# Worked example: solar local time and the coupled solar–lunar target

## Uncoupled solar local time (circle target)

Fix a global instant \(s\). Let \(Σ_s\) be the Earth surface slice at that instant, and let \(P=86400\) seconds.

Define the solar local-time phase observable \(Φ_☉ : Σ_s → ℝ/Pℝ\).
On any fixed-latitude loop \(E ⊂ Σ_s\) parameterized by longitude \(λ \in S^1\), the restriction has degree one:
\[
Φ_☉(λ) = c_s ⊕ \frac{P}{2π}λ \quad (∀ λ ∈ S^1),
\]
so \(Φ_☉(E)=ℝ/Pℝ\). This is a loop-degree witness (LDW) for coverage on the circle target.

## Coupled solar–lunar local time (CST target)

Let \(Φ_☉\) be the solar phase and let \(Φ_☽\) be a lunar phase observable on the same slice.
In the SOL coupling model used in this bundle, the phases satisfy
\[
Φ_☽ ⊖ Φ_☉ = Ψ(s),
\]
so the correct target is the diagonal translate (a coupled subtorus coset)
\[
\mathrm{CST}_s = \{(a,b) \in (ℝ/Pℝ)^2 : b ⊖ a = Ψ(s)\}.
\]
Coverage and any baseline measure statements are asserted on \(\mathrm{CST}_s\), with coupled Haar as baseline.

## Shipped artifacts and audits

- Global witness example (AWT): `engine/artifacts/WSC-G-AWT-0001.json`
- Coupled-target example (CMT-SOL): `engine/artifacts/WSC-C-CMT-0001.json`

Audit commands:

- `python3 engine/audit/audit_wsc.py engine/artifacts/WSC-G-AWT-0001.json`
- `python3 engine/audit/audit_wsc.py engine/artifacts/WSC-C-CMT-0001.json`

Thresholds and margins used by audits are documented in `engine/contracts/constants_thresholds.md`.
