\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,microtype}
\usepackage[colorlinks=true,linkcolor=black,urlcolor=black]{hyperref}
\usepackage{enumitem}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0.7em}
\title{A certified de Bruijn–Newman upper bound of 0.1729}
\author{Kenan}
\date{2026-07-20}

\begin{document}
\maketitle

\begin{abstract}
In the standard Polymath15 normalization, the de Bruijn–Newman constant satisfies $\Lambda\le 1729/10000=0.1729$. The certificate combines the published verified zeta-zero height, a cutoff-uniform sparse four-prime finite canopy, complementary analytic tails, and an 844-rectangle argument-principle barrier. It improves the established bound $\Lambda\le 11/50=0.22$ by exactly $471/10000=0.0471$ and replaces the earlier under-audit $0.175$ target.
\end{abstract}

\[
\Lambda \le \dfrac{1729}{10000} = 0.1729
\]

\section{The statement}

Let $H_t$ and $\Lambda$ have the normalization of the pinned Polymath15 source. Set $X=6000000185827$, $t_0=3377/20000$, and $y_0=9/100$. Polymath15 Theorem 1.2 converts the three certified zero-free regions into the displayed bound.

Relative to the established Polymath15 theorem $\Lambda\le 11/50$, the exact decrease is $471/10000=0.0471$.

\[
\Lambda\le t_0+\frac12y_0^2=\frac{1729}{10000}=0.1729
\]
\section{The three zero-free regions}

The initial-time strip uses the published S-0001 theorem only for critical-line location through height $3000175332800$. The required heights $X/2$ and $(X+1)/2$ leave reserves $175239886.5$ and $175239886$; no zero-simplicity input is used.

At $t=t_0$, 1,649 low-$y$ cells and 107 high-$y$ cells cover the finite ray. Four Euler primes are used through $N=770000$, three through $N=2000000$, and two thereafter, followed by complementary analytic tails.

The intermediate barrier is covered by a pinned V3 stored-sum matrix and 844 adaptive argument-principle rectangles. The winding enclosure contains only the integer zero, and the exact-model boundary margin exceeds $0.95116$.
\section{The cutoff repair}

The sparse checker evaluates the literal state at the initial cutoff and after every reachable divisor threshold before taking termwise maxima. This repairs the terminal-cutoff substitution defect identified by O-0177.

The package-native regression enumerates all 451 reachable zero-through-four-prime state patterns, including all 353 four-prime patterns. On an actual theorem cell it rejects an omitted-initial-state counterfeit that passes the older bounded regression, closing the fail-open boundary identified by O-0184.

\[
N_1=690988,\quad N_2=692000,\quad r=691110,\quad \text{states }(32767,65535)
\]
\section{Independent finite evidence}

Local arm64 and decorrelated x86 FLINT 3.6.0 evidence cover all 1,756 finite cells in canonical order. The weakest low-$y$ and high-$y$ endpoints are $0.0016881143766\ldots$ and $0.0241907009721\ldots$, both above the raw threshold $0.00114$.

The complete effective-model transfer cost is at most $0.00113438368318\ldots$, leaving strict positive reserve. The default verifier authenticates the package, reruns every analytic seam and state-pattern gate, checks both architecture evidence layers, and fails closed on source, runtime, grid, matrix, or transcript drift.

\section{Pinned certificate}
The pinned verifier authenticates the full package inventory, theorem and S-0001 source layers, all local and x86 finite-cell evidence, the parameter and error-transfer calculations, both analytic tails, the complete four-prime pattern-space regression, scalar-versus-polynomial parity, counterfeit controls, and a semantic V3 matrix read. The release-strength full-barrier mode also reruns and byte-compares all 844 argument-principle rectangles.

\subsection*{Run command}
\begin{verbatim}
nice -n 19 uv run --frozen python canon/witnesses/C-0130/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0130/verify.py}
\item \path{canon/witnesses/C-0130/PROOF.md}
\item \path{canon/witnesses/C-0130/PIN.md}
\item \path{canon/witnesses/C-0130/DEPENDENCIES.md}
\item \path{canon/witnesses/C-0130/MANIFEST.sha256}
\end{itemize}

\section{Scope}
The certificate proves $\Lambda\le 1729/10000=0.1729$ in the standard Polymath15 normalization. It does not claim optimality, and it gives no lower bound on $\Lambda$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0130-certified-debruijn-newman-upper-bound-01729.md}
\item Proof: \path{canon/witnesses/C-0130/PROOF.md}
\item Pinned witness: \path{canon/witnesses/C-0130/PIN.md}
\item Published theorem source: D.H.J. Polymath, Effective approximation of heat flow evolution of the Riemann xi function, and a new upper bound for the de Bruijn–Newman constant, Res. Math. Sci. 6 (2019), article 31. \url{https://doi.org/10.1007/s40687-019-0193-1}
\item Published finite-height input: D. J. Platt and T. S. Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, Bull. Lond. Math. Soc. 53 (2021), 792–797. \url{https://doi.org/10.1112/blms.12460}
\end{itemize}

\end{document}
