The Electromagnetic Coupling at Unification: α_EM(GUT) = 1/64 from the Weinberg Angle and SU(5) Normalisation in a G2 Framework

v2 adds a comparison of the structural boundary value 1/64 with the standard grand-unified expectation (1/64.8), agreeing to 1.2%.

The fine-structure constant is among the most famous unexplained numbers in physics. We do not claim to derive its low-energy value 1/137.036 from first principles; we report instead the quantity the framework does fix exactly — the electromagnetic coupling at the grand-unification scale. Within a G2 algebraic framework established in prior work, two independently grounded quantities combine to give α_EM(GUT) = (1/24) × (3/8) = 1/64 exactly. The factor 1/24 is the SU(5) normalisation, identified through the operator relation val(HM) = 48 = 2 × dim(SU(5)); the factor 3/8 is the weak mixing angle at unification, derived previously from the SO(8) spinor embedding. Their product, 3/192 = 1/64, is the electromagnetic coupling at the unification scale. We show this boundary value is consistent with the observed low-energy fine-structure constant under standard renormalisation-group running, while stating clearly that the precise running correction is not fixed by the framework and remains an open calculation. The exact result of this paper is the boundary condition 1/64; the recovery of 1/137.036 is a consistency check, not a derivation.

Keywords: fine-structure constant, electromagnetic coupling, grand unification, Weinberg angle, SU(5), G2 algebra, renormalisation group, gauge coupling unification