{
  "schema_version": 1,
  "problem_number": "AMR-103-0231",
  "title": "A Torsion-Rank Obstruction to Turaev's Alexander Polynomial Realization Conjecture",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give counterexamples to the sufficiency assertion in Turaev's Conjecture 12.26 in Ohtsuki's problem list. If a closed connected oriented 3-manifold has first homology Z direct sum T and its Alexander polynomial is nonzero modulo a prime p, its order of vanishing at t=1 modulo p is at least dim_Fp(T/pT). An integral square presentation from a primitive dual surface proves this bound. Consequently, whenever T has p-rank at least three, the pair (Z direct sum T, t+t^(-1)+|T|-2) satisfies the conjecture's symmetry and augmentation conditions but is not realizable. The concrete example T=(Z/2)^3 has polynomial t+t^(-1)+6. A Smith-normal-form ideal condition strengthens the obstruction, and explicit torus bundles show that the same polynomial is realizable with cyclic torsion of the same order. This is a complete negative answer to the sufficiency assertion in Conjecture 12.26, recorded as AMR-103-0231 in ulamai/UnsolvedMath v1.6.0. It is not a classification of all realizable pairs. The cut-surface and determinant methods are classical and credited. No absolute priority claim is made. The five-page English preprint is by Alper Ferudun, Mercury Software GmbH. It is AI-assisted, self-audited and unrefereed; no independent peer review or proof-assistant certification is claimed. Portable exact regression checks accompany the source and do not replace the general proof. Version 1.0, 15 October 2026. CC BY 4.0.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "keywords": [
    "Alexander polynomial",
    "Turaev conjecture",
    "3-manifold",
    "torsion rank",
    "AMR-103-0231"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-103-0231/",
  "pdf_url": "https://eulersolve.org/papers/amr-103-0231/paper.pdf?v=d426dab18354",
  "doi": "10.5281/zenodo.23384960",
  "zenodo_record_url": "https://zenodo.org/records/23384960",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete negative answer to the sufficiency assertion of Turaev Conjecture 12.26 (AMR-103-0231). No classification of all realizable pairs or sufficiency of the Smith-ideal condition is asserted. Classical cut-surface and determinant methods are credited. Self-audited, unrefereed and AI-assisted; no absolute novelty, independent review or formal verification claimed.",
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      "sha256": "d426dab183549b20b9cad9171cf43b02057d056fe1f9981fbf9c673aa6a1e120"
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    "source.zip": {
      "sha256": "c9d833a087e06abb06ed542f10378d9a028a06fa9861346af9af336f2c36f53f"
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    "verification_report.md": {
      "sha256": "7bb51c64388bf120b94233febea6a33d6c838cde824b665973957ac8b6dbf5ca"
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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."
}