Stokes in two signatures — a holonomy equals an enclosed area

Scope. Closed-loop results where a holonomy equals an enclosed area: ∮ω = ∬dω. Evicted from the vocabulary (v4 §11) as a resident — a specific theorem placed at the bridge coordinate, not an axis itself.

Four instances. The Thomas rotation (hyperbolic area in velocity space), the Weyl phase (symplectic area in phase space), odd elasticity's work per cycle, and Carnot work as the (T, S) area.

Naming the theorem showed the instances are not the same kind of object. In three of the four, dω is a constant 2-form. Only the Thomas rotation has dω a genuine curvature — so the founding slogan ("noncommutativity is curvature") holds for Thomas and is Stokes-with-a-constant-form for the rest.

The fifth instance is the same theorem Wick-rotated. The Wilson loop's area law, ⟨W⟩ ~ e^(−σ·Area), is the Weyl phase e^(i·area) rotated: e^(iS) → e^(−S_E). Confinement is the statement that the Euclidean coefficient is nonzero. Here dω is the Yang–Mills field strength, which is a curvature — restoring the tally to two curvatures and three constant forms.

Placement. A bridge instance; the bridge coordinate lives in rung-vocabulary-v5.md §8.1, the trace-identity family it neighbours is [[CASE-16]].