# Pass J — Paper Skeleton and Submission-Ready Draft Plan

## Purpose

Produce a coherent paper skeleton that:

- states the canonical Syncre law (SCL) and symbol \(\mathscr{S}_{\!\mathrm{r}}(P)\)
- clearly separates known ingredients from the project’s additive contributions
- integrates witness catalog, invariants/moduli, coupling, diagnostics, and dynamics
- provides an appendix plan for technical proofs and examples

This is a draft blueprint intended for LaTeX expansion.

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## Working title

**Syncre Form Theory: Coverage Phase Fields on Sliced Arenas — Witness Templates, Robust Moduli, Couplings, and Transport Dynamics**

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## Abstract (draft)

We formalize a phenomenon: at a single global instant, a spatial arena can contain points whose local phase labels span an entire cycle. We encode this as a slice-wise coverage law for a phase field \(\Phi:M\to\mathbb{T}_P\) on a sliced arena \(M=\bigsqcup_s\Sigma_s\). We introduce Syncre as the class \(\mathscr{S}_{\!\mathrm{r}}(P)\) of frames satisfying the coverage law on every slice. We give reusable witness templates (phase-coordinate, equivariant circle action, and loop-degree certification) and show that rotating bodies, orbital rings, traveling waves, and order-parameter windings yield explicit Syncre witnesses. To move beyond qualitative coverage, we define quantitative diagnostics: a coverage modulus and a uniformity defect via pushforward measures on the codomain. We extend the framework to multi-cycle torus fields with coupling laws, including a synodic offset constraint, and define the correct coupled-subtorus convergence target. Finally, we propose a transport dynamics for lifts \(\Phi=F\bmod P\) and prove preservation of winding and holonomy under admissible dynamics, yielding sufficient conditions for coverage and modulus persistence. The framework provides a unified language for phase coverage, robustness, coupling, and dynamics across diverse physical realizations.

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## 1. Introduction

### 1.1 Motivation

## Universality thesis (scope)

Universality in SFT means universality of mechanism: whenever physical structure supplies a nontrivial periodic observable (an \(S^1\)-phase) with nonzero winding/holonomy across a spatial subsystem, Syncre coverage holds on that subsystem and admits a stable witness.


- The intuitive Earth picture: local time-of-day is a positional label of a single global instant
- The abstraction: this is a phase-field phenomenon, not Earth-specific
- The goal: a framework that produces witnesses and supports nonlinear deformation and analysis

### 1.2 Contribution summary

Organize claims by tier:

- Tier 0 (known tools, used as ingredients): degree of circle maps, equivariance, torus rotations
- Tier 1 (Syncre packaging): canonical law SCL, symbol \(\mathscr{S}_{\!\mathrm{r}}(P)\), witness templates, coupled-subtorus targets
- Tier 2 (additive results): quantitative moduli, robustness criteria, preservation under dynamics (state what is proved in this paper vs what is programmatic)

### 1.3 Paper outline

Briefly list sections.

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## 2. Definitions and canonical law

### 2.1 Phase codomain and slicing

- \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- slicing \(M=\bigsqcup_s\Sigma_s\)

### 2.2 Syncre class and symbol

State the definition from `SYNCRE_SYMBOL_DEFINITION.md`:

\[
(M,\{\Sigma_s\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P)
\iff
\forall s,\ \Phi(\Sigma_s)=\mathbb{T}_P.
\]

### 2.3 Canonical Syncre equation

State SCL from `SYNCRE_CANONICAL_EQUATION.md` as the project’s canonical identity.

---

## 3. Witness templates (coverage theorems)

### 3.1 Phase-coordinate template

State and prove Theorem A1.

### 3.2 Equivariant circle-action template

State and prove Theorem A2.

### 3.3 Loop-degree witness

State and prove Lemma B1; include degree definition and basic surjectivity lemma (A3).

### 3.4 Robustness of witness certification

State degree stability lemma (B2) and robust coverage theorem (B3).

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## 4. Universe witness catalog (flagship models)

Integrate the four flagship witnesses from Pass E:

- rotating rigid body
- orbital ring/circular orbit
- traveling wave on periodic medium
- order-parameter phase winding (embedded loop witness)

For each:

- specify \(M,\Sigma_s,\Phi\)
- cite which template applies
- conclude membership in \(\mathscr{S}_{\!\mathrm{r}}(P)\) or in an embedded-witness variant

---

## 5. Quantitative diagnostics: moduli and uniformity

### 5.1 Pushforward measures

Define \(\mu_s=(\Phi_s)_\#\nu_s\).

### 5.2 Coverage modulus and uniformity defect

Define SCM and SUD (Pass C).

### 5.3 First quantitative theorem (anti-collapse)

State and prove the stretch-bound lower mass theorem (C3). Discuss how it yields SCM lower bounds.

This section is where the paper starts to look new: it ties topological winding to distributional lower bounds.

---

## 6. Lift calculus, holonomy, and nonlinear admissibility

### 6.1 Lift law

State \(\Phi=F\bmod P\).

### 6.2 Lift atlas and obstruction class

Introduce cocycle \(n_{ij}\) and SOC; explain when a global lift exists.

### 6.3 Connection form and holonomy

Define \(\alpha=dF\) locally and holonomy \(\int_\gamma \alpha\). State gauge invariance.

### 6.4 Admissible future direction and bounded rates

Define AFD, SMF, and SBR; connect to “nonlinear time” without time reversal.

---

## 7. Multi-cycle torus fields and coupling laws

### 7.1 Torus codomain

Define \(\mathbb{T}_{\mathbf{P}}\) and STF.

### 7.2 Coupling constraints and coupled subtori

Define CPL and CST.

### 7.3 Synodic offset law

State SOL and prove the coupled coverage theorem (F2). Identify the correct Haar target on CST.

---

## 8. Convergence program and stroboscopic sampling

### 8.1 Stroboscopic schedules

Define \(s_n=s_0+n\Delta\) and induced sequences \(\mu^{(n)}\).

### 8.2 Baseline equidistribution results

State circle and torus equidistribution theorems under RNF nonresonance.

### 8.3 Coupled-subtorus convergence target

State how EQT changes under SOL coupling.

---

## 9. Syncre dynamics (transport) and preservation

### 9.1 Syncre transport dynamics

State STD: \(\partial_s F + v\cdot\nabla F=\omega\).

### 9.2 Preservation theorems

- degree preservation on \(S^1\) (DPR)
- holonomy preservation under \(d\omega=0\) (HPR)
- modulus preservation under stretch control (MPR) as a sufficient condition

This section makes “fluid Syncre” precise.

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## 10. Discussion and future work

State the “breakthrough targets” as next steps (BP-1, BP-2, BP-3):

- fully quantitative SCM/SUD robustness in SADC
- coupled-subtorus convergence bounds in diagnostics metrics
- explicit SCM degradation bounds under dynamics with flow distortion constants

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## Appendix plan

- A. Degree and lift proofs in full detail (for completeness)
- B. Lift atlas and cocycle obstruction details
- C. Stretch-bound theorem details and variants
- D. Equidistribution theorem statements (with citations in a real draft)
- E. Full computations for each flagship witness
- F. Dynamics preservation computations (characteristic method and holonomy evolution)

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## Acceptance gates

- All definitions match the bundle’s canonical references (SCL, SLL, \(\mathscr{S}_{\!\mathrm{r}}\))
- Every witness is certified by a template or loop-degree lemma
- At least one quantitative theorem (C3) is included as a proved result
- Coupling and dynamics sections clearly distinguish “proved here” vs “programmatic next”


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## Universality section (copy-ready)

Paste in `universality/UNIV_PASS_10_PAPER_SECTION.md` as the universality section, replacing any earlier informal universality wording.


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## Appendix: SFT Engine Artifacts

Include the engine contract, schemas, hypotheses menu, thresholds ledger, audit script, and example artifacts.

- `engine/contracts/engine_contract.md`
- `engine/schemas/`
- `engine/contracts/analytic_hypotheses_menu.md`
- `engine/contracts/constants_thresholds.md`
- `engine/audit/audit_wsc.py`
- `engine/artifacts/`

## Appendix: Worked Example

- `appendix/DI_PASS_I_WORKED_EXAMPLE.md`
