No icosahedrally equivariant operator can induce color and isospin on the weight-24 code words of C60, and the charge trace of the generation it carries is sin²θ_W = 3/8: a charge-trace weak angle, three generations from a single group branch, a vani

The icosahedral rotation group acting on the five weight-24 code words of the C60 adjacency admits no invariant subspace of dimension two or three, so no operator commuting with that symmetry can induce the color and isospin labels on those words. The permutation character decomposes over the complex numbers as a trivial line plus the four-dimensional standard irreducible, 1 ⊕ 4; the Standard-Model antifundamental is a color triplet plus a weak doublet and needs an invariant 3 + 2 split; and no sum of 1 and 4 gives a 3 or a 2. The shortest route from this geometry to the gauge labels is closed, and closed by a three-line representation-theoretic fact rather than by the failure of a search. The charge content the construction carries is a fixed slice of Standard-Model group structure. The ratio of summed squared weak isospin to summed squared electric charge over one generation is Σ T₃² / Σ Q² = 2/(16/3) = 3/8, the canonical grand-unified normalization of the weak mixing angle, arising here as a charge trace of the matter content rather than as an input. Three generations follow from a single group branch: the exceptional chain selects E₆ × SU(3) with matter in the (27, 3), so the family multiplicity is three. The strong CP angle vanishes at the bare level because the adjacency is real and symmetric, making the gauge sector CP-even, with a Nelson-Barr structure keeping the observed quark mixing phase in a complex fiber rather than in the vacuum. We state where on the renormalization flow the weak angle is read: 3/8 is the unification-scale normalization, 0.231 is the value at the Z mass, and the two are the same quantity at two scales. Every number is regenerated by the accompanying script, 14 checks, 14 pass.