\section{Location map for the earlier documents}\label{app:integration}

This revision adds explanatory material and diagrams to the integrated
draft, without changing the proposed argument. It combines the
38-page referee draft, version~1, and
the 13-page simultaneous-realization supplement. This appendix is a
navigation aid for readers of those versions; the mathematical arguments
are contained in the preceding sections. The octahedron, formerly
Figure~1 of the integrated draft, is now Figure~\ref{fig:octahedron},
with the local web pictures included at its vertices.

\begin{center}\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.39\textwidth}>{\raggedright\arraybackslash}p{0.55\textwidth}@{}}
\toprule
Earlier location & Integrated location\\
\midrule
Referee-draft subsection 4.3, inverse excision
& Section~\ref{loc:torus-geometry} and
Lemma~\ref{loc:matching-orbit}: the joint torus model and matching-orbit
bijection\\[5pt]
Referee-draft Verification requirement 4.1
& Proposition~\ref{loc:family-data}, with realization supplied by
Theorem~\ref{en:simultaneous-data}\\[5pt]
Referee-draft Propositions 4.2 and 4.3
& Propositions~\ref{loc:constructor} and~\ref{loc:primitive}:
the local chain homotopies and common source primitive\\[5pt]
Referee-draft Hypothesis 7.4
& Theorem~\ref{en:simultaneous-data} and its proof\\[5pt]
Supplement Section 2, collar regularity
& Proposition~\ref{en:collar} and Lemma~\ref{en:temporal-uc}\\[5pt]
Supplement Section 3, marking calculation
& Section~\ref{loc:torus-geometry}, including the fixed product bundle,
restriction groups and cap framings\\[5pt]
Supplement Section 4, torus compactification
& Lemma~\ref{en:torus-gap} and Proposition~\ref{en:torus-gluing}\\[5pt]
Supplement Section 5, simultaneous choices and budget
& Proposition~\ref{en:coherence} and
Theorem~\ref{en:simultaneous-data}, within Section~\ref{en:section}\\[5pt]
Referee-draft Section 8 and supplement Section 6
& Section~\ref{conclusion:section}: the combined proposed proof and
its review dependencies\\
\bottomrule
\end{tabular}
\end{center}

\subsection{The correction carried through the manuscript}

The earlier one-generator torus notation suppressed a correlation.
The actual quotient on the two torus boundaries is diagonal, with
states $\mathfrak a=[00]=[11]$ and $\mathfrak b=[01]=[10]$.
The flat cylinder and the two aligned caps both select
$Q_{\mathfrak a}$ by a multiplicity-one marking-orbit identification.
The four inverse-comparison identities retain their algebraic form
with this common term. No vanishing of $Q_{\mathfrak b}$ is asserted.
The same cap framings and bundle identifications are used in the
analytic gluing and topological index sections.

The local-model proofs, finite matrix inputs and capped geometric
calculations have been retained. The principal additional arguments
are now placed where they are used: the marking proof in
Section~\ref{loc:section}, the normalization choice order in
Sections~\ref{op:section} and~\ref{en:section}, and the regularity,
gluing and quantitative choice proofs in Section~\ref{en:section}.
The finite supporting code and its recorded verification results
are unchanged.

\subsection{A continuous review path}

A review of the realization step can follow the actual Hopf marking
and product bundle through the joint torus quotient, both matching
fillings, and the four inverse homotopies. It can then check their
use in the local chain homotopies and source primitive, the fixed
all-$L$ normalization, and the capped scalar expansion.
The last connection is to the absolute normal data and index
calculation, not merely to the formal type of each operator.
This review path complements examination of the perfect local
models and the geometric matrix inputs; it is not a replacement
for either.