Location-Mixed Flea Profiles and Born-State Convergence: Robustness of the Two-Scale Classification in the Two-State Flea Model

Abstract

We study the robustness of the two-scale Born-state convergence classification in the exact two-state flea model when the fixed scaled source profile is replaced by genuinely h-dependent location mixtures. For a fixed normalized kernel k in L1(R), we consider profiles g_h = k * nu_h, where the mixing measures nu_h may vary arbitrarily with h, including atomic, singular, diffuse, splitting, and translation-escaping families. We first prove a translation-uniform exact-gap dephasing estimate and combine it with uniform basis localization to obtain Born-diagonal tracking for physically concentrating location mixtures satisfying w_h Y_h -> 0 in probability. The resulting universal property obeys the same sharp two-coordinate criterion as in the fixed-profile model: Delta_h / w_h -> 0 and t_h w_h / h -> infinity. The necessity direction is understood at the universal source-class level and is inherited from the fixed-profile classification. We also exhibit physically concentrating location-mixture families whose scaled profiles are not relatively compact in L1, showing that the result is not a compactness corollary. Conversely, a physically concentrating exact-gap resonant family outside the structured location-mixture class retains nonvanishing coherence despite both asymptotic scale conditions. Thus physical concentration alone does not control moving-profile resonance; the location-mixture structure supplies a distinct robustness mechanism.