The C60 cage is genus zero, its continuum limit is a sphere with twelve conical defects, and its two natural refinements are flow-inequivalent: twelve pentagons forced two ways, a defect charge of one sixth of a full turn per pentagon, and a symbolic

The graph of the fullerene C60 is genus zero, forced two independent ways, and its symmetry-preserving continuum limit is a 2-sphere carrying exactly twelve conical defects, not a torus and not any higher-genus surface. The combinatorial route is Euler's formula: for a trivalent map built only from pentagons and hexagons, edge counting gives f₅ = 6χ, so a sphere forces exactly twelve pentagons, which C60 has, while a torus would force a pentagon-free map. The energetic route is independent. The twelve pentagon centers sit at the vertices of a regular icosahedron, which is also the minimum-energy configuration of twelve mutually repelling charges on a sphere, the Thomson problem at twelve points, with energy 49.165253. In the Goldberg-Coxeter continuum limit the hexagonal regions flatten and the whole curvature concentrates at the twelve pentagon sites, each an angle deficit of π/3, one sixth of a full turn, the twelve summing to 4π as Gauss-Bonnet requires. The defect charge is 1/6 per pentagon in units of a full turn. A model that treats the C60 carrier as a torus is therefore not supported by the geometry. Separately, the two refinements the cage carries, the four-child area refinement of the sphere and a scale ladder that changes length by a fixed factor per level, are flow-inequivalent as shifts of finite type: their signed Bowen-Franks groups are Z/3 and 0, so no flow equivalence identifies one refinement level with a fixed number of ladder steps, and the registration between them is a free normalization. Every number is regenerated by the accompanying script, 14 checks, 14 pass.