Mass Hierarchies Without Mixing: Abelian Froggatt-Nielsen Models with Uncharged Left-Handed Doublets

Abstract

Abelian flavor charges on right-handed fermions produce left-handed anarchy: we prove that all abelian discrete Froggatt-Nielsen models with uncharged left-handed doublets yield Haar-random PMNS and CKM matrices, regardless of $\mathbb{Z}_N$ group order, charge assignment, or Majorana mass structure. Scanning $\mathbb{Z}_3$ through $\mathbb{Z}_7$ with 12 charge assignments and $10^5$ Monte Carlo samples each, we demonstrate that the mass spectrum failure previously identified for $\mathbb{Z}_3$ -- the seesaw over-suppression mechanism that pushes $\Delta m^2_{21}/\Delta m^2_{31}$ to $\sim 10^{-11}$ -- is specific to $\mathbb{Z}_3$ and avoidable for $N \geq 4$. The mixing angle failure, however, is universal and irreducible. The PMNS angles from every abelian model are statistically consistent with Haar-random unitary matrices, with median $\sin^2\theta_{12} \approx \sin^2\theta_{23} \approx 0.50$ and $\sin^2\theta_{13} \approx 0.31$ across all models tested. The same applies to the CKM: the joint probability of achieving CKM-like mixing from generic $O(1)$ coefficients is $< 2 \times 10^{-6}$. We identify the algebraic origin of this obstruction: abelian groups have only one-dimensional representations, so each generation transforms as an independent singlet with 18 free parameters for three Dirac mass matrices -- far exceeding the 10 physical observables. The transition to non-abelian flavor symmetries such as $A_4$, whose triplet representation reduces free parameters to 4 at leading order, is required specifically for mixing structure. This obstruction applies to the well-motivated subclass of models where left-handed fields are uncharged; models that assign abelian charges to both left- and right-handed fields can evade it.