\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{longtable}
\usepackage{booktabs}
\usepackage{graphicx}
\usepackage{xurl}
\usepackage{tabularx}
\usepackage{microtype}
\geometry{margin=1in}

\setlength{\emergencystretch}{3em}
\Urlmuskip=0mu plus 1mu
\newcommand{\codeid}[1]{\url{#1}}
\hypersetup{hidelinks}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}

\title{Syncr\'e Form Theory III:\\ Target Inference and Certificate Construction from Data}
\author{Drew Higgins}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This paper turns the SFT decision contract into a data-to-artifact workflow.
Given samples of phase vectors together with a declared model class and tolerance, the task is to infer the correct target (full torus versus coupled subtorus coset), estimate witness integers (degree or holonomy multiple) on an observed subsystem \cite{hatcher2002,munkres2000}, and attach robustness margins and convergence certificates only when justified \cite{walters1982,einsiedler2011,pavliotis2014}.
The resulting output type is exhaustive inside COC: a witness stability certificate (WSC) on the correct target, or a canonical obstruction report (WOB) with evidence.
\end{abstract}

\section*{Global universality meaning}
\input{COC_definition.tex}

\input{WELL_POSEDNESS_SPINE.tex}

\section*{Universality-from-cyclic-structure schema}
\input{COC_universality_schema.tex}

\input{COC_rotation_proof.tex}

\input{COC_embedded_loop_proof.tex}
\input{COC_multicycle_proof.tex}

\section{Data model and declared context}

A declared context specifies:
\begin{itemize}[leftmargin=*]
\item a slice or subsystem to be observed (global slice, embedded loop, or coupled target),
\item a phase observable valued in a circle or torus, with a chosen period vector $\mathbf P$,
\item a coupling hypothesis class (none, affine diagonal, or integer-matrix constraints),
\item a perturbation model and desired margins (LCM/HMR),
\item whether any statistical claims are intended (RNF or NDB+MXC).
\end{itemize}

Data consist of samples of phase values, either along a loop parameter (for winding) or over time (for drive increments and coupling inference).

\section{Obstruction exhaustiveness by predicate exhaustion}

Data are allowed to produce certificates only when the declared regime validators hold.
When a validator fails, the correct output is a finite obstruction code with auditable evidence.
The predicate exhaustion lemma suite fixes this boundary formally.

\input{PREDICATE_EXHAUSTION_LEMMAS.tex}

\input{VALIDATOR_PREDICATE_SPEC.tex}

\input{WOB_CATALOG_COMPLETENESS.tex}

\section{Target inference: full torus versus coupled target}

\subsection*{CST inference theorem (small-integer constraints with stability)}

The practical bottleneck for multi-cycle systems is deciding whether the correct target is the full torus or a coupled subtorus coset.
SFT treats this as a bounded integer-relation problem with a stability margin and an obstruction fallback \cite{lenstra1982,ferguson1992pslq}.

\begin{theorem}[CST inference with stability or obstruction]
Fix a sampling model and a tolerance corresponding to the declared perturbation scale.
There is a conservative procedure that, given sampled phase vectors in $\mathbb{T}_{\mathbf P}$, outputs exactly one of:
\begin{itemize}[leftmargin=*]
\item \textbf{CST output.} A candidate small-integer constraint description $A y=\Psi$ together with a quantitative residual bound, constraint rank $r:=\mathrm{rank}(A)$, and an explicit stability margin within the declared tolerance.
\item \textbf{Full-torus output.} A declaration that no constraint was found in the declared bounded search, together with an explicit quantitative gap to the best relation in that search.
\item \textbf{Obstruction output.} A canonical WOB code with evidence indicating why target inference is ill-posed at the declared tolerance (typically ambiguity, insufficient excitation, or resonance masking).
\end{itemize}
\end{theorem}

\input{TARGET_IDENTIFIABILITY_THEOREM.tex}

\begin{definition}[Target inference problem]
Given sampled phase vectors $y(t)\in \mathbb{T}_{\mathbf P}$, determine whether the image behaves as:
\begin{itemize}[leftmargin=*]
\item full-rank on $\mathbb{T}_{\mathbf P}$, or
\item restricted to a coupled subtorus coset $\mathrm{CST}$ described by integer constraints $A y=\Psi$.
\end{itemize}
\end{definition}

\section{Witness integer inference from samples}

\subsection*{Loop samples (degree)}
If a loop is sampled by phases $p_0,p_1,\dots,p_N\in [0,P)$, lift-unwrapping provides a lifted sequence $\tilde p_i$ with small jumps, and the degree estimate is
\[
k \approx \frac{\tilde p_N-\tilde p_0}{P}\in \mathbb{Z}.
\]
This is stable under the LCM margin: if two lifts remain within $\pi$, the integer cannot change \cite{hatcher2002}.

\subsection*{Holonomy samples}
If the observable is realized via a local lift atlas or a connection, holonomy inference uses integrated loop data to estimate the multiple $m$ such that the loop integral equals $mP$. Stability is controlled by the HMR margin.

\section{Certificate construction}

A data-driven WSC records:
\begin{itemize}[leftmargin=*]
\item the target (full torus or CST) and baseline rule,
\item the witness integer $k$ (degree) or $m$ (holonomy multiple),
\item stability margins (LCM/HMR),
\item pointers to the data used and the inference method used to compute the fields.
\end{itemize}

A data-driven WOB is the correct output when inference detects a finite obstruction: no subsystem, trivial winding, coupling mismatch, resonance masking, and so on.

\section{Convergence certificates from data}

SFT treats statistical claims as artifacts:
\begin{itemize}[leftmargin=*]
\item \textbf{RNF certificate.} Provide evidence that increments are nonresonant (up to declared tolerance) and state Haar convergence on the correct target \cite{walters1982,einsiedler2011,kuipers1974}.
\item \textbf{NDB+MXC certificate.} Declare a noise-to-diffusion bridge and record a verified mixing inequality with a rate from the thresholds ledger \cite{pavliotis2014,evans2010,meyn2009}.
\end{itemize}

\bibliographystyle{plain}
\bibliography{sft3_bibliography}
\end{document}
