% UNCONDITIONAL_CONTRACTION_K.tex
\section{Contraction from a Certified Doeblin Word: the Finite Recurrence Branch}\label{sec:unconditional-contraction-k}

Once the finite obstruction taxonomy produces a certified Doeblin word on an aperiodic tight SCC, there is only one further way
for ``Doeblin $\Rightarrow$ contraction'' to fail on \emph{all} tight trajectories: the Doeblin word might be avoidable.
This is exactly OG4, witnessed by a directed cycle in the avoidance graph.
When OG4 does not occur, the Doeblin word is \emph{unavoidable} on the induced subshift over the SCC, hence appears with bounded gaps on every tight path,
and uniform projective contraction follows with fully explicit constants.

We also record a second, quantitative route: along a general minimizing path, bounded gaps need not hold globally because tightness holds only in density.
Nevertheless, if the minimizer spends positive lower density of tight time in the aperiodic SCC, then unavoidable recurrence forces a uniform positive lower density
of Doeblin blocks, which is enough for an explicit exponential contraction rate.


\subsection{Hilbert metric and Birkhoff contraction}

\begin{definition}[Hilbert (projective) metric]
For $x,y\in\mathbb{R}^S_{>0}$, define
\[
d_H(x,y):=\log\Bigl(\max_{i\in S}\frac{x_i}{y_i}\Bigr)+\log\Bigl(\max_{i\in S}\frac{y_i}{x_i}\Bigr)
=\log\Bigl(\max_{i,j\in S}\frac{x_i y_j}{x_j y_i}\Bigr).
\]
This metric is invariant under positive scalar rescaling: $d_H(cx,dy)=d_H(x,y)$ for all $c,d>0$.
\end{definition}

\begin{lemma}[Birkhoff contraction from entry bounds]\label{lem:birkhoff}
Let $A\in\mathbb{R}^{S\times S}_{>0}$ satisfy $\min_{i,j\in S}A_{ij}\ge \varepsilon$ and $\max_{i,j\in S}A_{ij}\le U$.
Set
\[
\Delta:=2\log\frac{U}{\varepsilon},\qquad \tau:=\tanh\Bigl(\frac{\Delta}{4}\Bigr)=\tanh\Bigl(\frac12\log\frac{U}{\varepsilon}\Bigr)\in(0,1).
\]
Then for all $x,y\in\mathbb{R}^S_{>0}$,
\[
d_H(Ax,Ay)\le \tau\, d_H(x,y).
\]
\end{lemma}

\begin{remark}
The constant $\tau$ is the standard Birkhoff contraction coefficient corresponding to the projective diameter bound $\Delta$ induced by entry ratio control; see Birkhoff \cite{birkhoff_1957} and modern treatments such as Bushell \cite{bushell_1973} or Lemmens--Nussbaum \cite{lemmens_nussbaum_2012}.
The certificate pipeline will provide explicit values of $(\varepsilon,U)$ for Doeblin words.
\end{remark}

\begin{definition}[Recurrence input for a Doeblin word]\label{def:rec-input}
Fix an infinite tight path $\pi$ in a tight SCC $H$ and a Doeblin word $w$ of length $|w|$ on $H$.
Let $N_w(n;\pi)$ denote the number of occurrences of $w$ starting among the first $n$ edges of $\pi$.
Let $N_w^{\mathrm{dis}}(n;\pi)$ denote the maximal number of pairwise disjoint occurrences of $w$
whose starting indices lie in $\{0,\dots,n-1\}$; equivalently, the greedy selection rule yields
$N_w^{\mathrm{dis}}(n;\pi)\ge \lfloor N_w(n;\pi)/|w|\rfloor$.
We say that $\pi$ satisfies one of the following recurrence inputs.
\begin{itemize}[leftmargin=2em]
\item \textbf{(RH1) Syndetic recurrence:} there exists $L_0\in\mathbb{N}$ such that every length-$L_0$ subword of $\pi$ contains an occurrence of $w$.
\item \textbf{(RH2) Positive lower density:} there exists $\rho>0$ such that $\liminf_{n\to\infty} \frac{N_w(n;\pi)}{n}\ge \rho$.
\end{itemize}
\end{definition}

\begin{corollary}[Exponential projective contraction from Doeblin blocks]\label{cor:gap-free-regularity}
Let $w$ be a Doeblin word on a face $S$ whose face-restricted product $B(w)|_S$ has entry bounds $(\varepsilon,U)$.
Let $\tau\in(0,1)$ be the corresponding Birkhoff coefficient from Lemma~\ref{lem:birkhoff}.
Then along any tight path $\pi$ in $H$, the Hilbert distance between two positive face vectors transported by the tight product contracts exponentially provided $\pi$ satisfies either recurrence input from Definition~\ref{def:rec-input}.
More precisely:
\begin{itemize}[leftmargin=2em]
\item Under \textbf{(RH1)} with modulus $L_0$, one has $d_H(x_n,y_n)\le \tau^{\lfloor n/L_0\rfloor}\, d_H(x_0,y_0)$ and hence $d_H(x_n,y_n)\le C e^{-\kappa n} d_H(x_0,y_0)$ with $\kappa:=\frac{1}{L_0}(-\log\tau)$ and $C:=\tau^{-1}$.
\item Under \textbf{(RH2)} with density $\rho$, let $N_w^{\mathrm{dis}}(n;\pi)$ be as in Definition~\ref{def:rec-input}.
Then $d_H(x_n,y_n)\le \tau^{N_w^{\mathrm{dis}}(n;\pi)} d_H(x_0,y_0)$.
Moreover, extracting disjoint occurrences by the greedy rule gives $N_w^{\mathrm{dis}}(n;\pi)\ge \lfloor N_w(n;\pi)/|w|\rfloor$, so the density bound implies an explicit exponential form
$d_H(x_n,y_n)\le C e^{-\kappa n} d_H(x_0,y_0)$ with $\kappa:=\frac{\rho}{|w|}(-\log\tau)$ and $C:=\tau^{-1}$.
\end{itemize}
\end{corollary}

\subsection{A tight-path version (uniform on the induced subshift over the tight SCC)}

\begin{theorem}[Finite recurrence dichotomy and contraction on tight paths]\label{thm:unconditional-tight}
Fix an aperiodic tight SCC $H$ of $G^{(0)}$ and assume OG2 and OG3 do not occur on $H$, so that a consistent active face $S(H)$ exists and
an explicit Doeblin word $w$ supported in $H$ is certified with $B(w)$ strictly positive on $S(H)\times S(H)$.

Then exactly one of the following holds.
\begin{itemize}
\item[(OG4)] The avoidance graph $\mathcal{A}_{\neg w}(H)$ contains a directed cycle.
Equivalently, there exists a periodic tight path in $H$ that avoids $w$ entirely (Proposition~\ref{prop:og4-witness}).

\item[(\textbf{UC})] The avoidance graph $\mathcal{A}_{\neg w}(H)$ is acyclic.
Then every infinite tight path in $H$ contains $w$ with bounded gaps, with explicit gap modulus
\[
L_{\mathrm{syn}}:=L_{\mathrm{syn}}(H,w)\le |V(H)|\cdot |w|
\]
(Lemma~\ref{lem:avoidance-syndetic}).
Consequently, for every tight path $\pi$ supported in $H$ and all $x,y\in\mathbb{R}^{S(H)}_{>0}$,
\[
d_H\big(A^{(n)}(\pi)x,\,A^{(n)}(\pi)y\big)\ \le\ C\,e^{-\kappa n}\, d_H(x,y)\qquad(n\ge 1),
\]
with explicit constants
\[
\begin{aligned}
\varepsilon &:= m_*^{|w|}, &
U &:= (dM_*)^{|w|},\\
\tau &:= \tanh\!\Big(\tfrac12\log(U/\varepsilon)\Big)\in(0,1), &
C &:= \tau^{-1},\\
\kappa &:= \frac{-\log\tau}{L_{\mathrm{syn}}+|w|}. &&
\end{aligned}
\]
\end{itemize}
\end{theorem}

\begin{proof}
If $\mathcal{A}_{\neg w}(H)$ contains a directed cycle, OG4 holds by Definition~\ref{def:og4}.
If $\mathcal{A}_{\neg w}(H)$ is acyclic, Lemma~\ref{lem:avoidance-syndetic} yields syndetic recurrence of $w$ on \emph{every} infinite path in $H$ with gap bound $L_{\mathrm{syn}}$.
Since $B(w)$ is strictly positive on $S(H)\times S(H)$, the FTG bounds imply the window $\varepsilon\le B(w)_{ij}\le U$ on $S(H)\times S(H)$, hence the Birkhoff/Hilbert contraction factor is at most $\tau$.
Syndeticity implies at least $\lfloor n/(L_{\mathrm{syn}}+|w|)\rfloor$ contraction events by time $n$, so
\(d_H(\cdot)\le \tau^{\lfloor n/(L_{\mathrm{syn}}+|w|)\rfloor}d_H(x,y)\),
and the exponential form follows as in Corollary~\ref{cor:gap-free-regularity}.
\end{proof}

\subsection{A mean-minimizer version (forced frequency from positive core time)}

The preceding theorem is uniform on the induced subshift over $H$.
For a general minimizing path, tightness holds only in density, so bounded gaps for $w$ need not hold on the full path.
However, if a minimizing path spends positive lower density of (tight) time in the aperiodic SCC $H$,
then the finite recurrence modulus $L_{\mathrm{syn}}$ forces \emph{positive lower density} of Doeblin blocks along that minimizer,
which is sufficient for an explicit exponential contraction rate.

\begin{theorem}[Contraction along minimizers in the Doeblin regime]\label{thm:unconditional-mean}
Fix an aperiodic tight SCC $H$ and a certified Doeblin word $w$ on $S(H)$ as in Theorem~\ref{thm:unconditional-tight}.
Assume OG4 does not occur for $(H,w)$, so $\mathcal{A}_{\neg w}(H)$ is acyclic and $L_{\mathrm{syn}}(H,w)$ is defined.
Let $L_0:=L_{\mathrm{syn}}(H,w)+|w|$.

Let $\pi$ be any one-sided minimizing path such that
\[
\theta:=\liminf_{n\to\infty}\frac{T_H(n;\pi)}{n}\ >\ 0,
\]
where $T_H(n;\pi)$ counts times $t<n$ for which $e_t$ is tight and its tail vertex lies in $H$ (cf.\ Proposition~\ref{prop:tight-dichotomy}).
Then the Doeblin word $w$ occurs along $\pi$ with uniform positive lower density
\[
\rho\ :=\ \liminf_{n\to\infty}\frac{N_w(n;\pi)}{n}\ \ge\ \frac{\theta}{L_0}
\]
(Corollary~\ref{cor:forced-positive-density}).
Consequently, for all $x,y\in\mathbb{R}^{S(H)}_{>0}$,
\[
d_H\big(A^{(n)}(\pi)x,\,A^{(n)}(\pi)y\big)\ \le\ C\,e^{-\kappa n}\, d_H(x,y)\qquad(n\ge 1),
\]
with explicit constants
\[
\begin{aligned}
\varepsilon &:= m_*^{|w|}, &
U &:= (dM_*)^{|w|},\\
\tau &:= \tanh\!\Big(\tfrac12\log(U/\varepsilon)\Big), &
C &:= \tau^{-1},\\
\kappa &:= \frac{\rho}{|w|}(-\log\tau)\ \ge\ \frac{\theta}{L_0|w|}(-\log\tau). &&
\end{aligned}
\]
\end{theorem}

\begin{proof}
Because OG4 does not occur, \textbf{(RC1)} holds for $(H,w)$ and Corollary~\ref{cor:forced-positive-density} yields $\rho\ge \theta/L_0$.
Positive lower density of Doeblin blocks is precisely recurrence input (RH2) from Definition~\ref{def:rec-input}.
Apply Corollary~\ref{cor:gap-free-regularity} with this $\rho$.
\end{proof}

\begin{remark}[What is ``gap-free'' here]
Theorems~\ref{thm:unconditional-tight} and \ref{thm:unconditional-mean} remove recurrence as a \emph{hypothesis}:
the only alternative to contraction is the explicit finite-witness obstruction OG4.
For minimizing paths, the positive-density condition $\theta>0$ is guaranteed whenever an aperiodic tight SCC is selected in the tight-core dichotomy
(Proposition~\ref{prop:tight-dichotomy}).
\end{remark}
