{
  "abstract": "FINDING: Golden ratio (φ) appears in quantum calculus via Fibonacci divisors and in CHSH game optimality proofs, but no direct φ–√2–CHSH success-probability identity is established in these sources. | MATH: φ = (1+√5)/2 ≈ 1.618; φ⁻¹ = φ−1 ≈ 0.618; φ² = φ+1 ≈ 2.618; √2 ≈ 1.414 (silver ratio's irrational base); CHSH quantum success probability = cos²(π/8) ≈ 0.8536 (Tsirelson bound), classical = 0.75. The arXiv paper (2410.04169v2) defines quantum calculus with two bases: q₁ = φ, q₂ = 1+√2 (silver ratio), linking Fibonacci divisor derivatives to Binet-form number operators in Fock space. | CONNECTION: φ and √2 are both quadratic irrationals with continued fractions [1;1,1,…] and [1;2,2,…] respectively — they generate distinct root-system-like lattices (φ → pentagonal/quasicrystalline order; √2 → square/tetragonal order). The CHSH optimal angle π/8 = 22.5° relates to octagonal symmetry (π/4 halved), not directly to φ. However, cos²(π/8) = (2+√2)/4 ≈ 0.8536, which involves √2, not φ. No evi\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,
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  "author_total": 1,
  "authors": [
    {
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        0
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      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/3me28p09jnr17.bib",
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  "doi": "10.5281/zenodo.23658191",
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  "id": "3me28p09jnr17",
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  "published": "2026-10-26",
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  "ris": "https://research.draconitic.com/paper/3me28p09jnr17.ris",
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      "first_seen": "2026-11-09T12:21:38Z",
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  "title": "Golden Ratio and Silver Ratio in Quantum Calculus and CHSH Games — E8 Intelligence Research",
  "title_text": "Golden Ratio and Silver Ratio in Quantum Calculus and CHSH Games — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:38Z",
  "url": "https://research.draconitic.com/paper/3me28p09jnr17",
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