{
  "schema_version": 1,
  "problem_number": "OWR-1587-003",
  "title": "The Dimension Bound for Graph Algebraic Immunity Is Attainable",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let the graph algebraic immunity of a Boolean map f:F_2^n -> F_2^m be the least degree of a nonzero reduced Boolean polynomial vanishing on its graph. We prove that the dimension bound, the least d with sum_{j=0}^d binomial(n+m,j)>2^n, is attained for every n,m>=1. Translation invariance gives a rank-density inequality for every subset of the Boolean cube. Rado's classical independent-representative theorem, applied after adjoining dummy vectors, yields a basis using distinct input fibres. Completing those points to a graph proves the result. The proof is self-contained and answers the unrestricted existence question in Oberwolfach Report 31/2007. No balancing, permutation, or additional cryptographic property is imposed. Exact finite certificates accompany the argument. This is an unrefereed preprint using classical matroid tools, with no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.IT",
    "cs.CR"
  ],
  "keywords": [
    "OWR-1587-003",
    "math.CO",
    "math.IT",
    "cs.CR"
  ],
  "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-1587-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-1587-003/paper.pdf?v=f61cba6c4c46",
  "doi": "10.5281/zenodo.23587039",
  "zenodo_record_url": "https://zenodo.org/records/23587039",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every n,m>=1 there exists an arbitrary Boolean map F_2^n to F_2^m whose graph algebraic immunity is the least d for which sum_{j<=d} binomial(n+m,j)>2^n. This proves the entire frozen 1<=m<=n existence conjecture. Balance, bijectivity and other cryptographic criteria are not imposed. 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": "f61cba6c4c46dee54e7ba7eb404a475255729bb5b447597075417025a1feff06"
    },
    "source.zip": {
      "sha256": "83cba729d91bdef94b9c2aa93023da72608c95a2aeedb4a1031758f173c0759a"
    },
    "verification_report.md": {
      "sha256": "c621ce308e8aee5e773a41c0a07ebc3b64058d511f2abab00d87edad913203da"
    }
  },
  "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."
}