{
  "abstract": "FINDING: The search results are a mix of popular expositions (PBS, Numberphile, Sybermath) and one substantive arXiv paper (2410.04169v2) linking golden/silver ratios to Fibonacci divisors, quantum calculus, and supersymmetric oscillators. No direct CHSH–golden-ratio relation is established in these sources. | MATH: arXiv paper: quantum calculus with two bases \\( q_\\phi = \\phi = (1+\\sqrt{5})/2 \\) and \\( q_\\sigma = 1+\\sqrt{2} \\) (silver ratio); Fibonacci divisor derivative \\( D_q f(x) = [f(qx)-f(x)]/[(q-1)x] \\); Binet form \\( F_n = (\\phi^n - (-\\phi)^{-n})/\\sqrt{5} \\) extended to operator \\( \\hat{F}_N \\) acting on Fock space; energy spectrum \\( E_n \\propto \\phi^n \\) or \\( \\sigma^n \\) for golden/silver oscillators. Nested radical identity: \\( \\phi = \\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}} \\). | CONNECTION: Golden ratio \\( \\phi \\approx 1.618 \\) and its inverse \\( \\phi^{-1} \\approx 0.618 \\) appear as base of quantum calculus; silver ratio \\( 1+\\sqrt{2} \\approx 2.414 \\) is related to \\( \\sqrt{2} \\\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/9gypb6nbk5pgr.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658269",
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  "id": "9gypb6nbk5pgr",
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  "language": "eng",
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  "licence_label": "CC BY 4.0",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gypb6nbk5pgr.ris",
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      "first_seen": "2026-11-09T12:21:37Z",
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  "title": "Golden Ratio and Silver Ratio in Quantum Calculus: A Search Result Review — E8 Intelligence Research",
  "title_text": "Golden Ratio and Silver Ratio in Quantum Calculus: A Search Result Review — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:37Z",
  "url": "https://research.draconitic.com/paper/9gypb6nbk5pgr",
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