Capacity, Coherence, and the Thin Radical in Equivariant Categories

Let G be a transitive permutation group on a finite set X. How many essentially different G-equivariant category structures exist on X with finite group endomorphism monoid M_0? We identify a combinatorial invariant — the thin radical D of the Schurian coherent configuration CC(G,X) — and conjecture that the isomorphism classes biject with H^2(D, Z(M_0)). We prove this classification at two structural extremes and verify it computationally in nine additional cases. At one extreme, G = Sym(X): every non-diagonal orbit product is thick; D is trivial, H^2 = 0, and only one structure exists. This is the s_M universality theorem (Theorem 5.1): every valid datum with |M_1| ≥ 2 is isomorphic to s_{M_0}, where all five composition maps equal the group operation. The proof establishes freeness and transitivity of the M_0-actions, determines all compositions from a single central parameter d in Z(M_0), and constructs an explicit isomorphism. At the other extreme, G = Z/n: every product is thin; D = Z/n, and the isomorphism classes biject with H^2(Z/n, Z(M_0)) via standard group cohomology (Theorem 7.1). Between these extremes, the general thin-radical classification is Conjecture 6.2. The proposed mechanism is associativity: thin orbit products propagate composition parameters via cocycle conditions, thick orbit products destroy them via norm relations and extension splitting, and D captures exactly the orbits whose parameters survive. We verify the conjecture in 9 distinct (G, M_0) pairs — including the mixed case D_4, where one thick product kills exactly one degree of freedom — with no failures. This paper is the focused equivariant-category specialization of the broader fibered-descent framework developed in (Badkur & Dak, 2026d, DOI: 10.5281/zenodo.20334628). Appendix A documents the computational verification protocol for independent reproduction.