{
  "abstract": "In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel $\\mathcal{F}(Δ,z) = {}_2F_1(Δ,Δ;2Δ;z)$ is proved to be strictly totally positive of order infinity for $Δ> 0$ and $z \\in (0,1)$. Second, the sharp lower $Δ$-parameter threshold for the associated one-dimensional $SL(2,\\mathbb{R})$ conformal block $G_Δ(z) = z^Δ\\mathcal{F}(Δ,z)$ is shown to be $1/2$: the conformal-block kernel $G_Δ(z)$ is strictly totally positive of order infinity for $Δ\\geq 1/2$. For every $τ\\in (0,1/2)$, there exists a strictly negative odd-order minor of $G_Δ(z)$ such that all its $Δ$-values are in $(τ,1/2)$ and all its $z$-values can be chosen arbitrarily close to one. Consequently, no restriction $z > z_0$ with $z_0 < 1$ can restore total positivity of order infinity over all $Δ> 0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21210",
  "author_total": 2,
  "authors": [
    {
      "name": "Frédéric Ouimet"
    },
    {
      "name": "Donald Richards"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6x0f806svp0ft.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21210",
  "fulltext_url": null,
  "id": "6x0f806svp0ft",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21210",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.21210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6x0f806svp0ft.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.21210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional $SL(2,\\mathbb{R})$ Conformal Block",
  "title_text": "Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional SL(2,ℝ) Conformal Block",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/6x0f806svp0ft",
  "venue": "arXiv",
  "versions": []
}