% APPROX_CERTIFICATION_RU_E.tex
\section{Approximate Certification on Parameter Regions (Robust Neighborhood Cover)}\label{sec:ru-e}

This section upgrades pointwise certificates to \emph{region certification}.
The guiding observation is topological and quantitative:
all certificate predicates used by the pipeline are \emph{open} conditions once the robust margins are recorded.
Therefore a finite verification on an $\varepsilon$-net certifies an entire parameter region
except near an explicit exceptional set where margins vanish.

\subsection{Local openness packaged as a single statement}

Fix a parameter instance $\theta$ (costs and templates) and assume the pipeline produces a robust certificate object
with robustness radii $\varepsilon_{\mathrm{tight}},\varepsilon_{\mathrm{Doeblin}},\varepsilon_{\mathrm{rec}}$ and
\(\varepsilon_* = \min\{\varepsilon_{\mathrm{tight}},\varepsilon_{\mathrm{Doeblin}},\varepsilon_{\mathrm{rate}},\varepsilon_{\mathrm{rec}}\}\)
as in Sections~\ref{sec:robust-certificate-ru-a}--\ref{sec:ru-d}.

\begin{theorem}[Robust neighborhood invariance]\label{thm:ru-neighborhood-invariance}
Let $\theta$ admit a robust certificate with radius $\varepsilon_* > 0$.
Then every perturbed instance $\theta'$ with perturbation size $\le \varepsilon_*$ satisfies:
\begin{itemize}[leftmargin=2em]
\item the tight graph $G^{(0)}$ and its SCC/period decomposition are unchanged;
\item the stored Doeblin word $w$ remains a valid Doeblin witness on the same active face, with explicit margin $\delta'(\varepsilon_*)>0$;
\item the recurrence layer is unchanged in Mode (combinatorial) (same RC1/OG4 outcome and witnesses),
or remains a stated hypothesis in Mode (hypothesis);
\item the projective contraction constants degrade explicitly according to the formulas stored in the certificate.
\end{itemize}
In particular, every theorem in this paper that is invoked through certificate fields at $\theta$
holds verbatim at $\theta'$ with constants replaced by the certificate's degraded constants.
\end{theorem}

\begin{proof}
The tight-core invariance is RU-B.
Doeblin survival and the explicit $\delta'(\varepsilon)$ / $\tau'(\varepsilon)$ bounds are RU-C.
Recurrence-layer stability in Mode (combinatorial) is RU-D.
All remaining constants are explicit functions of these margins and the FTG bounds, hence are stable under the same inequalities.
\end{proof}

\subsection{Finite net certification of a compact region}

Let $(\Omega,d)$ be a compact parameter set with a product metric compatible with the perturbation model
(e.g. $d(\theta,\theta') = \max\{\|\Delta c\|_\infty,\|\Delta A\|_{\infty,\text{entry}}\}$ in absolute-entrywise mode).

\begin{definition}[$\varepsilon$-net]\label{def:eps-net}
A finite set $\mathcal{N}\subset \Omega$ is an $\varepsilon$-net if every $\theta\in\Omega$ lies within distance $\varepsilon$ of some $\theta_i\in\mathcal{N}$.
\end{definition}

\begin{theorem}[Region certification by robust balls]\label{thm:ru-region-certification}
Suppose that for each net point $\theta_i\in\mathcal{N}$ the pipeline produces a robust certificate with radius $\varepsilon_*(\theta_i)>0$.
Define the certified region
\[
\Omega_{\mathrm{cert}}:=\bigcup_{\theta_i\in\mathcal{N}} B\big(\theta_i,\varepsilon_*(\theta_i)\big)\cap \Omega.
\]
Then every instance $\theta\in\Omega_{\mathrm{cert}}$ inherits the same certified conclusions as its center $\theta_i$
(in the sense of Theorem~\ref{thm:ru-neighborhood-invariance}).
Moreover, if $\mathcal{N}$ is an $\varepsilon$-net and $\min_i \varepsilon_*(\theta_i)\ge \varepsilon$,
then $\Omega_{\mathrm{cert}}=\Omega$.
\end{theorem}

\begin{proof}
If $\theta\in\Omega_{\mathrm{cert}}$ then $\theta\in B(\theta_i,\varepsilon_*(\theta_i))$ for some center $\theta_i$.
Apply Theorem~\ref{thm:ru-neighborhood-invariance} at $\theta_i$.
The final claim follows directly from the definition of an $\varepsilon$-net.
\end{proof}

\subsection{Exceptional set description}

The only ways robustness can fail are the explicit margin-boundary events where $\varepsilon_*=0$.
Accordingly, region mode reports uncovered points as being near one of the following certificate-boundary loci:
\begin{itemize}[leftmargin=2em]
\item \textbf{Tightness degeneracy:} $\gamma=0$ (tight/non-tight separation vanishes), so the tight graph can change.
\item \textbf{Doeblin degeneracy:} $\delta=0$ for the stored word (or no Doeblin word exists), so minorization can fail.
\item \textbf{Recurrence boundary:} the RC1/OG4 dichotomy changes only when the tight graph changes (hence is subsumed by $\gamma=0$),
while Mode (hypothesis) has no derived recurrence boundary because recurrence is not asserted.
\end{itemize}

\subsection{Pipeline: region mode outputs}

In region mode, the pipeline:
\begin{itemize}[leftmargin=2em]
\item samples or constructs a finite net $\mathcal{N}$ of parameter points;
\item runs the pointwise certificate pipeline at each $\theta_i$;
\item outputs the list of certified balls $\big(\theta_i,\varepsilon_*(\theta_i)\big)$ and their conclusions;
\item reports uncovered points and diagnoses them in terms of the margin boundaries above.
\end{itemize}

A recommended machine-readable artifact is a \texttt{robust\_region\_report.json} containing:
centers with their radii $\varepsilon_*$, the induced outcome labels (OG1--OG4 or contraction), and an uncovered-set diagnostic summary.
