Quark Mixing from a Lattice Flavon Model: A Four-Magnitude Parameterization

We present the quark weak-mixing component of a Froggatt–Nielsen program, with one flavon and three messengers, in which a single hierarchy parameter $B$ (with $\epsilon \equiv 1\mathrm{/}B$) and a rational-exponent ``$B$-lattice’’ organize fermion Yukawa textures. Building on companion mass-fit work, we translate the lattice into predictions for quark mixing. The four-magnitude parameterization serves as a practical interface between the flavon Yukawa textures and quark weak mixing observables. The lattice exponent structure, together with a four-phase multi-messenger coefficient model, simultaneously reproduces all four CKM magnitudes, the full $3\times 3$ CKM matrix, and the CP-violating Jarlskog invariant, providing a stringent overdetermination test of the single-$B$ hypothesis. The same lattice parameter that organizes quark mixing also fixes the charged-lepton mass ratio $m_{\mu }\mathrm{/}{m}_{\tau }=c_{\mu }{\epsilon }^{5\mathrm{/}3}$, yielding the striking integer-exponent identity $\mathrm{\mid }{V}_{ub}\mathrm{\mid }\simeq (m_{\mu }\mathrm{/}{m}_{\tau }{)}^2$, which connects the smallest CKM magnitude to the charged-lepton spectrum with no free parameters and agrees with the measured value at about the $6\mathrm{\%}$ level. An equivalent cross-sector form $\mathrm{\mid }{V}_{ub}\mathrm{\mid }\simeq {\sin }^3{\theta }_{13}$ ties the smallest CKM magnitude directly to the smallest PMNS magnitude; the two routes agree at the few-percent level, providing complementary determinations of the same ninths-lattice quantity from the quark and lepton sectors.