{
  "schema_version": 1,
  "problem_number": "OWR-1740-002",
  "title": "Unbounded Gaps in the Tropical Secant Dimension Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The optimized affine winner-cell bound gives a combinatorial lower bound for the dimension of a secant cone of a monomial image. We show that it need not equal the actual dimension, answering negatively the unrestricted monomial formulation of a question in Draisma's Oberwolfach report. For the six cubic monomials x^3, y^3, z^3, x^2 y, x^2 z, y z^2, the optimized two-cell bound is exactly five, whereas the two-point secant cone is all of C^6. The upper bound follows from a planar separation obstruction; a Jacobian determinant of 216 proves the actual dimension. Disjoint coordinate blocks give, for every r >= 1, a degree-three monomial cone with optimized bound 5r and secant dimension 6r. The affine cones used here are not normal. No counterexample for normal toric embeddings or complete Segre-Veronese systems is claimed. This is an unrefereed preprint; no absolute priority claim is made.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "keywords": [
    "OWR-1740-002",
    "math.AG",
    "math.CO"
  ],
  "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-1740-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-1740-002/paper.pdf?v=cc54d57d9fa9",
  "doi": "10.5281/zenodo.23587509",
  "zenodo_record_url": "https://zenodo.org/records/23587509",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The unrestricted monomial-map equality in Draisma's Theorem 7 and Question 1 is false: the six cubic monomials have optimized two-cell bound 5 and actual secant dimension 6; disjoint blocks give 5r versus 6r for every r>=1. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "cc54d57d9fa95fd5e6fd3607e375fd5701c7c6abd9e037ccdcb8578a5a661373"
    },
    "source.zip": {
      "sha256": "2278863b3527c5b64bff733e64a3a0e8488975e3e8e3f94c34166e02ed9ff6e5"
    },
    "verification_report.md": {
      "sha256": "97504240029e75d10e5eb9dcf8b427cb5e53823391080d5ce23582c51e1ec0f1"
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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."
}