Abstract. The One-Octonion Brane-Bulk (OOB) framework predicts the solar-to-atmospheric mass-squared ratio Δm²31/Δm²21 = 33.796, matching the experimental central value 33.80 to 0.013%, via the Class III Gudermannian blend mi = fIII·sinh²(φi) + (1−fIII)·sin²(φi) at the three Fano-edge angles φi = (0°,45°,90°) − θG2. We strengthen this on two fronts. (1) The weight fIII = 1 − 4/π² = 0.594715 is derived as the brane-coupling probability of the e7 brane–bulk transit: a two-leg path that projects onto the e7 node (cosθ) and exits orthogonally into the bulk (so only |cosθ| survives), giving the coherent loop-average amplitude A = (1/2π)∮|cosθ|dθ = 2/π, which is independent of the e7 orientation; Born squaring gives |A|² = 4/π². This corrects the Born-rule misstep of Paper CCLXXI. (2) The historical +77 ppm residual between the inverse-solved blend fraction (0.594793) and fIII is shown to arise because the two bulk-transiting neutrinos ν2, ν3 carry a hyperbolic tilt θG2 → gd⁻¹(θG2) = arctanh(sinθG2) — the Gudermannian Euclidean→hyperbolic map as the state crosses from brane (circular) to bulk (hyperbolic) geometry — while ν1 (the brane short-root state) does not. Applied parameter-free, this reduces the residual to +4.9 ppm (16×). Honest scope: the derivation rests on two stated premises; the 77→5 ppm correction is internal-consistency at 0.0003σ, not a data tension; the ~5 ppm tail is left unfit (a BCH commutator does not account for it at a forced coefficient). The forward test is JUNO: ratio 33.796, normal hierarchy mandatory, no sterile neutrinos.
Publication Date: 2026-06-05