# Univ Pass 7 — Mixing Universality (When Convergence to Haar is a Universal Outcome)

## Purpose

Perfect the part of the universality argument that concerns *uniformity*:

- deterministic transport alone is not universally mixing
- convergence to Haar becomes universal once a defensible noise/diffusion mechanism is present
- state the exact conditions (diffusion dominates shear) as a mixing certificate
- align with SFT diagnostics (FUD/SUD) and stability signatures (SSG/CSSG)

This prevents overclaiming while still providing a strong universal mechanism statement.

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## Program language (mixing universality)

| Term | Tag | Meaning |
|---|---|---|
| Mixing Certificate | **MXC** | explicit inequality guaranteeing exponential decay of defects |
| Noise-to-Diffusion Bridge | **NDB** | modeling step: observational/noise mechanism induces diffusion |
| Universal Uniformity Regime | **UUR** | parameter region where convergence holds robustly |

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## 1) The honest boundary

SFT does not claim:

- “all transport dynamics mix”
- “all phase distributions converge to Haar”

SFT claims:

- when a physically justified noise/diffusion mechanism is present, convergence is a universal consequence with explicit rates.

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## 2) Diffusion-driven mixing (circle)

Under DSD:
\[
\partial_s p=\kappa\,\partial_{yy}p,
\]
nonzero Fourier modes decay exponentially (Breakthrough Pass 4). Therefore FUD decays exponentially.

This is a universal mechanism statement: diffusion on a compact cycle enforces uniformity.

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## 3) Advection–diffusion mixing certificate (circle)

Under SADM:
\[
\partial_s p + \partial_y(u p)=\kappa\,\partial_{yy}p,
\]
Breakthrough Pass 5 gives an energy-based condition:

Let \(\mathrm{FDR}=\kappa(2\pi/P)^2\) and \(\mathrm{SHC}=\|\partial_y u\|_\infty\). If
\[
\boxed{\;2\,\mathrm{FDR}>\mathrm{SHC},\;}
\]
then \(\|p-1/P\|_{L^2}\) decays exponentially. This yields a mixing certificate (MXC).

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## 4) Coupled mixing (SOL)

Under SOL, the correct target is coupled Haar on CST (Univ Pass 5). Breakthrough Pass 6 shows that diffusion along the free coordinate yields exponential convergence to that coupled Haar target.

Thus mixing universality in coupled settings is:

- universal along the free coordinate under diffusion/noise,
- full-torus mixing requires additional sweep (stroboscopic nonresonance) or diffusion transverse to CST.

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## 5) Noise-to-diffusion bridge (NDB)

To make diffusion physically meaningful rather than arbitrary, SFT adopts NDB as a modeling bridge:

- measurement noise, phase jitter, unresolved microstructure, or random sampling increments
- induce an effective diffusion term in the phase-density evolution

NDB is not a theorem inside SFT unless derived from a specific physical model; it is an explicit modeling assumption. Once adopted, the mixing theorems become rigorous consequences.

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## Acceptance gates

- the nonclaim boundary is explicit (transport ≠ universal mixing)
- diffusion-based mixing is presented as the universal mechanism
- MXC is stated as a checkable inequality with explicit constants
- coupled mixing targets are aligned to CST + coupled Haar
- NDB is stated explicitly as an assumption, not smuggled in
