\section*{Reader guide, notation, and what is proved}

\subsection*{Core objects}

A claim in SFT is stated on a \emph{declared context}:

\begin{itemize}[leftmargin=*]
\item a sliced arena $M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s$ and a declared slice or subsystem on which phases are observed,
\item a phase observable into a circle $\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}$ or a torus $\mathbb{T}_{\mathbf P}$,
\item a target rule that selects the correct target set (circle, full torus, or a coupled subtorus coset $\mathrm{CST}_s$) together with the baseline convention (Haar on the target, or coupled Haar on $\mathrm{CST}_s$),
\item a witness mechanism (action, loop degree, holonomy, or coupled-target mechanism) and a perturbation model that fixes stability meaning.
\end{itemize}

\subsection*{Certificate-or-obstruction meaning}

Within the Cyclic Observability Class (COC), every declared claim resolves to exactly one audited object:

\begin{itemize}[leftmargin=*]
\item \textbf{WSC (Witness Stability Certificate):} coverage on the correct target with explicit robustness margins,
\item \textbf{WOB (Witness Obstruction):} a canonical obstruction code (first failed validator predicate) with evidence pointers.
\end{itemize}

\subsection*{Acronyms}

\begin{center}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
\textbf{Token} & \textbf{Meaning} \\
\midrule
SCL & Syncr\'e Coverage Law: slice-wise surjectivity of a phase field on the declared target \\
COC & Cyclic Observability Class: the declared regime where predicates and witness engines apply \\
WSC / WOB & Witness Stability Certificate / Witness Obstruction (certificate-or-obstruction output types) \\
CST / $\mathrm{CST}_s$ & Coupled subtorus coset (possibly slice-dependent), the correct target under coupling \\
AWT & Action witness template (equivariant circle action forcing) \\
LDW & Loop degree witness (nonzero degree on an embedded loop) \\
HW & Holonomy witness (nontrivial holonomy multiple on a loop) \\
CTP / CHT & Correct-target principle / coupled Haar target baseline convention \\
LCM / HMR & Loop-closure margin / holonomy-margin rule (robustness margins stored in WSC) \\
RNF / NDB / MXC & Nonresonant increment hypothesis / noise-to-diffusion bridge / mixing certificate inputs \\
\bottomrule
\end{tabularx}
\end{center}

\subsection*{What is proved}

The technical content combines standard forcing mechanisms with a new verifiable decision contract.

\begin{itemize}[leftmargin=*]
\item Forcing theorems for coverage from equivariant circle actions (AWT), loop degree (LDW), and holonomy (HW).
\item Target-correctness axioms and a coupled-target calculus: under coupling, coverage and baselines are asserted on $\mathrm{CST}_s$, not on the ambient product torus.
\item Predicate-exhaustion exhaustiveness: inside COC, every declared claim admits either a WSC or a WOB, with WOB codes determined by first predicate failure.
\item Conservative target-inference theorem: from data and a declared tolerance, either produce a stable CST description (with margins) or return a finite obstruction (ambiguity, insufficient excitation, resonance masking).
\end{itemize}
