Paper CCLXXI: Closing the Class III Wick Weight -- pi-Rotation of Paired Neutrino Scars at the e_7 Node Forces f_III = 1 - 4/pi^2

In the OOB neutrino sector (Papers CCLII-CCLV), the three active-neutrino masses arise from a Class III blend of compact (sin^2) and non-compact (sinh^2) potentials evaluated at Fano-edge angles phi_i = (0, pi/4, pi/2) - Theta_tilt with the blend weight f_III = 1 - 4/pi^2 = 0.5947. The resulting Delta m^2_31/Delta m^2_21 = 33.7957 matches the JUNO/PDG value 33.80 to 1.3 parts in 10^4 (0.013%). Until this paper the half-wave-rectified Wick prescription underlying f_III was a structural axiom. This paper closes that axiom via five forced steps: (1) Cayley-Dickson H -> O introduces e_7 as the brane-bulk separator (Paper CXLIX); (2) the doubling pairs every prior basis element with itself * e_7, so neutrino scars passing through e_7 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/pi^2, complement f_III = 1 - 4/pi^2. No axiom: five independently anchored steps determine the result uniquely. Three independent framework anchors triangulate the same e_7 half-wave structure: CCLV |delta_CP| = 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 f_III recovers the (8/pi^2) - (32/pi^4) 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 m^2 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 e_0...e_3 minus the scalar e_0 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 f_III = 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 f_required for exact JUNO match (0.594793) and f_III = 1 - 4/pi^2 (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.