Exponential Type and the Law of the Iterated Logarithm for Non-Holomorphic Bird Map Borel Coefficients (Paper 27 in the Non-Holomorphic Fractal Series)

Paper 27 in the Non-Holomorphic Fractal Series. Papers 19–26 established that the Feigenbaum singularity is destroyed, not displaced, in the Gamma–Borel atlas of non-holomorphic Bird maps: for any perturbation strength ε ∈ (0, 1] (verified numerically down to ε = 10^{−10}) and any bounded observable h ∈ {exp(y)−1, tanh(y), sin(y)}, no Padé-stable pole exists near 1/δ_F ≈ 0.2142 on the real Borel axis (Outcome B). Paper 27 supplies the underlying dynamical mechanism. The orbit-multiplier coefficients {a_n(ε)} satisfy a Law of the Iterated Logarithm (LIL) for the ergodic sum φ_n = ∑ h(y_k), and their exponential type lim sup |a_n|^{1/n} = K/σ ≈ 0.0942 is strictly less than 1/δ_F ≈ 0.2142. Since the Gamma–Borel transform therefore converges in a disk |s| < σ/K ≈ 10.53 that strictly contains 1/δ_F, no singularity of the transform can exist at that location: the Feigenbaum singularity is constructively destroyed. The distance floor d_min(ε, h) ≥ 73.34 is confirmed numerically for all three observable families and satisfies d(ε) = 73.47 − 0.122ε + O(ε²) with R² = 0.998.