{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09262",
  "author_total": 1,
  "authors": [
    {
      "name": "Vernon Barger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9arcwxpyrw86.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09262",
  "fulltext_url": null,
  "id": "e9arcwxpyrw86",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09262",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ex"
    ],
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Particle physics theoretical and experimental studies"
    ]
  ],
  "openalex": "W7212145369",
  "pdf_url": "https://arxiv.org/pdf/2609.09262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9arcwxpyrw86.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09262"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ex",
      "subfield": "high-energy"
    }
  ],
  "title": "Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses",
  "title_text": "Level-Four Modular Symmetry Selected by the Harmonic Pattern of Quark and Lepton Masses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:12Z",
  "url": "https://research.draconitic.com/paper/e9arcwxpyrw86",
  "venue": "arXiv",
  "versions": []
}