\section{Related work and positioning}

SFT is built from standard ingredients:

\begin{itemize}[leftmargin=*]
\item \textbf{Degree and winding.} Coverage forcing on a circle target is governed by degree of loop maps $S^1\to S^1$ and homotopy invariance \cite{hatcher2002,munkres2000}.
\item \textbf{Compact group actions and equivariance.} Action witnesses are compact transformation group arguments: equivariant observables reduce forcing to the induced circle homomorphism and orbit structure \cite{bredon1972}.
\item \textbf{Torus translations and equidistribution.} When convergence to Haar is asserted, background facts include unique ergodicity of irrational torus translations and standard equidistribution criteria \cite{walters1982,einsiedler2011,kuipers1974}.
\item \textbf{Noise, diffusion, and mixing.} Quantitative convergence in the presence of noise is modeled via Markov or diffusion operators on compact manifolds under explicit spectral-gap or minorization conditions \cite{pavliotis2014,stroock2000,meyn2009,doeblin1937}.
\item \textbf{Integer relations and constraint discovery.} Coupled-target inference uses bounded small-integer relation search or lattice reduction methods; stability is expressed through explicit margins \cite{lenstra1982,ferguson1992pslq}.
\end{itemize}

\subsection*{What SFT adds}

SFT contributes a verifiable contract for phase-coverage statements rather than a new topological degree theorem. The distinctive content is the target-correct, audited interface:

\begin{itemize}[leftmargin=*]
\item \textbf{Correct-target discipline.} Coupling changes the target and the baseline; certificates are rejected when they silently use Haar on the ambient product torus while the correct target is a coupled coset.
\item \textbf{Certificate-or-obstruction decision objects.} Every declared claim in COC resolves to a finite object: WSC with margins or WOB with a canonical code and evidence pointers.
\item \textbf{Predicate-exhaustion exhaustiveness.} The finite validator predicate suite makes refusal modes explicit and exhaustive inside COC.
\item \textbf{Reproducible artifact bundle.} Theorems are linked to schemas, audits, benchmarks, and hash-verified builds so readers can independently validate claims.
\end{itemize}
