Commensurate Structure of Quark and Lepton Mixing: Eighteenth Powers of One Parameter and a Digital-Clock Quantization of the Unitarity Triangles

Abstract

We identify two commensurate structures in the measured quark and lepton mixing data, one multiplicative and one angular, and their exact relation. With the hierarchy parameter $\epsilon \equiv |V_{ub}{|}^{3/10}=0.1869\pm 0.0018$, the measured Fritzsch--Xing (FX) angles satisfy $\sin {\theta }_u={\epsilon }^{26/18}$, $\sin {\theta }_d={\epsilon }^{17/18}$, and $\sin \theta ={\epsilon }^{34/18}$ with coefficients within $2.5\%$ of unity, and the phase is maximal, ${\varphi }_{\mathrm{FX}}=92.3^{\circ }\pm 2.7^{\circ }$ against $\pi /2$. Unit coefficients reproduce the remaining moduli at the percent level and fix the derived Jarlskog invariant, $J={\epsilon }^{111/18}\sin {\varphi }_{\mathrm{FX}}$, matching the measured $3.08(14)\times 10^{-5}$. The rephasing-invariant image of this structure is a unitarity triangle quantized on the $\pi /24$ lattice, a digital clock, $(\alpha ,\beta ,\gamma )=(12,3,9)\times 7.5^{\circ }$, consistent with data at $1\sigma$ through $\alpha \simeq {\varphi }_{\mathrm{FX}}$ and the theorem $\tan \beta \simeq \sqrt{\epsilon }$; degree-level $\gamma$ confronts $67.5^{\circ }$--$68^{\circ }$. For the leptons, where only the angular structure can act, its PMNS analog, formed from columns $1$ and $3$, free of Majorana phases, on the same lattice requires ${\delta }_{\mathrm{CP}}\simeq 296^{\circ }$ in the upper octant or $244^{\circ }$ in the lower, testable at DUNE and Hyper-Kamiokande. The quark and lepton triangles share $\beta =22.5^{\circ }$, measured for quarks, predicted for leptons, and differ by a transfer of two lattice units, with corollary ${\delta }_{\mathrm{CP}}\simeq -\delta \simeq 295^{\circ }$. The same $\epsilon$ sets the neutrino mass ratio, $r\equiv m_2/m_3={\epsilon }^{19/18}$, and the PMNS first row follows in powers of $r$, $(r^{1/9},r^{1/3},\frac{\sqrt{3}}{2}r)$, both relations at $0.2\sigma$ and testable at JUNO. The analysis is confined to the parameterization level; no dynamics is assumed.