Abstract

We present a projector-defined electroweak normalization in which the gauge factors gL, gZ, the neutral mixing angle θW , and the electromagnetic coupling e are determined from a specified coherent pre-geometric background. A quadratic expansion about the self-dual background two-form Σ produces a background stiffness τ(ρ0); projection of the induced Yang–Mills block onto SU(2)L and the Cartan U(1)Z fixes gL−2 = ωLeff  /τ(ρ0) and gZ−2 = 1/τ(ρ0) in component-canonical trace. The photon is the massless neutral mode of the kinetic/mass system, so e follows from the neutral kinetic block and α(μ) = e2(μ)/(4π). Two-loop effective-field-theory running with threshold matching gives, for the product-UV isotropic ledger (ωLeff , ωZ) = (1, 1), the forward prediction α-2(MZ ) ≃ 140.8 and sin2 θW (MZ ) ≃ 0.258, identifying the isotropic coherent-background kernel as the source of the normalization offset. A coupled-oscillator morton x-pair supplies an explicit anisotropic mixed-branch benchmark whose declared spectral scale gives τem = 0.199506 and aw = 0.734159, yielding electroweak values near the observed pair when propagated through the same running pipeline. The parent-response scale analysis does not yet select this mixed branch and spectral scale self-consistently. The anisotropic result is therefore a deterministic conditional benchmark, with parent selection of the coherent branch and scale remaining as the closure condition.

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Keywords: ab initio electroweak normalization, electroweak mixing origin, emergent electroweak symmetry, composite stiffness parameters, pre-geometric background, coherence field / scalar coherence,  Weinberg angle from first principles, derivation of the fine-structure constant,  geometric origin of U(1)EM,  canonical SU(2)L× U(1)Z reduction, neutral boson mixing from scalar alignment, nonperturbative electroweak boundary conditions, gauge unification without GUTs, emergent gauge fields, pre-metric field theory, projection geometry, coherently-induced symmetry breaking, non-Abelian to Abelian reduction,  electroweak boson stiffness,  fine-structure constant (α) prediction,  Weinberg angle constraint, axis-model electroweak sector,  scalar-filtered gauge projections,  geometric gauge mixing,  UV boundary condition from coherence