# Pass 04 — Flagship Theorem Chain (Certificate ⇒ Robust Witness ⇒ Quantitative Diagnostic Bound)

## Purpose

Deliver a single coherent theorem chain that reads as genuinely additive:

- not just “degree implies surjective”
- but a **certificate-driven theorem** that outputs:
  - witness tier and mechanism,
  - robustness margins (LCM/HMR),
  - correct target under coupling (CST),
  - and a quantitative diagnostic bound (coverage profile lower bound)

This is the closest thing to a “flagship theorem” for SFT because it ties together:

- universality engine,
- robustness,
- coupling targets,
- and quantitative diagnostics,

all in one chain.

---

## Objects and assumptions

Fix \(P>0\). Let \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\).

Let \(M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s\) and \(\Phi:M\to\mathbb{T}_P\) be a phase field.

On a fixed slice \(\Sigma=\Sigma_s\), let \(E\hookrightarrow \Sigma\) be an embedded loop witness \(E\cong S^1\) with parameterization \(\gamma:S^1\to \Sigma\).

Let \(f:=\Phi\circ\gamma:S^1\to \mathbb{T}_P\cong S^1\).

Assume:

- (A) **Loop degree witness:** \(\deg(f)=k\neq 0\).
- (B) **Robustness margin (LCM):** there exists a lift \(g:\mathbb{R}\to\mathbb{R}\) of \(f\) and for any perturbed observable \(\widetilde\Phi\) with restricted lift \(\widetilde g\),
  \[
  \|g-\widetilde g\|_\infty < \pi
  \]
  on a fundamental domain (this implies degree stability).
- (C) **Branch-certified quantitative regularity (MBC + BJB):** on the witness loop, the lifted map is a degree-\(|k|\) covering that decomposes into \(|k|\) monotone branches, and on each branch
  \[
  \partial_\theta g(\theta)\le b
  \]
  for a constant \(b>0\). (This is the Monotone Branch Condition plus a branch Jacobian bound.)

Define the pushforward phase distribution on the loop under uniform measure \(d\theta/(2\pi)\):
\[
\mu := f_\#\left(\frac{d\theta}{2\pi}\right).
\]

Define the coverage profile:
\[
\mathrm{CP}(\rho):=\inf_{y\in\mathbb{T}_P}\mu(B_\rho(y)),\qquad 0\le\rho\le P/2,
\]
where \(B_\rho(y)\) is the arc-neighborhood of radius \(\rho\).

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## Theorem 1 (Flagship: ESW + WSC ⇒ robust embedded coverage + quantitative anti-collapse)

Under (A)–(C):

1. **Embedded coverage:** \(f(S^1)=\mathbb{T}_P\). Hence \(\Phi\) admits an embedded Syncre witness (ESW) on \(\Sigma\).

2. **Robust persistence:** for any perturbed observable \(\widetilde\Phi\) whose restriction to the loop satisfies the LCM condition (B), the degree remains \(k\neq 0\), hence \(\widetilde\Phi\) admits the same embedded witness.

3. **Quantitative anti-collapse on the witness:** for every \(0\le\rho\le P/2\),
\[
\boxed{\;\mathrm{CP}(\rho)\ \ge\ \frac{|k|}{2\pi\,b}\,(2\rho).\;}
\]

### Proof

(1) Degree nonzero implies surjectivity, so \(f(S^1)=\mathbb{T}_P\).

(2) Degree stability under lift closeness \(\|g-\widetilde g\|_\infty<\pi\) implies \(\deg(\widetilde f)=\deg(f)=k\neq 0\), hence the embedded witness persists.

(3) Under MBC, each output arc of length \(2\rho\) has one preimage interval on each branch. The branch Jacobian bound \(\partial_\theta g\le b\) implies each preimage interval has length at least \((2\rho)/b\). Summing \(|k|\) branches gives total preimage length at least \(|k|(2\rho)/b\), and normalizing by \(2\pi\) yields the stated lower bound.

∎

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## Coupled version (correct targets under SOL/affine coupling)

Suppose instead \(\Phi=(\Phi_1,\Phi_2):M\to\mathbb{T}_P^2\) and a coupling constraint defines a coupled subtorus \(\mathrm{CST}_s\) parameterized by the free coordinate \(a=\Phi_1\).

If the same loop witness produces \(\deg(\Phi_1\circ\gamma)=k\neq 0\), then:

- the correct target is \(\mathrm{CST}_s\) (CTP),
- coverage of the free coordinate implies coverage of the coupled target (CMT),
- and the same anti-collapse bound applies to the free-coordinate pushforward on the loop (and hence to coupled neighborhoods measured in the parameter coordinate).

This is the unified “witness + target correctness + quantitative bound” story in the coupled setting.

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## What makes this theorem chain additive

Each component is standard alone, but the chain is not:

- it is a single statement that starts from witness data and ends with a quantitative diagnostic bound,
- it explicitly includes stability margins,
- it remains correct under coupling by selecting the right target.

This is the SFT signature: mechanism + certificate + correct target + measurable bound.

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## Acceptance gates

- one theorem ties together: ESW, WSC margins, coupling target correctness, and a quantitative diagnostic bound
- MBC is explicitly included as the hypothesis needed for the quantitative bound
- the coupled variant explicitly states that CST is the correct target and that coverage is inherited from the free coordinate
