# Pass E — Universe Witness Catalog (Flagship Models)

## Purpose

Make the “exists in nature” claim rigorous by writing fully checkable witnesses.

Each witness below specifies:

- arena \(M\)
- global instants \(\{\Sigma_s\}\) (an admissible slicing)
- phase field \(\Phi:M\to\mathbb{T}_P\)
- a proof of coverage using a template from Pass A (phase coordinate form, equivariant circle action) or a loop-degree witness from Pass B

Notation:
- \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- \((M,\{\Sigma_s\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P)\) means the Syncre Coverage Law holds on every slice

---

## Witness 1 — Rotating rigid body with a distinguished direction source

### Data

- Arena: \(M=\mathbb{R}\times S^2\)
- Slicing: \(\Sigma_t=\{t\}\times S^2\)
- Period: \(P>0\) (use \(P=86400\) seconds for a day-cycle interpretation)
- Phase field: choose an “azimuth coordinate” \(\lambda\in S^1\) along each latitude circle and define
  \[
  \Phi(t,(\lambda,\varphi)) = \Bigl(c(t,\varphi)\oplus \frac{P}{2\pi}\lambda\Bigr)\in\mathbb{T}_P
  \]
  where \(c(t,\varphi)\in\mathbb{T}_P\) is any slice-dependent offset that may depend on latitude \(\varphi\)

This is the abstract version of “local solar phase,” where longitude shifts phase additively.

### Coverage proof (phase coordinate template)

Fix \(t\) and fix any latitude \(\varphi_0\). Restrict to the embedded loop
\[
\gamma(\lambda) := (t,(\lambda,\varphi_0))\in \Sigma_t.
\]
Then
\[
(\Phi\circ\gamma)(\lambda)=c(t,\varphi_0)\oplus \frac{P}{2\pi}\lambda.
\]
The map \(\lambda\mapsto \frac{P}{2\pi}\lambda\ (\bmod P)\) is the standard degree-1 wrap of \(S^1\) onto \(\mathbb{T}_P\). Therefore \(\Phi\circ\gamma\) is surjective, and by the loop-witness lemma (Pass B) the image \(\Phi(\Sigma_t)\) contains \(\mathbb{T}_P\). Hence \(\Phi(\Sigma_t)=\mathbb{T}_P\).

Because \(t\) was arbitrary, the Syncre Coverage Law holds for all slices:
\[
(M,\{\Sigma_t\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P).
\]

### Physical instantiations

- Earth: longitude + local solar time-of-day phase
- any rotating planet or star with a preferred illumination or reference direction
- any rigid rotor where phase depends additively on azimuth

---

## Witness 2 — Orbital ring / circular orbit phase

### Data

- Arena: \(M=\mathbb{R}\times S^1\)
- Slicing: \(\Sigma_t=\{t\}\times S^1\)
- Phase field:
  \[
  \Phi(t,\theta)=\theta\oplus c(t)\in \mathbb{T}_{2\pi}\cong S^1
  \]
  or, after choosing a period \(P\),
  \[
  \Phi(t,\theta)=\Bigl(c(t)\oplus \frac{P}{2\pi}\theta\Bigr)\in\mathbb{T}_P.
  \]

### Coverage proof (phase coordinate template)

For fixed \(t\), \(\theta\mapsto \frac{P}{2\pi}\theta\ (\bmod P)\) is surjective onto \(\mathbb{T}_P\). Adding the constant \(c(t)\) is a bijection, so \(\Phi(\Sigma_t)=\mathbb{T}_P\). Hence
\[
(M,\{\Sigma_t\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P).
\]

### Physical instantiations

- particles in a planetary ring
- any circular orbit parameterized by true anomaly or azimuth angle
- any system whose observable phase is the spatial angle coordinate itself

---

## Witness 3 — Traveling wave on a periodic medium

### Data

Let the medium be a periodic domain \(S^1\) with coordinate \(x\in \mathbb{R}/2\pi\mathbb{Z}\).

- Arena: \(M=\mathbb{R}\times S^1\)
- Slicing: \(\Sigma_t=\{t\}\times S^1\)
- Phase field (traveling-wave phase):
  \[
  \Phi(t,x)=\big(kx-\omega t\big)\bmod 2\pi
  \]
  which is a map into \(\mathbb{T}_{2\pi}\cong S^1\)

### Coverage proof (degree template)

Fix \(t\). The slice map is
\[
\Phi_t(x)=kx-\omega t\ (\bmod 2\pi).
\]
As a circle map \(x\mapsto kx\ (\bmod 2\pi)\) has degree \(k\). If \(k\neq 0\), it is surjective. Therefore \(\Phi(\Sigma_t)=S^1\) for every \(t\), and
\[
(M,\{\Sigma_t\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(2\pi).
\]

### Physical instantiations

- phase of a wave on a ring resonator
- phase of a traveling mode on a periodic lattice
- any periodic medium with a coherent traveling phase observable

---

## Witness 4 — Order-parameter phase with nontrivial winding

Many physical theories include an order parameter \(\Psi\) with a phase:
\[
\Psi(x)=|\Psi(x)|e^{i\theta(x)},
\quad \theta(x)\in S^1.
\]

To keep the witness checkable, we focus on a loop subsystem where winding is defined.

### Data

- Choose a loop \(\Gamma\cong S^1\) in the material region (for instance, a loop encircling a vortex line)
- Arena: \(M=\Gamma\) or \(M=\mathbb{R}\times \Gamma\) if time-dependent
- Slice: if time-dependent, \(\Sigma_t=\{t\}\times\Gamma\)
- Phase field: \(\Phi=\theta\bmod 2\pi\) restricted to \(\Gamma\)

### Coverage proof (loop-degree witness)

Assume the phase winds nontrivially along \(\Gamma\). Concretely, the induced circle map
\[
f:\Gamma\cong S^1\to S^1,\quad f(x)=e^{i\theta(x)}
\]
has degree \(n\neq 0\). By the nonzero-degree lemma, \(f\) is surjective, so \(\theta(\Gamma)=S^1\). Hence \(\Phi(\Gamma)=S^1\), which is the Syncre Coverage Law on this subsystem slice.

This provides a rigorous “embedded witness” form: Syncre coverage is certified on a physically meaningful loop subsystem.

### Physical instantiations

- vortices in superfluids and superconductors
- phase winding in coherent wavefunctions or complex scalar fields
- any system with an \(S^1\)-valued order parameter and a loop with nonzero winding

---

## Summary: what Pass E proves

These flagship witnesses show that Syncre coverage is not an Earth-specific coordinate trick.

- rotating bodies, orbits/rings, and traveling waves produce full-phase coverage by the same degree/equivariance mechanism
- order-parameter phases provide embedded loop witnesses with a natural holonomy/winding invariant

This is the core “exists in the known physical universe” step of the proof program: coverage is forced whenever a system supplies a surjective phase coordinate or a nontrivial winding loop.
