PMNS Mixing, Cauchy Duality, Unified CP Phases, Neutrino Mass Splitting, and the Higgs Mass Ratio from the Rendering Algebra R12

Abstract

The CKM quark mixing matrix and the PMNS lepton mixing matrix are derived from opposite sectors of the rendering algebra R12 ∼= M3(C) ⊗ M4(C) through the Cauchy duality: confinement selects the multiplicative (exponential) Cauchy equation for quarks and the additive (linear) equation for leptons, explaining why CKM angles are small and PMNS angles are large.
The three PMNS mixing angles are derived from the tensor structure M4 = M2 ⊗ M2:

(1) sin2 θ23 = (1 + Z)/Tw from M2 ↔ M2 exchange symmetry breaking (0.6%)

(2) sin2 θ12 = Zπ from the SU(2) maximal torus measure (0.4%), and

(3) sin2 θ13 = Zπ/(2b0) from the Riemann-suppressed torus (0.8%). A unified CP phase formula δ = base×(n+1)/n governs both matrices: δPMNS = 13π/12 (n = N = 12, base = π, 0.15%) and δCKM = 4(1−Z)/3 (n = Ngen = 3, base = 1−Z, 0.54%). The CKM phase satisfies δCKM =4/3 (1 − Z − Z 3/3) = ζ(3) to 3 ppm. Additionally, the Higgs-to-VEV mass ratio mH/v = 1/Tw +Z/N (0.10%) yields a Higgs self-coupling prediction λ = 0.1291 (0.20%), and the W boson mass MW = 80.39 GeV refutes the CDF anomaly in favour of the PDG value. Thirteen results from zero free parameters.