A Geometric Periodic Table of Particles: Classification by Angle and Overtone on the Three-Sphere

We present a classification of all known elementary particles using two geometric parameters on S³ equipped with the Hopf fibration: the angle θ between an excitation and the electromagnetic fiber direction, and the radial overtone number n. The classification organises every fermion, boson, and composite particle into a periodic table whose structure makes falsifiable predictions. The fermion sector is arranged as a five-column grid (angle) × three-row grid (generation). The 0° angle admits two distinct excitation types — charged (Q = −1, fiber winding ≠ 0) and neutral (Q = 0, zero winding) — giving the sterile neutrinos their own dedicated column. This layout makes the structural predictions visually immediate: the neutral 0° column is entirely predicted, requiring three sterile neutrinos (νs1, νs2, νs3) as dark matter candidates. The 61.2° column is entirely excluded (no consistent quantum numbers yield integer charge for fermions). A fourth generation row is forbidden (above the spectral gap on S³). Three closed-form expressions involving only π and integers fit the measured coupling constants to sub-σ precision with zero free parameters (postdictive): - α⁻¹ from the balance equation 1/α + α/D = G_π, where G_π = 4π³ + π² + π and D = 24 − 1/(8π): α⁻¹ = 137.035999213 (0.8σ from Rb 2020)- sin²θ_eff^lept = π/(4π+1) = 0.231572 (0.26σ from Z-pole world average)- α_s(M_Z) = 1/(3π−1) = 0.118697 (0.8σ from PDG 2024) The charged lepton masses are recovered to