P-7: A Geometric Proof of Electroweak Unification — From Geometric Integrals of the Spatial Orientation Field to Renormalization Group Running

Abstract: Starting from the first principles of Space Ontology—the spatial orientation field equation—we prove that the unification of electroweak interactions originates from the intrinsic geometric structure of space itself. By introducing a complexified spatial orientation field, its internal symmetry group is shown to be isomorphic to the electroweak gauge group SU(2)_L × U(1)_Y of the Standard Model. The gauge coupling constants g2 and g1 are given by geometric integrals over the soliton solution, determined entirely by the sole free parameter κ ≈ 0.017 of the spatial orientation field equation and the characteristic soliton width a_N ≈ 0.48 fm. Evaluating these geometric integrals, we obtain for the first time from first principles the bare Weinberg angle sin²θ_W^(bare) ≈ 0.704 at the low-energy scale μ_0 = 1/a_N ≈ 0.41 GeV. Applying two-loop renormalization group equations to evolve this value to the Z boson mass scale M_Z = 91.1876 GeV, we obtain the theoretical prediction sin²θ_W(M_Z) = 0.237 ± 0.008 (confidence ★★★★☆), which is consistent with the experimental value 0.2313 ± 0.0003 published by the Particle Data Group to within approximately 2.5%. This result demonstrates that electroweak unification is not a fundamental postulate, but an inevitable consequence of the geometric dynamics of space itself. This paper is the seventh in the first series (17 papers total) of Space Ontology, establishing the geometric proof of electroweak unification.