Two-over-Two Lattice Flavor from a Single Flavon with Three Messenger Chains

Abstract

Flavor hierarchies are organized by a single parameter $B\simeq 5.357$ in a single-flavon Froggatt--Nielsen (FN) framework, in which each effective Yukawa entry arises from the sum of \emph{three} unit-magnitude messenger chains. We present benchmark complex $O(1)$ Yukawa matrices that reproduce quark and charged-lepton masses at $M_Z$ as powers of $\epsilon\equiv 1/B$. The organizing principle is a two-over-two (2/2) lattice of quadrilateral mass ratios, which maps directly to a rational lattice of FN exponents. Sequential dominance preserves the leading-power exponent matrices, while subleading messenger chains generate entry-dependent complex $O(1)$ coefficients and provide a UV-friendly origin for CP violation. Neutrino masses are discussed at the level of eigenvalues within the same $B^n$ counting.