Structural Limits of Conformal Padé–Borel on the Class C Orbit Multiplier: A Resolution Ceiling for the Conformal Atlas of the Non-Holomorphic Bird Map (Paper 24 in the Non-Holomorphic Fractal Series)

Paper 24 in the Non-Holomorphic Fractal Series. Paper 23 introduced a conformal Padé–Borel variant φ(s) = s/(1 + s/s_c) and deployed it across the full atlas. Five of six ladders passed G4 calibration; the lone exception was the Class C orbit multiplier. Paper 24 resolves this asterisk by constructing a genuine Taylor-coefficient generator for the Class C orbit multiplier at C = 0.3605, extending the conformal Borel engine to these true coefficients, and testing both the standard map (s_c = 1.0) and a Feigenbaum-centred map (s_c = s_F). For all tested truncation orders and conformal centres the planted G4 pole cannot be recovered: Padé systems in the conformal variable are singular or blind, while the underlying real data remain Feigenbaum-negative. We trace this failure to the intrinsic growth rate of the Class C orbit coefficients, which compresses the conformal re-expansion into a narrow interval where low-order Padé approximants are structurally ill-conditioned. The conformal Padé–Borel family has a provable resolution ceiling on this observable at accessible orders (N ≤ 80): the negative Feigenbaum verdict is robust, but the Class C orbit multiplier is conformal-blind at these orders.