{
  "abstract": "Let $S_{26}$ denote the supremum of the sum of the radii of 26 pairwise interior-disjoint closed disks of positive radius contained in the unit square. We prove $2.635983084917607783\\ldots \\le S_{26}\\le \\sqrt{\\frac{2577}{2500}+\\frac{11}{\\sqrt3}}=2.716919756099889566\\ldots$. The lower bound is accompanied by an exact rational feasibility certificate. For the upper bound, we assign the disks their clipped power cells, convert cell area into a reciprocal exterior-angle cost, and combine this local inequality with Euler side counts. The extremal case for the resulting relaxation has four boundary transitions and a four-cell boundary cycle; its continuous angle allocation is reduced to a concave one-variable problem. All finite numerical inequalities are rigorously enclosed by a reproducible certificate. The argument does not determine $S_{26}$ exactly, but gives a nontrivial global upper bound for a benchmark whose best-known packing is asymmetric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13962",
  "author_total": 1,
  "authors": [
    {
      "name": "Ayush Agarwal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rracqnxcjfgt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13962",
  "fulltext_url": null,
  "id": "1rracqnxcjfgt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13962",
  "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.MG"
    ],
    [
      "openalex",
      "2209|Optimization and Packing Problems"
    ]
  ],
  "openalex": "W7221789855",
  "pdf_url": "https://arxiv.org/pdf/2610.13962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rracqnxcjfgt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.13962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square",
  "title_text": "A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1rracqnxcjfgt",
  "venue": "arXiv",
  "versions": []
}