Response-Controlled Dressed Dynamics Beyond Static Isolation in a Semiclassical Double Well

We study a random flea perturbation of the semiclassical quartic double well in a regime where the perturbation-to-harmonic-gap ratio need not remain bounded. For a fixed smooth nonnegative barrier-supported perturbation, relative-form control and an exact invariant-graph/Riccati reduction yield a response-controlled C1 approximation of the dressed bottom cluster, with error governed by Ξ_h + r_h(1+r_h)Θ_h, where r_h = w_h/h. This control permits exact spectral-gap and projector tracking, branchwise push-forward of every fixed normalized L1 amplitude profile, and Riemann-Lebesgue dephasing without a time-amplified phase-replacement estimate. On the domain w_h→0 and r_h(1+r_h)Θ_h→0, universal full-space convergence to the localized-basis Born diagonal holds exactly when Δ_h/(w_hγ_h)→0 and t_h w_h γ_h/h→∞. The domain strictly contains bounded perturbation-to-gap-ratio paths and includes an explicit response-adapted family for which w_h/h→∞. Thus static isolation is not the controlling validity scale for this full-model classification; the relevant sufficient condition is response-controlled. We do not claim that this condition is necessary or maximal.