Ab-initio Electroweak Normalization and Mixing from a Coherent Pre-Geometric Background

This paper develops an ab-initio electroweak normalization scheme where the SU(2)_L and U(1)_Z gauge couplings, and therefore the fine-structure constant alpha, are derived from a coherent geometric background rather than fitted to data. A single background stiffness parameter, together with fixed geometric projection weights that select the weak and Abelian directions inside the neutral sector, defines a neutral kinetic metric. The physical photon appears as the unique massless eigenstate of this metric, and its coupling e is extracted with no adjustable knobs.

These “boxed” UV couplings are imposed at a matching scale and evolved down to experimental energies using a full two-loop renormalization-group analysis with proper threshold matching. In the canonical “product–UV” background (weak weight equal to 1 and stiffness about 0.178 in the paper’s units), the framework predicts alpha^{-1}(M_Z) ≈ 140.8 and sin^2(theta_W)(M_Z) ≈ 0.258. This differs from experiment by about 10 percent at the Z pole (and about 4 percent at very low momentum), so the undeformed product–UV background is empirically ruled out.

A diagnostic inversion shows that reproducing the observed pair {alpha^{-1}(M_Z), sin^2(theta_W)(M_Z)} would require an effective neutral kernel with a weak weight of about 0.30 and an electromagnetic stiffness of about 0.128. Appendix J.8 shows that this requirement can be stated entirely in terms of the eigenvalues and eigenvectors of the neutral stiffness matrix, so the remaining UV work reduces to a finite and well-posed microphysical problem rather than an open ambiguity. The paper also includes a first-principles lattice computation of hadronic vacuum polarization in an internal S[theta,Z] theory (used as a controlled analogue of QCD). All data and code used to generate the figures and numerical results are available in the accompanying repository.