{
  "abstract": "FINDING: Ramanujan tau function congruences extend to arbitrary-weight modular forms, including half-integral weights, via generalized moduli and arithmetic progressions. | MATH: The tau function τ(n) is defined by q∏(1−q^k)^24 = Σ τ(n)q^n. Ramanujan's congruences: τ(n) ≡ σ₁₁(n) mod 691; τ(n) ≡ n σ₅(n) mod 5; τ(n) ≡ n² σ₃(n) mod 7. Generalization: for weight k modular forms (including k ∈ ℤ+½), congruences arise modulo primes p dividing the denominator of B_k/(2k) (Bernoulli numbers), with moduli p^a and arithmetic progressions n ≡ r (mod p^b). | CONNECTION: The weight 12 (Δ) has Fourier coefficients τ(n) linked to the Leech lattice (24 dimensions, root system of E₈³). The exponent 24 and the product ∏(1−q^k)^24 encode the 24-cell / D₄ lattice structure. Half-integral weights connect to theta functions of lattices — crystallographic symmetry (e.g., E₈, Leech) is implicit. The prime 691 = 2·3·5·23 + 1, and 24 = 4! — base-60 resonance: 691 ≡ 31 (mod 60), and 60 = lcm(1,2,3,4,5). | DEPTH:\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/7hrcn55r0hq7p.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23616666",
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  "id": "7hrcn55r0hq7p",
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  "language": "eng",
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  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hrcn55r0hq7p.ris",
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    {
      "first_seen": "2026-11-07T12:20:49Z",
      "native_id": "23616666",
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  "subjects": [],
  "title": "Generalized Ramanujan Congruences for Arbitrary-Weight Modular Forms — E8 Intelligence Research",
  "title_text": "Generalized Ramanujan Congruences for Arbitrary-Weight Modular Forms — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:49Z",
  "url": "https://research.draconitic.com/paper/7hrcn55r0hq7p",
  "venue": "Zenodo",
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