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\title{Syncr\'e Form Theory II:\\ The Syncr\'e Stability Modulus and Classification of Cyclic Phase-Field Systems}
\author{Drew Higgins}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This paper introduces a compact modulus that records the operational ingredients needed to certify forced coverage, stability under perturbations, correct targets under coupling, and (when explicitly justified) convergence rates \cite{hatcher2002,walters1982,pavliotis2014}.
The modulus supports a classification program that organizes cyclic systems by target type and obstruction type, turning diverse realizations into a single decision architecture: within COC, either a stable witness certificate exists on the correct target, or a finite obstruction is reported with evidence.
\end{abstract}

\section*{Global universality meaning}
\input{COC_definition.tex}

\input{WELL_POSEDNESS_SPINE.tex}

\section*{Universality-from-cyclic-structure schema}
\input{COC_universality_schema.tex}

\input{COC_rotation_proof.tex}

\input{COC_embedded_loop_proof.tex}
\input{COC_multicycle_proof.tex}

\section{Obstruction completeness by predicate exhaustion}

Inside COC, refusal to certify is implemented by a fixed validator predicate suite and canonical first-failure WOB codes.
The predicate-exhaustion lemmas and catalog completeness statements formalize this boundary.

\input{PREDICATE_EXHAUSTION_LEMMAS.tex}

\input{VALIDATOR_PREDICATE_SPEC.tex}

\input{WOB_CATALOG_COMPLETENESS.tex}

\section{The Syncr\'e Stability Modulus}

\begin{definition}[Syncr\'e Stability Modulus]
On a declared target (full torus or coupled target), define
\[
\mathfrak{S}=(k,\ \mathrm{LCM},\ \mathrm{HMR},\ b,\ r,\ \lambda),
\]
where $k$ is the witness integer (degree/holonomy/free-coordinate degree) \cite{hatcher2002,munkres2000}, $\mathrm{LCM}$ and $\mathrm{HMR}$ are stability margins, $b$ is a distortion bound used for anti-collapse bounds, $r$ is effective target rank, and $\lambda$ is a certified convergence rate when a diffusion/mixing model is adopted \cite{walters1982,einsiedler2011,pavliotis2014,evans2010,meyn2009,doeblin1937}.
\end{definition}

\section{Three theorem statements phrased in $\mathfrak{S}$}

\begin{theorem}[Persistence threshold]
If $k\neq 0$ and $\mathrm{LCM}<\pi$ (and/or $\mathrm{HMR}<P/2$ in the holonomy setting), then the witness tier persists under the declared perturbation model. In engine terms, a WSC exists with the same witness integer and target.
\end{theorem}

\begin{theorem}[Anti-collapse threshold]
Assume the witness loop admits a monotone-branch decomposition and a branch bound $b$. Then the coverage profile admits an explicit lower bound proportional to $(|k|/b)$ on arc neighborhoods. In modulus terms, larger $|k|$ and smaller $b$ force stronger quantitative coverage.
\end{theorem}

\begin{theorem}[Convergence threshold when justified]
If a noise-to-diffusion bridge is declared and a verified mixing inequality holds, then defects from the baseline decay exponentially at rate $\lambda$ recorded in the thresholds ledger \cite{pavliotis2014,evans2010,meyn2009}. In modulus terms, $\lambda$ becomes a certified component of $\mathfrak{S}$.
\end{theorem}

\section{Stability transfer in the modulus}

\begin{theorem}[Modulus stability transfer]
Fix a declared perturbation model. Suppose a WSC exists with stability modulus $\mathfrak{S}=(k,\mathrm{LCM},\mathrm{HMR},b,r,\lambda)$.
\begin{itemize}[leftmargin=*]
\item If perturbations remain within the stated margins (LCM/HMR), the witness integer $k$ cannot change.
\item Under any frame equivalence preserving slice correspondence and phase meaning (FEQ), the modulus components are invariant up to the declared identification rules.
\item If the model includes a dynamics that preserves the invariants used by the certificate, the witness tier persists and the same $k$ remains valid along evolution.
\end{itemize}
\end{theorem}

\section{Classification program}

SFT classifies systems by target type (full circle/torus versus coupled target) and obstruction type (finite WOB class) when certification fails. This yields canonical classes: forced full coverage, forced coupled coverage, embedded-only forcing, and no forcing in the declared regime.

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