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

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\maketitle

\begin{abstract}
Syncre 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 Syncre 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 Syncre 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 and recommended next moves.
\end{abstract}


\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{../pass_03_contributions_novelty/contributions_table.tex}

\section{Foundations: RFP, PIP, and the Syncre 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 Syncre 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 Syncre 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 (Syncre Coverage Law)}
\[
\boxed{\ \forall s\in\mathbb{R},\qquad \Phi(\Sigma_s)=\mathbb{T}_P.\ }
\]

\subsection*{SLL (Syncre 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 surjective 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 any full orbit $O\cong S^1$ satisfies $\phi(O)=\mathbb{T}_P$, hence $\phi(\Sigma)=\mathbb{T}_P$.

\subsection{Embedded witnesses (ESW)}
Many physical realizations expose phase structure on embedded subsystems. An embedded Syncre 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)}

\subsection*{Universality thesis}
Universality in SFT means universality of mechanism, not universality of outcome. SFT does not claim every physical system satisfies SCL. Instead:

\begin{quote}
Whenever physical structure supplies a nontrivial cycle-valued phase observable together with nonzero winding or holonomy on a spatial subsystem, full-cycle observability holds on that subsystem and admits a stable finite witness.
\end{quote}

\subsection*{Copy-ready universality section}
The project includes a fully scoped, copy-ready universality section (developed in Univ Pass 10). For the draft, we incorporate it verbatim-style as the authoritative statement of scope, targets, robustness, coupling, dynamics, and nonclaims. See the companion file in this bundle:
\begin{itemize}[leftmargin=*]
\item \texttt{universality/UNIV\_PASS\_10\_PAPER\_SECTION.md}
\end{itemize}


\section{Coupled Atlas (targets and baselines)}

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}
\label{tab:coupled-atlas}
\input{../pass_07_coupled_atlas/coupled_atlas_table.tex}
\end{center}


In multi-cycle settings $\Phi:M\to\mathbb{T}_{\mathbf{P}}$, coupling constraints restrict the image to a coupled subtorus $\mathrm{CST}_s$. The correct coverage target is $\Phi(\Sigma_s)=\mathrm{CST}_s$ and the correct uniformity baseline is Haar on $\mathrm{CST}_s$ (a pushforward from free coordinates), not Haar on the full torus.

\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, integer value, perturbation model, and explicit stability margins (LCM, HMR).
\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 (DMG) and are summarized below.

\begin{center}
\input{../pass_08_dynamics/dynamics_boundary_table.tex}
\end{center}

\section{Dynamics and mixing: preservation vs convergence}

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 (NDB) is adopted and mixing certificates (MXC) are satisfied.

\section{Appendix A: SFT engine artifacts}

\input{../pass_05_artifacts_appendix/ARTIFACTS_APPENDIX.tex}


This draft is designed to ship with artifact contracts and example objects:

\begin{itemize}[leftmargin=*]
\item Engine contract: \texttt{engine/contracts/engine\_contract.md}
\item Schemas: \texttt{engine/schemas/}
\item Hypotheses menu: \texttt{engine/contracts/analytic\_hypotheses\_menu.md}
\item Threshold ledger: \texttt{engine/contracts/constants\_thresholds.md}
\item Audit: \texttt{engine/audit/audit\_wsc.py}
\item Example artifacts: \texttt{engine/artifacts/}
\end{itemize}

\section{Appendix B: worked example}

% Worked example pack



A worked example pack (Earth local time + SOL coupling) is provided:
\begin{itemize}[leftmargin=*]
\item \texttt{appendix/DI\_PASS\_I\_WORKED\_EXAMPLE.md}
\end{itemize}

\section*{Appendix: Referee Map}
\input{../pass_06_referee_map/REFEREE_MAP.tex}

\section*{Appendix: Worked Examples}

See `score10/pass_09_examples/EXAMPLES_APPENDIX.md` and the corresponding artifacts in `engine/artifacts/`.

\section{References}

SFT relies on standard tools from degree theory, group actions, torus translations/equidistribution, and diffusion/advection--diffusion on compact manifolds. A bibliography file is included for consolidation and expansion.

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