Lepton and Quark Mass Ratios from SU(3) Braiding: The Koide Angle as a Conformal Dimension

The charged lepton masses satisfy the Koide relation Q = 2/3 and are parametrizedby a single Brannen phase δexp = 0.22222(5) ≈ 2/9. We prove that this parametrizationis the exact eigenvalue structure of a democratic element of the exceptionalJordan algebra J3(O), with cos(3δ) = −φ(V ) where φ is the G2 3-form. Fromthe Sumino SU(3)F family gauge symmetry, we derive Q = 2/3 (equivalent to thequantum dimension d□ = 2 of SU(3)3 WZW) and the amplitude A =√2 (proven).The phase δ = 2/9 – formerly an open conjecture – is now conditionally derivedfrom WZW braiding: the antisymmetric braiding label in SU(N)N satisfies|h(Λ2□) − 2h(□)|/h(□) = 2/(N − 1), which equals unity if and only if N = 3. Thesame braiding structure gives the down-quark phase δd = 1/9 from the symmetricchannel, and a conjectured quantization relation yields δu = 2/39. All six Brannenparameters (Q, δ) for leptons, down quarks, and up quarks are functions of N = 3alone, predicting mc = 617 MeV (0.2σ), ms = 93.1 MeV (0.04σ), and four additionalmass ratios within 1.1σ. Twenty distinct algebraic conditions select N = 3;the neutrino extension is decisively falsified. One hundred and eight alternativeapproaches are cataloged.