% COC_universality_schema.tex

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

\begin{theorem}[Universality-from-cyclic-structure (COC embedding schema)]
\label{thm:COC-schema}
Fix a declared model class and let $\Sigma$ denote the slice/subsystem on which a Syncr\'e claim is posed.
Assume the declaration provides:
\begin{itemize}[leftmargin=*]
\item \textbf{Phase observable (PO).} A cycle-valued observable $\phi:\Sigma\to\mathbb{T}_P$ or $\Phi:\Sigma\to\mathbb{T}_{\mathbf P}$ with the minimal regularity needed to define the intended winding/holonomy notion and to test nontriviality.
\item \textbf{Witness mechanism (WM).} At least one admissible mechanism from the SFT menu, in one of the following structural forms:
  \begin{itemize}[leftmargin=*]
  \item \emph{Action forcing.} An admissible continuous action $A:G\times D\to D$ on a domain $D\subseteq\Sigma$ with $G=S^1$ or $G=\mathbb{T}^k$, together with a continuous homomorphism $\chi:G\to H$ to the declared target group $H$ such that $\phi$ or $\Phi$ is $\chi$-equivariant on $D$ in the declared tolerance (exactly or with a stated defect bound). The action-forcing mechanism is \emph{nondegenerate} when the induced map on the declared target has nontrivial cyclic response (nonzero slope in the $S^1$ case; positive rank on the target coordinates in the $\mathbb{T}^k$ case).
  \item \emph{Embedded loop/holonomy.} An embedded loop $\gamma:S^1\to\Sigma$ (or loop in an embedded subsystem) together with either a nonzero degree witness $\deg(\phi\circ\gamma)\neq 0$ or a nonzero holonomy multiple in a declared lift/connection model.
  \end{itemize}
\item \textbf{Target rule (TR).} A declared target type (circle/torus/coupled subtorus) and baseline convention on that target (Haar on the target, coupled Haar on CST when coupling is present).
\end{itemize}
Then the declared context belongs to \COC.
Consequently, the SFT universality posture applies on this context: on the declared target, the correct engine output is either a Witness Stability Certificate (WSC) with explicit margins, or a finite Witness Obstruction (WOB) from the obstruction catalog with evidence.
\end{theorem}

\subsection*{Canonical cyclic families covered by the schema}
The theorem is a \emph{schema}: it reduces broad families to the same three-checklist declaration (PO/WM/TR).

The circle-action / periodic-forcing family admits a fully formal proof template in Theorem~\ref{thm:circle-action-template}.
Each line below is a structural criterion (not a physical promise): if the criterion holds in the adopted model class, it produces a COC embedding.

\begin{itemize}[leftmargin=*]
\item \textbf{Periodic forcing / time-shift symmetry.} If the declared subsystem admits a continuous periodic parameter (identified with $S^1$) acting by reparameterization on states, and the phase observable is equivariant under that shift with nonzero slope on the declared phase circle, then the action-forcing mechanism applies.

\item \textbf{Rotation / rigid cyclic response.} If the declared subsystem carries an admissible $S^1$-action (or a torus action in the multi-cycle setting) and the observable is equivariant with nontrivial induced homomorphism on the declared target, then forcing applies on the corresponding orbit-image (full circle when slope is nonzero; subtorus/coset when rank is deficient and the target rule selects CST).

\item \textbf{Wave-train / translation-phase reduction.} If a periodic direction induces an admissible $S^1$-translation action on a domain and the phase observable is equivariant under translation with nonzero slope, the situation is again an action-forcing context.

\item \textbf{Limit-cycle / periodic-orbit phase.} If the declared dynamics exhibits a periodic orbit equipped with a continuous phase coordinate (so that motion along the orbit defines an $S^1$-action on that orbit) and the observable responds with nonzero winding along that cycle, then either action-forcing or an embedded loop degree witness certifies cyclic observability.

\item \textbf{Defect loop / winding witness.} If the declaration provides a loop around a distinguished locus (a boundary component, defect core, or excluded set) on which the phase observable has nonzero winding (or nonzero holonomy multiple in a lift model), then the embedded witness mechanism applies.

\item \textbf{Coupled drives / multi-cycle response.} If multiple cyclic inputs generate an admissible $\mathbb{T}^k$-action and the observable is equivariant with integer-matrix response $\chi$, then the correct target is the image subtorus (or the inferred coupled subtorus CST), and the forcing/obstruction dichotomy is posed on that target.
\end{itemize}
