Generation as Compositeness: A Subconstituent Interpretation of the $B$-Lattice Flavor Hierarchy

Abstract

We interpret the $B$-lattice flavor framework as a compositeness hierarchy: all three fermion generations are elementary chiral fields, but third-generation Yukawa couplings are undressed ($Q=0$), while lighter generations acquire their Yukawa couplings through chains of spin-$0$ subconstituents (``hops'') whose depth is counted by the $\mathbb{Z}_9$ discrete gauge symmetry. The $\mathbb{Z}_9$ charge admits a two-index decomposition $q_9\mapsto(a,b)$ that identifies two hop species ($\alpha$, $\beta$) and organizes all fundamental scales from $v_{\rm EW}$ to $M_{\rm Pl}$ on a ``ninths ladder'' $\Lambda\times\epsilon^{n/9}$. The lattice structure yields the CKM and PMNS mixing parameterizations (with all mixing exponents expressible as charge differences $\Delta Q$ dressed by a universal Fritzsch--Xing phase shift of $\pm 1/9$), a seesaw benchmark $m_3\simeq 51$~meV, the axion mass window $m_a\sim 7$--$12\;\mu$eV, and the prediction $\tan\beta\simeq 10$--$16$ (from the chain internal factor combined with the DFSZ-II two-Higgs-doublet structure), all from two parameters ($\Lambda$ and $\epsilon = 14/75$). Generation-dependent Peccei--Quinn charges restore the axion--photon coupling ($C_{a\gamma}\simeq 0.6$--$1.0$) from ``back to invisible'' suppression. An illustrative UV realization in terms of hypercolor-confined scalars communicated to the SM by a gauge-invariant messenger chain is presented as an existence proof.