{
  "abstract": "Let a n be the number of closed linear λ-terms in β-normal form with n variable occurrences. We prove\n\nis the large-order constant of the Riccati-Airy sequence K m attached to the turning point of the WKB expansion. We show that the K m are exactly rescaled coefficients of the formal logarithmic derivative of the Airy asymptotic series, and deduce κ ∞ = 1/(2π) from the explicit Airy coefficients. Hence\n\nThe proof combines four further ingredients: (i) the WKB coefficients are polynomials with nonnegative coefficients in v = (1 -4w) -1/2 and satisfy an explicit positive recurrence; (ii) the linearisation of the WKB equation has a combinatorial meaning (λ-chain terms) and is a translation in the action coordinate s, whose singularity sits at the instanton action A 0 = ln 2 -1 2 ; (iii) an exact identity shows that, after conjugation by the transported WKB prefactor µ = r/(1+r) 2 , the remaining non-linear terms are either negative or of relative size O(m -2 ); (iv) under the conjugated linear dynamics the depth profile is dominated by its top coefficient. A finite bound on an explicit ratio sequence is obtained by exact computer arithmetic (the program is given in an appendix); it also yields a n ≤ 1.14736 f ′′ n (1) for all n, with f ′′ n (1) ∼ 5 144 (n -1)!λ n .",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Olivier Bodini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5d6eqkwd6ffnv.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "5d6eqkwd6ffnv",
  "issns": [],
  "landing_url": "https://hal.science/hal-05841200v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05841200/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5d6eqkwd6ffnv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:02Z",
      "native_id": "hal-05841200",
      "source": "hal",
      "url": "https://hal.science/hal-05841200v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Asymptotics of linear normal lambda terms: WKB structure, Airy turning point and a positive recurrence",
  "title_text": "Asymptotics of linear normal lambda terms: WKB structure, Airy turning point and a positive recurrence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:02Z",
  "url": "https://research.draconitic.com/paper/5d6eqkwd6ffnv",
  "venue": "HAL",
  "versions": []
}