A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data

We study the resurgence and isomonodromic structure of a constant Vquad ≈ 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); we identify, conjecturally, its space of initial conditions as 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 βexp=-1/(3√3), confirmed to eight significant figures by Domb–Sykes analysis and Richardson³ extrapolation. The Stokes constant S = 2πK = 0.45790662316901763611… is computed to 38 digits by the Dingle late-term formula in the canonical convention (Borel transform with Γ(n); 2πi discontinuity factor; full Stokes jump), the amplitude→Stokes factor 2π being calibrated against Euler's series (|S|=2π ⇒ K=1). 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 σconn, 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). Version 1.1 (Stokes-constant numeric correction). The Stokes constant is corrected from the v1.0 value S ≈ 0.43770528 to S = 2πK = 0.45790662316901763611… (38-digit certified). The v1.0 value used an incorrect Dingle prefactor Γ(βexp) = −6.00599 in place of the universal resurgence factor 2π = 6.28319 (the two agree to 4.4%, which camouflaged the error); the late-term amplitude K = 0.07287810255… is unchanged and triple-confirmed (Neville–Richardson, Berry–Howls, Domb–Sykes), calibrated against Euler's series. The conjectural connection parameter σconn moves correspondingly from 0.06087689 to 0.06335272. The Painlevé V / Sakai-surface-D5(1) classification (Conjecture 4.1), the Gevrey/Borel branch-point structure, the Wallis-discriminant exclusion argument, and the closed-form non-identification (re-verified against S = 2πK) are all unchanged. This version supersedes only the reported value of S and the one paragraph stating it. Version 1.2 (classification demoted to conjecture; δ normalization made explicit). Two editorial corrections, with no change to mathematical substance. (1) The Painlevé V / Sakai-surface-D5(1) classification, formerly stated as a theorem, is restated as a labeled Conjecture (Conjecture 4.1). The parameters α=1/6, β=0, γ=0 are derived from the WKB and apparent-singularity data; the surface-type determination D5(1), however, rests on a match against the Jimbo–Miwa–Ueno classification table and is not yet corroborated by an independent Lax-pair or Hamiltonian re-derivation. Pending that re-derivation, the classification is stated as a well-supported conjecture rather than a theorem; only the epistemic label changes, the supporting analysis is unchanged. (2) The δ-parameter line is corrected for internal consistency: the relation δ = −½σ+² applied to the raw WKB exponent σ+ = 1/√3 gives δraw = −1/6, and the canonical value δ = −1/2 (DLMF §32.2; Gromak–Laine–Shimomura) follows by the standard rescaling z ↦ √3·z, under which δ ↦ c²δ (c=√3) while α, β, γ=0 stay invariant. The value δ ≠ 0 holds in every scale, so the PV (non-PIII) determination, the surface type, and all numerical results are unchanged.