{
  "abstract": "We prove three results about Runge--Kutta methods with non-negative coefficients. First, every $s$-stage implicit Runge--Kutta method of order at least two has SSP coefficient at most $2s$; the bound is sharp and is attained by a known diagonally implicit method, so full implicitness gives no advantage over DIRK at second order. This extends the second-order DIRK result of Kocsis \\& Németh to full coefficient matrices and settles a conjecture of Ketcheson, Macdonald & Gottlieb. These proofs were found with the assistance of AI tools, which uncovered an interesting relationship between the Butcher matrices for SSP methods and phase-type distributions, and led to the proof of the second result: a bound on the SSP coefficient of $s$-stage implicit Runge--Kutta method of order at least three. The third result in this work is a proof that a third-order Runge--Kutta method with entrywise non-negative $A$ and $b$ cannot be algebraically stable. This algebraically intensive proof was also AI-assisted. This paper describes the iterative process of AI-assisted results, and presents an approach that centers human understanding and creativity in the AI-assisted mathematical endeavor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07981",
  "author_total": 6,
  "authors": [
    {
      "name": "Sigal Gottlieb"
    },
    {
      "name": "Zachary Grant"
    },
    {
      "name": "Lior Horesh"
    },
    {
      "name": "Lajos Lóczi"
    },
    {
      "name": "Avraham Shinnar"
    },
    {
      "name": "Barry Trager"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0v0sqpc4n11rr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07981",
  "fulltext_url": null,
  "id": "0v0sqpc4n11rr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07981",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0v0sqpc4n11rr.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs",
  "title_text": "A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/0v0sqpc4n11rr",
  "venue": "arXiv",
  "versions": []
}