% DOEBLIN_STABILITY_RU_C.tex
\section{Doeblin Stability and Robust Rate Pack (RU-C)}\label{sec:ru-c-doeblin-stability}

RU-A records a Doeblin word witness $w=w_1\cdots w_L$ and its unperturbed margin
\(\delta=\min_{i,j}(P_w)_{ij}>0\) on the relevant core/face block, where
\(P_w=A^{(w_L)}\cdots A^{(w_1)}\).
RU-C upgrades this to a \,\emph{robust rate pack}: the \emph{same witness word} persists under small perturbations,
and the associated Hilbert/Birkhoff contraction constants degrade by an explicit computable formula.

Throughout this section we work under the perturbation metric fixed in the robust certificate.
We write $\varepsilon_A$ for the template perturbation radius and assume the tight core is held fixed
throughout the neighborhood by RU-B (so the witness word remains admissible in the same tight SCC).

\subsection{Product perturbation bounds and a surviving Doeblin margin}

Assume the certificate stores a uniform entry bound $M\ge 1$ such that
\(\max_{k,i,j} A^{(k)}_{ij}\le M\) on the block where the Doeblin witness is evaluated.

\begin{lemma}[Entrywise product perturbation bound]\label{lem:ru-c-product-perturbation}
Assume \emph{absolute entrywise} perturbations
\(\|\Delta A\|_{\infty,\mathrm{ent}}\le \varepsilon_A\).
Let $\tilde P_w := \tilde A^{(w_L)}\cdots \tilde A^{(w_1)}$.
Then
\[
\|\tilde P_w-P_w\|_{\infty,\mathrm{ent}}
\le
L\,(M+\varepsilon_A)^{L-1}\,\varepsilon_A.
\]
Consequently, the perturbed Doeblin margin satisfies
\[
\delta'(\varepsilon_A):=\min_{i,j} (\tilde P_w)_{ij}
\ge
\delta- L\,(M+\varepsilon_A)^{L-1}\,\varepsilon_A.
\]
\end{lemma}

\begin{proof}
This is Lemma~\ref{lem:open-doeblin}, restated for later reuse.
\end{proof}

\begin{remark}[Relative entrywise perturbations]\label{rem:ru-c-relative}
For \emph{relative entrywise} perturbations $|\Delta A^{(k)}_{ij}|\le \varepsilon_A A^{(k)}_{ij}$,
one may convert to an absolute bound on the relevant block:
\(\|\Delta A\|_{\infty,\mathrm{ent}}\le \varepsilon_A M\).
All formulas below remain valid after replacing $\varepsilon_A$ by $\varepsilon_A M$
in the right-hand sides and recording this choice in the certificate.
\end{remark}

\subsection{A robust diameter bound and Birkhoff contraction coefficient}

In addition to $\delta$, the robust certificate stores an \emph{upper} bound
\(U:=\max_{i,j} (P_w)_{ij}\) on the same block (or any certified upper envelope used by the pipeline).

\begin{lemma}[Upper envelope under perturbations]\label{lem:ru-c-upper-envelope}
Under the assumptions of Lemma~\ref{lem:ru-c-product-perturbation},
\[
U'(\varepsilon_A):=\max_{i,j} (\tilde P_w)_{ij}
\le
U + L\,(M+\varepsilon_A)^{L-1}\,\varepsilon_A.
\]
\end{lemma}

\begin{proof}
Entrywise, $|(\tilde P_w)_{ij}-(P_w)_{ij}|\le\|\tilde P_w-P_w\|_{\infty,\mathrm{ent}}$.
Taking maxima and applying Lemma~\ref{lem:ru-c-product-perturbation} gives the claim.
\end{proof}

\begin{lemma}[Robust projective diameter bound for a positive block]\label{lem:ru-c-diameter}
Assume $\delta'(\varepsilon_A)>0$ and let $U'(\varepsilon_A)$ be as in Lemma~\ref{lem:ru-c-upper-envelope}.
Then the Hilbert projective diameter of the image of the positive cone under $\tilde P_w$ satisfies
\[
\Delta(\varepsilon_A) \;\le\; 2\,\log\!\Big( \frac{U'(\varepsilon_A)}{\delta'(\varepsilon_A)} \Big).
\]
In particular, the Birkhoff contraction coefficient of $\tilde P_w$ satisfies
\[
\tau'(\varepsilon_A) \;\le\; \tanh\!\Big( \frac{\Delta(\varepsilon_A)}{4} \Big).
\]
\end{lemma}

\begin{proof}
If all entries of a positive matrix lie in $[\delta',U']$, then
\(\max_{i,j,k,\ell} \frac{B_{ik}B_{j\ell}}{B_{i\ell}B_{jk}} \le (U'/\delta')^2\).
The standard diameter bound is $\Delta(B)\le \log\!\big(\max \frac{B_{ik}B_{j\ell}}{B_{i\ell}B_{jk}}\big)$,
giving $\Delta\le 2\log(U'/\delta')$.
The Birkhoff bound then yields $\tau\le \tanh(\Delta/4)$.
\end{proof}

\subsection{Robust rate extraction under a recurrence certificate}

RU-C does \emph{not} assert recurrence. Instead, it states: once recurrence is certified (RU-D) or assumed as a named hypothesis,
the contraction rate survives perturbation with explicit degradation.

\begin{proposition}[Robust contraction per Doeblin occurrence]\label{prop:ru-c-robust-per-block}
Assume $\delta'(\varepsilon_A)>0$.
Whenever the Doeblin word $w$ occurs (so the block product applied at that time is $\tilde P_w$),
the Hilbert distance contracts by at least the factor $\tau'(\varepsilon_A)$ from Lemma~\ref{lem:ru-c-diameter}.
\end{proposition}

\begin{proof}
This is the Birkhoff contraction principle applied to the positive block $\tilde P_w$.
\end{proof}

\begin{corollary}[Rate degradation under syndetic recurrence]\label{cor:ru-c-robust-rate}
Assume a recurrence certificate provides a bounded-gap modulus $B$ for occurrences of $w$ along the trajectory under study.
Let $L=|w|$ and suppose $\delta'(\varepsilon_A)>0$.
Then for all times $n$,
\[
d_H(x_n,y_n)\le C\,(\tau'(\varepsilon_A))^{\lfloor n/(L+B)\rfloor}\,d_H(x_0,y_0),
\]
so an admissible exponential rate is
\[
\kappa'(\varepsilon_A)= -\frac{\log\tau'(\varepsilon_A)}{L+B}.
\]
\end{corollary}

\begin{proof}
Partition the trajectory into consecutive windows of length at most $L+B$,
each containing an occurrence of $w$.
Each window contributes at least one contraction by Proposition~\ref{prop:ru-c-robust-per-block};
absorbing the inter-block projective nonexpansive bounds into a prefactor yields the stated inequality.
\end{proof}

\subsection{Pipeline-facing fields added by RU-C}

RU-C requires the robust certificate to store enough information to compute:
\begin{itemize}
\item the surviving Doeblin margin function $\delta'(\varepsilon_A)$ and a computed threshold \texttt{eps\_doeblin} ensuring $\delta'(\varepsilon_A)>0$,
\item an upper envelope $U$ for the unperturbed Doeblin block and a formula for $U'(\varepsilon_A)$,
\item a diameter bound $\Delta(\varepsilon_A)$ and a contraction factor bound $\tau'(\varepsilon_A)$,
\item an optional \texttt{eps\_rate} threshold under which a target contraction quality is guaranteed
(e.g., $\delta'(\varepsilon_A)\ge \delta/2$ and $U'(\varepsilon_A)\le 2U$).
\end{itemize}

These enforce the hard acceptance gate:
\begin{center}
\emph{No statement may claim robust survival of a Doeblin witness or a rate without citing \texttt{doeblin.delta}, \texttt{doeblin.U\_bound}, and the computed thresholds \texttt{eps\_doeblin} and (when used) \texttt{eps\_rate}.}
\end{center}
