{
  "schema_version": 1,
  "problem_number": "OWR-15957-003",
  "title": "The Maximal Finite CP-Plus-Rank Equals the Maximal CP-Rank",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The cp-plus-rank is the smallest number of columns in a strictly positive Gram factor. We prove that, in every order n, its largest finite value equals the maximal cp-rank p_n. A strictly positive factor stays positive under a sufficiently small contraction along the Perron eigendirection. Together with the positive Perron lift of Bomze, Dickinson and Still, this realizes every finite cp-plus-rank as an ordinary cp-rank in the same order and matrix rank, including singular matrices. This determines the relation between the two maxima, not a numerical formula for p_n or the maximum pointwise gap. The AI-assisted preprint is self-audited and unrefereed; no absolute priority, independent review or formal verification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OC",
    "math.RA"
  ],
  "keywords": [
    "completely positive matrix",
    "cp-rank",
    "cp-plus-rank",
    "Perron-Frobenius",
    "positive Gram factor"
  ],
  "manuscript_version_date": "2026-10-13",
  "publication_date": "2026-10-13",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-13",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15957-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-15957-003/paper.pdf?v=d727573c2b43",
  "doi": "10.5281/zenodo.23350364",
  "zenodo_record_url": "https://zenodo.org/records/23350364",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem on the equality of the two maxima, including finite cp-plus-rank on singular matrices. The source's separate maximum pointwise-gap question remains unresolved. Bomze-Dickinson-Still's lift and interior theorem are credited. No numerical formula for general p_n or certified novelty is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "d727573c2b43edc1acba59007e172c6745a248c6466769d21b205d96b0a1657e"
    },
    "source.zip": {
      "sha256": "f21945c142468e806b952350675f2f478c3ae3ac318e6e9ce0f87b2ec44ad5d0"
    },
    "verification_report.md": {
      "sha256": "facc4f261bcc6899cc04da36763583b6a70902be73e57c1256854dfcb5e33eee"
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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."
}