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.