\section*{The five primitives and the engine theorem}

SFT is built on five primitives. Everything else is a refinement of these objects.

\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 and gauge equivalence.} Different descriptions that preserve slice correspondence and phase meaning are treated as equivalent (FEQ), so the theory is not a coordinate trick.
\item \textbf{Witness engine.} Coverage claims are not rhetorical: the output is a stable witness certificate (WSC) or a named obstruction report (WOB) with evidence.
\item \textbf{Target correctness.} Under coupling, the correct target is a coupled subtorus coset $\mathrm{CST}_s$ and the correct baseline is coupled Haar on $\mathrm{CST}_s$; otherwise the target is the full torus with Haar baseline.
\end{itemize}

\subsection*{SFT Engine Theorem (decision architecture)}

Given a declared context consisting of $(M,\{\Sigma_s\},\Phi)$ together with any stated coupling constraints and an admissible perturbation model, exactly one of the following holds within the declared hypothesis regime:

\begin{itemize}[leftmargin=*]
\item \textbf{Witness Stability Certificate (WSC).} A witness exists (global, embedded, or coupled), the correct target is specified, and explicit margins certify stability under perturbations.
\item \textbf{Witness Obstruction (WOB).} A witness cannot be certified in the declared regime, and at least one finite obstruction class occurs with evidence and recommended next moves.
\end{itemize}

Forcing theorems are sufficient conditions that guarantee WSC. Obstruction theorems are completeness statements that force WOB when WSC is impossible. Robustness theorems propagate margins and quantitative bounds once WSC exists.
