{
  "abstract": "A Leech tree of order $n$ is a tree with positive integral edge weights whose $n(n-1)/2$ pairwise weighted distances are precisely $1,2,\\ldots,n(n-1)/2$. This paper gives a computer-assisted proof that no Leech tree of order $18$ exists. The argument has three layers. First, a development in Lean 4 verifies the structural facts used in the paper. These facts reduce every putative example to one of eight local configurations and justify several necessary conditions. Second, conventional mathematical arguments prove a component-pair whole-block exact-cover condition and the completeness of a recursive search. Third, exhaustive computations close all eight configurations. The computation records exact coverage, source and input hashes, terminal receipts, and checked exact-zero results. The structural layer is kernel-checked, but the search program, its execution, and the certificate checker have not been formalized in Lean. The result is therefore a computer-assisted proof, not an end-to-end Lean proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20492",
  "author_total": 1,
  "authors": [
    {
      "name": "Maseeh Ghodsi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/402tzmvmct9aa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20492",
  "fulltext_url": null,
  "id": "402tzmvmct9aa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20492",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213581353",
  "pdf_url": "https://arxiv.org/pdf/2609.20492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/402tzmvmct9aa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nonexistence of a Leech Tree of Order 18: A Computer-Assisted Proof",
  "title_text": "Nonexistence of a Leech Tree of Order 18: A Computer-Assisted Proof",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/402tzmvmct9aa",
  "venue": "arXiv",
  "versions": []
}