KIR-QER-Nezirov - Supplement IV WZ Boson Mass from σ_c and σ*

This Supplement IV derives the fundamental electroweak mass parameters — the Weinberg angle, the W/Z boson mass ratio, the gauge coupling constant, and the vacuum expectation value of the σ-field — from the two critical points of the dual channel structure established in Supplement III: the dominance transition σ_c = ½ and the SNR inflection point σ* = ⅓. The derivation is organized in three epistemically distinct stages of increasing rigor, supplemented by an exact relation between the correction factor ξ and the one-loop running of sin²θ_W, and a stationary solution for the electroweak scale from the nonlinear Κ-potential. Stage 1 (Dimensional Analysis) establishes the characteristic length and energy scales of the channel transition from the nuclear σ-field (r_N = 1.46 fm, Supplement I). At σ = σ*, the characteristic interaction radius is r* = 3r_N = 4.38 fm, corresponding to an energy scale E* ≈ 45 MeV. The hierarchy factor N = m_W/E* ≈ 1784 quantifies the known electroweak-nuclear scale separation and locates the hierarchy problem within the KIR framework. Stage 2 (Exact from KIR Axioms) constitutes the core contribution. The Weinberg angle is identified parameter-free with the rate channel weight at the inflection point: sin²θ_W(tree) = w_R(σ*) = σ* = ⅓. The mass ratio follows from the saddle point condition of the channel overlap measure Κ(σ) = σ(1−σ) at σ_c: m_W/m_Z(tree) = cos θ_W = √(2/3) ≈ 0.8165. Both relations are necessary consequences of the channel structure and require no free parameters. Stage 3 (Perturbative) introduces a single correction factor ξ = 3·sin²θ_W(exp) = 0.6937, which brings all structural predictions into quantitative agreement with experiment: sin²θ_W(pert) = ξ/3 = 0.2312 (deviation < 0.01%), m_W/m_Z(pert) = √(1−ξ/3) ≈ 0.877 (deviation 0.5%), g_W(pert) ≈ 0.629 (deviation < 0.1%). The perturbation theory of the nonlinear channel coefficients (Supplement III, Section 8.6) traces ξ back to the structural formula ξ = 1 + ε·F(a₂, b₃), establishing its origin in the σ-field nonlinearity rather than leaving it as a free fit parameter. Section V establishes the exact relation C = √ξ · √η, where C² = 3·sin²θ_W(tree) and η = 0.9304 is the well-known one-loop running rate of sin²θ_W between Q² = 0 and Q² = m_Z². This relation is exact in the limit of vanishing radiative corrections (η → 1) and constitutes a non-trivial consistency check of the framework. Section VI derives the stationary solution for the electroweak vacuum expectation value from the Κ-potential U(σ) = Λ⁴·σ(1−σ). Expanding around the saddle point σ_c yields a fluctuation field δσ = σ − ½ with negative effective mass squared, establishing the structural analogy to electroweak symmetry breaking. The Higgs mass fixes the energy scale to Λ = m_H/√2 ≈ 88.6 GeV. The correct direct route v_ew = 2m_W/g_W(pert) ≈ 255 GeV (deviation 3.7% from the Fermi scale 246.2 GeV) supersedes a previously erroneous detour via an implicit quartic coupling assumption. The nonlinear correction of the Κ-potential connects the SM quartic coupling λ to resonance damping at the saddle point, with the condition v_ew = 246 GeV fixing ε(a₂ + b₃) ≈ −0.47, comfortably within the stability bound. The Standard Model requires two independent input parameters (sin²θ_W and v) for the same electroweak observables. The KIR reduces this to one (ξ), with ξ itself structurally traced to the nonlinear channel coefficients. The absolute mass scale and the individual nonlinear coefficients a₂ and b₃ remain explicitly open as research objectives and are not concealed. KIR-QER-Nezirov: Kinetically Induced Spacetime and Quantum Field Emergence of Reality — Complete Theoretical Framework with Supplements I–V. Zenodo. https://doi.org/10.5281/zenodo.18943890