The Weinberg Angle from Foam Geometry: sin²θ_W = 3/8 at the GUT Scale

The weak mixing angle sin²θ_W = 3/8 at the GUT scale is derived as an exact theorem from the mode dimension ratio of the face Laplacian of the truncated octahedron (Kelvin cell). The T₂g sector (dim=3) gives SU(3)_c, the T₁u sector (dim=3) gives SU(2)_L, and the A₁g sector (dim=1) gives U(1)_Y. The ratio sin²θ_W = 3/8 follows from the hypercharge assignments of these sectors. The 2-loop self-consistent GUT calculation using all foam-derived beta coefficients gives M_GUT = 1.50×10¹⁶ GeV, α_GUT = 1/46, and reproduces α_s(M_Z) = 0.1179 exactly from the GUT chain.