E₆ GUT Cascade Model: 3 Generations = 3 Gauge Breakings, Normal Ordering derived, E6 twisting from cubic invariant

Abstract

GUT model based on the chain E8 → E6 → SO(10) → SU(5) → G_SM where each breaking generates a generation. The number 3 is a theorem of Lie classification (ANA_TEO_785): among all maximal subgroups H of E8 such that the branching E8 → H produces chiral representations containing the 16 of SO(10), the unique one is H = E6 × SU(3). The family symmetry SU(3)_F has fundamental of dimension 3, giving exactly 3 generations. Normal Ordering is Derived (ANA_TEO_807): the cascade imposes (ε₁², ε₁, 1) in M_D(ν), shared with the quark sector via Y₁₀, Y₁₂₆, Y₁₂₀; the seesaw produces m₁ ≪ m₂ ≪ m₃ by construction, and Inverted Ordering would require inverting the cascade in M_D(ν), breaking SO(10) unification (Y₁₀ would give mass to the 1st generation instead of the 3rd, contradicting m_t ≫ m_u). The E6 twisting ratio a/b = 2/√5 is Derived from the cubic invariant 27³ (ANA_TEO_824): the ratio is computed from Clebsch–Gordan coefficients of 27³ → 1 with no free parameters. The twisting formula sin(θ) = a·v_R / √(a²v_R² + b²v_S²) is Derived from the Higgs potential (ANA_TEO_825); with v_R/v_S = h(E6)/h(E8) = 2/5 (Plausible, ANA_TEO_826), sin(θ) = 0.337 (error 8.5%), or sin(θ) = 0.352 with back-reaction (Conditional, error 4.2%). 13 Q/L parameters + 6 M_R = 19 total (vs 26 in the SM), 16/18 observables at >96%. Falsifiable predictions: δ_CP^ν = 180° (Derived, structural J=0), m_ββ = 2.9 meV, Σm_ν = 0.059 eV, τ(p → e⁺π⁰) ~ 6×10³³ yr (tension with Super-K, factor ~4×; threshold corrections do not resolve, ANA_TEO_829), d_e^PMNS = 0 (falsifiable by ACME), r₁₂₀ nEDM constraint Archived. NO-GO: reduction below 13 parameters impossible. Structural tension: V_ub 2.8× off (120_H sector-dependent does not fix). θ₁₃ resolved: was optimizer artifact, baseline gives 100%.