An explicit exponential-period representation of the V_quad connection coefficient

Abstract

Correction notice (v1.3). This version withdraws a claim. Through v1.2 the Discussion stated that this paper “completes part (ii)(a) for d=2” of the master conjecture of the Sakai stratification (10.5281/zenodo.20694840). Part (ii)(a) concerns the Edge–Borel coefficient, a different object from the connection coefficient C studied here; the Sakai stratification records the distinction in its Observation on p. 4. That sentence is replaced by one stating what Theorem 1.1 establishes, and part (ii)(a) is recorded as open. This is a withdrawal, not an additive clarification. Surface type. The previous version described C as “the rank-one standalone closure on the Sakai surface D₅⁽¹⁾ (Painlevé V)” without qualification. That classification is Conjecture 4.1 of the companion paper (10.5281/zenodo.20455089, read at v1.2), and this version carries it as conjectural throughout. Build script. The bundled build.py carried a reproducibility target naming the v1.0 PDF, which it retained through v1.2. A reproduction run therefore compared a freshly built PDF against superseded bytes. The target now names this version’s PDF. Unchanged. Theorem 1.1 and its proof; the 46-digit agreement of the three independent verifications; every numerical value; the differential-Galois and resurgence results; and the conditional transcendence statement. No computation is affected. Sequencing. The same correction will be carried into the journal manuscript at revision. A reader arriving here from the manuscript’s data-availability statement may, until that revision appears, find a manuscript whose corresponding sentences have not yet been corrected; this record is the corrected text. We prove that the connection coefficient C of the V_quad polynomial continued fraction—the rank-one standalone closure, whose Sakai surface type is conjectured to be D₅⁽¹⁾ (Painlevé V) (companion paper, 10.5281/zenodo.20455089, read at v1.2)—admits an explicit exponential-period representation C=(|Γ(β)|/2π)∫_γ e^ξ B̂(ξ) dξ=|Γ(β)| K, where B̂ is the Borel transform of the V_quad asymptotic series, holonomic of order 4 (the Borel–Laplace dual of the order-2 operator annihilating the series) with coefficients in the real quadratic field ℚ(√3); γ is an explicit Hankel rapid-decay cycle on (−∞,−2/√3]; and β=−1/(3√3). The identity is verified by three structurally independent methods—differential-equation/operator duality, Borel–Laplace contour deformation, and Stokes-data—agreeing to 46 digits. The order-2 operator annihilating the asymptotic series has differential Galois group SL₂(ℂ) by Kovacic's algorithm. As a by-product, holonomicity of the Borel transform forces a finite resurgent structure. Finally, conditional on the Fresán–Jossen period conjecture for exponential motives and on a stated motivic-comparison hypothesis, C is transcendental over ℚ̄. This record accompanies the VQUAD-PERIODREP reproducibility bundle (verification scripts, certificates, and a byte-reproducible build), which reproduces every numerical claim above, including the 46-digit agreement. The work was produced under the SIARC governance methodology; per-stage provenance and AEAL-level claim ledgers are linked from the bundle's SIARC_PROVENANCE.md. The identity continues the V_quad companion paper (concept DOI 10.5281/zenodo.20455089) within the Sakai Stratification of PCF Transcendence program. Changes in version 1.1 (2026-07-17) — editorial/metadata corrections only. The mathematical content is unchanged from v1.0: all definitions, theorems, proofs, constants, numerical values (46-digit agreement) and the bibliography are identical. Changes in this version: 1. Fixed a duplicated "References" heading in the bibliography (removed a redundant manual \section*{References} that duplicated the heading generated by the thebibliography environment). 2. Added an AI-use disclosure ("Declaration on the use of AI tools") documenting that AI assistants (large language models) were used in the research and manuscript preparation under the author's governance framework, with all mathematical claims independently verified by the exact-computation certificates in the reproducibility appendix; the author takes full responsibility for the content. (Added to align with publisher AI-use policy.) 3. Added MSC 2020 classification and keywords to the front matter — Primary 34M55; Secondary 34M40, 34M56, 34M30, 33E17, 11J81; keywords: Painlevé V, connection coefficient, exponential periods, Stokes constants, polynomial continued fractions, isomonodromic deformation. Changes in version 1.2 (2026-07-17) — editorial/compliance additions only. The mathematical content is unchanged from v1.0/v1.1. This version adds three standard declarations required by the target journal, before the reference list: 1. A Conflict-of-interest statement ("On behalf of all authors, the corresponding author states that there is no conflict of interest"). 2. A Funding statement ("The author received no funding for this work"). 3. A Data-availability statement pointing to this Zenodo reproducibility deposit (concept DOI 10.5281/zenodo.20719042, which resolves to the latest version). No results, claims, or numerical values were altered.