{
  "schema_version": 1,
  "problem_number": "OWR-13494-006",
  "title": "A Sharp Linear Computability Gap for Direct Sums of Binary Cubics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Over an algebraically closed field of characteristic zero, the sum F_m of m disjoint binary cubics x_i y_i^2 has Waring rank 3m, but the largest linear-computability colon bound is 2m+1. An exact inverse-system calculation covers every linear ideal and identifies the unique maximizer. In particular, xy^2+zw^2 is a complete counterexample to closure under disjoint sums of linearly computable forms, answering the question on page 503 of Oberwolfach Report 9/2015. The rank formula is known and credited. The paper includes its elementary proof and a standard-library exact checker. The latter passes 1264 quotient-length calculations on 629 distinct sampled subspaces and scope controls; the universal statement is proved analytically. General linear elements and degree-one-generated ideals are essential. English unrefereed preprint, CC BY 4.0, 21 October 2026. No absolute priority, independent peer-review or formal-verification claim. OpenAI Codex assisted with research and preparation.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AC",
    "math.AG"
  ],
  "keywords": [
    "OWR-13494-006",
    "math.AC",
    "math.AG"
  ],
  "manuscript_version_date": "2026-10-21",
  "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-13494-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-13494-006/paper.pdf?v=3a14d7db2ec2",
  "doi": "10.5281/zenodo.23586458",
  "zenodo_record_url": "https://zenodo.org/records/23586458",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Over an algebraically closed field of characteristic zero, F_m=sum_i x_i y_i^2 has Waring rank 3m and maximum general-linear colon quotient length 2m+1 over every degree-one-generated ideal. The maximizing linear generator subspace is uniquely span(X_1,...,X_m). Thus xy^2+zw^2 is not linearly computable although both disjoint binary summands are. This completely answers the source closure question negatively; rank additivity remains true. General q and generation in degree one are essential. The rank value is known; novelty and absolute priority of the computability gap are not certified. 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": "3a14d7db2ec23ba2b17ed9513cb8dc185379077e7bd8bf4c3b6325530ff932ec"
    },
    "source.zip": {
      "sha256": "e1a0d38db78f4d9fdc6441a30a7fb73010364b56e5199c70458cafc5294f3a50"
    },
    "verification_report.md": {
      "sha256": "c2059fe43c3bd18fc4a598e22ec77b1a56e96d1a38516d0242ca347c5c91a8cc"
    }
  },
  "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."
}