\section{Finite validation boundaries}
\label{app:finite}

The executable binary panel has four epistemic roles: it verifies the
left-rotation and gradient conventions; exercises negative-ray,
strict-crossing, and zero-sector cases; replays every legal firing and
terminal invariant; and rejects convention mutations.  It neither
interpolates in $k$ nor tests the alphabet-general theorem.

\begin{table}[ht]
\centering
\small
\begin{tabular}{rrrrrrr}
\toprule
$k$ & PCRs & zero PCRs & stable zero & paired zero & $|\supp\chi|$ & firings\\
\midrule
2  & 3   & 0  & 0 & 0 & 0   & 0\\
4  & 6   & 3  & 1 & 1 & 2   & 1\\
6  & 14  & 5  & 1 & 2 & 10  & 5\\
8  & 36  & 6  & 2 & 2 & 52  & 26\\
10 & 108 & 9  & 1 & 4 & 220 & 110\\
12 & 352 & 19 & 5 & 7 & 940 & 470\\
\bottomrule
\end{tabular}
\caption{Exact binary convention-assurance panel.  PCRs use distinct
rotations, and ``paired zero'' counts PCR pairs.}
\label{tab:validation}
\end{table}

The total is 612 legal firings.  Runtime and envelope metadata are excluded
from the stable scientific payload.  The paper proof, not this table or its
hash, is authority for every all-order statement.