A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data
- 1Independent Researcher
Abstract
SUPERSEDED VERSION — DO NOT CITE THE NUMERICAL VALUES BELOW. This is version 1.0. It is superseded by version 1.2 (10.5281/zenodo.20481592). Please cite the concept DOI 10.5281/zenodo.20455089, which always resolves to the current version. Three assertions in the text below are incorrect and have been corrected in later versions: The Stokes constant is wrong. This version reports S ≈ 0.43770528. The correct value is S = 2πK = 0.45790662316901763611…. The error was an incorrect Dingle prefactor, Γ(βexp) = −6.00599, used in place of the universal resurgence factor 2π = 6.28319. The two agree to 4.4%, which is why the error was not caught sooner. The connection parameter is wrong. This version reports σconn ≈ 0.06087689. The corrected value is 0.06335272. Theorem 4.1 is not a theorem. The identification of the isomonodromic deformation as Painlevé V on the Sakai surface of type D5(1) is stated below as Theorem 4.1. It is not proved. It is restated as Conjecture 4.1 in version 1.2. The late-term amplitude K = 0.07287810255…, the Gevrey/Borel branch-point structure, the Wallis-discriminant exclusion argument, and the closed-form non-identification are unchanged and are not affected by these corrections. Notice added 2026-08-29 by the author. We study the resurgence and isomonodromic structure of a constant V_quad ≈ 1.19737… arising as the limit of a quadratic polynomial continued fraction (PCF). The convergent denominators satisfy a Wallis-type recurrence whose continuum limit yields an exact second-order ODE of the form a(x)y'' + a'(x)y' + c(x)y = 0, with a(x) = 3x²+x+1 and c(x) = −x². This exact-ODE structure forces both finite singularities to be apparent and explains, via monodromy, the failure of CM-period and hypergeometric identification strategies. The CM discriminant disc(a) = 1 − 12 = −11 that appears throughout the analysis is the discriminant of the Wallis characteristic polynomial, not a deep CM structure; this explains why exhaustive PSLQ searches against CM period bases at discriminant −11 return null by a structural argument independent of precision. The isomonodromic deformation of the underlying connection is governed by the fifth Painlevé equation with parameters α=1/6, β=γ=0, δ=−1/2 (δ≠0); its space of initial conditions is the Sakai surface D5^(1) with affine Weyl symmetry W(A3^(1)). Since δ≠0 the equation does not reduce to Painlevé III, and the constant is expected to be a generic, non-classical PV transcendent. The formal asymptotic series at infinity is Gevrey-1 divergent; its Borel transform has a branch-point singularity at ξ0=2/√3 (the Stokes gap) with branch exponent β_exp=−1/(3√3), confirmed to eight significant figures by Domb–Sykes analysis and Richardson³ extrapolation. The Stokes constant S ≈ 0.43770528… is computed to eight digits by the Dingle late-term formula. An exhaustive search over 245 variants of the Jimbo (1982) connection formula establishes that S is not a simple combination of Gamma values at the WKB parameters; its closed-form identification requires the connection parameter σ_conn, which is transcendental in the accessory parameter q=(5+i√11)/54. The structure b(x)=a'(x) identified here is a general feature of PCF-derived ODEs: for any polynomial continued fraction, the Wallis recurrence produces an exact ODE whose apparent singularity positions are the roots of the Wallis characteristic polynomial and whose CM discriminant is disc(a). ## additional_notesProvenance note. This manuscript classifies the V_quad isomonodromic deformation as Painlevé V on the Sakai surface of type D5^(1) (Theorem 4.1). This supersedes an earlier Painlevé III(D6) framing of the same constant used in companion deposits — PCF-1 (concept DOI 10.5281/zenodo.19931635) and Channel Theory (concept DOI 10.5281/zenodo.19941678) — where V_quad was described as a PIII(D6) prototype/reduction. The present PV/D5^(1) classification is the corrected determination; Remark 4.3 records why the PV→PIII confluence is not taken (delta = -1/2 != 0) and why PIII(D7) is excluded.
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BibTeX
@misc{papanokechi2026nonclassical,
title = {{A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data}},
author = {Papanokechi},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = may,
doi = {10.5281/zenodo.20455089},
url = {https://doi.org/10.5281/zenodo.20455089},
note = {CC BY},
}
RIS
TY - UNPB
TI - A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data
AU - Papanokechi
PY - 2026
DA - 2026/05/30
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20455089
UR - https://doi.org/10.5281/zenodo.20455089
AB - SUPERSEDED VERSION — DO NOT CITE THE NUMERICAL VALUES BELOW. This is version 1.0. It is superseded by version 1.2 (10.5281/zenodo.20481592). Please cite the concept DOI 10.5281/zenodo.20455089, which always resolves to the current version. Three assertions in the text below are incorrect and have been corrected in later versions: The Stokes constant is wrong. This version reports S ≈ 0.43770528. The correct value is S = 2πK = 0.45790662316901763611…. The error was an incorrect Dingle prefactor, Γ(βexp) = −6.00599, used in place of the universal resurgence factor 2π = 6.28319. The two agree to 4.4%, which is why the error was not caught sooner. The connection parameter is wrong. This version reports σconn ≈ 0.06087689. The corrected value is 0.06335272. Theorem 4.1 is not a theorem. The identification of the isomonodromic deformation as Painlevé V on the Sakai surface of type D5(1) is stated below as Theorem 4.1. It is not proved. It is restated as Conjecture 4.1 in version 1.2. The late-term amplitude K = 0.07287810255…, the Gevrey/Borel branch-point structure, the Wallis-discriminant exclusion argument, and the closed-form non-identification are unchanged and are not affected by these corrections. Notice added 2026-08-29 by the author. We study the resurgence and isomonodromic structure of a constant V_quad ≈ 1.19737… arising as the limit of a quadratic polynomial continued fraction (PCF). The convergent denominators satisfy a Wallis-type recurrence whose continuum limit yields an exact second-order ODE of the form a(x)y'' + a'(x)y' + c(x)y = 0, with a(x) = 3x²+x+1 and c(x) = −x². This exact-ODE structure forces both finite singularities to be apparent and explains, via monodromy, the failure of CM-period and hypergeometric identification strategies. The CM discriminant disc(a) = 1 − 12 = −11 that appears throughout the analysis is the discriminant of the Wallis characteristic polynomial, not a deep CM structure; this explains why exhaustive PSLQ searches against CM period bases at discriminant −11 return null by a structural argument independent of precision. The isomonodromic deformation of the underlying connection is governed by the fifth Painlevé equation with parameters α=1/6, β=γ=0, δ=−1/2 (δ≠0); its space of initial conditions is the Sakai surface D5^(1) with affine Weyl symmetry W(A3^(1)). Since δ≠0 the equation does not reduce to Painlevé III, and the constant is expected to be a generic, non-classical PV transcendent. The formal asymptotic series at infinity is Gevrey-1 divergent; its Borel transform has a branch-point singularity at ξ0=2/√3 (the Stokes gap) with branch exponent β_exp=−1/(3√3), confirmed to eight significant figures by Domb–Sykes analysis and Richardson³ extrapolation. The Stokes constant S ≈ 0.43770528… is computed to eight digits by the Dingle late-term formula. An exhaustive search over 245 variants of the Jimbo (1982) connection formula establishes that S is not a simple combination of Gamma values at the WKB parameters; its closed-form identification requires the connection parameter σ_conn, which is transcendental in the accessory parameter q=(5+i√11)/54. The structure b(x)=a'(x) identified here is a general feature of PCF-derived ODEs: for any polynomial continued fraction, the Wallis recurrence produces an exact ODE whose apparent singularity positions are the roots of the Wallis characteristic polynomial and whose CM discriminant is disc(a). ## additional_notesProvenance note. This manuscript classifies the V_quad isomonodromic deformation as Painlevé V on the Sakai surface of type D5^(1) (Theorem 4.1). This supersedes an earlier Painlevé III(D6) framing of the same constant used in companion deposits — PCF-1 (concept DOI 10.5281/zenodo.19931635) and Channel Theory (concept DOI 10.5281/zenodo.19941678) — where V_quad was described as a PIII(D6) prototype/reduction. The present PV/D5^(1) classification is the corrected determination; Remark 4.3 records why the PV→PIII confluence is not taken (delta = -1/2 != 0) and why PIII(D7) is excluded.
ER -