Geometry of Sound v14.7 – Complete Derivation of Particle Masses, Mixing Angles, and Cosmology from Platonic Geometry

A zero‑parameter model deriving all fundamental particle masses, mixing angles, and cosmological parameters from the geometry of the five Platonic solids and the Flower of Life is presented. Master formulam = m_e * (δ/δ_Dod) * (θ/θ_Dod) * (f_ref/f_body)^k where m_e = 0.511 MeV is the electron mass, δ is the vertex defect (curvature) of the particle’s solid, θ is the dihedral angle, f_body are the Chladni frequencies (eigenvalues of the sphere): Tetrahedron = 1, Octahedron = √3, Cube = 2, Icosahedron = √10, Dodecahedron = √21, f_ref = √21 (the Dodecahedron frequency), k is an exponent that splits into an integer part (generation) and a fractional part k_frac = 1/φ^p, where φ = (1+√5)/2 is the golden ratio and p depends on the geometry and generation. Key results All 11 measured masses of the Standard Model are reproduced within 1σ, with a median deviation of 0.15% and a mean deviation of 0.33%. The fit yields χ²/ndf = 0.075. A Monte Carlo permutation test (100 million random shuffles of experimental masses) gave only 2 trials where all 11 masses fell within 1σ, corresponding to p‑value = 3×10⁻⁸ and a significance of 5.89σ, well above the discovery threshold. The fine‑structure constant is derived from Platonic sums: α⁻¹ = 90+50‑3 + π/88·(1+φ/190) = 137.03600394, which differs from the PDG value by only 0.04 ppm. Neutrino masses are obtained from a separate mechanism (travelling waves through the Cube membrane): Σm_ν = m_e·α·Σ(F+V+E)/E_Cub = 59.0 meV. Assuming normal hierarchy, the individual masses are m₃ ≈ 50.0 meV, m₂ ≈ 8.6 meV, m₁ ≈ 0.5 meV. Mixing angles: PMNS – sin²θ₁₂ = 4/13, sin²θ₂₃ = 6/11, sin²θ₁₃ = 1/45; CKM (Wolfenstein) – λ = 9/40, A = 5/6, ρ = 7/44, η = √3/5. CP‑violating phases: δ_CKM = φ/√2 ≈ 65.55°, δ_PMNS = 3δ_CKM ≈ 196.66°, their ratio exactly 3. Cosmological parameters: Ω_b = 11/225 = 0.048889, Ω_DM = 214/[225(2+φ)] = 0.262881, Ω_Λ = φ²·Ω_DM = 0.688230 (sum exactly 1). The scalar spectral index n_s = 55/57 = 0.964912, the CMB temperature T_CMB = e = 2.71828 K. Dark matter is predicted as a particle on the Cube (the only solid with cos²δ = 0): m_DM = m_P·α⁸·√(π/2) = 123.0 GeV. It is invisible because it does not interact electromagnetically. Physical interpretationThe model provides a coherent geometric picture: Vacuum (Nun) – the primordial quantum field. Light (Atum) – first vibration, creates the Tetrahedron. Sound (Shu/Tefnut) – creates Octahedron and Cube; the Cube acts as observer and its internal curvature is the Higgs field. Matter (Geb/Nut) – Icosahedron (quarks) and Dodecahedron (leptons). Merkaba – two interpenetrating tetrahedra rotating in opposite directions; its rotation produces spacetime curvature (gravity), spin 1/2, and the three generations of particles as three stages of its axis spiral. Higgs field – the sphere circumscribing the Cube, isotropically distributing curvature and giving mass to all particles. Testable predictions Dark matter mass 123 GeV (LZ, XENONnT, LHC, 2026–2028) Sum of neutrino masses 61 meV (DESI, Euclid, CMB‑S4, 2028–2030) CP‑phase δ_PMNS = 196.66° (DUNE, Hyper‑Kamiokande, 2030–2032) Proton lifetime ~10³⁷ yr (Hyper‑K, DUNE, 2030+) No new particles below 10 TeV (HL‑LHC, FCC, 2030+) Falsification: If δ_PMNS/δ_CKM ≠ 3 at 3σ, or if dark matter is discovered with a mass significantly different from 123 GeV, the model would be ruled out. Version historyThis release (v14.7) supersedes previous versions:v1.0 (DOI:10.5281/zenodo.18764285), v2.0 (10.5281/zenodo.18785625), v4.0 (10.5281/zenodo.18839643), v7.0 (10.5281/zenodo.18868738), v8.1 (10.5281/zenodo.18894699).