1.p1.6.m6.1.1.2.3.1"></times><ci id="A1.p1.6.m6.1.1.2.3.2.cmml" xref="A1.p1.6.m6.1.1.2.3.2">𝑖</ci><ci id="A1.p1.6.m6.1.1.2.3.3.cmml" xref="A1.p1.6.m6.1.1.2.3.3">𝑗</ci></apply></apply><apply id="A1.p1.6.m6.1.1.3.cmml" xref="A1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.p1.6.m6.1.1.3.1.cmml" xref="A1.p1.6.m6.1.1.3">subscript</csymbol><ci id="A1.p1.6.m6.1.1.3.2.cmml" xref="A1.p1.6.m6.1.1.3.2">𝑛</ci><ci id="A1.p1.6.m6.1.1.3.3.cmml" xref="A1.p1.6.m6.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.6.m6.1c">n_{ij}=n_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.6.m6.1d">italic_n start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and dropping the <math alttext="i" class="ltx_Math" display="inline" id="A1.p1.7.m7.1"><semantics id="A1.p1.7.m7.1a"><mi id="A1.p1.7.m7.1.1" xref="A1.p1.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.p1.7.m7.1b"><ci id="A1.p1.7.m7.1.1.cmml" xref="A1.p1.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.p1.7.m7.1d">italic_i</annotation></semantics></math> indices, the sum to be evaluated is <math alttext="\sum_{j}^{S}P(\lambda_{j}|n)" class="ltx_Math" display="inline" id="A1.p1.8.m8.1"><semantics id="A1.p1.8.m8.1a"><mrow id="A1.p1.8.m8.1.1" xref="A1.p1.8.m8.1.1.cmml"><msubsup id="A1.p1.8.m8.1.1.2" xref="A1.p1.8.m8.1.1.2.cmml"><mo id="A1.p1.8.m8.1.1.2.2.2" xref="A1.p1.8.m8.1.1.2.2.2.cmml">∑</mo><mi id="A1.p1.8.m8.1.1.2.2.3" xref="A1.p1.8.m8.1.1.2.2.3.cmml">j</mi><mi id="A1.p1.8.m8.1.1.2.3" xref="A1.p1.8.m8.1.1.2.3.cmml">S</mi></msubsup><mrow id="A1.p1.8.m8.1.1.1" xref="A1.p1.8.m8.1.1.1.cmml"><mi id="A1.p1.8.m8.1.1.1.3" xref="A1.p1.8.m8.1.1.1.3.cmml">P</mi><mo id="A1.p1.8.m8.1.1.1.2" xref="A1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="A1.p1.8.m8.1.1.1.1.1" xref="A1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="A1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="A1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.p1.8.m8.1.1.1.1.1.1" xref="A1.p1.8.m8.1.1.1.1.1.1.cmml"><msub id="A1.p1.8.m8.1.1.1.1.1.1.2" xref="A1.p1.8.m8.1.1.1.1.1.1.2.cmml"><mi id="A1.p1.8.m8.1.1.1.1.1.1.2.2" xref="A1.p1.8.m8.1.1.1.1.1.1.2.2.cmml">λ</mi><mi id="A1.p1.8.m8.1.1.1.1.1.1.2.3" xref="A1.p1.8.m8.1.1.1.1.1.1.2.3.cmml">j</mi></msub><mo fence="false" id="A1.p1.8.m8.1.1.1.1.1.1.1" xref="A1.p1.8.m8.1.1.1.1.1.1.1.cmml">|</mo><mi id="A1.p1.8.m8.1.1.1.1.1.1.3" xref="A1.p1.8.m8.1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="A1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="A1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.8.m8.1b"><apply id="A1.p1.8.m8.1.1.cmml" xref="A1.p1.8.m8.1.1"><apply id="A1.p1.8.m8.1.1.2.cmml" xref="A1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.p1.8.m8.1.1.2.1.cmml" xref="A1.p1.8.m8.1.1.2">superscript</csymbol><apply id="A1.p1.8.m8.1.1.2.2.cmml" xref="A1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.p1.8.m8.1.1.2.2.1.cmml" xref="A1.p1.8.m8.1.1.2">subscript</csymbol><sum id="A1.p1.8.m8.1.1.2.2.2.cmml" xref="A1.p1.8.m8.1.1.2.2.2"></sum><ci id="A1.p1.8.m8.1.1.2.2.3.cmml" xref="A1.p1.8.m8.1.1.2.2.3">𝑗</ci></apply><ci id="A1.p1.8.m8.1.1.2.3.cmml" xref="A1.p1.8.m8.1.1.2.3">𝑆</ci></apply><apply id="A1.p1.8.m8.1.1.1.cmml" xref="A1.p1.8.m8.1.1.1"><times id="A1.p1.8.m8.1.1.1.2.cmml" xref="A1.p1.8.m8.1.1.1.2"></times><ci id="A1.p1.8.m8.1.1.1.3.cmml" xref="A1.p1.8.m8.1.1.1.3">𝑃</ci><apply id="A1.p1.8.m8.1.1.1.1.1.1.cmml" xref="A1.p1.8.m8.1.1.1.1.1"><csymbol cd="latexml" id="A1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.1">conditional</csymbol><apply id="A1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.p1.8.m8.1.1.1.1.1.1.2.1.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.2">subscript</csymbol><ci id="A1.p1.8.m8.1.1.1.1.1.1.2.2.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.2.2">𝜆</ci><ci id="A1.p1.8.m8.1.1.1.1.1.1.2.3.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="A1.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="A1.p1.8.m8.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.8.m8.1c">\sum_{j}^{S}P(\lambda_{j}|n)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.8.m8.1d">∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT italic_P ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_n )</annotation></semantics></math>. The sum of PDFs is given by their convolution. Defining the convolution <math alttext="f" class="ltx_Math" display="inline" id="A1.p1.9.m9.1"><semantics id="A1.p1.9.m9.1a"><mi id="A1.p1.9.m9.1.1" xref="A1.p1.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.p1.9.m9.1b"><ci id="A1.p1.9.m9.1.1.cmml" xref="A1.p1.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="A1.p1.9.m9.1d">italic_f</annotation></semantics></math> with itself <math alttext="S-1" class="ltx_Math" display="inline" id="A1.p1.10.m10.1"><semantics id="A1.p1.10.m10.1a"><mrow id="A1.p1.10.m10.1.1" xref="A1.p1.10.m10.1.1.cmml"><mi id="A1.p1.10.m10.1.1.2" xref="A1.p1.10.m10.1.1.2.cmml">S</mi><mo id="A1.p1.10.m10.1.1.1" xref="A1.p1.10.m10.1.1.1.cmml">−</mo><mn id="A1.p1.10.m10.1.1.3" xref="A1.p1.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.10.m10.1b"><apply id="A1.p1.10.m10.1.1.cmml" xref="A1.p1.10.m10.1.1"><minus id="A1.p1.10.m10.1.1.1.cmml" xref="A1.p1.10.m10.1.1.1"></minus><ci id="A1.p1.10.m10.1.1.2.cmml" xref="A1.p1.10.m10.1.1.2">𝑆</ci><cn id="A1.p1.10.m10.1.1.3.cmml" type="integer" xref="A1.p1.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.10.m10.1c">S-1</annotation><annotation encoding="application/x-llamapun" id="A1.p1.10.m10.1d">italic_S - 1</annotation></semantics></math> times as <table class="ltx_equation ltx_eqn_table" id="A1.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="f^{\{S\}}=\underbrace{f*(f*(f*\cdots*(f*f)))}_{S-1\text{ convolutions}}," class="ltx_Math" display="block" id="A1.E10.m1.3"><semantics id="A1.E10.m1.3a"><mrow id="A1.E10.m1.3.3.1" xref="A1.E10.m1.3.3.1.1.cmml"><mrow id="A1.E10.m1.3.3.1.1" xref="A1.E10.m1.3.3.1.1.cmml"><msup id="A1.E10.m1.3.3.1.1.2" xref="A1.E10.m1.3.3.1.1.2.cmml"><mi id="A1.E10.m1.3.3.1.1.2.2" xref="A1.E10.m1.3.3.1.1.2.2.cmml">f</mi><mrow id="A1.E10.m1.1.1.1.3" xref="A1.E10.m1.1.1.1.2.cmml"><mo id="A1.E10.m1.1.1.1.3.1" stretchy="false" xref="A1.E10.m1.1.1.1.2.cmml">{</mo><mi id="A1.E10.m1.1.1.1.1" xref="A1.E10.m1.1.1.1.1.cmml">S</mi><mo id="A1.E10.m1.1.1.1.3.2" stretchy="false" xref="A1.E10.m1.1.1.1.2.cmml">}</mo></mrow></msup><mo id="A1.E10.m1.3.3.1.1.1" xref="A1.E10.m1.3.3.1.1.1.cmml">=</mo><munder id="A1.E10.m1.3.3.1.1.3" xref="A1.E10.m1.3.3.1.1.3.cmml"><munder accentunder="true" id="A1.E10.m1.2.2" xref="A1.E10.m1.2.2.cmml"><mrow id="A1.E10.m1.2.2.1" xref="A1.E10.m1.2.2.1.cmml"><mi id="A1.E10.m1.2.2.1.3" xref="A1.E10.m1.2.2.1.3.cmml">f</mi><mo id="A1.E10.m1.2.2.1.2" lspace="0.222em" rspace="0.222em" xref="A1.E10.m1.2.2.1.2.cmml">∗</mo><mrow id="A1.E10.m1.2.2.1.1.1" xref="A1.E10.m1.2.2.1.1.1.1.cmml"><mo id="A1.E10.m1.2.2.1.1.1.2" stretchy="false" xref="A1.E10.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.E10.m1.2.2.1.1.1.1" xref="A1.E10.m1.2.2.1.1.1.1.cmml"><mi id="A1.E10.m1.2.2.1.1.1.1.3" xref="A1.E10.m1.2.2.1.1.1.1.3.cmml">f</mi><mo id="A1.E10.m1.2.2.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="A1.E10.m1.2.2.1.1.1.1.2.cmml">∗</mo><mrow id="A1.E10.m1.2.2.1.1.1.1.1.1" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="A1.E10.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.E10.m1.2.2.1.1.1.1.1.1.1" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="A1.E10.m1.2.2.1.1.1.1.1.1.1.3" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="A1.E10.m1.2.2.1.1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.2.cmml">∗</mo><mi id="A1.E10.m1.2.2.1.1.1.1.1.1.1.4" mathvariant="normal" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.4.cmml">⋯</mi><mo id="A1.E10.m1.2.2.1.1.1.1.1.1.1.2a" lspace="0.222em" rspace="0.222em" xref="A1.E10.m1.2.2.1.1.1.1.1.1.1.2.cmml">∗</mo><mrow id="A1.E10.m1.2.2.1.1.1.1.1.1.1.1.1" 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xref="A1.E10.m1.2.2.2.cmml">⏟</mo></munder><mrow id="A1.E10.m1.3.3.1.1.3.2" xref="A1.E10.m1.3.3.1.1.3.2.cmml"><mi id="A1.E10.m1.3.3.1.1.3.2.2" xref="A1.E10.m1.3.3.1.1.3.2.2.cmml">S</mi><mo id="A1.E10.m1.3.3.1.1.3.2.1" xref="A1.E10.m1.3.3.1.1.3.2.1.cmml">−</mo><mrow id="A1.E10.m1.3.3.1.1.3.2.3" xref="A1.E10.m1.3.3.1.1.3.2.3.cmml"><mn id="A1.E10.m1.3.3.1.1.3.2.3.2" xref="A1.E10.m1.3.3.1.1.3.2.3.2.cmml">1</mn><mo id="A1.E10.m1.3.3.1.1.3.2.3.1" xref="A1.E10.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mtext id="A1.E10.m1.3.3.1.1.3.2.3.3" xref="A1.E10.m1.3.3.1.1.3.2.3.3a.cmml"> convolutions</mtext></mrow></mrow></munder></mrow><mo id="A1.E10.m1.3.3.1.2" xref="A1.E10.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E10.m1.3b"><apply id="A1.E10.m1.3.3.1.1.cmml" xref="A1.E10.m1.3.3.1"><eq id="A1.E10.m1.3.3.1.1.1.cmml" xref="A1.E10.m1.3.3.1.1.1"></eq><apply id="A1.E10.m1.3.3.1.1.2.cmml" xref="A1.E10.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="A1.E10.m1.3.3.1.1.2.1.cmml" 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xref="A1.E10.m1.3.3.1.1.3.2.3"><times id="A1.E10.m1.3.3.1.1.3.2.3.1.cmml" xref="A1.E10.m1.3.3.1.1.3.2.3.1"></times><cn id="A1.E10.m1.3.3.1.1.3.2.3.2.cmml" type="integer" xref="A1.E10.m1.3.3.1.1.3.2.3.2">1</cn><ci id="A1.E10.m1.3.3.1.1.3.2.3.3a.cmml" xref="A1.E10.m1.3.3.1.1.3.2.3.3"><mtext id="A1.E10.m1.3.3.1.1.3.2.3.3.cmml" mathsize="70%" xref="A1.E10.m1.3.3.1.1.3.2.3.3"> convolutions</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E10.m1.3c">f^{\{S\}}=\underbrace{f*(f*(f*\cdots*(f*f)))}_{S-1\text{ convolutions}},</annotation><annotation encoding="application/x-llamapun" id="A1.E10.m1.3d">italic_f start_POSTSUPERSCRIPT { italic_S } end_POSTSUPERSCRIPT = under⏟ start_ARG italic_f ∗ ( italic_f ∗ ( italic_f ∗ ⋯ ∗ ( italic_f ∗ italic_f ) ) ) end_ARG start_POSTSUBSCRIPT italic_S - 1 convolutions 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">(10)</td> </tr></tbody> </table> and rewriting <math alttext="P(\lambda_{j}|n)" class="ltx_Math" display="inline" id="A1.p1.11.m1.1"><semantics id="A1.p1.11.m1.1a"><mrow id="A1.p1.11.m1.1.1" xref="A1.p1.11.m1.1.1.cmml"><mi id="A1.p1.11.m1.1.1.3" xref="A1.p1.11.m1.1.1.3.cmml">P</mi><mo id="A1.p1.11.m1.1.1.2" xref="A1.p1.11.m1.1.1.2.cmml">⁢</mo><mrow id="A1.p1.11.m1.1.1.1.1" xref="A1.p1.11.m1.1.1.1.1.1.cmml"><mo id="A1.p1.11.m1.1.1.1.1.2" stretchy="false" xref="A1.p1.11.m1.1.1.1.1.1.cmml">(</mo><mrow id="A1.p1.11.m1.1.1.1.1.1" xref="A1.p1.11.m1.1.1.1.1.1.cmml"><msub id="A1.p1.11.m1.1.1.1.1.1.2" xref="A1.p1.11.m1.1.1.1.1.1.2.cmml"><mi id="A1.p1.11.m1.1.1.1.1.1.2.2" xref="A1.p1.11.m1.1.1.1.1.1.2.2.cmml">λ</mi><mi id="A1.p1.11.m1.1.1.1.1.1.2.3" xref="A1.p1.11.m1.1.1.1.1.1.2.3.cmml">j</mi></msub><mo fence="false" id="A1.p1.11.m1.1.1.1.1.1.1" xref="A1.p1.11.m1.1.1.1.1.1.1.cmml">|</mo><mi id="A1.p1.11.m1.1.1.1.1.1.3" xref="A1.p1.11.m1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="A1.p1.11.m1.1.1.1.1.3" stretchy="false" xref="A1.p1.11.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.11.m1.1b"><apply id="A1.p1.11.m1.1.1.cmml" xref="A1.p1.11.m1.1.1"><times id="A1.p1.11.m1.1.1.2.cmml" xref="A1.p1.11.m1.1.1.2"></times><ci id="A1.p1.11.m1.1.1.3.cmml" xref="A1.p1.11.m1.1.1.3">𝑃</ci><apply id="A1.p1.11.m1.1.1.1.1.1.cmml" xref="A1.p1.11.m1.1.1.1.1"><csymbol cd="latexml" id="A1.p1.11.m1.1.1.1.1.1.1.cmml" xref="A1.p1.11.m1.1.1.1.1.1.1">conditional</csymbol><apply id="A1.p1.11.m1.1.1.1.1.1.2.cmml" xref="A1.p1.11.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.p1.11.m1.1.1.1.1.1.2.1.cmml" xref="A1.p1.11.m1.1.1.1.1.1.2">subscript</csymbol><ci id="A1.p1.11.m1.1.1.1.1.1.2.2.cmml" xref="A1.p1.11.m1.1.1.1.1.1.2.2">𝜆</ci><ci id="A1.p1.11.m1.1.1.1.1.1.2.3.cmml" xref="A1.p1.11.m1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="A1.p1.11.m1.1.1.1.1.1.3.cmml" xref="A1.p1.11.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.11.m1.1c">P(\lambda_{j}|n)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.11.m1.1d">italic_P ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_n )</annotation></semantics></math> as <math alttext="f(\lambda_{j})" class="ltx_Math" display="inline" id="A1.p1.12.m2.1"><semantics id="A1.p1.12.m2.1a"><mrow id="A1.p1.12.m2.1.1" xref="A1.p1.12.m2.1.1.cmml"><mi id="A1.p1.12.m2.1.1.3" xref="A1.p1.12.m2.1.1.3.cmml">f</mi><mo id="A1.p1.12.m2.1.1.2" xref="A1.p1.12.m2.1.1.2.cmml">⁢</mo><mrow id="A1.p1.12.m2.1.1.1.1" xref="A1.p1.12.m2.1.1.1.1.1.cmml"><mo id="A1.p1.12.m2.1.1.1.1.2" stretchy="false" xref="A1.p1.12.m2.1.1.1.1.1.cmml">(</mo><msub id="A1.p1.12.m2.1.1.1.1.1" xref="A1.p1.12.m2.1.1.1.1.1.cmml"><mi id="A1.p1.12.m2.1.1.1.1.1.2" xref="A1.p1.12.m2.1.1.1.1.1.2.cmml">λ</mi><mi id="A1.p1.12.m2.1.1.1.1.1.3" xref="A1.p1.12.m2.1.1.1.1.1.3.cmml">j</mi></msub><mo id="A1.p1.12.m2.1.1.1.1.3" stretchy="false" xref="A1.p1.12.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.12.m2.1b"><apply id="A1.p1.12.m2.1.1.cmml" xref="A1.p1.12.m2.1.1"><times id="A1.p1.12.m2.1.1.2.cmml" xref="A1.p1.12.m2.1.1.2"></times><ci id="A1.p1.12.m2.1.1.3.cmml" xref="A1.p1.12.m2.1.1.3">𝑓</ci><apply id="A1.p1.12.m2.1.1.1.1.1.cmml" xref="A1.p1.12.m2.1.1.1.1"><csymbol cd="ambiguous" id="A1.p1.12.m2.1.1.1.1.1.1.cmml" xref="A1.p1.12.m2.1.1.1.1">subscript</csymbol><ci id="A1.p1.12.m2.1.1.1.1.1.2.cmml" xref="A1.p1.12.m2.1.1.1.1.1.2">𝜆</ci><ci id="A1.p1.12.m2.1.1.1.1.1.3.cmml" xref="A1.p1.12.m2.1.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.12.m2.1c">f(\lambda_{j})</annotation><annotation encoding="application/x-llamapun" id="A1.p1.12.m2.1d">italic_f ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT 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xref="A1.E11.m1.2.2.1.1.1.1.1.3">𝑗</ci></apply></apply></apply><apply id="A1.E11.m1.3.3.2.cmml" xref="A1.E11.m1.3.3.2"><times id="A1.E11.m1.3.3.2.2.cmml" xref="A1.E11.m1.3.3.2.2"></times><apply id="A1.E11.m1.3.3.2.3.cmml" xref="A1.E11.m1.3.3.2.3"><csymbol cd="ambiguous" id="A1.E11.m1.3.3.2.3.1.cmml" xref="A1.E11.m1.3.3.2.3">superscript</csymbol><ci id="A1.E11.m1.3.3.2.3.2.cmml" xref="A1.E11.m1.3.3.2.3.2">𝑓</ci><set id="A1.E11.m1.1.1.1.2.cmml" xref="A1.E11.m1.1.1.1.3"><ci id="A1.E11.m1.1.1.1.1.cmml" xref="A1.E11.m1.1.1.1.1">𝑆</ci></set></apply><apply id="A1.E11.m1.3.3.2.1.1.1.cmml" xref="A1.E11.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="A1.E11.m1.3.3.2.1.1.1.1.cmml" xref="A1.E11.m1.3.3.2.1.1">subscript</csymbol><ci id="A1.E11.m1.3.3.2.1.1.1.2.cmml" xref="A1.E11.m1.3.3.2.1.1.1.2">𝜆</ci><ci id="A1.E11.m1.3.3.2.1.1.1.3.cmml" xref="A1.E11.m1.3.3.2.1.1.1.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E11.m1.3c">\sum_{j}^{S}~{}f(\lambda_{j})=f^{\{S\}}(\lambda_{j})</annotation><annotation encoding="application/x-llamapun" id="A1.E11.m1.3d">∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT italic_f ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = italic_f start_POSTSUPERSCRIPT { italic_S } end_POSTSUPERSCRIPT ( italic_λ 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">(11)</td> </tr></tbody> </table> If the decay chain was simulated as a whole and not by segment, <math alttext="P(\lambda|n)" class="ltx_Math" display="inline" id="A1.p1.13.m1.1"><semantics id="A1.p1.13.m1.1a"><mrow id="A1.p1.13.m1.1.1" xref="A1.p1.13.m1.1.1.cmml"><mi id="A1.p1.13.m1.1.1.3" xref="A1.p1.13.m1.1.1.3.cmml">P</mi><mo id="A1.p1.13.m1.1.1.2" xref="A1.p1.13.m1.1.1.2.cmml">⁢</mo><mrow id="A1.p1.13.m1.1.1.1.1" xref="A1.p1.13.m1.1.1.1.1.1.cmml"><mo id="A1.p1.13.m1.1.1.1.1.2" stretchy="false" xref="A1.p1.13.m1.1.1.1.1.1.cmml">(</mo><mrow id="A1.p1.13.m1.1.1.1.1.1" xref="A1.p1.13.m1.1.1.1.1.1.cmml"><mi id="A1.p1.13.m1.1.1.1.1.1.2" xref="A1.p1.13.m1.1.1.1.1.1.2.cmml">λ</mi><mo fence="false" id="A1.p1.13.m1.1.1.1.1.1.1" xref="A1.p1.13.m1.1.1.1.1.1.1.cmml">|</mo><mi id="A1.p1.13.m1.1.1.1.1.1.3" xref="A1.p1.13.m1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="A1.p1.13.m1.1.1.1.1.3" stretchy="false" xref="A1.p1.13.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.13.m1.1b"><apply id="A1.p1.13.m1.1.1.cmml" xref="A1.p1.13.m1.1.1"><times id="A1.p1.13.m1.1.1.2.cmml" xref="A1.p1.13.m1.1.1.2"></times><ci id="A1.p1.13.m1.1.1.3.cmml" xref="A1.p1.13.m1.1.1.3">𝑃</ci><apply id="A1.p1.13.m1.1.1.1.1.1.cmml" xref="A1.p1.13.m1.1.1.1.1"><csymbol cd="latexml" id="A1.p1.13.m1.1.1.1.1.1.1.cmml" xref="A1.p1.13.m1.1.1.1.1.1.1">conditional</csymbol><ci id="A1.p1.13.m1.1.1.1.1.1.2.cmml" xref="A1.p1.13.m1.1.1.1.1.1.2">𝜆</ci><ci id="A1.p1.13.m1.1.1.1.1.1.3.cmml" xref="A1.p1.13.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.13.m1.1c">P(\lambda|n)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.13.m1.1d">italic_P ( italic_λ | italic_n )</annotation></semantics></math> would be given by Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5.Ex1" title="V Promoting single values to distributions ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V</a> but with <math alttext="S=1" class="ltx_Math" display="inline" id="A1.p1.14.m2.1"><semantics id="A1.p1.14.m2.1a"><mrow id="A1.p1.14.m2.1.1" xref="A1.p1.14.m2.1.1.cmml"><mi id="A1.p1.14.m2.1.1.2" xref="A1.p1.14.m2.1.1.2.cmml">S</mi><mo id="A1.p1.14.m2.1.1.1" xref="A1.p1.14.m2.1.1.1.cmml">=</mo><mn id="A1.p1.14.m2.1.1.3" xref="A1.p1.14.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.14.m2.1b"><apply id="A1.p1.14.m2.1.1.cmml" xref="A1.p1.14.m2.1.1"><eq id="A1.p1.14.m2.1.1.1.cmml" xref="A1.p1.14.m2.1.1.1"></eq><ci id="A1.p1.14.m2.1.1.2.cmml" xref="A1.p1.14.m2.1.1.2">𝑆</ci><cn id="A1.p1.14.m2.1.1.3.cmml" type="integer" xref="A1.p1.14.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.14.m2.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="A1.p1.14.m2.1d">italic_S = 1</annotation></semantics></math> and <math alttext="n=Sn_{j}" class="ltx_Math" display="inline" id="A1.p1.15.m3.1"><semantics id="A1.p1.15.m3.1a"><mrow id="A1.p1.15.m3.1.1" xref="A1.p1.15.m3.1.1.cmml"><mi id="A1.p1.15.m3.1.1.2" xref="A1.p1.15.m3.1.1.2.cmml">n</mi><mo id="A1.p1.15.m3.1.1.1" xref="A1.p1.15.m3.1.1.1.cmml">=</mo><mrow id="A1.p1.15.m3.1.1.3" xref="A1.p1.15.m3.1.1.3.cmml"><mi id="A1.p1.15.m3.1.1.3.2" xref="A1.p1.15.m3.1.1.3.2.cmml">S</mi><mo id="A1.p1.15.m3.1.1.3.1" xref="A1.p1.15.m3.1.1.3.1.cmml">⁢</mo><msub id="A1.p1.15.m3.1.1.3.3" xref="A1.p1.15.m3.1.1.3.3.cmml"><mi id="A1.p1.15.m3.1.1.3.3.2" xref="A1.p1.15.m3.1.1.3.3.2.cmml">n</mi><mi id="A1.p1.15.m3.1.1.3.3.3" xref="A1.p1.15.m3.1.1.3.3.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.15.m3.1b"><apply id="A1.p1.15.m3.1.1.cmml" xref="A1.p1.15.m3.1.1"><eq id="A1.p1.15.m3.1.1.1.cmml" xref="A1.p1.15.m3.1.1.1"></eq><ci id="A1.p1.15.m3.1.1.2.cmml" xref="A1.p1.15.m3.1.1.2">𝑛</ci><apply id="A1.p1.15.m3.1.1.3.cmml" xref="A1.p1.15.m3.1.1.3"><times id="A1.p1.15.m3.1.1.3.1.cmml" xref="A1.p1.15.m3.1.1.3.1"></times><ci id="A1.p1.15.m3.1.1.3.2.cmml" xref="A1.p1.15.m3.1.1.3.2">𝑆</ci><apply id="A1.p1.15.m3.1.1.3.3.cmml" xref="A1.p1.15.m3.1.1.3.3"><csymbol cd="ambiguous" id="A1.p1.15.m3.1.1.3.3.1.cmml" xref="A1.p1.15.m3.1.1.3.3">subscript</csymbol><ci id="A1.p1.15.m3.1.1.3.3.2.cmml" xref="A1.p1.15.m3.1.1.3.3.2">𝑛</ci><ci id="A1.p1.15.m3.1.1.3.3.3.cmml" xref="A1.p1.15.m3.1.1.3.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.15.m3.1c">n=Sn_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.15.m3.1d">italic_n = italic_S italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. The functional form of <math alttext="f(\lambda_{j})" class="ltx_Math" display="inline" id="A1.p1.16.m4.1"><semantics id="A1.p1.16.m4.1a"><mrow id="A1.p1.16.m4.1.1" xref="A1.p1.16.m4.1.1.cmml"><mi id="A1.p1.16.m4.1.1.3" xref="A1.p1.16.m4.1.1.3.cmml">f</mi><mo id="A1.p1.16.m4.1.1.2" xref="A1.p1.16.m4.1.1.2.cmml">⁢</mo><mrow id="A1.p1.16.m4.1.1.1.1" xref="A1.p1.16.m4.1.1.1.1.1.cmml"><mo id="A1.p1.16.m4.1.1.1.1.2" stretchy="false" xref="A1.p1.16.m4.1.1.1.1.1.cmml">(</mo><msub id="A1.p1.16.m4.1.1.1.1.1" xref="A1.p1.16.m4.1.1.1.1.1.cmml"><mi id="A1.p1.16.m4.1.1.1.1.1.2" xref="A1.p1.16.m4.1.1.1.1.1.2.cmml">λ</mi><mi id="A1.p1.16.m4.1.1.1.1.1.3" xref="A1.p1.16.m4.1.1.1.1.1.3.cmml">j</mi></msub><mo id="A1.p1.16.m4.1.1.1.1.3" stretchy="false" xref="A1.p1.16.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.16.m4.1b"><apply id="A1.p1.16.m4.1.1.cmml" xref="A1.p1.16.m4.1.1"><times id="A1.p1.16.m4.1.1.2.cmml" xref="A1.p1.16.m4.1.1.2"></times><ci id="A1.p1.16.m4.1.1.3.cmml" xref="A1.p1.16.m4.1.1.3">𝑓</ci><apply id="A1.p1.16.m4.1.1.1.1.1.cmml" xref="A1.p1.16.m4.1.1.1.1"><csymbol cd="ambiguous" id="A1.p1.16.m4.1.1.1.1.1.1.cmml" xref="A1.p1.16.m4.1.1.1.1">subscript</csymbol><ci id="A1.p1.16.m4.1.1.1.1.1.2.cmml" xref="A1.p1.16.m4.1.1.1.1.1.2">𝜆</ci><ci id="A1.p1.16.m4.1.1.1.1.1.3.cmml" xref="A1.p1.16.m4.1.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.16.m4.1c">f(\lambda_{j})</annotation><annotation encoding="application/x-llamapun" id="A1.p1.16.m4.1d">italic_f ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> must be such that the sum of the PDFs of the segments equals the PDF of the full decay chain: <table class="ltx_equation ltx_eqn_table" id="A1.E12"> <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 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start_POSTSUPERSCRIPT { italic_S } end_POSTSUPERSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = italic_λ start_POSTSUPERSCRIPT italic_S italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(12)</td> </tr></tbody> </table> Taking the Laplace transform and working simultaneously on both sides yields: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx4"> <tbody id="A1.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathscr{L}\left[f^{\{S\}}(\lambda_{j})\right](t)" class="ltx_Math" display="inline" id="A1.Ex6.m1.3"><semantics id="A1.Ex6.m1.3a"><mrow 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xref="A1.Ex6.m2.2.2.1.1.1.1.3.2">𝑒</ci><apply id="A1.Ex6.m2.2.2.1.1.1.1.3.3.cmml" xref="A1.Ex6.m2.2.2.1.1.1.1.3.3"><minus id="A1.Ex6.m2.2.2.1.1.1.1.3.3.1.cmml" xref="A1.Ex6.m2.2.2.1.1.1.1.3.3"></minus><ci id="A1.Ex6.m2.2.2.1.1.1.1.3.3.2.cmml" xref="A1.Ex6.m2.2.2.1.1.1.1.3.3.2">𝜆</ci></apply></apply></apply></apply><ci id="A1.Ex6.m2.1.1.cmml" xref="A1.Ex6.m2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex6.m2.2c">\displaystyle=\mathscr{L}\left[\lambda^{Sn_{j}}e^{-\lambda}\right](t)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex6.m2.2d">= script_L [ italic_λ start_POSTSUPERSCRIPT italic_S italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ end_POSTSUPERSCRIPT ] ( italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f(\lambda_{j})" class="ltx_Math" display="inline" id="A1.Ex8.m1.1"><semantics id="A1.Ex8.m1.1a"><mrow id="A1.Ex8.m1.1.1" xref="A1.Ex8.m1.1.1.cmml"><mi id="A1.Ex8.m1.1.1.3" xref="A1.Ex8.m1.1.1.3.cmml">f</mi><mo id="A1.Ex8.m1.1.1.2" xref="A1.Ex8.m1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex8.m1.1.1.1.1" xref="A1.Ex8.m1.1.1.1.1.1.cmml"><mo id="A1.Ex8.m1.1.1.1.1.2" stretchy="false" xref="A1.Ex8.m1.1.1.1.1.1.cmml">(</mo><msub id="A1.Ex8.m1.1.1.1.1.1" xref="A1.Ex8.m1.1.1.1.1.1.cmml"><mi id="A1.Ex8.m1.1.1.1.1.1.2" xref="A1.Ex8.m1.1.1.1.1.1.2.cmml">λ</mi><mi id="A1.Ex8.m1.1.1.1.1.1.3" xref="A1.Ex8.m1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="A1.Ex8.m1.1.1.1.1.3" stretchy="false" 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encoding="application/x-tex" id="A1.Ex8.m2.3c">\displaystyle=\sqrt[S]{(Sn_{j})!}~{}\mathscr{L}^{-1}\left[\frac{1}{(t+1)^{n_{j% }+1/S}}\right](\lambda_{j})</annotation><annotation encoding="application/x-llamapun" id="A1.Ex8.m2.3d">= nth-root start_ARG italic_S end_ARG start_ARG ( italic_S italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ! end_ARG script_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT [ divide start_ARG 1 end_ARG start_ARG ( italic_t + 1 ) start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + 1 / italic_S end_POSTSUPERSCRIPT end_ARG ] ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - 1 + 1 / italic_S end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG roman_Γ ( italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + 1 / italic_S ) 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">(13)</td> </tr></tbody> </table> Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#A1.Ex6" title="Appendix A Efficiency distribution prior derivation ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">A</a> has the same functional form as Eq. <a class="ltx_ref" href="https://arxiv.org/html/2408.06786v1#S5.Ex1" title="V Promoting single values to distributions ‣ An assay-based background projection for the Majorana Demonstrator using Monte Carlo Uncertainty Propagation">V</a> up to a constant. Therefore when <math alttext="n_{ij}" class="ltx_Math" display="inline" id="A1.p1.17.m1.1"><semantics id="A1.p1.17.m1.1a"><msub id="A1.p1.17.m1.1.1" xref="A1.p1.17.m1.1.1.cmml"><mi id="A1.p1.17.m1.1.1.2" xref="A1.p1.17.m1.1.1.2.cmml">n</mi><mrow id="A1.p1.17.m1.1.1.3" xref="A1.p1.17.m1.1.1.3.cmml"><mi id="A1.p1.17.m1.1.1.3.2" xref="A1.p1.17.m1.1.1.3.2.cmml">i</mi><mo id="A1.p1.17.m1.1.1.3.1" xref="A1.p1.17.m1.1.1.3.1.cmml">⁢</mo><mi id="A1.p1.17.m1.1.1.3.3" xref="A1.p1.17.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.p1.17.m1.1b"><apply id="A1.p1.17.m1.1.1.cmml" xref="A1.p1.17.m1.1.1"><csymbol cd="ambiguous" id="A1.p1.17.m1.1.1.1.cmml" xref="A1.p1.17.m1.1.1">subscript</csymbol><ci id="A1.p1.17.m1.1.1.2.cmml" xref="A1.p1.17.m1.1.1.2">𝑛</ci><apply id="A1.p1.17.m1.1.1.3.cmml" xref="A1.p1.17.m1.1.1.3"><times id="A1.p1.17.m1.1.1.3.1.cmml" xref="A1.p1.17.m1.1.1.3.1"></times><ci id="A1.p1.17.m1.1.1.3.2.cmml" xref="A1.p1.17.m1.1.1.3.2">𝑖</ci><ci 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