{
  "abstract": "FINDING: Clebsch–Gordan coefficients are the unitary transformation matrix elements between uncoupled and coupled angular momentum bases, directly encoding SU(2) tensor product decomposition into irreducible representations. | MATH: For \\(j_1 \\otimes j_2 = \\oplus_j (j)\\), the coefficients \\(\\langle j_1 m_1 j_2 m_2 | J M\\rangle\\) satisfy orthogonality: \\(\\sum_{m_1,m_2} \\langle j_1 m_1 j_2 m_2 | J M\\rangle \\langle j_1 m_1 j_2 m_2 | J' M'\\rangle = \\delta_{JJ'}\\delta_{MM'}\\). Recursion via ladder operators: \\(\\sqrt{(J\\mp M)(J\\pm M+1)}\\langle j_1 m_1 j_2 m_2 | J, M\\pm 1\\rangle = \\sqrt{(j_1\\pm m_1)(j_1\\mp m_1+1)}\\langle j_1, m_1\\mp 1, j_2 m_2 | J M\\rangle + \\sqrt{(j_2\\pm m_2)(j_2\\mp m_2+1)}\\langle j_1 m_1, j_2, m_2\\mp 1 | J M\\rangle\\). Racah formula gives closed form as sum over integers \\(z\\): \\(\\langle j_1 m_1 j_2 m_2 | J M\\rangle = \\delta_{M,m_1+m_2} \\sqrt{(2J+1)\\frac{(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!}{(J+j_1+j_2+1)!}} \\times \\sqrt{(J+M)!(J-M)!(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!}\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/20k8ns000dw2a.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23616458",
  "fulltext_url": null,
  "id": "20k8ns000dw2a",
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  "landing_url": "https://zenodo.org/records/23616458",
  "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/",
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  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20k8ns000dw2a.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T12:20:53Z",
      "native_id": "23616458",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23616458"
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  ],
  "subjects": [],
  "title": "Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research",
  "title_text": "Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:53Z",
  "url": "https://research.draconitic.com/paper/20k8ns000dw2a",
  "venue": "Zenodo",
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}