% RETURN_COST_H.tex
\section{Return Cost for Excursions Out of a Tight SCC}\label{sec:return-cost}

This section isolates a second forcing mechanism that complements Section~\ref{sec:slack-forcing}:
even if a trajectory attempts to avoid a certified Doeblin word by \emph{leaving} the relevant tight SCC,
any later \emph{return} to that SCC forces reduced-cost slack.
This turns repeated excursions (a natural avoidance strategy) into a measurable slack burden.

\subsection{A structural fact: tight SCCs do not admit tight returns from outside}

Fix the calibrated potential $\phi$ and reduced costs $c_\phi$.
Let $G^{(0)}$ be the tight graph.
Let $H$ be an SCC of $G^{(0)}$.

\begin{definition}[Excursion and return]\label{def:excursion-return}
Let $\pi=(e_0,e_1,\dots)$ be a one-sided path in the full graph $G$ with vertex sequence $(v_t)_{t\ge 0}$, where $e_t:v_t\to v_{t+1}$.
A time $t\ge 1$ is called a \emph{return time to $H$} if $v_t\in H$ and $v_{t-1}\notin H$.
An \emph{excursion} is a maximal time interval during which $v_t\notin H$ following a departure from $H$ and ending at the next return time (if it exists).
\end{definition}

\begin{lemma}[Returning to a tight SCC forces slack]\label{lem:return-forces-slack}
Let $\gamma=(e_0,\dots,e_{L-1})$ be a finite path segment in $G$ with vertex sequence $(v_0,\dots,v_L)$.
Assume $v_0\in H$ and $v_L\in H$, and assume that $v_t\notin H$ for at least one intermediate time $t$ with $0<t<L$.
Then $\gamma$ contains at least one non-tight edge, i.e.\ $\mathrm{SlackCount}(\gamma)\ge 1$.
In particular, if $\alpha<\infty$ (as in \eqref{eq:alpha-min-positive-slack}), then $\mathrm{SlackMass}(\gamma)\ge \alpha$.
\end{lemma}

\begin{proof}
Suppose for contradiction that every edge of $\gamma$ is tight, so $\gamma$ is a path in the tight graph $G^{(0)}$.
Then in $G^{(0)}$ there exists a tight path from $H$ to a vertex outside $H$ (since some $v_t\notin H$ occurs) and a tight path back from outside to $H$ (since $v_L\in H$).
This implies that in the SCC condensation DAG of $G^{(0)}$ there is a directed path from the SCC $H$ to a different SCC and a directed path back to $H$,
creating a directed cycle in the condensation.
But the SCC condensation is a DAG, so such a cycle is impossible.
Therefore $\gamma$ must contain a non-tight edge, proving $\mathrm{SlackCount}(\gamma)\ge 1$.
The bound on $\mathrm{SlackMass}$ follows from $\mathrm{SlackMass}(\gamma)\ge \alpha\,\mathrm{SlackCount}(\gamma)$.
\end{proof}

\subsection{Quantifying slack per return}

\begin{proposition}[Slack lower bound per number of returns]\label{prop:slack-per-return}
Let $\pi=(e_0,e_1,\dots)$ be any one-sided path in $G$ and let $R_n(H)$ be the number of return times to $H$ among $\{1,\dots,n\}$ (Definition~\ref{def:excursion-return}).
Then for every $n\ge 1$,
\begin{equation}\label{eq:return-count-slackcount}
R_n(H)\ \le\ \sum_{t=0}^{n-1}\mathbf{1}_{\{e_t\notin E^{(0)}\}},
\end{equation}
and consequently, if $\alpha<\infty$,
\begin{equation}\label{eq:return-count-slackmass}
\sum_{t=0}^{n-1} c_\phi(e_t)\ \ge\ \alpha\cdot R_n(H).
\end{equation}
\end{proposition}

\begin{proof}
Each return time to $H$ completes an excursion that starts in $H$, leaves $H$, and ends in $H$.
Apply Lemma~\ref{lem:return-forces-slack} to the finite path segment spanning that excursion.
Different excursions are edge-disjoint in time, so each return time accounts for at least one distinct non-tight edge.
This yields \eqref{eq:return-count-slackcount}, and \eqref{eq:return-count-slackmass} follows from \eqref{eq:alpha-min-positive-slack}.
\end{proof}

\subsection{Connection to word-avoidance strategies}

Let $H$ be a tight SCC and let $w$ be a tight word supported in $H$.
If \textbf{(RC1)} holds for $(H,w)$, then Section~\ref{sec:slack-forcing} shows that long word-avoidance windows \emph{inside} $H$ force linear slack.
Proposition~\ref{prop:slack-per-return} shows that word-avoidance by repeated \emph{excursions} out of $H$ also forces slack.

\begin{remark}[Excursions as avoidance]\label{rem:excursions-as-avoidance}
Assume \textbf{(RC1)} holds for $(H,w)$ and let $L_0$ be as in \eqref{eq:L0-def}.
If a path visits $H$ frequently but attempts to avoid $w$, then it must do at least one of the following on most long windows:
\begin{itemize}
\item introduce slack edges while staying in $H$, which is quantitatively penalized by Proposition~\ref{prop:slack-forcing-window};
\item leave $H$ before a tight run of length $L_0$ can occur and later return, which is penalized by Proposition~\ref{prop:slack-per-return}.
\end{itemize}
This is the mechanism used in the forcing dichotomy that follows: either $w$ recurs with bounded gaps on the relevant tight core, or slack density is bounded below by a positive constant (contradicting asymptotic tightness for mean-minimizers).
\end{remark}
