{
  "schema_version": 1,
  "problem_number": "AMR-020-0202",
  "title": "Global Symplectic Equivalence Does Not Determine a Lagrangian Torus Fibration",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Consider the projection of the four-torus onto its last two coordinates, equipped with a constant magnetic symplectic form with real parameter c. We compare global symplectic equivalence with equivalence preserving the Lagrangian fibration. Fibre-preserving equivalence identifies exactly parameters equal up to sign and an integer shift. Global equivalence is equality of the period subgroups Z+cZ: for rational parameters it is equality of reduced denominators, while for irrational parameters it coincides with fibre-preserving equivalence. The parameters 1/5 and 2/5 therefore have globally symplectomorphic total spaces but no fibre-preserving equivalence over any base diffeomorphism; an explicit integral matrix realizes the global map. This disproves the unrestricted global-versus-fibre-preserving implication recorded in Leung-Symington, Section 7.2, p.185. The magnetic construction, twisting invariant and integral alternating normal-form method are classical and credited. The contextual boundary-domain version of AMR-020-0202 is not claimed solved. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.SG",
    "math.DS"
  ],
  "keywords": [
    "symplectic geometry",
    "Lagrangian fibrations",
    "four-torus",
    "magnetic twisting",
    "integral alternating forms",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-020-0202/",
  "pdf_url": "https://eulersolve.org/papers/amr-020-0202/paper.pdf?v=6cce98494b8a",
  "doi": "10.5281/zenodo.23308492",
  "zenodo_record_url": "https://zenodo.org/records/23308492",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete equivalence classification for the specified real magnetic four-torus family, including a counterexample to the unrestricted global-versus-fibre-preserving implication recorded in Leung-Symington, Section 7.2, p.185. This does not close the contextual boundary-domain AMR-020-0202 problem or assert that every qualification of Mishachev's original article has been checked. Classical constructions and normal forms are credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
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    "source.zip": {
      "sha256": "bcf8a4161efc2b5f210b904b1c82bf573eef575147f5d00f65bfad9c6bfd6134"
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    "verification_report.md": {
      "sha256": "ebe1b966e901ab4028ae881d67207f0e370ca7be9bbddda50164eddd1ba8b5f6"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}