% SHARPNESS_M.tex
\section{Sharpness and Realizability of the Obstruction Taxonomy}\label{sec:sharpness-m}

This section records explicit, finite examples showing that each obstruction class OG1--OG4 is \emph{genuinely necessary}.
We also give a concrete ``good regime'' example where OG1--OG4 do not occur and the forcing mechanism (Sections~\ref{sec:slack-forcing} and \ref{sec:forcing-ion}) yields a nonvacuous
unconditional contraction rate.

Throughout, we present examples at the reduced-graph/template level:

\begin{itemize}
\item vertices represent descriptor states,
\item each directed edge $e:u\to v$ carries a nonnegative template matrix $T_e$,
\item the canonical support sets $S(\cdot)$ are specified explicitly and checked by support propagation
(Definition~\ref{def:face-map}).
\end{itemize}

\subsection{OG1 is realizable (no aperiodic tight SCC)}

Let $V=\{a,b\}$ and let the tight graph have exactly the edges $a\to b$ and $b\to a$.
This SCC has period $2$, hence it is imprimitive.
Set $d=1$ and take $T_{a\to b}=T_{b\to a}=[1]$.
Then every edge is tight, the unique SCC has period $2$, and therefore OG1 holds.
No aperiodic tight SCC exists, so no Doeblin/minorization regime can be entered.

\subsection{OG2 is realizable (face-map inconsistency cycle)}

Let $V=\{u,v\}$, $d=2$.
Define support sets
\[
S(u)=\{1\},\qquad S(v)=\{1\}.
\]
Let the reduced graph contain both edges $u\to v$ and $v\to u$ (so it is an SCC).
Assign the template on the edge $e:u\to v$ by
\[
T_e=\begin{pmatrix}1 & 0\\ 1 & 0\end{pmatrix}.
\]
Then for $x\in\mathbb{R}^{S(u)}_{>0}$ we have
$T_ex=(x_1,x_1)$, so coordinate $2$ is activated.
But $2\notin S(v)$, hence the ``no spurious activation'' condition in Definition~\ref{def:face-map} fails.
By Lemma~\ref{lem:og2-witness-cycle}, the edge $e$ together with the marked indices $(i,j)=(2,1)$ and any return path $v\leadsto u$
produces a finite inconsistency cycle witness.
Thus OG2 is realizable.

\subsection{OG3 is realizable (no Doeblin word on the active face)}

We record two standard sharp shapes corresponding to the two OG3 witness types.

\subsubsection{OG3(cut) realizable: block-triangular support}

Let $V=\{h\}$ (one-vertex SCC), $d=2$, $S(h)=\{1,2\}$.
Let the unique edge be a self-loop $e:h\to h$ with
\[
T_e=\begin{pmatrix}1 & 0\\ 1 & 1\end{pmatrix}.
\]
Then $S(\cdot)$ is a face map and $S(H)=\{1,2\}$.
However, there is no arrow $2\to 1$ in the coordinate digraph $\Gamma(H)$ (Definition~\ref{def:coord-digraph}),
so $\Gamma(H)$ is not strongly connected.
Equivalently, letting $A=\{1\}$ and $B=\{2\}$, we have $(T_e)_{1,2}=0$ and hence $(B(w))_{1,2}=0$ for every word $w$.
Thus no word can be strictly positive on $S(H)\times S(H)$ and OG3 holds with a cut witness
(Lemma~\ref{lem:og3-witness-cut}).

\subsubsection{OG3(product-automaton) realizable: permutation-only regime}

Let $V=\{h\}$, $d=2$, $S(h)=\{1,2\}$.
Let the allowed generator templates be the two permutation matrices
\[
P=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},\qquad
Q=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}
\]
and let the reduced graph have two self-loop edges at $h$ labeled by $P$ and $Q$.
Then $\Gamma(H)$ is strongly connected (each coordinate reaches the other under $Q$),
but every product is still a permutation matrix, so no word yields strict positivity on $S(H)\times S(H)$.
This is exactly the ``strongly connected but not Doeblin'' counter-shape in Remark~\ref{rem:gamma-not-sufficient},
and the Boolean-pattern product automaton returns a finite non-reachability certificate.
Thus OG3 is realizable even in the presence of strong connectivity.

\subsection{OG4 is realizable (Doeblin exists but is avoidable)}

We now realize OG4 in the regime where OG1--OG3 fail.
Let $V=\{h,x,y\}$ and consider the tight SCC $H$ supported on these vertices.
Take $d=2$ and $S(v)=\{1,2\}$ for all $v$.
Let there be edges
\[
 h\to x,\quad x\to h,\quad x\to y,\quad y\to x.
\]
This SCC is aperiodic (it contains a $2$-cycle $h\to x\to h$ and a $3$-cycle $x\to y\to x\to h\to x$).
Assign templates:
\[
T_{h\to x}=\begin{pmatrix}1&1\\1&1\end{pmatrix} \quad\text{(strictly positive)},
\qquad
T_{x\to h}=I,
\qquad
T_{x\to y}=Q,
\qquad
T_{y\to x}=Q,
\]
where $I$ is the identity and $Q$ is the swap matrix from the preceding example.
Then the single-edge word $w=(h\to x)$ is a Doeblin word (already strictly positive), so OG3 fails.
However, the periodic tight path alternating $x\to y\to x\to y\to\cdots$ avoids the vertex $h$ entirely,
so it avoids $w$ entirely.
In particular, the avoidance graph $\mathcal{A}_{\neg w}(H)$ has a directed cycle and OG4$(H,w)$ holds
(Definition~\ref{def:og4}).

This shows that even in the ``good'' structural regime where a Doeblin word exists,
bounded-gap recurrence of that \emph{chosen} word is not automatic.

\subsection{Nonvacuous good regime: OG1--OG4 fail and forcing succeeds}

Finally, we give a compact example where the forcing mechanism is nontrivial.
Let $V=\{h,x,y\}$ and edges
\[
 h\to x,\quad x\to h,\quad x\to y,\quad y\to h.
\]
Assume these are the only outgoing edges from $h$ (so $h$ has outdegree $1$).
The SCC $H$ on $\{h,x,y\}$ is strongly connected and aperiodic:
there is a cycle of length $2$ ($h\to x\to h$) and a cycle of length $3$ ($h\to x\to y\to h$).
Let $d=2$ and $S(v)=\{1,2\}$ for all $v$.
Assign templates
\[
T_{h\to x}=\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad
T_{x\to h}=I,\qquad
T_{x\to y}=I,\qquad
T_{y\to h}=I.
\]
Then $w=(h\to x)$ is a Doeblin word.
Moreover, because $h$ has a unique outgoing edge, \emph{every} infinite path in $H$ contains the edge $h\to x$
with gaps at most $3$ (in fact at most the maximum return time to $h$).
Equivalently, the avoidance graph $\mathcal{A}_{\neg w}(H)$ is acyclic, so RC1 holds and OG4 fails.
Therefore the forcing dichotomy applies (Corollary~\ref{cor:forced-positive-density}), and Theorem~\ref{thm:unconditional-mean} yields a uniform exponential contraction rate
on all tight minimizers supported in $H$.

\begin{remark}[What this demonstrates]
This example is intentionally simple: the Doeblin block is a single positive generator and recurrence is forced
by a finite outdegree constraint.
The point is not complexity but \emph{nonvacuity}: the ``OG1--OG4 fail $\Rightarrow$ contraction'' regime is populated,
and the forcing lemmas deliver explicit rates from finite witnesses.
\end{remark}
