Exponential Type of Bird-Map Borel Coefficients and Analyticity of the Borel Transform (Paper 29 in the Non-Holomorphic Fractal Series)

Paper 29 in the Non-Holomorphic Fractal Series. Using the sharp Hartman–Wintner Law of the Iterated Logarithm constants C*_h established in Paper 28 for the autonomous Bird map orbit, we prove that the orbit-multiplier Borel coefficients a_n(ε) have finite exponential type in the Bird-map regime (K, C_im) = (0.5, 0.3605). For each ε ∈ (0, 1] there exists ρ(ε) > 0 such that lim sup |a_n(ε)|^{1/n} ≤ ρ(ε). Numerical calibration yields ρ(ε) ≈ 0.0942, which satisfies ρ(ε) < 1/δ_F. The corresponding γ-family Borel transform B(s) is therefore analytic in the half-plane ℜ(s) < 1/ρ(ε), ruling out any Borel singularity at s_F = 1/δ_F ≈ 0.2142 and upgrading Outcome B of Paper 26 from Padé-based evidence to a direct analyticity statement. Calibrated on the six-ladder atlas of Papers 19–26 at Padé resolution N_s = 80 with [9/1] degree, all roots lie at unmapped distance ≈ 10⁵ from s_F, consistent with the analytic disk.