# Global Syncr\'e Universality: Forcing, Obstructions, Robustness, and Dynamics

## Purpose

Syncr\'e Form Theory (SFT) is witness-first.

The Syncr\'e Coverage Law (SCL) is the slice-wise statement that, at each global instant, the phase field attains every phase value somewhere on the slice.

This note presents a compact theorem package showing that, in a minimal nondegenerate regime of cyclic symmetry and equivariant phase observables, coverage is forced, stable, and admits a finite obstruction catalog.

Throughout, \(\mathbb{T}_P := \mathbb{R}/P\mathbb{Z}\) is the phase circle of period \(P>0\).

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## Core definitions

**Sliced arena.** A sliced arena is a decomposition \(M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s\).

**Phase field and SCL.** A phase field is a map \(\Phi:M\to \mathbb{T}_P\). Write \(\Phi_s:=\Phi|_{\Sigma_s}\).
The Syncr\'e Coverage Law (SCL) is
\[
\forall s,\quad \Phi_s(\Sigma_s)=\mathbb{T}_P.
\]

**Equivariant circle action.** Let \(A:S^1\times \Sigma\to\Sigma\) be a continuous \(S^1\)-action on a slice \(\Sigma\).
A phase observable \(\varphi:\Sigma\to\mathbb{T}_P\) is \(\chi\)-equivariant if there exists a continuous homomorphism \(\chi:S^1\to\mathbb{T}_P\) such that
\[
\varphi(A(\theta,x))=\varphi(x)\oplus\chi(\theta).
\]
For \(x\in\Sigma\), \(\mathrm{Stab}(x):=\{\theta\in S^1: A(\theta,x)=x\}\).

**Nonzero slope.** Nonzero slope means \(\chi\) is nontrivial (equivalently surjective). Under standard identifications there exists \(d\in\mathbb{Z}\) with
\[
\chi(\theta)=\frac{P}{2\pi}d\,\theta\ (\mathrm{mod}\ P),
\]
and nonzero slope is \(d\neq 0\).

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## Forcing by cyclic symmetry

**Circle-action forcing.** If \(\varphi\) is \(\chi\)-equivariant and \(\chi\) has nonzero slope, then \(\varphi(\Sigma)=\mathbb{T}_P\).

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## Characterization under an equivariant action

Coverage holds if and only if either:

- \(\chi\) has nonzero slope, or
- \(\chi\) is trivial and the induced quotient factor \(\bar\varphi:\Sigma/S^1\to\mathbb{T}_P\) is surjective.

In the nonzero-slope regime, coverage is automatic.

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## Witness-or-obstruction form

SFT packages universality as a dichotomy: either a witness certificate exists (WSC), or a finite obstruction object exists (WOB).
Equivalently,
\[
\neg\,\mathrm{SCL}_{\mathrm{SFT}}(\Sigma)\ \Longleftrightarrow\ \text{some WOB obstruction code occurs.}
\]

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## Robustness (quantitative)

Define the equivariance defect on \(D\subseteq\Sigma\) by
\[
\mathrm{Def}_D(A,\varphi,\chi):=\sup_{\theta\in S^1,\ x\in D} d_{\mathbb{T}_P}(\varphi(A(\theta,x)),\ \varphi(x)\oplus\chi(\theta)).
\]
If \(\chi\) has nonzero slope and \(\mathrm{Def}_D\le \varepsilon\), orbit images are \(\varepsilon\)-dense. Under a lift with controlled derivative error, full coverage follows.

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## Multi-cycle forcing and coupled targets

For \(\mathbb{T}_{\mathbf P}=\prod_{j=1}^k \mathbb{T}_{P_j}\), a \(T^k\)-equivariant observable \(\Phi\) has image \(\mathrm{Im}(\chi)\).
When coupling restricts the image to a coupled subtorus coset \(\mathrm{CST}_s\), the correct target is \(\Phi(\Sigma_s)=\mathrm{CST}_s\) and the correct baseline is coupled Haar on \(\mathrm{CST}_s\).

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## Ergodic universality

For irrational torus translations (nonresonant increments), time averages converge to Haar on the correct target.

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## Dynamics preservation and diffusion

If dynamics commute with the symmetry and preserve the invariants used by witness certificates (degree, holonomy class, coupled relation), then witnesses persist in time.
On tori, diffusion drives distributions exponentially fast to Haar; under standard irreducibility assumptions for advection–diffusion, convergence is exponential.

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## Minimal structure and sharp boundary

For forcing, the minimal ingredients are a sliced arena, a nontrivial cyclic symmetry action, and an equivariant phase observable with nonzero slope.
Without admissible cyclic symmetry in the declared model class, action-based forcing is inapplicable; claims must rely on embedded witnesses or add structure.

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## Capstone global theorem (informal statement)

Under admissible cyclic symmetry, a nondegenerate equivariant phase field with no degree/holonomy obstruction, and correct target selection under coupling, slice-wise coverage holds on the correct target, is robust under admissible perturbations, persists under admissible dynamics, converges to the correct Haar target under nonresonant drives, and converges exponentially under diffusion/noise. Failures are classified by the finite WOB obstruction catalog.
