{
  "abstract": "We study probability measures minimizing the energy with kernel $|x\\cdot y|^p$ on the real unit sphere. For every positive rational $p=a/b$, each minimizer admits a finitely supported minimizing replacement. If $a/b$ is reduced and $b\\ge d$, every minimizer on $\\mathbb S^{d-1}$ is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32973",
  "author_total": 1,
  "authors": [
    {
      "name": "Josiah Park"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fcx6d2hqxtvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32973",
  "fulltext_url": null,
  "id": "4fcx6d2hqxtvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32973",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7214720085",
  "pdf_url": "https://arxiv.org/pdf/2609.32973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fcx6d2hqxtvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Discreteness, finite rank, and stability of p-frame energy minimizers",
  "title_text": "Discreteness, finite rank, and stability of p-frame energy minimizers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4fcx6d2hqxtvt",
  "venue": "arXiv",
  "versions": []
}