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Syncre Form Theory II:\\ The Syncre Stability Modulus and the Classification of Cyclic Phase-Field Systems
Project Scope
The Stability Modulus package is the quantitative layer of Syncre Form Theory.
In Syncré, a single global instant can be treated as a field of phase labels; stability asks how that field-level claim behaves under perturbation.
It answers a specific question that the foundational paper intentionally leaves as a separate artifact:
- when coverage is certified by a witness, how does the certificate degrade under controlled perturbations
This matters because it turns “robust” into a measurable object that can be compared across arenas, not merely asserted.
If you want significance and what changed in the current release, open:
Syncre Form Theory II: The Syncre Stability Modulus and the Classification of Cyclic Phase-Field Systems
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.
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.
The Syncre Stability Modulus
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.
Three theorem statements phrased in $\mathfrak{S}$
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.
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.
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 $\mathfrak{S}$.
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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