{
  "abstract": "We study the chiral zero-mode spectrum of the compact six-manifold $X_6=S^2\\times \\mathbb{C}P^2$ in a ten-dimensional Kaluza-Klein setting. We review the monopole Dirac index on $S^2$ and the $\\mathrm{Spin}^{c}$ index analysis, showing that $\\mathbb{C}P^2$ admits a minimal unit-index chiral sector. Dolan and Nash established the $\\mathrm{Spin}^c$ index mechanism on complex projective spaces and showed that $\\mathbb{C}P^2$ can supply a one-family chiral building block with Standard-Model-like quantum numbers. The step taken here is to place the $\\mathbb{C}P^2$ sector on the six-dimensional product $S^2\\times\\mathbb{C}P^2$ and use an independently quantized monopole sector on $S^2$ to control the multiplicity. Unlike Calabi--Yau model-building constructions, where the family number is tied to vector-bundle and quotient data, this homogeneous product gives the three-family count through the factorized Dirac-index formula: $ N_{\\rm fam} = {\\rm index} D_{S^2 \\times \\mathbb{C}P^2}=\\bigl({\\rm index} D_{S^2,m}\\bigr) \\,\\bigl({\\rm index} D_{\\mathbb{C}P^2,n}\\bigr)=m\\,\\frac{n^2-1}{8}$. The canonical complex $\\mathrm{Spin}^{c}$ structure on $\\mathbb{C}P^2$ has $n=3$ and unit index. We show that this index is saturated by exactly one chiral zero mode and no opposite-chirality vectorlike partner. Thus, two flux sectors produce exactly three chiral internal zero modes: $(m,n)=(3,3)$ and $(m,n)=(1,5)$. Saturation is proved for all odd $n\\geq 3$. This homogeneous product isolates the family number in an elementary geometric formula. For a 10D Weyl fermion valued in a visible $\\mathbf{16}$ of $\\mathrm{Spin}(10)_G$, Lorentz branching shows these internal zero modes yield three left-handed 4D $\\mathbf{16}_G$ multiplets. The analysis is restricted to the spectral and index-theoretic problem; moduli stabilization and phenomenological dynamics are deferred to a companion paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02907",
  "author_total": 1,
  "authors": [
    {
      "name": "Edward J. Shaya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3n6kq185r86ke.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02907",
  "fulltext_url": null,
  "id": "3n6kq185r86ke",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02907",
  "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",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7208779862",
  "pdf_url": "https://arxiv.org/pdf/2609.02907",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3n6kq185r86ke.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2609.02907",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02907"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3106|Black Holes and Theoretical Physics",
      "subfield": "high-energy"
    }
  ],
  "title": "Three Chiral Families from a Monopole Product over $S^2\\times\\mathbb{C}P^2$",
  "title_text": "Three Chiral Families from a Monopole Product over S^2×ℂP^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:31Z",
  "url": "https://research.draconitic.com/paper/3n6kq185r86ke",
  "venue": "arXiv",
  "versions": []
}