
\section{Parameter Chambers and Generic Cycle Selection}\label{sec:param-chambers}

This section formalizes what is generic in linear parameter families in a literal, checkable sense: uniqueness of the minimizing cycle mean off a finite hyperplane arrangement,
together with a finite chamber decomposition of parameter space on which the minimizing set is constant.
No claim of ``generic aperiodicity'' is made, since periodic/aperiodic is a combinatorial property of the selected critical SCC.

\subsection{Linear parameter model and hyperplane arrangement}

Fix a finite directed graph $G=(V,E)$ (after reduction) and a feature map $\psi:E\to\mathbb{Z}^k$.
For $\theta\in\mathbb{R}^k$ define edge costs
\[
c_\theta(e)=\langle \theta,\psi(e)\rangle,
\]
and for a directed cycle $\gamma$ define its mean cost
\[
\mu_\theta(\gamma):=\frac{1}{|\gamma|}\sum_{e\in\gamma} c_\theta(e).
\]
Let $\mathcal{C}$ be the finite set of simple directed cycles in $G$ and let
\[
\lambda_*(\theta):=\min_{\gamma\in\mathcal{C}} \mu_\theta(\gamma)
\]
be the minimal cycle mean.

\begin{definition}[Degeneracy hyperplanes]\label{def:degeneracy-hyperplanes}
For distinct cycles $\gamma\neq\gamma'$ define the affine functional
\[
F_{\gamma,\gamma'}(\theta):=\mu_\theta(\gamma)-\mu_\theta(\gamma').
\]
The \emph{degeneracy arrangement} is the finite union of hyperplanes
\[
\mathcal{D}:=\bigcup_{\gamma\neq\gamma'} \{\theta:\ F_{\gamma,\gamma'}(\theta)=0\}.
\]
\end{definition}

\begin{lemma}[Unique minimizing cycle off $\mathcal{D}$]\label{lem:unique-cycle-off-D}
If $\theta\notin\mathcal{D}$, then there exists a unique cycle $\gamma_*(\theta)\in\mathcal{C}$ with $\mu_\theta(\gamma_*(\theta))=\lambda_*(\theta)$.
\end{lemma}

\begin{proof}
If two distinct cycles both attained the minimum, then $F_{\gamma,\gamma'}(\theta)=0$ for that pair, so $\theta\in\mathcal{D}$.
\end{proof}

\subsection{Chamber decomposition and critical SCC stability}

\begin{definition}[Chambers]\label{def:chambers}
A \emph{parameter chamber} is a connected component of $\mathbb{R}^k\setminus\mathcal{D}$.
\end{definition}

\begin{proposition}[Finite chamber structure]\label{prop:finite-chambers}
$\mathbb{R}^k\setminus\mathcal{D}$ is a finite union of open polyhedral chambers.
On each chamber, the minimizing cycle $\gamma_*(\theta)$ is constant.
\end{proposition}

\begin{proof}
$\mathcal{D}$ is a finite arrangement of affine hyperplanes, so its complement has finitely many connected components,
each described by a finite system of strict linear inequalities $F_{\gamma,\gamma'}(\theta)\lessgtr 0$.
Within a fixed component, the sign pattern of all $F_{\gamma,\gamma'}$ is constant, hence the unique minimizer $\gamma_*$ is constant by Lemma~\ref{lem:unique-cycle-off-D}.
\end{proof}

\begin{corollary}[Computable periodic/aperiodic regime on chambers]\label{cor:regime-on-chambers}
Fix a chamber $\mathcal{U}$ and let $\gamma_*$ be its minimizing cycle.
Then the essential critical SCC of the tight graph is the SCC generated by $\gamma_*$ (in particular, it is periodic).
More generally, on lower-dimensional strata $\theta\in\mathcal{D}$ where multiple cycles tie, the tight critical subgraph may contain multiple cycles.
It may be periodic or aperiodic depending on the gcd of cycle lengths in the resulting critical SCC.
All of these properties are decidable from the finite tight geometry.
\end{corollary}

\begin{theorem}[Boundary type is constant on chambers]\label{thm:boundary-type-constant-on-chambers}
Fix the graph $G$ and feature map $\psi$ as above, and consider the induced linear cost family $c_\theta$.
For each chamber $\mathcal{U}$ of $\mathbb{R}^k\setminus\mathcal{D}$, the following data are constant over $\theta\in\mathcal{U}$:
\begin{itemize}
  \item the minimizing cycle $\gamma_*$ and minimal mean $\lambda_*(\theta)$ (hence the critical/tight SCC);
  \item the periodic/aperiodic regime of the essential critical SCC;
  \item the boundary classification output of the Tight-Core Boundary Theorem on that SCC (imprimitive trapping, or in the aperiodic case:
  certificate/Doeblin/contraction versus OG2/OG3 obstructions, together with the associated witness objects).
\end{itemize}
Consequently, the boundary type can change only when crossing the degeneracy arrangement $\mathcal{D}$.
\end{theorem}

\begin{proof}
On a chamber $\mathcal{U}$ the sign pattern of all cycle-mean differences $F_{\gamma,\gamma'}$ is fixed, so the minimizer $\gamma_*$ is constant by Proposition~\ref{prop:finite-chambers}.
In the unique-minimizer case, the critical/tight subgraph is generated by the edges of $\gamma_*$, hence the essential critical SCC (and its periodicity) is constant.
All subsequent boundary outputs in the Tight-Core Boundary Theorem are computed from finite graph data on the essential SCC and its induced tight geometry
(certificate graph construction, SCC type, and the witness-producing obstruction tests), so they are constant once that finite geometry is fixed.
Crossing a hyperplane in $\mathcal{D}$ is the only way the minimizer set can change, hence it is the only place where the induced tight/critical geometry and boundary type can change.
\end{proof}

\paragraph{Edge cases on strata $\theta\in\mathcal{D}$.}
On the degeneracy arrangement, ties produce a critical subgraph with multiple cycles.
The same finite pipeline applies, but the tight geometry must be formed from the tied minimizing set.
For convenience, the most common tie patterns can be summarized as follows.

\begin{center}
\begin{tabular}{|p{0.25\linewidth}|p{0.68\linewidth}|}
\hline
\textbf{Situation on $\mathcal{D}$} & \textbf{What is decidable (and what to report)} \\
\hline
Ties among cycles \emph{within one} critical SCC & Compute the gcd of cycle lengths in that SCC.
If the gcd is $>1$, the regime is periodic (trapping). If the gcd is $1$, the SCC is aperiodic and the Tight-Core Boundary Theorem applies (certificate/Doeblin/contraction or OG2/OG3 with witnesses). \\
\hline
Ties split across \emph{multiple} critical SCCs & The tight/critical graph decomposes into SCCs.
Run the boundary classification on each essential SCC and report the resulting regime/witnesses; the overall boundary type is the finite list of SCC outcomes. \\
\hline
Higher-codimension ties (many cycles) & Form the tight subgraph induced by the minimizing set and apply the same SCC + certificate/obstruction pipeline.
The output remains finite and witness-producing. \\
\hline
\end{tabular}
\end{center}

\begin{remark}[What this does (and does not) justify]
In linear cost families, uniqueness of the minimizing cycle is generic in the precise sense ``off $\mathcal{D}$.''
That generic situation typically yields a periodic critical SCC.
Accordingly, any claim of ``generic aperiodicity'' would require additional structure beyond the linear cycle-mean model.
The robust message supported here is different: the periodic/aperiodic and certificate/obstruction boundary is finitely computable from the tight geometry,
and by Theorem~\ref{thm:boundary-type-constant-on-chambers} the boundary type is constant on chambers and can change only across $\mathcal{D}$.
\end{remark}
