{ "method": "eta-product Fourier coefficients and split Mellin transform", "rigour": "heuristic comparison; the modular determinant identities are not used as certification", "skill_manifest": "agents/runs/20261101T095610Z-build-skills/manifest.json", "controls": { "Euler_pentagonal_through_100": true, "nonzero_Mellin_normalization_error": "1.5274682e-151" }, "values": { "5": { "at_130": "0.5811267442172664750118554387666488776575434280960615560745639247643905231242909364995471087268572851856968277606468828997693887631", "at_180": "0.581126744217266475011855438766648877657543428096061556074563924764390523124290936499547108726857285185696827760646882899769388763075836397131702687186885939500853725884117432083731", "terms": [ 320, 420 ], "relative_discrepancy": "4.1580607e-131" }, "6": { "at_130": "0.6365248546872152815752897344523518436337827180703738214825087140325630612464512964150241232906190384971141402855340213510208382523", "at_180": "0.636524854687215281575289734452351843633782718070373821482508714032563061246451296415024123290619038497114140285534021351020838252289281332600771647590156283159220913858315688609104", "terms": [ 320, 420 ], "relative_discrepancy": "1.6839354e-131" }, "8": { "at_130": "4.940392397461092408551432019611251206067077233820884095099615446766174498418504274179786803629407373677295539639409939001902936701", "at_180": "4.94039239746109240855143201961125120606707723382088409509961544676617449841850427417978680362940737367729553963940993900190293670145561710363933425256812579647815637890017316835415", "terms": [ 320, 420 ], "relative_discrepancy": "9.2222857e-131" } }, "eta_checks": { "5": [ [ "4.5824045e-151", "1.2219745e-150" ], [ "8.401075e-151", "8.401075e-151" ] ], "6": [ [ "3.0549364e-151", "1.0692277e-150" ], [ "6.1098727e-151", "7.6373409e-151" ] ], "8": [ [ "7.6373409e-152", "2.5966959e-150" ], [ "1.0692277e-150", "3.5895502e-150" ] ] }, "stored_document": { "Title": "Broadhurst\u2013Mellit vacuum determinants", "Definition": "The Broadhurst\u2013Mellit vacuum determinants are $V_n=\\det\\begin{pmatrix}B_n(0,1)&C_n(0)\\\\B_n(2,1)&C_n(2)\\end{pmatrix}$ for $n\\in\\{5,6,8\\}$, where $C_n(a)=B_n(a,3)$ for $n=5,6$ and $C_8(a)=B_8(a,3)-2B_8(a,5)$ CITE{Zhou} CITE{Broadhurst}.", "Parameters": { "n": { "type": "Z", "title": "total number of Bessel factors", "display": "$n$", "constraints": "$n\\in\\{5,6,8\\}$ for the three constructions defined here" } }, "Tags": [ "integral", "period", "special values", "physics", "number theory" ], "References": { "Zhou": { "bib": "Yajun Zhou, Wro\u0144skian factorizations and Broadhurst\u2013Mellit determinant formulae (2018), Theorem 1.3, equations (1.9) and (1.10), and Section 5.", "arxiv": "1711.01829" }, "Broadhurst": { "bib": "David Broadhurst, Feynman integrals, L-series and Kloosterman moments (2016), Sections 7.2, 7.3 and 7.8; equation (175) specifies the eight-factor matrix.", "arxiv": "1604.03057" } }, "Links": { "ZhouText": { "title": "Full text of Zhou\u2019s determinant theorem", "url": "https://arxiv.org/html/1711.01829v2" } }, "Data properties": { "type": "R", "rigour": "heuristic (agreement-checked)", "complete": "no", "complete-note": "it holds the three documented vacuum constructions at $n=5,6,8$. The five- and six-factor determinants use the monomial columns in Zhou\u2019s Theorem 1.3, and the eight-factor determinant uses Broadhurst\u2019s specified polynomial column. This finite catalogue covers both constructions without inventing matrices at other factor counts or arbitrary minors", "rigour details": "The defining moments were computed by tanh-sinh quadrature at 150 and 200 working decimal digits, retaining 100 significant digits. Arb evaluates the Bessel functions with their relative radii checked before their midpoints enter quadrature. The largest relative discrepancy between the determinant runs is below $3.8\\times10^{-151}$; cancellation in the final two-product subtraction loses less than $0.245$ decimal digits. The integral is split at $1,4,16,64$ and truncated at $3(w+60)$, where $w$ is the working precision. The generator gives an analytic exponential upper-tail estimate. The quadrature itself is not a certified enclosure, so the rigour is agreement-checked.\n\nBefore the table moments, the nonzero controls $\\int_0^\\infty tK_0(t)\\,dt=1$ and $\\int_0^\\infty tK_0(t)^2\\,dt=1/2$ were tested. The five- and six-factor full-period determinants were also compared with Zhou\u2019s proved products. Independent calculations from eta-product Fourier coefficients and split Mellin transforms at 130 and 180 working digits agree with all three vacuum determinants to more than 180 relative digits. Those computations share no Bessel evaluation or moment quadrature code with the generator. Their Euler-product coefficients, Mellin normalization and imaginary-axis modular transformations have separate controls. The modular determinant relations in Broadhurst\u2019s paper are used as checks, not as certificates for the stored digits.\n\nThe attached argument-selection.md explains the finite catalogue. The generator, standalone checks and numerical evidence record the methods, dependency versions, precision comparisons and archived skill manifest." }, "Display properties": { "number-header": "$V_n$" }, "Numbers": { "5": "0.5811267442172664750118554387666488776575434280960615560745639247643905231242909364995471087268572852", "6": "0.6365248546872152815752897344523518436337827180703738214825087140325630612464512964150241232906190385", "8": "4.940392397461092408551432019611251206067077233820884095099615446766174498418504274179786803629407374" }, "Comments": { "moments": "The unit-scale HREF{Bessel_moments}[Bessel moment] is $B_n(a,r)=\\int_0^\\infty I_0(t)^aK_0(t)^{n-a}t^r\\,dt=\\operatorname{IKM}(a,n-a;r)$, where HREF{Values_of_modified_Bessel_functions_of_the_first_kind}[$I_0$] and HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[$K_0$] are the standard order-zero modified Bessel functions on the positive real axis. Every factor has argument $t$. No loop factors, powers of $\\pi$ or factorials are absorbed into the moments.", "conventions": "The determinant is signed, with the vacuum row $a=0$ first and the row $a=2$ second; the $r=1$ column comes first. For $n=8$, the second column integrates $t^3-2t^5$. It is not the monomial-column determinant obtained by continuing the five- and six-factor construction.", "convergence": "All moments converge without regularization. At zero the integrands are $O(t^r(1+|\\log t|)^{n-a})$ with $r\\geq1$; at infinity they are $O(t^{r-n/2}e^{-(n-2a)t})$ with $n>2a$. The polynomial combination at $n=8$ retains this exponential decay.", "notation": "Zhou denotes $V_5$ and $V_6$ by $\\det\\check{\\mathbf M}_2$ and $\\det\\check{\\mathbf N}_2$ in Theorem 1.3 of CITE{Zhou}; its proof is available in CITE{ZhouText}. Broadhurst defines $V_8=\\det\\mathcal M_2$ in equation (175) of CITE{Broadhurst} and states its modular $L$-value evaluation as Conjecture 6 in that paper. The notation $V_n$ here means a determinant, whereas Zhou also uses $V_n$ for the scalar vacuum integral $B_n(0,1)$.", "boundary": "These are the three explicitly specified vacuum-containing constructions, not a definition at every factor count. The HREF{T647}[full-period Broadhurst\u2013Mellit determinants] instead use consecutive positive-$a$ rows and monomial columns." }, "Formulas": { "mahler": "$V_5=\\frac{2\\pi^3}{15\\sqrt{15}}\\mu_5$ and $V_6=\\frac{\\pi^4}{96}\\mu_6$, where HREF{Mahler_measures_of_(short_random_walks)#5}[$\\mu_5$] and HREF{Mahler_measures_of_(short_random_walks)#6}[$\\mu_6$] are the logarithmic Mahler measures $\\mu_n=m(1+x_1+\\cdots+x_{n-1})$ CITE{Zhou}, Theorem 1.3." }, "Similar tables": [ { "table": "HREF{Bessel_moments}[Bessel moments]", "relation": "the individual unit-scale integrals used in the determinant matrices; the five- and six-factor entries occur there with parameters $(a,n-a,r)$" }, { "table": "HREF{T647}[Broadhurst\u2013Mellit determinants of Bessel moments]", "relation": "the full-period matrices with consecutive positive first-kind exponents and monomial columns" }, { "table": "HREF{Mahler_measures_of_(short_random_walks)}[Mahler measures of sums of monomials]", "relation": "the five- and six-term logarithmic measures related to the vacuum determinants by Zhou\u2019s theorem" } ], "Keywords": [ "Broadhurst Mellit vacuum determinants", "vacuum Bessel determinants", "IKM", "equal-mass Feynman integrals", "noncritical modular L-values" ], "Programs": { "python": { "language": "Python (mpmath)", "code": "from functools import lru_cache\nfrom mpmath import mp\nmp.dps = 50\n\n@lru_cache(maxsize=None)\ndef bessel(t):\n return mp.besseli(0, t), mp.besselk(0, t)\n\ndef B(n, a, r):\n def integrand(t):\n if not t:\n return mp.mpf(0)\n i, k = bessel(t)\n return i**a * k**(n-a) * t**r\n return mp.quad(integrand, [0, 1, 4, 16, 64, mp.inf])\n\ndef V(n):\n if n not in (5, 6, 8):\n raise ValueError('The defined factor counts are 5, 6, 8')\n def C(a):\n return B(n, a, 3) - (2*B(n, a, 5) if n == 8 else 0)\n return B(n, 0, 1)*C(2) - B(n, 2, 1)*C(0)\n\nfor n in (5, 6, 8):\n print(n, mp.nstr(V(n), 30))\n" } } }, "stored_modular_differences_in_last_places": { "5": "0.143031722394", "6": "0.0288585971447", "8": "0.322704460361" }, "stored_Mahler_differences_in_last_places": { "5": "0.255771297151", "6": "0.449210567565" }, "Mahler_limitation": "T283 has 50-digit conjectural eta-integral evaluations at n=5,6; these are comparisons, not certificates." }