Phase Transition and Basin Reorganization in the Newton Dynamics of C_alpha(z) = z - exp(-alpha/z)

Paper 20 in The Geometry of the Critical Line programme. We study the Newton dynamics associated with the transcendental family C_α(z) = z − exp(−α/z), α > 0, with particular attention to how the geometry of Newton basins changes as the parameter α varies. Using high-resolution numerical sweeps, we identify two distinct structural events: a primary bifurcation at α = 1/e consistent with the Lambert W branch-point transition, and a reproducible global basin reorganization near α = 1/2 characterized by an entropy valley followed by steep complexity increase, confirmed across six independent metrics, two spatial scales, and multiple robustness controls. A direct mechanism test based on the critical orbit z₀ = α/2…