{
  "abstract": "Let $X=\\{x_1,\\ldots,x_N\\}\\subset\\Sph^{d-1}$ be a spherical $t$-design, $t\\ge2$, and let $P_X\\subset\\R^d$ be its Minkowski polytope. If $h_i=h_{P_X}(x_i)$, then \\[ P_X^\\circ=\\conv\\{x_i/h_i:1\\le i\\le N\\}, \\] so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of $P_X$. Our main results concern self-polarity. If \\[ P_X^\\circ=cUP_X,\\qquad U\\in O(d), \\] we obtain the structured slack factorization \\[ A=c\\,hh^T-X^TU^TX,\\qquad \\rank A=d+1, \\] whose zero pattern records the facet--vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio $r/R$. Combining this with the one-dimensional moment problem underlying the Fazekas--Levenshtein covering bound, we prove \\[ \\frac rR\\geη_{t,d}, \\] with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss--Radau quadrature rule; in particular, the quantities $Nλ_k$ become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if \\[ δ=\\frac rR-η_{t,d} \\] is small, then each row is $O_{d,t}(\\sqrtδ)$-close in $W_1$ to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02439",
  "author_total": 1,
  "authors": [
    {
      "name": "Congpei An"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tbhh4kdgnmng.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02439",
  "fulltext_url": null,
  "id": "9tbhh4kdgnmng",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02439",
  "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.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7207595512",
  "pdf_url": "https://arxiv.org/pdf/2609.02439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tbhh4kdgnmng.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure",
  "title_text": "Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9tbhh4kdgnmng",
  "venue": "arXiv",
  "versions": []
}