The Periodic Table of Elementary Particles

The Periodic Table of Elementary Particles is a Music!What if all the particles in the universe — their masses, their interactions, even the strength of electromagnetism — come from the shape of a single geometric object? We started with a simple question: can the topology of the 3-sphere S³ (a higher-dimensional analog of the surface of a ball) explain why particles have the masses they do? The answer turned out to be far bigger than we expected. THE KEY IDEA S³ naturally contains three nested resonating structures — like Russian dolls — each producing a different family of particles: • A circle S¹ (1D) → generates hadron masses (J/ψ, Ω⁻, B mesons. . . ) • A torus T² (2D) → generates lepton masses (electron, muon, tau) and mixing angles • The full sphere S³ (3D) → generates boson masses (W, Higgs, Z) AND coupling constants All three come from one topological construction: the Hopf fibration. No parameters are adjusted. The only inputs are the electron mass and the muon mass. Everything else is geometry. WHAT WE FOUND From just mₑ = 0. 511 MeV and m_μ = 105. 66 MeV, the framework predicts: ✦ The tau lepton mass — to 0. 006% (6 parts in 100, 000) ✦ The W boson mass — to 0. 03% (via MW = 2⁸ × μ², where 2⁸ = dim Cl (8) ) ✦ The Higgs boson mass — to 0. 12% (via MH/MW = j₃/j₁, a ratio of Bessel roots) ✦ The J/ψ meson mass — to 0. 019% (via π-harmonics on the circle) And the headline discovery of this paper: ✦ THE FINE-STRUCTURE CONSTANT α = √2 / (j₁ · j₂ · j₃) Predicted: 1/α (MZ) = 127. 97 Experiment: 1/α (MZ) = 127. 952 Accuracy: 0. 011% This is a PURE NUMBER derived from PURE GEOMETRY — the product of three Bessel function roots (the resonance frequencies of the three quaternionic axes of S³), divided by √2 (the amplitude of the Clifford torus). No masses, no scales, no adjustable parameters. ✦ The Weinberg angle: sin²θW = 3/8 − 1/j₃ = 0. 2319 (exp: 0. 2312, accuracy 0. 29%) In total: 25 parameter-free predictions spanning five classes of observables — fermion masses, boson masses, mixing angles, coupling constants, and the number of particle generations (= 3, derived exactly). WHY THIS MATTERS For 50+ years, the ~20 free parameters of the Standard Model have been measured but never explained. Nobody has derived the fine-structure constant (1/137 at low energy, 1/128 at high energy) from first principles. Our formula gives it from the resonance structure of S³ — the same geometry that gives the particle masses. The number 2⁸ = 256 appearing as the bridge between the flavor scale and the electroweak scale is the dimension of the Clifford algebra Cl (8), connected to octonions and Bott periodicity — hinting at a deep algebraic origin of the mass hierarchy. This is not a "theory of everything" — it is a MAP. Like Mendeleev's periodic table organized chemistry before quantum mechanics explained it, our particle map organizes the Standard Model spectrum from geometry, pointing the way toward a deeper dynamical theory. TESTABLE PREDICTIONS The S³ resonator has a complete eigenvalue spectrum indexed by quantum numbers (l, n) — angular momentum and radial overtone. Three cells are occupied by known particles; the rest are predictions: THE BESSEL RESONANCE GRID (mass = Λₑw × j_ (l, n), Λₑw = 17. 9 GeV): n=1 n=2 n=3 n=4 l=0 ★ 56 GeV ★ 112 GeV ✓ 169 (top) ★ 225 GeV l=1 ✓ 80 (W) ★ 138 GeV ★ 195 GeV ★ 252 GeV l=2 ★ 103 GeV ★ 163 GeV ★ 220 GeV ★ 277 GeV l=3 ✓ 125 (H) ★ 186 GeV ★ 245 GeV ★ 303 GeV ✓ = known particle ★ = predicted new resonance Key: the top quark IS the (l=0, n=3) Bessel mode — j₀, ₃ = 3π! THE DARK π-HARMONIC TOWER (mass = m_χ × πⁿ, m_χ = 53. 85 MeV): n = −2: 5. 46 MeV ★ (nuclear anomaly searches) n = −1: 17. 14 MeV ★ X17 boson — Atomki reports 17. 01 ± 0. 16 MeV! n = 0: 53. 85 MeV ★ dark scalar χ (target for NA64, LDMX) n = +1: 169. 2 MeV ★ (near-pion region) n = +2: 531. 5 MeV ★ (near-η′ region) n = +3: 1670 MeV ✓ Ω⁻ baryon (exp: 1672. 5 MeV — confirmed!) n = +4: 5245 MeV ✓ B± meson (exp: 5279 MeV — confirmed!) HEAVY COMPOUND RESONANCES: • MW × j₁² = 1. 62 TeV ★ (matches ATLAS Wγ excess at ~1. 6 TeV) • MW × j₁j₂ = 2. 08 TeV ★ (accessible at HL-LHC) • MW × j₂j₃ = 3. 24 TeV ★ (accessible at HL-LHC) NEUTRINO MASSES (from Clifford torus duality, Paper I): • m₁ = 0. 36 meV, m₂ = 8. 62 meV, m₃ = 50. 1 meV • Σm_ν = 0. 059 eV (Planck bound: < 0. 12 eV — testable by DESI + CMB-S4!) TOTAL: ~34 confirmed quantities + ~29 predicted new states = ~63 entries in the periodic table. The three most decisive near-term tests: 1. Σm_ν = 0. 059 eV — DESI + CMB-S4 (~2027–2028) 2. New scalar at 56 or 103 GeV — LHC Run 3 diphoton searches 3. Dark scalar χ at 53. 85 MeV — NA64 / LDMX experiments Any one of these would confirm the geometric origin of particle masses. The empty cells are waiting.