{
  "abstract": "Gardam's counterexample to the Kaplansky unit conjecture is a unit of $\\mathbb{F}_2[P]$, $P$ the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over $\\mathbb{F}_2$ the unit equation is a parity condition on a pair of supports, searchable over word balls. We upgrade two statements to machine-checked certificates: no support pair of sizes $\\ge2$ lies in the radius-$3$ ball (a DRAT proof of a ball form of the Craven--Pappas theorem), and every nontrivial unit with supports in the radius-$4$ ball has total support at least $42$, confirming Gardam's expectation in ball-limited form. The relatives $H_4=F(3,4)$ and the Nielsen--Soelberg groups are swept with certificates; $G_3$ is Gardam's amalgam $S$, whose unit we localize into its radius-$4$ ball. $H_4$ resists the twisted-unitary ansatz through radius $6$ for every non-identity dihedral twist. An effective localization principle makes the least support of a nontrivial unit of $\\mathbb{F}_2[P]$ computable in principle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17559",
  "author_total": 1,
  "authors": [
    {
      "name": "Moe Tabei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/989zs7c99vb55.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17559",
  "fulltext_url": null,
  "id": "989zs7c99vb55",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17559",
  "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.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7213529633",
  "pdf_url": "https://arxiv.org/pdf/2609.17559",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/989zs7c99vb55.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.17559",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17559"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives",
  "title_text": "Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/989zs7c99vb55",
  "venue": "arXiv",
  "versions": []
}