Sharp Laws of the Iterated Logarithm for the Bird-Map Orbit (Paper 28 in the Non-Holomorphic Fractal Series)

Paper 28 in the Non-Holomorphic Fractal Series. We establish sharp Hartman–Wintner Laws of the Iterated Logarithm for the autonomous y-orbit of the non-holomorphic Bird map under bounded observables, combining exponential decay of correlations with ASIP results of Gouëzel, Melbourne–Nicol, and Cuny–Merlevède–Peligrad. Paper 27 used a conservative universal gate C = 3.0 for the LIL bound on ergodic sums. Paper 28 identifies the sharp Hartman–Wintner constants C*_h = √(2σ²_h), where σ²_h is the asymptotic variance from the CLT/ASIP. For the Bird map parameter regime (K = 0.5, C_im = 0.3605, ε = 1), the calibrated constants are C*_tanh ≈ 3.84 × 10⁻³ and C*_sin ≈ 8.3 × 10⁻⁵, yielding sharp constants approximately 780× and 36000× smaller than the conservative gate C = 3.0 used in Paper 27. Exponential mixing is confirmed numerically with rates θ ≈ 0.516 for h = tanh and θ ≈ 0.37 for h = sin.