{
  "abstract": "Csernák and Soukup introduced the stacking number of a graph: the least integer $t\\ge2$ such that every configuration of $t$ pebbles can be transformed by pebbling moves into a configuration supported on one vertex. They conjectured an explicit formula for the stacking number of a tree in terms of the degrees of its vertices and their distances from a chosen root. We prove this conjecture for every finite tree with at least two vertices. We also show that whether a configuration on a tree can be stacked at a given vertex is decided exactly by the sign of an integer score built from recursive branch messages; consequently, stackability can be tested with a linear number of arithmetic operations. The proof combines this criterion with an explicit zero-score obstruction and a weighted cancellation argument for arbitrary nonstackable configurations. The main theorem and the stackability criterion are formalized in Lean 4, and the main theorem has passed Palomar mechanical verification; it is publicly registered as PALOMAR-2026-09-25-000010.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31811",
  "author_total": 1,
  "authors": [
    {
      "name": "John Fairfax-Ball"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5wq664bje9xnq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31811",
  "fulltext_url": null,
  "id": "5wq664bje9xnq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31811",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214715641",
  "pdf_url": "https://arxiv.org/pdf/2609.31811",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5wq664bje9xnq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.31811",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31811"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The stacking number of a tree",
  "title_text": "The stacking number of a tree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/5wq664bje9xnq",
  "venue": "arXiv",
  "versions": []
}