{
  "schema_version": 1,
  "problem_number": "OWR-1538-001",
  "title": "A Cubic Heisenberg Centralizer and Non-flat Diamond Argument-Shift Maps",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compute two small examples separating maximality, maximal transcendence degree and flatness in symmetric Poisson algebras. For the four-dimensional diamond Lie algebra we classify the regular argument shifts whose natural morphisms are flat. The shift at x* is not flat despite satisfying the full codimension-three and invariant-degree hypotheses of the Panyushev-Yakimova maximality theorem. The underlying example and its shift algebra are due to Ooms. For the five-dimensional Heisenberg algebra, the homogeneous cubic F=2z^2t+zp^2-xq^2 has Poisson centralizer k[z,F], a maximal commutative polynomial algebra of transcendence degree two, below the maximal value three. An elementary weighted-degree obstruction for the Airy derivation proves the centralizer formula in all degrees. Ooms's earlier unrestricted negative answer to the maximality/dimension question is explicitly credited. This is an unrefereed manuscript, with AI assistance disclosed in the text and no absolute priority claim. Source dataset record: OWR-1538-001 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.RA",
    "math.AG",
    "math.RT"
  ],
  "keywords": [
    "OWR-1538-001",
    "Poisson centralizer",
    "argument shift",
    "Heisenberg algebra",
    "flat morphism"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1538-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-1538-001/paper.pdf?v=629d5d4c19af",
  "doi": "10.5281/zenodo.23621691",
  "zenodo_record_url": "https://zenodo.org/records/23621691",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For the algebraic diamond Lie algebra [t,x]=-x, [t,y]=y, [x,y]=z, the full regular argument-shift algebra is k[z,xy-tz,alpha*y+beta*x-gamma*t]. Its natural morphism is flat exactly when gamma!=0 or alpha*beta!=0; in those cases it is faithfully flat. At xi=x*, base change by (z,y) gives the nonzero k[C]-module k[t,x] annihilated by C, proving nonflatness under the full codimension-three and invariant-degree hypotheses. For the algebraic nilpotent Heisenberg algebra h5 with [x,t]=[q,p]=z, the homogeneous cubic F=2*z^2*t+z*p^2-x*q^2 has centralizer k[z,F], a graded maximal Poisson-commutative subalgebra of transcendence degree 2<b(h5)=3. An all-degree Airy-derivation obstruction proves the centralizer formula. The unrestricted answer to the first frozen question and the diamond setup/shift were already given by Ooms and are explicitly credited. Ooms already supplied an unrestricted negative answer to the first source question and the diamond setup. This paper gives explicit centralizer and all-shift flatness calculations; it does not claim two newly solved open problems. 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": "629d5d4c19afc6de2dd26fc57a9ace56e090e1e1278d114cd91c561eda1d2f3a"
    },
    "source.zip": {
      "sha256": "feaff74839ab6fc5bfbb969495535378732522a75d7e36a2bff3e46333891d8a"
    },
    "verification_report.md": {
      "sha256": "eb6f3c8857bf82343d57bfb7224a1e69cc3a32ec4d78c41450a3edcb22e1c8a2"
    }
  },
  "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."
}