% NEW_OBSTRUCTION_J.tex
\section{Recurrence Obstruction: Doeblin-Avoidance Cycle (RC1 Failure)}\label{sec:new-obstruction-j}

The obstruction taxonomy OG1--OG3 controls the existence of an aperiodic tight SCC, a consistent active-face map,
and a strictly positive Doeblin block on that active face.
Even when none of these obstructions occurs, a logically correct gap-free contraction statement must still address one further issue:
a certified Doeblin word need not recur along \emph{every} tight trajectory unless its recurrence is forced by finite-state structure.

We therefore isolate a finite recurrence predicate \textbf{(RC1)} for a fixed SCC and a fixed Doeblin word,
and we record its failure as a fourth obstruction class OG4 with an explicit finite witness.
This keeps recurrence out of the hypotheses while preserving a complete, decidable boundary taxonomy.

\subsection{Definition and finite witness schema}

\begin{definition}[OG4 / recurrence obstruction: Doeblin-avoidance cycle]\label{def:og4}
Let $H$ be a tight SCC of $G^{(0)}$ and let $w$ be a finite edge-word supported in $H$.
We say that the obstruction \textbf{OG4$(H,w)$} holds if the avoidance graph
$\mathcal{A}_{\neg w}(H)$ (Definition~\ref{def:avoidance-graph})
contains a directed cycle.
Equivalently, \textbf{(RC1)} fails for $(H,w)$ (Definition~\ref{def:rc1}).
\end{definition}

A finite witness for OG4$(H,w)$ is simply a directed cycle in $\mathcal{A}_{\neg w}(H)$,
which can be produced by standard cycle-detection on a finite directed graph.

\subsection{Why OG4 is a genuine obstruction}

\begin{proposition}[OG4 produces a tight minimizing trajectory with no Doeblin blocks]\label{prop:og4-witness}
If OG4$(H,w)$ holds, then there exists a periodic tight path $\pi$ supported in $H$
that avoids $w$ entirely.
In particular, no theorem can guarantee occurrence of $w$ along \emph{all} tight minimizers in $H$.
\end{proposition}

\begin{proof}
By Definition~\ref{def:og4}, $\mathcal{A}_{\neg w}(H)$ contains a directed cycle.
Lemma~\ref{lem:avoidance-cycle-witness} converts such a cycle into an explicit periodic tight path in $H$
that avoids $w$ entirely.
Since the path is tight, Lemma~\ref{lem:tight-path-mean} shows that it has asymptotic mean cost $\lambda_*$,
so it is a minimizing trajectory, yet it contains no occurrences of $w$.
\end{proof}

\begin{remark}[Relationship to OG1--OG3]\label{rem:og4-vs-og123}
OG4 can occur even when OG1--OG3 do not:
an aperiodic tight SCC may exist, the face map may be consistent, and a strictly positive Doeblin word $w$ may be certified,
but $w$ may still be avoidable by some tight trajectories in $H$.
Accordingly, OG4 is the correct finite obstruction to record whenever \textbf{(RC1)} is not satisfied for the chosen $(H,w)$.
\end{remark}
