% Copyright 2026 Ricardo Pinho. CC BY 4.0.
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\title[An upper trace certificate]{A trace upper-bound certificate with complex-root auxiliaries}
\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{30 October 2026; 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 complex-root auxiliaries},pdfauthor={Ricardo Pinho}}
\begin{document}
\begin{abstract}
A rational piecewise-linear measure with 23 auxiliary polynomials gives
$C_{86}\le1.80697883$ for the Schur--Siegel--Smyth trace constant.
The proof uses Smith's arithmetic-measure criterion and converse theorem.
Two degree-eight auxiliaries have nonreal roots; both polynomials already
occur in Flammang (2009) and McKee (2011). Their use in this upper-bound
certificate, rather than their discovery, is the change described here.
This note is unrefereed, with no independent human review or formalization.
\end{abstract}
\maketitle

\section{Statement and provenance}
Let $C_{86}$ be the smallest limit point of
$\operatorname{Tr}(\alpha)/[\mathbb Q(\alpha):\mathbb Q]$ over totally
positive algebraic integers $\alpha$.
\begin{theorem}
\[
 C_{86}\le\frac{135523412047086487}{75000000000013500}
 <1.80697883.
\]
\end{theorem}
The construction follows Smith, Orloski--Talebizadeh Sardari, and
Orloski--Sardari--Smith \cite{smith,os,oss}, using enlarged polynomial sets from Koshta and
David Niedbala Giraudin's PR~270 \cite{giraudin}.
Tomar's public piecewise-linear certificate gives
$1.8070808325882437\ldots$ \cite{tomar}; the supplied review reproduced
that certificate and the smaller value above. This is a comparison with
identified prior work, not an exhaustive priority claim.

The two auxiliaries in question are
\begin{align*}
 Q_1(x)&=x^8-14x^7+78x^6-221x^5+339x^4-277x^3+111x^2-19x+1,\\
 Q_2(x)&=x^8-14x^7+78x^6-222x^5+345x^4-289x^3+120x^2-21x+1.
\end{align*}
Each has six real roots and one nonreal conjugate pair. Both occur in
Flammang (2009), Table 1, rows 22 and 23, and McKee (2011), Table 1,
rows 21 and 23 \cite{flammang,mckee}. Complex-root auxiliaries in this
subject are therefore not new. Their earlier use for lower bounds differs
from their role in this feasible upper-bound measure. No optimizer formula
requiring real-rooted auxiliaries is invoked.

\section{A sufficient certificate and its proof}
Let $\mu$ have a bounded continuous nonnegative piecewise-linear density
supported on a compact subset $S$ of the positive real axis. Put
\[
 m=\mu(\mathbb R)>0,\quad T=\int x\,d\mu(x),\quad
 U_\mu(z)=\int\log|z-x|\,d\mu(x).
\]
Let $Q_j$ be distinct primitive irreducible integer polynomials with
positive leading coefficients, and $b_j\ge0$. It suffices to check
\begin{gather}
 \beta:=\sum_jb_j\deg Q_j<m,\qquad
 \int\log|Q_j|\,d\mu\ge0\quad\hbox{for every }j,\label{direct}\\
 F(x):=U_\mu(x)-\sum_jb_j\log|Q_j(x)|\ge0
 \quad(x\in S).\label{potential}
\end{gather}
At auxiliary roots the inequality has its extended-real meaning.
The logarithmic potential is continuous because the density is bounded.
Off $S$ and the auxiliary roots, $F$ is harmonic; at an auxiliary root
with positive weight it tends to $+\infty$. At infinity
$F(z)=(m-\beta)\log|z|+O(1)\to+\infty$.
Excising small disks about those roots and truncating at a large circle,
the minimum principle extends \eqref{potential} to every complex $z$.
This includes gaps in $S$ and nonreal roots.

For a primitive irreducible integer polynomial $P$, distinct from the
$Q_j$, with leading coefficient $c>0$ and roots $\alpha_k$, resultants give
\begin{align*}
 \int\log|P|\,d\mu
 &=m\log c+\sum_k U_\mu(\alpha_k)\\
 &\ge(m-\beta)\log c+
       \sum_jb_j\log|\operatorname{Res}(P,Q_j)|\ge0.
\end{align*}
Each resultant is a nonzero integer. This identity includes the leading
coefficient of every $Q_j$ in $\log|Q_j|$, so no monicity assumption
has been suppressed. The selected polynomials satisfy \eqref{direct};
factorization and integer content cover every nonzero integer polynomial.
Thus $\mu/m$ is an arithmetic probability measure in Smith's terminology
\cite[Proposition 5.15]{smith}.

In the present certificate $S\subset[27/500,5227/1000]$. Take the ambient
interval $[1/100,6]$, whose logarithmic capacity is $599/400>1$.
Smith's Theorem 1.5 gives distinct totally positive algebraic integers
whose conjugate measures converge weakly to $\mu/m$ in this interval.
The continuous test function $x$ shows that their normalized traces
converge to $T/m$. Hence $C_{86}\le T/m$.

\section{The finite input and what has been checked}
The exact rational node heights, edges, 23 polynomials and weights are in
\path{complex-auxiliaries/certificate.json} of \cite{certificate}, also
included in the source archive. Their stored exact values are
\begin{gather*}
 m=\frac{50000000000009}{50000000000000},\qquad
 T=\frac{135523412047086487}{75000000000000000},\\
 \beta=\frac{43742583595997}{50000000000000}.
\end{gather*}
The certificate SHA-256 is
\begin{center}\scriptsize
\texttt{f59be9a136ad96930f20d4f9a7a6ff019d9a30ab9c9081c0c05582a3f8e0ea43}.
\end{center}
The supplied \texttt{verify\_linear.py} checks the rational structure and
irreducibility, encloses complex roots, evaluates logarithmic integrals
analytically in ball arithmetic, and covers every closed support cell.
Its retained receipt records positive integrals for all 23 polynomials and
a positive potential lower bound on 5587 pieces. A separate AI review
used another implementation and an endpoint/adjacency audit; its refined
9764-piece cover proves $F>10^{-6}$ on all 1556 nonzero density cells.
These are stored results, not new verifier runs for this note.
They prove feasibility, not optimality of the auxiliary pool or density.

The certificate citation deliberately retains \texttt{COMMIT\_C86} as an
unresolved revision placeholder. It is not an available immutable public
link until that token is replaced; the attached local source archive fixes
the present input bytes.

This is an unrefereed preprint. There has been no independent human review
and no formalization. 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}{9}
\bibitem{smith} A. Smith, Algebraic integers with conjugates in a prescribed
distribution, \emph{Annals of Mathematics} 200 (2024), Theorem 1.5 and
Proposition 5.15; \url{https://arxiv.org/html/2111.12660v2}.
\bibitem{os} B. Orloski and N. Talebizadeh Sardari, A quantitative converse
of Fekete's theorem; \url{https://arxiv.org/abs/2304.10021}.
\bibitem{giraudin} D. Niedbala Giraudin, trace upper-bound certificate,
PR 270; \url{https://github.com/teorth/optimizationproblems/pull/270}.
\bibitem{tomar} T. Tomar, Schur--Siegel--Smyth trace bounds, unrefereed
preprint (21 October 2026), certificate repository;
\url{https://github.com/tejasvi/schur-siegel-smyth-trace-bounds}.
\bibitem{oss} B. Orloski, N. Talebizadeh Sardari and A. Smith,
trace-bound construction, \emph{Mathematics of Computation} 94 (2025),
2005--2040; \url{https://arxiv.org/abs/2401.03252}.
\bibitem{flammang} V. Flammang, Trace of totally positive algebraic integers
and integer transfinite diameter, \emph{Mathematics of Computation} 78
(2009), 1119--1125, Table 1, rows 22--23;
\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,
Table 1, rows 21 and 23; \url{https://doi.org/10.1090/S0025-5718-2010-02424-X}.
\bibitem{certificate} R. Pinho, \emph{C86 trace certificate}, directory
\texttt{complex-auxiliaries}, revision placeholder \texttt{COMMIT\_C86};
\url{https://github.com/rpinho/c86-trace-certificate/tree/e4d9c640ebaf922b8869a81f275c798c09ab89b1/complex-auxiliaries}.
\end{thebibliography}
\end{document}