{
  "abstract": "The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article reviews an algebraic framework in which an intrinsic $S_3$ family symmetry relates three gauge-equivalent fermion sectors. The framework is motivated by division- and Clifford-algebra constructions in which the complex octonions and $\\mathbb{C}\\ell(6)$ organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley--Dickson sequence, the sedenions provide an intrinsic $S_3$ automorphism structure, while their complexified left-multiplication operators generate an associative algebra isomorphic to $\\mathbb{C}\\ell(8)$. In the $\\mathbb{C}\\ell(8)$ formulation, the order-three family action generates three linearly independent fermion sectors while leaving a single $\\mathfrak{su}(3)_C\\oplus\\mathfrak{u}(1)_{\\mathrm{em}}$ gauge algebra. Extension to $\\mathbb{C}\\ell(10)$ incorporates a single $\\mathfrak{su}(3)_C\\oplus\\mathfrak{su}(2)_L\\oplus\\mathfrak{u}(1)_Y$ gauge algebra, with each generation including a right-handed neutrino sterile under the SM gauge interactions. We place the construction in context by comparing it with representative algebraic and family-symmetry approaches to three generations, including triality-based proposals, and clarify why the family symmetry used here is distinct from standard $\\operatorname{Spin}(8)$ triality despite the common appearance of $\\mathbb{C}\\ell(8)$ and $S_3$. The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family-symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10569",
  "author_total": 1,
  "authors": [
    {
      "name": "Niels Gresnigt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tmfxdmptc3mg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10569",
  "fulltext_url": null,
  "id": "6tmfxdmptc3mg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10569",
  "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",
      "physics.gen-ph"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7212283292",
  "pdf_url": "https://arxiv.org/pdf/2609.10569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tmfxdmptc3mg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.10569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10569"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.gen-ph",
      "subfield": "general"
    }
  ],
  "title": "Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review",
  "title_text": "Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:26Z",
  "url": "https://research.draconitic.com/paper/6tmfxdmptc3mg",
  "venue": "arXiv",
  "versions": []
}