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\title{Gap-Free Rigidity, Robust Certificates, and Chamberwise Global Cover/Atlas Universality for Extremal Growth in Finite Positive Templates}
\author{Drew Alexander Higgins}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We present a gap-free rigidity mechanism for extremal (minimal) growth in finite families of positive template operators.
The core outcome is a finite certificate dichotomy:
for every reduced descriptor instance, either one produces a concrete obstruction witness (OG1--OG4), or one produces an explicit Doeblin word and a recurrence modulus (RC1) that together yield uniform projective contraction along all tight trajectories.
We then upgrade the pipeline to a robust form: each certificate is accompanied by explicit perturbation margins and a computed stability radius $\varepsilon^*$ so the decision is stable under small changes of template entries.
The resulting project yields a deterministic, checkable route from reduction to quantitative Hilbert-metric contraction, with constants that can be read off from the certificate itself.
Finally, on compact normalized parameter families (within the certified class) we obtain a witness-producing global cover/atlas decider: either a finite set of robust certificates covers the region with a single uniform robustness radius and uniform rate pack (good-global contraction universality), or the procedure returns explicit obstruction/degeneracy data locating where and why uniform contraction cannot hold, packaged as a boundary-atlas report.
\end{abstract}

\section{Overview}

This paper is built for two complementary goals.

\noindent\textbf{Minimal mental model.} A finite family of positive templates induces a reduced tight core capturing all extremal (minimal-growth) behavior. Either there is a finite obstruction witness (OG1--OG4), or an explicit Doeblin word on an aperiodic tight component together with a recurrence predicate (RC1) yields uniform projective contraction along all tight trajectories. Robust certificates attach explicit margins and a computed stability radius $\boldsymbol{\varepsilon}^*$, so the conclusion is stable under small perturbations. On any compact normalized parameter family $\Theta$ within the certified class, this openness becomes a finite cover: the outcome is always \emph{cover-or-atlas}---either a single uniform good-global contraction certificate, or a boundary-atlas report localizing the degeneracy where uniform contraction cannot hold.

\begin{itemize}[leftmargin=2em]
\item \textbf{Structural goal:} explain why ``optimality forces regularity'' in finite positive-template growth problems.
\item \textbf{Operational goal:} give a finite certificate contract that either returns a specific obstruction (OG1--OG4) or returns quantitative contraction data that can be verified directly.
\end{itemize}

Technically, the contraction step is a quantitative use of Hilbert's projective metric and Birkhoff's contraction coefficient \cite{birkhoff_1957,bushell_1973,lemmens_nussbaum_2012}.
Conceptually, extremal itineraries are modeled by a finite-type shift and its sofic refinements \cite{lind_marcus_1995}, and the resulting rate packs interface naturally with the language of linear cocycles, random matrix products, and Lyapunov exponents \cite{furstenberg_kesten_1960,bougerol_lacroix_1985,viana_2014}.
The motivating growth viewpoint aligns with transfer-matrix product growth systems in statistical mechanics and related positive-operator settings \cite{baxter_1982}.

The technical spine proceeds in three layers.

\begin{itemize}[leftmargin=2em]
\item \textbf{Reduction and tight core:} the descriptor reduction produces a tight subgraph $G^{(0)}$ and local face data.
\item \textbf{Gap-free forcing and contraction:} calibrated minimizers concentrate on the tight core; slack and return-cost arguments yield positive-density forcing; Doeblin and recurrence then give projective contraction on minimizers.
\item \textbf{Certificates and robustness:} each step is packaged as a finite certificate with explicit robustness margins.
\end{itemize}

\subsection*{Where to find what}
\begin{center}
\small
\begin{tabular}{p{0.33\linewidth}p{0.60\linewidth}}
\toprule
If you are looking for & See \\
\midrule
Decidable-island framing and scope limits & \S\ref{sec:decidable-island-jsr} and \S\ref{sec:open-problem-resolutions} \\
Reduction, normalization, and the induced tight-core boundary system & \S\ref{sec:core-truth} and \S\ref{sec:aubry-core} (with computation in \S\ref{sec:calib-alg}) \\
Obstruction taxonomy (OG1--OG4) and witness objects & \S\ref{sec:new-obstruction-j} and \S\ref{sec:sharpness-m} \\
Doeblin-word decision and witnessed word extraction & \S\ref{sec:doeblin-word-search} \\
Recurrence dichotomy, forcing, and syndetic recurrence tools & \S\ref{sec:forcing-ion} and \S\ref{sec:syndetic-recurrence} \\
Contraction from Doeblin plus recurrence (including disjoint-occurrence counting in RH2) & \S\ref{sec:unconditional-contraction-k} \\
Rate-pack extraction (constants, decay rates, and bounds) & \S\ref{sec:rate-pack-l} \\
Robustness radii and the stability layers (RU-A to RU-D) & \S\ref{sec:robust-certificate-ru-a}, \S\ref{sec:ru-b-tight-core-stability}, \S\ref{sec:ru-c-doeblin-stability}, \S\ref{sec:ru-d} \\
Global-on-$\Theta$ cover/atlas guarantee and output objects & \S\ref{sec:ru-e}, \S\ref{sec:global-universality-thms}, \S\ref{sec:global-universality-decider}, and \S\ref{sec:global-universality-artifacts} (with the $\Theta$-family contract in \S\ref{sec:theta-can}) \\
\bottomrule
\end{tabular}
\end{center}

\noindent\textbf{Compilation.} The LaTeX entry point is \texttt{rigidity\_gapfree\_global\_universality.tex}. A standard build is \texttt{latexmk -pdf} on that file (or two runs of \texttt{pdflatex}).

\noindent\textbf{Schema validation.} Schema files (JSON Schema manuscript 2020-12) are included for the report and certificate artifacts. Validation is optional and can be performed with any standards-compliant validator; for example, one may run \texttt{python -m jsonschema -i <file.json> <schema.json>} for the included \texttt{GlobalUniversalityReport}, \texttt{GlobalUniversalityCertificate}, and \texttt{RobustCertificate} objects.

\input{jsr_decidability_frontier.tex}

\input{open_problem_resolution_map.tex}

\input{canonical_normalization.tex}
\input{theta_family_canonical_example.tex}
\input{aubry_tight_core.tex}
\input{calibration_algorithm.tex}
\input{calibrated_minimizers.tex}
\input{minimality_limitations.tex}
\input{slack_forcing.tex}
\input{return_costs_and_excursions.tex}
\input{forcing_positive_density.tex}

\input{descriptor_face_certificates.tex}
\input{doeblin_word_search.tex}
\input{syndetic_recurrence_tools.tex}

\input{obstruction_taxonomy.tex}
\input{contraction_from_doeblin.tex}
\input{explicit_rate_pack.tex}

\input{boundary_theorem.tex}
\input{certificate_pipeline_contract.tex}

\input{robust_certificate_schema.tex}
\input{tight_core_stability.tex}
\input{doeblin_stability.tex}
\input{recurrence_robustness.tex}
\input{region_net_certification.tex}

\input{global_universality_theorems.tex}
\input{global_universality_verification.tex}
\input{global_universality_completion.tex}
\input{global_universality_artifacts.tex}

\input{sharpness_examples.tex}
\input{minimality_sharpness_notes.tex}
\input{parameter_chambers_notes.tex}

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