% MINIMALITY_ROUTE_F.tex
\section{Minimality Route: what can and cannot be forced on the tight core}\label{sec:minimality-route}

This section clarifies the ``minimality'' upgrade route for the minimizing dynamics on the tight core.
The conclusion is sharp and finite-witness: because the boundary carrier $\Sigma_{\mathcal{A}}$ is a shift of finite type,
true topological minimality (and hence linear recurrence) occurs only in the purely periodic case.
Accordingly, minimality is not an automatic consequence of calibration or tightness;
the correct replacement for gap-control is the \emph{finite} recurrence certificate layer (RC1) and the slack-forcing route developed later.

\subsection{Minimal SFTs are periodic}

We record the elementary structural fact that blocks the naive minimality upgrade on SFT carriers.

\begin{definition}[Minimality]
A two-sided subshift $(X,\sigma)$ is \emph{minimal} if every orbit is dense in $X$.
Equivalently, $X$ has no proper nonempty closed $\sigma$-invariant subsets.
\end{definition}

\begin{lemma}[A minimal edge shift is a single cycle]\label{lem:minimal-sft-periodic}
Let $H$ be a finite directed graph and let $X_H$ be its two-sided edge shift.
If $(X_H,\sigma)$ is minimal, then $X_H$ consists of a single periodic orbit.
Equivalently, $H$ is a directed cycle (every vertex has in-degree and out-degree equal to $1$ in the active component).
\end{lemma}

\begin{proof}
If $X_H$ is nonempty, then $H$ contains a directed cycle, hence $X_H$ contains a periodic point $\omega$ obtained by repeating that cycle.
The orbit closure of a periodic point is finite.
If $X_H$ were minimal, the orbit closure of $\omega$ would have to equal all of $X_H$.
Therefore $X_H$ is finite.
An edge shift over a finite graph is finite if and only if the underlying active component is a directed cycle,
because any branching (a vertex with two distinct outgoing edges or two distinct incoming edges) produces at least two distinct infinite paths.
Thus $H$ must be a directed cycle and $X_H$ is a single periodic orbit.
\end{proof}

\subsection{Implication for the minimizing carrier}

Recall that the boundary carrier $\Sigma_{\mathcal{A}}$ defined in Definition~\ref{def:boundary-subshift} is the two-sided edge shift of the finite graph $G^{\mathcal{A}}$.
Hence it is a shift of finite type.

\begin{proposition}[Minimality on $\Sigma_{\mathcal{A}}$ is a periodic obstruction]\label{prop:minimality-obstruction}
If the boundary carrier $(\Sigma_{\mathcal{A}},\sigma)$ is minimal, then every SCC of $G^{\mathcal{A}}$ is a directed cycle.
In particular, the only minimal regime for the tight-core carrier is the purely periodic regime.

Conversely, if a tight SCC $H\subseteq G^{\mathcal{A}}$ is a directed cycle, then its edge shift is minimal and every tight trajectory supported in $H$ is periodic.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:minimal-sft-periodic} to each nonempty irreducible component.
If $\Sigma_{\mathcal{A}}$ is minimal and nonempty, it cannot decompose into multiple invariant components,
so it coincides with the shift of a single SCC.
That SCC must be a directed cycle, and the conclusions follow.
The converse direction is immediate for a directed cycle.
\end{proof}

\subsection{The exact missing property and the correct replacement}

The previous proposition isolates the precise missing property:
without an explicit \emph{finite} obstruction witness forcing the tight SCC to be a directed cycle,
minimality (and linear recurrence) cannot be deduced from calibration alone.
In particular, tightness is compatible with:

\begin{itemize}
\item multiple tight cycles in the same tight SCC,
\item branching tight choices (out-degree $\ge 2$ along tight edges),
\item aperiodic tight SCCs that are topologically transitive but far from minimal.
\end{itemize}

These shapes are not pathologies; they occur naturally and are consistent with the obstruction taxonomy OG1--OG3.
Therefore, the minimality route is not the right vehicle for gap-free recurrence on minimizers.

\begin{remark}[Finite recurrence certificate as the correct ``minimality substitute'']\label{rem:minimality-substitute}
What is actually used in the contraction upgrade is not minimality of the entire tight SCC,
but a finite return-control statement for a \emph{specific certified Doeblin word} $w$.
This is exactly the role of the recurrence certificate layer (RC1):
for a fixed tight SCC $H$ and a candidate word $w$ supported in $H$, one either
\begin{itemize}
\item produces an explicit bounded-gap modulus $L_{\mathrm{syn}}$ guaranteeing syndetic returns of $w$ along \emph{every} infinite tight path in $H$, or
\item returns a finite avoidance-cycle witness showing that some periodic tight path avoids $w$ entirely.
\end{itemize}
This finite dichotomy is the sharp replacement for minimality in the gap-free regime.
\end{remark}
