{
  "abstract": "We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\\arcsin x)/x$ on $(-1,1)$. A positive-kernel formula and the classical Schur algorithm give these signs, and a para-orthogonal transformation yields positivity in every degree. We also compute the positive density representing the negative of the Schur function as a Hausdorff moment generating function. After rescaling their indices to $[0,1]$, the normalized critical vectors converge weakly to the arcsine law, while Chenevier's critical scale is asymptotic to $8π/n$. At the critical boundary, a single nonzero effective integral vector is negative for every admissible test function exactly in odd degree and in degree zero. Finally, we prove an exact first-variation formula for exponential perturbations of the Legendre measure and derive its asymptotics for endpoint cusps. For the perturbation leading to Chenevier's weight, the derivative at the Legendre measure has a $(\\log n)/(π^2n^2)$ term and an explicit constant at order $n^{-2}$. The corresponding nonlinear asymptotic remains conjectural.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10328",
  "author_total": 1,
  "authors": [
    {
      "name": "Shisong Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74wx1kq2p4gyd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10328",
  "fulltext_url": null,
  "id": "74wx1kq2p4gyd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10328",
  "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.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7212193664",
  "pdf_url": "https://arxiv.org/pdf/2609.10328",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74wx1kq2p4gyd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10328",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10328"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positivity and Asymptotics for Chenevier's Orthogonal Polynomials",
  "title_text": "Positivity and Asymptotics for Chenevier's Orthogonal Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/74wx1kq2p4gyd",
  "venue": "arXiv",
  "versions": []
}