# Meta-Witness Theorem from an \(S^1\)-Action (Universality Engine)

## Purpose

Replace a catalog of examples with a single forcing mechanism theorem.

The mechanism is:

- a cyclic symmetry action on a slice
- an equivariant phase observable with nonzero slope

From these, full-cycle observability follows.

---

## Setting and notation

- Period \(P>0\), phase circle \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- Slice \(\Sigma\)
- Circle action \(A:S^1\times \Sigma\to\Sigma\)
- Phase observable \(\phi:\Sigma\to\mathbb{T}_P\)

A phase observable is equivariant if there exists a continuous group homomorphism
\[
\chi:S^1\to\mathbb{T}_P
\]
so that for all \(\theta\in S^1\) and \(x\in\Sigma\),
\[
\boxed{\ \phi(A(\theta,x))=\phi(x)\oplus\chi(\theta).\ }
\]

For \(x\in\Sigma\), its stabilizer is
\(
\mathrm{Stab}(x):=\{\theta\in S^1: A(\theta,x)=x\}.
\)
The orbit is \(O_x\cong S^1/\mathrm{Stab}(x)\).

### Structural lemma (stabilizers are forced into the kernel)

If \(\phi\) is \(\chi\)-equivariant, then \(\mathrm{Stab}(x)\subseteq\ker\chi\) for every \(x\in\Sigma\).

*Reason:* if \(A(\theta,x)=x\), then equivariance forces \(\chi(\theta)=0\).

This is why an extra "free orbit" hypothesis is not needed: in the nonzero-slope regime, \(\ker\chi\) is finite, so orbits in an equivariance domain are automatically circle-type.

---

## Structural fact: nonzero slope is the same as surjective response

Any continuous homomorphism \(\chi:S^1\to\mathbb{T}_P\cong S^1\) is either trivial or surjective. In the standard identifications \(S^1\cong \mathbb{R}/2\pi\mathbb{Z}\) and \(\mathbb{T}_P\cong \mathbb{R}/P\mathbb{Z}\), there exists an integer \(d\in\mathbb{Z}\) such that
\[
\chi(\theta)=\tfrac{P}{2\pi}d\,\theta\ \bmod P.
\]

- \(d=0\) is slope zero and \(\chi\) is trivial
- \(d\neq 0\) is nonzero slope and \(\chi\) is surjective

---

## Action Witness Template (AWT)

### Theorem (AWT: action witness implies coverage)

Assume:

- \(A\) is a continuous \(S^1\)-action on \(\Sigma\)
- \(\phi\) is \(\chi\)-equivariant
- \(\chi\) has nonzero slope (equivalently, \(\chi(S^1)=\mathbb{T}_P\))

Then
\[
\boxed{\ \phi(\Sigma)=\mathbb{T}_P.\ }
\]

#### Proof

Pick any \(x\in\Sigma\). By equivariance,
\(
\phi(A(\theta,x))=\phi(x)\oplus\chi(\theta).
\)
Since \(\chi\) is surjective, \(\phi(O_x)=\phi(x)\oplus\chi(S^1)=\mathbb{T}_P\).
Because \(O_x\subseteq\Sigma\), \(\phi(\Sigma)=\mathbb{T}_P\). \(\square\)

This theorem is the universality engine.

---

## Embedded-orbit variant

Sometimes the observable is only physically meaningful on a subsystem loop.

### Theorem (embedded orbit witness)

Assume \(\phi\) is \(\chi\)-equivariant with nonzero slope. For any circle-type orbit \(O\subseteq\Sigma\) on which equivariance holds,
\[
\boxed{\ \phi(O)=\mathbb{T}_P.\ }
\]

This is the correct theorem form for loop-level phase phenomena.

---

## Necessary and sufficient conditions for coverage under equivariant circle actions

### Theorem (necessary and sufficient)

Let \((\Sigma,A,\phi,\chi)\) be as above with \(\phi\) \(\chi\)-equivariant. Then \(\phi(\Sigma)=\mathbb{T}_P\) holds if and only if at least one of the following mechanisms holds:

- Nonzero-slope regime: \(\chi\) has nonzero slope
- Quotient regime: \(\chi\) is trivial and the induced map \(\bar\phi:\Sigma/S^1\to\mathbb{T}_P\) is surjective

In particular, in the nonzero-slope regime, coverage is automatic.

---

## Checklist: how flagship physical models fit

- Rotating body
  - \(\Sigma\) is the spatial slice (surface or a latitude circle)
  - \(A\) is rotation about an axis
  - \(\phi\) is an hour-angle style phase observable
  - nonzero slope is the statement that rotation advances the phase

- Orbital ring
  - \(\Sigma\cong S^1\)
  - \(A\) is angle shift
  - \(\phi\) is orbital phase

- Traveling wave on a periodic medium
  - \(\Sigma\cong S^1\)
  - \(A\) is translation
  - \(\phi\) is wave phase
  - nonzero slope is nonzero wavenumber

The burden becomes verifying hypotheses, not debating interpretation.
