{
  "schema_version": 1,
  "problem_number": "OWR-734-003",
  "title": "An Odd-Divisor Obstruction for Cyclically Invariant Grassmannian Initial Ideals",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Over a field of characteristic zero, we prove that the Pluecker ideal of Gr(k,n) has no monomial initial ideal invariant under cyclic rotation whenever gcd(k,n) has an odd divisor greater than one. More generally, for n=mq and k=rq with q>1 odd and 1<=r<m, a permutation of cycle type q^m has trace -binomial(binomial(m,r),2) on the quadratic relations. This negative trace also excludes every invariant monomial basis of the degree-two coordinate ring. No squarefreeness, quadratic-generation or Gorenstein assumption on the putative initial ideal is required. The four-page proof uses classical standard monomial theory and the two-column Schur identity, and credits Santos--Stump--Welker's known Gr(3,6) flag-complex obstruction. Exact rational relation-matrix checks and a reproducibility script accompany the paper. This is a complete auxiliary symmetry theorem, not a solution of the broader Gorenstein-initial-ideal question in UnsolvedMath OWR-734-003; without the symmetry requirement, the Grassmannian existence case is already known. This unrefereed preprint was developed with AI assistance and is self-audited, without formal proof-assistant verification or an absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AC",
    "math.CO",
    "math.AG"
  ],
  "keywords": [
    "OWR-734-003",
    "Grassmannian",
    "Pluecker ideal",
    "monomial initial ideal",
    "cyclic symmetry",
    "character obstruction"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-734-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-734-003/paper.pdf?v=f5e7fc693681",
  "doi": "10.5281/zenodo.23680206",
  "zenodo_record_url": "https://zenodo.org/records/23680206",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Characteristic-zero Pluecker algebra: for n=mq, k=rq, q>1 odd and 1<=r<m, a permutation of cycle type q^m has trace -binomial(binomial(m,r),2) on the quadratic relations. No invariant monomial basis of the degree-two quotient and no invariant monomial initial ideal exist. Full cyclic invariance is excluded whenever gcd(k,n) has an odd divisor greater than one. This complete auxiliary symmetry theorem does not close the general Gorenstein-initial-ideal question. Grassmannian existence without symmetry is already known; the Gr(3,6) flag-complex obstruction is credited to Santos--Stump--Welker. 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": "f5e7fc6936810d3aff2677e5c61ea0b67e6d27dee7d50594ce6148b3834716f4"
    },
    "source.zip": {
      "sha256": "8455d0c944c354dadaa3a81cb739088f785a15d5ea4532c5ad8593920dd5a809"
    },
    "verification_report.md": {
      "sha256": "495bbf6af2be990f8d418283534de9e72abf281f242582f687b39e20c8e4e293"
    }
  },
  "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."
}