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 in which each of the three breakings above G_SM generates one fermion generation. The number 3 is presented as a classification theorem: among the maximal subgroups of E8 whose branching produces chiral representations containing the 16 of SO(10), the unique one is E6 × SU(3), and the family SU(3) has a 3-dimensional fundamental representation. Normal Ordering of the neutrino spectrum is derived from the cascade structure (shared with the quark sector via SO(10) unification) and is falsifiable by DUNE (2028+). The E6 twisting ratio a/b = 2/√5 is derived from the cubic invariant 27³ with no free parameters. The model uses 19 parameters (vs 26 in the SM) for 18 observables, with 16/18 reproduced at >96%. Falsifiable outputs: δ_CP = 180° (structural J = 0), m_ββ = 2.9 meV (GUT fit; 4.1 meV for the companion Coxeter spectrum), Σm_ν ≈ 57–59 meV, d_e = 0 (ACME), and τ(p→e⁺π⁰) ~ 3×10³⁶ yr. Declared structural tension: V_ub off by 2.8×. Version 3 — correction notice. The v1/v2 proton-lifetime formula M⁴/(α²·m_p⁴) is dimensionless and cannot be a lifetime; the correct expression M⁴/(α²·m_p⁵), with the paper's own inputs (M_GUT = 10¹⁶ GeV, α_GUT = 1/32π), gives 2.9×10³⁶ yr rather than 6×10³³ — a factor 483. The previously claimed "4× tension with Super-K" was therefore backwards: the estimate sits ~10² above the bound, and the genuine unresolved problem is gauge-coupling unification (the non-SUSY SM does not unify; threshold corrections do not rescue M_GUT ≥ 10^16.3 GeV). Also corrected: SU(2)×SU(3)×SU(6) is not a maximal subgroup of E8 (it descends from the non-prime mark 6; E8 has five maximal-rank maximal subalgebras); the α1 = α3 crossing is at 10^14.4 GeV; b0 for a gauged SU(3)_F is 17/3 (with an anomaly note; the anomaly-free E8 content gives −7); the Δm² ratio deviation is −2.5%; the M_R cascade eigenvalues are 10¹⁵/10¹¹/10⁷ GeV; parameter/DOF counts and Σm_ν quotations are harmonized throughout.