# Embedded Loop Witnesses

## Purpose

Provide a small canonical library of Embedded Syncre Witness (ESW) templates that are reusable across physical scopes.

Each template includes:

- subsystem specification (what \(E\) is and how it sits in \(\Sigma\))
- witness mechanism (LDW or HW)
- certificate recipe: WCT fields + WSC margin recipe (LCM/HMR)

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## Template C1 — Latitude loop on a rotating body (LDW)

- Slice: \(\Sigma_s=S^2\) (surface at global instant \(s\))
- Subsystem: latitude circle \(E\cong S^1\) at fixed colatitude \(\varphi_0\neq 0,\pi\)
- Embedding: \(i(\lambda)=(\lambda,\varphi_0)\)
- Phase observable: \(\Phi_s(\lambda,\varphi)=c_s(\varphi)\oplus \frac{P}{2\pi}\lambda\)

**Witness:** \(f(\lambda)=\Phi_s\circ i(\lambda)\) has degree \(1\).

**WCT recipe**
- tier: WTR-E
- mechanism: LDW
- witness integer: \(k=1\)

**WSC margin recipe**
- use PM-2 (lift sup metric) on the loop lift
- ensure lift error \(<\pi\) (LCM)

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## Template C2 — Ring/orbit loop (LDW)

- Slice: \(\Sigma_s=S^1\)
- Subsystem: \(E=\Sigma_s\)
- Phase observable: \(\Phi_s(\theta)=c_s\oplus \frac{P}{2\pi}\theta\)

**Witness:** degree \(1\) (or degree \(k\) if scaled).

**WCT/WSC:** same as C1.

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## Template C3 — Traveling wave on periodic medium (LDW)

- Slice: \(\Sigma_s=S^1\) coordinate \(x\)
- Phase observable: \(\Phi_s(x)=kx\oplus c_s\) in \(\mathbb{T}_{2\pi}\)

**Witness:** degree \(k\neq 0\).

**WSC margin recipe**
- PM-2 with LCM \(<\pi\) is enough to preserve degree.

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## Template C4 — Vortex loop in an order-parameter phase (HW/LDW)

- Slice: \(\Sigma_s\) contains a loop \(E\cong S^1\) encircling a defect
- Phase observable: \(\Phi_s=\theta\) (order parameter phase mod \(2\pi\))
- Lift atlas exists locally; connection form \(\alpha=dF\) defines holonomy

**Witness (holonomy):** \(\int_E \alpha = m(2\pi)\), \(m\neq 0\).

**WCT recipe**
- tier: WTR-E
- mechanism: HW
- witness integer: \(m\neq 0\)
- evidence: loop integral computation or measured phase winding

**WSC margin recipe**
- PM-4 (connection perturbation)
- require \(|\int_E(\alpha-\tilde\alpha)|<P/2\) (HMR)

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## Template C5 — Coupled loop (SOL) on a subsystem (WTR-C)

- Codomain: \(\mathbb{T}_P^2\)
- Coupling: \(b\ominus a=\Psi(s)\)
- Subsystem: any loop where \(a\) covers \(\mathbb{T}_P\)

**Witness:** free coordinate degree \(1\) implies CST coverage on that slice.

**WSC recipe**
- store SOL assumption + target CST
- LCM or HMR margins depending on measurement type

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## Acceptance gates

- at least four canonical ESW templates are given
- each includes a WCT recipe and a WSC margin recipe
- templates cover rotation, orbit, wave, and order-parameter winding settings
