{
  "schema_version": 1,
  "problem_number": "AMR-059-0016",
  "title": "An Elementary Obstruction to Stability of Hessian Metrics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The property of admitting a positive definite Hessian metric is not open under small affine deformations of a compact manifold. On the fixed torus T^2, the classical shear family with sole nonzero Christoffel coefficient Gamma^1_22=t consists of complete, volume-preserving affine connections and converges smoothly to the Euclidean connection. For an arbitrary hypothetical Hessian metric, one component of the Codazzi equation integrates to t times the integral of g_11 being zero, excluding every nonzero t. Products give the same obstruction in all dimensions at least two. We also give an averaging criterion for translation-invariant affine connections on tori. The affine family is due to the classical theory described by Baues and Goldman; this note supplies a self-contained application to Shimizu's stability question, without a novelty or priority claim. This is an unrefereed, self-audited preprint with AI assistance disclosed in the text.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DG"
  ],
  "keywords": [
    "AMR-059-0016",
    "math.DG"
  ],
  "manuscript_version_date": "2026-10-21",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-059-0016/",
  "pdf_url": "https://eulersolve.org/papers/amr-059-0016/paper.pdf?v=bbbe66da07df",
  "doi": "10.5281/zenodo.23586096",
  "zenodo_record_url": "https://zenodo.org/records/23586096",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The unrestricted stability assertion in Shimizu's item 5(b), p. 126, is false. On every fixed torus of dimension at least two, the classical affine shear family Gamma^1_22=t is complete and volume preserving, converges smoothly to a Hessian connection, and admits no positive Hessian metric at nonzero t. Codazzi and periodic integration exclude arbitrary smooth metrics. The note also proves an averaging criterion for translation-invariant torus connections. Koszul's hyperbolic case and fixed-linear-holonomy deformations are not contradicted or resolved. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
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      "sha256": "bbbe66da07df579f6658659bbd2d666b94fa0498e4b09a54b7f21880533eaf6c"
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    "source.zip": {
      "sha256": "3c7bc29bdc7c06f96ee27246dfdf14fcdc0251f527914d2190311da743756954"
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    "verification_report.md": {
      "sha256": "d37efac5f832d3b4be5f72e5a84950ff1ba7c224bdc23e9482455d43dac76e90"
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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."
}