{
  "abstract": "FINDING: The 3j-symbols of SU(2) encode the Clebsch–Gordan decomposition of the A₁ root system, with the adjoint representation (j=1) forming an octahedral (O_h) symmetry via the double cover Spin(3)→SO(3). | MATH: 3j-symbols: \\(\\begin{pmatrix} j_1 & j_2 & j_3 \\\\ m_1 & m_2 & m_3 \\end{pmatrix}\\) = \\((-1)^{j_1-j_2-m_3}(2j_3+1)^{-1/2}\\langle j_1 m_1; j_2 m_2 | j_3\\,{-m_3}\\rangle\\). Adjoint rep: \\(j=1\\), generators \\(t^a = -i\\epsilon^{abc}\\) (structure constants of su(2) = A₁). Double cover: \\(SU(2)/\\{\\pm I\\} \\cong SO(3)\\); octahedral group \\(O \\subset SO(3)\\) lifts to binary octahedral group \\(2O \\subset SU(2)\\) of order 48. | CONNECTION: A₁ root system has two roots \\(\\pm\\alpha\\) with Cartan matrix [[2]] — its Weyl group is \\(S_2\\) (order 2), but the *adjoint* SU(2) action on \\(\\mathfrak{su}(2)\\) gives the spherical harmonics \\(Y_{1m}\\) (l=1), whose nodal surfaces form octahedral symmetry (the three axes). The 3j-symbols for \\(j_1=j_2=j_3=1\\) yield the structure constants \\(\\epsilon^{abc\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/a86scwqxh23nn.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635280",
  "fulltext_url": null,
  "id": "a86scwqxh23nn",
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  "landing_url": "https://zenodo.org/records/23635280",
  "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,
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  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a86scwqxh23nn.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:56Z",
      "native_id": "23635280",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23635280"
    }
  ],
  "subjects": [],
  "title": "Octahedral Symmetry in SU(2) 3j-Symbols via A₁ Root System Decomposition — E8 Intelligence Research",
  "title_text": "Octahedral Symmetry in SU(2) 3j-Symbols via A₁ Root System Decomposition — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:56Z",
  "url": "https://research.draconitic.com/paper/a86scwqxh23nn",
  "venue": "Zenodo",
  "versions": []
}