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

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 ($α$, $β$) and organizes all fundamental scales from $v_{\rm EW}$ to $M_{\rm Pl}$ on a ninths ladder’’ $\mathrm{\Lambda }\times {\epsilon }^{n\mathrm{/}9}$. The lattice structure yields the CKM and PMNS mixing parameterizations (with all mixing exponents expressible as charge differences $\mathrm{\Delta }Q$ dressed by a universal Fritzsch–Xing phase shift of $\pm 1\mathrm{/}9$), 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 ($\mathrm{\Lambda }$ and $\epsilon =14\mathrm{/}75$). 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.