Structural 9CG Fermion Coefficients and an Internal PMNS Parametrization

Abstract

This manuscript gives a bounded public formulation of the Steps 6–7 fermion block of the TTOE framework. The first part records a Coxeter/branching-motivated selection of nine rational coefficients, denoted 9CG, along the chain E8 → E6 → D5 → A4 → G_SM. These coefficients are compared with a historical Clebsch/GUT target table CG_req; that target table is treated as an external input, not as a mass table derived from E8. A finite-table audit included as supplementary material supports only a local rarity statement inside the same formal grammar, not a global p-value and not a derivation of fermion masses. The second part records an internal PMNS parametrization and a type-I seesaw mass scale under the explicit input Y_ν = 1. The reactor angle sin²θ₁₃ = 19/900 is presented as a structural selection inside a finite class; θ₁₂ and θ₂₃ remain internal Coxeter ratios outside current experimental precision. The CKM sector, previously outside the scope of this manuscript, has since been developed in a companion paper (PAP_TEO_10) using the w₀-breaking spurion framework: a reproducible CKM pipeline with mean error 1.5% and CP phase γ = 66.3° vs 65.7° (0.2σ) is now available. Additionally, the theorem w₀ = −id proven in PAP_TEO_10 provides a structural explanation for why the Coxeter reflection pairs m ↔ 30 − m appear complete in the 9CG table (p = 0.0001): the central element of the Weyl group exchanges the chiral representations 27 ↔ 27̄, and the 9CG denominators systematically select the broken side (m > h/2 = 15) of this reflection. New in v1.2: Normal Ordering of the neutrino mass spectrum is Derived from the cascade structure (ANA_TEO_807): the Coxeter chain imposes (ε₁², ε₁, 1) in M_D(ν), shared with the quark sector via SO(10) unification; the seesaw produces m₁ ≪ m₂ ≪ m₃ by construction. 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). This is falsifiable by DUNE (2028+, >5σ).