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

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 P₂, ₁₎ₔ₍₂₄, relay-row entropy HC, 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.