\section*{The Syncre Stability Modulus}

To give SFT a single intrinsic ``state'' variable that can be compared across systems, we package the operational ingredients of stability and convergence into one modulus.

\subsection*{Definition (Syncre Stability Modulus)}
On a declared target (full torus or coupled target), define the Syncre Stability Modulus
\[
\mathfrak{S}=(k,\ \mathrm{LCM},\ \mathrm{HMR},\ b,\ r,\ \lambda),
\]
where:
\begin{itemize}[leftmargin=*]
\item $k$ is the witness integer (degree or holonomy multiple).
\item $\mathrm{LCM}$ is the lift-closeness margin guaranteeing degree stability.
\item $\mathrm{HMR}$ is the holonomy margin guaranteeing holonomy class stability.
\item $b$ is a distortion/branch bound used for quantitative anti-collapse on a witness loop (when applicable).
\item $r$ is the effective target rank (full torus rank or free-coordinate rank on $\mathrm{CST}_s$).
\item $\lambda$ is a certified convergence rate when a diffusion/mixing model is adopted (otherwise $\lambda$ is left unspecified).
\end{itemize}

\subsection*{How the modulus is used}
SFT statements can be phrased as threshold claims on $\mathfrak{S}$:

\begin{itemize}[leftmargin=*]
\item \textbf{Existence and persistence.} If $k\neq 0$ and $\mathrm{LCM}<\pi$ (and/or $\mathrm{HMR}<P/2$), then the witness tier persists under the perturbation model.
\item \textbf{Quantitative non-collapse.} If the witness loop satisfies a monotone-branch structure and branch bound $b$, then the coverage profile admits explicit lower bounds proportional to $(|k|/b)$.
\item \textbf{Convergence when justified.} If a diffusion/noise bridge is adopted and the mixing certificate holds, then $\lambda$ is read directly from the thresholds ledger and recorded as part of the certificate.
\end{itemize}

The modulus is not a new physical constant. It is a compact way to record what SFT needs in order to certify inevitability, stability, and (when appropriate) convergence.
