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

Abstract

We show that quark weak mixing follows from the same one-flavon, three-messenger Froggatt-Nielsen construction, with a single hierarchy parameter $B$ ($\hat{I}\mu \equiv 1/B$) and a rational-exponent "$B$-lattice", that organizes the fermion mass spectrum in companion work. The lattice exponents, together with a four-phase messenger coefficient model, account for the four measured CKM magnitudes and, through the Fritzsch-Xing reconstruction, the full $3\times 3$ CKM matrix and the Jarlskog invariant $J$, providing a consistency test of the single-$B$ hypothesis. A benchmark Yukawa pair on the lattice reproduces, under direct diagonalization, the light-quark masses to better than 5%, all nine CKM magnitudes to better than 1.2%, and $J$ within $0.3\ddot{I}ƒ$ of the PDG global fit, with order-unity coefficients; a $10^7$-configuration phase scan shows the CKM hierarchy to be generic once the quark masses are fit, while percent-level agreement across all four magnitudes is not. The near-maximal CP phase admits a geometric interpretation: equal-weight messenger interference fixes the phase at exactly $\ddot{I}€/2$ for any individual phase values, and the fitted $\ddot{I}{•}_{\mathrm{FX}}\simeq 93^{\circ }$ measures the departure from this limit. The same lattice parameter fixes the charged-lepton mass ratio $m_{\hat{I}}¼/m_{\ddot{I}}„=c_{\hat{I}}¼\hat{I}{\mu }^{5/3}$, whose exponent is the second-generation lepton charge sum; this yields $|V_{ub}|\simeq (m_{\hat{I}}¼/m_{\ddot{I}}„{)}^2$, agreeing with the measured value at about the 6% level, and the equivalent form $|V_{ub}|\simeq {\sin }^3\hat{I}{¸}_{13}$ ties the smallest CKM magnitude to the smallest PMNS magnitude, the two routes furnishing complementary determinations of the same ninths-lattice quantity from the quark and lepton sectors.