{
  "abstract": "FINDING: Platonic solids are uniquely constrained by Euler's formula and angular defect; q-continued fractions diverge on dense unit-circle sets, linking modular forms to geometric boundaries. MATH: - Euler: V − E + F = 2 → for regular {p,q}: 1/p + 1/q = 1/2 + 1/E → only (3,3), (3,4), (4,3), (3,5), (5,3). - Angle defect per vertex: δ = 2π − q·(interior angle) → tetrahedron δ=π/3, cube δ=π/2, octahedron δ=2π/3, dodecahedron δ=π/5, icosahedron δ=π/3 (sum = 4π, Gauss-Bonnet discrete). - q-continued fraction divergence: uncountable set on |q|=1 where Rogers–Ramanujan and Ramanujan–Selberg CFs fail to converge in general sense (arXiv:1812.10873). CONNECTION: - Platonic solids ↔ root systems A₃, B₃, H₃ (crystallographic vs icosahedral) — H₃ symmetry yields golden ratio φ = (1+√5)/2 ≈ 1.618, and its inverse 0.618 appears in dodecahedron/icosahedron edge ratios. - Angle defect ratios: δ_tetra/δ_octa = 1/2, δ_cube/δ_icosa = 3/2 — no direct 0.382/0.786, but φ appears in H₃ Coxeter\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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      ],
      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/5k9kqt68d2epz.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23409885",
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  "published": "2026-10-16",
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  "ris": "https://research.draconitic.com/paper/5k9kqt68d2epz.ris",
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      "first_seen": "2026-10-30T12:18:24Z",
      "native_id": "23409885",
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  "title": "Euler's Formula and Angular Defect Uniquely Constrain Platonic Solids — E8 Intelligence Research",
  "title_text": "Euler's Formula and Angular Defect Uniquely Constrain Platonic Solids — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:21Z",
  "url": "https://research.draconitic.com/paper/5k9kqt68d2epz",
  "venue": "Zenodo",
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