# Univ Pass 6 — Dynamics Universality (What Preserves Witnesses, What Breaks Them)

## Purpose

Perfect the universality argument under evolution:

- identify dynamic classes that preserve witness tiers (global/embedded/coupled)
- give explicit “boundary statements”: what SFT does and does not claim under dynamics
- connect dynamics preservation to certificate objects (WSC / SSG / CSSG)

This prevents the common pushback: “your phenomenon is coordinate-static; does it survive evolution?”

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## Program language (dynamics universality)

| Term | Tag | Meaning |
|---|---|---|
| Witness-Preserving Dynamics | **WPD** | dynamics preserving degree/holonomy witnesses |
| Witness-Breaking Dynamics | **WBD** | dynamics that can destroy witnesses (tears, discontinuities) |
| Dynamics Margin | **DMG** | quantitative margin ensuring witness stability under evolution |
| Slice-Compatible Evolution | **SCE** | evolution consistent with an admissible slicing |

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## 1) Canonical dynamics models in SFT

### Lift transport dynamics (STD)

On \(M=\mathbb{R}\times\Sigma\):
\[
\partial_s F + v\cdot\nabla F = \omega.
\]
Phase is \(\Phi=F\bmod P\).

### Phase-density advection–diffusion (SADM)

On \(\mathbb{T}_P\):
\[
\partial_s p + \partial_y(u p)=\kappa\,\partial_{yy}p.
\]

These two models serve different roles:
- STD preserves topological structure (witness invariants) but may not mix
- SADM provides a provable mixing mechanism under noise/diffusion assumptions

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## 2) Witness-Preserving Dynamics (WPD)

### WPD-A: Degree preservation on \(S^1\)

Under periodic data and sufficient regularity, winding \(k\) is preserved (Pass H). Therefore:

- if \(k\neq 0\) at some time, global or embedded LDW persists
- this is universality under evolution for phase wrapping systems

### WPD-B: Holonomy preservation under \(d\omega=0\)

If \(\omega\) is spatially constant (or \(d\omega=0\)), holonomy classes are preserved (Pass H). Therefore:

- embedded holonomy witnesses persist
- SOC/SCF/SHF structure remains stable

### WPD-C: Coupled witness preservation under SOL

If SOL holds structurally (constraint maintained by the physics/model), then the coupled target CST persists. If the free coordinate witness persists, coupled coverage persists (Univ Pass 5).

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## 3) Witness-Breaking Dynamics (WBD)

SFT explicitly does not claim witness persistence when:

- regularity fails (tears/discontinuities break degree/holonomy definitions)
- coupling constraints are not preserved (CST target changes)
- evolution changes the admissible slicing class (non-ADS behavior) without a controlled FEQ relation

These are not flaws; they define the boundary of universality.

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## 4) Dynamics margins (DMG) and certificate linkage

Witness Stability Certificates (WSC) can include a dynamics margin:

- for degree: ensure a lift-closeness margin remains \(<\pi\) on witness loops through time
- for holonomy: ensure loop integrals remain within \(P/2\) of the intended multiple

Stability Signatures (SSG/CSSG) store the parameters controlling:

- shear bounds
- forcing bounds
- diffusion dominance conditions (when mixing claims are made)

This turns “universality under dynamics” into a certifiable statement: preservation holds within a verified regime.

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## Acceptance gates

- WPD vs WBD boundary is stated clearly
- Preservation mechanisms are tied to previously proved theorems (Pass H; Breakthrough passes)
- Coupled persistence is linked to CST + free coordinate witness (Univ Pass 5)
- Dynamics margins are connected to WSC/SSG/CSSG certificate objects
