{
  "abstract": "We establish a complete information-theoretic framework for the minimum-energy configurations of the two-dimensional Thomson problem on a hard-wall disk, based on the Shannon entropy of the Voronoi topological charge distribution. Two exact identities are proven and verified to machine precision for all studied systems of size $N=12-10^5$. First, an entropy balance law decomposes the total entropy into bulk, boundary, and mixing contributions without approximation. Second, a topological charge sum rule fixes the total charge at twice the number of boundary particles plus six; together with the disjointness of bulk and boundary coordination sets $-$ justified analytically via the Delaunay empty-circumcircle criterion and elastic disclination splitting $-$ it relates the three- and four-coordinated boundary populations directly to the bulk charge. Applying Euler's theorem to the outermost Delaunay annulus yields an exact shell-mismatch identity: the four-coordinated boundary population equals the adjacent interior shell population, whereas the three-coordinated population equals their step mismatch, directly detecting third-shell nucleation at $N=30$. Furthermore, we prove that the boundary entropy is bounded by $\\ln2$ and that its saturation is equivalent to the bulk charge per boundary particle approaching $-1/2$, a regime restricted self-consistently to boundaries of at least $12$ particles. Numerically, we locate bulk nucleation at $N=56$, maximal global disorder at $N=146$, and boundary saturation above $N\\sim10^3$, with a direct extension to spherical geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15428",
  "author_total": 2,
  "authors": [
    {
      "name": "Georgiy K. Lavrov"
    },
    {
      "name": "Eduard G. Nikonov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30qcscqmx3dak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15428",
  "fulltext_url": null,
  "id": "30qcscqmx3dak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15428",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ],
    [
      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7213348259",
  "pdf_url": "https://arxiv.org/pdf/2609.15428",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30qcscqmx3dak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2609.15428",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15428"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation",
  "title_text": "Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:35Z",
  "url": "https://research.draconitic.com/paper/30qcscqmx3dak",
  "venue": "arXiv",
  "versions": []
}