{
  "abstract": "We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\\mathcal{I}$ be the standard icosian maximal order over $K=\\mathbb{Q}(\\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\\mathcal{I}$-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of $\\operatorname{Aut}_{\\mathcal{I}}(L_{\\mathrm T})\\cong2.J_2$. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to $P\\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22842",
  "author_total": 1,
  "authors": [
    {
      "name": "Gerald Höhn"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/02w8we4grn2sm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22842",
  "fulltext_url": null,
  "id": "02w8we4grn2sm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22842",
  "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",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214012190",
  "pdf_url": "https://arxiv.org/pdf/2609.22842",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/02w8we4grn2sm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22842",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22842"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Hall-Janko Simple Group and the Icosian Leech Lattice",
  "title_text": "The Hall-Janko Simple Group and the Icosian Leech Lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/02w8we4grn2sm",
  "venue": "arXiv",
  "versions": []
}