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\title{Axler's global fourth- and fifth-order prime intervals improve sharply}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
For every real $x\ge2$ and $n\in\{4,5\}$, there is a prime in $\left(x,x(1+B_n/\log^n x)\right]$ with $B_4=68.499418523333785\ldots$ and $B_5=126484.607970195861\ldots$. The order-four value is the actual least full-range coefficient, forced by the prime gap $1327<1361$. The order-five value is exact for the corrected Fiori--Kadiri--Swidinsky step-envelope splice, but is not claimed globally least for the primes.
\end{abstract}

\[
x<p\le x\left(1+\frac{B_n}{\log^n x}\right),\quad n\in\{4,5\},\quad B_4=\frac{34}{1327}\log^4(1327),\ B_5=\frac{252969215940000000000}{1999999999996903}
\]

\section{What is proved}

The theorem improves Axler's fourth- and fifth-order prime intervals on the identical range $x\ge2$. The interval is open at $x$ and closed at its right endpoint.

\[
B_4=\frac{34}{1327}\log^4(1327)=68.4994185233337850616\ldots
\]

\[
B_5=\frac{252969215940000000000}{1999999999996903}=126484.6079701958614154\ldots
\]

\[
x<p\le x\left(1+\frac{B_n}{\log^n x}\right),\qquad n\in\{4,5\}
\]
\section{The sharp order-four gap}

For $G_n(x)=x(1+B_n/\log^n x)$, the order-four derivative has one low critical point inside the gap $11<x<13$. The certificate proves $G_4(x)>13$ there and strict increase afterward.

An exhaustive sieve through $17{,}051{,}887$ compares every consecutive prime gap. The unique largest requirement occurs at $1327<1361$; the runner-up $113<127$ is lower by more than $6.6216$.

\[
G_n'(x)=1+B_n\frac{\log x-n}{(\log x)^{n+1}}
\]

\[
\max_{p<q\ \mathrm{consecutive}}\frac{q-p}{p}\log^4p=\frac{34}{1327}\log^4(1327)=B_4
\]
\section{Why B4 is globally least}

At $x=1327$, the right endpoint is exactly $1361$, so the proposed interval reaches the next prime with equality. Any smaller coefficient leaves the entire interval below $1361$.

This single prime-free interval proves actual full-range optimality for $B_4$, independently of the later analytic splice.

\[
1327\left(1+\frac{B_4}{\log^4(1327)}\right)=1361
\]
\section{Bridges and numerical cells}

Axler's order-three theorem carries the order-four interval from the finite sieve into the corrected Fiori--Kadiri--Swidinsky range. Once order four is established, it similarly carries order five into that range.

For a two-sided theta-error row $(L_i,\epsilon_i)$ and $y=x+B_nx/L^n$, positivity of $\theta(y)-\theta(x)$ follows from an exact cell inequality. The first Table 2 theta exponent is repaired from Proposition 17 before all 236 cells are compared.

\[
\theta(y)-\theta(x)\ge x\left(\frac{(1-\epsilon_i)B_n}{L^n}-2\epsilon_i\right)
\]

\[
B_n>\frac{2\epsilon_iL_{i+1}^n}{1-\epsilon_i}
\]
\section{The active cell and analytic tail}

For both orders the unique active numerical cell is $2075\le L<2100$. Its order-four requirement is below $B_4$, while its order-five right-end supremum is exactly $B_5$. Because the cell is open at $L=2100$, that supremum is not attained; the next row gives a strict positive knot margin.

Beyond $L=10^8$, C-0053's absolute $\psi$ estimate and the standard prime-power decomposition give an absolute theta error. The certificate proves the error and its $L^n$ multiples decrease, so the prime-producing theta increment remains positive forever.

\[
B_{4,\mathrm{cell}}=60.230765700093265\ldots<B_4
\]

\[
B_{5,\mathrm{cell}}=\frac{252969215940000000000}{1999999999996903}=B_5
\]

\[
\epsilon_{\mathrm{abs}}(10^8)<3.323858\times10^{-3669}
\]

\section{Pinned certificate}
The certificate combines an exhaustive low prime-gap proof, exact source bridges, all corrected two-sided FKS cells, and a monotone absolute-theta tail derived from C-0053; it separately establishes global sharpness only for B4.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0062/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0062/verify.py}
\item \path{canon/witnesses/C-0062/PIN.md}
\item \path{canon/witnesses/C-0047/verify.py}
\item \path{canon/witnesses/C-0047/PIN.md}
\item \path{canon/witnesses/C-0053/verify.py}
\item \path{canon/witnesses/C-0053/PIN.md}
\end{itemize}

\section{Scope}
The order-four coefficient is the least full-range value. The order-five coefficient is exact for the corrected Fiori--Kadiri--Swidinsky step-envelope splice.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0062-axler-prime-interval-log45-improved-vector.md}
\item Certificate pin: \path{canon/witnesses/C-0062/PIN.md}
\item Existing proof: \path{scratch/adjacent-unconditional--axler-prime-interval-log45-improved-vector/PROOF.md}
\item Independent reconstruction: \path{scratch/adjacent-unconditional--axler-prime-interval-log45-improved-vector/INDEPENDENT-REVIEW.md}
\item C-0047 dependency: \path{canon/claims/C-0047-broadbent-global-theta-log3-coefficient-0-0143406585.md}
\item C-0053 dependency: \path{canon/claims/C-0053-fks-global-psi-amplitude-9-2202181.md}
\end{itemize}

\end{document}
