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

\subsection{The kernel, Gram and arithmetic identities}

The proof of rank in Section~\ref{sec:gram} applies to every $m$
and $B$. The accompanying computations check its formulas and
normalization, together with the denominator and dilation identities.
They use exact integer and rational arithmetic.

For the Gram entries, the residue formula and the moment integral
provide two calculations. With every fraction reduced, put
\[
 \Delta=\prod_{\alpha=v/d\in\Lambda}d^2,\qquad
 P_*(x)=\prod_{v/d\in\Lambda\setminus\mathcal A}(d^2x+v^2).
\]
Since $\mathcal A\subseteq\Lambda$,
\[
 \frac{\prod_{\alpha\in\Lambda}(x+\alpha^2)}{Q(x)}
                         =\frac{P_*(x)}{\Delta}.
\]
For $0\le j<S$, the elementary beta integral gives
\begin{equation}\label{eq:moments}
 \frac1{2\pi}\int_{-\infty}^{\infty}
                \frac{y^{2j}}{(y^2+4)^S}\,dy
 =2^{2j-4S+2}\frac{(2j)!(2S-2j-2)!}{j!(S-j-1)!(S-1)!}.
\end{equation}
Thus $M_{b,c}/C_1$ is $\sigma/\Delta^m$ times the moment
functional in \eqref{eq:moments} applied to
$x^aP_*(x)^m h_b(x)h_c(x)$. Its degree is at most
$m(DL/2+N)-a<S$, so all the moments converge.

The checker uses $m=1,3,5$, $B=1,2,3,4$, $r=3$ and
$S=10mD+1$ for the seed calculations. For complete prime
dilations it uses $(m,B)=(1,1),(3,1),(5,1),(3,2)$ with the
same rule for $S$, and primes $17,19$.
These smaller parameters test identities; decay is checked at
the theorem's displayed endpoints.
\begin{center}
\begin{tabular}{@{}lr@{}}
\toprule
Exact check&Number passed\\
\midrule
Differentiated-kernel evaluations&120\\
Gram entries, residues versus moments&90\\
Positive leading principal minors&30\\
Integer seed-vector identities&12\\
Selected-pole congruences&800\\
Other-pole valuation bounds&6800\\
Constant congruences&14\\
Grouped integrality, seed and prime indices&6564\\
\bottomrule
\end{tabular}
\end{center}

Run \path{checks/checks.py} for these calculations and the rational
inequalities in Section~\ref{sec:endpoints}.
Run \path{checks/endpoints.py} for the five individual intervals
and a direct all-$m$ check at $B=1$, $S=67m$.
Both 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 5 November 2026
found no substantive mathematical error or gap in the underlying
plain-text proof, identified by its hash in the accompanying
audit record. The audit covered the general arguments for arbitrary
allowed parameters and independently reproduced the finite checks
without running or importing the supplied programs. It also checked
a prime dilation with seed index $\ell=3$, prime $17$ and index
$n=51$.
The audit was not formal verification or certification of priority.