\section{Introduction and contributions}

Phase observables occur whenever a system carries a clock-like quantity: angle, time-of-day, wave phase, oscillator phase, or the phase of a periodic drive. At a fixed global instant, such a phase can vary across space or across a declared subsystem, producing a \emph{phase field}. SFT formalizes a common slice-wise question:

\begin{quote}
On a declared slice or subsystem, does the phase field attain every phase value on the correct target, and is this statement stable under a declared perturbation model?
\end{quote}

Two difficulties force discipline.

\begin{itemize}[leftmargin=*]
\item \textbf{Forcing must be explicit.} Coverage is not inferred from intuition. SFT requires a declared witness mechanism (action, degree, holonomy, or coupled-target forcing) that can be audited.
\item \textbf{Targets must be correct under coupling.} In multi-cycle settings, coupling can restrict the image to a coupled subtorus coset $\mathrm{CST}_s$. Coverage and any baseline distribution must then be asserted on $\mathrm{CST}_s$, not on the ambient product torus.
\end{itemize}

SFT packages these requirements as a verifiable decision contract. Given a declared context, the engine produces exactly one audited object type inside COC:

\begin{itemize}[leftmargin=*]
\item a \textbf{WSC} proving coverage on the correct target with explicit stability margins, or
\item a \textbf{WOB} naming the first failed validator predicate in a finite codebook, with evidence pointers.
\end{itemize}

\subsection*{What is new in this work}

The forcing mechanisms themselves rely on standard pillars (degree theory, compact group actions, torus dynamics, diffusion on compact manifolds). The contribution is the way these pillars are assembled into a target-correct, auditable certificate system with a sharp boundary.

\begin{itemize}[leftmargin=*]
\item \textbf{Target-correct certificate semantics under coupling.} Coupling is treated as part of meaning: certificates and baselines are stated on $\mathrm{CST}_s$ when constraints force the image into a coupled coset.
\item \textbf{Predicate-exhaustion exhaustiveness.} Inside COC, the validator predicate suite makes outcomes exhaustive: a declared claim admits either a WSC or a WOB, with canonical first-failure codes.
\item \textbf{Quantitative stability margins as first-class objects.} WSC artifacts store margins (LCM/HMR and derived modulus fields) that make stability claims checkable and portable.
\item \textbf{Conservative target inference with obstruction fallback.} A data-to-certificate workflow either outputs a stable coupled-target description with margins or returns a finite obstruction (ambiguity, insufficient excitation, resonance masking).
\item \textbf{Artifact-backed reproducibility.} The release bundle contains schemas, audits, benchmarks, and hash-verified build scripts linking theorems to checkable objects.
\end{itemize}
