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\title{RU Gap-Free Global Universality as a Certificate Engine for Finite Host Structures}
\author{Drew Higgins}
\date{}

\begin{document}
\maketitle

\section*{Purpose}
This paper describes a single technology: a certificate engine for universality and obstruction phenomena in finite constrained products of nonnegative operators.

The project center is:

\begin{itemize}
\item RU is the universality engine.
\item Host structures supply inputs only.
\item Strong claims are carried by auditable certificate objects.
\end{itemize}

String-theory-adjacent language appears only as one source of host inputs (finite worldsheet or CFT sectors, boundary constraints, fusion or channel systems). No physical validation claim is made.

\section*{Abstract}
Given finite data (a directed constraint graph, nonnegative template operators on a cone, and an optional cost model), the RU pipeline performs canonical normalization to extract a tight core and then either certifies projective contraction with explicit quantitative rates and robustness tolerances or returns a finite obstruction witness explaining why universality is not forced.
All outcomes are encoded as JSON artifacts validated against schemas.
The project includes theorem sources that justify the certificate predicates and the contraction chain, plus reproducibility tools that regenerate, validate, and sanity-audit the audited suite.

\section{Input contract}
An RU instance is specified by a finite \emph{host instance}:
\begin{itemize}
\item a finite directed graph $G=(V,E)$,
\item a cone $\mathcal{K}\subset \mathbb{R}^d$,
\item a family of nonnegative templates $\{T_e\}_{e\in E}$ acting on $\mathcal{K}$,
\item optionally an additive cost model $c:E\to\mathbb{R}$ used to define extremal regimes.
\end{itemize}

In the project, the standardized input contract is encoded as:
\begin{itemize}
\item \texttt{sources/host\_instance\_schema.json}
\item \texttt{sources/host\_instances/*.json}
\end{itemize}
The host-instance format is the main interface. String theory is only one source of such host instances.

\section{Certificate predicates as formal hypotheses}
The engine emits a \emph{robust certificate object} whose fields are the formal hypotheses used by the theorem chain.

Schema:
\begin{itemize}
\item \texttt{sources/ru/robust\_certificate\_schema.json}
\end{itemize}

A robust certificate instance is a proof-carrying object only to the extent that its fields satisfy the theorem predicates.

We state the predicates at the level of the schema.

\subsection{Tight core predicate}
Fields:
\begin{itemize}
\item \texttt{tight\_core.tight\_edges}
\item \texttt{tight\_core.tight\_scc\_summary}
\item \texttt{tight\_core.tight\_gap\_gamma}
\end{itemize}

Meaning:
canonical normalization and reduced costs determine which edges are tight, and the tight subgraph determines the core combinatorial structure used for recurrence and obstruction classification.

Theorem sources:
\begin{itemize}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/canonical\_normalization.tex}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/aubry\_tight\_core.tex}
\end{itemize}

\subsection{Doeblin predicate}
Fields:
\begin{itemize}
\item \texttt{doeblin.word}, \texttt{doeblin.L}
\item \texttt{doeblin.delta}, \texttt{doeblin.U\_bound}
\item \texttt{doeblin.delta\_prime\_formula}, \texttt{doeblin.U\_prime\_formula}
\end{itemize}

Meaning:
a finite word produces a uniform minorization (positivity) on the active cone with explicit margin $\delta>0$, and the formulas encode robust degradation under perturbations.

Theorem sources:
\begin{itemize}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/doeblin\_word\_search.tex}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/doeblin\_stability.tex}
\end{itemize}

\subsection{Recurrence predicate and obstructions}
Fields:
\begin{itemize}
\item \texttt{recurrence.details.rc1\_holds}
\item \texttt{recurrence.details.gap\_bound\_B}
\item \texttt{recurrence.details.og4\_avoidance\_cycle} (when rc1\_holds is false)
\item periodicity recorded in \texttt{tight\_core.tight\_scc\_summary.periods} (OG1)
\end{itemize}

Meaning:
recurrence converts existence of a mixing block into unavoidable occurrence along tight paths.
When recurrence fails or the core is periodic, the engine records explicit finite obstruction witnesses.

Theorem sources:
\begin{itemize}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/global\_universality\_verification.tex}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/global\_universality\_theorems.tex}
\end{itemize}

\subsection{Contraction and rate-pack predicate}
Fields:
\begin{itemize}
\item \texttt{rate\_pack.projective\_diameter\_bound}
\item \texttt{rate\_pack.tau\_unperturbed}
\item \texttt{rate\_pack.kappa\_prime\_formula}
\end{itemize}

Meaning:
once Doeblin and recurrence predicates hold on the active face, the theorem chain yields an explicit Hilbert-metric contraction and an explicit exponential rate.

Theorem sources:
\begin{itemize}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/contraction\_from\_doeblin.tex}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/explicit\_rate\_pack.tex}
\end{itemize}

\subsection{Global report predicate}
Schema:
\begin{itemize}
\item \texttt{sources/ru/global\_universality\_report\_schema.json}
\end{itemize}

A global report object records:
\begin{itemize}
\item region description,
\item chamber coverage and certificate index,
\item explicit boundary obstruction chambers.
\end{itemize}

Theorem sources:
\begin{itemize}
\item \texttt{sources/theorem\_submission\_v19/rigidity\_gapfree\_global/global\_universality\_artifacts.tex}
\end{itemize}

\section{The theorem chain as implications}
At the level of engine logic, the certificate predicates support the following implication chain:

\begin{center}
Tight core extraction $\Rightarrow$ active combinatorial core
\end{center}
\begin{center}
Doeblin witness on active face $+$ stability margins $\Rightarrow$ bounded projective diameter
\end{center}
\begin{center}
Recurrence (unavoidable mixing) $\Rightarrow$ repeated Doeblin blocks along tight paths
\end{center}
\begin{center}
Repeated Doeblin blocks $+$ Hilbert metric contraction $\Rightarrow$ exponential projective contraction with explicit rate
\end{center}

Each arrow corresponds to a named theorem or lemma component in the theorem submission project listed in the sections above.

If an arrow fails, the engine does not assert universality. It returns a finite obstruction witness (OG1 to OG4), recorded in a robust certificate object plus a host-model witness file.

\section{Audited real-host outcomes}
The project ships multiple real host datasets and audited outcomes.

\subsection{Real-host contraction success}
Fibonacci fusion host data:
\begin{itemize}
\item \texttt{sources/host\_data/fibonacci\_fusion\_data.json}
\end{itemize}

Audited certificate:
\begin{itemize}
\item \texttt{sources/ru/robust\_certificate\_fibonacci\_realhost\_contraction.json}
\end{itemize}

\subsection{Real-host obstructions}
Ising fusion (OG3 and OG2):
\begin{itemize}
\item \texttt{sources/host\_data/ising\_fusion\_data.json}
\end{itemize}

Toric code fusion (OG3):
\begin{itemize}
\item \texttt{sources/host\_data/toric\_code\_fusion\_data.json}
\end{itemize}

The obstruction artifacts are recorded as:
\begin{itemize}
\item robust certificates under \texttt{sources/ru/robust\_certificate\_*.json}
\item bridge wrappers under \texttt{sources/ru\_bridge/bridge\_certificate\_*.json}
\item witness models under \texttt{sources/ru\_bridge/*\_model.json}
\end{itemize}

\section{Reproducibility and mechanical audits}
The project includes a self-check pipeline:
\begin{itemize}
\item \texttt{python tools/ci\_selfcheck.py}
\end{itemize}
This regenerates and schema-validates built-in examples, runs mechanical sanity audits, builds the artifact index, and produces a chamberwise global report.

\section{Scope and non-claims}
This project does not claim physical validation of string theory and does not claim that every string construction fits the finite positivity template.
It claims a reusable certificate technology: on the class of finite host reductions that fit the input contract, universality statements become auditable certificates and failures become explicit finite obstruction witnesses.

\section{References}
See \texttt{docs/References.md} for the bibliographic list used by the project.
\end{document}
