# Breakthrough Pass 3 — Quantitative Uniformity via Bounded Distortion and Fourier Control

## Purpose

Push beyond “anti-collapse” into a controlled notion of **uniformity** that is:

- computable from geometric/analytic bounds on the lift \(F\)
- stable under small perturbations
- compatible with the Syncre diagnostics (SUD/SDF)

This pass does **not** claim mixing that is not present. Instead it proves the strongest true statement available under bounded distortion:

> If the pushforward density is bounded above and below (equivalently, the map has controlled stretch both ways), then the phase distribution is absolutely continuous with a density that is uniformly comparable to Haar. This yields explicit bounds on total variation and on a Fourier-based uniformity defect.

It also records a clean “route to real decay” (true convergence to Haar) which requires additional mixing or diffusion assumptions.

---

## Setting

- Period \(P>0\), phase circle \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- Spatial circle \(S^1=\mathbb{R}/2\pi\mathbb{Z}\) with coordinate \(\lambda\)
- Uniform measure on \(S^1\): \(d\nu=\frac{d\lambda}{2\pi}\)
- Lift \(F(s,\lambda)\) with winding \(k\neq 0\) at time \(s\):
  \[
  F(s,\lambda+2\pi)=F(s,\lambda)+kP
  \]
- Phase map \(\Phi_s(\lambda)=F(s,\lambda)\bmod P\)
- Pushforward \(\mu_s=(\Phi_s)_\#\nu\)

We work at a fixed slice \(s\) and suppress \(s\) when convenient.

---

## Bounded distortion hypothesis (BDH)

Assume \(F(\cdot)\) is \(C^1\) in \(\lambda\) and satisfies two-sided stretch bounds:
\[
0<a\le \partial_\lambda F(\lambda)\le b\quad \text{for all }\lambda.
\]

Define the distortion ratio:
\[
\boxed{\;\mathrm{BD}:=\frac{b}{a}\ge 1.\;}
\]

This is the minimal analytic assumption that upgrades “anti-collapse” to “controlled uniformity.”

---

## Theorem 1 (Uniform comparability to Haar)

Under BDH, \(\mu\) is absolutely continuous with respect to Haar measure \(\mu_{\mathrm{H}}\), with density \(f=\frac{d\mu}{d\mu_{\mathrm{H}}}\) satisfying:
\[
\boxed{\;\frac{|k|P}{2\pi b}\ \le\ f(y)\ \le\ \frac{|k|P}{2\pi a}\quad\text{for a.e. }y.\;}
\]

This is the density sandwich already proved in Breakthrough Pass 1, rewritten here as the core uniformity input.

---

## Theorem 2 (Total-variation uniformity bound in terms of distortion)

Let \(f\) be the Haar-density from Theorem 1. Then
\[
\boxed{\;\|\mu-\mu_{\mathrm{H}}\|_{\mathrm{TV}}\ \le\ \frac12\int_{\mathbb{T}_P}|f(y)-1|\,d\mu_{\mathrm{H}}(y)\ \le\ \frac12\max\!\left(\frac{|k|P}{2\pi a}-1,\ 1-\frac{|k|P}{2\pi b}\right).\;}
\]

In particular, if \(|k|=1\) and \(a\) and \(b\) are close to the “ideal” values \(P/(2\pi)\), then TV distance is small.

### Interpretation

- This does not claim decay in time.
- It gives a **slice-wise** upper bound on how non-uniform the phase distribution can be, purely from bounded distortion.

---

## A Syncre-native uniformity defect based on Fourier modes

Total variation is strong but can be pessimistic. On a circle, an alternative diagnostic is a Fourier-mode defect.

Represent \(\mathbb{T}_P\) as \([0,P)\) with periodic identification. Let
\[
\widehat{\mu}(n):=\int_{\mathbb{T}_P} e^{-2\pi i n y/P}\,d\mu(y),\qquad n\in\mathbb{Z}.
\]
For Haar measure, \(\widehat{\mu_{\mathrm{H}}}(0)=1\) and \(\widehat{\mu_{\mathrm{H}}}(n)=0\) for \(n\neq 0\).

### Definition (Fourier uniformity defect, FUD)

Define
\[
\boxed{\;\mathrm{FUD}_N(\mu):=\max_{1\le |n|\le N} |\widehat{\mu}(n)|.\;}
\]
Small \(\mathrm{FUD}_N\) means no strong periodic bias at low harmonics.

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## Theorem 3 (Fourier control from bounded variation of the density)

Assume BDH and additionally that the Haar-density \(f\) has bounded variation:
\[
\mathrm{Var}(f)\le V.
\]
Then for \(n\neq 0\),
\[
\boxed{\;|\widehat{\mu}(n)| = \left|\int f(y)e^{-2\pi i n y/P}\,d\mu_{\mathrm{H}}(y)\right|\ \le\ \frac{V}{2\pi |n|}.\;}
\]
Consequently,
\[
\boxed{\;\mathrm{FUD}_N(\mu)\ \le\ \frac{V}{2\pi} \cdot \frac{1}{1} = \frac{V}{2\pi}\quad\text{and more sharply}\quad \mathrm{FUD}_N(\mu)\le \frac{V}{2\pi N}.\;}
\]

### Proof sketch

For a BV function on a circle, Fourier coefficients decay like \(O(1/n)\). The stated inequality is a standard integration-by-parts/BV estimate. ∎

### Syncre meaning

If your deformation class produces densities with controlled variation, then low-mode biases are uniformly controlled. This is a meaningful “uniformity” statement that is stronger than anti-collapse but weaker than mixing.

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## How to get real decay (true convergence to Haar)

To prove \(\mathrm{FUD}_N(\mu_s)\to 0\) or \(\mathrm{UD}(s)\to 0\) as \(s\to\infty\), you need one of:

- **mixing dynamics** on the phase map (expanding map / Markov structure)
- **diffusion/regularization** (e.g., add a small Laplacian term in a PDE for \(F\) or for a density)
- **randomness / noise** in sampling increments (nondegenerate stochastic perturbations)

This is not a weakness; it is a clean boundary between what is true from bounded distortion alone and what requires additional dynamical hypotheses.

This boundary is exactly where Syncre becomes genuinely new: the project can specify admissible perturbations and prove sharp decay rates in terms of Syncre diagnostics.

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## Acceptance gates

- A distortion-based uniformity bound is stated cleanly and tied to BDH
- A Fourier-mode defect diagnostic (FUD) is defined as Syncre program language
- A BV-to-Fourier bound is stated to give an actionable route to uniformity control
- The document explicitly separates slice-wise control from true time-decay claims
