% PIPELINE_CONTRACT_N.tex
\section{Pipeline Contract (recurrence derived, finite-witness branches)}\label{sec:pipeline-contract-n}

This section states the end-to-end contract of the gap-free pipeline once the finite recurrence layer and the rate extraction layer are included.
The key upgrade is that \emph{recurrence is no longer an external hypothesis}.
Instead, the pipeline returns either a \emph{finite obstruction witness} (OG1--OG4) or an \emph{explicit contraction certificate with rate}.

\subsection{Inputs and canonical objects}

The pipeline assumes the standard reduced-graph data already used by the obstruction checks:

\begin{itemize}
\item a finite reduced directed graph $G=(V,E)$,
\item for each edge $e:u\to v$ a nonnegative template (or fixed-length block) $T_e\in\mathbb{R}^{d\times d}_+$,
\item the calibrated potential $\phi$ and reduced costs $c_\phi(e)\ge 0$ (so the tight edge set $E^{(0)}=\{c_\phi=0\}$ is defined),
\item the canonical support sets $S(v)\subseteq\{1,\dots,d\}$ per descriptor state.
\end{itemize}

From these, the pipeline constructs:

\begin{itemize}
\item the tight graph $G^{(0)}=(V^{(0)},E^{(0)})$,
\item SCC decomposition and periods $\mathrm{per}(H)$,
\item face-map validation and SCC face $S(H)=\bigcap_{v\in H}S(v)$ when consistent (Definition~\ref{def:face-map}),
\item the coordinate digraph $\Gamma(H)$ on $S(H)$ (Definition~\ref{def:coord-digraph}),
\item the Boolean-pattern product automaton for Doeblin-word search (Algorithm~\ref{alg:doeblinword}),
\item for a returned Doeblin word $w$, the avoidance graph $\mathcal{A}_{\neg w}(H)$ and (if acyclic) the modulus $L_{\mathrm{syn}}(H,w)$ (Algorithm~\ref{alg:syndetic}).
\end{itemize}

\subsection{Branch logic and output artifacts}

The pipeline returns exactly one of the following mutually exclusive outcomes.

\begin{itemize}
\item[\textbf{(OG1 witness)}] No aperiodic SCC exists in $G^{(0)}$.
Return the SCC/period table certifying that every tight SCC is imprimitive.

\item[\textbf{(OG2 witness)}] An aperiodic tight SCC $H$ exists, but the proposed face sets fail to form a face map on $H$.
Return the inconsistency cycle witness from Lemma~\ref{lem:og2-witness-cycle}.

\item[\textbf{(OG3 witness)}] An aperiodic tight SCC $H$ with a valid face map exists, but no Doeblin word exists on $S(H)$.
Return either:
\begin{itemize}
\item a cut witness $(H,S(H),A,B)$ from Lemma~\ref{lem:og3-witness-cut}, or
\item a product-automaton non-reachability certificate (Corollary~\ref{cor:doeblin-from-cert}).
\end{itemize}

\item[\textbf{(OG4 witness)}] A Doeblin word $w$ on $S(H)$ is certified, but RC1 fails for $(H,w)$.
Return a directed cycle in $\mathcal{A}_{\neg w}(H)$, which yields an explicit periodic tight minimizer avoiding $w$
(Proposition~\ref{prop:og4-witness}).

\item[\textbf{(Contraction certificate)}] A Doeblin word $w$ on $S(H)$ is certified and RC1 holds for $(H,w)$.
Return:
\begin{itemize}
\item the witness word $w$ and the modulus $L_{\mathrm{syn}}(H,w)$,
\item the FTG window parameters $(\varepsilon,U)$ for the Doeblin block on $S(H)$,
\item the Hilbert contraction factor $\tau$ and the exponential rate parameters $(C,\kappa)$ as in Theorem~\ref{thm:unconditional-tight}.
\end{itemize}
This certificate implies uniform exponential projective contraction along every tight minimizer supported in $H$.
\end{itemize}

\subsection{How general minimizers are handled}

For a general one-sided minimizing path $\pi$, the tight edges need not appear with bounded gaps.
The pipeline therefore records the correct forcing upgrade:

\begin{itemize}
\item if $\pi$ has nontrivial tight time in the chosen SCC $H$ (measured by $\theta=\liminf T_H(n;\pi)/n>0$) and slack density $0$,
then the forcing lemma yields a \emph{positive lower density} of Doeblin blocks (Corollary~\ref{cor:forced-positive-density}),
\item this density supplies the recurrence input needed by the contraction module, yielding an explicit rate
(Theorem~\ref{thm:unconditional-mean} and the formulas packaged in Section~\ref{sec:rate-pack-l}).
\end{itemize}

Accordingly, the pipeline contract records two rate modes:

\begin{itemize}
\item a \emph{tight-minimizer} rate $(C,\kappa)$ that depends only on finite witnesses $(H,w)$ and the FTG bounds,
\item a \emph{mean-minimizer} rate that additionally depends on the measurable parameter $\theta$ of the specific trajectory.
\end{itemize}

\begin{remark}[What is now fully finite]
All branching decisions up through OG4 and the tight-minimizer contraction certificate are entirely finite and witness-producing.
The only non-finite input that can remain in applications is an estimate of how much time a given minimizer spends in the SCC $H$
(the parameter $\theta$).
This is the correct remaining interface: it is a property of the chosen trajectory, not a hidden structural assumption.
\end{remark}
