Concentrating Flea Distributions and Born-State Convergence: A Two-Scale Classification in the Two-State Flea Model

The method of arbitrary functions has been applied to a two-state approximation of the perturbed double-well “flea” model by treating the flea parameter as random and averaging the corresponding unitary evolutions. The source two-state result and the fixed-distribution predecessor analysis keep the probability measure fixed as the semiclassical parameter tends to zero. Here the distribution itself is allowed to concentrate, with concentration width w tending to zero. An exact rescaling reduces the problem to two dimensionless quantities: kappa = tunneling splitting divided by distribution width, and lambda = observation time times distribution width divided by hbar. We prove that Born-state convergence for every fixed admissible profile and every normalized initial superposition holds if and only if kappa tends to zero and lambda tends to infinity. The two conditions express distinct mechanisms. Failure of lambda to diverge prevents sufficient ensemble phase mixing. Finite positive limiting kappa produces complete long-time dephasing in a limiting energy basis that remains tilted relative to the outcome basis, with an explicit positive trace-norm defect. In the large-kappa regime, the finite-channel family has no unique extension to the naive boundary at finite positive lambda because inequivalent phase classes can produce different limiting channels. The conclusions are stable under L1-asymptotic self-similarity of the concentrating densities. The analysis is confined to the exact two-state ensemble model and does not constitute a general derivation of the Born rule or a solution of the measurement problem.