% GLOBAL_UNIVERSALITY_COMPLETION.tex
\section{A Global Cover/Atlas Decider on Compact Normalized Families}\label{sec:global-universality-decider}

The robust-certificate contract makes every local claim
(tight-core structure, Doeblin persistence, recurrence mode, and rate pack)
an \emph{open} property with an explicit stability radius.
This section turns that openness into a \emph{global} statement on a fixed compact normalized family $\Theta$ within the certified class.
The outcome is a finite, witness-producing decision procedure:
either the family is certified globally by a single paired robustness radius and uniform rate pack, or the procedure returns
an explicit obstruction/degeneracy report identifying where and why a global certificate cannot hold.

\subsection{Setup and the property being decided}

Fix a compact parameter set $\Theta$ together with a normalization guaranteeing uniform bounds
on all template entries and reduced-data constants appearing in the robust certificate.
For each $\theta\in\Theta$, the instance induces reduced graph data and the certificate pipeline
returns exactly one of the outcomes in Section~\ref{sec:pipeline-contract-n}.

\begin{definition}[Good-global contraction universality on $\Theta$]\label{def:global-universality-on-Theta}
We say \emph{good-global contraction universality} holds on $\Theta$ if there exist
\(\boldsymbol{\varepsilon}_{\mathrm{global}}=(\varepsilon_{c,\mathrm{global}},\varepsilon_{A,\mathrm{global}})\) and a uniform rate pack \((\tau_{\mathrm{global}},C_{\mathrm{global}},\kappa_{\mathrm{global}})\)
\emph{such that for every $\theta\in\Theta$} the pipeline returns a contraction certificate and, moreover,
the certificate remains valid under all perturbations satisfying
$\|\Delta c\|\le \varepsilon_{c,\mathrm{global}}$ and $\|\Delta A\|\le \varepsilon_{A,\mathrm{global}}$
(in the chosen perturbation model), yielding uniform exponential projective contraction with the stated rate pack.
We also write the conservative scalar floor
\(\varepsilon_{\min,\mathrm{global}}:=\min\{\varepsilon_{c,\mathrm{global}},\varepsilon_{A,\mathrm{global}}\}\)
when a single-radius parameter-space cover statement is sufficient.
\end{definition}

\begin{definition}[Boundary-global regime atlas on $\Theta$]\label{def:boundary-global-atlas}
A \emph{boundary-global regime atlas} on $\Theta$ is a finite, witness-producing report that either certifies
Definition~\ref{def:global-universality-on-Theta}, or else exhibits explicit obstruction witnesses and/or a nonempty
uncovered set together with diagnostics placing it near the named degeneracy loci
\(\{\gamma=0\}\cup\{\delta=0\}\cup\{\mathrm{RC1\ failure}\}\).
\end{definition}

The atlas viewpoint is the honest global completion.
It never asserts uniform contraction on a region that intersects the degeneracy boundary,
and it never leaves the exceptional set implicit.

\subsection{Decision procedure and completeness}

The proof mechanism has two ingredients already established:
\begin{itemize}[leftmargin=2em]
\item the robust certificate predicates are open with explicit stability radii
(Sections~\ref{sec:robust-certificate-ru-a}, \ref{sec:ru-b-tight-core-stability}, \ref{sec:ru-c-doeblin-stability}, \ref{sec:ru-d});
\item the region-cover route: finitely many certificates cover $\Theta$ by neighborhoods of radius $\varepsilon_{\min}^*(\theta_i)$
(Section~\ref{sec:ru-e}).
\end{itemize}

We now project these into a theorem whose conclusion is \emph{algorithmic and dichotomic}.

\begin{theorem}[Global cover/atlas decider on $\Theta$]\label{thm:global-universality-decider}
Assume $\Theta$ is compact and that the normalization chosen in Section~\ref{sec:core-truth}
provides uniform bounds sufficient to evaluate every constant appearing in the robust certificate fields.
Run the region-mode procedure of Section~\ref{sec:ru-e} with adaptive refinement.
The stopping criterion is: either the certified neighborhoods close coverage (uncovered set empty), or the remaining uncovered set is certified to lie inside the declared degeneracy neighborhood; in both cases the \emph{GlobalUniversalityReport} is the output object.
Then the procedure terminates and outputs a \emph{GlobalUniversalityReport}
matching the schema in \path{global_universality_report_schema.json}, with exactly one of the following outcomes.

\begin{itemize}
\item[\textbf{(Certified good-global contraction universality on compact normalized $\\Theta$ within the certified class)}]
The uncovered set is empty.
Then Definition~\ref{def:global-universality-on-Theta} holds on $\Theta$ with
\(
\boldsymbol{\varepsilon}_{\mathrm{global}}:=\bigl(\min_i \varepsilon_c^*(\theta_i),\;\min_i \varepsilon_A^*(\theta_i)\bigr)
\)
and a uniform rate pack obtained by taking conservative maxima/minima over the finitely many center certificates.

For downstream use, this certified case can also be emitted as a standalone \emph{GlobalUniversalityCertificate} object (a compressed form of the report) matching \path{global_universality_certificate_schema.json}.

\item[\textbf{(Boundary-global atlas)}]
The output contains either
\begin{itemize}
\item a finite obstruction witness (one of OG1--OG4) at some sampled parameter,
or
\item a nonempty uncovered set together with diagnostics placing it inside a neighborhood of the explicit degeneracy surfaces
\(\{\gamma=0\}\cup\{\delta=0\}\cup\{\mathrm{RC1\ failure}\}\)
as defined in Section~\ref{sec:global-universality-thms}.
\end{itemize}
In this case, no choice of a single paired radius can certify the entire region by the robust-certificate method
without entering the reported degeneracy neighborhood.
\end{itemize}
\end{theorem}

\begin{proof}
For each parameter value $\theta$, the robust certificate defines an explicit paired radius $\boldsymbol{\varepsilon}^*(\theta)$
whenever the certificate branch conditions have positive margins.
By the openness lemmas (RU-A through RU-D), the same certificate remains valid throughout the corresponding perturbation neighborhood.
Compactness of $\Theta$ implies there exists a finite subcover of any open cover.
The region-mode algorithm either constructs such a finite subcover explicitly (yielding empty uncovered set),
or it refines until any remaining uncovered region lies within the declared degeneracy neighborhood,
since outside that neighborhood the margins are bounded away from zero by the Lipschitz-net verification route
of Lemma~\ref{lem:lipschitz_min_cert}.
Obstruction witnesses OG1--OG4 are finite and explicit by the pipeline contract.
The report combines these outputs.
\end{proof}

\begin{remark}[Certificate object for downstream use]
When the outcome is certified, the global information needed by a third party is typically smaller than the full region report.
The file \texttt{global\_universality\_certificate\_schema.json} records a minimal object containing:
$\boldsymbol{\varepsilon}_{\mathrm{global}}$, $\gamma_0$, $\delta_0$, a witness record (uniform or chamberwise),
and the uniform rate pack.
This is the natural carry-forward artifact for later proofs or numerical experiments.
\end{remark}

\begin{corollary}[When the decider must certify: degeneracy avoidance on finitely many witness chambers]
\label{cor:decider-certifies-under-degeneracy-avoidance}
Assume $\Theta$ is compact and is covered by finitely many frozen witness chambers in the sense of Definition~\ref{def:frozen-witness-project}.
If on each chamber the degeneracy sets of Definition~\ref{def:degeneracy_sets} are empty
(equivalently: the verified lower bounds in Proposition~\ref{prop:uniform_margins_via_lipschitz} are strictly positive and RC1 holds on that chamber),
then the region-mode procedure returns the certified outcome of Theorem~\ref{thm:global-universality-decider}.
In particular, good-global contraction universality holds on $\Theta$ with an explicit uniform paired radius
\(\boldsymbol{\varepsilon}_{\mathrm{global}}:=\min_{\text{chambers }\Omega}\boldsymbol{\varepsilon}_{\mathrm{global}}(\Omega)\) (componentwise minimum)
and a uniform rate pack.
\end{corollary}

\begin{proof}
On each frozen witness chamber $\Omega$, the emptiness of degeneracy sets implies uniform robust margins by Proposition~\ref{prop:global_from_empty_deg},
and the explicit net+Lipschitz route supplies certified lower bounds when desired (Proposition~\ref{prop:uniform_margins_via_lipschitz}).
Thus Theorem~\ref{thm:global_universality_uniform} applies chamberwise and produces an explicit chamberwise paired radius $\boldsymbol{\varepsilon}_{\mathrm{global}}(\Omega)$.
Finiteness of the chamber cover gives a positive global componentwise minimum, and compactness yields a finite cover by robust certificate neighborhoods.
Therefore the region-mode decider closes coverage and outputs the certified case.
\end{proof}

\subsection{Consequences for decidability islands}

Theorem~\ref{thm:global-universality-decider} converts a qualitative ``robustness in a neighborhood'' statement
into a concrete, checkable conclusion for parameter families.
In particular, it yields a sharp \emph{decidable island} statement for robust switching stability:
on the certified positive/tight/Doeblin subclass supporting the pipeline, the question
``does this compact normalized family admit a uniform robust contraction mechanism?''
reduces to a finite cover computation plus explicit degeneracy diagnostics.

\begin{remark}[What this does and does not claim]
The decider above is a \emph{family-level} completeness statement for the certified class.
It does not claim that good-global contraction universality holds for all possible parameter families,
nor that the general joint spectral radius decision problem is decidable.
Instead, it provides an explicit frontier: inside the certified class and under normalization,
the program decides the outcome by producing either a global certificate or explicit obstruction/degeneracy data.
\end{remark}
