\section{Assembly of the proposed proof}\label{conclusion:section}

\begin{proof}[Proposed proof of Theorem~\ref{intro:main}]
Propositions~\ref{alg:finite-input} and \ref{alg:alternating} identify
the finite diagram and the three scalar tests. The local models
of Proposition~\ref{lm:perfect}, together with
Theorem~\ref{en:simultaneous-data}, supply the geometric complexes with zero differential
and compatible data for the local chain homotopies of
Proposition~\ref{loc:constructor}.

In the inverse-excision comparison, the two torus limits are
identified by the matching-orbit bijection of
Lemma~\ref{loc:matching-orbit} and the low-dimensional gluing theorem,
Proposition~\ref{en:torus-gluing}. They both give $Q_{\mathfrak a}$,
with multiplicity one. Summing the four comparison homotopies
therefore gives the geometric inverse relation used by that
construction. The nonmatching coefficient $Q_{\mathfrak b}$ is not
asserted to vanish and is not selected by either flat filling.

Proposition~\ref{loc:primitive} gives a common primitive for the two
full source boundaries. Lemma~\ref{op:primitive-lemma} transports it
to the augmented all-$L$ construction. The normalization clause of
Theorem~\ref{en:simultaneous-data} preserves the chosen local sphere
and $S^{2,1}$ models and completes the counted operation with its
boundary data fixed. Proposition~\ref{op:formula} consequently
identifies the resulting scalar expression with $\ell h([c])$
for each required source and cap.

Every constituent has the topology and dot count of
Proposition~\ref{geo:allterms}. Choose the simultaneous data with
total action error strictly less than $1/16$. Propositions~\ref{en:uniform-bound}
and \ref{en:vanishing} exclude all zero-dimensional contributions,
including matched broken products. Thus the three scalar coefficients
vanish, and \eqref{intro:reduction} gives $\dim\J(D)=64$.
\end{proof}

\subsection{Dependencies and the scope of review}

The proof has three distinct inputs beyond its formal assembly.
The matrices obtained by evaluating closed foams supply
the octahedral reduction. The actual local perfect complexes
allow homology relations to become literal chain relations.
The marked families, their normalization, and the absolute index
calculation identify the scalar counts to which the action estimate
applies. None of these roles is supplied by the numerical agreement
of the final ranks.

The simultaneous-realization argument is now included as a proof,
with its permitted perturbation space, finite marking quotients,
torus strata and choice order specified. It should be checked
together with the constructions to which it is applied.
In particular, one should follow the product Hopf
bundle through the diagonal quotient and the matching cap framings,
then through the local chain homotopies and the fixed local models
used to normalize the all-$L$ triangle. The resulting four-dimensional
constituents must
be the same ones whose signed normal data and indices are computed
in Section~\ref{geo:section}.

The local-model claims of Proposition~\ref{lm:perfect} remain a
separate substantive part of that review, especially the cubic
model at the central cube orbit. The finite extension calculation
has an independently written implementation; the entire geometric
pairing evaluator has not been independently reimplemented.
Appendix~\ref{app:computation} records these limits.

The dimension-$64$ conclusion remains provisional pending expert
examination of the complete construction. If a step of the proposed
argument fails, the finite calculation still leaves both $60$ and $64$.