\section{Boundary cases, reusable interfaces, and exact nonclaims}
\label{common:sec-scope-boundary}

The final section records how the theorems behave at their parameter
boundaries and which interfaces can safely be reused.  It is a mathematical
scope statement: the nonclaims below prevent the local source relations from
being read as stronger global or operator-level coherence.

\subsection{Quantified boundary cases}

\begin{proposition}[Fixed-site boundary cases]
\label{fixed:prop-boundary-cases}
Theorem~\ref{fixed:thm-main-package} includes \(g=1\), \(k=1\), the all-zero
and all-one phase vectors, complementary phase vectors, empty phase words,
and arbitrary positive, zero, or negative active weights.  If
\(\epsilon=\eta\), then \(b=k\); if
\(\eta=\mathbf1-\epsilon\), where \(\mathbf1=(1,\ldots,1)\), then \(b=0\).
For \(k=1\), the four
preferred-weight entries of the local table are
\((\mathcal D,1,1,\mathcal D)\).
\end{proposition}

\begin{proof}
These cases are direct substitutions in
\[
 b=\#\{t:\epsilon_t=\eta_t\},
 \qquad
 \beta_1(M^{\epsilon,\eta})=2g+b,
\]
and in the four-entry local table.  The proofs of the Walker equalities are
integer identities involving only weight sums and vanishing Maslov
corrections, so zero and negative active coordinates require no separate
convention.
\end{proof}

\begin{proposition}[Slide-cube boundary cases]
\label{slide:prop-boundary-cases}
Theorem~\ref{slide:thm-complete-marked-disjoint-slide-cube} includes
\(g=1\), \(s=1\), \(k=2s\), every unused spectator when \(k>2s\), the
empty phase word, zero and negative base or phase weights, the reference
vertex \(A=\varnothing\), and the terminal vertex \(A=[s]\), without an
additional hypothesis.
\end{proposition}

\begin{proof}
 The slide boundary cases reduce to the already proved one-edge identities when
\(s=1\).  Extra spectators lie in the identity exterior, while zero or negative
weights remain covered by the same integer identities.  The reference and
terminal vertices are constructed directly, and the edge quantifier itself
explains why the terminal vertex has no outgoing edge.

The old summand is used only through \(g\geq1\).  If \(s=1\), there is one
slide order; the assertions involving two distinct directions are vacuous,
whereas order independence and telescoping reduce to the proved one-edge
identities.  If \(k=2s\), the unused-spectator sum is empty.  If \(k>2s\),
the identity exterior fixes every extra site, its complete marking, and its
homology plane pointwise.

For the zero phase assignment, the supported phase product is the
weight-zero identity cylinder, so the all-phase identities reduce literally
to their phase-zero forms.  Every weight calculation is an identity in
\(\mathbb Z\), hence permits zero and negative coordinates.  The completed
construction defines the reference and terminal vertices directly.  Edges
are asserted only for \(a\notin A\), so the terminal vertex has no outgoing
edge, exactly as the quantifiers require.
\end{proof}

\subsection{Input and output interfaces}

The fixed-site part of this paper ends with the completed forward and reverse
traces, their literal order independence, the source-reversal formulas, and
the match-count topology of their double.  These outputs are used at the
reference slide vertex only.  A fixed-site attachment arrow is never renamed
as a slide edge.

The local positive-slide input from
\cite[Theorem~2.2 and Appendix~F]{CompleteMarkedPentagons2026} ends with one
constructed two-site map and
its complete geometric calibration.  The identification in
\cref{slide:subsec-model-identification} transports that map into a supplied
protected sector and verifies that the incoming boundary, old region, other
sectors, and unused handles lie in its identity exterior.  Spectator
insertion is a conclusion of that identification and of the separation
hypotheses; it is not a conclusion of the imported two-site construction.
That construction contributes no phase, Walker-weight, several-slide,
square, or finite-order assertion.

The reusable slide output is precisely the directly completed Boolean family
\(\{\mathfrak C_A\}_{A\subseteq[s]}\), its supported positive edges and
signed integral homology actions, the common Lagrangian, phase-zero and
all-phase Walker covariance, literal disjoint two-faces, and the
order-independent terminal cylinder with its telescoped endpoint identities.
A later application may cite this result only after supplying the same
complete marking and pairwise-disjoint protected-sector hypotheses.

\subsection{Exact nonclaims}

The two main theorems do not assert any of the following:
\begin{enumerate}
\item a negative slide or the opposite band orientation;
\item isotopy independence for arbitrary choices of slide band;
\item shared-label, overlapping, intersecting, or nested slide sectors;
\item a braid relation for overlapping slides;
\item an arbitrary slide sequence, slide groupoid, or higher coherence;
\item a vertex or arrow with only part of a local slide completed;
\item a slide cylinder of nonzero Walker weight;
\item birth, death, cancellation, stabilization, or Cerf relations;
\item a cap-adjoined functor, a cap-absorption statement for a slide edge, or
      any same-fixed-cap absorption beyond the fixed-site identities in
      \cref{fixed:thm-main-package};
\item a realization-kernel, faithfulness, fullness, or completeness theorem;
\item a finite refined carrier or a physical slide operator;
\item unrestricted Heegaard or Reidemeister--Singer descent;
\item a gluing, sewing, monoidal, or TQFT theorem;
\item a closed three-manifold invariant;
\item a WRT comparison; or
\item a comparison with \(\widehat Z\) or with Habiro completion.
\end{enumerate}

The inverse \(F_{a\mid A}^{-1}\) is available only as the literal inverse of
the constructed positive edge, with its stated source and target complete
markings.  It is not an arbitrary negative-band transition.  Likewise, the
adjacent-replacement symbols used in the slide-cube proof name substitutions
of literal Walker equalities between order expressions.  They are not
mapping cylinders, comparison isotopies, retained swap maps, or generators
of a permutation-group action.

The disjoint cube and the overlapping-slide pentagon answer different
coherence questions.  The present square comes from supported maps whose
protected closures are disjoint.  The companion theorem treats a prescribed
overlapping-support configuration.  Neither result, alone or together,
extends the present theorem to shared labels, intersecting bands, or an
unrestricted handleslide calculus.

\subsection{Geometric conclusion}

Within those boundaries, the paper supplies two complete source-level
coherence mechanisms.  At one fixed handle site, doubling reduces to a
match-count calculation and a corresponding Walker-weight identity.  For
pairwise-disjoint slide sectors, the completed vertices form a Boolean
family whose supported maps commute pointwise, whose Walker faces commute
literally, and whose terminal cylinder is independent of the order of the
slides.  These two mechanisms are the reusable geometric output: one controls
local phase choices, and the other controls disjoint global assembly without
replacing the separate overlapping-slide pentagon.