{
  "abstract": "FINDING: Golden ratio appears as a dilatation eigenvalue on invariant subsurfaces of nonorientable mapping class groups, and as a modular symmetry-breaking parameter in A5 lepton mixing models. | MATH: On nonorientable surfaces of odd genus g≥5, there exists a mapping class whose dilatation (stretch factor) on an invariant subsurface equals Φ = (1+√5)/2 ≈ 1.618. In the A5 modular model, two modular symmetries are broken to a diagonal A5 subgroup; the golden ratio enters via modular form weight and the Weinberg operator, yielding a \"Golden Ratio 2\" mixing pattern (likely sin²θ₁₂ = 1/(2+Φ) ≈ 0.276 or related to 1/√(2+Φ)). | CONNECTION: Φ = 1+1/Φ = 2cos(π/5); its reciprocal φ = Φ−1 = 0.618; φ² = 0.382. The dilatation Φ on a surface corresponds to a pseudo-Anosov map — the simplest hyperbolic dynamics on a surface, directly tied to the modular group PSL(2,Z) and its action on the upper half-plane. The A5 symmetry is the icosahedral group — the largest crystallographic point group in 3D (or\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/29krnhzz0hrbn.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23634954",
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  "id": "29krnhzz0hrbn",
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  "landing_url": "https://zenodo.org/records/23634954",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29krnhzz0hrbn.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:22:01Z",
      "native_id": "23634954",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23634954"
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  "title": "Golden Ratio Unifies Mapping Class Dilatation and A5 Modular Symmetry Breaking — E8 Intelligence Research",
  "title_text": "Golden Ratio Unifies Mapping Class Dilatation and A5 Modular Symmetry Breaking — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:22:01Z",
  "url": "https://research.draconitic.com/paper/29krnhzz0hrbn",
  "venue": "Zenodo",
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}