Destruction, Not Displacement, of the Feigenbaum Singularity: Feigenbaum Singularity Loss in Non-Holomorphic Bird Maps (Paper 26 in the Non-Holomorphic Fractal Series)

Paper 26 in the Non-Holomorphic Fractal Series. Papers 19–25 established a complete six-ladder negative Borel atlas for the non-holomorphic Bird map: no Padé-stable, N-stable Feigenbaum singularity exists on the positive real or complex Borel axis. Paper 25 (Outcome B) showed that the singularity is absent, not merely displaced off the real axis. Paper 26 explains why. We prove that the non-holomorphic perturbation g(Im z) ≠ 1 destroys the Feigenbaum Borel singularity via a perturbative mechanism rooted in the Generalized Decoupling Lemma: the chaotic modulation of the local multiplier K g(y_n) breaks the exact geometric recursion a_n^(0) ~ δ_F^{-n}, reducing the exponential type of the Borel transform from 1/δ_F to zero. The destruction is sharp in the numerical sense: for the Padé resolution N ≤ 80 tested here, the singularity appears only at ε = 0, and is absent for all ε > 0 down to ε = 10^{-10}. We verify this across twelve orders of magnitude in ε.