% WELL_POSEDNESS_SPINE.tex

\providecommand{\COC}{\ensuremath{\mathrm{COC}}}

\section*{Well-posedness spine: what a universality claim means in SFT}

SFT treats universality as a statement with an explicit boundary.
The boundary is not a philosophical warning; it is part of the formal interface.
Every claim is posed in a \emph{declared context} and every answer is a finite object.
This is the only sense in which SFT uses the word \emph{meaning}: a claim is meaningful exactly when it has an auditable certificate-or-obstruction output.

\begin{definition}[Declared SFT claim]
A declared SFT claim is a tuple
\[\mathsf{q}=(\mathsf{c},\ \mathsf{T},\ \mathsf{S}),\]
where $\mathsf{c}$ is a declared context (with a phase observable and an admissible witness mechanism), $\mathsf{T}$ is the declared \emph{correct target} (circle/torus/coupled target) together with its baseline convention (Haar on the target; coupled Haar on CST when coupling is present), and $\mathsf{S}$ is the concrete Syncr\'e statement being tested (coverage, stability, and any declared upgrade).
\end{definition}

\begin{quote}
\textbf{Well-posedness axiom (WSC/WOB meaning).}
For every declared claim $\mathsf{q}$, the SFT engine returns exactly one of the following two outputs.
\begin{itemize}[leftmargin=*]
\item \textbf{WSC (Witness Stability Certificate).} A finite certificate proving the stated claim on the declared correct target, together with explicit margins that make the claim stable in the declared perturbation model.
\item \textbf{WOB (Witness Obstruction).} A finite obstruction report from a named catalog, with evidence, asserting that certification is blocked or that the claim is inapplicable in the declared regime.
\end{itemize}
The claim is \emph{meaningful in SFT} if and only if it is posed as such an engine query and therefore admits a WSC/WOB output.
\end{quote}

The sharp boundary can be read in one glance.

\begin{center}
\begin{tabular}{@{}p{0.46\textwidth}p{0.50\textwidth}@{}}
\toprule
\textbf{Where the claim lives} & \textbf{Correct output} \\
\midrule
Inside \COC\ (phase observable + witness route + target rule) & WSC or a finite WOB (structural failure mode) \\
Outside \COC\ (missing phase / missing witness route / wrong target rule) & Boundary-type WOB (inapplicable or ill-posed) \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{How to read any ``universality'' sentence in this project.}
\begin{itemize}[leftmargin=*]
\item First check whether the context is declared as belonging to \COC\ (the checklist is part of the definition).
\item Then check the target: circle, full torus, or a coupled target $\mathrm{CST}$.
\item Then check the object-level output: a WSC (with margins) or a WOB (with a named obstruction code and evidence).
\end{itemize}

This spine is the project's scope wall in mathematical form.
It prevents universality from becoming vague: every use of the word is anchored to the named class (\COC) defined by the interface conditions and to an explicit certificate-or-obstruction output.

\medskip
\input{TARGET_CORRECTNESS_AXIOMS.tex}

\input{TARGET_DECISION_FLOW.tex}
