Paper CXXVIII · One-Octonion Brane-Bulk Framework 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: (P_RB1) the symmetric H :H mass ratio is exactly 1:3; (P_RB2) the ⁺ ⁻ Genesis tilt θG₂ measures the departure from the perfect 1:3 rolling contact; (P_RB3) N_c = 3 QCD colors is the rolling ratio denominator, not a free parameter; (P_RB4) the confinement threshold γconf = 1/3 is the rolling ratio itself; (P_RB5) 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.