What question did this study set out to answer?

This work aims to propose a transcendence predicate based on the balanced discriminant of degree-two polynomial continued fractions.

June 1, 2026Open Access

Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions

Key Points

This work aims to propose a transcendence predicate based on the balanced discriminant of degree-two polynomial continued fractions.
The study investigates the PSLQ detectability of 30 degree-two families based on their discriminants.
Employs the Spectral classification framework and adds components to fix cross-paper inconsistencies.
Conducted empirical tests using specific recurrence-parameter channels and Borel-resummation channels.
24 families show PSLQ-detectability against a basis, suggesting a positive signal for transcendence.
6 candidates demonstrate PSLQ non-detection, indicating challenges in establishing closed forms.
The V_quad family is confirmed as a non-classical Painleve V transcendent, expanding the understanding of its structure.

Abstract

Version 1. 6 (2026-05-30) — Vquad surface-type correction, complete. The isomonodromic deformation of the Vquad family (Delta = -11) is corrected from Painleve III (D6) to a non-classical Painleve V transcendent on the Sakai surface of type D5^ (1) (W (A3^ (1) ), delta = -1/2 != 0, so not a Painleve III reduction), per the companion paper "A non-classical Painleve V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data" (Zenodo DOI 10. 5281/zenodo. 20455090). This corrects the abstract, Conjecture A part (iv), and the body section "Vquad as the Explicit Prototype", so the surface-type determination is consistent throughout. No other theorem, conjecture, or numerical result is changed: the Delta>0 / Delta0 F (2, 4) Trans-stratum families is corrected from "admit elementary closed forms" to its supported sense: PSLQ-detectability against the working bilinear-pi / special-value basis (220 digits, integer bound 10¹0). Detectability against a fixed basis is a positive heuristic signal, not an exhibited elementary closed form; the analytic class is left open. A targeted closed-form search on the cleanest non-degenerate Delta=+1 representative QL09 (bₙ = (n+1) (2n+1), aₙ = 5n) fails to exhibit an elementary closed form to >= 360 decimal places, and the negative persists under the canonical alternative numerators over the same bₙ (the obstruction is a property of the quadratic-b structure, not the numerator). No theorem, conjecture, or numerical result is retracted: Conjecture A is unchanged (its dichotomy is a statement about detection outcomes; part (i) asserts only the Delta= 13 matching digits across all six Delta<0 families. Conjecture A part (iv) is restated as channel-scoped: Vquad is the unique sporadic instance with an explicit Painleve determination in the connection-coefficient / Stokes-constant ODE channel; the channel structure of the other five families remains open. Two new open problems are stated. The 0/6 result in the L (t) and Borel channels is presented as a strengthening of the conjecture, not a refutation: it sharpens the channel-specificity of the Painleve question and motivates a forthcoming "Channel theory" outline as the structural prerequisite for the SIARC Master Conjecture.

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