% AUBRY_CORE_D.tex
\section{Aubry--Tight Core and the Induced Boundary Subshift}\label{sec:aubry-core}

This section pins down the precise meaning of \emph{core} used throughout the paper.
The definition is finite, canonical, and built from the normalized reduced costs.
It isolates the locally optimal transitions and the induced symbolic system that supports calibrated minimizers.

\subsection{Calibrated subactions and local optimality}

Fix the reduced graph $G=(V,E)$ with costs $c:E\to\mathbb{R}$ and the extremal mean $\lambda_*$.

\begin{definition}[Calibrated subaction]\label{def:calibrated-subaction}
A function $u:V\to\mathbb{R}$ is a \emph{calibrated subaction} (for $c$ at level $\lambda_*$) if for every edge $e\in E$,
\begin{equation}\label{eq:subaction-ineq}
 u(\mathrm{head}(e))\ \le\ u(\mathrm{tail}(e)) + c(e) - \lambda_*.
\end{equation}
Equivalently, the \emph{reduced cost}
\begin{equation}\label{eq:reduced-cost-u}
 c_u(e):= c(e)-\lambda_*+u(\mathrm{tail}(e))-u(\mathrm{head}(e))
\end{equation}
satisfies $c_u(e)\ge 0$ for all $e\in E$.
\end{definition}

\begin{remark}\label{rem:subaction-is-potential}
The potential $\phi$ produced by Theorem~\ref{thm:canonical-normalization} is a calibrated subaction in the sense of Definition~\ref{def:calibrated-subaction},
with $c_\phi=c_u$.
Conversely, any calibrated subaction yields a valid canonical normalization.
\end{remark}

\begin{lemma}[Tight edges are exactly locally optimal edges]\label{lem:tight-local-optimal}
Let $u$ be any calibrated subaction.
Then an edge $e\in E$ is tight (i.e. $c_u(e)=0$) if and only if equality holds in \eqref{eq:subaction-ineq}:
\[
 u(\mathrm{head}(e)) = u(\mathrm{tail}(e)) + c(e) - \lambda_*.
\]
In particular, the tight edge set $E^{(0)}=\{e: c_u(e)=0\}$ is exactly the set of one-step transitions that are locally optimal with respect to the calibrated subaction.
\end{lemma}

\begin{proof}
By definition, $c_u(e)=c(e)-\lambda_*+u(\mathrm{tail}(e))-u(\mathrm{head}(e))$.
The inequality \eqref{eq:subaction-ineq} is equivalent to $c_u(e)\ge 0$.
Thus $c_u(e)=0$ holds exactly when \eqref{eq:subaction-ineq} holds with equality.
\end{proof}

\subsection{Aubry--tight core}

Fix a calibrated subaction $u$ (for example $u=\phi$).
Let $G^{(0)}=(V^{(0)},E^{(0)})$ denote the tight subgraph, i.e. the subgraph of locally optimal edges.

\begin{definition}[Aubry--tight core]\label{def:aubry-core}
The \emph{Aubry--tight core} (or simply \emph{the core}) is the union of the strongly connected components of $G^{(0)}$ that contain at least one directed cycle.
Equivalently, it is the set of vertices that lie on some tight directed cycle.
We denote this vertex set by $\mathcal{A}\subseteq V^{(0)}$ and write $G^{\mathcal{A}}$ for the induced subgraph of $G^{(0)}$ on $\mathcal{A}$.
\end{definition}

\begin{remark}\label{rem:core-terminology}
Throughout the paper, when we refer to a \emph{tight core}, \emph{core SCC}, or \emph{aperiodic core}, we mean an SCC of $G^{\mathcal{A}}$ as in Definition~\ref{def:aubry-core}.
No statement later relies on any stronger ``eventual confinement'' property for arbitrary mean-minimizers.
\end{remark}

\subsection{Induced boundary subshift}

The tight core defines a finite-type symbolic system capturing calibrated two-sided minimizers.

\begin{definition}[Boundary subshift induced by the tight core]\label{def:boundary-subshift}
Let $\Sigma_{\mathcal{A}}\subset (E^{(0)})^{\mathbb{Z}}$ be the set of bi-infinite edge sequences
$\omega=(\dots,e_{-1},e_0,e_1,\dots)$ such that consecutive edges match and every vertex visited lies in $\mathcal{A}$.
Equivalently, $\Sigma_{\mathcal{A}}$ is the two-sided edge shift of the finite directed graph $G^{\mathcal{A}}$. See, e.g., Lind and Marcus \cite{lind_marcus_1995} for standard background on shifts of finite type and edge shifts.
We call $(\Sigma_{\mathcal{A}},\sigma)$ the \emph{boundary subshift} induced by the Aubry--tight core.
\end{definition}

\begin{lemma}[Tight paths have extremal mean]\label{lem:tight-path-mean}
Let $(e_0,\dots,e_{n-1})$ be a length-$n$ path consisting only of tight edges in $E^{(0)}$.
Then
\[
\frac{1}{n}\sum_{t=0}^{n-1} c(e_t) = \lambda_* + \frac{u(v_n)-u(v_0)}{n},
\]
where $v_0$ is the initial vertex, $v_n$ is the terminal vertex, and $u$ is any calibrated subaction.
In particular, along any one-sided infinite tight path, the asymptotic average cost equals $\lambda_*$.
\end{lemma}

\begin{proof}
For each tight edge $e_t:v_t\to v_{t+1}$ we have $c_u(e_t)=0$, i.e.
$c(e_t)-\lambda_*+u(v_t)-u(v_{t+1})=0$.
Summing from $t=0$ to $n-1$ yields
$\sum_{t=0}^{n-1} c(e_t) = n\lambda_* + u(v_n)-u(v_0)$.
Divide by $n$.
\end{proof}

\begin{remark}\label{rem:boundary-subshift-role}
The boundary subshift $\Sigma_{\mathcal{A}}$ is the natural symbolic carrier of calibrated (two-sided) minimizers.
The gap-free boundary classification later in the paper is performed SCC-by-SCC on this core subshift.
When working with a specific one-sided minimizing trajectory, the relevant input is the set of \emph{tight times} (Proposition~\ref{prop:asymptotic-tight})
and the positive-density core selection (Proposition~\ref{prop:tight-dichotomy}); no stronger trapping statement is assumed.
\end{remark}
