Born-Diagonal Tracking Beyond Phase-Accurate Two-Level Transfer in a Semiclassical Double Well

We distinguish two asymptotic questions for a random flea perturbation of the one-dimensional quartic double well: whether the full Schrödinger ensemble tracks the localized-basis Born diagonal, and whether its long-time dynamics is phase-accurately transferable to an ideal two-state propagator. The perturbation is a fixed nonnegative strict right-barrier shape multiplied by a shrinking random amplitude w_h x, with x drawn from one fixed normalized L¹ profile, and initial data are restricted to the unperturbed bottom doublet. Using the exact continued bottom spectral cluster, we obtain C¹ control in the scalar amplitude of the dressed 2×2 block. This yields branchwise control of its exact spectral gap and projectors and permits an exact gap push-forward of the ensemble measure; Riemann–Lebesgue cancellation then suppresses the oscillatory cross terms without a phase comparison carrying a time-amplified block error. We prove direct finite-κ full-model dephasing and, inside the static-isolation domain w_h/h → 0, the sharp universal criterion Δ_h/(w_h γ_h) → 0, t_h w_h γ_h/h → ∞. Finally, an explicit family of paths satisfies this Born-tracking criterion while lying outside the domain controlled by the preceding phase-accurate transfer estimate. The result is a separation of proved theorem domains, not a proof that two-level transfer itself fails there.