# Breakthrough Pass 7 — Rigorous Anti-Collapse Certification (Branch Condition Patch)

## Purpose

Repair and strengthen the anti-collapse bounds used in earlier breakthrough passes by making the “multiple preimage branches” step fully rigorous.

This pass introduces a minimal additional hypothesis—called the **Monotone Branch Condition (MBC)**—under which the lower mass bounds (and hence coverage-profile bounds) are valid using only an upper stretch bound. It also records a safe fallback: two-sided stretch (bounded distortion) implies MBC.

This closes a key referee tripwire: it ensures every quantitative anti-collapse statement rests on checkable hypotheses.

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## Setting

- \(S^1=\mathbb{R}/2\pi\mathbb{Z}\) with coordinate \(\lambda\)
- \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- lift \(F(s,\lambda)\) with winding \(k\neq 0\):
  \[
  F(s,\lambda+2\pi)=F(s,\lambda)+kP
  \]
- phase map \(\Phi_s(\lambda)=F(s,\lambda)\bmod P\)
- pushforward \(\mu_s=(\Phi_s)_\#\nu\) under uniform \(\nu(d\lambda)=d\lambda/(2\pi)\)

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## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Monotone Branch Condition | **MBC** | a degree-\(|k|\) covering decomposes into \(|k|\) monotone branches |
| Branch Jacobian Bound | **BJB** | an upper bound on \(\partial_\lambda F\) on each branch |
| Certified Anti-Collapse | **CAC** | a verified lower bound for the coverage profile \(\mathrm{CP}_s(\rho)\) |

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## MBC: the missing structural hypothesis

### Definition (MBC)

Fix \(s\). The phase map \(\Phi_s:S^1\to\mathbb{T}_P\) satisfies **MBC** if there exists a partition of the circle into \(|k|\) disjoint intervals
\[
S^1 = I_1\cup\cdots\cup I_{|k|}
\]
(up to endpoints) such that:

- on each \(I_j\), a chosen lift representative of \(F(s,\cdot)\) is strictly monotone
- the restriction \(\Phi_s|_{I_j}\) is a bijection onto \(\mathbb{T}_P\)

Intuition: the degree-\(|k|\) wrapping is realized as \(|k|\) monotone “covering branches.”

This is standard for \(C^1\) maps with \(\partial_\lambda F>0\) everywhere; it can also be verified in piecewise-monotone settings.

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## Theorem 1 (Anti-collapse from MBC + upper stretch)

Assume MBC holds at slice \(s\). Assume further that on each branch,
\[
\partial_\lambda F(s,\lambda)\le b(s)\quad\text{for a.e. }\lambda\in I_j.
\]
Then for any arc \(I\subseteq\mathbb{T}_P\) of length \(|I|\),
\[
\boxed{\;\mu_s(I)\ \ge\ \frac{|k|}{2\pi\,b(s)}\,|I|.\;}
\]

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

### Proof

Fix an arc \(I\). By MBC, on each branch \(I_j\), the restricted map is a bijection onto \(\mathbb{T}_P\), hence \(I\) has a unique preimage interval \(J_j\subseteq I_j\).

On branch \(j\), the upper derivative bound implies that producing an output arc of length \(|I|\) requires input length at least \(|I|/b(s)\):
\[
|J_j|\ge \frac{|I|}{b(s)}.
\]
Summing across \(|k|\) branches yields total preimage length \(\sum_j |J_j|\ge |k||I|/b(s)\). Normalizing by \(2\pi\) gives the stated measure bound. The profile bound is the special case \(|I|=2\rho\). ∎

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## Theorem 2 (Two-sided stretch implies MBC)

Assume \(F(s,\cdot)\in C^1(S^1)\) and satisfies:
\[
0<a\le \partial_\lambda F(s,\lambda)\le b\quad\text{for all }\lambda.
\]
Then MBC holds (with all branches strictly increasing) and Theorem 1 applies.

### Proof sketch

A uniform positive lower bound implies strict monotonicity of the lift on each fundamental wrap, and the winding relation forces the degree-\(|k|\) covering structure. Standard covering-map arguments give a \(|k|\)-branch decomposition. ∎

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## Patch note for earlier passes

- Any result using the “\(|k|\) preimage branches” step with only an upper stretch bound should be read as requiring **MBC** (or the stronger two-sided stretch assumption).
- Under two-sided stretch, the stronger density sandwich and TV/Fourier bounds also hold (Breakthrough Pass 1 and 3).

This restores a fully checkable hypothesis set for all anti-collapse and degradation claims.

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## Acceptance gates

- MBC is defined as a verifiable branch decomposition condition
- The lower mass bound is proved rigorously under MBC + upper stretch
- A safe sufficient condition (two-sided stretch) is proved to imply MBC
- A patch note clarifies how earlier statements should be interpreted under the corrected hypotheses
