Paper CXXIX: · Genesis Tilt and Twist from First Principles: θ_{G₂} = 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 θ_{G₂} = 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 δ_CP). 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. θ_{G₂} = arctan(2√3/49) = 4.0439° (observed: 4.054°, error 0.25%) FROM: (N_{H }−N_{H+})/N × (2/N) × √3 = (1/7) × (2/7) × √3 ⁻ NATURE: geometric. Encodes rolling ratio 1:3 via Fano 4:3 node count. Writhe = τ/θ_{G₂} = 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.