# Univ Pass 4 — Robustness Layer for Universality (Witness Stability Certificates)

## Purpose

Make the universality argument robust:

- define the perturbation models relevant to SFT (phase noise, field perturbation, frame changes)
- state and prove “witness stability” results for global and embedded witnesses
- formalize the Witness Certificate (WCT) as a stability object with explicit margins
- connect robustness to universality: if the mechanism is physical, it must persist under small perturbations

This pass turns “exists in nature” into “exists stably,” which is much closer to what people mean by “universal principle.”

---

## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Witness Stability Certificate | **WSC** | A WCT with explicit perturbation radii that preserve the witness |
| Perturbation Model | **PM** | A specified metric on fields/slicings (what “small change” means) |
| Lift-Closeness Margin | **LCM** | A bound ensuring degrees match under perturbations (Pass B) |
| Holonomy Margin | **HMR** | A buffer preventing holonomy class change (Pass C/D) |
| Frame Robustness | **FRB** | invariants survive frame equivalence / slicing changes (Pass C) |

---

## 1) Perturbation models (PM)

Universality needs a notion of “small change.” SFT uses three.

### PM-1: Phase-field sup metric on a subsystem

For \(\Phi,\widetilde\Phi:E\to\mathbb{T}_P\), define
\[
d_\infty(\Phi,\widetilde\Phi)=\sup_{x\in E} d_{\mathbb{T}_P}(\Phi(x),\widetilde\Phi(x)).
\]

This is the right metric for observational noise on a phase.

### PM-2: Lift sup metric (when lifts exist)

If \(\Phi=F\bmod P\) and \(\widetilde\Phi=\widetilde F\bmod P\) with chosen lifts on a subsystem \(E\),
\[
\|F-\widetilde F\|_\infty:=\sup_{x\in E}|F(x)-\widetilde F(x)|.
\]

This is the metric needed for degree stability via lift closeness.

### PM-3: Frame perturbation / slicing perturbation

Use FEQ (Pass C) as the baseline: slicings are considered equivalent if they are related by admissible reparameterization and slice-bijection that respects \(\Phi\) up to gauge.

Small deviations not exactly FEQ can be treated as perturbations if they produce a small phase-field change on witnesses. For universality, FEQ invariance is the primary requirement.

---

## 2) Robustness for loop-degree witnesses (LDW)

### Theorem 1 (Degree stability under lift closeness)

Let \(f,\tilde f:S^1\to S^1\) admit lifts \(g,\tilde g\) with
\[
\sup_\theta |g(\theta)-\tilde g(\theta)|<\pi.
\]
Then \(\deg(f)=\deg(\tilde f)\).

This is Pass B’s core stability lemma.

### Corollary 2 (Embedded witness stability under LCM)

Let \(\gamma:S^1\to\Sigma\) be an embedded witness loop. Suppose \(f=\Phi\circ\gamma\) has \(\deg(f)=k\neq 0\).

If \(\widetilde\Phi\circ\gamma\) admits a lift within \(\pi\) of a lift of \(\Phi\circ\gamma\), then \(\deg(\widetilde\Phi\circ\gamma)=k\neq 0\), hence \(\widetilde\Phi\) has the same embedded witness.

Define the **Lift-Closeness Margin (LCM)** as any \(\delta\) with:
\[
\|F-\widetilde F\|_\infty<\delta<\pi
\]
on the loop lift representatives.

---

## 3) Robustness for holonomy witnesses (HW)

### Lemma 3 (Holonomy continuity under connection perturbation)

Let \(\alpha\) and \(\widetilde\alpha\) be connection 1-forms on \(\Sigma\). For any loop \(\gamma\),
\[
\left|\int_\gamma \alpha - \int_\gamma \widetilde\alpha\right|\le \int_\gamma |\alpha-\widetilde\alpha|.
\]
So holonomy changes continuously under small connection perturbations.

### Corollary 4 (Holonomy witness stability under HMR)

If \(\int_\gamma \alpha = mP\) with \(m\neq 0\) and the perturbation satisfies
\[
\left|\int_\gamma (\alpha-\widetilde\alpha)\right|<\frac{P}{2},
\]
then \(\int_\gamma \widetilde\alpha\) remains within \(P/2\) of the same integer multiple of \(P\), hence the holonomy class does not change. This defines a **Holonomy Margin (HMR)**.

---

## 4) Witness Stability Certificate (WSC)

A witness becomes a stability object by storing the margins needed for persistence.

### Definition (WSC)

A **Witness Stability Certificate** includes:

- witness tier (global or embedded)
- subsystem specification (loop parameterization or embedding)
- witness value (degree \(k\) or holonomy multiple \(m\))
- perturbation model (PM-1/PM-2/PM-3)
- stability margin:
  - LCM \(<\pi\) for degree witnesses, and/or
  - HMR \(<P/2\) for holonomy witnesses
- optional dynamics margin (if using preservation theorems)

This is the universality-friendly version of “it exists in nature”: it exists with margins.

---

## 5) Frame robustness (FRB)

Under FEQ, witness values are invariant:

- degree does not change under reparameterization of the loop and gauge shift
- holonomy is gauge invariant on closed loops
- thus WSC survives frame equivalence

This ensures universality does not depend on a fragile coordinate choice.

---

## Acceptance gates

- Three perturbation models (PM-1/2/3) are defined
- Degree witness stability is stated as an LCM condition
- Holonomy witness stability is stated as an HMR condition
- WSC is defined as a certificate object capturing margins and witness tier
- FRB is stated linking witness invariance to FEQ

---

## 6) Quantitative forcing bounds (Lipschitz and Fourier defects)

Robustness also applies to the action-forcing mechanism itself.

### Equivariance defect bound

Given \((A,\phi,\chi)\) on a slice \(\Sigma\), define the equivariance defect
\[
\mathrm{Def}(A,\phi,\chi):=\sup_{\theta\in S^1,\ x\in\Sigma} d_{\mathbb{T}_P}\big(\phi(A(\theta,x)),\phi(x)\oplus\chi(\theta)\big).
\]

If \(\mathrm{Def}(A,\phi,\chi)\le \varepsilon\) and \(O\subseteq\Sigma\) is a circle-type orbit, then restricting to \(O\cong S^1\) yields
\[
\phi(O)\ \text{is}\ \varepsilon\text{-dense in }\mathbb{T}_P.
\]
In regimes where the target metric admits an explicit arc-gap bound, choosing \(\varepsilon\) below that bound upgrades \(\varepsilon\)-density to surjectivity.

### Lift-derivative sufficient condition

On a circle-type orbit \(O\cong S^1\), write \(\phi|_O\) as a lift \(f(t)\bmod P\) with \(t\in\mathbb{R}/2\pi\mathbb{Z}\). If
\[
\big\|f'(t) - \tfrac{dP}{2\pi}\big\|_{\infty}<\tfrac{|d|P}{2\pi}
\]
for the integer slope \(d\neq 0\), then \(f\) is strictly monotone on a lift interval and winds exactly \(|d|\) times, forcing \(\phi(O)=\mathbb{T}_P\).

### Fourier defect sufficient condition

If a lift decomposes as
\[
 f(t)=\tfrac{dP}{2\pi}t + \psi(t)
\]
with \(\psi\) \(2\pi\)-periodic and \(\|\psi'\|_{\infty}<\tfrac{|d|P}{2\pi}\), the same conclusion holds.

Bounding \(\|\psi'\|_{\infty}\) can be done by Fourier tails:

- if \(\psi(t)=\sum_{n\neq 0} a_n e^{int}\), then \(\|\psi'\|_{\infty}\le \sum_{n\neq 0} |n a_n|\)

Thus a quantitative Fourier-defect bound \(\sum_{n\neq 0}|n a_n|<\tfrac{|d|P}{2\pi}\) yields robust surjectivity.
