\section*{Certification dichotomy (declared regime)}

\subsection*{Declared regime}
Fix a regime consisting of:
\begin{itemize}[leftmargin=*]
\item a sliced arena and a declared observation domain (global slice, embedded loop, or coupled target),
\item a circle- or torus-valued phase observable in $\mathbb{T}_{\mathbf P}$,
\item either (i) a genuine $S^1$/$\mathbb{T}^k$ symmetry action with an equivariance relation, or (ii) an embedded loop/defect subsystem supporting degree/holonomy,
\item a declared perturbation model with explicit margins (LCM/HMR),
\item in coupled settings, a declared target-correctness rule selecting $\mathrm{CST}_s$ and coupled Haar baseline.
\end{itemize}

\subsection*{Theorem (soundness and completeness of the interface)}
For any stated claim target within the declared regime, the engine returns exactly one of the following outputs, and each output is independently checkable:

\begin{itemize}[leftmargin=*]
\item \textbf{Certificate.} A witness stability certificate (WSC) on the correct target, recording a witness integer (degree/holonomy/free-coordinate degree), explicit stability margins, and the correct statistical baseline. The WSC validator predicates imply the stated coverage claim, and the margins quantify robustness under the declared perturbation model.
\item \textbf{Obstruction.} A witness obstruction (WOB) whose code belongs to the finite catalog and whose evidence satisfies that code's checklist. Within the declared regime, this evidence rules out the existence of any WSC satisfying the same regime validators for the stated target.
\end{itemize}

\subsection*{Proof skeleton (predicate exhaustion)}
The completeness content is carried by a finite suite of validator predicates.
Each predicate corresponds to one canonical obstruction code, and each predicate failure forces its code.
Within the declared regime, any failed validator yields a WOB whose evidence is auditable.
If all validators hold, then the forcing templates and target-correctness rules permit a WSC.

\input{PREDICATE_EXHAUSTION_LEMMAS.tex}

\input{VALIDATOR_PREDICATE_SPEC.tex}

\input{WOB_CATALOG_COMPLETENESS.tex}


\subsection*{Corollary (well-posed universality questions)}
A universality question is well-posed precisely when its target and hypotheses are stated in a form that lies inside the declared regime, so that the engine can return either a WSC or a WOB on the correct target.
