A first-principles derivation of the Weinberg angle sin²θ_W = 1/φ³ from the group theory of the FCC vacuum lattice. The 12-dimensional edge representation decomposes as A₁ ⊕ E ⊕ T₁ ⊕ 2T₂ under the octahedral group. The Coxeter projection of the cuboctahedron fixes the ratio of the electromagnetic (A₁) and weak (E) subspace norms, giving a geometric prediction within 2% of the measured value.