% FORCING_ION.tex
\section{Forcing Dichotomy: Doeblin Frequency versus Slack Density}\label{sec:forcing-ion}

This section upgrades the qualitative ``tight-density'' information available for general minimizers
into a quantitative forcing statement.
It combines:

\begin{itemize}
\item the \textbf{RC1} avoidance certificate for a word $w$ inside a fixed tight aperiodic SCC $H$
(Definition~\ref{def:rc1}), and
\item the \textbf{return-cost} principle for excursions out of $H$
(Proposition~\ref{prop:slack-per-return}).
\end{itemize}

The output is a dichotomy that is strong enough to feed the Doeblin$\Rightarrow$Hilbert-contraction chain:
either the Doeblin word occurs with \emph{uniform positive lower density} along the trajectory,
or the reduced-cost slack has \emph{positive lower density} (hence the trajectory cannot be minimizing).

\subsection{Setup}

Fix the calibrated potential $\phi$ and reduced costs $c_\phi(e)\ge 0$ as in \eqref{eq:reduced-cost-u}.
Let $E^{(0)}=\{e\in E: c_\phi(e)=0\}$ be the tight edge set and let $G^{(0)}=(V^{(0)},E^{(0)})$ be the tight graph.

Fix a tight SCC $H$ of $G^{(0)}$ and a finite edge-word $w$ supported in $H$ such that \textbf{(RC1)} holds for $(H,w)$.
Let $L_{\mathrm{syn}}(H,w)$ be the explicit syndetic modulus from Lemma~\ref{lem:avoidance-syndetic} and set
\begin{equation}\label{eq:L0-RC1}
L_0:=L_{\mathrm{syn}}(H,w)+|w|
\end{equation}
as in \eqref{eq:L0-def}.
Let
\begin{equation}\label{eq:alpha-RC1}
\alpha := \min\{c_\phi(e): e\in E\setminus E^{(0)}\}\in(0,\infty]
\end{equation}
as in \eqref{eq:alpha-min-positive-slack}.

For a one-sided path $\pi=(e_0,e_1,\dots)$ in $G$ define the length-$n$ slack mass
\begin{equation}\label{eq:slack-mass-n}
\mathrm{SlackMass}_n(\pi):=\sum_{t=0}^{n-1} c_\phi(e_t).
\end{equation}
Define the tight time count in $H$ by
\begin{equation}\label{eq:Th-def}
T_H(n;\pi):=\sum_{t=0}^{n-1}\mathbf{1}_{\{e_t\in E^{(0)}\text{ and } e_t \text{ is supported in }H\}},
\end{equation}
and let $R_n(H)$ be the number of return times to $H$ among $\{1,\dots,n\}$ as in Definition~\ref{def:excursion-return}.

Finally, let $N_w(n;\pi)$ denote the number of starting indices $0\le i\le n-1$ at which the word $w$
occurs along $\pi$ (i.e.\ $e_i e_{i+1}\cdots e_{i+|w|-1}=w$).

\subsection{A counting lemma on maximal tight runs}

\begin{definition}[Maximal tight-in-$H$ runs]\label{def:Ht-runs}
For fixed $n$, consider the set of indices $t\in\{0,\dots,n-1\}$ for which $e_t$ is tight and supported in $H$.
The maximal contiguous intervals of such indices determine disjoint finite subpaths
\[
\gamma^{(1)},\dots,\gamma^{(m)}
\]
which we call the \emph{maximal tight-in-$H$ runs} of $\pi$ up to time $n$.
Write $\ell_j:=|\gamma^{(j)}|$ for the length (number of edges) of the $j$-th run.
Then $\sum_{j=1}^m \ell_j = T_H(n;\pi)$ and $m-1\le R_n(H)$.
\end{definition}

\begin{lemma}[Each long tight run forces many occurrences of $w$]\label{lem:run-forces-w-count}
Assume \textbf{(RC1)} holds for $(H,w)$ and let $L_0$ be as in \eqref{eq:L0-RC1}.
Let $\gamma$ be a finite tight-in-$H$ run of length $\ell$.
Then $\gamma$ contains at least $\lfloor \ell/L_0\rfloor$ occurrences of $w$.
\end{lemma}

\begin{proof}
Partition $\gamma$ into consecutive disjoint blocks of length $L_0$,
discarding at most a final remainder block of length $<L_0$.
Each full block is a tight path in $H$ of length $L_0$.
By Lemma~\ref{lem:long-tight-run-impossible}, no such block can avoid $w$,
so each full block contains an occurrence of $w$ supported entirely inside that block.
Disjoint blocks cannot share an occurrence, hence the number of occurrences is at least the number
of full blocks, which is $\lfloor \ell/L_0\rfloor$.
\end{proof}

\begin{lemma}[Occurrences versus returns]\label{lem:occ-vs-returns}
For every $n\ge 1$ and every path $\pi$,
\begin{equation}\label{eq:occ-lower}
N_w(n;\pi)\ \ge\ \frac{T_H(n;\pi)}{L_0}\ -\ R_n(H)\ -\ 1.
\end{equation}
\end{lemma}

\begin{proof}
Let $\gamma^{(1)},\dots,\gamma^{(m)}$ be the maximal tight-in-$H$ runs up to time $n$
(Definition~\ref{def:Ht-runs}) with lengths $\ell_1,\dots,\ell_m$.
By Lemma~\ref{lem:run-forces-w-count},
\[
N_w(n;\pi)\ \ge\ \sum_{j=1}^m \Big\lfloor \frac{\ell_j}{L_0}\Big\rfloor
\ \ge\ \sum_{j=1}^m \Big(\frac{\ell_j}{L_0}-1\Big)
\ =\ \frac{T_H(n;\pi)}{L_0}-m.
\]
Since $m\le R_n(H)+1$, we obtain \eqref{eq:occ-lower}.
\end{proof}

\subsection{The forcing dichotomy}

\begin{proposition}[Doeblin frequency vs slack density]\label{prop:forcing-ion}
Assume \textbf{(RC1)} holds for $(H,w)$ and let $L_0$ and $\alpha$ be as in \eqref{eq:L0-RC1}--\eqref{eq:alpha-RC1}.
Then for every $n\ge 1$ and every one-sided path $\pi$ in $G$,
\begin{equation}\label{eq:forcing-master}
N_w(n;\pi)\ \ge\ \frac{T_H(n;\pi)}{L_0}\ -\ \frac{\mathrm{SlackMass}_n(\pi)}{\alpha}\ -\ 1,
\end{equation}
with the convention that $\mathrm{SlackMass}_n(\pi)/\alpha=0$ if $\alpha=\infty$.
Equivalently,
\begin{equation}\label{eq:slack-lower-from-missing-w}
\mathrm{SlackMass}_n(\pi)\ \ge\ \alpha\cdot\Big(\frac{T_H(n;\pi)}{L_0}-N_w(n;\pi)-1\Big)_+ .
\end{equation}
\end{proposition}

\begin{proof}
Combine Lemma~\ref{lem:occ-vs-returns} with the slack-per-return bound
\eqref{eq:return-count-slackmass} from Proposition~\ref{prop:slack-per-return},
which yields $R_n(H)\le \mathrm{SlackMass}_n(\pi)/\alpha$ (for $\alpha<\infty$).
If $\alpha=\infty$, then $E\setminus E^{(0)}=\emptyset$ and $\mathrm{SlackMass}_n(\pi)=0$ identically,
so \eqref{eq:forcing-master} reduces to \eqref{eq:occ-lower}.
\end{proof}

\begin{corollary}[Forced positive density of the Doeblin word]\label{cor:forced-positive-density}
Assume \textbf{(RC1)} holds for $(H,w)$ and let $L_0$ be as in \eqref{eq:L0-RC1}.
Let $\pi$ be any path such that
\[
\liminf_{n\to\infty}\frac{T_H(n;\pi)}{n}\ \ge\ \theta>0
\qquad\text{and}\qquad
\lim_{n\to\infty}\frac{\mathrm{SlackMass}_n(\pi)}{n}\ =\ 0.
\]
Then
\[
\liminf_{n\to\infty}\frac{N_w(n;\pi)}{n}\ \ge\ \frac{\theta}{L_0}.
\]
In particular, $w$ occurs along $\pi$ with a uniform positive lower density depending only on the forcing constants.
\end{corollary}

\begin{proof}
Divide \eqref{eq:forcing-master} by $n$ and take $\liminf_{n\to\infty}$, using the hypotheses.
\end{proof}

\begin{remark}[Interpretation]\label{rem:forcing-ion-interpret}
Corollary~\ref{cor:forced-positive-density} is the promised forcing upgrade:
in the regime where a minimizer has nontrivial tight mass in a fixed aperiodic tight SCC $H$ and reduced-cost slack density $0$,
\textbf{(RC1)} forces a quantitative supply of Doeblin blocks along the trajectory.
This is sufficient for explicit exponential projective contraction (see the rate pack in Section~\ref{sec:rate-pack-l}), without needing global bounded-gap recurrence.
\end{remark}
