{
  "abstract": "In this Letter, we develop a graph-geometric framework and prove that the pentagonal bipyramid (PBP) is the unique global minimiser of both the seven-particle Lennard-Jones energy in $\\mathbb{R}^3$ and the seven-charge Thomson energy on $S^2$, up to rigid symmetries and particle relabelling. For the Thomson problem, this strengthens a recent deterministic global-optimality certificate to numerical tolerance into an exact global-optimality and uniqueness theorem. A weak-interaction graph, defined by thresholding pair geometry, organises configuration space without imposing symmetry or discarding any pair interaction. For Lennard-Jones, pair-energy counting, graph classification, subset bounds, and force balance reduce all possible minimisers to 29 bounded domains; exact geometric certificates exclude 28, localise the survivor near the PBP, and a uniform restoring-energy estimate proves uniqueness. The Thomson proof follows the same progression: graph reduction, geometric localisation, and rigidity, but finalises through whole-domain coercivity and a global localisation theorem. The same fivefold order therefore emerges in two very different physical settings: an unconstrained cluster with a preferred bond length and purely repelling charges confined to a sphere, showing a common geometric organising principle for seven particles that is not tied to either interaction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.09925",
  "author_total": 1,
  "authors": [
    {
      "name": "Li Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hegbshnqd03d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.09925",
  "fulltext_url": null,
  "id": "7hegbshnqd03d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.09925",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.atm-clus"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.09925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hegbshnqd03d.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:23Z",
      "native_id": "2611.09925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.09925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.atm-clus",
      "subfield": "amo"
    }
  ],
  "title": "On the Seven-particle Clusters of Lennard-Jones and Thomson",
  "title_text": "On the Seven-particle Clusters of Lennard-Jones and Thomson",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:33Z",
  "url": "https://research.draconitic.com/paper/7hegbshnqd03d",
  "venue": "arXiv",
  "versions": []
}