{
  "abstract": "Every group has a canonical coloring of its unordered pairs by inverse pairs of differences. We prove that this coloring can be coarsened to four colors without changing its color-preserving permutation group, for groups of arbitrary cardinality. Four is the least uniform bound. A related construction applies to finite homogeneous coherent configurations: if every basic relation has valency at most $d$, where $d\\geq 2$, a coarsening with at most $2d+1$ off-diagonal colors in the symmetric case, or $2d+3$ off-diagonal arc colors in general, has the original configuration as its coherent closure. Two-dimensional Weisfeiler-Leman refinement recovers that partition within $2\\lceil\\log_2 n\\rceil$ rounds on $n$ points, with the specified atomic initialization. Affine-plane schemes show that linear dependence on valency is necessary. We apply the group realization to classify groups admitting naturalizing invariant metrics and show that, for a suitable such metric, naming the identity determines the compatible multiplication uniquely. Finally, using additional Cayley-realization and finite-classification inputs and explicit certificates, we determine the least number of metric values for every group. The direct universal construction and the finite coherent-reconstruction theorem do not depend on those computations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33938",
  "author_total": 1,
  "authors": [
    {
      "name": "Alex J. Sutherland"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bj7xp4tp1yhgq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33938",
  "fulltext_url": null,
  "id": "bj7xp4tp1yhgq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33938",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214769613",
  "pdf_url": "https://arxiv.org/pdf/2609.33938",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bj7xp4tp1yhgq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.33938",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33938"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Finite-color reconstruction of groups and coherent configurations",
  "title_text": "Finite-color reconstruction of groups and coherent configurations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/bj7xp4tp1yhgq",
  "venue": "arXiv",
  "versions": []
}