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

Scoped supersession notice (22 September 2026). This record is part of the source research consolidated in Paper A12, Dynamics of the Lambert W Newton Iteration: Basin Geometry, Routing Transitions, and Finite-Rank Period-Three Resonance (DOI: 10.5281/zenodo.21845665). For the Newton-dynamics, routing, conjugation-parity, and finite-rank period-three results stated in that article, it supersedes the corresponding earlier programme presentation and is the current consolidated reference. Supersession is limited to that scope; other results, programme interpretations, and open problems in this record are not thereby replaced or resolved. In particular, the finite-matrix results do not assert a continuum resonance. The original description below and deposited files are retained for provenance. 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 yields a negative result. The leading interpretation is a global redistribution of convergence hesitation across Lambert branch territories. All claims are restricted to the dynamical system; the Riemann connection is treated as conjectural motivation only.