\section{Scope and further comparison problems}

Both transport theorems are fibrewise at one fixed \(K\).  The cube uses
pairwise-disjoint positive standard slides with distinct active labels; it
does not cover overlapping sectors, an arbitrary slide groupoid, or Cerf
completeness.  The pentagon uses one selected fully marked three-handle
witness.  Proposition~\ref{prop:comp-q-existence} supplies its comparison
core, and Theorem~\ref{thm:crosswalk-choice-comparison} compares auxiliary
choices by intertwiners rather than literal equality of coordinate matrices.
It proves neither relations among several pentagons nor phase and trace
extensions for the five pentagon edges.

\subsection{Existence and choice control for pentagon comparison maps}

\begin{description}
\item[Existence]  Lemma~\ref{lem:rational-lagrangian-cap} and Proposition
\ref{prop:comp-q-existence} construct a compatible comparison core for every
selected geometric pentagon in the stated range.  The construction uses the
literal selected vertex-zero compression body directly, by choosing its cap
and complementary handlebody and applying the normalized reduced pairing.

\item[Choice control]  Given two comparison maps with the same TQFT target,
Theorem~\ref{thm:crosswalk-choice-comparison} defines the correctly typed
comparison \(Q_\beta^{-1}Q_\alpha:\mathcal M_\alpha\to\mathcal M_\beta\).
All transported endocobordisms intertwine, and comparisons satisfy the
groupoid cocycle law.  If the source module is already literally fixed, this
becomes conjugacy.
\end{description}

The comparison need not be the identity or centralize the operator image, and
it does not identify different geometric witness data.

\subsection{Further comparison problems}

The disjoint-cube branch uses
\cite[Theorem~1.2]{MixedPhaseDisjointHandleslideCubes2026}.
The independent pentagon branch uses
\cite[Theorem~4.1, Proposition~12.3, Theorems~13.5 and~14.1,
and Section~15]{CompleteMarkedPentagons2026}.
These are external geometric source theorems; their geometry is not reproved
in the present paper.

Thus Papers~1 and~2 supply the source relations, while this paper supplies
their normalized transport, coefficient and anomaly control, and typed
comparison intertwiners.  It constructs no cross-level object, refined
theory, closed invariant, or WRT--Habiro--\(\widehat Z\) comparison.  Those
questions require new stabilization, completion, cap, or disk-comparison
input.