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\title{Fiori--Kadiri--Swidinsky's global absolute psi decay improves to 0.8817882}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
For every real $x>2$, the global absolute Chebyshev-$\psi$ bound holds with amplitude $9.2202181$ and decay $0.8817882$. This keeps the amplitude and full range of the preceding certified bound while increasing its decay from $0.88178$, and it improves the published Fiori-Kadiri-Swidinsky decay $0.8476836$ in the same displayed form.
\end{abstract}

\[
|\psi(x)-x|<9.2202181x(\log x)^{3/2}e^{-0.8817882\sqrt{\log x}}
\]

\section{What is proved}

Writing $L=\log x$, the theorem gives one explicit inequality for the complete range $x>2$. Because the amplitude is unchanged from the preceding $0.88178$ result, the larger decay makes the right-hand side strictly smaller at every point in the range.

\[
\frac{|\psi(x)-x|}{x}<9.2202181L^{3/2}e^{-0.8817882\sqrt L}\quad(x>2)
\]
\section{Supply an admissible zero-free input}

The upstream classical zero-free theorem gives an open region with denominator $4.8568$. Fiori-Kadiri-Swidinsky formulate their input on a closed boundary, so the proof uses the slightly larger exact denominator $R_*=4.85681$.

That closed region lies strictly inside the proved open region. The padding is therefore a logical interiorization, not a numerical improvement of the zero-free theorem.

\[
R_*=4.85681>4.8568
\]

\[
\zeta(\sigma+it)\ne0\quad\text{for }\sigma>1-\frac{1}{4.8568\log t}
\]
\section{Recompute the asymptotic envelope}

At the handoff $L=2008$, the certificate recomputes all $99\,999$ cells in the Fiori-Kadiri-Swidinsky amplitude formulas using $R_*$. Normalization produces a tail coefficient $C_*$ and decay $d_*$.

Since $d_*>0.8817882$, conversion to the target decay becomes easier as $L$ increases. It is enough to check the tail at the handoff.

\[
C_*=28.6643229697\ldots,\qquad d_*=\frac{2}{\sqrt{R_*}}=0.9075163092\ldots
\]

\[
E_\psi(x)\le C_*L^{3/2}e^{-d_*\sqrt L}\quad(L\ge2008)
\]
\section{Splice the low and high ranges}

For $0<L\le2008$, the published low-range envelope has coefficient $2$ and decay $0.8476836$. Converting it to the target decay requires the largest coefficient at $L=2008$.

For $L\ge2008$, the recomputed asymptotic envelope requires its largest converted coefficient at the same endpoint. Directed arithmetic proves positive margins on both sides, so the two bounds splice into one all-range inequality.

\[
d_{\rm low}=0.8476836+\frac{\log(9.2202181/2)}{\sqrt{2008}}=0.8817882015416\ldots
\]

\[
9.2202181-C_*e^{-(d_*-0.8817882)\sqrt{2008}}>0.1702763
\]
\section{Role of the certificate}

The verifier hash-pins and reruns the C-0096 zero-free certificate and the C-0067 Fiori-Kadiri-Swidinsky reconstruction, including their source checks. It then restores directed precision, checks the open-to-closed padding, recomputes every amplitude cell, and certifies both splice margins.

The proof therefore covers the absolute value and every real $x>2$ rather than only the asymptotic range.

\section{Pinned certificate}
The certificate reruns the pinned zero-free and prior psi-bound dependencies, interiorizes the open zero-free radius, recomputes the complete Fiori-Kadiri-Swidinsky amplitude at the new input, and proves positive directed margins for both sides of the $\log x=2008$ splice.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0098/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0098/verify.py}
\item \path{canon/witnesses/C-0098/PIN.md}
\item \path{canon/witnesses/C-0096/verify.py}
\item \path{canon/witnesses/C-0067/verify.py}
\item \path{canon/claims/C-0096-global-classical-zeta-zero-free-denominator-4-8568.md}
\item \path{canon/claims/C-0067-fks-yang-global-psi-decay-0-88178.md}
\end{itemize}

\section{Scope}
The bound holds for every real $x>2$ with amplitude $9.2202181$. The active low-range splice fixes the same-amplitude decay cap at $0.8817882015416\ldots$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0098-fks-c0096-global-psi-decay-0-8817882.md}
\item Certificate pin: \path{canon/witnesses/C-0098/PIN.md}
\item Zero-free dependency: \path{canon/claims/C-0096-global-classical-zeta-zero-free-denominator-4-8568.md}
\item Prior global psi bound: \path{canon/claims/C-0067-fks-yang-global-psi-decay-0-88178.md}
\end{itemize}

\end{document}
