\section{Minimality and sharpness: why the boundary is tight}

This section records short ``edge-case'' constructions that explain why the hypotheses and obstruction gates in the boundary theorem
cannot be substantially weakened without changing the conclusion.
The goal is not to catalog counterexamples, but to isolate the minimal mechanisms that force the regime split.

\subsection{Why tight-core reduction is not optional}

The calibration potential $\phi$ and the induced tight graph $G^{(0)}$ are not merely a technical convenience.
They are the device that makes the extremal problem finite and local.
Without passing to tight edges, statements about minimizers are typically false if one reads them directly on the unreduced descriptor graph.

\begin{remark}[A two-cycle tie shows why one must reduce]
Consider a finite graph with two disjoint cycles $\gamma_1$ and $\gamma_2$ that have the same cycle mean $\lambda_*$.
Assume there are additional edges of slightly higher mean that connect the cycles in both directions.
Then minimizing measures may be supported on either cycle, or on mixtures that switch between the two cycles with sublinear frequency.
On the unreduced graph there is no canonical ``core'' and no unique active face.
After calibration, the tight subgraph separates the true minimizer-supporting structure from slack edges;
the subsequent obstruction tests are meaningful only on this tight reduction.
\end{remark}

\subsection{How contraction fails: realizable obstructions}

The obstruction gates OG2 and OG3 are designed to be witness-producing and to capture genuinely different failure modes.
Both occur in simple finite-template systems.

\begin{itemize}[leftmargin=*]
\item \textbf{OG2 (face-map inconsistency) is a real obstruction.}
It is possible for tight edges to force incompatible coordinate support requirements.
Concretely, one can have two tight words $u$ and $v$ in the same tight SCC such that $u$ maps mass from an active face into a strict subface,
while $v$ maps mass into a different strict subface, and the SCC allows alternating concatenations.
The witness produced by Lemma \texttt{lem:og2-witness-cycle} is exactly the finite inconsistency cycle that certifies that no consistent face map exists.

\item \textbf{OG3 (no Doeblin word) is strictly stronger than ``$\Gamma(H)$ not strongly connected.''}
Even when the union support digraph $\Gamma(H)$ is strongly connected, the allowed face-restricted patterns may generate a semigroup
that never contains an all-positive block.
The clean counter-shape is the ``permutation-pattern'' family: if the admissible patterns on the face are permutation matrices
that generate a transitive action, then $\Gamma(H)$ is strongly connected but every product remains a permutation pattern,
so no Doeblin word exists.
This is why the certificate used in the paper is the Boolean-pattern product-automaton test:
Doeblin existence is equivalent to reachability of the all-ones pattern state.
\end{itemize}

\subsection{When imprimitive trapping is the right conclusion}

The regime split treats aperiodicity (period $1$) and imprimitivity (period $p>1$) differently because the projective geometry is different.
In an imprimitive tight SCC, the index set splits into cyclic classes and products respect that cyclic structure.
Even with strong tightness and perfect calibration, one should not expect a single stationary projective section on the full face.

\begin{remark}[Imprimitive tight cores can force cycling]
Let $H$ be a tight SCC with period $p>1$.
Then there is a cyclic decomposition of states into classes $C_0,\dots,C_{p-1}$ such that every edge advances the class modulo $p$.
Along an admissible tight trajectory, the support of mass on the active face can rotate among these classes.
This produces a genuine periodic trapping phenomenon at the level of projective directions.
Accordingly, the boundary theorem outputs ``imprimitive trapping'' (periodic regime) rather than uniform contraction on the full face.
\end{remark}

Taken together, these examples explain the sharpness of the theorem chain:
the tight-core reduction is needed to make minimizers finite and local;
OG2 and OG3 isolate distinct algebraic failures on the active face;
and the imprimitive branch is the correct endpoint when the tight core has period $>1$.
