{
  "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 the lattice's deep-hole structure, unifying Conway's original Leech lattice moonshine with a broader classification. | MATH: Niemeier lattices: 24 even unimodular positive-definite lattices in R^24; 23 have non-trivial root systems (A24, D24, E6^4, A12^2, A8^3, D16E8, etc.). Root system Weyl vector ρ satisfies ρ² = 2(h∨−1) for Coxeter number h∨. Umbral group G^X = Aut(N_X)/W_X (automorphism quotient by Weyl group). Mock modular forms H_g(τ) with q-expansions whose coefficients are traces of g on the lattice's \"umbral\" module. Deep holes: points in N_X ⊗ R at maximal distance from lattice points; their stabilizers are the umbral groups. | CONNECTION: Root systems A24, D24, E6^4 are crystallographic — their Cartan matrices encode base-60-like symmetries (E6 has Coxeter number 12,\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/7v300zf9k1gxt.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23451163",
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  "id": "7v300zf9k1gxt",
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  "landing_url": "https://zenodo.org/records/23451163",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7v300zf9k1gxt.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:10Z",
      "native_id": "23451163",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23451163"
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  "subjects": [],
  "title": "Umbral Moonshine: Unifying Niemeier Lattices and Finite Groups via Mock Modular Forms — E8 Intelligence Research",
  "title_text": "Umbral Moonshine: Unifying Niemeier Lattices and Finite Groups via Mock Modular Forms — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:10Z",
  "url": "https://research.draconitic.com/paper/7v300zf9k1gxt",
  "venue": "Zenodo",
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}