# Pass B — Robustness and Stability of Syncre Coverage

## Purpose

Turn Syncre coverage from a “one-shot” statement into a stable, perturbation-resistant property.

This pass builds proof tools that will be reused in later passes:

- coverage from a nonzero-degree loop witness
- stability of degree under homotopy and under small perturbations (via lifts)
- quantitative “almost coverage” diagnostics using pushforward measures

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## Recap: coverage via a loop witness

Let \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\cong S^1\). Fix a slice \(\Sigma\) (suppress the index \(s\)).

### Definition (Loop witness)

A **loop witness** for a phase field \(\Phi:\Sigma\to \mathbb{T}_P\) is a continuous map \(\gamma:S^1\to \Sigma\) such that the circle map
\[
f := \Phi\circ\gamma : S^1\to \mathbb{T}_P\cong S^1
\]
has nonzero degree.

### Lemma (Nonzero-degree loop implies coverage)

If \(\Phi:\Sigma\to \mathbb{T}_P\) admits a loop witness \(\gamma\) with \(\deg(\Phi\circ\gamma)\neq 0\), then
\[
\Phi(\Sigma)=\mathbb{T}_P.
\]

#### Proof

By the nonzero-degree lemma from Pass A, \(f=\Phi\circ\gamma\) is surjective, so \(f(S^1)=\mathbb{T}_P\). Since \(\gamma(S^1)\subseteq\Sigma\), we have \(\mathbb{T}_P\subseteq \Phi(\Sigma)\). Therefore \(\Phi(\Sigma)=\mathbb{T}_P\). ∎

This is the most useful “robust coverage certificate” for universe-scale witnesses: it reduces coverage to checking a degree on a loop.

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## Degree stability: the basic engine of robustness

The degree of a continuous circle map is homotopy invariant. This already implies a robust statement:

- once you have a nonzero-degree loop witness, any perturbation that does not change the degree keeps coverage

To make that usable, we formalize two stability tools.

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## Tool 1: degree is invariant under homotopy

### Lemma (Homotopy invariance of degree)

Let \(f_0,f_1:S^1\to S^1\) be continuous. If there exists a continuous homotopy \(H:[0,1]\times S^1\to S^1\) with \(H(0,\cdot)=f_0\) and \(H(1,\cdot)=f_1\), then
\[
\deg(f_0)=\deg(f_1).
\]

#### Proof sketch

Use lifts \(g_u:\mathbb{R}\to\mathbb{R}\) of \(H(u,\cdot)\) that vary continuously in \(u\). The increment \(g_u(\theta+2\pi)-g_u(\theta)\) is constant in \(\theta\) and integer-multiple of \(2\pi\). Continuity in \(u\) forces that integer to be constant, hence degrees agree. ∎

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## Tool 2: degree is stable under small perturbations (via lifts)

To state a quantitative condition, fix the standard identification \(S^1=\{e^{i\theta}\}\). For a continuous map \(f:S^1\to S^1\), choose a lift \(g:\mathbb{R}\to\mathbb{R}\) with
\[
f(e^{i\theta})=e^{ig(\theta)},\qquad g(\theta+2\pi)=g(\theta)+2\pi k,
\]
and \(\deg(f)=k\).

### Lemma (Lift stability criterion)

Let \(f,h:S^1\to S^1\) be continuous. Suppose there exist lifts \(g,\tilde g:\mathbb{R}\to\mathbb{R}\) of \(f\) and \(h\) such that
\[
\sup_{\theta\in\mathbb{R}} |g(\theta)-\tilde g(\theta)| < \pi.
\]
Then \(\deg(f)=\deg(h)\).

#### Proof

For each \(\theta\),
\[
(g(\theta+2\pi)-g(\theta))-(\tilde g(\theta+2\pi)-\tilde g(\theta))
\]
is an integer multiple of \(2\pi\). But the left-hand side has absolute value bounded by
\[
|g(\theta+2\pi)-\tilde g(\theta+2\pi)| + |g(\theta)-\tilde g(\theta)| < 2\pi.
\]
The only integer multiple of \(2\pi\) with magnitude \(<2\pi\) is \(0\). Hence the increments match and degrees agree. ∎

### Corollary (Degree stability under a uniform phase error)

If \(h\) is obtained from \(f\) by a perturbation that admits a continuous lift-error bounded by \(<\pi\), then \(h\) has the same degree, hence the same surjectivity behavior.

This makes “small perturbations preserve coverage” precise when perturbations can be controlled in a lifted coordinate.

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## Robust coverage for Syncre slices

### Theorem (Robust coverage via loop degree)

Let \(\Phi:\Sigma\to \mathbb{T}_P\) be a phase field and let \(\gamma:S^1\to\Sigma\) be a loop witness with \(\deg(\Phi\circ\gamma)\neq 0\).

Let \(\widetilde\Phi:\Sigma\to \mathbb{T}_P\) be another phase field such that the circle maps
\[
f=\Phi\circ\gamma,\qquad \tilde f=\widetilde\Phi\circ\gamma
\]
admit lifts \(g,\tilde g\) satisfying \(\sup|g-\tilde g|<\pi\).

Then \(\widetilde\Phi(\Sigma)=\mathbb{T}_P\).

#### Proof

By the lift stability criterion, \(\deg(\tilde f)=\deg(f)\neq 0\). Therefore \(\tilde f\) is surjective, so \(\mathbb{T}_P\subseteq \widetilde\Phi(\Sigma)\), hence \(\widetilde\Phi(\Sigma)=\mathbb{T}_P\). ∎

This is the principal robustness theorem we will use in later passes.

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## Quantitative diagnostics: “almost coverage” and uniformity

Coverage is a yes/no statement. For analysis and convergence, it helps to track *how* phase labels are distributed.

Let \((\Sigma,\nu)\) be a measured slice (e.g., uniform longitude measure on \(S^1\), surface area on \(S^2\), or any chosen weighting).

### Definition (Pushforward phase distribution)

Define the phase distribution at slice \(\Sigma\) as
\[
\mu := \Phi_\# \nu,
\]
the pushforward measure on \(\mathbb{T}_P\).

- If \(\Phi(\Sigma)=\mathbb{T}_P\), then \(\mathrm{supp}(\mu)=\mathbb{T}_P\).
- If \(\mu\) is close to the uniform (Haar) measure, then phase labels are not only present but evenly represented.

### Definition (Coverage radius)

Define the **coverage radius** \(\rho(\Phi,\nu)\) as the largest \(\rho\ge 0\) such that for every phase point \(y\in\mathbb{T}_P\),
\[
\mu\big(B_\rho(y)\big) > 0,
\]
where \(B_\rho(y)\) is the arc-neighborhood of radius \(\rho\).

- \(\rho>0\) implies a strong quantitative version of coverage: every phase neighborhood occurs with positive weight.
- For continuous \(\Phi\) on compact \(\Sigma\) with full coverage, \(\rho\) is positive whenever \(\nu\) has full support and \(\Phi\) does not collapse entire regions to sets of measure zero.

This diagnostic will be used later to discuss convergence and “mixing” of phase labels.

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## Acceptance gates

- A loop witness definition is stated and used to imply coverage
- Degree stability is proved under homotopy and under a quantitative lift perturbation bound
- A robust coverage theorem is stated and proved using the loop witness plus lift stability
- Pushforward phase distribution \(\mu=\Phi_\#\nu\) and at least one quantitative coverage diagnostic are defined clearly
