The Koide relation from the Z3 symmetry of the trefoil: a topological origin of Q=2/3, a parameter-free prediction of mₜau, and the charged-lepton spectrum at 0. 4% (TQNT)

Koide's relation Q= (mₑ+mₘu+mₜau) / (sqrt (mₑ) +sqrt (mₘu) +sqrt (mₜau) ) ² = 2/3 is one of the most precise numerical coincidences in particle physics (satisfied to 9. 2 ppm), 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) =3₁, the Z3 symmetry of the trefoil enforces exactly the three-mode structure of the Koide parametrisation sqrt (mₖ) =sqrt (m₀) (1+r cos (delta+2*pi*k/3) ). The cube-root-of-unity identities make the Koide form an exact consequence, and the topological amplitude r=sqrt (2) (the spin-1/2 loop measure, braid index b=2) gives Q=2/3 exactly. Parameter-free prediction of mₜau from mₑ and mₘu: mₜau = 1776. 97 MeV vs 1776. 86 MeV measured (+61 ppm). Full charged-lepton spectrum at ~0. 4% from one scale m₀=mₚ/3 and two topological numbers r=sqrt (2) and delta=2/cr²=2/9 (cr=3 crossings of the trefoil): e (-0. 4%), mu (-0. 4%), tau (-0. 3%). The tau Z3-angle extracted from data equals 2/9 to 0. 05%; the extracted scale equals mₚ/3 to 0. 35%. Honest epistemic separation. DERIVED: the Koide form from the Z3 cube-root identities and the algebraic equivalence r=sqrt (2) Q=2/3. POSTULATED: generations = Z3 modes, r=sqrt (2), delta=2/9, m₀=mₚ/3. CONFIRMED: Q=2/3 (9. 2 ppm), mₜau (61 ppm, parameter-free), the three masses (0. 4%). A numerical coincidence cannot be excluded; the honest probability that this structure is fundamental rather than an a posteriori fit is of order 15-25%. It is, by a wide margin, the most constraining, predictive and falsifiable result of the framework. Bundle contents. English (arXiv-ready, with references to Koide, Foot, Brannen, Sumino) and French LaTeX sources and PDFs. CC BY 4. 0. --- (FR) La relation de Koide Q=2/3 est vérifiée à 9, 2 ppm et inexpliquée par le Modèle Standard. En TQNT, le lepton chargé est le trèfle 3₁ ; sa symétrie Z3 impose exactement la forme de Koide, et l'amplitude topologique r=sqrt (2) donne Q=2/3 de façon exacte. Il en résulte une prédiction PARAMÈTRE-LIBRE de mₜau (depuis mₑ, mₘu) à +61 ppm, et le spectre complet des trois leptons à ~0, 4% depuis l'échelle mₚ/3 et deux nombres topologiques (r=sqrt (2), delta=2/9). Séparation honnête dérivé/postulé/confirmé ; coïncidence non exclue (probabilité ~15-25% que ce soit fondamental). C'est le résultat le plus contraignant et falsifiable du cadre.