{
  "abstract": "We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids $\\mathcal{G}_{n}$ such that the forking instability in rank $n$ is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in $\\mathcal{G}_{n}$. Our contribution is to show that this set of matroids $\\mathcal{G}_{n}$, determining based forking stability in rank $n$, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks $n > 3$: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.09069",
  "author_total": 1,
  "authors": [
    {
      "name": "Scott Mutchnik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cb2y0cpxf4gy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.09069",
  "fulltext_url": null,
  "id": "5cb2y0cpxf4gy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.09069",
  "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.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7168202127",
  "pdf_url": "https://arxiv.org/pdf/2607.09069",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cb2y0cpxf4gy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2607.09069",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.09069"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Reducing stable forking dependence to finitely many pregeometries",
  "title_text": "Reducing stable forking dependence to finitely many pregeometries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/5cb2y0cpxf4gy",
  "venue": "arXiv",
  "versions": []
}