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 ZN 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 S4, 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 ε=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 |Vtd| close the CKM matrix, giving |Vus|=√(md/ms)=0.2255, |Vcb|=0.04197, |Vub|=0.003721, and δ=1.138 against the measured 1.139±0.023, with no continuous parameter. The unitarity triangle follows, with apex $(0.161,0.348)$ against the measured (0.161±0.010,0.347±0.010), and J=3.11×10−5 against (3.09±0.07)×10−5. The same structure carries the neutrinos. A single theta insertion gives sinθ13=θ2(τ)m2/m3, which holds at 0.7σ on the 207-day JUNO data, and the harmonic condition completes a normal-ordered spectrum with m1=0.25~meV, Σmν=0.0589~eV, a CP-conserving phase δCP=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.
Related
- Simple Modular S3 Models for Lepton Masses and Mixing
- Quasi-Phase-Matched Second Harmonic Generation in Poling-Free Lithium Niobate Microrings with Surface Molecular Pattern
- Modular multiscale approach to modelling high-harmonic generation in gases
- Parity-selected Bethe-Salpeter equation: symmetry-adapted regularization of the exciton-phonon self-energy in polar semiconductors
- Quark Mixing Model with Three Higgs Doublets and S3 Modular Symmetry at τ=i
BibTeX
@misc{barger2026levelfour,
title = {{Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses}},
author = {Vernon Barger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09262},
eprint = {2609.09262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses
AU - Vernon Barger
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09262
UR - https://arxiv.org/abs/2609.09262
AB - 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 ε=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|=√(m_d/m_s)=0.2255, |V_cb|=0.04197, |V_ub|=0.003721, and δ=1.138 against the measured 1.139±0.023, with no continuous parameter. The unitarity triangle follows, with apex $(0.161,0.348)$ against the measured (0.161±0.010,0.347±0.010), and J=3.11×10^−5 against (3.09±0.07)×10^−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.
ER -