The Particle Mass Spectrum from Torus Knot Mode-Locking: 56 Masses from Four Quantum Numbers

We show that the masses of 56 particles—including hadrons, mesons, leptons, tetraquarks, and pentaquarks—are accurately reproduced by a single formula with four quantum numbers and no continuously adjustable parameters beyond the electron mass. In the Null Worldtube (NWT) framework, each particle is a nonlinear electromagnetic mode propagating as a (p,q) torus knot on a superfluid vacuum torus with aspect ratio κ = π². The mass formula combines a topological factor (p²+q²), a geometric factor from phase closure (β and ln(8β)), and a confinement factor n_q^q arising from a mode-locking cascade: each of q poloidal windings compounds the interaction among n_q quarks, giving a geometric (compound interest) enhancement. The confinement condition gcd(n_q, q) = 1 ensures quarks are incommensurate with the poloidal periodicity and cannot escape. Quark flavor maps to the poloidal winding number: light quarks at q=5 (2⁵=32), charm at q=7 (2⁷=128), and bottom at q=9 (2⁹=512). The charmonium excitation tower emerges as the (2,7) mode with incrementing phase closure integer m, with level spacing ~600–700 MeV. All 56 predictions fall within 3% of experiment, with 46 within 1% and median absolute error 0.40%. The τ lepton admits an alternative interpretation as a 'stealth baryon'—topologically a (3,4) state with n_q=3 confined constituents (+0.73% error), appearing as a lepton because perfect confinement (gcd(3,4)=1) suppresses strong interactions. The physical coupling K ~ 10⁻⁶ places the system in the deeply perturbative regime of Arnold tongue theory, where n_q^q is consistent with the leading-order tongue-width scaling. The Standard Model mass spectrum is thus the perturbative mode-locking structure of torus knot vortices in a superfluid vacuum.