{
  "abstract": "FINDING: The search results are mostly pedagogical or tangential (Markov chains ≠ Markov numbers; Fibonacci explainers). The only substantive mathematical item is an arXiv paper linking the golden ratio to number-theoretic functions (Möbius, Euler totient, natural logarithm). No direct source on Markov numbers' golden-ratio growth or tree asymptotics was retrieved.\n\nMATH: From the arXiv abstract (1109.3216v4): identities involving φ = (1+√5)/2 and its reciprocal φ⁻¹ = φ−1 = 0.618…, expressed via sums/products over the Möbius function μ(n), Euler totient φ(n), and ln n. The paper proves, e.g., ∑_{n=1}^∞ μ(n) [ln(φ^n − (−φ)^{−n})] / n = … (exact form not in abstract), and analogous identities for φ⁻¹. Also: φ satisfies φ² = φ + 1, φ⁻¹ = φ − 1. No Markov number triple equation (x²+y²+z² = 3xyz) or its growth constant (≈2.618… per level, conjectured) appears in the retrieved content.\n\nCONNECTION: The golden ratio φ = 1.618… and its reciprocal 0.618… are explicitly central. The recipr\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/8wrne75p62c4q.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23616349",
  "fulltext_url": null,
  "id": "8wrne75p62c4q",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23616349",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8wrne75p62c4q.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T12:20:53Z",
      "native_id": "23616349",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23616349"
    }
  ],
  "subjects": [],
  "title": "Golden Ratio Ties to Number-Theoretic Functions, Not Markov Numbers — E8 Intelligence Research",
  "title_text": "Golden Ratio Ties to Number-Theoretic Functions, Not Markov Numbers — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:53Z",
  "url": "https://research.draconitic.com/paper/8wrne75p62c4q",
  "venue": "Zenodo",
  "versions": []
}