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\title{Axler's Table 8 coefficient-1.071 row fails and has a sharp repair}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
Axler's published row $\pi(x)<x/(\log x-1.071)$ for $x\ge22{,}078{,}017$ fails at the prime $22{,}078{,}033$. The exact same-range threshold is $a_*=\log(22{,}078{,}033)-22{,}078{,}033/1{,}393{,}895$, so $1.07100036$ is a valid strict repair. If the coefficient $1.071$ is retained, the smallest valid integer start is $22{,}078{,}034$.
\end{abstract}

\[
\pi(x)<\frac{x}{\log x-1.07100036}\ (x\ge22078017),\qquad \pi(x)<\frac{x}{\log x-1.071}\ (x\ge22078034)
\]

\section{The false row and its repair}

Put $x_0=22{,}078{,}017$, $p_*=22{,}078{,}033$, and $n_*=1{,}393{,}895$. At $p_*$ the published right-hand side is smaller than $\pi(p_*)$, so the strict inequality fails inside the stated range.

The normalized value at that prime determines the exact non-strict same-range coefficient.

\[
a_*=\log p_*-\frac{p_*}{n_*}=1.07100035826576367645708887648\ldots
\]

\[
\pi(p_*)-\frac{p_*}{\log p_*-1.071}=0.0315286257171953\ldots>0
\]

\[
\pi(x)<\frac{x}{\log x-1.07100036}\qquad(x\ge x_0)
\]
\section{Why only endpoints matter}

Define $A(x)=\log x-x/\pi(x)$. On an interval between consecutive primes, $\pi(x)$ is constant and $A$ decreases. At the next prime it jumps upward.

Therefore a global maximum on the published half-line can occur only at the actual starting endpoint or at a prime. This turns the finite part of the problem into an exact comparison of discrete candidates.

\[
A'(x)=\frac1x-\frac1n<0\qquad(p_n\le x<p_{n+1})
\]

\[
A(p_{n+1})-\lim_{x\uparrow p_{n+1}}A(x)=\frac{p_{n+1}}{n(n+1)}>0
\]
\section{Finite maximizer and independent tail}

The certificate enumerates the primes through $H=45{,}399{,}074$ and compares $1{,}347{,}034$ candidates. The unique maximum is $A(p_*)=a_*$; the published start is the unique runner-up.

Beyond $H$, the proof no longer relies on neighboring appendix rows. It first shows $\operatorname{li}(x)<x/(\log x-1.070)$, then uses Buethe's strict $\pi(x)<\operatorname{li}(x)$ computation through $10^{19}$ and an Axler Theorem 2 reduction with coefficient $1.0343$ for the remaining tail.

\[
A(p_*)-A(x_0)=0.0000006092657002968\ldots>0
\]

\[
A(x)<1.070\qquad(x\ge H)
\]
\section{Sharp coefficient classification}

The finite maximum and the independent tail prove that $a_*$ is the unique global maximum of $A$ on $x\ge x_0$. Hence the non-strict bound with coefficient $a_*$ has equality only at $p_*$.

For real coefficients $a<\log x_0$, the strict upper bound on the published range holds exactly when $a>a_*$. The rounded repair $1.07100036$ lies above $a_*$ by a certified positive margin.

\[
\pi(x)\le\frac{x}{\log x-a_*}\qquad(x\ge x_0)
\]

\[
1.07100036-a_*=1.7342363235\ldots\times10^{-9}>0
\]
\section{Keeping the coefficient 1.071}

On the prime-free interval beginning at $p_*$, the crossing equation can be solved exactly with the $W_{-1}$ branch of the Lambert $W$ function. The principal branch is irrelevant because it gives a solution below $n_*$.

The published inequality fails from $p_*$ through the crossing and holds afterward. Since the crossing lies strictly between $22{,}078{,}033$ and $22{,}078{,}034$, the latter is the smallest valid integer start.

\[
r=-n_*W_{-1}\!\left(-\frac{e^{1.071}}{n_*}\right)=22{,}078{,}033.53303818298\ldots
\]

\[
\pi(x)<\frac{x}{\log x-1.071}\qquad(x\ge22{,}078{,}034)
\]

\section{Pinned certificate}
The pinned certificate checks the published row against exact prime counts, proves the unique finite maximizer, closes the tail independently of neighboring rows, certifies the rounded coefficient repair, and encloses the Lambert-W crossing.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0057/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0057/verify.py}
\item \path{canon/witnesses/C-0057/PIN.md}
\item \path{canon/witnesses/C-0057/source/repository/original-verify.py}
\item \path{canon/witnesses/C-0057/source/repository/independent-check.py}
\item \path{canon/witnesses/C-0057/source/repository/INDEPENDENT-REVIEW.md}
\end{itemize}

\section{Scope}
The repair concerns Axler's coefficient-$1.071$ row on $x\ge22{,}078{,}017$ within the one-parameter denominator $x/(\log x-a)$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0057-axler-table8-1071-counterexample-sharp-repair.md}
\item Certificate pin: \path{canon/witnesses/C-0057/PIN.md}
\item Axler journal source: \path{canon/witnesses/C-0057/source/primary/axler-integers-24-A34.pdf}
\item Axler arXiv source: \path{canon/witnesses/C-0057/source/primary/axler-2203.05917.pdf}
\item Buethe source: \path{canon/witnesses/C-0057/source/primary/buthe-1511.02032.pdf}
\item Independent reconstruction: \path{canon/witnesses/C-0057/source/repository/INDEPENDENT-REVIEW.md}
\end{itemize}

\end{document}
