# Arena Universality

## Purpose

Lift the universality engine from “single slice” language to a full sliced-arena statement compatible with SFT’s frame equivalence (FEQ) and gauge invariance.

This makes the main universality claim read as a genuine theory statement, not a collection of slice facts.

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## Data

- Sliced arena: \(M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s\)
- Period: \(P>0\), phase circle \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- Phase field: \(\Phi:M\to \mathbb{T}_P\)
- Slice restriction: \(\Phi_s:=\Phi|_{\Sigma_s}\)

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## Arena Action Witness Template (A-AWT)

### Hypotheses (slice-uniform)

Assume there exists a family of \(S^1\)-actions on slices:
\[
A_s:S^1\times \Sigma_s\to \Sigma_s
\]
and a single phase response homomorphism \(\chi:S^1\to\mathbb{T}_P\) such that:

- (H1) **Equivariance on every slice:** for all \(s\), \(\theta\), and \(x\in\Sigma_s\),
 \[
 \Phi_s(A_s(\theta,x))=\Phi_s(x)\oplus\chi(\theta).
 \]
- (H2) **Surjective response:** \(\chi(S^1)=\mathbb{T}_P\).
- (H3) **Full-orbit condition on every slice:** for each \(s\), there exists \(x_s\in\Sigma_s\) such that the orbit \(O_s=\{A_s(\theta,x_s)\}\) is homeomorphic to \(S^1\).

### Theorem B1 (Arena-level universality engine)

Under (H1)–(H3),
\[
\boxed{\;\forall s\in\mathbb{R},\qquad \Phi(\Sigma_s)=\mathbb{T}_P.\;}
\]
Equivalently, \((M,\{\Sigma_s\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P)\).

#### Proof

Fix \(s\). Apply the slice-level AWT theorem (Univ ) to \((\Sigma_s,\Phi_s)\) with action \(A_s\) and response \(\chi\). The full orbit \(O_s\) implies \(\Phi_s(O_s)=\mathbb{T}_P\), hence \(\Phi(\Sigma_s)=\mathbb{T}_P\). Since \(s\) was arbitrary, the conclusion holds for all slices. ∎

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## FEQ invariance statement (frame robustness)

Let \((M,\{\Sigma_s\},\Phi)\) and \((M,\{\widetilde\Sigma_t\},\widetilde\Phi)\) be FEQ-equivalent frames (): there is a slice-bijection and reparameterization mapping slices to slices, and \(\widetilde\Phi\) differs from \(\Phi\) only by admissible gauge shifts.

### Proposition B2 (FEQ invariance of SCL)

If SCL holds for one FEQ representative, then it holds for all FEQ-equivalent representatives.

#### Reason

FEQ maps each slice bijectively onto a slice and gauge shifts are bijections of \(\mathbb{T}_P\), so image coverage is preserved.

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## Embedded arena universality (ESW tier)

If (H3) holds only on a subsystem \(E_s\hookrightarrow \Sigma_s\) (for example a loop orbit), then the same argument produces an embedded witness on each slice rather than global coverage. This is the honest “universality of embedded observability” variant.

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## Acceptance gates

- Arena-level AWT hypotheses are stated as slice-uniform conditions.
- Theorem B1 is proved by reduction to slice-level AWT.
- FEQ invariance is stated clearly.
- The embedded arena variant is included to support loop-level physical observability.
