% RECURRENCE_ROBUSTNESS_RU_D.tex
\section{Recurrence Robustness Layer (Robust Mode)}\label{sec:ru-d}

The gap-free upgrade separates \emph{existence} of an algebraically contracting block (a Doeblin word on the active face)
from \emph{recurrence} of that block along trajectories.
A correct finite recurrence layer is obtained by separating two objects:
\textbf{(RC1)} is a purely combinatorial predicate for tight trajectories in a fixed tight SCC $H$,
while OG4$(H,w)$ records the explicit failure mode via an avoidance-cycle witness.

This section enforces that separation in a robust, checkable way.
All recurrence statements are permitted only in one of two modes, each with an explicit robustness threshold.

\subsection{Two recurrence modes with margins}

Fix an aperiodic tight SCC $H$ of the tight graph $G^{(0)}$ and a certified Doeblin word $w$ supported in $H$.

\begin{definition}[Recurrence modes for a robust certificate]\label{def:ru-recurrence-modes}
A \emph{robust certificate} records recurrence information in exactly one of the following modes.
\begin{itemize}[leftmargin=2em]
\item \textbf{Mode (combinatorial).}
The certificate records that \textbf{(RC1)} holds for $(H,w)$ (Definition~\ref{def:rc1}),
with an explicit syndetic modulus $L_{\mathrm{syn}}(H,w)$ (Lemma~\ref{lem:avoidance-syndetic}).
Equivalently, the avoidance graph $\mathcal{A}_{\neg w}(H)$ is acyclic.
\item \textbf{Mode (hypothesis).}
The certificate records a user-supplied recurrence hypothesis on a \emph{chosen} minimizing trajectory,
for example a bounded-gap hypothesis (gap bound $B$) or a lower-density hypothesis (density margin $\eta>0$).
In this mode, the pipeline does \emph{not} certify the truth of the hypothesis; it only certifies that
all constants depending on $w$ and the template family degrade explicitly and continuously under perturbation.
\end{itemize}
\end{definition}

\subsection{Robustness threshold for recurrence statements}

The point of robust mode is that the same finite witness objects remain valid on an explicit neighborhood.
The neighborhood size is computed from earlier margins.

\begin{definition}[Recurrence robustness radius]\label{def:eps-recurrence}
Let $\varepsilon_{\mathrm{tight}}$ be the tight-core stability threshold from RU-B (Theorem in Section~\ref{sec:ru-b-tight-core-stability})
and let $\varepsilon_{\mathrm{Doeblin}}$ be the Doeblin-survival threshold from RU-C (Section~\ref{sec:ru-c-doeblin-stability}).
Define
\[
\varepsilon_{\mathrm{rec}}:=\min\{\varepsilon_{\mathrm{tight}},\,\varepsilon_{\mathrm{Doeblin}}\}.
\]
A recurrence statement is said to be \emph{robustly certified} on the neighborhood of size $\varepsilon_{\mathrm{rec}}$
if it is in Mode (combinatorial) and its finite witness objects are unchanged throughout that neighborhood.
\end{definition}

\begin{proposition}[Stability of the RC1/OG4 dichotomy]\label{prop:ru-rc1-stable}
Assume $\varepsilon\le \varepsilon_{\mathrm{tight}}$.
Then the tight graph $G^{(0)}$ and its SCC decomposition are unchanged under the perturbation model,
so the avoidance graph $\mathcal{A}_{\neg w}(H)$ is unchanged as a finite directed graph.
Consequently, the dichotomy
\[\text{(RC1) holds for $(H,w)$}\quad\text{or}\quad\text{OG4$(H,w)$ holds}\]
(and any explicit witness: either $L_{\mathrm{syn}}$ or an avoidance cycle) is invariant under all such perturbations.
\end{proposition}

\begin{proof}
Under $\varepsilon\le \varepsilon_{\mathrm{tight}}$, RU-B guarantees that the tight-edge set and hence the tight SCC $H$ are unchanged.
The avoidance graph $\mathcal{A}_{\neg w}(H)$ is constructed purely from the tight edge set of $H$ and the fixed word $w$
(Definition~\ref{def:avoidance-graph}), so its vertex/edge set is unchanged.
Therefore acyclicity (RC1) and the existence of an avoidance cycle (OG4) are unchanged, and any explicit witness carries over.
\end{proof}

\begin{remark}[Why $\varepsilon_{\mathrm{rec}}$ uses $\varepsilon_{\mathrm{Doeblin}}$]
Proposition~\ref{prop:ru-rc1-stable} is purely combinatorial and only needs $\varepsilon_{\mathrm{tight}}$.
However, in the contraction pipeline, recurrence is meaningful only in tandem with the Doeblin minorization property of $w$.
Thus robust certificates record $\varepsilon_{\mathrm{rec}}=\min\{\varepsilon_{\mathrm{tight}},\varepsilon_{\mathrm{Doeblin}}\}$
so that ``recurrence + Doeblin'' survives together.
\end{remark}

\subsection{What robust mode does and does not claim}

\begin{itemize}[leftmargin=2em]
\item In Mode (combinatorial), recurrence is \emph{derived} for \emph{all} tight trajectories supported in the same SCC $H$,
with an explicit gap bound $L_{\mathrm{syn}}$.
This derived recurrence is stable for all perturbations with $\varepsilon\le \varepsilon_{\mathrm{rec}}$.
\item In Mode (hypothesis), recurrence is \emph{not} asserted without the stated hypothesis.
Robustness refers only to the \emph{constants} in the contraction estimate (diameter bounds, Birkhoff factor, and rate degradation)
when the same hypothesis is assumed along the perturbed instance.
\end{itemize}

\paragraph{Enforcement rule.}
No theorem, corollary, or remark may state a recurrence conclusion unless it cites
\emph{either} a Mode (combinatorial) certificate field (with $L_{\mathrm{syn}}$ and $\varepsilon_{\mathrm{rec}}$)
\emph{or} a Mode (hypothesis) field (with its explicit margin and the disclaimer that the hypothesis is assumed).
