The Weinberg Angle from C5 = C3 ⊕ C2: A Geometric Origin for Electroweak Mixing

The Weinberg angle θ_W, which characterizes electroweak mixing in the Standard Model, has the experimentally measured value sin²θ_W ≈ 0.231 at the Z-boson mass scale. While grand unified theories predict sin²θ_W = 3/8 at the unification scale, the origin of this value has remained geometrically obscure. Building on our geometric framework based on C₅ = C₃ ⊕ C₂, we demonstrate that the ratio g²/g′² = 5/3 between gauge couplings is purely geometric, equal to the dimension ratio dim(C₅)/dim(C₃) = 5/3. This immediately yields sin²θ_W = 1/(1 + 5/3) = 3/8 at the unification scale. Renormalization group evolution then produces the observed low-energy value. We also identify a direct low-energy formula, sin²θ_W ≈ (dim C₃ × dim C₂)/(dim C₅)² = 6/25 = 0.24, which matches experiment to within 4 percent. These results demonstrate that the Weinberg angle, like the Koide formula for charged-lepton masses, emerges from the dimensional structure of C₅.