The Weinberg Angle sin²θW = 3/8 as an Exact Algebraic Identity of the Exceptional Lie Algebra G2

The weak mixing angle (Weinberg angle) is one of the nineteen free parameters of the Standard Model, with no value predicted by the theory itself. We show that within a 14-element exceptional Lie algebra (G2) framework established in prior work, the weak mixing angle at the grand-unification scale is fixed exactly by the representation structure: sin² W = 3/8 = 0. 375. This is the same value predicted by SU (5) θ grand unified theory, but here it arises from the embedding of the electroweak sector in the eightdimensional spinor of SO (8) rather than from an assumed unifying gauge group. The value 3/8 is the high-scale boundary condition; renormalisation-group running to the Z pole gives the measured lowenergy value sin² W (MZ) ≈ 0. 231 through standard physics. We present the algebraic derivation, state θ the relationship to the measured value explicitly, and note that the same framework fixes the unified coupling and the electromagnetic coupling boundary condition, giving a consistent picture of the electroweak sector at unification. Keywords: Weinberg angle, weak mixing angle, electroweak unification, grand unification, G2 algebra, SU (5), SO (8), exceptional Lie algebras