Paper LXXXVII (v2) Neutrino Mass Eigenvalues from the G2 Structure Constants Projected onto S6 = G2/SU(3):

Paper LXXXVI of the framework series identifies the three neutrino mass eigenstates with three Fano lattice edges through the sump-projection node e7: e1, e3, e6, e2, e4, e5 (in standard Cayley-Dickson convention), respectively the matter- matter, cross-sector, and colour-colour pairs. Mass values follow from the angles ϕi of each edge to the brane propagation direction (e7 axis), with ϕ1 ≈θG2 = 4. 04◦ (matter edge, nearly parallel to brane), ϕ2 = 45◦exact (cross-sector, forced by the e7 valve geometry), and ϕ3 ≈90◦−θG2 = 85. 96◦ (colour edge, nearly perpendicu- lar). Paper LXXXVI gives the lowest-order ansatz mνi ∝sin2 ϕi, producing ∆m2 31/∆m2 21 ≈4. 0 versus the observed 33. 8, off by a factor of ∼8 and explicitly flagged in that paper as requiring refinement. The present Paper LXXXVII reads the required refinement from the G2 Gram matrix (Paper LXVIII, omnibus §5. 6): G = − (1 + γ) I + γJ with γ ≈0. 155 for the matter sector. The S6 = G2/SU (3) projection introduces a curvature correction mνi ∝sin ϕi/ q 1 −niγ2 sin2 ϕi with a sector-dependent integer ni counting the colour-singlet projections. We report the explicit ni from G2 representation theory (n1 = 0, n2 = 4, n3 = 8 for the matter, cross, and colour edges) and the corresponding refined mass ratios. The Paper LXXXVI qualitative predictions (normal hierarchy, three flavours, two large PMNS angles and one small) survive intact; the quantitative refinement reduces the ratio residual from ∼8× to ∼2×, narrowing the gap but not closing it. The remaining residual is attributable to two-loop G2 holonomy corrections, which we leave for a future companion paper. Six falsifiable predictions registered. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.