\section*{What this paper proves}

\begin{itemize}[leftmargin=*]
\item A precise coverage law for cycle-valued phase fields on sliced arenas, including the correct lift/holonomy interfaces needed for nonlinear deformation.
\item Forcing results: under minimal cyclic symmetry plus nondegenerate equivariance (or nonzero winding/holonomy on an embedded loop), full-cycle coverage is structurally forced on the appropriate target.
\item A finite obstruction taxonomy: when a coverage claim fails in the declared regime, at least one named structural obstruction must occur, and the correct response is an obstruction report with evidence.
\item Robustness: witnesses persist under explicit quantitative margins, and quantitative anti-collapse bounds follow under stated regularity hypotheses.
\item Correct target selection under coupling: when constraints restrict the image, coverage and uniformity must be stated on the coupled target (CST) with the coupled Haar baseline.
\item Convergence upgrades when justified: under nonresonant drives, time averages converge to Haar on the correct target \cite{walters1982,einsiedler2011}; under explicit diffusion/noise models, uniformity improves at certified rates \cite{pavliotis2014,evans2010}.
\end{itemize}
