An explicit exponential-period representation of the V_quad connection coefficient

Correction notice (v1.4). This version withdraws the main identity and records the completed withdrawal of the journal submission. The text of version 1.3 follows this notice unchanged, except that the identity and the v1.3 list of unchanged results are marked in place. Journal submission withdrawn. The submission of this paper to Letters in Mathematical Physics (manuscript MATH-D-26-00523) has been withdrawn. The editorial office confirmed on 23 September 2026 that the withdrawal had been processed. Theorem 1.1 is false as stated. Its display (6), C = (|Γ(β)|/2π) ∫_γ e^ξ B̂(ξ) dξ = |Γ(β)| K (p. 3), does not hold, and neither does the recentred form (18) of Theorem 4.1, e^{ξ₀} ∫_γ e^ξ B̂(ξ) dξ = S with S = 2πK (p. 12). The exact relation, in modulus, is |∫_γ e^ξ B̂(ξ) dξ| = S e^{−ξ₀} · C(−1), C(−1) = 0.9308581…, where C(−1) is the Borel sum, in the direction π and at the point −1, of the formal series obtained from the V_quad asymptotic series by applying the conjugation √3 ↦ −√3 of ℚ(√3) to its coefficients. The letter C in C(−1) does not denote the constant C of Theorem 1.1. The statement on p. 3 that the thimble integral equals S e^{−ξ₀} at leading order is correct; the identity omits the factor C(−1), and the form (6) omits e^{−ξ₀} as well: (|Γ(β)|/2π) |∫_γ e^ξ B̂(ξ) dξ| = |Γ(β)| K · e^{−ξ₀} C(−1) = 0.2933… · |Γ(β)| K, not |Γ(β)| K. Correction to the v1.3 notice. The v1.3 notice’s list of unchanged results is superseded. It included Theorem 1.1 and its proof, which are withdrawn; the conditional transcendence statement, Corollary 1.4, which is obtained by interpreting (6) (p. 3) and loses its stated basis with it; and the differential-Galois results, two statements of which are corrected below. The three verifications of Section 5 are not re-examined here; their agreement does not establish (6). The values of K, S = 2πK and C = |Γ(β)| K printed on p. 12 agree in all 45 of their printed significant digits with the values computed for the superseding paper. Superseded. A new paper, which states and proves the corrected relation above, supersedes this paper’s main identity. It will be named here, with its identifier, when it has one. Two further statements. Theorem 1.2 gives Lφ the Galois group SL₂(ℂ); its argument concerns the reduced equation u″ = r u, whose group is SL₂(ℂ), while the group of Lφ itself is GL₂(ℂ), because the Wronskian of Lφ is not algebraic (it has an essential singularity at z = 0). Proposition 2.4 and Lemma 2.7 (pp. 8–9) call ξ = 0 an apparent singular point of L_V; L_V has a logarithm there, although B̂ itself is holomorphic at 0 on its principal sheet. Cluster statement (read with this notice). A dependency analysis of the 61-record deposit corpus finds this record inside a group of 18 records that cite one another in both directions across successive versions. The deposit metadata cannot establish an ordering within that group, so it cannot be determined from the record which way influence ran between its members. The group contains all three records whose headline claims are withdrawn or under review: this record and CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089) This notice therefore does not assert that the other members of that group are unaffected. Corrections issued together. This notice is issued in one batch with the corrections to the records below. Each carries its own notice, which this one does not summarise. CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635; correction notice, this batch) T2B — Two arithmetic classes of degree-(2,1) Trans-stratum continued fractions: a Birkhoff–Trjitzinsky / Gauss-continued-fraction dichotomy (concept DOI 10.5281/zenodo.19783311; description correction, this batch) EBR-Ib — The EBR Amplitude as a Connection Coefficient: Characterization and a Rigidity Dividing-Line Conjecture (concept DOI 10.5281/zenodo.20566465; erratum, this batch) EBR-I — An Edge–Borel Radius Theorem for Positive Polynomial Continued Fractions (concept DOI 10.5281/zenodo.20564079; scope correction, this batch) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089; corrected in its version 1.3, already deposited) 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. [v1.4: this list is superseded; see the correction notice above.] 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, [v1.4: false as stated; see the correction notice above.] 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.