Scaled-Time Classification of Born-State Convergence in the Two-State Flea Model

Abstract

Cicchella et al. study a two-state approximation to the perturbed double-well “flea” model and prove that an ensemble-averaged state converges to the Born-rule mixture under the successive limits of long time followed by the semiclassical limit. We keep the same finite-dimensional Hamiltonian, the same arbitrary normalized initial superposition, and the same class of fixed flea distributions that are absolutely continuous with respect to Lebesgue measure. We then classify coupled semiclassical/time scalings by the scaled-time parameter sℏ = tℏ/ℏ. If sℏ → ∞, the ensemble state converges to the Born state for every fixed admissible distribution. This yields, as corollaries, both the reverse iterated limit and genuine path-independent convergence along every path with tℏ → ∞ and ℏ → 0. If instead sℏ → s < ∞, the limiting density matrix retains an off-diagonal term determined exactly by the characteristic function of the flea distribution at s. Consequently, sℏ → ∞ is necessary and sufficient for Born convergence at the universal source-class level, although it is not necessary for every individual distribution or initial state. The analysis is restricted to the finite-dimensional Proposition-2 surface and makes no claim to a general derivation of the Born rule, a solution of the measurement problem, or the full double-well theorem