{
  "abstract": "FINDING: Weyl group orbit structure on Kac-Moody root systems is governed by Cartan matrix transpose symmetries, linking long/short root ratios to Langlands dual data. MATH: For a root system with Cartan matrix \\(A_{ij} = \\frac{2(\\alpha_i,\\alpha_j)}{(\\alpha_j,\\alpha_j)}\\), the dual system uses \\(A^T\\). Long/short root length ratio squared = \\(2\\) for simply-laced (all roots equal), but for \\(B_n\\): long²/short² = 2; for \\(C_n\\): ratio = 2 (dual swap); for \\(G_2\\): long²/short² = 3. Weyl group \\(W\\) acts on \\(\\Delta\\), with orbit structure determined by \\(W\\)-invariant bilinear form on \\(\\mathfrak{h}^*\\). The Kac-Moody generalization preserves the orbit count via \\(A^T\\) — e.g., for affine \\(A_1^{(1)}\\), the imaginary root \\(\\delta\\) has \\(W\\)-orbit size 1, while real roots split into long/short orbits with multiplicities tied to \\(A^T\\) entries. CONNECTION: Root length ratios {1, √2, √3} map to crystallographic constraints. The ratio √2 appears in \\(B_n/C_n\\) — related to 0.707 (ha\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/5sxb5qjrjwfaj.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23553857",
  "fulltext_url": null,
  "id": "5sxb5qjrjwfaj",
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  "landing_url": "https://zenodo.org/records/23553857",
  "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/",
  "metadata_licence": null,
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  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5sxb5qjrjwfaj.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:21:16Z",
      "native_id": "23553857",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23553857"
    }
  ],
  "subjects": [],
  "title": "Weyl Orbits and Cartan Transpose Symmetries in Kac-Moody Root Systems — E8 Intelligence Research",
  "title_text": "Weyl Orbits and Cartan Transpose Symmetries in Kac-Moody Root Systems — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-05T12:21:16Z",
  "url": "https://research.draconitic.com/paper/5sxb5qjrjwfaj",
  "venue": "Zenodo",
  "versions": []
}