Response-Matched Dressed Control for Vanishing Flea Amplitudes in a Semiclassical Double Well

Abstract

We study a fixed smooth nonnegative strict right-barrier flea in the one-dimensional semiclassical quartic double well, multiplied by a signed random scalar coefficient and an amplitude w_h -> 0. A preceding response-controlled analysis established exact dressed C^1 validity only under the sufficient condition r_h(1+r_h)Theta_h -> 0, where r_h = w_h/h and Theta_h measures off-block response relative to the diagonal flea response. We ask whether that response-domain hypothesis is required by the exact low-cluster reduction. The main result is that it is not. A forbidden-region weighted one-resolvent estimate, combined with the strict-flea response injection ||W^(1/2)P_h||^2 = O(gamma_h), converts the natural O(h^(-1)) weighted squared-resolvent bound into the directional estimate ||R_Q Q_h W P_h|| = O(sqrt(gamma_h/h)). This controls the moving-energy self-energy derivative and, through weighted Sylvester estimates for the differentiated Riccati equation, yields ||M_h - M_h^id||_{C^1([-R,R])} <= C_R(Xi_h + w_h) -> 0 for every fixed R and every w_h -> 0. The stronger uniform O(1) squared-resolvent bound is neither assumed nor proved. The inherited exact-gap and fixed-L^1 push-forward mechanism then gives the sharp full-space universal Born-diagonal criterion on the complete amplitude-vanishing class: Delta_h/(w_h gamma_h) -> 0 and t_h w_h gamma_h/h -> infinity. This is a removal of a sufficient response-domain hypothesis by a response-matched control argument; no maximality or strict enlargement of the preceding response domain is claimed.