The Koide relation from the Z3 symmetry of the trefoil: Q=2/3 with r=sqrt(2) carried by the sense doubling (conditional), Koide's 1981 relation for m_tau, the phase decomposed as pi/12 - epsilon, and m_e computed to ±3% (TQNT)

Abstract

Koide's relation Q=(m_e+m_mu+m_tau)/(sqrt(m_e)+sqrt(m_mu)+sqrt(m_tau))^2 = 2/3 is one of the most precise numerical coincidences in particle physics (satisfied to 3.3 ppm on the 2024–25 masses), yet the Standard Model offers no explanation: the three charged-lepton masses are three free Yukawa couplings. Result. Within the Topological Knot Quantum Theory (TQNT), where the charged lepton is the trefoil knot T(2,3), the Z3 symmetry of the trefoil enforces exactly the three-mode Koide parametrisation (the cube-root-of-unity identities make the Koide form an exact consequence). One recovers Koide's 1981 relation predicting the tau mass from m_e and m_mu: m_tau = 1776.97 MeV vs 1776.93 ± 0.09 measured (+22 ppm, 0.4 sigma). New in v2 (the rail-magnetism chain, monograph consolidation of 2026-07-27). (i) The amplitude r = sqrt(2) — postulated in v1 after documented failed derivation attempts — now has a carrier (conditional): the lepton doublet is the doublet of the two circulation senses of the rail current (degeneracy theorem: the C3-invariant inductance kernel leaves the classes exactly degenerate; the T-odd sense is the minimal breaker, the mirror exchanges mu and tau), and two channels against one give r = d_1/2(k=2) = sqrt(2) — the same sqrt(2) as the exchange excess d−1 of the heavy sector. (ii) The phase, an orphan hint in v1 (delta = 2/9), is decomposed: delta = pi/12 − epsilon, where pi/12 = 2pi/24 is one step of the trefoil's minimal lattice circuit (24 steps, Diao) and epsilon = 0.0396 ≈ sqrt(m_e/m_0) is the quasi-nodality of the electron — the electron's lightness is a nodal protection. (iii) The scale is identified: m_0 = (3/4) sqrt(sigma_chi) = 313.85 MeV vs 313.85 extracted (1e-5). (iv) The chain closes with a computation: m_e = retarded-Green energy of the framing-line regrouping mode of the ideal trefoil tube (n=3, m=+1 selected by line-current conservation) = 0.518–0.522 MeV, i.e. m_e to +1.4/+2.2%, zero parameters (declared ±3% budget). Epistemic honesty. Every statement carries its status (derived / conditional / postulated / confirmed). The v1 caveat stands: the naive spin-1/2 loop-measure argument for sqrt(2) is wrong (it is the spin-1 ladder coefficient); the carrier is of a different nature (channel doubling of the k=2 ribbon), and every link of the chain is conditional on a declared identification — none is a fit. The distinctive parameter-free content remains Q=2/3 (3.3 ppm) and m_tau (+22 ppm, 0.4 sigma); v2 added the independent m_e computation. New in v3 (2026-08-31). Masses updated to PDG 2024/2025 (S. Navas et al., Phys. Rev. D 110, 030001 (2024) and 2025 update): m_tau = 1776.93 ± 0.09 MeV. Every headline number improves: Q_exp = 0.6666645 (−3.3 ppm, was −9.2), the 1981 prediction m_tau = 1776.97 now sits at +22 ppm = 0.4 sigma (was +61 ppm), and the extracted scale matches (3/4)sqrt(sigma_chi) at the 1e-5 level. Notably, the 2024 world-average update (driven by Belle II) moved the measurement toward the 1981 prediction. Conceptual-gap pass: the extracted phase is now written delta_exp (the orphan theta_tau symbol removed), epsilon ≈ sqrt(m_e/m_0) (the scale, not the muon mass), and the mode-selection section introduces the framing lines and Călugăreanu's relation Lk = Tw + Wr explicitly (Lk = +1, Wr = 3.42, Tw = −2.42). Bundle contents. FR + EN LaTeX sources and PDFs (v3, 6 pages each).