The Innocent Lepton: How to Be in the Emergence Tree
Kibrom Kidane · · DOI 10.5281/zenodo.19932394
Abstract
When compact G2 symmetry breaks to SU(3), one confined branch closes on an SU(3)3 topological phase. In this framework the charged-lepton mass amplitude is represented by a Z3-equivariant Bogoliubov–de Gennes (BdG) operator on the three family labels; the physical masses are mk = ∆2 k. The direct finite G2 →SU(3) path retains the center harmonic v= [1,− 1 1 2 ,− 2 ] and therefore keeps a microscopic sector label. The charged-lepton branch used here is the cyclic closure: [∆,S] = 0, so the mass matrix is circulant and has eigenvalues ∆k = A+ Bcos θ+ 2πk 3. On this branch the chiral SU(3)3 modular tensor category and the octonionic G2/SU(3) Clebsch–Gordan data fix the amplitude ratio B/A= √2; the fundamental conformal weight gives the phase θ= h(3) = 2/9. The same branch defines the UV coupling αG2 (MPl) = |Z3|/(2πD2 tot) = 1/(24π), and the projected pair two-point norm gives the trace-lift normalization ceff = 1/2. One-loop dimensional transmutation then gives mk = 1 2 MPl e−9π 2 /2 1 + √2 cos 2 9 + 2πk 3 , k= 0,1,2, which reproduces the electron, muon, and tau masses with a common 0.12% scale residual, of the same order as α(0)/(2π), from the branch constants above. The Koide relation Q= 2/3 follows as an algebraic consequence of |B/A|= √2 on the positive branch. The mass formula depends only on the charged-lepton-layer closure algebra: the cyclic BdG form and the chiral data shared by the three SU(3)3 modular invariants. The D(6) modular invariant is the full-RCFT certificate of this condition, with M1,J = 1 and M8,8 = 3. Calculation-level details are collected in the Supplemental Material
Keywords
charged-lepton masses·Koide relation·modular tensor category·SU(3)3 Wess–Zumino– Witten model·octonions·Bogoliubov–de Gennes spectrum