{
  "abstract": "Let $Γ$ be the Cayley graph of Thompson's group $F$ with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density $7/2$, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with \\[ \\operatorname{dens}(Γ)>3.50074529. \\] The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed $a\\ge0$, the optimal limiting density of a rule depending on the window $[-a,1]$ is exactly $7/2$. Thus arbitrary finite left context together with one right-hand category does not improve the Belk--Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of $\\mathbb Q(\\sqrt3,\\sqrt{2\\sqrt3-1})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12290",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Prellberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/69z0g3432atd2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12290",
  "fulltext_url": null,
  "id": "69z0g3432atd2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12290",
  "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"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212823255",
  "pdf_url": "https://arxiv.org/pdf/2609.12290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/69z0g3432atd2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.12290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cayley-graph density of Thompson's group F: local deletions and finite-window bounds",
  "title_text": "Cayley-graph density of Thompson's group F: local deletions and finite-window bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/69z0g3432atd2",
  "venue": "arXiv",
  "versions": []
}