Song of the Bubble · a CRR foam composition

CRR and Plateau's Laws: Geometry as Precision Saturation

A foam at equilibrium obeys Plateau's laws: three films meet along an edge at 120°, and four edges meet at a vertex at the tetrahedral angle (≈109.47°). These aren't empirical fits; they are the angles where the surface-area functional is stationary. That same stationarity has a CRR reading, and it is where the framework contributes something real to foam geometry rather than just paraphrasing known physics.

The precision argument

For a soap film, surface area is proportional to the film's Fisher information about the position of the surfactant monolayer. A foam minimises its total area because it minimises positional precision loss — equivalently, it maximises the coherence of the surfactant arrangement. So the Plateau angles are the saturation geometry of a 2D Fisher-Rao manifold embedded in 3D space.

This is the CRR-honest claim: the angles are fixed points of the precision gradient, so a foam relaxing toward equilibrium is a system accumulating coherence along its Fisher-Rao geodesics until C saturates. The rate of relaxation is a slider in the panel so you can see this convergence directly in the measured ⟨θ₃⟩ and ⟨θ₄⟩ readouts.

T1 events as localised rupture

When a small bubble loses a face and its neighbours rearrange (a T1 topological transition), the local coherence saturates: C·Ω = 1 at that vertex. The film can't accommodate further compression without changing topology. This maps T1 events onto CRR's Dirac delta — localised ruptures in a field that globally persists. The foam doesn't fail; one piece of it reorganises.

Markov blanket across the cluster

When bubbles form a connected cluster, they share a Markov blanket. The canonical multi-agent CRR result gives:

The √n scaling reflects that an ensemble of n coherent members gains collective coherence sublinearly — the right scaling for a Fisher-information ensemble, matching what jammed foams actually do at the liquid→solid transition around 36% gas fraction. Members receive coherence boosts that slow their drainage and reduce rupture risk, which is why foam bubbles last longer than isolated ones (sometimes by orders of magnitude).

What's still Maxwell, not CRR

The iridescence is still computed from Cauchy's n(λ) and the Airy formula across 31 spectral samples, as in the Rainbow piece. The Young-Laplace pressure balance determining contact disc radius is classical fluid statics. CRR doesn't derive these. It provides the temporal grammar — when films accumulate, where they saturate, how memory shapes the next bubble — and the geometric saturation identity for the angles.