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

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 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.