G2 Geometry Predicts All Standard Model Parameters: A Zero-Free-Parameter Framework

This preprint presents a theoretical framework in which all Standard Model particle masses, mixing angles, and coupling constants are derived geometrically from a single G2 octonion field undergoing a topological vortex condensation in the early universe. The condensation produces exactly six active vacuum orientations at 60-degree intervals, from which precisely three stable particle loop sizes emerge — corresponding to the three observed generations of matter — with no free parameters beyond the Planck mass. The framework yields over 30 numerical predictions matched against PDG experimental values. Three are independently verifiable with a pocket calculator and are presented on pages 11-12: the proton mass (predicted 939.0 MeV, measured 938.3 MeV, error 0.08%), the neutrino mixing angle theta_13 (predicted pi/21 = 8.571 degrees, measured 8.57 degrees, error 0.02%), and the hydrogen ionisation energy (predicted 13.6057 eV, measured 13.6057 eV, error 0.000%). Additional predictions include new particles accessible to existing LHC searches: a chi familon resonance at 94.6 GeV in the dijet channel, and supersymmetric partners at 391 GeV and 684 GeV. The document is structured in four parts: an accessible introduction with the three verifiable predictions (Part 1, pages 1-14), the physical and geometric foundation (Parts 2 and 3), and the full mathematical derivations with the complete prediction table (Part 4) V5 Update — June 30, 2026 (electromagnetic / meson sector) This revision closes or derives the four electromagnetic meson observables that the previous version left open or only approximated. The underlying framework, scales, and all other sectors are unchanged. π⁺ − π⁰ mass splitting — previously the leading-order EM formula gave −0.77% and was described as irreducible at current order. The splitting is now derived to −0.04% via the inter-quark Coulomb energy of the charged pion, Q_u · Q_d̄ = 2/9. The self-conjugate π⁰ pays no Coulomb cost, which explains why only the charged pion is shifted. The factor 2/9 coincides with the Koide angle φ_K = τ₀²/8 = 2/9; this is shown to follow from the parameter-free relation 2·h_dual = N_c² − 1 (= 8) among the dual Coxeter numbers of the forced G2/SU(3) structure — a genuine, group-rooted connection, specific to that pair rather than a general theorem. ρ⁰ → e⁺e⁻ — previously the largest single gap (−30.5%). Now closed to +0.3% by the 13/9 photon-visibility factor (N_c + τ₀)/N_c: the photon couples only to the SU(3) colour part of the ρ geometry, not the τ₀ twist. η, η′ → γγ — previously off by −53% / +127% under a single-angle scheme attributed to an unresolved NLO effect. The failure is now traced to a missing octet–singlet decay-constant splitting. With f₀ = √(τ₀)·f_π (proved) and the strange-quark-raised octet f₈ > f₀, the required two-angle scheme closes both rates to ~1–3%. The one remaining band-limited input is f_K (refinement, not gap). π⁰ → γγ — unchanged (+0.43%, Fano 3-form anomaly A = 1), restated for completeness. After this update the electromagnetic meson sector is closed, derived, or essentially closed via explicit mechanisms, with every near-fit and band-limited input flagged.