In the OOB neutrino sector (Papers CCLII-CCLV), the three active-neutrino masses arise from a Class III blend of compact (sin²) and non-compact (sinh²) potentials evaluated at Fano-edge angles phiᵢ = (0, pi/4, pi/2) - Thetaₜilt with the blend weight fIII = 1 - 4/pi² = 0. 5947. The resulting Delta m²₃1/Delta m²₂1 = 33. 7957 matches the JUNO/PDG value 33. 80 to 1. 3 parts in 10⁴ (0. 013%). Until this paper the half-wave-rectified Wick prescription underlying fIII was a structural axiom. This paper closes that axiom via five forced steps: (1) Cayley-Dickson H -> O introduces e₇ as the brane-bulk separator (Paper CXLIX) ; (2) the doubling pairs every prior basis element with itself * e₇, so neutrino scars passing through e₇ inherit a pair structure; (3) the pair relative-orientation angle alpha is identified modulo pi (the alpha -> alpha + pi map is the label-swap gauge redundancy), giving a physical configuration space 0, pi rather than 0, 2 pi; (4) the brane amplitude projection is |cos alpha| (the mass is a Hermitian positive operator, so |cos| -- not signed cos -- is required) ; (5) Born squaring of the average projection gives the brane probability (2/pi) ² = 4/pi², complement fIII = 1 - 4/pi². No axiom: five independently anchored steps determine the result uniquely. Three independent framework anchors triangulate the same e₇ half-wave structure: CCLV |deltaCP| = pi/2 from the same Cayley-Dickson doubling phase; CCLXIX 2/pi as leading Fourier coefficient at the Klein quartic boundary; Paper L diffusion factor combined with fIII recovers the (8/pi²) - (32/pi⁴) half-wave-rectified sine spectral fingerprint. The OOB Mathematical Provenance Audit (2026-05-30) is updated: 11 of 11 framework functions now have explicit first-principles or Lagrangian bases; 0 unforced axiomatic choices in the Class III neutrino blend. Three predictions follow: CCLXXI-P1 sharpened JUNO Delta m² ratio at 0. 05% theoretical precision; CCLXXI-P2 no light sterile neutrinos at short baselines (pair structure on the four pre-existing quaternion basis elements e₀. . . e₃ minus the scalar e₀ gives exactly three active-neutrino pairs, no sterile partner) ; CCLXXI-P3 cross-sector 2/pi signature at any future brane-bulk junction where Born squaring of an amplitude-projection average is the relevant operation (candidate sectors: GW polarization residual at the T2* horizon, FRB spectrum low-energy plateau, J-PET coincidence DC offset). Clinical link: the same fIII = 0. 5947 governs the T2* envelope in high-field 7T MRI (CCLXIII-P3), where a Born-squared 2/pi spectral signature is testable on existing CMRR instrumentation. Honest scope: 77 ppm residual between fᵣequired for exact JUNO match (0. 594793) and fIII = 1 - 4/pi² (0. 594715) is unchanged from CCLII and is attributed to higher-order tilt corrections. Framework count: 272 numbered papers. 3 pp. 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.
Bharathi Jagadeesan (Wed,) studied this question.