{
  "abstract": "FINDING: The A₁ root system (rank-1 simple Lie algebra, sl(2,K)) is the foundational algebraic structure for spin-1/2 qubit Hilbert space, with its principal elements and embeddings into sl(m,K) revealing universal Frobenius/Lie-algebraic constraints. | MATH: A₁ ≅ sl(2,K) ≅ su(2)⊗ℂ; root system Φ = {±α}, α² = 2; Cartan matrix [2]; Weyl group ℤ₂; fundamental rep: 2-dimensional Hilbert space ℂ² (qubit); Pauli matrices σ_x, σ_y, σ_z generate su(2) with [σ_i, σ_j] = 2iε_ijk σ_k; Casimir invariant C₂ = (1/2)σ·σ = (3/4)I₂; principal nilpotent element e (Gerstenhaber–Giaquinto) satisfies ad(e)² ≠ 0, ad(e)³ = 0 in sl(2); embedding theorem: any Lie algebra with left-symmetric (pre-Lie) structure embeds into sl(m,K) — for A₁, m=2 minimal. | CONNECTION: A₁ root system is the crystallographic root system of type A₁ — the only rank-1 case, with root length ratio 1:1 (no longer/shorter roots), corresponding to the single simple root α with Cartan integer 2. The Weyl group order |W| = 2 = 2¹, matchin\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/3wjy7hxrrfg67.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635256",
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  "id": "3wjy7hxrrfg67",
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  "landing_url": "https://zenodo.org/records/23635256",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wjy7hxrrfg67.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:56Z",
      "native_id": "23635256",
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      "url": "https://zenodo.org/records/23635256"
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  ],
  "subjects": [],
  "title": "A₁ Root System as the Universal Algebraic Foundation of Qubit Hilbert Space — E8 Intelligence Research",
  "title_text": "A₁ Root System as the Universal Algebraic Foundation of Qubit Hilbert Space — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:56Z",
  "url": "https://research.draconitic.com/paper/3wjy7hxrrfg67",
  "venue": "Zenodo",
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}