% RATE_PACK_L.tex
\section{Rate Extraction without Recurrence Inputs} \label{sec:rate-pack-l}

This section is a clean ``rate pack'' that converts the forcing outcomes (Sections~\ref{sec:slack-forcing}, \ref{sec:return-cost}, and \ref{sec:forcing-ion}) and the avoidance certificate \textbf{(RC1)}
into explicit exponential contraction constants.
It is designed to be referenced directly by the pipeline contract in Section~\ref{sec:pipeline-contract-n}.

\subsection{Constants from a Doeblin witness}

Fix an aperiodic tight SCC $H$ of $G^{(0)}$ with a consistent active face $S(H)$ and a certified Doeblin word $w$ supported in $H$.
Write $L:=|w|$ and let $B(w)$ be the block product.
By certification, $B(w)$ is strictly positive on $S(H)\times S(H)$.

Define the explicit FTG window and Birkhoff factor
\begin{equation}\label{eq:ratepack-window}
\varepsilon:=m_*^{L},\qquad U:=(dM_*)^{L},\qquad
\tau:=\tanh\!\Big(\tfrac12\log(U/\varepsilon)\Big)\in(0,1).
\end{equation}
Then every occurrence of $w$ contracts Hilbert distance on the active face by factor at most $\tau$
(Lemma~\ref{lem:birkhoff} and Corollary~\ref{cor:gap-free-regularity}).

\subsection{Rate from syndetic recurrence (RC1 on a tight tail)}

\begin{proposition}[Rate from bounded gaps] \label{prop:rate-from-syndetic}
Assume OG4 fails for $(H,w)$, so the avoidance graph $\mathcal{A}_{\neg w}(H)$ is acyclic and the explicit gap modulus
$L_{\mathrm{syn}}:=L_{\mathrm{syn}}(H,w)$ is defined.
Let $\pi$ be any infinite path whose tail is a path in $H$ (in particular, any tight minimizer supported in $H$).
Then the occurrences of $w$ along $\pi$ are syndetic with gap bound $L_{\mathrm{syn}}$.
Consequently, for all $x,y\in\mathbb{R}^{S(H)}_{>0}$ and all $n\ge 1$,
\[
d_H\big(A^{(n)}(\pi)x,\,A^{(n)}(\pi)y\big)\ \le\ C\,e^{-\kappa n}\,d_H(x,y)
\]
with
\begin{equation}\label{eq:rate-syndetic}
C:=\tau^{-1},\qquad \kappa:=\frac{-\log\tau}{L_{\mathrm{syn}}+L}.
\end{equation}
\end{proposition}

\begin{proof}
If $\mathcal{A}_{\neg w}(H)$ is acyclic, Lemma~\ref{lem:avoidance-syndetic} provides syndetic recurrence on every infinite path in $H$.
The contraction estimate is exactly the (RH1) branch of Corollary~\ref{cor:gap-free-regularity}, using \eqref{eq:ratepack-window}.
\end{proof}

\subsection{Rate from forced positive density (mean-minimizers with zero slack density)}

\begin{proposition}[Rate from forced word density] \label{prop:rate-from-density}
Assume OG4 fails for $(H,w)$, and let $L_0:=L_{\mathrm{syn}}(H,w)+L$.
Let $\pi$ be any one-sided path and define
\[
\theta:=\liminf_{n\to\infty}\frac{T_H(n;\pi)}{n},\qquad
\eta:=\limsup_{n\to\infty}\frac{\mathrm{SlackMass}_n(\pi)}{n}.
\]
If $\theta>0$ and $\eta=0$, then
\begin{equation}\label{eq:rho-lower}
\rho:=\liminf_{n\to\infty}\frac{N_w(n;\pi)}{n}\ \ge\ \frac{\theta}{L_0}
\end{equation}
(Corollary~\ref{cor:forced-positive-density}).
Consequently, for all $x,y\in\mathbb{R}^{S(H)}_{>0}$ and all $n\ge 1$,
\[
d_H\big(A^{(n)}(\pi)x,\,A^{(n)}(\pi)y\big)\ \le\ C\,e^{-\kappa n}\,d_H(x,y)
\]
with
\begin{equation}\label{eq:rate-density}
C:=\tau^{-1},\qquad \kappa:=\frac{\rho}{L}(-\log\tau)\ \ge\ \frac{\theta}{L_0 L}(-\log\tau).
\end{equation}
\end{proposition}

\begin{proof}
Equation \eqref{eq:rho-lower} is Corollary~\ref{cor:forced-positive-density}.
Given a lower density $\rho$ of starting indices, extract a disjoint subfamily of occurrences by the standard greedy rule.
Among the first $n$ steps this produces at least $\lfloor \rho n/L\rfloor$ disjoint $w$-blocks.
Each disjoint block contracts by factor at most $\tau$, yielding
\(d_H(\cdot)\le \tau^{\lfloor \rho n/L\rfloor}d_H(x,y)\le \tau^{-1}e^{-(\rho/L)(-\log\tau)n}d_H(x,y)\),
which is \eqref{eq:rate-density}.
\end{proof}

\begin{remark}[Where $\theta$ enters]
The parameter $\theta$ measures the asymptotic lower density of tight time spent in the chosen aperiodic SCC $H$.
For tight minimizers supported in $H$ one has $\theta=1$, so \eqref{eq:rate-density} reduces to an explicit path-independent lower bound.
For general minimizers, $\theta$ is recorded as a trajectory-level input (or estimated in applications) while the constants
$\tau$ and $L_0$ are fully witness-derived.
\end{remark}
