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\title{Syncr\'e Form Theory (SFT): Coverage Phase Fields, Witness Engines, Coupled Targets, and Stability Signatures}
\author{Drew Higgins}
\date{}

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\begin{document}
\maketitle

\begin{abstract}
Syncr\'e Form Theory (SFT) provides a target-correct, auditable semantics for phase-coverage statements on slices or declared subsystems.
A declared context specifies a slice/subsystem, a circle- or torus-valued phase observable, a target rule (including coupling conventions), a witness mechanism, and a perturbation model.
Within the Cyclic Observability Class (COC), the SFT engine resolves a declared claim into exactly one checkable object:
a Witness Stability Certificate (WSC) proving coverage on the correct target with explicit robustness margins, or a Witness Obstruction (WOB) naming the first failed validator predicate in a finite codebook with evidence pointers.
The theory packages standard forcing mechanisms (equivariant circle actions, loop degree, holonomy) together with a coupled-target calculus for multi-cycle observables and a conservative target-inference theorem producing coupled subtorus descriptions or obstructions at a declared tolerance.
The release bundle includes schemas, audits, benchmarks, a witness atlas tied to shipped artifacts, and hash-verified reproducibility scripts.
\end{abstract}

\input{FRONT_MATTER.tex}

\input{ESSAY_INTRO.tex}

\input{DIRECTION_HIGHER_THEOREM.tex}

\input{PROVES_BLOCK.tex}

\input{WHY_THIS_IS_UNIVERSAL.tex}

\input{SCOPE_WALL.tex}

\input{WELL_POSEDNESS_SPINE.tex}

\input{PRIMITIVES_ENGINE_THEOREM.tex}

\input{ENGINE_COMPLETENESS.tex}

\input{STABILITY_MODULUS.tex}

\input{MODULUS_LAWS.tex}

\section*{Modulus stability law (summary)}

When a certificate records the Syncr\'e Stability Modulus $\mathfrak{S}$, its meaning is operational: within the declared perturbation model, the witness integer cannot change unless a stability margin is crossed. Under frame equivalence (FEQ) and dynamics preserving the recorded invariants, the certificate class is unchanged. When margins are crossed or assumptions fail, the correct output is an obstruction report rather than a modified certificate.

\input{CLASSIFICATION_CHART.tex}

\input{CANONICAL_QUESTIONS.tex}
\input{DIAGNOSTICS_TABLE.tex}
\input{OBSTRUCTION_TABLE.tex}

\section{Contribution summary table}

\input{contributions_table.tex}

\input{RELATED_WORK.tex}

\section{Definitions: sliced arenas and the Syncr\'e Coverage Law}

Fix $P>0$ and $\mathbb{T}_P:=\mathbb{R}/P\mathbb{Z}$ \cite{munkres2000}. A \emph{sliced arena} is a decomposition
\[
M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s,
\]
with slice parameter $s$ representing a chosen global-instant labeling.

A \emph{phase field} is a map $\Phi:M\to\mathbb{T}_P$. Write $\Phi_s:=\Phi|_{\Sigma_s}$.

\subsection*{SCL (Syncr\'e Coverage Law)}
\[
\boxed{\ \forall s\in\mathbb{R},\qquad \Phi(\Sigma_s)=\mathbb{T}_P.\ }
\]

\subsection*{SLL (Syncr\'e Lift Law)}
A lift interface is specified by $\Phi = F \bmod P$ for a real-valued lift $F$ on the relevant domain (globally or via a lift atlas), enabling nonlinear deformation while keeping analysis in $\mathbb{R}$.

\section{Witness engines: action and loop mechanisms}

SFT is witness-first: coverage claims are supported by finite witness data.

\subsection{Slice-level action witness template (AWT)}
If a slice $\Sigma$ carries a continuous $S^1$-action $A:S^1\times\Sigma\to\Sigma$ and there exists an equivariant phase observable $\phi:\Sigma\to\mathbb{T}_P$ with nonzero-slope phase response $\chi:S^1\to\mathbb{T}_P$ \cite{bredon1972},
\[
\phi(A(\theta,x))=\phi(x)\oplus \chi(\theta),
\qquad \chi(S^1)=\mathbb{T}_P,
\]
then equivariance forces stabilizers into the kernel of $\chi$, so in the nonzero-slope regime the action is circle-type on the forcing domain. In particular, $\phi(\Sigma)=\mathbb{T}_P$.

\subsection{Embedded witnesses (ESW)}
Many realizations expose phase structure on embedded subsystems. An embedded Syncr\'e witness is an embedding $i:E\hookrightarrow\Sigma_s$ such that $\Phi\circ i(E)=\mathbb{T}_P$. For $E\cong S^1$, this is certified by:
\begin{itemize}[leftmargin=*]
\item \textbf{LDW (Loop degree witness).} $\deg(\Phi\circ \gamma)\neq 0$ for a loop $\gamma:S^1\to\Sigma_s$ \cite{hatcher2002}.
\item \textbf{HW (Holonomy witness).} With a connection form $\alpha$ (from a lift atlas), $\int_{\gamma}\alpha = mP$ with $m\neq 0$.
\end{itemize}

\section{Universality (scope and meaning)}

\input{UNIVERSALITY_SECTION.tex}

\input{UNIVERSALITY_SCORECARD.tex}

The next step is to keep the target correct when multiple cycles interact: coupling changes what coverage and uniformity mean.

\section{Coupled Atlas (targets and baselines)}

\section*{Necessity in the presence of frame equivalence and coupling}

Two correctness principles prevent hidden assumptions.

First, frame or gauge changes that preserve slice correspondence and phase meaning do not change the content of a Syncr\'e claim. Forcing, obstruction, and robustness statements are therefore expressed on equivalence classes rather than coordinate choices.

Second, coupling changes the meaning of coverage. When constraints restrict the image to a coupled target $\mathrm{CST}_s$, necessity statements refer to surjectivity and stability \emph{on $\mathrm{CST}_s$}, not on the full torus. Any statistical baseline must be coupled Haar on $\mathrm{CST}_s$.

Coupling constraints restrict the image to a coupled subtorus $\mathrm{CST}_s$. The correct target and baseline must be selected using CTP/CHT. Table \ref{tab:coupled-atlas} summarizes the coupling classes used in SFT.

\begin{center}
\input{coupled_atlas_table.tex}
\end{center}

\section{Robustness, certificates, and obstructions}

Observed principles must persist under perturbations. SFT packages witness claims as artifacts:

\begin{itemize}[leftmargin=*]
\item \textbf{WSC (Witness Stability Certificate):} stores witness tier, mechanism, value, perturbation model, and explicit stability margins.
\item \textbf{WOB (Witness Obstruction):} stores a named failure mode with evidence and recommended next moves.
\end{itemize}

\section{Dynamics boundary (preservation vs breaking)}

SFT separates witness preservation from mixing. Preservation conditions are encoded as certificate margins and are summarized below.

\begin{center}
\input{dynamics_boundary_table.tex}
\end{center}

\section{Dynamics and mixing: preservation vs convergence}

\input{CONVERGENCE_CERTIFICATES.tex}

Deterministic transport can preserve witnesses (degree and, under suitable conditions, holonomy), but transport alone is not asserted to be universally mixing. Convergence to Haar is conditional and becomes theorem-level once a noise-to-diffusion bridge is adopted and mixing certificate hypotheses are satisfied \cite{walters1982,einsiedler2011,pavliotis2014}.

\section{Appendix A: SFT engine artifacts}

\input{ARTIFACTS_APPENDIX.tex}

The paper ships with artifact contracts and example objects in the accompanying bundle:

\begin{itemize}[leftmargin=*]
\item Contracts: \texttt{artifacts/\allowbreak contracts/\allowbreak }
\item Schemas: \texttt{artifacts/\allowbreak schemas/\allowbreak }
\item Example artifacts: \texttt{artifacts/\allowbreak artifacts/\allowbreak }
\item Audit script: \texttt{artifacts/\allowbreak audit/\allowbreak audit\_\allowbreak wsc.py}
\end{itemize}

\section{Appendix B: worked example}

A worked example pack (Earth local time plus coupled solar--lunar constraint) is provided in the bundle.

\section*{Appendix: Referee Map}
\input{REFEREE_MAP.tex}

\section*{Appendix: Worked Examples}
\input{EXAMPLES_APPENDIX.tex}

\input{CAPSTONE_COC_ENGINE.tex}

\def\SFTINMAIN{1}
\input{GLOBAL_SYNCRE_UNIVERSALITY_THEOREM.tex}

\section{References}

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\end{document}
