# Breakthrough Pass 4 — A Genuine Decay Mechanism: Diffusive Syncre Smoothing and Exponential Fourier Damping

## Purpose

Close the most important remaining gap between “slice-wise bounds” and “true convergence to Haar” by introducing a minimal, mathematically clean decay mechanism:

- add a small diffusion term to the lift dynamics (or equivalently, to the phase-density evolution)
- prove exponential decay of Fourier-mode defects (hence convergence in common metrics)

This is the first place where “Syncre convergence” becomes a theorem with an explicit rate, rather than an assumption about rotations.

This pass is weighed for real value:

- diffusion is a standard analytical tool, but within Syncre it becomes a principled *regularization mechanism* that yields provable uniformity improvement
- the result is stated in Syncre diagnostics language (FUD / SUD) and gives a clear rate constant

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## Setting

Work on \(\Sigma=S^1=\mathbb{R}/2\pi\mathbb{Z}\). Let \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\).

We track phase distributions \(\mu_s\) on \(\mathbb{T}_P\). In a “density” regime, write
\[
d\mu_s(y) = p(s,y)\,dy,\qquad p\ge 0,\quad \int_0^P p(s,y)\,dy=1.
\]

Haar density is \(p_{\mathrm{H}}(y)=1/P\).

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## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Diffusive Syncre Dynamics | **DSD** | evolution with a Laplacian term that smooths phase density |
| Fourier Mode Defect | **FUD** | \(\mathrm{FUD}_N(\mu)=\max_{1\le |n|\le N} |\widehat{\mu}(n)|\) |
| Exponential Damping Rate | **EDR** | explicit decay constant for \(|\widehat{\mu}(n)|\) |
| Syncre Mixing Certificate | **SMC** | a condition guaranteeing decay of nonzero modes |

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## Diffusive Syncre Dynamics (DSD) for the phase density

A minimal decay mechanism is the heat equation on the phase circle:

\[
\boxed{\;\partial_s p(s,y) = \kappa\,\partial_{yy} p(s,y),\qquad y\in\mathbb{T}_P,\ \kappa>0.\;}
\]

This is the simplest PDE that drives any initial density toward uniformity.

In more physical settings, you can add transport:
\[
\partial_s p + \partial_y\big(u(s,y)\,p\big) = \kappa\,\partial_{yy}p,
\]
but the pure diffusion case already delivers the core “breakthrough” theorem: exponential decay of nonzero Fourier modes.

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## Theorem 1 (Fourier exponential damping under DSD)

Let \(p(s,\cdot)\) solve \(\partial_s p=\kappa\,\partial_{yy}p\) on \(\mathbb{T}_P\). Let
\[
\widehat{p}(n,s):=\int_0^P p(s,y)\,e^{-2\pi i n y/P}\,dy,\qquad n\in\mathbb{Z}.
\]

Then:

- \(\widehat{p}(0,s)=1\) for all \(s\)
- for \(n\neq 0\),
\[
\boxed{\;\widehat{p}(n,s)=\widehat{p}(n,0)\,\exp\!\left(-\kappa\left(\frac{2\pi n}{P}\right)^2 s\right).\;}
\]

### Proof

Expand \(p\) in the Fourier basis on \(\mathbb{T}_P\). The Laplacian acts diagonally: \(\partial_{yy}\) multiplies mode \(n\) by \(-\left(\frac{2\pi n}{P}\right)^2\). Therefore each nonzero mode obeys a scalar ODE with the stated solution. ∎

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## Corollary 2 (Exponential decay of Fourier uniformity defects)

Using the Syncre Fourier defect (Pass 3),
\[
\mathrm{FUD}_N(\mu_s)=\max_{1\le |n|\le N}\left|\frac{\widehat{p}(n,s)}{\widehat{p}(0,s)}\right|
=\max_{1\le |n|\le N}|\widehat{p}(n,s)|.
\]

Then
\[
\boxed{\;\mathrm{FUD}_N(\mu_s)\ \le\ \mathrm{FUD}_N(\mu_0)\,\exp\!\left(-\kappa\left(\frac{2\pi}{P}\right)^2 s\right).\;}
\]
(The rate is governed by the slowest nonzero mode \(n=\pm1\).)

This is an explicit **EDR**:
\[
\mathrm{EDR}=\kappa\left(\frac{2\pi}{P}\right)^2.
\]

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## Corollary 3 (Convergence to Haar in \(L^2\) and \(L^1\))

Let \(q(s,y):=p(s,y)-1/P\). Then the Fourier series of \(q\) has only nonzero modes. By Theorem 1, each nonzero mode decays exponentially, hence:

- \(\|q(s,\cdot)\|_{L^2}\) decays exponentially with the same base rate
- by standard inequalities on a compact domain, \(\|q(s,\cdot)\|_{L^1}\) also decays (possibly with a different constant)

Therefore \(p(s,\cdot)\to 1/P\) and \(\mu_s\to \mu_{\mathrm{H}}\) as \(s\to\infty\).

This is a true convergence theorem, not a stroboscopic assumption.

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## Syncre meaning and how it integrates with the main program

This pass provides a **Syncre Mixing Certificate (SMC)**:

- if your model includes an effective diffusive smoothing mechanism in phase (or if phase noise can be modeled as diffusion),
- then uniformity defects decay at a provable exponential rate.

This can be combined with earlier passes:

- Breakthrough Pass 1/2 give anti-collapse bounds under deterministic transport, guaranteeing the distribution never develops “holes” while it evolves.
- DSD then guarantees that any residual nonuniformity is exponentially damped.

This is a clean, defensible claim boundary:

- deterministic transport preserves structure but may not mix
- adding diffusion yields genuine mixing and provable convergence

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## Acceptance gates

- DSD is stated as a phase-density dynamics on \(\mathbb{T}_P\)
- Fourier exponential damping is proved with an explicit rate constant
- FUD decay is stated as a Syncre diagnostic theorem
- Convergence to Haar is stated in standard norm terms and tied back to Syncre SUD language
