{
  "abstract": "FINDING: Markov numbers form a tree of solutions to x²+y²+z²=3xyz, with growth rates linked to the golden ratio; Fibonacci numbers are a special case. | MATH: Markov equation: x²+y²+z² = 3xyz. Fibonacci solutions: (1, F₂ₙ₋₁, F₂ₙ₊₁) satisfy the equation. Tree structure via Vieta involutions: (x,y,z) → (x,y,3xy−z). Growth rate of Markov numbers along the tree: asymptotic to C·φ^(3n) where φ = (1+√5)/2 ≈ 1.618 (known result: Markov numbers grow exponentially with exponent related to log φ). | CONNECTION: The golden ratio φ appears as the dominant eigenvalue of the Markov tree's growth — the tree's branching structure is isomorphic to the Farey tree / Stern-Brocot tree, which encodes rational approximations and is deeply tied to modular group PSL(2,Z) and its cusp geometry. The ratio 1/φ ≈ 0.618 and φ² = φ+1 ≈ 2.618 appear in the recurrence structure. The Markov tree's vertices correspond to simple closed geodesics on the modular torus — a crystallographic/lattice symmetry (the modular gro\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/22kw8w43rd7dh.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635357",
  "fulltext_url": null,
  "id": "22kw8w43rd7dh",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23635357",
  "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": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22kw8w43rd7dh.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:54Z",
      "native_id": "23635357",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23635357"
    }
  ],
  "subjects": [],
  "title": "Markov Numbers' Golden Ratio Growth and Fibonacci Connections — E8 Intelligence Research",
  "title_text": "Markov Numbers' Golden Ratio Growth and Fibonacci Connections — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:54Z",
  "url": "https://research.draconitic.com/paper/22kw8w43rd7dh",
  "venue": "Zenodo",
  "versions": []
}