\section*{Modulus laws, composition, and sharp failure}

The stability modulus \(\mathfrak{S}\) is the native invariant for the engine.

\begin{theorem}[FEQ invariance]
If two witness contexts are frame-equivalent (FEQ) and the witness integer \(k\) and declared margins (LCM/HMR) coincide, then the engine outputs are equivalent up to the same FEQ transformation: a certificate remains in the same certificate class and carries the same modulus \(\mathfrak{S}\).
\end{theorem}

\begin{theorem}[Margin crossing forces obstruction]
Fix a perturbation model.
If a perturbation exceeds the declared margin recorded in \(\mathfrak{S}\) (for example, \(\mathrm{LCM}\ge \pi\) for degree stability or \(\mathrm{HMR}\ge P/2\) for holonomy stability), then the correct engine output is not a modified certificate: it is a boundary report.
In artifact form this is a WOB whose code matches the first failed validator predicate.
\end{theorem}

\begin{theorem}[Coupled targets are meaning]
If coupling restricts the image to a coupled subtorus \(\mathrm{CST}_s\), then any statistical baseline on the full torus is ill-posed.
In artifact form, the validator predicate \(\mathrm{V06}\) enforces that the baseline is coupled Haar on \(\mathrm{CST}_s\), and failure produces \texttt{WOB-06-coupling-mismatch}.
\end{theorem}

\subsection*{Composition and degradation (artifact rules)}

Certificate composition is treated as an explicit transformation on artifacts.

\begin{itemize}[leftmargin=*]
\item Composition rules are listed in \codeid{engine/contracts/certificate_composition_rules.md}.
\item Degradation rules and their audit interface are listed in \codeid{engine/contracts/certificate_degradation_rules.md}.
\end{itemize}

When a certificate declares a degradation inequality, the audit script verifies it numerically.

