Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses

Abstract

The mass ratios of the quarks and charged leptons follow a simple pattern. The logarithmic generation steps come in a two to one ratio, a note to its octave, and the ordering is mirrored between the up-down and down-lepton comparisons. The pattern, read as charge counting on a $Z_N$ clock, is realized at level four and at no level that is not a multiple of four, which motivates modular level four and the finite group $S_4$, with the generations in the triplet at $T$ charges $(3,1,0)$ and the mirror in the sign singlet. The charged-lepton mass ratios lie on a theta-dressed lattice at $0.02\%$, which sets the base $\varepsilon=0.18664$, taken as $14/75$, and the harmonic modulus $τ=1.0685i$. Five postulates in the weight one-half theta constants, identified on the same data, reproduce the six quark masses from three lepton masses and the electroweak scale. Two more and one empirical relation for $|V_{td}|$ close the CKM matrix, giving $|V_{us}|=\sqrt{m_d/m_s}=0.2255$, $|V_{cb}|=0.04197$, $|V_{ub}|=0.003721$, and $δ=1.138$ against the measured $1.139\pm0.023$, with no continuous parameter. The unitarity triangle follows, with apex $(0.161,0.348)$ against the measured $(0.161\pm0.010,0.347\pm0.010)$, and $J=3.11\times10^{-5}$ against $(3.09\pm0.07)\times10^{-5}$. The same structure carries the neutrinos. A single theta insertion gives $\sinθ_{13}=θ_2(τ)m_2/m_3$, which holds at $0.7σ$ on the 207-day JUNO data, and the harmonic condition completes a normal-ordered spectrum with $m_1=0.25$~meV, $Σm_ν=0.0589$~eV, a CP-conserving phase $δ_{CP}=π$, and effective Majorana masses quantized at $1.3$ to $3.9$~meV. The ultraviolet completion belongs to the magnetized torus class, with Pati--Salam the natural gauge embedding.