{
  "schema_version": 1,
  "problem_number": "OWR-14218-004",
  "title": "Higher Koszul Homology of Square-Zero Algebras in Positive Characteristic",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compute the higher Koszul homology, defined by the Euler cap differential, of A=k plus V with V squared zero and finite-dimensional V. In each total weight, the calculation is the natural map from invariants to coinvariants of a signed cyclic rotation. A word orbit contributes higher classes exactly when its sign is compatible and its length is divisible by the characteristic. An explicit necklace formula gives every bigraded dimension. In every positive characteristic, k[x,y]/(x,y)^2 is Koszul but has nonzero higher Koszul homology; a three-term characteristic-three cycle and boundary witness are displayed. The first nonzero positive homological degree is p-1 in characteristic p. This is a complete counterexample to a characteristic-free extension of Koszul acyclicity and the comparison with left Koszul homology. It does not resolve the characteristic-zero converse associated with UnsolvedMath record OWR-14218-004. Classical cyclic-word decomposition is credited. The English, AI-assisted preprint includes a complete proof, a self-audit and105 exact diagnostic checks. It is unrefereed; no absolute priority or formal verification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.RA"
  ],
  "keywords": [
    "OWR-14218-004",
    "higher Koszul homology",
    "Koszul algebra",
    "Euler cap differential",
    "positive characteristic",
    "square-zero algebra",
    "necklace formula"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14218-004/",
  "pdf_url": "https://eulersolve.org/papers/owr-14218-004/paper.pdf?v=57a180f1d74b",
  "doi": "10.5281/zenodo.23669461",
  "zenodo_record_url": "https://zenodo.org/records/23669461",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete all-degree higher Koszul homology formula for A=k plus V, V squared zero, over every field. In each positive characteristic p, the Koszul algebra k[x,y]/(x,y)^2 has nonzero higher Koszul homology; an explicit F_3 cycle and boundary witness are given. This refutes the characteristic-free acyclicity implication and characteristic-free comparison with left Koszul homology. Complete all-degree formula and positive-characteristic counterexamples to a characteristic-free acyclicity statement. The characteristic-zero converse associated with OWR-14218-004 remains unresolved; this is not counted as closure of the whole source problem. Classical cyclic-word decomposition is credited. No absolute-priority or formal-verification claim is made. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "57a180f1d74b73d8342382a051050497f368f88d4371e894ca37f5df8c8d4853"
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    "source.zip": {
      "sha256": "5655a9b001f97c300f53906d7212c4e37fb253399afa0e5492acf3ca4f75a1df"
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    "verification_report.md": {
      "sha256": "d39c190d3f00631737a1c7a96c908989cf53b1515d0668cec90dbb3f29bc7b88"
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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."
}