Geometry of Sound v20: Final Theory — From Platonic Dualities to the Standard Model, Cosmology, and Gravity

The Geometry of Sound (GoS) v20 presents a complete geometric framework in which 19 Standard Model parameters, the CKM and PMNS mixing matrices, neutrino masses and CP phases, and cosmological densities are derived from the geometry of Platonic solids and two duality classes. The fundamental chain: dim R³ = 3 → dim SO(3) = 3 → 3 duality classes → 3 Merkabas → all physics All results follow from two numbers √3 and φ, both derived from dualities of Platonic solids. No free parameters. KEY RESULTS: 1. All 19 SM parameters (masses, mixing angles, gauge couplings) derived from face numbers F = {4, 6, 8, 12, 20} with precision < 0.1%. 2. CKM matrix: all 5 parameters from π, φ, √2, √3, √6. 3. PMNS matrix: θ₁₂ = arctan(1/√2) − 3/φ, θ₂₃ = arcsin√(4/7), θ₁₃ = π/21, δ_CP = −137.578° (from dynamics). 4. Neutrino masses: m_i = m₀·R_i^(π/2), R_i = 1, 3, 9. 5. Cosmology: Ω_b, Ω_DM, Ω_Λ all within 0.1% of Planck 2018. 6. Gravity: Chern-Simons with α_CS = 4√6, derived from the octupole of the tetrahedron. 7. Look-elsewhere closed: each G-factor has exactly one unique symbolic formula; all 11 control constants have none. 8. Monte Carlo test (100M iterations): p < 2×10⁻⁸ for both permutation tests. STATUS: - Core results (particle masses, mixing, cosmology): complete. - Open problems: quantum corrections (Coleman-Weinberg shifts minima by 2–7%), UV completion (Landau pole ~207 GeV), dark sector (production mechanism). - The core predictions are independent of these open problems. PREDICTIONS: 1. δ_CP^PMNS = −137.58° (DUNE, Hyper-K). 2. Σm_ν ≈ 60.5 meV (CMB-S4, DESI). 3. Dark matter 123.04 GeV (LHC, XENONnT). 4. Normal hierarchy of neutrino masses. 5. sin²θ₂₃ = 4/7 = 0.5714. This work is the continuation of: - GoS Empirical v.19 (DOI: 10.5281/zenodo.22650356) - GoS v20 Dynamics — Extension of v19 (DOI: 10.5281/zenodo.22718063)