{
  "abstract": "Koide's charged-lepton relation suggests that $(\\sqrt{m_e},\\sqrt{m_μ},\\sqrt{m_τ})$ is the natural family vector. We construct an effective compact-cycle realization in which this vector is sampled from a real amplitude $Z(φ)$ on an internal circle, while the masses arise from quadratic chiral-profile overlaps. The coherent branch-real rank-one sector is an explicit assumption of the effective theory and realizes the equal singlet--doublet norm associated with Koide's $45^\\circ$ cone. The empirical proximity of the $Z_3$ phase to $2/9$ predates the present work. The new ingredient proposed here is an endpoint-dipole Green response for one elementary family-shift link. The compact harmonic sum fixes the geometric factor $γ_{C_3}=2λ_B/9$, while $λ_B$ remains an undetermined matching coefficient, and the determinant anisotropy gives $θ_\\ell^{(0)}=-2λ_B/9$. Thus $λ_B=1$ yields $θ_\\ell^{(0)}=-2/9$ only as a unit-normalized benchmark, not as an exact first-principles prediction. With the current pole masses, this benchmark requires a relative finite matching correction of $9.83\\times10^{-6}$ in $m_e/m_μ$; no microscopic theoretical uncertainty for that correction is derived here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.10245",
  "author_total": 1,
  "authors": [
    {
      "name": "Kirill Shulga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2sh9t7fzzkcmp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.10245",
  "fulltext_url": null,
  "id": "2sh9t7fzzkcmp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.10245",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.10245",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2sh9t7fzzkcmp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2605.10245",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.10245"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle",
  "title_text": "Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/2sh9t7fzzkcmp",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1088/1361-6471/aea4ce",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}