{
  "abstract": "FINDING: Umbral moonshine establishes a precise correspondence between the 23 Niemeier lattices (rank-24 even unimodular lattices with non-trivial root systems) and finite groups whose mock modular forms encode their symmetries, extending monstrous moonshine beyond the Leech lattice. | MATH: Niemeier lattices: 24 even unimodular positive-definite lattices in dimension 24; 23 have non-trivial root systems (the 24th is the Leech lattice with empty root system). Root systems present: A24, D24, E6^4, A12^2, A8^3, D16E8, A24, D12^2, A15D9, E8^3, A17E7, D10E7^2, A11D7E6, A9^2D6, D6^4, A7^2D5^2, A5^4D4, A3^8, A1^24, A2^12, A4^6, D4^6, A6^4. Umbral group for lattice X: G^X = Aut(N_X)/W_X (automorphism group modulo Weyl group of root system). McKay–Thompson series: T_g^X(τ) = mock modular forms of weight 1/2 (or 3/2) with specific shadow functions. Key constants: 24 (dimension/rank), 23 (non-trivial root system lattices), 2 (umbral moonshine groups of order 2 for A1^24 and A2^12). | CONNECTION:\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fsymrjjv2djfq.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23465044",
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  "id": "fsymrjjv2djfq",
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  "landing_url": "https://zenodo.org/records/23465044",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fsymrjjv2djfq.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:49Z",
      "native_id": "23465044",
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      "url": "https://zenodo.org/records/23465044"
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  "subjects": [],
  "title": "Umbral Moonshine and the 23 Niemeier Lattices — E8 Intelligence Research",
  "title_text": "Umbral Moonshine and the 23 Niemeier Lattices — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-02T12:21:49Z",
  "url": "https://research.draconitic.com/paper/fsymrjjv2djfq",
  "venue": "Zenodo",
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}