One-Octonion Brane-Bulk Framework - Paper CCCXLI: The Very-Nearly-pi/8 Unitarity Triangle from sqrt(7) - the Exact Right Angle, the Bulk-Transit Warp, and a Texture-Level Confrontation with Arkani-Hamed-Figueiredo-Hall-Manzari
Authors: Jagadeesan, Bharathi Dasan
Type: Journal article
Published: 2026-08-03
Keywords: octonion, G2 algebra, Fano plane, CKM matrix, unitarity triangle, CP violation, Yukawa textures, flavor physics, quark mixing, Cabibbo angle, brane-bulk cosmology, standard model

<p><strong>A rival has arrived on the unitarity triangle, and the confrontation
is unusually clean.</strong> Arkani-Hamed, Figueiredo, Hall and Manzari (AHFHM,
arXiv:2607.27315) observe that the CKM unitarity-triangle angles lie close to simple
fractions of π — (α, β, γ) ≈ (π/2, π/8, 3π/8)
— and propose sparse “9-link” Yukawa textures with a spontaneously
broken CP phase quantized in multiples of π/8. This note shows the One-Octonion
Brane-Bulk (OOB) framework already contains the pattern, with a different and sharper
origin.</p>

<p><strong>(i) The flat OOB triangle is exactly right-angled.</strong> With apex
(ρ̅, η̅) = (1/7, √6/7) fixed by the √7 Dehn twist, the
apex sides are exactly orthogonal (−6/49 + 6/49 = 0): α<sub>flat</sub> =
π/2 identically, with the right angle split as tan γ<sub>flat</sub> = √6,
so γ<sub>flat</sub> = arccos(1/√7) = 67.79° = 3π/8 + 0.29°. The
proximity to the π/8 lattice needs no discrete symmetry: an irrational Gudermannian
angle grazes it.</p>

<p><strong>(ii) One bulk-transit depth warps the observed triangle.</strong>
α<sub>0</sub> = 0.10673 gives (α, β, γ) = (92.01°,
22.63°, 65.36°) — each within 0.4σ of PDG 2026, in a statistical
dead heat with the AHFHM anchors today.</p>

<p><strong>(iii) The texture-level cross-match.</strong> Written in AHFHM's own
parametrization (up-frame diagonal; down sector sparsified with the right-handed
U(3)<sub>dR</sub> freedom to a canonical RQ chart), the OOB flavor point lands exactly
on their 3π/8-family texture #29 — identical zero pattern and phase slot
Y<sup>d</sup><sub>12</sub> — as its canonical hierarchical representative. The
single rephasing invariant is φ = 65.409° = γ + 0.045°: it tracks
γ exactly as their leading-order theorem requires, with the next-order texture
correction computed here. The same texture chart thus carries two incompatible phase
laws: AHFHM-quantized #29 predicts γ ≈ 3π/8 − 0.045° =
67.455°; OOB predicts γ = 65.364°. The mirrored (lower-triangular) gauge
reproduces AHFHM's anomalous π/4 histogram peak. Next-generation LHCb/Belle II
determinations of γ (sub-degree, 2030s) decide.</p>

<p><strong>Framing (stated honestly):</strong> the texture identification is a
canonical-coordinate statement — the RQ chart's loop phase is a determined,
rephasing-invariant function of Y<sub>d</sub>Y<sub>d</sub><sup>†</sup> —
not a dynamical derivation of the texture; the group-theoretic (Ursa-Major S<sub>4</sub>
Clebsch-Gordan) route to the magnitudes remains open. No new parameter and no new
prediction number is introduced; the note sharpens the framework's long-standing
γ stake (Papers CXLIX, CCCII, CCCIII) to the texture level.</p>

<p><strong>Verification:</strong> one Python gate script and 19 independent Wolfram
gates (including symbolic proofs of the exact right angle and tan γ = √6),
all passing (supplementary files); one Fugu cross-model pre-publication audit with all
confirmed findings repaired. The results are strictly contingent on the established
OOB framework — the G<sub>2</sub> = Aut(O) reduction, the √7 Ursa-Major
twist and apex closure, the Wolfenstein closures λ = √3/(e<sup>a*</sup>+6)
and A = 8/π<sup>2</sup>, and the Class-I transit depth α<sub>0</sub> —
none of which is re-derived here. Full symbolic proofs and postulates are consolidated
in the BraneBulk Omnibus (concept DOI 10.5281/zenodo.19185171).</p>