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 snap from an asymmetric regime (k=0 at ~85%, k=−1 at ~5%) to approximate equipartition (~50/50). These changes are permanent: the post-branch corridor geometry and conjugate symmetry persist through the entire later routing arc. A distinct post-branch traffic anomaly peaks at α ≈ 0. 42, representing the traffic law's first strong response to the newly opened corridors. This anomaly is not a simple decay tail of the root-separation collapse but operates within the geometry that the branch point created. The Newton dynamics therefore exhibit a five-act structure: branch-point reorganisation, post-branch precursor, routing floor, activation/peak, and de-activation (empirical classification). Part of the Geometry of the Critical Line programme.