% CLOSURE_ROUTES_C.tex
%
% Purpose.
% This module logs three plausible routes for closing the remaining gap between
% (i) finite certificate outputs (tight core + Doeblin word existence) and
% (ii) a strongest theorem variant where contraction along minimizers is derived
% without smuggling a recurrence hypothesis.
%
% The goal of this module is NOT to claim these routes are already proved.
% It is to pin down the exact closure target and to prevent future drift.

\section{Closure routes and the chosen primary direction}
\label{sec:closure-routes}

\subsection{Closure target (what remains to be closed)}
\label{subsec:closure-target}

The pipeline developed so far yields a finite classification on the tight-core side:
for a certified aperiodic essential tight SCC $H$ (or witness of OG1), one can (i)
certify face-map consistency (or an OG2 witness), and (ii) certify existence of a
Doeblin word $w$ on the active face of $H$ (or an OG3 witness).

What is deliberately \emph{not} claimed is that an arbitrary mean-minimizing
trajectory must exhibit bounded-gap (or even positive-density) recurrence of $w$.
The remaining closure problem is therefore:

\begin{quote}
\emph{Closure problem.}  Under the same finite template hypotheses that produce
$H$ and $w$, strengthen the theory so that either
(a) recurrence of $w$ (in some quantitative sense) is \emph{derived} for minimizers,
leading to explicit Hilbert contraction rates, or else
(b) an additional obstruction condition is exposed (with a finite witness schema).
\end{quote}

The following three routes are logically distinct ways to close this problem.
We state them so that later theorem statements cannot accidentally assume
a recurrence input that has not been proved.

\subsection{Three closure routes}
\label{subsec:three-routes}

\paragraph{Route 1 (Primary): Optimality induces projective regularity.}
\label{par:route1}

This route seeks to derive recurrence/contraction \emph{on the minimizing set}
from a strengthened selection principle:

\begin{itemize}
\item Reduce to a recurrent aperiodic \emph{core} (tight SCC $H$) supporting a
calibrated structure (subaction / potential) that makes local optimality
compatible with tightness.
\item Use the certified Doeblin word $w$ to obtain a \emph{uniform projective
contraction block} on the active face (a Doeblin block implies a Birkhoff
contraction coefficient $\tau<1$ on that face).
\item Prove a \emph{return/selection principle}: along minimizers (or along
calibrated representatives of minimizing classes), one obtains a calibrated
projective section so that visits to the aperiodic core force the Doeblin block
to occur with a frequency sufficient for exponential contraction.
\end{itemize}

Conceptually, the route is: \emph{reduction} $\Rightarrow$ \emph{recurrent aperiodic
core} $\Rightarrow$ \emph{Doeblin block} $\Rightarrow$ \emph{uniform projective
contraction on the core} $\Rightarrow$ \emph{contraction along minimizers}
(via a selection/return mechanism).

This is the best fit for the current framework because it aligns with the
existing separation of (i) finite certification on the tight graph and (ii)
Hilbert metric contraction on an active face.  It also leverages the strongest
available structural principle: the minimizing set should exhibit regularity
in projective coordinates once optimality restricts dynamics to a recurrent
aperiodic core.

\paragraph{Route 2 (Secondary): Finite descriptor bound $+$ induced SFT/sofic boundary.}
\label{par:route2}

This route turns the tight core into a symbolic-dynamics object whose recurrence
properties can be attacked via finite-type arguments.

\begin{itemize}
\item Use the finite template structure and reduction to build an induced
symbolic system (SFT or sofic shift) describing admissible tight behaviors
on the boundary/minimizing set.
\item Prove that minimizers correspond to invariant measures or calibrated
orbits in this induced shift, with a \emph{finite descriptor bound} controlling
the complexity.
\item Derive recurrence properties (minimality, linear recurrence, or at least
uniform return bounds for key words) using symbolic dynamics tools.
\end{itemize}

This direction is highly compatible with the existing automata and SCC machinery.
It is the most promising fallback if Route~1 stalls, because it keeps the closure
problem within finite-state combinatorics rather than analytic slack estimates.

\paragraph{Route 3 (Tertiary/least-promising): Slack-forcing (avoidance costs slack).}
\label{par:route3}

This route tries to force recurrence by showing that avoiding the Doeblin word
$w$ for a long interval necessarily incurs a quantitative reduced-cost slack
penalty that contradicts mean-minimality.

\begin{itemize}
\item Prove a lower bound $g(L)$ such that if a trajectory avoids $w$ for length
$L$, then its accumulated reduced-cost slack is at least $g(L)$.
\item Show $g(L)$ grows fast enough (ideally linearly) to force bounded-gap
recurrence (or positive lower density) of $w$ along minimizers.
\end{itemize}

While conceptually clean, this route is technically risky in this framework:
it demands a structural mechanism that converts a \emph{combinatorial avoidance}
constraint into a \emph{uniform quantitative} slack penalty across all admissible
trajectories.  Without an additional strong hypothesis, one expects adversarial
patterns with sublinear slack accumulation, which is insufficient to force
bounded gaps.

\subsection{Decision: selected primary route and logged fallbacks}
\label{subsec:decision}

\begin{center}
\begin{tabular}{|l|p{0.62\linewidth}|}
\hline
\textbf{Route} & \textbf{Status and rationale}\\
\hline
Route 1: Optimality $\Rightarrow$ projective regularity & \textbf{Selected as primary.}
It uses the strongest structural hypothesis already implicit in the project:
optimality collapses degrees of freedom on a recurrent aperiodic core,
producing uniform projective contraction blocks and a return/selection principle
on the minimizing set.\\
\hline
Route 2: Finite descriptor $+$ induced SFT/sofic boundary & \textbf{Logged as secondary fallback.}
Most compatible with the existing finite-state certification infrastructure
(automata, SCC decomposition).  Pursue if the projective selection mechanism
in Route~1 does not yield a clean unconditional theorem.\\
\hline
Route 3: Slack-forcing (avoidance costs slack) & \textbf{Logged as tertiary/least promising.}
May require extra assumptions to make $g(L)$ robust.  Keep only as a controlled
attempt to isolate an explicit missing property if Routes~1--2 do not close.\\
\hline
\end{tabular}
\end{center}

\paragraph{What is proved vs. what is open (after this decision).}
All results already in the manuscript remain valid: OG1--OG3 are finite,
witness-producing; Doeblin implies explicit projective diameter contraction
on the active face; and any \emph{assumed} recurrence input (syndetic or
lower-density) yields explicit exponential contraction.

The open closure work is now pinned to a concrete target:
\emph{implement Route~1 to upgrade contraction from a recurrence-input theorem
into a derived property on the minimizing set}, or else isolate a new obstruction
with a finite witness schema.

