{
  "abstract": "Let \\(N_t\\) denote the least number of nodes in a positive cubature formula of degree \\(t\\) on \\(S^2\\). We prove that a formula of degree \\(2m+1\\) cannot have exactly \\((m+1)(m+2)+1\\) nodes for any \\(m\\ge2\\). Excluding equality in the Fisher bound then gives \\[ N_{2m+1}\\ge (m+1)(m+2)+2 \\] for \\(m\\ge3\\), and known constructions yield the exact values \\(N_7=22\\) and \\(N_9=32\\). For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient \\(13/8\\), parity-dependent linear terms, and an \\(O(m^{2/3})\\) remainder. We determine the sharp constant \\(49\\sqrt[3]{3}/72\\) for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP--Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40210",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhuo Cheng"
    },
    {
      "name": "Deyu Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1z4xx4c574wrp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40210",
  "fulltext_url": null,
  "id": "1z4xx4c574wrp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40210",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7215041941",
  "pdf_url": "https://arxiv.org/pdf/2609.40210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1z4xx4c574wrp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive cubature on $S^2$:low-degree rigidity and uniform bounds",
  "title_text": "Positive cubature on S^2:low-degree rigidity and uniform bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1z4xx4c574wrp",
  "venue": "arXiv",
  "versions": []
}