Z₉ Yukawa Coefficients: Rationalization and UV Completion

The Z9 Froggatt-Nielsen model reproduced all nine charged fermion masses using an expansion parameter epsilon = 2/9 and nine dimensionless Yukawa coefficients in the range [0.49, 2.09]. We show that all nine coefficients are expressible as exact rational numbers constructed exclusively from the Z9 structural constants. Statistical analysis establishes this correspondence at >4 sigma significance. We investigate the representation-theoretic origin and establish partial UV completion: Z9 emerges from modular invariance at the SL(2,Z) fixed point. This eliminates the nine free Yukawa parameters from the Standard Model while establishing the UV framework for Z9 flavor symmetry.