# Breakthrough Pass 6 — Coupled-Cycle Mixing: Convergence on a Coupled Subtorus with Diffusion

## Purpose

Extend the “true convergence” mechanism (diffusion-driven mixing) to the **coupled** multi-cycle setting introduced in Pass F, in a way that respects the Synodic Offset Law (SOL).

This pass delivers a clean, defensible statement:

- if coupling constrains the image to a 1D coupled subtorus, then diffusion along the *free coordinate* yields convergence to Haar on that subtorus
- if the synodic offset itself varies, the family of coupled subtori can sweep a higher-dimensional region; we state the correct time-indexed target and a stroboscopic equidistribution principle

This is weighed for value: it avoids claiming full-torus Haar when physics constrains support.

---

## Setting

Assume a shared period \(P>0\) and work on \(\mathbb{T}_P\).

We consider a two-cycle torus codomain:
\[
\mathbb{T}_P^2=\mathbb{T}_P\times \mathbb{T}_P
\]
with coordinates \((a,b)\) representing (solar phase, lunar phase).

### Synodic Offset Law (SOL)

On each slice \(\Sigma_s\), assume:
\[
\boxed{\;b \ominus a = \Psi(s)\;}
\]
for a global synodic offset \(\Psi(s)\in\mathbb{T}_P\).

Thus the slice image lies in the coupled subtorus (a diagonal translate):
\[
\boxed{\;\mathrm{CST}_s=\{(a,\ a\oplus \Psi(s)):\ a\in\mathbb{T}_P\}.\;}
\]

---

## Part I — Correct convergence target under SOL

Because the support lies in \(\mathrm{CST}_s\), the correct “uniform” target at fixed \(s\) is Haar on that circle.

### Definition (Coupled Haar target)

Let \(\eta_s:\mathbb{T}_P\to \mathbb{T}_P^2\) be the parameterization
\[
\eta_s(a)=(a,\ a\oplus\Psi(s)).
\]
Define the **coupled Haar measure** by pushforward:
\[
\boxed{\;\mu_{\mathrm{H},s}:=(\eta_s)_\# \mu_{\mathrm{H}}.\;}
\]

This is the natural uniform measure on \(\mathrm{CST}_s\).

---

## Part II — Diffusion along the free coordinate yields convergence on \(\mathrm{CST}_s\)

We model the induced distribution along the free coordinate \(a\) by a density \(p(s,a)\) on \(\mathbb{T}_P\) and push it forward via \(\eta_s\).

### Model (Coupled diffusion)

Assume \(p(s,\cdot)\) evolves by the heat equation on \(\mathbb{T}_P\):
\[
\boxed{\;\partial_s p(s,a)=\kappa\,\partial_{aa}p(s,a),\qquad \kappa>0.\;}
\]

Define the coupled measure on \(\mathbb{T}_P^2\) by
\[
\boxed{\;\mu_s := (\eta_s)_\#(p(s,a)\,da).\;}
\]

### Theorem 1 (Exponential convergence to coupled Haar at fixed \(s\))

Let \(q(s,a)=p(s,a)-1/P\). Then
\[
\|q(s,\cdot)\|_{L^2(\mathbb{T}_P)}\le \|q(0,\cdot)\|_{L^2}\exp\!\left(-\kappa\left(\frac{2\pi}{P}\right)^2 s\right).
\]
Consequently, \(\mu_s\) converges to the coupled Haar target \(\mu_{\mathrm{H},s}\) in any metric dominated by \(L^2\) (and, by standard compactness inequalities, also in \(L^1\)/TV).

#### Proof

This is the standard Fourier-mode damping on the circle, identical to Breakthrough Pass 4, applied to the coordinate \(a\). Pushforward by \(\eta_s\) preserves convergence because \(\eta_s\) is continuous and measure-preserving up to embedding. ∎

### Syncre interpretation

Under SOL, “uniformity” means uniformity along the free coordinate \(a\) on the coupled subtorus. Diffusion along \(a\) forces exponential convergence to that uniform state.

---

## Part III — If \(\Psi(s)\) varies, what can be claimed

When \(\Psi(s)\) changes with \(s\), the coupled subtorus itself moves inside \(\mathbb{T}_P^2\). There are two distinct analysis modes:

- **Instantaneous mode:** at each slice \(s\), compare \(\mu_s\) to \(\mu_{\mathrm{H},s}\) (time-dependent target).
- **Stroboscopic mode:** sample \(s_n=s_0+n\Delta\) and study how the family \(\mathrm{CST}_{s_n}\) sweeps the torus.

### Theorem 2 (Stroboscopic sweep criterion in the linear model)

Assume a stroboscopic model:
\[
a_n=a_0\oplus n\alpha,\qquad \Psi_n=\Psi_0\oplus n\beta,
\]
and observe
\[
x_n=(a_n,\ a_n\oplus \Psi_n).
\]
If \(\alpha/P\) and \(\beta/P\) are rationally independent, then \(\{x_n\}\) is equidistributed in \(\mathbb{T}_P^2\).

This restates the key coupled equidistribution result (Breakthrough Pass 2) as the correct “moving-subtorus” principle.

### Honest boundary

- Diffusion along \(a\) guarantees uniformity on each \(\mathrm{CST}_s\).
- Full-torus uniformity requires either:
  - \(\Psi(s)\) to sweep phases nonresonantly (stroboscopic mechanism), and/or
  - diffusion/noise acting in directions that move support off a 1D subtorus

This cleanly separates “coupled instantaneous uniformity” from “global torus mixing.”

---

## Part IV — A coupled Stability Signature

We extend the Stability Signature (SSG) to the coupled setting.

### Definition (Coupled Stability Signature, CSSG)

\[
\boxed{\;\mathrm{CSSG}:=\big(P,\ \kappa,\ \mathrm{SHC}_a,\ \mathrm{FDR},\ \mathrm{RNF}(\alpha,\beta)\big)\;}
\]
where
- \(\kappa\) controls diffusion along the free coordinate \(a\)
- \(\mathrm{FDR}=\kappa(2\pi/P)^2\)
- \(\mathrm{SHC}_a\) bounds shear in the \(a\)-direction if advection is included in \(a\)
- \(\mathrm{RNF}(\alpha,\beta)\) records nonresonance for stroboscopic sweeping

### Certificate statements

- Instantaneous convergence to \(\mu_{\mathrm{H},s}\) at rate \(\mathrm{FDR}\) (Theorem 1)
- If RNF is nonresonant, stroboscopic sweep yields full-torus equidistribution (Theorem 2)

This makes the coupled program measurable: one certificate for within-subtorus mixing, one flag for cross-subtorus exploration.

---

## Acceptance gates

- The correct coupled Haar target \(\mu_{\mathrm{H},s}\) is defined under SOL
- A diffusion-based exponential convergence theorem is proved for uniformity on \(\mathrm{CST}_s\)
- The moving-subtorus stroboscopic sweep criterion is stated cleanly and scoped honestly
- A coupled Stability Signature is defined with explicit constants and a nonresonance flag
