We report an empirical formula for charged lepton mass ratios in terms of torus knot invariants and the fine structure constant: m (K) /me = pq × (4αfs) ^−c/3, where K = T (p, q) is the torus knot assigned to each lepton, c is its crossing number, and αfs ≈ 1/137. With c₀ = 3 (the trefoil crossing number, not fitted), the formula reproduces the muon and tau mass ratios to 0. 59% and 3. 9% respectively; fitting c₀ = 3. 013 gives 2. 1% and 1. 3%. The formula was identified by systematic elimination: Beltrami eigenvalue ratios on knotted tubes approach unity; helicity ratios fall short by a factor of 26. 3; a scan over eight invariants and twenty-nine coupling functions isolated pq and (4αfs) ^−c/3 as the unique viable combination within the tested space. Like Balmer's formula for hydrogen, this empirical relation organises the data and predicts new states—but is not derived from first principles. The formula predicts a fourth mass eigenvalue at 26–27 GeV associated with T (2, 7) ; since sufficient centre-of-mass energy is necessary but not sufficient for producing topologically non-trivial states, the experimental status of this prediction depends on production dynamics not contained in the mass relation itself.
Daniel Edward Gruenberg (2026) studied this question.