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\title{The Schur--Siegel--Smyth trace problem:\\ the lower bound $1.8022246$}
\author{David Niedbala Giraudin\\ \small Independent researcher, France --- ORCID 0009-0009-1526-1178}
\date{27 October 2026 (this version: 10 November 2026)}
\begin{document}
\maketitle
\begin{abstract}
We prove $\lambda_{\rm SSS}\ge 9011123/5000000=1.8022246$ for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The previous best bound is $1.8022225$ (Tomar). The certificate consists of fourteen integer polynomials with rational multipliers and a continuous piecewise-linear density with $8\,678$ rational hat weights; its dual inequality is verified on $(0,\infty)$ in ball arithmetic. For the same family of polynomials the value of the underlying optimisation problem is $1.8022247$ to within $10^{-9}$ (numerical), so the bound is within $1.1\cdot10^{-7}$ of what this family can give.
\end{abstract}

\section{Statement}
For a totally positive algebraic integer $\alpha$ of degree $d$ write $\overline{\rm tr}(\alpha)=\mathrm{tr}(\alpha)/d$, and let $\lambda_{\rm SSS}=\liminf\overline{\rm tr}(\alpha)$.

\begin{theorem}[PROVED, computer-assisted]\label{thm}
$\lambda_{\rm SSS}\ \ge\ \dfrac{9011123}{5000000}=1.8022246.$
\end{theorem}

\noindent Previous lower bounds: $1.80203$ by Orloski, Talebizadeh Sardari and Smith~\cite{OSS}; $1.80220$ in~\cite{NG}; $1.8022225$ by Tomar~\cite{Tomar}. Several upper bounds obtained from Smith's theorem~\cite{Smith} have been announced since October 2026, for instance $1.8070809$~\cite{Tomar} and $1.80560940$~\cite{Pinho}.

\section{The criterion}
\begin{lemma}[\cite{OSS}; Proposition~2 of \cite{NG}; Lemma~2.1 of \cite{Tomar}]\label{lem}
Let $Q_1,\dots,Q_k\in\mathbb Z[x]$ be nonzero, $\lambda_{Q}\ge0$, and let $g\ge0$ be a bounded density with compact support and mass $s>0$. Put $U_g(x)=\int\log|x-y|\,g(y)\,dy$ and $I(g)=\int U_g\,g$. If
\[
F(x)=x-\lambda-\sum_Q\lambda_Q\log|Q(x)|-2U_g(x)+\frac{I(g)}{s}\ \ge\ 0
\]
for every $x>0$ that is not a root of one of the $Q$, then $\lambda_{\rm SSS}\ge\lambda$.
\end{lemma}

\section{The certificate}
The polynomials and multipliers are listed in Table~\ref{tab}; the polynomial $x^3-6x^2+9x-1$ receives multiplier $0$, so fourteen polynomials are used. The set is the one of~\cite{Tomar}. The density is
\[
g=\sum_{k=1}^{8678}\rho_k h_k,
\]
where $h_k$ is the hat function of mass one on $[e_{k-1},e_{k+1}]$ with peak at $e_k$, for nodes $e_0<\dots<e_{8679}$ that are rationals with denominator $2^{44}$ in $[0.045,5.369]$, and $\rho_k$ are rationals with denominator $2^{64}$. Exactly $6\,067$ weights are nonzero; the support of $g$ is contained in $[0.04888,5.35200]$, and
\[
s=0.535259605221817\ldots,\qquad I(g)=-8.42875692345304\ldots\cdot10^{-8}.
\]
The exact data are in the ancillary file \texttt{cert\_SSS\_1802224600.json} (format of~\cite{Tomar}: keys \texttt{A}, \texttt{lamQ}, \texttt{nodes}, \texttt{rho}, \texttt{lambda}, \texttt{x\_lo}$=1/100$, \texttt{x\_hi}$=8$).

\begin{table}[h]\centering\small
\begin{tabular}{ll}\toprule
$Q$ & $\lambda_Q$ \\ \midrule
$x$ & $0.503429120622$\\
$x-1$ & $0.437992939258$\\
$x-2$ & $0.042387356574$\\
$x^2-3x+1$ & $0.156351499433$\\
$x^3-5x^2+6x-1$ & $0.058154222706$\\
$x^4-7x^3+13x^2-7x+1$ & $0.017297099538$\\
$x^4-7x^3+14x^2-8x+1$ & $0.017604850341$\\
$x^5-9x^4+28x^3-35x^2+15x-1$ & $0.002319719944$\\
$x^5-9x^4+26x^3-29x^2+11x-1$ & $0.000660717275$\\
$x^5-9x^4+27x^3-32x^2+13x-1$ & $0.000388680112$\\
$x^3-6x^2+9x-1$ & $0$\\
$x^5-9x^4+27x^3-31x^2+12x-1$ & $0.002585133289$\\
$x^{10}-18x^9+134x^8-538x^7+1273x^6-1822x^5+1560x^4-766x^3+200x^2-24x+1$ & $0.000413411866$\\
$x^{10}-18x^9+135x^8-549x^7+1320x^6-1920x^5+1662x^4-813x^3+206x^2-24x+1$ & $0.000395645259$\\
$x^{10}-18x^9+134x^8-537x^7+1265x^6-1798x^5+1526x^4-743x^3+194x^2-24x+1$ & $0.000392591884$\\ \bottomrule
\end{tabular}
\caption{Polynomials and multipliers (decimals rounded; exact rationals in the ancillary file).}\label{tab}
\end{table}

\section{Verification}
Since $g$ is continuous and piecewise linear, $g''=\sum_n C_n\delta_{e_n}$ with exact rational $C_n$, and
\[
U_g(x)=\sum_n C_n\,G(x-e_n),\qquad I(g)=\sum_{n,m}C_nC_m\,\Psi(e_n-e_m),
\]
with $G(t)=\tfrac{t^2}{2}\log|t|-\tfrac34t^2$ and $\Psi(t)=\tfrac{t^4}{24}\log|t|-\tfrac{25}{288}t^4$ ($G''=\Psi''''=\log|t|$). The program \texttt{check\_sss.py} (ancillary) computes everything in arb ball arithmetic at 192 bits, starting from the exact rational data:
\begin{itemize}
\item the roots of the polynomials are certified real by Sturm sequences and enclosed in rational intervals of width $2^{-90}$;
\item $I(g)$ is computed from the closed form above, its polynomial part exactly;
\item on $(0,1/100]$ and $[8,\infty)$, $F$ is bounded below by monotonicity: the bounds are $0.48677\ldots$ and $0.53568\ldots$;
\item $[1/100,8]$ is cut at every node, at the root enclosures and into blocks; on each block the contribution of distant nodes is a Taylor polynomial of degree~$24$ with an explicit remainder; on each cell $F$ is bounded below by a second-order bound, and by chords or tangents for the terms singular at the cell ends; failing cells are bisected.
\end{itemize}
Result: $98\,710$ cell checks, all positive; total time $4.6$ minutes on two cores. Verdict: $F\ge0$ on $(0,\infty)$ with $\lambda=9011123/5000000$, which proves Theorem~\ref{thm}.

\paragraph{Controls (VERIFIED).}
\begin{enumerate}
\item With $\lambda=1.80222462$ (two units of $10^{-8}$ higher) the same program returns certified negative values, e.g.\ $F(0.049625)=-1.550\cdot10^{-8}$ and $F(0.0615)=-1.190\cdot10^{-8}$ (radii below $10^{-16}$). The certificate therefore proves $1.8022246$ and nothing above $1.80222462$.
\item On the certificates of~\cite{Tomar} the program reproduces the published values of $I(g)$ to all printed digits, returns VERIFIED for his bounds $1.8021892\ldots$ and $1.8022225$, and returns certified negative values at $\lambda+3\cdot10^{-6}$ for his first certificate at $x=0.0996570746$ and $x=0.1308260950$, the points reported by his own program.
\item The values of $F$ at five points of $[0.05,3.7]$ were recomputed by direct numerical integration of $\int\log|x-y|g(y)\,dy$ cell by cell (mpmath, 40 digits), without the closed forms; they agree with the ball values to twelve digits.
\end{enumerate}

\section{Remarks (numerical)}
\begin{remark}
For the fourteen polynomials of Table~\ref{tab}, the best bound obtainable from Lemma~\ref{lem} equals the value of the primal problem: minimise $\int x\,d\mu$ over probability measures $\mu$ on $[0,\infty)$ with $\int\log|Q|\,d\mu\ge0$ and $I(\mu)\ge0$. On a mesh of $8\,680$ nodes its value is $1.8022247019$, stable to $10^{-9}$ under refinement. The bound of Theorem~\ref{thm} is $1.02\cdot10^{-7}$ below it.
\end{remark}
\begin{remark}
For the optimal measure $\mu$ of this problem, every irreducible monic $P\in\mathbb Z[x]$ with all roots in $[0,5.3]$, degree at most $7$ and trace at most twice the degree ($883$ polynomials), and every such $P$ of degree $8$ with roots in $[0,5.4]$ ($3\,402$ polynomials), satisfies $\int\log|P|\,d\mu\ge0$. Outside Table~\ref{tab} the smallest value is $+0.0407$ (degree $7$) and $+0.160$ (degree $8$). Since $\mu$ satisfies these constraints, adding any of these polynomials leaves the optimal value unchanged.
\end{remark}

\section*{Acknowledgements}
This work was carried out with the assistance of an AI system (Claude, Anthropic). Every claim has been checked by computation; the verification program was written from Lemma~\ref{lem} and the certificate format alone.

\begin{thebibliography}{9}
\bibitem{NG} D.~Niedbala Giraudin, \emph{A new lower bound for the Schur--Siegel--Smyth trace problem}, arXiv:2609.29298 [math.NT] (2026).
\bibitem{OSS} B.~J.~Orloski, N.~Talebizadeh Sardari and A.~Smith, \emph{New lower bounds for the Schur--Siegel--Smyth trace problem}, Math. Comp. \textbf{94} (2025), no.~354, 2005--2040, doi:10.1090/mcom/4004; arXiv:2401.03252.
\bibitem{Smith} A.~Smith, \emph{Algebraic integers with conjugates in a prescribed distribution}, Ann. of Math. (2) \textbf{200} (2024), no.~1, 71--122, doi:10.4007/annals.2024.200.1.2.
\bibitem{Tomar} T.~S.~Tomar, \emph{Upper and lower bounds for the Schur--Siegel--Smyth trace problem}, version 4, Zenodo, doi:10.5281/zenodo.23563968 (2026).
\bibitem{Pinho} R.~Pinho, \emph{A trace upper-bound certificate with classical auxiliaries on an adaptive density}, Hexagon 2610.00695v3 (2026).
\end{thebibliography}
\end{document}