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\title{The global classical zeta zero-free denominator improves to 4.81}
\author{Kenan}
\date{2026-07-20}

\begin{document}
\maketitle

\begin{abstract}
The classical global zero-free region for the Riemann zeta-function is widened to denominator $4.81$ for every $t\ge2$. The proof combines a sharp exact vertical boundary, an everywhere-positive rational degree-$16$ detector, exact prime-phase certificates, a directed interval induction, and the published Platt--Trudgian finite-height theorem.
\end{abstract}

\[
\zeta(\sigma+it)\ne0\quad\left(t\ge2,\ \sigma>1-\frac1{4.81\log t}\right)
\]

\section{The global region}

For every real $t\ge2$, the zeta-function has no zero to the right of $1-1/(4.81\log t)$. A smaller denominator gives a wider zero-free region.

The same-range certified denominator improves from $4.824$ to $4.81$. Yang's published denominator is $4.862$.

\[
\zeta(\sigma+it)\ne0\qquad\left(\sigma>1-\frac1{4.81\log t}\right)
\]
\section{A sharp accessible boundary}

The former vertical one-sign boundary $151/153$ is inaccessible at the target starting height $10^{10}$. The proof replaces it with an exact rational boundary whose degree-six positivity polynomial has no nonnegative root.

At the target height the sharp boundary has positive access margin, while the former boundary has negative margin. A premise-matched counterfeit restores only the former boundary and is rejected.

\[
c_*=\frac{46110325513857}{46729244180480}
\]

\[
d_0-c_*>4.57\times10^{-5},\qquad d_0-\frac{151}{153}<0
\]
\section{Exact detector and source gains}

An inward rationalization of the optimized degree-$16$ cosine detector is proved positive for every real phase. For each prime below $100$, the verifier encloses every root of the corresponding exact degree-$240$ Chebyshev gap polynomial.

The resulting prime packet, cutoff correction, and sign-directed digamma estimate provide the positive source gains used by the induction.

\[
P(x)>\frac1{2\,000\,000}\quad\text{for every real }x
\]

\[
\sum_{p<100}(\log p)\min_xG_p(1,x)>0.2447566
\]
\section{Finite induction and global splice}

The shifted explicit-formula losses are enclosed by seven certified quadrature packets and outward-rounded to $26.421\eta^2+247.723\eta^3$.

All $2048$ closed induction boxes have positive directed margin. Exact finite steps reach denominator $4.81$, Yang's Littlewood region supplies the high-height handoff, and the published Platt--Trudgian theorem covers the low-height range.

\[
\frac{B(\mu,\eta)}D-\frac1{4.81}>3.9690\times10^{-5}
\]

\[
3000175332800-3000000000000=175332800
\]
\section{What the certificate checks}

The public package hash-pins the theorem, source, counterfeit, propagation, polynomial, and quadrature layers. It separates the exact published critical-line endpoint from the rounded simplicity statement and uses no zero-simplicity premise.

The theorem, counterfeit, propagation, and propagation-counterfeit modes all pass at their declared bounds. Optimized Python mode refuses because the certificate requires assertions.

\section{Pinned certificate}
The pinned package proves the exact detector floor, all prime-phase root enclosures, the sharp boundary, seven shifted error packets, every induction box, both global splices, and the complete noncircular downstream propagation.

\subsection*{Run command}
\begin{verbatim}
nice -n 19 gtimeout 1200s env PYTHONDONTWRITEBYTECODE=1 OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1 NUMEXPR_NUM_THREADS=1 uv run --frozen python canon/witnesses/C-0122/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0122/verify.py}
\item \path{canon/witnesses/C-0122/rational_candidate.py}
\item \path{canon/witnesses/C-0122/prime_phase.py}
\item \path{canon/witnesses/C-0122/boundary.py}
\item \path{canon/witnesses/C-0122/error_envelope.py}
\item \path{canon/witnesses/C-0122/counterfeit.py}
\item \path{canon/witnesses/C-0122/propagation_verify.py}
\item \path{canon/witnesses/C-0122/propagation_counterfeit.py}
\item \path{canon/witnesses/C-0122/PIN.md}
\item \path{canon/claims/C-0122-global-classical-zeta-zero-free-denominator-4-81.md}
\end{itemize}

\section{Scope}
The certificate proves denominator $4.81$ for every real $t\ge2$. It does not claim the denominator is globally optimal.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0122-global-classical-zeta-zero-free-denominator-4-81.md}
\item Proof: \path{canon/witnesses/C-0122/PROOF.md}
\item Source and verifier pin: \path{canon/witnesses/C-0122/PIN.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}
