{
  "abstract": "FINDING: A1 root system is the minimal rank-1 case whose Weyl group is S₂ (order 2), and its representation theory (Clebsch-Gordan / 3j-symbols) is the algebraic skeleton underlying rank-2 topological phase classifications.\n\nMATH: - A₁ root system: roots {α, −α}, Cartan matrix [2], Weyl group W(A₁) ≅ S₂ = {1, s_α}, |W| = 2. - Reflection: s_α(λ) = λ − 2(λ,α)/(α,α) · α; for A₁, (α,α)=2 ⇒ s_α(λ) = λ − (λ,α)α. - Fundamental weight ω = α/2; ρ = ω = α/2. - SU(2) irreps: dim V_j = 2j+1; CG decomposition V_{j₁}⊗V_{j₂} = ⊕_{j=|j₁−j₂|}^{j₁+j₂} V_j. - 3j-symbol selection rule: (j₁ j₂ j₃; m₁ m₂ m₃) ≠ 0 iff m₁+m₂+m₃=0 and |j₁−j₂| ≤ j₃ ≤ j₁+j₂ (triangle inequality = A₁ fusion rule). - Wigner–Eckart / 3j orthogonality: Σ_{m₁m₂} (j₁ j₂ j₃; m₁ m₂ m₃)(j₁ j₂ j₃'; m₁ m₂ m₃') = δ_{j₃j₃'} δ_{m₃m₃'} /(2j₃+1). - Quantum dimension of SU(2)_k: d_j = sin((2j+1)π/(k+2)) / sin(π/(k+2)); for k=1, d_{1/2} = φ = (1+√5)/2 = 1.618… (Fibonacci / Ising anyon limit). - Rank-2 root systems A₁×A₁, A₂, B₂≅C₂, G₂ — Weyl group\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/9fbxj8777gdmy.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658856",
  "fulltext_url": null,
  "id": "9fbxj8777gdmy",
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  "landing_url": "https://zenodo.org/records/23658856",
  "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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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fbxj8777gdmy.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:31Z",
      "native_id": "23658856",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23658856"
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  ],
  "subjects": [],
  "title": "A₁ Weyl Group S₂ as Algebraic Skeleton for Rank-2 Topological Phases — E8 Intelligence Research",
  "title_text": "A₁ Weyl Group S₂ as Algebraic Skeleton for Rank-2 Topological Phases — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:31Z",
  "url": "https://research.draconitic.com/paper/9fbxj8777gdmy",
  "venue": "Zenodo",
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}