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\title{A certified Polya-frequency order of 9,425,328,785,004}
\author{Kenan}
\date{2026-07-18}

\begin{document}
\maketitle

\begin{abstract}
The evenness of $\xi(1/2+w)$ makes $\xi_1(z)=\xi(1/2+\sqrt z)$ a real entire function of order $1/2$. A verified zero height $T$ constrains every zero of $\xi_1$ to lie outside a large sector about the positive real axis. Schoenberg's converse sector theorem converts that zero-free sector into finite Polya-frequency order. Platt and Trudgian's published height $T=3,000,175,332,800$ gives $\xi_1\in PF_{9,425,328,785,004}$ through the stated worst-case sector bound.
\end{abstract}

\[
\xi\!\left(\tfrac12+\sqrt z\right)\in PF_{9425328785004}
\]

\section{What is proved}

Set $\xi_1(z)=\xi(1/2+\sqrt z)$. The theorem proves that its Maclaurin coefficient sequence is Polya-frequency of every order up to $9,425,328,785,004$.

The reusable statement is parameterized: if all nontrivial zeta zeros through height $T$ lie on the critical line, then $\xi_1\in PF_m$ whenever $\pi/(m+1)\ge2\arctan(1/(2T))$.

\[
RH(T)\quad\Longrightarrow\quad \xi_1\in PF_m\ \text{ for }\ \frac{\pi}{m+1}\ge2\arctan\!\left(\frac1{2T}\right)
\]
\section{Moving zeta zeros into the z-plane}

Write a nontrivial zeta zero as $\rho=1/2+u+i\gamma$. Under the square map used to define $\xi_1$, it becomes $z_\rho=(u+i\gamma)^2$.

Zeros with $|\gamma|\le T$ lie on the critical line by hypothesis, so they map to the negative real axis. For the remaining zeros, the critical strip gives $|u|<1/2$, and their angular defect from the negative axis is bounded by $2\arctan(1/(2T))$.

\[
z_\rho=(\rho-\tfrac12)^2=(u+i\gamma)^2
\]

\[
\pi-|\arg z_\rho|=2\arctan\!\left(\frac{|u|}{|\gamma|}\right)<2\arctan\!\left(\frac1{2T}\right)
\]
\section{The sector-to-PF transfer}

Because $\xi_1$ has order $1/2$, its zeros admit a genus-zero product with summable reciprocal moduli. Truncating by complete negative-real factors and conjugate pairs produces real polynomials with positive constant term and the same zero-free sector.

Schoenberg's converse theorem says that a real polynomial with positive constant term and no zero in $|\arg z|<\pi m/(m+1)$ belongs to $PF_m$. Local uniform convergence of the real partial products gives coefficientwise convergence, and every fixed Toeplitz minor remains nonnegative in the limit.

\[
P_N(z)\ne0\ \text{ for }\ |\arg z|<\frac{\pi m}{m+1}\quad\Longrightarrow\quad P_N\in PF_m
\]
\section{The certified endpoint}

Platt and Trudgian's published finite-height theorem supplies $T=3000175332800$. The pinned interval calculation places $\pi/[2\arctan(1/(2T))]$ strictly between $9,425,328,785,005$ and $9,425,328,785,006$.

The condition is on $m+1$, so the largest integer order certified by this particular worst-case sector inequality is $m=9,425,328,785,004$.

\[
9425328785005<\frac{\pi}{2\arctan(1/(2T))}<9425328785006
\]

\[
\xi_1\in PF_{9425328785004}
\]
\section{What the certificate checks}

The witness pins Schoenberg's sector theorem through Katkova's published application and checks the endpoint arithmetic using Platt and Trudgian's published height. It also checks the square-map multiplicity argument, the real conjugate-orbit truncation, and closure of each fixed Toeplitz minor under the entire-function limit.

It also records two repairs to the earlier printed application. A displayed sector of $43\pi/44$ meets the $PF_{43}$ threshold, not the $PF_{44}$ threshold, and arbitrary zero truncations need not be real. Grouping complete conjugate orbits repairs the latter issue.

\[
\frac{43\pi}{44}=\frac{\pi\cdot43}{43+1}
\]

\section{Pinned certificate}
The certificate pins the converse sector theorem and the finite-height dependency, checks the zero-square angular geometry, real partial-product construction, limit closure, corrected order indexing, and the interval arithmetic selecting the stated finite order.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0026/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0026/PIN.md}
\item \path{canon/witnesses/C-0026/verify.py}
\item \path{canon/witnesses/C-0026/source/schoenberg-1955/publisher-page.html}
\item \path{canon/witnesses/C-0026/source/katkova-2007/arxiv-v1.tex}
\item \path{canon/witnesses/C-0026/source/katkova-2007/journal.pdf}
\item \path{canon/claims/C-0026-verified-zeta-height-implies-xi1-pf9425328785007.md}
\end{itemize}

\section{Scope}
The stated Polya-frequency order is the largest integer certified by the displayed worst-case sector inequality at Platt and Trudgian's published height $3,000,175,332,800$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0026-verified-zeta-height-implies-xi1-pf9425328785007.md}
\item Witness pin: \path{canon/witnesses/C-0026/PIN.md}
\item Katkova source text: \path{canon/witnesses/C-0026/source/katkova-2007/arxiv-v1.tex}
\item Finite-order literature audit: \path{literature/2026-07-18-xi1-pf-verified-height-search.md}
\item Published finite-height theorem: D. J. Platt and T. S. Trudgian, “The Riemann hypothesis is true up to $3\cdot10^{12}$,” Bulletin of the London Mathematical Society 53 (2021), 792–797. \url{https://doi.org/10.1112/blms.12460}
\end{itemize}

\end{document}
