Dressed Two-Level Transfer for Concentrating Flea Ensembles in a Semiclassical Double Well

We consider the one-dimensional quartic symmetric double well under a fixed nonnegative strict right-barrier flea shape W with signed random scalar coefficient x, so that the physical perturbation is w_h x W, with x drawn from a fixed normalized L1 profile. The dynamics is the full Schrödinger dynamics on L2(R), while initial data are restricted to the unperturbed bottom doublet. The principal result is a quantitative dressed transfer theorem on growing observation times: after continuation of the isolated perturbed doublet and a domain-safe graph/Riccati reduction, the full ensemble is asymptotically equivalent in trace norm to an ideal two-state ensemble along paths in an explicitly defined transfer-validity domain, which requires both w_h/h → 0 and a vanishing growing-time transfer error budget. The strict flea introduces a nontrivial response hierarchy. If γ_h is the left-right diagonal response, ν_h the off-block coupling, and b_h the within-doublet cross term, then −h log γ_h → 2A_R(W), −h log ν_h → A_R(W), while b_h is controlled by the full interwell action. Consequently the operative two-state amplitude width is a_h = w_h γ_h, not the bare physical width w_h. Inside the independently imposed transfer-validity domain, the exact two-state classification of the preceding work can then be imported: universal full-space Born-diagonal tracking holds if and only if Δ_h/a_h → 0 and t_h a_h/h → ∞. These transfer assumptions are sufficient validity conditions for the reduction and are not claimed to be necessary conditions for Born-diagonal tracking.