Branch-Point Reorganization and Post-Branch Routing Precursor Near α = 1/e

Paper 28 in the "Geometry of the Critical Line" programme. Papers 21–27 characterised the routing transition near α ≈ 0.48 and its four-regime lifecycle in the Newton dynamics of C_α(z) = z − exp(−α/z), but the relationship to the Lambert W branch-point at α = 1/e remained unexplored. This paper sweeps α ∈ [0.35, 0.46] with dense sampling around 1/e and discovers that the branch-point transition is the structural genesis of the later routing arc. At α = 1/e, the k=0 and k=−1 branches collide analytically; numerically, the sampled root separation reaches a minimum of 0.021. Mean convergence time spikes to 13.9 iterations, relay-exit throat widths jump from sub-unit values (~0.9) to ~3.6, and basin populations…