% ENGINE_COMPLETENESS_PAGE.tex

\section*{Engine completeness (one-page inevitability)}

The global universality claim in SFT is not a slogan.
It is an engine statement:

\begin{itemize}[leftmargin=*]
\item inside \COC, the engine must return either a \emph{Witness Stability Certificate} (WSC) or a \emph{Witness Obstruction} (WOB),
\item and there is no third outcome.
\end{itemize}

The reason this is definitive is that the engine is gated by a finite predicate suite.

\begin{definition}[Predicate gate]\label{def:predicate-gate}
Fix the canonical ordered list of validator predicates
\[
V_{\mathrm{PO}},V_{\mathrm{WM}},V_{\mathrm{CT}},V_{\mathrm{ROB}},\dots
\]
as specified in \texttt{VALIDATOR\_PREDICATE\_SPEC.tex} and the machine list \texttt{engine/contracts/validator\_predicates.json}.
A declared context \(\mathsf{c}\) satisfies the gate if every predicate required for its declared target type and mechanism holds.
\end{definition}

\begin{theorem}[Completeness of the obstruction catalog]\label{thm:catalog-completeness}
Fix the declared-regime contract (target type, perturbation model, and admissible mechanisms).
For any declared context \(\mathsf{c}\in\COC\), exactly one of the following holds:

\begin{itemize}[leftmargin=*]
\item \textbf{All required predicates hold.} Then a WSC is admissible and the engine is permitted to certify coverage on the correct target, with explicit margins and a predicate ledger.
\item \textbf{Some required predicate fails.} Let \(V_{\mathrm{XX}}\) be the \emph{first} failed predicate in the canonical order. Then the correct output is a WOB with obstruction code \texttt{WOB-XX-...}, and this code lies in the finite canonical codebook.
\end{itemize}

In particular, the WOB catalog is complete for in-regime failures: every in-\COC refusal to certify has a canonical code, and every canonical code corresponds to a predicate failure.
\end{theorem}

\noindent\textbf{Pointers to the full chain.}
The proof is the combination of:

\begin{itemize}[leftmargin=*]
\item predicate exhaustion logic: \texttt{PREDICATE\_EXHAUSTION\_LEMMAS.tex},
\item the canonical catalog: \texttt{WOB\_CATALOG\_COMPLETENESS.tex},
\item the predicate specification: \texttt{VALIDATOR\_PREDICATE\_SPEC.tex},
\item the capstone dichotomy theorem: \texttt{CAPSTONE\_COC\_ENGINE.tex}.
\end{itemize}
