An explicit exponential-period representation of the V_quad connection coefficient

We prove that the connection coefficient C of the V_quad polynomial continued fraction—the rank-one standalone closure on the Sakai surface D₅⁽¹⁾ (Painlevé V)—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.