{
  "abstract": "We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal $G$-module $W$ with $\\mathbb D=\\operatorname{End}_G(W)\\in\\{\\mathbb R,\\mathbb C,\\mathbb H\\}$, we consider the multiplicity-two representation $W\\oplus W$ and retain the failure of the $2$-design equation $M^*M=\\frac12 I_2$ as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension $r\\geq 1$, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point $W(E_6)$-invariant spherical $4$-design in $S^{11}$ and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25032",
  "author_total": 2,
  "authors": [
    {
      "name": "Kuan-Cheng Chien"
    },
    {
      "name": "Ming-Hsuan Kang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/acey6em3qd138.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25032",
  "fulltext_url": null,
  "id": "acey6em3qd138",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25032",
  "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"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214103368",
  "pdf_url": "https://arxiv.org/pdf/2609.25032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/acey6em3qd138.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Quadratic-Defect Completions of Spherical $2$-Design Orbits",
  "title_text": "Quadratic-Defect Completions of Spherical $2$-Design Orbits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/acey6em3qd138",
  "venue": "arXiv",
  "versions": []
}