% COC_definition.tex

\providecommand{\COC}{\ensuremath{\mathrm{COC}}}

\begin{definition}[Cyclic Observability Class (COC)]
The \emph{Cyclic Observability Class} (\COC) is the class of \emph{declared} SFT contexts $\mathsf{c}$ satisfying all of the following three conditions (and it is maximal by definition, since membership is exactly these conditions):
\begin{itemize}[leftmargin=*]
\item a \emph{phase observable} is declared on the relevant slice/subsystem: a cycle-valued map (circle/torus/coupled target) with the minimal regularity needed to define the intended winding/holonomy notion and to test nontriviality;
\item an \emph{admissible witness mechanism} is declared (symmetry action, loop/holonomy, embedded subsystem witness, defect loop, or another explicitly listed mechanism), so the engine has a prescribed route to either a WSC or to a named obstruction code;
\item a \emph{correct target rule} is satisfied: the target type (circle/torus/CST) and any baseline statements are defined on the correct target (Haar on the target, coupled Haar on CST when coupling is present).
\end{itemize}
\end{definition}

\begin{definition}[Universality on a class of contexts]
Fix a class of contexts $\mathcal{C}$. The SFT engine is \emph{universal on $\mathcal{C}$} if for every declared context $\mathsf{c}\in\mathcal{C}$ and every slice/subsystem where the claim is posed, the engine returns exactly one of:
\begin{itemize}[leftmargin=*]
\item a \emph{Witness Stability Certificate} (WSC) proving coverage on the correct target in $\mathsf{c}$ (with explicit margins), or
\item a \emph{Witness Obstruction} (WOB) belonging to a finite obstruction catalog (with evidence), certifying that the claim is blocked in $\mathsf{c}$.
\end{itemize}
\end{definition}

\paragraph{Global universality means: universal on \COC.}
In SFT, a statement labeled \emph{global universality} is a universality statement on \COC: for every context in \COC, the engine returns a WSC or a WOB on the correct target; outside \COC, claims are either inapplicable (target/predicates not well-posed) or produce a boundary-type obstruction rather than a forced-coverage theorem.

\input{COC_checklist.tex}
