{
  "abstract": "FINDING: The search results are dominated by pedagogical videos on symplectic geometry and group theory, with one arXiv paper on gravitational-wave anisotropy. No direct source addresses the invariant ring of Sp(2n)/U(n) or Langlands dual root system swaps. The mathematical core is the standard symplectic group structure and its role in Hamiltonian mechanics.\n\nMATH: - Symplectic group: Sp(2n) = {M ∈ GL(2n,ℝ) | MᵀJM = J}, with J = [[0, Iₙ], [−Iₙ, 0]]. - Symmetric space Sp(2n)/U(n): rank n, invariant ring generated by degrees 2,4,…,2n (Chevalley restriction; the ring is polynomial with fundamental invariants of those degrees). - Langlands dual swap: For Sp(2n), dual is SO(2n+1); root systems Cₙ ↔ Bₙ. The Weyl group order: |W(Sp(2n))| = 2ⁿ·n!, |W(SO(2n+1))| = 2ⁿ·n! (same). - No new equations or constants beyond standard symplectic structure appear in the search results.\n\nCONNECTION: - The invariant degrees 2,4,…,2n for Sp(2n)/U(n) are *even integers* — no golden-ratio or bas\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/fy2hx6kp7a8fv.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23587266",
  "fulltext_url": null,
  "id": "fy2hx6kp7a8fv",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23587266",
  "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,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy2hx6kp7a8fv.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:21:50Z",
      "native_id": "23587266",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23587266"
    }
  ],
  "subjects": [],
  "title": "Symplectic Group Structure and Hamiltonian Mechanics: A Pedagogical Overview — E8 Intelligence Research",
  "title_text": "Symplectic Group Structure and Hamiltonian Mechanics: A Pedagogical Overview — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-06T12:21:50Z",
  "url": "https://research.draconitic.com/paper/fy2hx6kp7a8fv",
  "venue": "Zenodo",
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}