\section{Computational specification and reproducibility}\label{app:computation}

The supporting bundle is part of the finite specification of this draft.
Its contents are sufficient to run the checks without the earlier research
archives. Python~3, NumPy and SymPy are required. The manuscript can be
typeset independently with a standard \LaTeX\ installation. The command
\begin{verbatim}
python3 reproduce.py
\end{verbatim}
from the bundle root runs the checks and writes a report and transcripts
under \texttt{verification/}. It does not attempt to verify any analytic
claim in this manuscript.

\subsection{The independently reconstructed finite calculation}

The directory \texttt{support/finite/} contains the new program
\texttt{check\_finite.py}. Its inputs are exact binary arrays with
explicit inclusion, projection and quotient-lift maps. It imports no
module from the earlier implementation. In particular it neither solves
in the stored $48$-dimensional Hom basis nor restricts extension
variables by the auxiliary degree labels. All $25\cdot37=925$ entries
of $h$ are variables.

The program constructs a presentation using eleven independent exterior
edges, different from the earlier chosen generator set. The presentation
has $102$ relations: $55$ commutators, $18$ quadratic vertex relations,
$18$ cubic vertex relations, and eleven cube relations. It builds the
extension and intertwining equations directly, eliminates the extension
variables first, and projects the solution space to the $h$ coordinates.
The recorded exact results are
\begin{center}
\begin{tabular}{lrr}
\toprule
Quantity & Lower extension & Upper extension\\
\midrule
Variables & 7,548 & 12,900\\
Nonzero equations & 60,154 & 110,274\\
Rank of complete system & 6,928 & 11,868\\
Rank of constraints on $h$ & 922 & 922\\
Dimension of projected space & 3 & 3\\
\bottomrule
\end{tabular}
\end{center}
Both projected spaces equal $\iota\operatorname{Alt}_3(\F)\rho$.
The program writes its newly computed bases to
\texttt{independent\_obstruction\_bases.npz}, and checks the normal
form against them after solving the systems. It also checks the module
relations, induced subquotient maps, ranks of $s,\gamma$, the source classes
and the functionals defined by reflected foams, and all eight ranks of the alternating matrix.
A separate recursive coloring enumeration within this program gives
Tait counts $60,48,36,72$ for $D,K_0,K_1,L_1$.

The independence is precisely at the level of equation construction
and elimination from the retained matrices. It is not an independent
derivation of those matrices from gauge theory. The file
\texttt{independent\_finite\_result.json} states that scope and records
input hashes. Running the script is an exact finite calculation over
$\F$, with no floating-point rank tests or random sampling.

\subsection{Geometric matrix inputs}

The retained evaluator in \texttt{support/movies/} constructs the
closed pairing movies from their half-foam reductions. Its certificate
records local tetrahedral cancellations, disk-facet peels and residual
sphere evaluations, or the nonbipartite seam obstruction. The mathematical
evaluation rule is explained in Section~\ref{alg:section}.
Running
\begin{verbatim}
python3 verify_foam_geometry.py --dots
\end{verbatim}
within that directory checks the three Gram matrices, the two capped-map
matrices, and all stored edge operators against this evaluator.
The aggregate script also runs \texttt{check\_faces.py} to check the
facial supports. These are retained implementations. The transcripts
supplied here record the runs accompanying referee-draft version~1;
the unchanged finite checks have not been rerun merely for this
integration. They are not newly independent evaluators.

The $97$ undotted movie templates supply many matrix entries by changing
their dot decorations. Their inventory, reduction commands and basis
order are supplied in the JSON files in this directory. Binary matrix
inputs are stored in NumPy's \texttt{.npz} format; shape and basis
conventions are described in the root \texttt{README.txt} and the
manuscript. The raw file names containing the word
\texttt{combinatorial} are retained to preserve provenance; that name
does not assert a general comparison between two homology theories.

\subsection{Local and global geometric bookkeeping}

The directory \texttt{support/local\_models/} contains exact checks of
the cube axis equations, terminal-model equations, the doubled
theta/prism boundaries, and the four terminal reduction routes.
The doubled triangle is checked as a six-vertex, nine-edge prism;
its tangent calculation has eighteen variables and rank fifteen.
These checks support the explicit geometric proofs in
Section~\ref{lm:section}. They do not prove elliptic reduction or
small-perturbation compactness.

The directory \texttt{support/geometry/} contains the ordered source
and cap movies, the spherical rotation data, a check of every propagated
facial move, and the exact signed port rotation and crossing-depth
calculation. The arguments placing the foams in a common three-manifold and
constructing their normal sections remain the written proofs in Section~\ref{geo:section}.

Finally, \texttt{support/assembly/term\_assembly\_check.py} checks the
expansion of the scalar formula, with the domain and codomain of
each map, the differential of the primitive and its terminal residual,
the source and cap data, the local boundary assigned to each relation,
one-parameter and dot counts, and the rational action arithmetic.
Its catalogue counts labelled occurrences, not distinct moduli spaces.
The number of optional continuation homotopies is immaterial; the
mathematical statement is that the finite expansion is complete and
each constituent has the stated type and topological index.

\subsection{What the files do not certify}

The finite checks do not establish that every operation in the
formulas is realized by a regular marked family of metrics, that its compactification
has the asserted ends, or that all its auxiliary data can be chosen
with the required uniform bound on the perturbation error. Those are exactly the
geometric and analytic claims addressed by the written arguments in
Sections~\ref{loc:section} and~\ref{en:section}.
The supporting readiness notes record the scope of the earlier
reconstructed sections. Their historical statements of open
requirements are superseded, for this integrated proposed argument,
by the proofs now included here. They are not additional hypotheses
or additional proofs of the analytic statements.