% TIGHT_CORE_STABILITY_RU_B.tex
\section{Tight-Core Stability Theorem (RU-B)}\label{sec:ru-b-tight-core-stability}

This section upgrades the openness lemma of RU-A into a purely combinatorial invariance theorem:
under an explicit margin condition, the \emph{tight edge set}, its SCC decomposition, and its periodicity labels
are unchanged throughout a perturbation neighborhood.

\subsection{Stability statement for the tight-edge sign test}

Fix a calibrated pair $(u,\lambda_*)$ and reduced costs $c_u$ as in \eqref{eq:reduced-cost-u}.
Let $H:=\{e\in E: c_u(e)=0\}$ be the tight edge set and let
\[
\gamma := \min\{c_u(e): e\in E\setminus H\}\in[0,+\infty)
\]
(with the convention that $\gamma:=0$ when $H=E$).

\begin{lemma}[Tight-set invariance under cost perturbations]\label{lem:ru-b-tight-set-invariance}
Let $\tilde c=c+\Delta c$ with $\|\Delta c\|_\infty\le \varepsilon_c$ and keep $(u,\lambda_*)$ fixed.
If $\varepsilon_c<\gamma/2$, then
\[
\tilde H:=\{e\in E: \tilde c_u(e)\le \gamma/2\}=H,
\]
and the inequalities
\[
 e\in H \implies \tilde c_u(e)\le \gamma/2,
 \qquad
 e\notin H \implies \tilde c_u(e)\ge \gamma/2
\]
hold.
\end{lemma}

\begin{proof}
This is exactly the sign-stability conclusion of Lemma~\ref{lem:open-tightness}, with the explicit convention
that $\gamma=0$ forces the stable radius to be $0$ in the degenerate all-tight case.
\end{proof}

\subsection{Combinatorial invariance: SCC structure and periodicity labels}

Let $G^{(0)}=(V,E^{(0)})$ be the directed graph obtained by restricting $G=(V,E)$ to the tight edges
$E^{(0)}:=H$.
Write $\mathrm{SCC}(G^{(0)})$ for its SCC decomposition.
For an SCC $C$ we write $\mathrm{per}(C)$ for its period (the gcd of lengths of directed cycles in $C$).

\begin{theorem}[Tight-core structural stability]\label{thm:ru-b-tight-core-stability}
Assume $\gamma>0$ and let $\tilde c=c+\Delta c$ with $\|\Delta c\|_\infty\le \varepsilon_c<\gamma/2$.
Define the perturbed tight set by the stable sign test
\[
\tilde H:=\{e\in E: \tilde c_u(e)\le \gamma/2\}.
\]
Then:
\begin{itemize}
\item $\tilde H=H$ (tight edge set is unchanged),
\item the SCC decomposition of the tight graph is unchanged:
$\mathrm{SCC}(\tilde G^{(0)})=\mathrm{SCC}(G^{(0)})$,
\item every SCC period is unchanged: for each tight SCC $C$,
\[
\mathrm{per}_{\tilde G^{(0)}}(C)=\mathrm{per}_{G^{(0)}}(C),
\]
so the periodic/aperiodic labels of SCCs are invariant.
\end{itemize}
\end{theorem}

\begin{proof}
The first bullet is Lemma~\ref{lem:ru-b-tight-set-invariance}.
Once the edge set of the tight subgraph is identical, the directed graph itself is identical,
so the SCC decomposition and the cycle-length sets within each SCC are identical.
Hence the gcd defining the period is unchanged.
\end{proof}

\begin{remark}[What this theorem does and does not claim]\label{rem:ru-b-scope}
Theorem~\ref{thm:ru-b-tight-core-stability} is a \emph{certificate-level} stability statement.
It asserts that all downstream logic that depends only on the tight graph (SCCs, periods, face-map checks,
Doeblin search, avoidance graphs, and the OG1--OG4 branching) is robust throughout the neighborhood
$\|\Delta c\|_\infty<\gamma/2$, \emph{provided the certificate keeps $(u,\lambda_*)$ fixed}.

It does \emph{not} assert that the globally optimal calibrated pair for the perturbed instance is the same,
nor that the minimizing set for the perturbed instance coincides with the unperturbed one.
Those are separate questions; the robustness contract is that the \emph{original witness project} remains valid
and produces valid conclusions for all perturbations within the certified radius.
\end{remark}

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

RU-B adds the following required fields to the \texttt{tight\_core} section of the robust certificate:
\begin{itemize}
\item \texttt{tight\_lipschitz\_Kc}: a constant $K_c\ge 1$ such that
$|\tilde c_u(e)-c_u(e)|\le K_c\,\|\Delta c\|_\infty$ for all edges when $(u,\lambda_*)$ is kept fixed
(in the basic cost-perturbation model, $K_c=1$),
\item \texttt{eps\_tight}: the computed stability threshold
\[
\texttt{eps\_tight}=\frac{\gamma}{2K_c},
\]
so that $\|\Delta c\|_\infty<\texttt{eps\_tight}$ implies tight-core invariance,
\item an optional \texttt{tight\_graph\_hash} (a fingerprint of the tight edge set) and
\texttt{tight\_scc\_summary} (counts and period labels) for auditability.
\end{itemize}

These fields enforce the hard acceptance gate:
\begin{center}
\emph{Every later robustness claim about the tight core must cite \texttt{tight\_gap\_gamma}, \texttt{tight\_lipschitz\_Kc}, and \texttt{eps\_tight}.}
\end{center}
