\section*{Primitives and the engine theorem}

SFT is organized around five primitives.

\begin{itemize}[leftmargin=*]
\item \textbf{Arena and slicing.} A system is modeled as an arena $M$ equipped with slices $\Sigma_s$ representing global instants.
\item \textbf{Phase field.} A cycle-valued observable $\Phi:M\to \mathbb{T}_{\mathbf P}$ assigns phase labels to points, with $\mathbb{T}_{\mathbf P}$ a circle or torus of periods.
\item \textbf{Frame equivalence (FEQ).} Descriptions that preserve slice correspondence and phase meaning are treated as equivalent so conclusions are invariant under admissible frame changes.
\item \textbf{Witness mechanism.} Coverage is asserted only via a declared forcing route (action, loop degree, holonomy, or coupled-target forcing) in the declared model class.
\item \textbf{Target correctness.} Under coupling, the correct target is a coupled subtorus coset $\mathrm{CST}_s$ with coupled Haar baseline; otherwise the target is the full torus with Haar baseline.
\end{itemize}

\subsection*{Engine theorem (interface soundness and exhaustiveness inside COC)}

Fix a declared context in the Cyclic Observability Class (COC), including a target rule, witness mechanism, and perturbation model.
Then the engine output type is exhaustive:

\begin{itemize}[leftmargin=*]
\item \textbf{WSC output.} A witness stability certificate (WSC) on the correct target, recording the witness integer data and explicit margins.
\item \textbf{WOB output.} A witness obstruction (WOB) whose code is the first failed validator predicate in the canonical predicate order, with evidence pointers meeting the code checklist.
\end{itemize}

The WSC predicates imply the stated coverage claim in the declared regime (soundness), and predicate exhaustion implies that a declared claim cannot fail without producing an explicit WOB code inside COC (exhaustiveness).
