Structural 9CG Fermion Coefficients and an Internal PMNS Parametrization

This manuscript gives a bounded public formulation of the Steps 6–7 fermion block of the TTOE framework. Part one records a Coxeter/branching-motivated selection of nine rational coefficients (9CG) along the chain E8 → E6 → D5 → A4 → G_SM, compared against a historical Clebsch/GUT target table that is treated as an external input, not as a mass table derived from E8; a finite-table audit supports only a local rarity statement, not a global p-value and not a derivation of fermion masses. Part two records an internal PMNS parametrization and a type-I seesaw mass scale under the explicit input Y_nu = 1: sin²θ13 = 19/900, Σm_ν = 57.3 meV, m_ββ = 4.1 meV. Normal Ordering is derived from the cascade structure (shared with the quark sector via SO(10)) and is falsifiable by DUNE (2028+). The theorem w0 = −id of the companion paper (doi:10.5281/zenodo.21574599) explains the one-sidedness of the 9CG denominators: the four E8 exponents used, {17,19,23,29}, all lie on the broken side m > 15, with no reflection pair completed (p ≈ 0.014–0.06). Version 4 — correction notice. This version supersedes v1.1–v1.3, which claimed "all 4 Coxeter reflection pairs appear complete in the 9CG table, p = 0.0001". That claim was false — the table uses only the four exponents {17,19,23,29}, so no pair is complete (0/4) — and it is withdrawn; the defensible regularity is one member per pair, always on the broken side. Further corrections: m_ββ = 4.1 meV (the earlier 2.9 meV is not reproducible from the paper's own masses, angles and Majorana phases); the mass relation is restated as m3²/m2² = 33 (the Δm²atm/Δm²sol form holds only for m1 → 0); δ_CP = 255° sits at +2.2σ from 197° ± 27°, not "in range"; the m_ν3 reference value is corrected to 50.28 meV (−6.7%); the seesaw convention is stated explicitly; and the companion-paper figures are aligned with PAP_TEO_10 v3.