Neutrino Masses as Compressed Fano Lattice Edges: Normal Hierarchy, PMNS Angles, and the Hyperbolic-to-Flat Decit Angle π/7 Paper LXXXVI of the Brane-Bulk Octonionic Series

Every massive particle identied in the series so far corresponds to a node of the Fano lattice a localized focal deceleration in the brane current. We identify neutri- nos as edges of the Fano lattice: extended objects connecting two nodes, carrying the cur- vature mismatch of the hyperbolic 7, 3-to-at 4D transition as inertia. The three Fano edges on lines through e7 e1, e2, e3, e4, e5, e6 sit at angles ϕ1 ≈θG2 = 4◦, ϕ2 = 45◦, ϕ3 ≈86◦relative to the brane propagation direction. Their compression from the decit angle δ = π/7 at each hyperbolic vertex gives masses proportional to sin2 ϕi. The mass ratio m1: m2: m3 ≈sin2 (4◦): sin2 (45◦): sin2 (86◦) ≈0. 005: 0. 500: 0. 997 gives the normal hierarchy, which is structurally forbidden from being inverted because θG2 < 45◦. The PMNS mixing angles θ12, θ23, θ13 are identied with the pairwise inner products of the three edge directions in the G2/SU (3) decomposition zero free pa- rameters. An edge travels at v < c because it cannot follow a hyperbolic geodesic in at spacetime: its mass is the curvature mismatch made manifest as inertia. 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.