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 adjustable parameters beyond two dimensionful anchors: the Planck mass (setting the scale sector) and the muon mass (setting the lepton sector via the Koide circle, itself derivable to ~0.35% from the Planck mass through the QCD scale).

The framework yields 34 numerical predictions matched against PDG experimental values, of which 32 are computed directly from the Planck mass and the G₂ group integers. A machine-checkable verification script (verify.py) recomputes every value from these inputs and confirms all 34 pass, with a mean absolute residual of 0.29% and χ²/dof = 0.40 against a 1% tree-level floor. Where a one-loop correction is applied on top of a tree-level prediction, it is in every case a fixed formula from standard field theory evaluated with framework-derived inputs and no adjustable coefficient.

Three predictions are independently verifiable with a pocket calculator and are presented on pages 11–12: the proton-to-tau mass ratio (predicted 0.52805, measured 0.52805, error 0.00003%, with the QCD scale cancelling), the neutrino mixing angle θ₁₃ (predicted π/21 = 8.571 degrees, measured 8.57 degrees, error 0.02%), and the hydrogen Rydberg energy (predicted 13.6057 eV, measured 13.6057 eV, error 0.000%). Additional predictions include new particles accessible to existing LHC searches: a χ familon resonance at 94.8 GeV in the dijet channel, and an inverted gaugino spectrum with a Bino near 391 GeV, gluino near 914 GeV, and wino near 1370 GeV.

The document is structured in four parts: an accessible introduction with the pocket-calculator 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 and verification script (Part 4).

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