{
  "abstract": "We disprove Kusner's 1983 conjecture that every equilateral set in $\\ell_p^n$ with $2<p<\\infty$ has at most $n+1$ points: there exist $58$ points in $\\mathbb{R}^{56}$ whose pairwise $\\ell_5$ distances are all equal, so the maximum equilateral-set size satisfies $e(\\ell_5^{56})\\ge58>57$. This is the first equilateral set of more than $n+1$ points in $\\ell_p^n$ for any finite $p\\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\\le p\\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14013",
  "author_total": 1,
  "authors": [
    {
      "name": "Logan R. Chalmers"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30k53gqzpm2k4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14013",
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  "id": "30k53gqzpm2k4",
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  "landing_url": "https://arxiv.org/abs/2608.14013",
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  "pdf_url": "https://arxiv.org/pdf/2608.14013",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30k53gqzpm2k4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.14013",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14013"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
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    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A counterexample to Kusner's conjecture on equilateral sets",
  "title_text": "A counterexample to Kusner's conjecture on equilateral sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/30k53gqzpm2k4",
  "venue": "arXiv",
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}