\subsection*{Universality inside COC}

SFT does not assert that every system satisfies slice-wise coverage.
Instead, SFT fixes a named regime (COC) where the question has a closed, auditable meaning:
a declared phase observable, a declared slice/subsystem, a target rule (including coupling conventions), and an admissible witness mechanism.

Within COC, the engine returns either a WSC proving coverage on the correct target or a WOB naming a finite obstruction code with evidence.

\subsection*{Universality-from-cyclic-structure schema}

\input{COC_definition.tex}

\input{COC_universality_schema.tex}

\input{COC_rotation_proof.tex}

\input{COC_embedded_loop_proof.tex}
\input{COC_multicycle_proof.tex}

\subsection*{Finite obstruction classification (witness-first)}

Inside COC, refusal to certify has a finite set of structural reasons.
These reasons are implemented as validator predicates and canonical WOB codes, so failure modes are explicit and auditable.
Examples include:

\begin{itemize}[leftmargin=*]
\item no nondegenerate response available in an equivariant action setting (slope-trivial response in the declared forcing domain),
\item no circle-type orbit or embedded loop supporting the declared witness,
\item equivariance or lift/holonomy consistency defect exceeding the declared tolerance,
\item coupling mismatch: a full-torus target is asserted when coupling forces a coupled coset target,
\item insufficient data to identify the coupled target stably at the declared tolerance (ambiguity, insufficient excitation, resonance masking).
\end{itemize}

\subsection*{Robustness margins}

Witnesses are packaged with explicit margins so that stability is checkable under a declared perturbation model:

\begin{itemize}[leftmargin=*]
\item degree witnesses persist under lift-closure margins $<\pi$,
\item holonomy witnesses persist under loop-integral margins $<P/2$,
\item action witnesses persist under quantified equivariance defect bounds recorded in the certificate.
\end{itemize}

\subsection*{Multi-cycle forcing and correct targets}

In multi-cycle settings, forcing is asserted on the correct target:
the response-image coset determined by the induced torus homomorphism and, when present, the coupled subtorus coset $\CST_s$ determined by constraints.
Baselines for statistical statements are Haar on the chosen target, or coupled Haar on $\CST_s$.

\subsection*{Dynamics and convergence upgrades}

SFT separates preservation from mixing.
Preservation hypotheses are recorded in certificates as margins; convergence to Haar is asserted only under explicit hypotheses (nonresonant increments for unique ergodicity, or a noise-to-diffusion bridge and a verified mixing inequality for exponential convergence).
