Paper CXXVIII: · The H –H Collision as a 1:3 Rolling Ball: Split G₂ Symmetry Breaking and the Origin of the Compact Octonion Structure

Baez and Huerta (2012) proved that the split real form of G₂ is the symmetry group of a ball of radius 1 rolling without slipping on a fixed ball of radius 3, and that this ratio 1: 3 is the unique ratio with G₂ symmetry. We identify the H –H collision of Paper I as precisely this rolling ball system: H (the ⁺ ⁻ ⁺ matter sector, radius 1) rolls on H (the antimatter sector, radius 3) in the 5D Anti-de Sitter bulk. This ⁻ gives the pre-nucleation universe the structure of the split octonions with their (2, 3, 5) Cartan contact distribution. The mass asymmetry δM/M ≈ 6. 1×10 ¹ breaks the perfect 1: 3 rolling symmetry, causing ⁻ ⁰ the split G₂ to break to the compact real form. The resulting brane carries the standard octonions with G₂ SO (7), the hyperbolic 7, 3 tiling, and the (2, 3, 7) Klein quartic triangle group structure. The ⊂ transition from (2, 3, 5) to (2, 3, 7) is the nucleation event itself, driven by the δ = 5 cross-sector G₂ roots. Five new predictions follow: (PRB1) the symmetric H: H mass ratio is exactly 1: 3; (PRB2) the ⁺ ⁻ Genesis tilt θ₆䃒 measures the departure from the perfect 1: 3 rolling contact; (PRB3) Nc = 3 QCD colors is the rolling ratio denominator, not a free parameter; (PRB4) the confinement threshold γ₂₎₍₅ = 1/3 is the rolling ratio itself; (PRB5) the pre-nucleation (2, 3, 5) contact structure leaves an imprint in the CMB pre-recombination spectrum. 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.