The Continuum — Part III: Three Generations, the Koide Relation, and the CKM/PMNS Angles from Z3 Symmetry

Building on the proposed formula for α_fs and the constant K_τ defined in Part II, we propose to identify (hypothesis; qualitative scheme) the Standard Model's three fermion generations with the three normal modes of the toroidal vortex with Z₃ symmetry. Identifying the three sub-vortices with the three colors, the algebra of traceless Hermitian observables on the three states is su(3), with 8 generators associated with the Gell-Mann matrices (a general fact for any three-state system; the part specific to the Continuum is the identification). We propose confinement as a mechanism suggested by the topology of the knot, with a string tension about 15% below the lattice QCD value (57% below, if the von Laue factor 2 discussed in Part I holds); the linear form of the potential remains a hypothesis.
 
Within this parametrization, which describes any three positive masses, we show that the Koide relation (Q=2/3, known experimentally since 1982) is exactly equivalent to a hypothesis of energy equipartition between the modes (B=√2 A): a reformulation of Koide, not a derivation of it from Z₃ symmetry. Imposing it, from m_e and m_μ follows m_τ=1776.97 MeV (0.4σ).
 
We propose formulas for the mixing angles of the CKM matrix in terms of K_τ, with chosen powers and coefficients: θ₁₃=K_τ=0.201° against 0.214° (PDG 2024, −2.7σ), θ₁₂=(3/2)K_τ^(1/3) (+1.2σ), θ₂₃=(p/q)K_τ^(1/2) (−3.5σ): the order of magnitude is captured, but only θ₁₂ is within the error. We also propose formulas for the unitarity triangle, in which β+δ=90° holds by construction (arcsin+arccos; observed 88.3°±1.6°, +1.1σ); the formulas for γ and δ are not compatible with unitarity. For the PMNS matrix we propose θ₁₂ (−1.3σ) and θ₂₃ by subtracting β_CKM from the knot's base angles, and three CKM–PMNS identities follow; the comparison of θ₂₃ and of the phase δ_PMNS=−112° with the data depends on the octant of θ₂₃, not yet determined. θ₁₃^PMNS is not obtained from the model: an open problem.
 
This work constitutes Part III of a series. Subsequent parts will complete particle physics (neutron, EDM, magnetic moments in Part IIIA; the neutrino in Part IIIB; the electroweak sector in Part IIIC) before addressing emergent gravity (Part IV).