\section{Exact checks and audit scope}\label{app:checks}

\subsection{The identities and endpoint certificates}

The finite checks compare the proof's formulas with separate exact
calculations. They supplement the arguments for arbitrary parameters;
the proof does not extrapolate from the checked cases.

For example, Gram entries can be calculated either from partial
fractions or from moments of the positive weight. With each
$\alpha=v/d$ reduced, put
\[
 \Delta=\prod_{v/d\in\Lambda}d^2,\qquad
 P_*(x)=\prod_{v/d\in\Lambda\setminus\mathcal A}(d^2x+v^2).
\]
Then
\[
 \frac{\prod_{\alpha\in\Lambda}(x+\alpha^2)}{Q(x)}
                           =\frac{P_*(x)}{\Delta}.
\]
The elementary beta integral gives, for $0\le j<a$,
\begin{equation}\label{eq:moments}
 \frac1{2\pi}\int_{-\infty}^{\infty}
       \frac{y^{2j}}{(1+y^2)^a}\,dy
 =\frac{(2j)!(2a-2j-2)!}
        {2^{2a-1}j!(a-j-1)!(a-1)!}.
\end{equation}
Thus $\mathcal M_{b,j}/\kappa_1$ is $\sigma/\Delta^m$ times the
moment functional in \eqref{eq:moments} applied to
$P_*^m h_bh_j$. Its degree is at most
$m(D(L+1)/2-1)<a$, so every required moment converges.

The structural checker uses nine seed cases with $m\in\{1,3,5\}$,
$B\in\{1,2,3,4\}$ and $r\in\{1,2\}$, and nine prime cases
with $3\le p\le17$. The complete parameter lists are in the
accompanying results file. These checks use $a=mDL+1$ to keep
the arithmetic small; decay is checked separately at the claimed
endpoints.
\begin{center}
\begin{tabular}{@{}lr@{}}
\toprule
Exact check&Number passed\\
\midrule
Differentiated-kernel evaluations&57\\
Gram entries, residues versus moments&49\\
Positive leading principal minors&19\\
Integer seed-vector identities&9\\
Rational-function reconstructions&54\\
Reflection identities&1308\\
Differentiated grid zeros&332\\
Grouped integrality&4516\\
Selected-pole congruences&404\\
Other-pole valuation bounds&1940\\
Constant congruences&19\\
Beta-coefficient valuation bounds&202\\
\midrule
Total&8909\\
\bottomrule
\end{tabular}
\end{center}

Run \path{checks/checks.py} for these identities.
Run \path{checks/endpoints.py} for the $21$ individual interval
certificates, the six uniform parameter choices recorded in its
output, and the rational comparisons in
Section~\ref{sec:endpoints} and Appendix~\ref{app:optimization}.
Floating point locates rational trial values of $u$; the inequalities
are then checked exactly over $\Q$.
The separate program \path{checks/worked_example.py} checks the
two-grid example in Section~\ref{sec:worked-gram}.
All three programs use only the Python standard library and should
be run without \texttt{-O}, since their assertions perform the checks.

\subsection{Independent audit}

An independent mathematical audit reported on 6 November 2026
found no substantive gap in the underlying plain-text proof,
identified by its hash in the accompanying audit record.
The audit covered the general Gram construction, denominator
clearing, decay, prime dilation, coefficient elimination and
uniform endpoints. It independently reproduced all $8909$
structural checks, all $21$ endpoint certificates, the six uniform
parameter records and the seven integer-$r$ comparisons in
Appendix~\ref{app:optimization}, without running or importing
the supplied programs.

The audit concerns the underlying proof and its certificates.
The subsequent typeset exposition, including the worked example,
was checked separately. The audit was not formal verification.