Module groupoid 

The action groupoid — what symmetry breaking means, categorically, as checked code.

Symmetry breaking is not a heuristic; it is the computation of π₀ of a groupoid. A symmetry group G acts on the space X of assignments (the “possible worlds”). The action groupoid X ⫽ G has the assignments as objects and the group elements as the (iso)morphisms α → g·α: two assignments are isomorphic iff related by a symmetry — bisimilar worlds. Symmetry breaking keeps ONE assignment per isomorphism class — it computes the skeleton, equivalently π₀(X ⫽ G) (the set of orbits = connected components). The exponential→polynomial collapse is exactly |X| → |π₀(X ⫽ G)|.

This X ⫽ G is a 1-groupoid (h-level 3 in the homotopy-level table). The ∞-tower above it — symmetries between symmetry-breakings, then between those, … — is the honest open direction. We build the first rung (this 1-groupoid) solidly and check that symmetry breaking is its π₀; the next rung (a groupoid of measures-and-their-morphisms) is sketched in the campaign notes. We do not claim to have built an ∞-groupoid — only the 1-truncation that the present theory occupies, and the precise statement of what climbing would mean.

Structs

ActionGroupoid

The action groupoid X ⫽ G: assignments over nv variables acted on by the group generated by gens (literal permutations). Small nv only — it enumerates all 2^nv objects, which is the point: it makes the orbit collapse visible and checkable.