Complex Borel-Plane Probes of the Non-Holomorphic Bird Map: Destruction, Not Displacement, of the Feigenbaum Singularity (Paper 25 in the Non-Holomorphic Fractal Series)

Paper 25 in the Non-Holomorphic Fractal Series. Papers 19–24 exhausted the positive real Borel axis for the non-holomorphic Bird map, establishing a six-ladder, two-family atlas (γ-family and conformal Padé–Borel) across Class C (C = 0.3605) and the Aether limit (α → ∞). Every calibrated probe found no Padé-stable, N-stable singularities in the Feigenbaum window [0.85, 1.15] on the positive real axis. Paper 25 asks whether the non-holomorphic perturbation g(Im z) pushed the Feigenbaum singularity off the real axis into the complex Borel plane, or destroyed it entirely. We extend the calibrated γ-family Padé–Borel engine to the complex s-plane by extracting all roots of the Padé denominator Q(s) via numpy.roots and applying complex-G1/G2 stability filters inside the neighbourhood |s − s_{F,mapped}| < 0.5. G4 synthetic calibration recovers both real and complex planted poles to machine precision. For every ladder and every (N, [L/M]) tested, no Padé-stable, N-stable complex poles appear in the Feigenbaum neighbourhood. A 200×200 heatmap of log|1/Q(s)| for the Class C orbit-multiplier confirms the Feigenbaum box is visually empty. This is Outcome (B): the Feigenbaum singularity is absent from the complex neighbourhood, not merely displaced off the real axis.