{
  "abstract": "We prove that every Voronoi cell of a unit-ball packing of $\\mathbb{R}^4$ has volume at least $8$, with equality only at the $D_4$ configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most $π^2/16$. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25120",
  "author_total": 4,
  "authors": [
    {
      "name": "Deep Bhattacharjee"
    },
    {
      "name": "Ushashi Bhattacharya"
    },
    {
      "name": "Priyabrata Mandal"
    },
    {
      "name": "Shounak Bhattacharya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjtxxvvs6anwk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25120",
  "fulltext_url": null,
  "id": "cjtxxvvs6anwk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25120",
  "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.NT"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7214128867",
  "pdf_url": "https://arxiv.org/pdf/2609.25120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjtxxvvs6anwk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.25120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture",
  "title_text": "The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/cjtxxvvs6anwk",
  "venue": "arXiv",
  "versions": []
}