{
  "abstract": "Building on the proposed formula for α_fs and the constant K_τ defined in Part II, we propose to identify (hypothesis; qualitative scheme) the Standard Model's three fermion generations with the three normal modes of the toroidal vortex with Z₃ symmetry. Identifying the three sub-vortices with the three colors, the algebra of traceless Hermitian observables on the three states is su(3), with 8 generators associated with the Gell-Mann matrices (a general fact for any three-state system; the part specific to the Continuum is the identification). We propose confinement as a mechanism suggested by the topology of the knot, with a string tension about 15% below the lattice QCD value (57% below, if the von Laue factor 2 discussed in Part I holds); the linear form of the potential remains a hypothesis.\n\nWithin this parametrization, which describes any three positive masses, we show that the Koide relation (Q=2/3, known experimentally since 1982) is exactly equivalent to a hypothesis of energy equipartition between the modes (B=√2 A): a reformulation of Koide, not a derivation of it from Z₃ symmetry. Imposing it, from m_e and m_μ follows m_τ=1776.97 MeV (0.4σ).\n\nWe propose formulas for the mixing angles of the CKM matrix in terms of K_τ, with chosen powers and coefficients: θ₁₃=K_τ=0.201° against 0.214° (PDG 2024, −2.7σ), θ₁₂=(3/2)K_τ^(1/3) (+1.2σ), θ₂₃=(p/q)K_τ^(1/2) (−3.5σ): the order of magnitude is captured, but only θ₁₂ is within the error. We also propose formulas for the unitarity triangle, in which β+δ=90° holds by construction (arcsin+arccos; observed 88.3°±1.6°, +1.1σ); the formulas for γ and δ are not compatible with unitarity. For the PMNS matrix we propose θ₁₂ (−1.3σ) and θ₂₃ by subtracting β_CKM from the knot's base angles, and three CKM–PMNS identities follow; the comparison of θ₂₃ and of the phase δ_PMNS=−112° with the data depends on the octant of θ₂₃, not yet determined. θ₁₃^PMNS is not obtained from the model: an open problem.\n\nThis work constitutes Part III of a series. Subsequent parts will complete particle physics (neutron, EDM, magnetic moments in Part IIIA; the neutrino in Part IIIB; the electroweak sector in Part IIIC) before addressing emergent gravity (Part IV).",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Eduart Desari",
      "orcid": "0009-0009-2334-3173"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f7wez3rhe7jw6.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23496745",
  "fulltext_url": null,
  "id": "f7wez3rhe7jw6",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23496745",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23496745/files/TheContinuum_PartIII_IT.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f7wez3rhe7jw6.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:46Z",
      "native_id": "23496745",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23496745"
    }
  ],
  "subjects": [],
  "title": "The Continuum — Part III: Three Generations, the Koide Relation, and the CKM/PMNS Angles from Z3 Symmetry",
  "title_text": "The Continuum — Part III: Three Generations, the Koide Relation, and the CKM/PMNS Angles from Z3 Symmetry",
  "type": "preprint",
  "updated_at": "2026-11-03T12:20:46Z",
  "url": "https://research.draconitic.com/paper/f7wez3rhe7jw6",
  "venue": "Zenodo",
  "versions": []
}