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

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}

\begin{document}
\maketitle

\input{FRONT_MATTER.tex}

\input{ESSAY_INTRO.tex}

\input{PROVES_BLOCK.tex}

\input{PRIMITIVES_ENGINE_THEOREM.tex}

\input{ENGINE_COMPLETENESS.tex}


\input{STABILITY_MODULUS.tex}


\section*{Modulus stability law (summary)}

When a certificate records the Syncre 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}

\section*{Well-posedness axiom and sharp boundary}

SFT treats universality claims as statements about \emph{forced and stable} structure. A claim belongs to SFT when it can be expressed on a correct target and certified by the engine.

\begin{quote}
\textbf{Well-posedness axiom (forced-and-stable regime).} A forced Syncre coverage claim is meaningful if and only if a witness stability certificate is producible on the correct target in the declared context. When no such certificate is producible, the correct output is an obstruction report identifying which finite obstruction class blocks certification.
\end{quote}

This axiom makes the boundary sharp. In the absence of cyclic symmetry or an observable cyclic subsystem (loop/orbit/defect) supporting a circle- or torus-valued phase, forced Syncre claims are inapplicable and no certificate exists. Conversely, when cyclic structure and nondegeneracy are present and no obstruction applies, coverage is forced on the appropriate target.





\input{DIAGNOSTICS_TABLE.tex}

\input{OBSTRUCTION_TABLE.tex}





\begin{abstract}
Syncr\'e Form Theory (SFT) formalizes a recurring observable structure: a single global instant can be expressed as a spatial field of cycle-valued phase labels. SFT begins with Syncr\'e as a principle-identical proxy (PIP) for a higher relational form aim (RFP), then strengthens the proxy through invariants, witnesses, obstruction taxonomy, quantitative diagnostics, coupling targets, and dynamics. The core Syncr\'e Coverage Law (SCL) is a slice-wise coverage statement for a phase field $\Phi:M\to\mathbb{T}_P$ on a sliced arena $M=\bigsqcup_{s}\Sigma_s$. SFT provides meta-witness templates (action-equivariance and loop-degree/holonomy witnesses), robustness margins under perturbations, correct target selection under coupling (coupled subtori and coupled Haar baselines), and conditional mixing certificates under explicit noise-to-diffusion bridges. The framework is packaged as an artifact-producing engine: given a witness context, it outputs either a Witness Stability Certificate (WSC) with explicit margins or a Witness Obstruction (WOB) with evidence.
\end{abstract}


\section*{Minimal template, forcing, and dichotomy (core statements)}

\subsection*{Minimal structure template }
A forced Syncre claim requires only: a sliced arena, a cyclic symmetry (or an embedded cyclic subsystem), and a nondegenerate cycle-valued observable (surjective response or nonzero winding/holonomy). This is the irreducible data behind SCL.

\subsection*{Minimal forcing }
Within the equivariant $S^1$-action regime, slice-wise coverage on the correct target is governed by the induced homomorphism $\chi:S^1\to\mathbb{T}_P$ and the response map $\phi$:
\begin{itemize}[leftmargin=*]
\item If $\chi$ has nonzero slope (equivalently, $\chi$ is surjective), then equivariance forces stabilizers into $\ker\chi$, so circle-type orbits are automatic on the equivariance domain and coverage reduces to $\phi(\Sigma)=\mathbb{T}_P$.
\item If $\chi$ is trivial, $\phi$ factors through the orbit space $\Sigma/S^1$; coverage is possible only by surjectivity of the induced map on $\Sigma/S^1$.
\end{itemize}

\subsection*{Obstruction dichotomy }
If a coverage claim fails within the declared hypothesis regime, then a finite obstruction class occurs and the engine outputs a WOB with evidence. Otherwise the engine outputs a WSC with explicit stability margins.



\section{Contributions and what is new}

SFT does not claim novelty in the classical ingredients (degree, equivariance, torus rotations, diffusion). The contribution is the unified, artifact-backed theory package: witness engines, correct coupled targets, certificate-or-obstruction discipline, explicit robustness margins, and scoped mixing certificates.

\input{contributions_table.tex}

\section{Foundations: RFP, PIP, and the Syncr\'e spine}

SFT is motivated by a higher relational aim:

\begin{quote}
A ``point'' (any reference marker, categorical object, or local datum) does not possess form in isolation; its form is inseparable from the structure in which it is embedded.
\end{quote}

Because this is difficult to formalize directly, SFT begins with Syncr\'e as a \emph{principle-identical proxy} (PIP): a minimal observable structure that preserves the invariant/gauge logic of the higher principle. The relational aim is summarized as:

\begin{itemize}[leftmargin=*]
\item \textbf{RFP (Relational Form Principle).} Form is an invariant of embeddedness: points are studied as $(M,p)$ rather than isolated $p$.
\item \textbf{PIP (Principle-Identical Proxy).} A minimal observable structure preserving invariant/gauge logic of the higher principle it proxies.
\end{itemize}

SFT proceeds by strengthening the proxy: refining admissible frames and slicings, enlarging invariant sets, classifying obstructions, quantifying collapse versus uniformity, and introducing dynamics with preservation and convergence theorems.

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

Fix $P>0$ and $\mathbb{T}_P:=\mathbb{R}/P\mathbb{Z}$. 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}

What follows isolates the smallest inputs that force coverage, then describes how those inputs can be certified in finite artifacts and how failures are classified.


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 $\chi:S^1\to\mathbb{T}_P$,
\[
\phi(A(\theta,x))=\phi(x)\oplus \chi(\theta),
\qquad \chi(S^1)=\mathbb{T}_P,
\]
then equivariance forces stabilizers into the kernel of $\chi$, so in the nonzero-slope regime the action is automatically 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 $\gamma:S^1\to\Sigma_s$.
\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}

The next step is to keep the target correct when multiple cycles interact: coupling changes what coverage and uniformity can even 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 Syncre 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. In particular, 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}

This certificate-or-obstruction discipline is the core engine form of the theory.

\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 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.

\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{GLOBAL_SYNCRE_UNIVERSALITY_THEOREM.tex}

\section{References}

SFT relies on standard tools from degree theory, group actions, torus translations and equidistribution, and diffusion or advection--diffusion on compact manifolds. The references below are sufficient for the background facts used in this note.

\begin{thebibliography}{9}

\bibitem{hatcher} A. Hatcher. \emph{Algebraic Topology}. Cambridge University Press, 2002.

\bibitem{walters} P. Walters. \emph{An Introduction to Ergodic Theory}. Springer, 1982.

\bibitem{katok-hasselblatt} A. Katok and B. Hasselblatt. \emph{Introduction to the Modern Theory of Dynamical Systems}. Cambridge University Press, 1995.

\bibitem{bredon} G. E. Bredon. \emph{Introduction to Compact Transformation Groups}. Academic Press, 1972.

\bibitem{pavliotis} G. A. Pavliotis. \emph{Stochastic Processes and Applications}. Springer, 2014.

\end{thebibliography}

\end{document}