# Univ Pass 9 — Referee Checklist (Claim → Assumption → Theorem Map)

## Purpose

Make the universality argument referee-hard by providing:

- a claim ladder (what is proved, what is conditional, what is not claimed)
- a mapping from each universality sentence to a theorem or hypothesis checklist
- explicit nonclaims to prevent overreach objections

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## 1) Claim ladder

### A. Proved implications (theorem-level)

- If an \(S^1\)-action and equivariant phase observable exist with surjective response and a full orbit, then coverage holds (Univ Pass 2, AWT).
- If an embedded orbit/loop exists with nontrivial winding/holonomy, then embedded coverage holds (Univ Pass 2–3).
- If SOL coupling holds and the free coordinate covers, then CST coverage holds (Univ Pass 5).
- If diffusion (DSD) holds on phase density, Fourier defects decay exponentially (Breakthrough Pass 4).
- If advection–diffusion holds and diffusion dominates shear, deviation from Haar decays exponentially (Breakthrough Pass 5).

### B. Conditional universality statements (explicit hypotheses)

- Witness persistence under perturbations requires LCM/HMR margins (Univ Pass 4).
- Witness persistence under dynamics requires WPD conditions (Univ Pass 6).
- Full torus equidistribution in coupled settings requires nonresonant sweep (Breakthrough Pass 2/6) or transverse diffusion.

### C. Nonclaims (explicit)

- SFT does not claim all systems satisfy SCL.
- SFT does not claim deterministic transport universally mixes.
- SFT does not claim full-torus Haar is the target when coupling is present.
- SFT does not claim NDB (noise-to-diffusion) is always physically justified; it is an explicit modeling bridge.

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## 2) Hypothesis checklists

### AWT checklist (Univ Pass 2)

- circle action exists
- equivariant phase observable exists
- phase response map surjective
- full orbit exists

### ESW checklist (Univ Pass 3)

- subsystem embedding specified
- degree/holonomy nontrivial computed
- witness certificate fields recorded

### WSC checklist (Univ Pass 4)

- perturbation model specified
- margin recorded (LCM < \(\pi\) and/or HMR < \(P/2\))
- witness persists within margin

### Coupled checklist (Univ Pass 5)

- coupling constraint stated (SOL)
- CST identified and parameterized
- correct coupled Haar target used
- free coordinate covers (or embedded witness provided)

### Mixing checklist (Univ Pass 7)

- DSD or SADM model specified
- diffusion constant \(\kappa\) identified
- shear bound \(\|\partial_y u\|_\infty\) identified (if advection present)
- dominance inequality checked (MXC)

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## 3) Suggested wording to avoid rejections

Replace:
- “Syncre is universal in reality”
with:
- “Syncre is a universal mechanism whenever a cyclic phase observable with nontrivial winding/holonomy is present; many broad physical families satisfy the hypotheses.”

Always attach “whenever” and cite AWT/ESW/WSC/CTP.

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## Acceptance gates

- proved vs conditional vs nonclaims are explicit
- every universality claim maps to a theorem and a hypothesis list
- coupled-target correctness is enforced (CTP)
- mixing claims are tied to explicit diffusion dominance conditions

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## 4) Boundary theorem statements to preempt triviality objections

A referee may ask whether the ``universality of mechanism'' claim is vacuous. It is not, because it has a sharp boundary.

- If a slice admits no nontrivial cyclic symmetry in the admissible automorphism class, then the AWT mechanism cannot apply.
- In strictly ordered linear systems with only monotone scalar observables and no cyclic symmetry, Syncré-style coverage cannot be forced by the SFT witness engines.

This boundary is part of the correctness discipline: Syncré is universal exactly on the cyclic-symmetry side of the boundary, and not universal elsewhere.
