{
  "abstract": "FINDING: A₁ root system is the minimal rank-1 case whose Weyl group is S₂ = ℤ₂; its representation theory (Clebsch–Gordan / 3j-symbols) is the algebraic skeleton underlying SU(2) angular momentum and, via Altland–Zirnbauer classes, topological phase classification.\n\nMATH: - A₁ root system: roots {±α}, Cartan matrix [2], Weyl group W(A₁) = S₂ = {1, s_α}, s_α(α) = −α. Rank 1, |W| = 2. - Clebsch–Gordan: V_{j₁} ⊗ V_{j₂} = ⊕_{J=|j₁−j₂|}^{j₁+j₂} V_J. For A₁ (SU(2)), multiplicity-free. - Wigner 3j-symbol relates to CG: ⟨j₁m₁j₂m₂|JM⟩ = (−1)^{j₁−j₂+M} √(2J+1) · (j₁ j₂ J; m₁ m₂ −M). - Selection rules (triangular): |j₁−j₂| ≤ J ≤ j₁+j₂; m₁+m₂+M = 0. - Orthogonality: Σ_{m₁m₂} (j₁ j₂ j₃; m₁ m₂ m₃)(j₁ j₂ j₃'; m₁ m₂ m₃') = δ_{j₃j₃'}δ_{m₃m₃'} Δ(j₁j₂j₃), where Δ is the triangle coefficient. - Symmetry: 3j invariant under even permutations of columns; odd permutations give phase (−1)^{j₁+j₂+j₃}. - AZ classification: 2 complex (A, AIII) + 8 real (AI, BDI, D, DIII, AII, CII, C, CI) classes ↔ symmetric spac\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"
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  ],
  "bibtex": "https://research.draconitic.com/paper/c3cczfps4jb5d.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658881",
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  "id": "c3cczfps4jb5d",
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  "landing_url": "https://zenodo.org/records/23658881",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c3cczfps4jb5d.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:30Z",
      "native_id": "23658881",
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      "url": "https://zenodo.org/records/23658881"
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  "subjects": [],
  "title": "A₁ Root System: Weyl Group S₂ Underlying SU(2) and Topological Phases — E8 Intelligence Research",
  "title_text": "A₁ Root System: Weyl Group S₂ Underlying SU(2) and Topological Phases — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:30Z",
  "url": "https://research.draconitic.com/paper/c3cczfps4jb5d",
  "venue": "Zenodo",
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