{
  "abstract": "We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\\cong L_2(7)$ acts as on $\\mathbf P^1(\\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25086",
  "author_total": 1,
  "authors": [
    {
      "name": "Gerald Höhn"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4z4w4dqbqrw3t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25086",
  "fulltext_url": null,
  "id": "4z4w4dqbqrw3t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25086",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214066592",
  "pdf_url": "https://arxiv.org/pdf/2609.25086",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4z4w4dqbqrw3t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25086",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25086"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The unique extremal threemodular lattice of rank 26, the generalized hexagon $(2,8)$, and the tight Cayley-plane $5$-design",
  "title_text": "The unique extremal threemodular lattice of rank 26, the generalized hexagon (2,8), and the tight Cayley-plane $5$-design",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/4z4w4dqbqrw3t",
  "venue": "arXiv",
  "versions": []
}