{
  "abstract": "Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18346",
  "author_total": 1,
  "authors": [
    {
      "name": "Moe Tabei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8k11rwfvaj2mx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18346",
  "fulltext_url": null,
  "id": "8k11rwfvaj2mx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18346",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7170034973",
  "pdf_url": "https://arxiv.org/pdf/2607.18346",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8k11rwfvaj2mx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.18346",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18346"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate",
  "title_text": "Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8k11rwfvaj2mx",
  "venue": "arXiv",
  "versions": []
}