\subsection*{Universality: from witness catalog to structural necessity}

Universality in Syncr\'e Form Theory (SFT) means universality of mechanism.
SFT does not claim that every system satisfies the Syncr\'e Coverage Law (SCL).
Instead, SFT proves that when cyclic symmetry and equivariance are present in the minimal nondegenerate way, coverage is forced, stable, and admits a finite certificate-or-obstruction description.

\subsection*{Minimal forcing on a slice (circle-action engine)}

Let $\Sigma$ be a slice and let $A:S^1\times\Sigma\to\Sigma$ be a continuous $S^1$-action. Let $\phi:\Sigma\to\mathbb{T}_P$ be an equivariant phase observable:
\[
\phi(A(\theta,x))=\phi(x)\oplus\chi(\theta),
\]
for a continuous homomorphism $\chi:S^1\to\mathbb{T}_P$.
A continuous homomorphism $S^1\to S^1$ is either trivial or surjective, so ``nonzero slope'' is equivalent to surjectivity of $\chi$.

A key structural fact is that equivariance forces stabilizers into the kernel:
if $A(\theta,x)=x$, then equivariance implies $\chi(\theta)=0$.
Thus in the nonzero-slope regime (finite kernel), orbits in an equivariance domain are automatically circle-type.

\begin{itemize}[leftmargin=*]
\item If $\chi$ is surjective and $\phi$ is genuinely $\chi$-equivariant on $\Sigma$, then $\phi(\Sigma)=\mathbb{T}_P$ (no separate ``free orbit'' hypothesis is needed).
\item If $\chi$ is trivial, $\phi$ factors through the orbit space $\Sigma/S^1$, and coverage is possible only by surjectivity of the induced map on the quotient.
\end{itemize}

This is the basic inevitability move: in the nonzero-slope equivariant regime, cyclic symmetry forces coverage.

\subsection*{Finite obstruction classification (witness-first)}

When a coverage claim fails in the declared model class, it is not mysterious.
SFT is certificate-driven: either a finite witness certificate exists (WSC), or a finite obstruction object exists (WOB).
The obstruction codes correspond to the finitely many ways the witness-engine hypotheses can fail:

\begin{itemize}[leftmargin=*]
\item slope zero (no nonzero-slope response available)
\item orbit collapse (no circle-type orbit in the claimed forcing domain; symmetry not realized on the domain)
\item equivariance defect (the observable is not genuinely equivariant in the adopted model class, or defect exceeds tolerance)
\item disconnected slice topology relative to the measurement domain
\item holonomy inconsistency (when the model uses lift/connection certification)
\item coupling mismatch / resonant locking (wrong target or rank deficiency in multi-cycle settings)
\end{itemize}

This yields the universality leverage: witnessed coverage or a named structural obstruction.

\subsection*{Robustness: coverage is not fragile}

Universality requires stability.
SFT packages witnesses with explicit margins:

\begin{itemize}[leftmargin=*]
\item degree witnesses persist under lift-closeness margins $<\pi$
\item holonomy witnesses persist under loop-integral margins $<P/2$
\item action-forcing is stable under small equivariance defect, producing $\varepsilon$-dense coverage on a circle-type orbit and exact surjectivity under explicit lift-derivative or Fourier-defect bounds
\end{itemize}

These margins are the mathematical content of ``exists stably.''

\subsection*{Multi-cycle forcing and correct coupled targets}

In multi-cycle settings $\Phi:M\to\mathbb{T}_{\mathbf{P}}$, coupling constraints typically restrict the image to a coupled subtorus $\mathrm{CST}_s\subseteq\mathbb{T}_{\mathbf{P}}$.
Universality must therefore target $\mathrm{CST}_s$, not the full torus.
Under the correct target selection and full-rank action on the target coordinates, forcing applies on $\mathrm{CST}_s$ and the correct uniform baseline is Haar on $\mathrm{CST}_s$.

\subsection*{Ergodic upgrade, dynamics preservation, and diffusion}

SFT separates preservation from mixing.

\begin{itemize}[leftmargin=*]
\item Under rational independence of drive frequencies, torus translations are uniquely ergodic and time averages converge to Haar on the correct target (full torus or coupled subtorus).
\item Deterministic transport can preserve witnesses (degree and, under suitable conditions, holonomy) but is not claimed to be universally mixing.
\item Once a defensible noise-to-diffusion bridge is adopted, diffusion (or advection--diffusion under standard irreducibility conditions) yields exponential convergence to Haar, strengthening uniformity over time.
\end{itemize}

\subsection*{Minimal structure and sharp boundary}

The forcing implication of SCL requires only:

\begin{itemize}[leftmargin=*]
\item a sliced arena
\item a nontrivial cyclic symmetry action in the admissible class
\item an equivariant phase observable with nonzero slope
\end{itemize}

If cyclic symmetry is absent in the admissible automorphism class, the forcing mechanism cannot apply.
This sharp boundary prevents triviality: Syncr\'e is universal exactly on the cyclic-symmetry side of the boundary.

\subsection*{Capstone statement}

Putting the layers together: on a sliced arena, if each slice admits admissible cyclic symmetry and the phase field is equivariant with nonzero slope without degree/holonomy obstruction (and with correct CST target selection in coupled settings), then slice-wise coverage holds on the correct target, is robust under admissible perturbations, is preserved under admissible dynamics, converges to the correct Haar target under nonresonant drives, and converges exponentially under diffusion.

A full theorem package and expanded proofs are provided in the bundle under \texttt{global\_universality/}.
