{
  "abstract": "FINDING: Langlands duality maps a root system to its dual via transposition of the Cartan matrix, exchanging long/short roots and Weyl group orbits; topological and mirror-symmetry incarnations extend this to 1-form symmetries and Higgs bundle moduli. | MATH: For simple Lie algebra 𝔤 with root system Φ and coroots Φ∨, the Langlands dual 𝔤∨ has roots Φ∨ (as vectors in dual space). Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j) → A^T_ij for 𝔤∨. Weyl group W(Φ) ≅ W(Φ∨) (isomorphic as abstract groups) but acts on dual lattices. Killing form scales: (·,·)∨ = (·,·)/c where c is the ratio of squared lengths (e.g., for SO(2n+1) vs Sp(2n): c=2, so long roots of SO become short of Sp). | CONNECTION: Root systems are crystallographic — their Weyl groups are finite reflection groups with Coxeter numbers h. For SO(2n+1) (B_n) and Sp(2n) (C_n), h = 2n. The duality exchanges root lengths: ratio of squared lengths = 2, which appears in the golden-ratio-adjacent context (2 = φ + 1/φ ≈ 2.236? No — exact 2, n\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/3fvmcq08h362d.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23526078",
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  "id": "3fvmcq08h362d",
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  "language": "eng",
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  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fvmcq08h362d.ris",
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    {
      "first_seen": "2026-11-04T12:21:51Z",
      "native_id": "23526078",
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  "title": "Langlands Duality: Root System Transposition, Symmetry Exchange, and Topological Mirror Extensions — E8 Intelligence Research",
  "title_text": "Langlands Duality: Root System Transposition, Symmetry Exchange, and Topological Mirror Extensions — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:51Z",
  "url": "https://research.draconitic.com/paper/3fvmcq08h362d",
  "venue": "Zenodo",
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}