Charged Lepton Masses from the Additive Cauchy Equation, Phase Equipartition, and Cauchy Duality in R12

Abstract

The three charged lepton masses (mτ , mµ, me) are derived from the rendering algebra R12 through the lepton algebra su(4) and the additive Cauchy functional equation. Because leptons are SU(3) singlets, they see M4 = M2 ⊗ M2 rather
than M3; the symmetry algebra is su(4), dim = 15. A unified generating function f(W) = dim(su(4)) · (W+1)/ dim(∧2(C4)) = 15(W+1)/6 produces the effective degrees of freedom for all three generations: f(3) = 10 = dim(Sym2(C4 )) (tau), f(2) = 15/2 (muon), f(1) = 5 (electron). The linearity of f is derived from the additive Cauchy equation: each generation transition adds the same number of su(4) degrees of freedom, giving f(W+1) − f(W) = 5/2 = const, whose unique solution is linear.