A Parameter-Free Relation Between the SU(2) and U(1) Gauge Couplings at High Scale

The Standard Model treats the gauge couplings as independent parameters. We report a relation between two of them that follows from a projection rule containing no adjustable quantities. Fields coupling to q internal axes project onto a two-dimensional substrate with integrand \sin^{2q-1}\theta; integration over the fundamental quadrant gives I(1)=1 and I(2)=2/3, whence \frac{g_2^2}{g_1^2} = \frac{I(2)}{I(1)} = \frac{2}{3}. Both integrals are forced by the rule; neither is chosen. Using PDG inputs and two-loop Standard Model running (top-Yukawa included, U(1) in GUT normalisation), the measured ratio at the Planck scale is 0.67247 \pm 0.00014, agreeing with 2/3 to 0.87%. What this means: The relation is not satisfied within experimental error. The propagated uncertainty is 0.022\%, so the residual is roughly forty experimental standard deviations. The meaningful uncertainty is theoretical—the assumption of no new physics across seventeen decades of running. The honest claim is percentage-level agreement of a forced value, not an exact identity. What this is not: A three-coupling unification. An earlier version asserted g_1^2:g_2^2:g_3^2 = 1:2/3:2/\pi with zero adjustable parameters. That claim is withdrawn. The factor I(3)=2/\pi required for g_3 cannot be obtained from any consistent extension of the rule that yields I(1) and I(2); among the consistent alternatives, only the value matching observation works. It is therefore a fitted choice, and no prediction for g_3, \alpha_s(M_P), or \Lambda_{\text{QCD}} is made here. The running is not imported. The one- and two-loop beta coefficients are counting formulae over matter content. The framework independently derives three generations, Standard Model representations, one Higgs doublet, and three gauge-singlet right-handed neutrinos, reproducing b_1=41/10, b_2=-19/6, b_3=-7 and the full two-loop matrix exactly. Falsifiability: Closure holds only if there is no new electroweak physics below M_P. A fourth generation would displace the ratio by 9\%, and TeV-scale supersymmetry by 13\%. Since the measured ratio matches the Standard-Model-only value, the relation is consistent with the framework's independent prediction that neither exists. A deliberate narrowing: A single ratio at the percent level is suggestive, not compelling. The substance is that the value is forced rather than chosen, and that it is falsifiable at accessible energies. Keywords: gauge couplings, SU(2), U(1), running, renormalisation group, Standard Model, parameter-free relation, falsifiability, high scale, Planck scale