Response-Matched Dressed Reduction and Sharp Born-Diagonal Tracking for Vanishing Flea Amplitudes in a Semiclassical Double Well

This preprint presents a self-contained semiclassical analysis of a one-dimensional symmetric quartic double well subjected to a fixed smooth nonnegative barrier perturbation multiplied by a signed random scalar coefficient and an amplitude w_h tending to zero. The effective response scale is a_h = w_h gamma_h, rather than the bare perturbation amplitude. We establish the model-specific response hierarchy and response injection, uniform isolation of the exact low-energy cluster for every w_h tending to zero, graph-norm differentiability of the invariant graph and the strong Riccati equation, and weighted resolvent/Sylvester estimates yielding an all-amplitude dressed C1 reduction. Using the exact dressed spectral gap as the oscillatory variable, we obtain a necessary-and-sufficient criterion for universal full-space Born-diagonal tracking: Delta_h/(w_h gamma_h) tends to zero, while t_h w_h gamma_h/h tends to infinity. The result concerns ensemble Born-diagonal tracking for this fixed strict-flea model; it is not an individual-outcome or collapse theorem. This version 1.0 preprint consolidates, repairs, and strengthens material developed in four earlier Zenodo preprints by the author: doi:10.5281/zenodo.22013490, doi:10.5281/zenodo.22017480, doi:10.5281/zenodo.22019379, and doi:10.5281/zenodo.22049293. It replaces the staged dependency structure by a self-contained proof chain and adds the all-amplitude isolated-graph closure, graph-norm C1 and strong-Riccati justification, quantitative quasimode-to-exact-state transfer, a direct finite-c necessity witness, a common h-independent coefficient-space formulation, and a time-uniform full-space trace-norm return lemma.