{
  "abstract": "FINDING: Gafni's restricted partition asymptotics connect density of allowed parts to exponential growth rates, with Dedekind eta function as the modular backbone. | MATH: For $\\mathcal{A}\\subset\\mathbb{N}$ with counting function $A(x)=\\alpha x + O(x^\\beta)$, $\\beta<1$, Gafni's result (arXiv:1407.4443) gives $\\log p_{\\mathcal{A}}(n) \\sim \\frac{\\pi}{\\sqrt{3}}\\sqrt{\\alpha n} + \\frac{1}{4}\\log n + O(1)$ — the leading term scales as $\\sqrt{\\alpha n}$, generalizing Hardy–Ramanujan ($\\alpha=1$). The generating function $\\sum p_{\\mathcal{A}}(n)q^n = \\prod_{a\\in\\mathcal{A}}(1-q^a)^{-1}$ and modularity enters via $\\eta(\\tau) = q^{1/24}\\prod_{n=1}^\\infty(1-q^n)$, where $q=e^{2\\pi i\\tau}$. The constant $\\pi/\\sqrt{3} \\approx 1.8138$ is the saddle-point residue from the Dedekind eta transformation $\\eta(-1/\\tau)=\\sqrt{-i\\tau}\\,\\eta(\\tau)$. | CONNECTION: The exponent $\\alpha$ in the density $A(x)\\sim\\alpha x$ directly multiplies $n$ inside the square root — this is a *linear scaling* of the modular\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/epm1c2j3td0td.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23431075",
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  "id": "epm1c2j3td0td",
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  "landing_url": "https://zenodo.org/records/23431075",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/epm1c2j3td0td.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:22:05Z",
      "native_id": "23431075",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23431075"
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  "subjects": [],
  "title": "Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research",
  "title_text": "Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-31T12:22:05Z",
  "url": "https://research.draconitic.com/paper/epm1c2j3td0td",
  "venue": "Zenodo",
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}