{
  "abstract": "We prove that no strongly regular graph with parameters $(266, 45, 0, 9)$ exists. The proof is formalized in Lean 4 and Mathlib without external infeasibility certificates or assumed classification theorems. A hypothetical graph gives a rank-$12$ integral Gram lattice with an integral centroid. A Lorentzian change of form, a marked $D_7$ gluing, and an explicit rank-six complement produce a positive-definite even unimodular lattice of rank $24$, together with the original indexed family of $220$ vectors. Harmonic theta identities and a root-isolation inequality force the root system $A_{11} \\perp D_7 \\perp E_6$. First and second moments then exclude the possible complements: the final case reduces to an impossible binary projection identity $4x + 4y - 2z = 50$. A type-$A$ subcase is closed by a separate classification-free proof of the known nonexistence of a quasi-symmetric $2$-$(56, 12, 9)$ design with intersections $0, 3$. That argument constructs a Krein graph and forces a Steiner $3$-$(12, 4, 1)$ design, contradicting its replication equation. The formal theorem depends only on the three standard Lean axioms and has also been checked independently with nanoda. The archived formalization is release v2.0.0.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08319",
  "author_total": 1,
  "authors": [
    {
      "name": "Kay Akiyama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b52atnjwz6bb0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08319",
  "fulltext_url": null,
  "id": "b52atnjwz6bb0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08319",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212016359",
  "pdf_url": "https://arxiv.org/pdf/2609.08319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b52atnjwz6bb0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08319"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nonexistence of a Strongly Regular Graph with Parameters (266,45,0,9): A Certificate-Free Lean Proof",
  "title_text": "Nonexistence of a Strongly Regular Graph with Parameters (266,45,0,9): A Certificate-Free Lean Proof",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/b52atnjwz6bb0",
  "venue": "arXiv",
  "versions": []
}