# Breakthrough Pass 2 — Forcing-Robust Anti-Collapse and Coupled-Subtorus Equidistribution

## Purpose

Strengthen the breakthrough layer in two ways:

- **BP-3 strengthening (forcing-robust):** extend the dynamics degradation bounds to allow spatially varying forcing \(\partial_\lambda \omega \neq 0\), while still obtaining a usable *anti-collapse* (coverage-profile) guarantee.
- **BP-2 strengthening (coupled):** provide a clean coupled-subtorus equidistribution theorem in the stroboscopic linear model under the Synodic Offset Law (SOL).

This pass is deliberately weighed toward real value:

- it avoids overpromising uniformity-to-Haar when lower stretch bounds are not preserved
- it provides the sharpest guarantees that remain true under the relaxed hypotheses

---

## Notation

- 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 \(S^1\): \(d\nu=\frac{d\lambda}{2\pi}\)
- Phase map at time \(s\): \(\Phi_s(\lambda)=F(s,\lambda)\bmod P\)
- Pushforward phase distribution: \(\mu_s=(\Phi_s)_\#\nu\)
- Coverage profile (from Pass 1): \(\mathrm{CP}_s(\rho)=\inf_{y}\mu_s(B_\rho(y))\)

We study the lift dynamics (STD) on \(S^1\):
\[
\partial_s F + v(s,\lambda)\,\partial_\lambda F = \omega(s,\lambda),
\]
with \(v,\omega\) \(2\pi\)-periodic in \(\lambda\).

---

## Part I — Anti-collapse bounds requiring only an upper stretch bound

The density-sandwich theorem in Breakthrough Pass 1 required a two-sided bound \(0<a\le \partial_\lambda F\le b\).  
For robustness under forcing, a uniform positive lower bound is hard to preserve without extra structure.

However, the **anti-collapse** mechanism (lower bounds on neighborhood mass) only needs an upper bound on stretch.

### Lemma 1 (Arc mass lower bound from an upper stretch bound)

Fix a time \(s\). Assume:

- winding \(k\in\mathbb{Z}\setminus\{0\}\): \(F(s,\lambda+2\pi)=F(s,\lambda)+kP\)
- \(F(s,\cdot)\) is absolutely continuous and satisfies an upper stretch bound
  \[
  \partial_\lambda F(s,\lambda)\le b(s)\quad\text{for a.e. }\lambda
  \]

Then for any arc \(I\subseteq\mathbb{T}_P\) of length \(|I|\),
\[
\boxed{\;\mu_s(I)\ \ge\ \frac{|k|}{2\pi\,b(s)}\,|I|.\;}
\]

#### Proof

For regular values, each arc has \(|k|\) preimage branches. On each branch, a length-\(|I|\) arc in output requires at least length \(|I|/b(s)\) in \(\lambda\) by the upper derivative bound. Summing \(|k|\) branches gives preimage length at least \(|k||I|/b(s)\). Normalize by circle length \(2\pi\). ∎

### Corollary 2 (Coverage-profile bound from \(b(s)\))

For every \(\rho\in[0,P/2]\),
\[
\boxed{\;\mathrm{CP}_s(\rho)\ \ge\ \frac{|k|}{2\pi\,b(s)}\,(2\rho).\;}
\]

So **collapse is quantitatively prevented** as long as the upper stretch bound stays controlled, even if a uniform positive lower stretch bound is not available.

---

## Part II — Controlling the upper stretch under forced STD dynamics

Let \(G=\partial_\lambda F\). Differentiating STD yields
\[
\partial_s G + v\,\partial_\lambda G + (\partial_\lambda v)\,G = \partial_\lambda \omega.
\]

Let
\[
B(s):=\|G(s,\cdot)\|_{L^\infty(S^1)}
\quad\text{(the upper stretch)}.
\]

### Assumptions for a clean bound

Assume there are constants \(C,D\ge 0\) such that for all \((s,\lambda)\),
\[
|\partial_\lambda v(s,\lambda)|\le C,\qquad |\partial_\lambda \omega(s,\lambda)|\le D.
\]

### Theorem 3 (Upper stretch growth bound under bounded shear and forcing)

If \(B(s_0)\le b_0\), then for all \(s\ge s_0\),
\[
\boxed{\;B(s)\ \le\ e^{C(s-s_0)}\,b_0\ +\ \frac{D}{C}\big(e^{C(s-s_0)}-1\big)\quad (C>0).\;}
\]
For \(C=0\), the bound becomes \(B(s)\le b_0 + D(s-s_0)\).

#### Proof sketch

Along characteristics, \(G\) satisfies the linear ODE
\[
\frac{d}{ds}G = -(\partial_\lambda v)\,G + (\partial_\lambda \omega).
\]
Taking absolute values and using bounds gives
\[
\frac{d}{ds}|G|\le C|G| + D.
\]
Applying Grönwall yields the stated bound for the supremum \(B(s)\). ∎

### Corollary 4 (Forced-dynamics degradation bound for the coverage profile)

Combine Corollary 2 with Theorem 3:

For \(C>0\) and all \(s\ge s_0\),
\[
\boxed{\;\mathrm{CP}_s(\rho)\ \ge\ \frac{|k|}{2\pi}\cdot \frac{2\rho}{\,e^{C(s-s_0)}b_0 + \frac{D}{C}(e^{C(s-s_0)}-1)\,}.\;}
\]

This is a forcing-robust version of BP-3: explicit, checkable, and does not rely on \(\partial_\lambda\omega=0\).

---

## What is and is not claimed

- These results guarantee **coverage strength** (no collapse) through \(\mathrm{CP}_s\) lower bounds.
- They do **not** guarantee closeness to Haar (SUD small) without additional lower-stretch or mixing assumptions.
- This is intentional: it keeps the program honest and makes later “uniformity” theorems meaningful rather than automatic.

---

## Part III — BP-2: Coupled-subtorus equidistribution under SOL in the linear stroboscopic model

We now address the coupled setting from Pass F.

Assume a shared period \(P\) for clarity. Let SOL hold on each slice:
\[
\Phi_{\leftmoon}(p)\ominus \Phi_\odot(p)=\Psi(s)\quad\text{for }p\in\Sigma_s.
\]
At a fixed \(s\), the coupled subtorus is the graph
\[
\mathrm{CST}_s=\{(a,a\oplus\Psi(s)):\ a\in\mathbb{T}_P\}.
\]

### Stroboscopic linear model

Sample at \(s_n=s_0+n\Delta\). Assume:

- solar coordinate evolves by a rotation:
  \[
  a_n = a_0 \oplus n\alpha
  \]
- synodic offset evolves by a rotation:
  \[
  \Psi_n := \Psi(s_n)=\Psi_0 \oplus n\beta
  \]

Then the observed pair is
\[
x_n := (a_n,\ a_n\oplus \Psi_n)\in \mathbb{T}_P\times \mathbb{T}_P.
\]

### Theorem 5 (Coupled equidistribution criterion)

If \(\alpha/P\) and \(\beta/P\) are rationally independent (no nontrivial integer relation \(m\alpha+n\beta\in P\mathbb{Z}\)), then \(\{x_n\}\) is equidistributed in the full torus \(\mathbb{T}_P^2\).

If \(\beta\equiv 0\ (\bmod P)\), then \(\{x_n\}\) is equidistributed on the fixed coupled subtorus \(\mathrm{CST}_{s_0}\) whenever \(\alpha/P\) is irrational.

#### Proof sketch

Write \(x_n = x_0 \oplus n(\alpha,\alpha+\beta)\). This is a torus rotation. Standard torus rotation theory gives equidistribution on the closure of the orbit. The closure is the full torus precisely when the increment vector has rationally independent components, which here is equivalent to rational independence of \(\alpha\) and \(\beta\) modulo \(P\). If \(\beta=0\), the orbit lies in the diagonal translate \(\{(a,a\oplus \Psi_0)\}\), and equidistribution reduces to the circle rotation by \(\alpha\). ∎

### Syncre interpretation

- SOL coupling does not prevent full-torus equidistribution over time if the offset \(\Psi_n\) itself cycles incommensurately with the solar rotation.
- At a fixed slice, the image is a 1D subtorus; across slices, the family of subtori can fill the 2D torus.

This is a concrete statement of how coupling and stroboscopic sampling interact.

---

## Acceptance gates

- Anti-collapse bounds are proved using only an upper stretch bound \(b(s)\)
- An explicit forced-dynamics bound for \(b(s)\) is proved under \(|\partial_\lambda v|\le C\), \(|\partial_\lambda \omega|\le D\)
- An explicit time-degraded lower bound for the coverage profile \(\mathrm{CP}_s(\rho)\) is derived
- A coupled-subtorus equidistribution theorem is stated and proved in the linear stroboscopic model under SOL


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## Errata and patch note (MBC)

Some anti-collapse bounds stated using only an upper stretch bound implicitly require a branch-structure hypothesis. See `breakthrough/BREAKTHROUGH_PASS_7.md` for the **Monotone Branch Condition (MBC)** under which the lower mass bounds are fully rigorous, and for a safe sufficient condition (two-sided stretch) that implies MBC.
