{
  "abstract": "We prove that every equal-weight spherical $4$-design on $\\mathbb{S}^2$ has at least twelve points. Since the regular icosahedron is a spherical $5$-design, this determines the exact minimum$$N_4(\\mathbb{S}^2)=12;$$equivalently, no such design has $9$, $10$ or $11$ points.The proof is part of a finite-defect theory. If a spherical $2m$-design has corank $c=N-\\dim P_m$, its Naimark complement consists of unit vectors $u_x\\in\\mathbb{S}^{c-1}$ forming a spherical $2$-design and satisfying the exact coupling$$u_x\\cdot u_y=-\\frac{K_m^{(d)}(x\\cdot y)}{c}\\qquad(x\\ne y).$$This gives a pairwise kernel bound, an antipodal lower bound, Cayley-Bacharach information, and uniform lower bounds for multiplicative-relation spaces. In corank one the design splits into two equal spherical $m$-designs. We derive a residue formula for its signed Schoenberg coefficients; the coefficient of degree $m+3$ is negative exactly when $3\\le d\\le m+1$, excluding corank one throughout that range. At strengths four and six the only examples in any dimension are the regular hexagon and octagon, respectively.In corank two the complement is a circle Gale frame. Multiplication by its phase forces at least $d-1$ linear-quadratic aliases and yields exact norm and socle identities in every dimension. For eleven nodes on $\\mathbb{S}^2$, two aliases produce a real harmonic cubic and a Hermitian quartic matrix. A matrix-valued Cayley-Bacharach argument eliminates the generic branch; the exceptional branch reduces to a Pauli normal form and contradicts the second moments. In dimensions $d\\ge4$ the corank-two problem remains open; we identify a forced quadratic socle as the obstruction to extending the present argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31689",
  "author_total": 2,
  "authors": [
    {
      "name": "Shalender Singh"
    },
    {
      "name": "Vishnupriya Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h6myab72rw12.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31689",
  "fulltext_url": null,
  "id": "6h6myab72rw12",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31689",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214803281",
  "pdf_url": "https://arxiv.org/pdf/2609.31689",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h6myab72rw12.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31689",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31689"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere",
  "title_text": "Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/6h6myab72rw12",
  "venue": "arXiv",
  "versions": []
}