{
  "abstract": "Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) bound the irrationality measure of pi with the same shape of complex integral. This note gives one closed-form law mu(a0,a1,b) for the whole family, shows that it reproduces all three published records, proves that it is exactly scale-invariant, and gives evidence that Bai's parameter point is the global optimum of the family - a statement Bai explicitly declines to make about her own result. The admissible configurations are forced by a Machin condition: zeros at 0, +-w, +-wbar and poles at +-P give a linear form in 1 and pi if and only if P^2 - N(w) = 2P Im(w). The law depends only on the two ratios a0/a1 and b/a1, so the infinite three-parameter lattice is exactly two-dimensional. The 2-adic saving is exactly 5(2a1-b)/2 per n, in two regimes, and no further saving exists. The decay rate is governed by the min-max value of the steepest-descent method, the lowest level at which the two endpoints of the integral become connected within a sublevel set of the integrand; the value obtained from the least critical level is a lower bound for mu and serves as a screening. Among the sixteen admissible configurations with P < 100, exactly one carries a triple that yields a bound, namely Salikhov's (P = 5, w = 1+2i), and the least mu over all sixteen and over all primitive triples with a0, a1 <= 10 and b <= 22 is 7.103205334137 at (2,2,3). Within that configuration the minimum is 7.101862832356 at (1857,1857,2785), which is Bai's point, the vertex sitting exactly at Q = 2a/(3a-2b) = 3714. Each statement carries an explicit label, PROVED or VERIFIED or CONJECTURE, and the reservations of scope are stated in the note. An ancillary file re-derives every numerical claim from scratch, with no external dependency.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37459",
  "author_total": 1,
  "authors": [
    {
      "name": "David Niedbala Giraudin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ea7289acsqgb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37459",
  "fulltext_url": null,
  "id": "8ea7289acsqgb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37459",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214949193",
  "pdf_url": "https://arxiv.org/pdf/2609.37459",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ea7289acsqgb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.37459",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37459"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A closed-form law for the Salikhov-Zeilberger-Zudilin-Bai family, and computational evidence that Bai's point is optimal",
  "title_text": "A closed-form law for the Salikhov-Zeilberger-Zudilin-Bai family, and computational evidence that Bai's point is optimal",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8ea7289acsqgb",
  "venue": "arXiv",
  "versions": []
}