% Copyright 2026 Ricardo Pinho. CC BY 4.0. \documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs} \usepackage{placeins} \usepackage[expansion=false]{microtype} \usepackage[hyphens]{url} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{2em} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \DeclareMathOperator{\Log}{Log} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \title[A trace certificate with classical auxiliaries]{A trace upper-bound certificate with classical auxiliaries on an adaptive density} \author{Ricardo Pinho} \address{Independent researcher} \thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. Licensed under CC BY 4.0.} \date{2 November 2026; version 3; unrefereed preprint} \subjclass[2020]{11R04, 11C08, 31A05} \keywords{totally positive algebraic integers, Schur--Siegel--Smyth trace constant, logarithmic potential, auxiliary polynomials, interval certificates} \hypersetup{pdftitle={A trace upper-bound certificate with classical auxiliaries on an adaptive density},pdfauthor={Ricardo Pinho}} \begin{document} \begin{abstract} An exactly specified rational piecewise-linear density gives $C_{86}\le1.80560940$ for the Schur--Siegel--Smyth trace constant. It uses 28 classical auxiliary polynomials, ten with nonreal roots, and 5,375 rational nodes. Their recorded list occurrences are: Flammang 2019 (all 28), Liang--Wu 2011 (26), Flammang 2009 (21), McKee 2011 (21), Tomar's public A22 pool (18), Aguirre--Peral 2007 (16), and Flammang 2024 (9). The density refines infrastructure originating in Tomar's adaptive-node certificate. We give the full deduction from finite potential and logarithmic-integral checks through Smith's arithmetic-measure criterion and converse theorem, and describe the rigorous continuous enclosures. The contribution is the explicit feasible measure and its smaller upper bound. This note is unrefereed, with no independent human review or formalization. \end{abstract} \maketitle \section{Result and antecedents} For a totally positive algebraic integer $\alpha$, its normalized trace is the mean of its conjugates, $\operatorname{Tr}(\alpha)/[\mathbb Q(\alpha):\mathbb Q]$. Write $C_{86}=\lambda_{\mathrm{SSS}}$ for the smallest limiting normalized trace, with finitely many exceptional algebraic integers disregarded, as in Smith \cite{smith}. \begin{theorem}\label{main} \[ C_{86}\le\frac{288897505229428473}{160000001279369600} <1.80560940. \] \end{theorem} The method is Smith's sufficient criterion for arithmetic measures, used in the measure optimization of Orloski and Talebizadeh Sardari \cite{smith,os}. The related work of Orloski, Talebizadeh Sardari and Smith develops the trace problem's measure and auxiliary-function framework \cite{oss}. Selected earlier upper bounds are listed below; the computational entries are unrefereed claims, not assertions of independent human validation. \begin{center} \begin{tabular}{ll} \toprule Source & Upper bound\\ \midrule Smith \cite{smith} & $1.89831$\\ Orloski--Talebizadeh Sardari \cite{os} & $1.8216$\\ Pinho, earlier step-density certificate \cite{history} & $1.8117251$\\ Dennis Koshta \cite{koshta} & $1.8084990$\\ Vidit Patankar, PR 270 \cite{giraudin} & $1.8077686$\\ Tejasvi Singh Tomar \cite{tomar} & $1.8070808326$\\ Pinho, version 2 of this note & $1.80697883$\\ Pinho, classical half-cell certificate \cite{half} & $1.80580911$\\ \bottomrule \end{tabular} \end{center} Tomar's exact certificate ratio is $81383685642728309283453/45035996273705240000000$; the displayed decimal is rounded upward. Tomar's piecewise-linear construction and adaptive nodes, Koshta's enlarged auxiliary pool and checking ideas, and Patankar's further pool and grid enlargement are computational antecedents \cite{tomar,koshta,giraudin}. Version 2 used 23 auxiliaries and emphasized a degree-eight pair with nonreal roots. Those polynomials already occur in Flammang's Table~1, rows 22 and 23, and McKee's Table~1, rows 21 and 23 \cite{flammang09,mckee}. Their discovery and the use of complex-root auxiliaries in the trace problem are prior work. The present measure instead combines more of the classical lists with a refined adaptive density. Recorded occurrences cover all 28 in Flammang's 2019 first trace list \cite{flammang19}, 26 in Liang--Wu's 2011 Table~1 \cite{liangwu11}, 21 each in Flammang's 2009 and McKee's 2011 Table~1 \cite{flammang09,mckee}, 18 in Tomar's public A22 pool \cite{tomar}, 16 in Aguirre--Peral's 2007 Table~2 \cite{aguirreperal07}, and 9 in Flammang's 2024 lists \cite{flammang24}. We claim the frozen feasible certificate and bound, without claiming new polynomials, a new sufficiency principle, or optimality. The value is filed, with its unrefereed qualification, in PR~298 \cite{pr298}. \section{Why a finite certificate suffices} We use the positive-log convention \[ U_\mu(z)=\int\log|z-x|\,d\mu(x). \] Smith's potential has the opposite sign. Let $\mu$ have a bounded, continuous, nonnegative piecewise-linear density, extended by zero, with compact support $S\subset(0,\infty)$. Put $m=\mu(\R)>0$ and $T=\int x\,d\mu(x)$. \begin{proposition}\label{criterion} Let $Q_j$ be distinct primitive irreducible integer polynomials with positive leading coefficients and $b_j>0$. Suppose that \begin{align} \beta:=\sum_j b_j\deg Q_j&0. \end{gather*} These follow by summing, with $\Delta_i=e_{i+1}-e_i$, \begin{align*} m&=\sum_i\frac{\Delta_i}{2}(h_i+h_{i+1}),\\ T&=\sum_i\frac{\Delta_i}{6} \big((2e_i+e_{i+1})h_i+(e_i+2e_{i+1})h_{i+1}\big), \end{align*} obtained by integrating the affine density and $x$ times that density on each cell. Thus no assumed unit mass enters the bound. The exact check of the final upward rounding can also be written \[ 1.80560940-\frac{T}{m} =\frac{3376916571357}{500000003998030000000}>0. \] There are 5,375 nodes and 3,791 cells with a positive endpoint height. Their closed union is $S$, whose hull is $[17/320,5237/1000]$; this does not assert that its intervening gaps carry density. For Proposition~\ref{criterion} we may use $\Sigma=[1/1000,13]$, of capacity $12999/4000>1$. All 28 selected polynomials are monic, primitive, distinct and irreducible, and every weight is positive. The largest degree is 21; ten polynomials have nonreal roots. Table~\ref{inputs} gives recorded source rows: all 28 occur in Flammang's 2019 first trace list \cite{flammang19}; 26 in Liang--Wu's 2011 Table~1 \cite{liangwu11}; 21 each in Flammang's 2009 and McKee's 2011 Table~1 \cite{flammang09,mckee}; 18 in Tomar's public A22 pool \cite{tomar}; and 16 in Aguirre--Peral's 2007 Table~2 \cite{aguirreperal07}. Flammang's 2024 Rhin-measure list contains 9 of them, and its sector lists contain 3 (among those 9) \cite{flammang24}; Flammang's 2019 second and third lists contain 13 and 8, respectively. Liang--Wu's matches include the nonreal-root auxiliaries $j=21,22,25,26,27$; only $j=23,24$ have no recorded occurrence outside Flammang's lists. The dated columns expose the earliest published occurrence recorded here, without asserting first discovery or exhaustive priority. The exact weights are integer multiples of $10^{-14}$. Complete coefficient arrays and all source matches are in \texttt{adaptive\_checked-provenance.json} \cite{adaptive}. \begin{table}[htbp] \centering \caption{Active inputs and recorded list matches: AP07 (16), F09 (21), M11 (21), LW11 (26), F19 (28), T (18), F24 (9). AP07: Aguirre--Peral 2007 Table~2; F09: Flammang 2009 Table~1; M11: McKee 2011 Table~1; LW11: Liang--Wu 2011 Table~1; F19: Flammang 2019 first trace list; T: Tomar's public A22 pool. Entries are source rows (pool entries for T); a dash means no recorded match in that source. A plus in F24 marks Flammang 2024 Rhin-list membership. The index $j$ starts at zero; $r_j$ counts nonreal roots.}\label{inputs} \small \setlength{\tabcolsep}{3pt} \begin{tabular}{@{}rrrrrrrrrcr@{}} \toprule $j$ & $\deg Q_j$ & $r_j$ & AP07 & F09 & M11 & LW11 & F19 & T & F24 & $10^{14}b_j$\\ \midrule 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & + & 4133119422362\\ 1 & 1 & 0 & 2 & 2 & 2 & 2 & 2 & 2 & + & 3447544095751\\ 2 & 1 & 0 & 3 & 3 & 3 & 3 & 3 & 3 & + & 2014805323618\\ 3 & 2 & 0 & 4 & 4 & 4 & 4 & 4 & 4 & + & 2239752443239\\ 4 & 3 & 0 & 7 & 7 & 7 & 7 & 7 & 5 & + & 1284972133134\\ 5 & 4 & 0 & 11 & 11 & 11 & 11 & 11 & 6 & + & 865418825428\\ 6 & 4 & 0 & 12 & 12 & 12 & 12 & 12 & 7 & + & 926684307119\\ 7 & 5 & 0 & 17 & 16 & 14 & 16 & 15 & 8 & + & 465206989929\\ 8 & 5 & 0 & 14 & 13 & 15 & 13 & 16 & 9 & -- & 550600969104\\ 9 & 5 & 0 & 16 & 15 & 16 & 15 & 13 & 10 & -- & 180658152586\\ 10 & 5 & 0 & 15 & 14 & 13 & 14 & 14 & 12 & -- & 561133439478\\ 11 & 10 & 0 & 36 & 27 & 27 & 29 & 28 & 13 & -- & 434034647635\\ 12 & 10 & 0 & 38 & 29 & 25 & 31 & 30 & 14 & -- & 323476166442\\ 13 & 10 & 0 & 37 & 28 & 26 & 30 & 29 & 15 & -- & 234508271836\\ 14 & 8 & 0 & -- & 25 & 20 & 25 & 24 & 16 & -- & 371665324449\\ 15 & 8 & 0 & -- & 24 & 22 & 24 & 23 & 17 & -- & 249638717832\\ 16 & 7 & 0 & 31 & 21 & 19 & 21 & 20 & 18 & -- & 239541442220\\ 17 & 2 & 0 & 6 & 6 & 5 & 6 & 6 & 19 & + & 719437546552\\ 18 & 8 & 2 & -- & 23 & 23 & 23 & 26 & -- & -- & 257823933007\\ 19 & 12 & 4 & -- & 33 & 29 & 35 & 33 & -- & -- & 701940569923\\ 20 & 12 & 4 & -- & 32 & 30 & 34 & 35 & -- & -- & 569338653568\\ 21 & 14 & 4 & -- & -- & -- & 49 & 47 & -- & -- & 403799833432\\ 22 & 14 & 2 & -- & -- & -- & 46 & 48 & -- & -- & 349477648447\\ 23 & 14 & 4 & -- & -- & -- & -- & 54 & -- & -- & 321626320155\\ 24 & 15 & 2 & -- & -- & -- & -- & 58 & -- & -- & 3148218\\ 25 & 16 & 4 & -- & -- & -- & 54 & 64 & -- & -- & 19981362\\ 26 & 17 & 2 & -- & -- & -- & 61 & 72 & -- & -- & 282758541948\\ 27 & 21 & 4 & -- & -- & -- & 77 & 83 & -- & -- & 21080332809\\ \bottomrule \end{tabular} \end{table} \FloatBarrier \section{Continuous checks and enclosure formulas} This section gives the analytic basis of the separately written standalone checker in \cite{adaptive}, following its \texttt{VERIFIER\_NOTES.md}. Exact rational arithmetic checks the nodes, nonnegative heights, zero exterior heights, moments, weights and degree deficit. FLINT exact factorization checks irreducibility; certified complex root balls account for every root, including nonreal ones. The analytic checks use Arb ball arithmetic, with acceptance only from a strictly positive lower endpoint. Floating diagnostics do not decide feasibility. \subsection{Potentials and direct logarithmic integrals} Let $s_i=(h_{i+1}-h_i)/(e_{i+1}-e_i)$, and set exterior slopes to zero. The slope jumps $J_i=s_i-s_{i-1}$ satisfy $\rho''=\sum_iJ_i\delta_{e_i}$ in the distributional sense. For real $t$ put \[ G(t)=\frac{t^2}{2}\log|t|-\frac{3t^2}{4},\qquad G(0)=0. \] Differentiation gives $G'(t)=t(\log|t|-1)$ and $G''(t)=\log|t|$ away from zero, with the latter singularity integrable. Twice integrating by parts, with vanishing exterior density and slope, gives the real-axis identity \begin{equation}\label{jumps} U_\mu(x)=\sum_i J_iG(x-e_i). \end{equation} The same integrations against $1$, $x$, and $x^2/2$ give the exact checks $\sum_iJ_i=\sum_iJ_ie_i=0$ and $\sum_iJ_ie_i^2/2=m$. For a separate computation of $I_j$, factor $Q_j(x)=\ell_j\prod_\alpha(x-\alpha)$ over its certified roots. On one density cell write $z=x-\alpha$ and $\rho(x)=sz+k$, where $k$ is the affine cell density evaluated at $\alpha$. A primitive for $\rho(x)\log|x-\alpha|$ is \begin{equation}\label{primitive} \operatorname{Re}\left[ \left(\frac{s z^2}{2}+kz\right)\Log z -\frac{s z^2}{4}-kz\right]. \end{equation} Its derivative is the desired integrand: the nonlogarithmic derivative terms cancel and $sz+k=\rho(x)$ is real on the integration segment. For nonreal $\alpha$ use a continuous logarithm on the horizontal segment; for real $\alpha$ use $\log|z|$ and the continuous limits of $z\log|z|$ and $z^2\log|z|$ at zero. Sum endpoint differences over cells and roots and add $m\log\ell_j$. This proves the formula used to enclose every $I_j$, with no numerical quadrature or omitted leading coefficient. \subsection{Closed-interval lower bounds} For a trial interval $[c-r,c+r]$, write $x=c+h$, $|h|\le r$, and use Taylor order $N=8$ in the separately written checker. For a knot with $t=c-e_i$ and $|t|>2r$, the coefficients for $G(t+h)$ are $G(t)$, $G'(t)$, $\log|t|/2$, and \[ \frac{(-1)^{n-3}}{n(n-1)(n-2)t^{n-2}}\quad(3\le n\le N). \] They follow by differentiating $G'''(t)=1/t$. The Lagrange remainder has absolute value at most \begin{equation}\label{tailG} \frac{r^{N+1}}{(N+1)N(N-1)(|t|-r)^{N-1}}, \end{equation} since $|G^{(N+1)}(u)|=(N-2)!/|u|^{N-1}$ and $|u|\ge|t|-r$ along the Taylor segment. Multiply this bound by $|J_i|$. Near a knot, $|t|\le2r$, the checker requires $|t|+r<1$. Concavity from $G''(v)=\log|v|\le0$ gives \begin{equation}\label{near} -2r^2(1-\log r) \le G(t+h)-G(t)-hG'(t)\le0, \end{equation} where $G'(0)=0$. To see the lower bound, the absolute integral remainder is at most $r$ times the integral of $-\log|v|$ on a containing interval of length $2r$. That integral is largest on $[-r,r]$, where it equals $2r(1-\log r)$. Thus a positive $J_i$ incurs the negative error in \eqref{near}; for a negative $J_i$, the affine tangent already gives a lower bound. This argument also covers knots at interval endpoints. For an auxiliary root $\alpha$ with $d=|c-\alpha|>2r$, expand \[ \log|c+h-\alpha|=\log d+ \operatorname{Re}\sum_{n=1}^{N} \frac{(-1)^{n+1}h^n}{n(c-\alpha)^n}+R. \] The logarithm's convergent power series bounds its tail by \begin{equation}\label{tailroot} |R|\le\frac{(r/d)^{N+1}}{(N+1)(1-r/d)}: \end{equation} each denominator in the tail is at least $N+1$, after which the geometric series sums explicitly. For a nearer root, the triangle inequality instead gives \[ -b_j\log|x-\alpha|\ge-b_j\log(d+r). \] It remains valid at the root, with the left side interpreted as $+\infty$. Include $-\sum_jb_j\log\ell_j$ separately. Summing these lower enclosures gives a polynomial in $h$ minus rigorous remainder bounds. Interval evaluation on the closed ball $[-r,r]$ yields a lower bound for $F$ on the entire trial interval. If it is not positive, bisect at the exact rational midpoint. The separately written checker uses 192-bit Arb arithmetic for these bounds. It saves accepted leaves with their exact rational endpoints; for each original support cell it checks that sorted leaves start and end at the cell endpoints and that adjacent endpoints coincide. Thus every point of the closed support, including every endpoint, is covered. Unsuccessful enclosures would leave the certificate unproved rather than certify a floating approximation. \subsection{Receipts and conclusion} For the fixed certificate, the saved receipt of the separately written checker reports positive direct integrals for all 28 polynomials and 15,332 accepted closed leaves covering all 3,791 support cells. Conservative downward summaries of its saved ball enclosures are \begin{equation}\label{margins} \min_j I_j>8.94\cdot10^{-8},\qquad F(x)>1.29\cdot10^{-10}\quad(x\in S). \end{equation} The receipt is \path{adaptive_checked-independent.json}. These numbers are obtained by comparing each saved lower ball with the displayed rational decimal; they are enclosure floors, not estimates of the true minima. The separate first checker, \texttt{verify\_linear.py}, has a positive receipt on 10,243 pieces. The retained closed-cover audit reports successful endpoint and hash checks; a further, separately written AI review checks the same bound and all active auxiliaries \cite{adaptive}. The certificate SHA-256, binding these data and receipts, is \begin{center}\scriptsize \texttt{10d68ec249f47c0504182d419fd8ddd1ca1c917e42a4bba59d944b9d1de0dad5}. \end{center} The rational identities above establish \eqref{degree} and the stated ratio, and the continuous enclosures establish \eqref{integrals} and \eqref{support}. Proposition~\ref{criterion} therefore proves Theorem~\ref{main}. This is a computer-assisted proof using the frozen data; the analytic receipts are existing runs, not new full checker runs made for this writing revision. Both standalone checkers can be run on the pinned JSON to regenerate the finite checks. \section{Scope and disclosure} The density search uses nonnegative rational triangular hats and adds knots and free heights to the inherited adaptive grid. Floating linear programming proposes a measure; the proof concerns only its exported rational node data and verified inequalities. Refining the density and changing the active classical set both enter the certificate. We do not infer a unique cause of the numerical gain or an optimal solution from finite LP diagnostics. The prior-art checks on 2 November 2026 covered the live Tao-table entry and open contributions, identified certificate repositories, selected public AI-result repositories, recent papers, and indexed community searches. Tomar's certificate is the strongest earlier external upper bound identified in those checks. Searches and list transcriptions are incomplete: in particular the source review records inaccessible or unaccepted Wang--Wu--Wu, Liang--Wu's lists for second and third powers (distinct from their 2011 Table~1 for the first power), Aguirre--Peral's 2008 trace-chapter list (distinct from their 2007 Farey-interval paper's Table~2), Wang--Pang--Wu and scanned Smyth lists. No exhaustive priority, auxiliary-pool coverage, or optimality claim follows. The pinned \texttt{METHOD.md} and \texttt{review/REVIEW.md} distinguish these search limits from the hypotheses of the frozen bound \cite{adaptive}. This is an unrefereed preprint. There has been no independent human review and no formalization. The result was found and checked by AI agents under Ricardo Pinho's direction. The AI checks do not formalize Smith's theorem or the complete analytic argument. Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checking software, and publication materials. No line-by-line human verification is claimed. \begin{thebibliography}{99} \raggedright \bibitem{smith} A. Smith, Algebraic integers with conjugates in a prescribed distribution, \emph{Annals of Mathematics} 200 (2024), 71--122; Theorem 1.5 and Proposition 5.15. \url{https://doi.org/10.4007/annals.2024.200.1.2}. Author manuscript: \url{https://arxiv.org/html/2111.12660v2}. \bibitem{os} B. Orloski and N. Talebizadeh Sardari, A quantitative converse of Fekete's theorem, preprint, version 2 (2024). \url{https://arxiv.org/abs/2304.10021v2}. \bibitem{oss} B. Orloski, N. Talebizadeh Sardari and A. Smith, New lower bounds for the Schur--Siegel--Smyth trace problem, \emph{Mathematics of Computation} 94 (2025), 2005--2040. \url{https://doi.org/10.1090/mcom/4004}. \bibitem{ransford} T. Ransford, \emph{Potential Theory in the Complex Plane}, London Mathematical Society Student Texts 28, Cambridge University Press, 1995; harmonic minimum principle and capacity of a real interval. \bibitem{flammang09} V. Flammang, Trace of totally positive algebraic integers and integer transfinite diameter, \emph{Mathematics of Computation} 78 (2009), 1119--1125. \url{https://doi.org/10.1090/S0025-5718-08-02120-0}. \bibitem{mckee} J. McKee, Computing totally positive algebraic integers of small trace, \emph{Mathematics of Computation} 80 (2011), 1041--1052. \url{https://doi.org/10.1090/S0025-5718-2010-02424-X}. \bibitem{flammang19} V. Flammang, Une nouvelle minoration pour la trace des entiers alg\'ebriques totalement positifs, preprint (2019), first polynomial list. \url{https://arxiv.org/abs/1907.09407v1}. \bibitem{liangwu11} Y. Liang and Q. Wu, The trace problem for totally positive algebraic integers, \emph{Journal of the Australian Mathematical Society} 90 (2011), 341--354; Table~1. \url{https://doi.org/10.1017/S1446788711001030}. \bibitem{aguirreperal07} J. Aguirre and J. C. Peral, The integer Chebyshev constant of Farey intervals, \emph{Publicacions Matem\`atiques} (2007), Proceedings of the Primeras Jornadas de Teor\'ia de N\'umeros, 11--27; Table~2. \url{https://ddd.uab.cat/pub/pubmat/pubmat_a2007vEXTRA/pubmat_a2007vextrap11.pdf}. \bibitem{flammang24} V. Flammang, To answer a question of Professor Georges Rhin, preprint (2024), Rhin-measure and sector lists. \url{https://arxiv.org/abs/2401.12951v1}. \bibitem{history} R. Pinho, C86 step-density certificate, historical account and data description, at the pinned repository commit of \cite{adaptive}. \url{https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86}. \bibitem{koshta} D. Koshta, C86 trace certificate, unrefereed computational result (2026), as recorded in the Tao table, entry 86a. \url{https://teorth.github.io/optimizationproblems/constants/86a.html}. \bibitem{giraudin} V. Patankar, Improved bounds for C16, C1a, C1c, C86, C89 (exact certificates), PR 270 (2026). \url{https://github.com/teorth/optimizationproblems/pull/270}. \bibitem{tomar} T. S. Tomar, Upper and lower bounds for the Schur--Siegel--Smyth trace problem, unrefereed preprint, first posted 21 October 2026. \url{https://www.mefyi.com/papers/schur-siegel-smyth-trace-bounds/}. \bibitem{half} R. Pinho, C86 classical half-cell certificate, exact data, checkers and receipts (2026), at the pinned commit of \cite{adaptive}. \url{https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86/classical-half-cells}. \bibitem{adaptive} R. Pinho, C86 adaptive-density certificate, exact data, proof, checkers, receipts and AI review (2026), commit \texttt{848f899c973666f64b73fb6f37760849a600285f}. \url{https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86/adaptive-cone}. \bibitem{pr298} R. Pinho, C86: upper bound 1.80560940 with classical auxiliaries on an adaptive density (unrefereed), PR 298 (2026). \url{https://github.com/teorth/optimizationproblems/pull/298}. \end{thebibliography} \end{document}