% theta_family_canonical_example.tex

\section{A concrete compact normalized target family \texorpdfstring{$\Theta_{\mathrm{can}}$}{Theta\_can} (ready-to-run)}
\label{sec:theta-can}

The global machinery of Sections~\ref{sec:global-universality-thms}--\ref{sec:global-universality-decider} applies to any compact normalized parameter family.
To remove ambiguity about what ``choose $\Theta$'' means in practice, we record a canonical, concrete choice that is broad enough to cover most finite-template models while remaining fully compatible with the robust-certificate contract.

\subsection{Raw parameters and normalization}

Fix once and for all:
\begin{itemize}[leftmargin=2em]
\item a finite reduced descriptor digraph $G=(V,E)$ together with a fixed labeling map $e\mapsto T_e$ into nonnegative $d\times d$ templates,
\item the \emph{support patterns} of all templates $T_e$ (which entries are structurally allowed to be positive),
\item and the perturbation metrics used in RU-A (entrywise for templates and a norm on costs).
\end{itemize}

A raw parameter $\theta$ consists of:
\begin{itemize}[leftmargin=2em]
\item \textbf{template entry weights:} for each structurally allowed entry $(i,j)$ of each label $T_e$, a positive real value $a_{e,ij}$,
\item \textbf{edge costs:} a real cost vector $c_\theta:E\to\mathbb{R}$ (interpreted as reduced costs after the canonical normalization of Section~\ref{sec:core-truth}).
\end{itemize}

To eliminate noncompact scaling degrees of freedom, we use the entrywise-max normalization:
for each label $e$, rescale the nonzero entries of $T_e(\theta)$ so that
\[
\max_{i,j} (T_e(\theta))_{ij} = 1.
\]
This normalization is continuous on the strictly positive locus and is compatible with the projective nature of the contraction conclusions.

\subsection{The canonical compact family}

Fix constants
\[
0<m_0\le 1,\qquad C_0>0.
\]
Define $\Theta_{\mathrm{can}}=\Theta_{\mathrm{can}}(m_0,C_0)$ to be the set of all normalized parameters $\theta$ such that:
\begin{itemize}[leftmargin=2em]
\item every structurally allowed entry of every template satisfies
\[
(T_e(\theta))_{ij}\in [m_0,1],
\]
\item the costs satisfy a mean-zero normalization and a uniform box bound
\[
\sum_{e\in E} c_\theta(e)=0,\qquad |c_\theta(e)|\le C_0\ \ \text{for all }e\in E.
\]
\end{itemize}

\begin{lemma}[Compactness of $\Theta_{\mathrm{can}}$]\label{lem:Theta-can-compact}
$\Theta_{\mathrm{can}}$ is compact in the product topology induced by the RU-A norms.
\end{lemma}

\begin{proof}
Each template-entry coordinate lies in the compact interval $[m_0,1]$ and the cost coordinates lie in $[-C_0,C_0]$ with one closed linear constraint $\sum c=0$.
Finite products of compact sets are compact, and closed subsets of compact sets are compact.
\end{proof}

\subsection{What ``completion'' means on \texorpdfstring{$\Theta_{\mathrm{can}}$}{Theta\_can}}

On $\Theta_{\mathrm{can}}$ the global decider of Theorem~\ref{thm:global-universality-decider} is now a concrete, fully specified procedure:
the input is the machine description of $\Theta_{\mathrm{can}}$ (box constraints plus normalization), and the output is a \texttt{GlobalUniversalityReport}.

The only remaining mathematical distinction is \emph{which outcome occurs}:
\begin{itemize}[leftmargin=2em]
\item If $\Theta_{\mathrm{can}}$ avoids the degeneracy loci (Definition~\ref{def:degeneracy_sets}) chamberwise, then the decider must certify (Corollary~\ref{cor:decider-certifies-under-degeneracy-avoidance}) and yields \emph{good-global contraction universality within the certified class on the compact normalized family $\Theta_{\mathrm{can}}$}.
\item Otherwise, the decider returns a \emph{boundary-global regime atlas} locating the obstruction/degeneracy neighborhood where uniform contraction cannot hold on all of $\Theta_{\mathrm{can}}$ without further restriction.
\end{itemize}

In both cases the output object is the report schema of \path{global_universality_report_schema.json}; in the certified case, it can be compressed into a \texttt{GlobalUniversalityCertificate} object.

\subsection{Making the net-certification inputs explicit (no hidden regularity)}

The ``global-on-\(\Theta\) by finite net'' route in Proposition~\ref{prop:uniform_margins_via_lipschitz} requires conservative continuity moduli.
To prevent any impression that these are being assumed implicitly, the parameter-family descriptor schema includes a dedicated contract block \texttt{net\_certification\_contract}.

Concretely, when one wants a certified uniform margin lower bound via Lemma~\ref{lem:lipschitz_min_cert}, the descriptor must declare:
\begin{itemize}[leftmargin=2em]
\item the parameter norm used to build an \(h\)-net;
\item a conservative Lipschitz constant \(L_c\) for \(\theta\mapsto c(\theta)\) in \(\|\cdot\|_{\infty}\);
\item a conservative Lipschitz constant \(L_A\) for \(\theta\mapsto A^{(k)}(\theta)\) in entrywise \(\|\cdot\|_{\infty,\mathrm{ent}}\);
\item a uniform entry bound \(M\ge \sup_{\theta\in\Theta}\max_{k,i,j}A^{(k)}_{ij}(\theta)\).
\end{itemize}

On a frozen witness chamber, these inputs yield explicit envelopes for \(L_{\gamma}\) and \(L_{\delta}\) via Lemmas~\ref{lem:lipschitz_gamma_explicit} and~\ref{lem:lipschitz_delta_explicit}.
The canonical file \path{theta_family_example.json} supplies this block for \(\Theta_{\mathrm{can}}\) in a coordinate chart where the normalized variables are the parameters.

\begin{remark}[Concrete machine description files]
The repository includes a canonical region descriptor file \path{theta_family_example.json} matching the schema \path{theta_family_schema.json}.
The example global report/certificate files \path{global_universality_report_ThetaCan_certified.json} and \path{global_universality_certificate_ThetaCan.json} illustrate the exact output structure on this target family.
\end{remark}
