The Fourth Regime: Bounce De-Activation at Large α in the Cathedral Pullback

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 27 in the "Geometry of the Critical Line" programme. Papers 21–26 established a localised routing transition near α ≈ 0.48 in the Newton dynamics of C_α(z) = z − exp(−α/z), with a fixed symbolic scaffold, a relay-row traffic-law redistribution, and a local corridor-accessibility mechanism. This paper extends the parameter sweep to α ∈ [0.40, 2.00] to determine whether the post-transition increase in bounce traffic persists indefinitely. It does not. The relay-row bounce exit probability P_{C,bounce}, relay-row entropy H_C, and routing ratio R(α) all rise sharply after the routing floor, reach their maxima near α ≈ 0.58–0.60, and then decline at larger α. Meanwhile, the coarse basin scaffold remains fixed at 41 basins, mean convergence time increases from approximately 10 to 21 iterations, and the slow-orbit fraction grows from 0.3% to 5.3%. The large-α regime is not a simple continuation of routing activation but a distinct fourth regime in which bounce competitiveness wanes while convergence slows. The de-activation regime is not a return to the pre-transition state: both regimes have low bounce competitiveness, but the pre-transition regime is fast while the de-activation regime is slow. The Newton arc has a full four-stage lifecycle in α: suppression, activation, peak bounce competitiveness, and de-activation (empirical classification). Part of the Geometry of the Critical Line programme.