{
  "abstract": "FINDING: Gafni's restricted partition asymptotics connect density α of allowed parts to Dedekind eta-function modular behavior, yielding a phase transition in growth rate. | MATH: For $\\mathcal A$ with counting function $A(x) \\sim \\alpha x$ (density α), Gafni's result (arXiv:1407.2918) gives: $p_{\\mathcal A}(n) \\sim \\frac{1}{2\\pi\\sqrt{2\\alpha n}} \\exp\\left( \\pi \\sqrt{\\frac{2\\alpha n}{3}} \\right)$ for $0<\\alpha<1$ (with error terms via saddle-point on the Dedekind eta $\\eta(\\tau)$). For α=1 (all integers), this reduces to Hardy–Ramanujan: $p(n) \\sim \\frac{1}{4n\\sqrt{3}} e^{\\pi\\sqrt{2n/3}}$. The generating function is $\\sum p_{\\mathcal A}(n)q^n = \\prod_{a\\in\\mathcal A}(1-q^a)^{-1} = \\frac{1}{\\eta(\\tau)} \\cdot \\text{(modular correction)}$ when $\\mathcal A$ is arithmetic progression-like. Key constant: the exponent $\\pi\\sqrt{2\\alpha/3}$ — linear in $\\sqrt{\\alpha}$, not α. | CONNECTION: The factor $\\sqrt{2/3}$ in the exponent — note $\\sqrt{2/3} \\approx 0.8165$, and the ratio $\\frac{\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/4fv0aqvnd4a3q.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23430998",
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  "id": "4fv0aqvnd4a3q",
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  "language": "eng",
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  "published": "2026-10-17",
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  "ris": "https://research.draconitic.com/paper/4fv0aqvnd4a3q.ris",
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    {
      "first_seen": "2026-10-31T12:22:08Z",
      "native_id": "23430998",
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  "subjects": [],
  "title": "Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research",
  "title_text": "Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-31T12:22:08Z",
  "url": "https://research.draconitic.com/paper/4fv0aqvnd4a3q",
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