{
  "schema_version": 1,
  "problem_number": "OWR-4086-008",
  "title": "A Counterexample to the Weighted Homogeneous Jacobian Jet Closure Conjecture",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Chen and Zuo conjectured that the rth jet closure of the Jacobian ideal of a weighted homogeneous polynomial equals that ideal plus the (r+1)st power of the maximal ideal. For f=z(x^2+y^3), we exhibit xy^3 in the fourth jet closure but outside J(f)+(x,y,z)^5. Membership follows from an explicit identity in universal jet coefficients, and nonmembership from a homomorphism to k[s]/(s^10). We also give an irreducible four-variable example and a transfer principle for principal-ideal jet defects. The construction uses the published cusp example of de Fernex, Ein and Ishii. Both examples are nonisolated and disprove the conjecture as written, not a version restricted to isolated singularities. The all-orders OWR-4086-008 question is already resolved by Mallory and is not claimed as a new result of this note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE_TO_ADJACENT_CONJECTURE",
  "categories": [
    "math.AC",
    "math.AG"
  ],
  "keywords": [
    "OWR-4086-008",
    "math.AC",
    "math.AG"
  ],
  "manuscript_version_date": "2026-10-22",
  "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/owr-4086-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-4086-008/paper.pdf?v=a1502dbac88d",
  "doi": "10.5281/zenodo.23587224",
  "zenodo_record_url": "https://zenodo.org/records/23587224",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The general Chen-Zuo weighted-homogeneous Jacobian finite-jet conjecture is false: for f=z(x^2+y^3), xy^3 lies in J(f)^(4-jc) but not in J(f)+m^5. The irreducible F=z(x^2+y^3)+w^7 has the same defect. Both examples are nonisolated; no isolated or two-variable variant is resolved. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
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      "sha256": "a1502dbac88d801fca6beea0cb32ea7010514215c7224e5893fe4fe1f064c6c0"
    },
    "source.zip": {
      "sha256": "38e8d1e8d831e372957c6e11eeac0f29ae0918832b7be532d40457eb1f9f699f"
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    "verification_report.md": {
      "sha256": "90f365645d85ab6a69e7588382b95e141c4a8646cc10afb2c6942ce61d6d29ee"
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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."
}