% COC_embedded_loop_proof.tex
%
% This file is intended to be \input{} into SFT I/II/III and the global universality pack.

\providecommand{\TorusP}{\ensuremath{\mathbb{T}_P}}
\providecommand{\COC}{\ensuremath{\mathrm{COC}}}
\providecommand{\ESW}{\ensuremath{\mathrm{ESW}}}
\providecommand{\LCM}{\ensuremath{\mathrm{LCM}}}
\providecommand{\HMR}{\ensuremath{\mathrm{HMR}}}

\subsection*{Canonical proof template: embedded loop / defect witnesses (degree and holonomy)}

\begin{theorem}[Embedded loop degree witness forces circle coverage]
\label{thm:COC-ESW-LDW}
Let $\Sigma$ be a slice and let $i:S^1\hookrightarrow \Sigma$ be a continuous embedding of a loop $E=i(S^1)$.
Let $\phi:\Sigma\to\TorusP$ be continuous and set $f:=\phi\circ i:S^1\to\TorusP$.
If $\deg(f)=k\neq 0$, then $f(S^1)=\TorusP$.
In particular, $E$ is an embedded Syncr\'e witness (\ESW) for circle-type coverage with witness integer $k$.
\end{theorem}

\begin{proof}
Identify $\TorusP=\mathbb{R}/P\mathbb{Z}$ with the standard circle $\mathbb{T}_{2\pi}$ by linear rescaling; degree is invariant under this identification.
Represent $S^1$ as $\mathbb{R}/2\pi\mathbb{Z}$ and lift $f$ to a continuous $\widehat f:\mathbb{R}\to\mathbb{R}$ satisfying
$\widehat f(t+2\pi)=\widehat f(t)+2\pi k$.
By continuity, $\widehat f([0,2\pi])$ is an interval whose endpoints differ by $2\pi k$.
Hence its image modulo $2\pi$ meets every residue class, so $f$ is surjective onto $\mathbb{T}_{2\pi}$, equivalently onto $\TorusP$ \cite{hatcher2002}.
\end{proof}

\begin{theorem}[Holonomy witness forces circle coverage]
\label{thm:COC-ESW-HW}
Let $i:S^1\hookrightarrow \Sigma$ and $\phi:\Sigma\to\TorusP$ be as above.
Assume there exists a lift atlas along $E$ with local lifts $F$ and associated connection form $\alpha$ such that for the loop $\gamma:=i$,
\[
\int_\gamma \alpha = m P \qquad \text{for some integer } m\neq 0.
\]
Then $\phi(E)=\TorusP$, and the induced map $f=\phi\circ i$ has nonzero degree (indeed degree $m$) under the standard circle identification.
\end{theorem}

\begin{proof}
On each chart in the lift atlas, the connection form is $\alpha=dF$ for a local lift $F$.
By additivity of integrals and the cocycle compatibility on overlaps, the loop integral of $\alpha$ equals the total increment of any lifted phase around one circuit of $\gamma$.
Thus the lifted phase changes by $mP$ on one circuit, i.e.\ the winding number is $m\neq 0$.
A nonzero winding number implies nonzero degree, so surjectivity follows from Theorem~\ref{thm:COC-ESW-LDW}.
\end{proof}

\begin{theorem}[Robustness of embedded witnesses under \LCM/\HMR]
\label{thm:COC-ESW-ROBUST}
Let $\tilde\phi:\Sigma\to\TorusP$ be a perturbation of $\phi$ and write $\tilde f:=\tilde\phi\circ i$.

\begin{itemize}[leftmargin=*]
\item \textbf{LDW stability (lift closeness).} Suppose $f$ and $\tilde f$ admit lifts $\widehat f,\widehat{\tilde f}:\mathbb{R}\to\mathbb{R}$ (after identifying $\TorusP$ with $\mathbb{T}_{2\pi}$) such that
\[
\sup_{t\in[0,2\pi]}|\widehat f(t)-\widehat{\tilde f}(t)|<\pi .
\]
Then $\deg(\tilde f)=\deg(f)$.

\item \textbf{HW stability (holonomy margin).} Suppose $\alpha$ and $\tilde\alpha$ are connection forms on $E$ representing the declared lift/connection model for $\phi$ and $\tilde\phi$, and
\[
\left|\int_\gamma(\alpha-\tilde\alpha)\right|<P/2 .
\]
Then the holonomy integer multiple is unchanged.
\end{itemize}

Consequently, any \ESW with nonzero witness integer remains an \ESW under perturbations satisfying the corresponding margin inequality (the engine records these as $\LCM<\pi$ and $\HMR<P/2$ in the Syncr\'e Stability Modulus).
\end{theorem}

\begin{proof}
For LDW, the lift-closeness bound provides a homotopy through lifted maps that never crosses a branch cut, so winding number is constant along the homotopy and degree is preserved.

For HW, the integral bound forces $\int_\gamma \tilde\alpha$ to remain within $P/2$ of the integer multiple $\int_\gamma \alpha=mP$, hence the nearest integer multiple is still $mP$ and the holonomy class is unchanged.
\end{proof}
