Three Entropy Projections of the Routing Transition: Methodology Alignment in the α ≈ 0.48 Neighbourhood

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 24 in the "Geometry of the Critical Line" programme Papers 21–23 identified a routing transition near α ≈ 0.481 in the Newton dynamics of Cα(z) = z − exp(−α/z). This paper determines how that transition appears in three independent entropy projections. Under Paper 20-like methodology (iteration cap 3000), H_time attains a valley at α ≈ 0.481, co-locating with slow-orbit word entropy H_word. Basin-identity entropy H_basin, computed here for the first time, attains a nearby but distinct minimum around α ≈ 0.478. A fixed-threshold entropy audit indicates that stricter symbolic minima may shift slightly leftward toward α ≈ 0.475 when the slow subset is defined by absolute iteration count rather than percentile, although the precise location is subject to finite-sample effects. A threshold-independent routing ratio observable R(α) = (C,−1,C)/(C,) provides a compact scalar diagnostic of the transition. The routing event is therefore best understood as a narrow neighbourhood near α ≈ 0.48, with different observables emphasizing slightly different parts of it. Part of the Geometry of the Critical Line programme.