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

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\begin{abstract}
This paper extends Syncre Form Theory (SFT) by introducing a single compact modulus that records the operational ingredients needed to certify forced coverage, stability under perturbations, correct targets under coupling, and (when justified) convergence rates. The modulus supports a classification program that organizes cyclic systems by target type and obstruction type, turning diverse physical realizations into a single decision architecture: either a stable witness certificate exists on the correct target, or a finite obstruction must be reported with evidence.
\end{abstract}

\section{The five primitives and the engine posture}

SFT is built from: sliced arenas, phase fields, frame/gauge equivalence, witness outputs (certificate or obstruction), and target correctness under coupling. The guiding posture is an engine theorem: given a declared context, the correct output is either a witness stability certificate or an obstruction report.

\section{The Syncre Stability Modulus}

\begin{definition}[Syncre 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, $\mathrm{LCM}$ and $\mathrm{HMR}$ are stability margins, $b$ is a distortion bound used for anti-collapse, $r$ is effective target rank, and $\lambda$ is a certified convergence rate when a diffusion/mixing model is adopted.
\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. In modulus terms, $\lambda$ becomes a certified component of $\mathfrak{S}$.
\end{theorem}


\section{Stability transfer and monotonicity in the modulus}

This section records the ``native law'' of the modulus: margins certify invariance, and admissible operations cannot change the witness integer unless a margin is crossed.

\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 or gauge equivalence preserving slice correspondence and phase meaning (FEQ), the modulus components transform trivially: the certificate class is invariant.
\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}
Equivalently, the only way to change the certified witness integer is to cross a declared stability margin or violate the declared regime assumptions, in which case the correct output becomes a WOB rather than a WSC.
\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 four canonical classes: forced full coverage, forced coupled coverage, embedded-only coverage, and no forced coverage in the declared regime.

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