Paper CXXIX · One-Octonion Brane-Bulk Framework Genesis Tilt and Twist from First Principles: θ₆䃒 = arctan (2√3/49) and τ = arccos (1/√7)

The One-Octonion Brane-Bulk Framework has carried two foundational geometric parameters since Paper I: the Genesis tilt θ₆䃒 = 4. 054° (the polar impact parameter of the H –H collision) and the ⁺ ⁻ Genesis twist τ = 67. 79° (the azimuthal phase, equal to the CP violation phase δCP). Until now, these were introduced as observed values. Paper CXXIX derives both from the Fano-rolling-ball geometry of Paper CXXVIII without additional inputs. τ = arccos (1/√7) = 67. 7923° FROM: eᵢ, ψ₀ = 1/√7 (any Fano node vs uniform collision axis) ⟨ ⟩ NATURE: topological invariant of Fano order N=7. Independent of δM/M. θ₆䃒 = arctan (2√3/49) = 4. 0439° (observed: 4. 054°, error 0. 25%) FROM: (N₇ −N₇+) /N × (2/N) × √3 = (1/7) × (2/7) × √3 ⁻ NATURE: geometric. Encodes rolling ratio 1: 3 via Fano 4: 3 node count. Writhe = τ/θ₆䃒 = arccos (1/√7) /arctan (2√3/49) = 16. 764 ≈ 17 (Dehn slope) ✓ Both derivations are purely from the internal geometry of the Fano-rolling-ball system. No observational input is required beyond the Fano order N = 7 and the rolling ratio 1: 3 (which is itself derived: Baez- Huerta require ratio 1: 3 for G₂ symmetry of a spinorial rolling ball). 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.