# Global Syncré Universality Package

## Purpose

Syncré Form Theory (SFT) is a witness-first framework for phase coverage on declared slices or subsystems.

The Syncré Coverage Law (SCL) is the slice-wise statement that, at each global instant, a phase field attains every phase value on the correct target.

This package collects the core theorem chain used throughout the bundle:

- cyclic symmetry + equivariance forces coverage
- loop degree and holonomy force coverage on embedded subsystems
- coupling determines the correct target (a coupled subtorus coset CST) and the correct baseline (coupled Haar)
- robustness is expressed through explicit margins stored in certificates
- when statistical statements are asserted, they are asserted on the correct target and require explicit hypotheses

Throughout, Τₚ := ℝ/Pℝ denotes the phase circle of period P>0.

## Well-posedness spine

SFT treats a universality question as well-posed only after the claim specifies:

- a phase observable (circle/torus-valued),
- a slice or subsystem where phases are observed,
- a target rule (circle, full torus, or CST) and baseline convention,
- a witness mechanism admissible in the declared model class.

### Cyclic Observability Class (COC)

The Cyclic Observability Class (COC) is the named applicability class where the above items are declared in a form that can be audited by the engine.

### WSC/WOB meaning

Within COC, the engine output is one of:

- **WSC:** coverage on the correct target with explicit stability margins
- **WOB:** a canonical obstruction code with evidence pointers

Outside COC, the correct output is a boundary-type WOB indicating which prerequisite for meaning is missing.

## Engine exhaustiveness inside COC

Inside COC, outcome types are exhaustive:

- a WSC is admissible only if all required validator predicates hold
- if any required predicate fails, the first failed predicate determines a canonical WOB code

Pointers:

- contract: `engine/contracts/engine_contract.md`
- predicate spec: `engine/contracts/VALIDATOR_PREDICATE_SPEC.md`
- predicate ordering: `engine/contracts/validator_predicates.json`
- codebook: `engine/contracts/wob_codebook.json`

The TeX form of this page is `global_universality/ENGINE_COMPLETENESS_PAGE.tex`.

## Core definitions

### Sliced arena

A sliced arena is a decomposition
\[
M = \bigsqcup_{s\in \mathbb{R}} \Sigma_s,
\]
with \(\Sigma_s\) representing a global instant.

### Phase field

A phase field is a map
\[
\Phi: M \to \mathbb{T}_P.
\]
Write \(\Phi_s := \Phi|_{\Sigma_s}\).

### Syncré Coverage Law (SCL)

\[
\boxed{\ \forall s,\quad \Phi_s(\Sigma_s) = \mathbb{T}_P.\ }
\]

### Equivariant circle action and phase response

Let \(A: S^1\times \Sigma \to \Sigma\) be a continuous \(S^1\)-action on a slice \(\Sigma\).
A phase observable \(\phi: \Sigma\to \mathbb{T}_P\) is equivariant if there exists a continuous homomorphism
\[
\chi: S^1 \to \mathbb{T}_P
\]
such that
\[
\boxed{\ \phi(A(\theta,x)) = \phi(x) \oplus \chi(\theta)\ \text{for all }\theta\in S^1,\ x\in\Sigma.\ }
\]

## Minimal forcing theorem

If \(\chi\) has nonzero slope (equivalently, \(\chi\) is surjective), equivariance forces slice-wise coverage on the circle target:
\[
\phi(\Sigma)=\mathbb{T}_P.
\]

The full theorem statement and proof are in `global_universality/GLOBAL_SYNCRE_UNIVERSALITY_THEOREM.tex` and the compiled PDF in `finish_ready/global_universality.pdf`.

## Coupled targets

In multi-cycle settings, coupling can restrict the image of the phase field to a coupled subtorus coset \(\mathrm{CST}_s\).
Coverage and any baseline distribution are asserted on \(\mathrm{CST}_s\), not on the ambient product torus.
This bundle includes:

- target correctness axioms: `global_universality/TARGET_CORRECTNESS_AXIOMS.tex`
- target decision flow: `global_universality/TARGET_DECISION_FLOW.tex`
- target identifiability theorem: `global_universality/TARGET_IDENTIFIABILITY_THEOREM.tex`

## Obstruction catalog

Within COC, failure to certify has a finite set of structural reasons implemented as validator predicates and canonical WOB codes.
Catalog completeness is stated in `global_universality/WOB_CATALOG_COMPLETENESS.tex`.
