  We introduce a randomized iterative fragmentation procedure for finite metric
spaces, which is guaranteed to result in a polynomially large subset that is
$D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed
target distortion. Since this procedure works for $D$ arbitrarily close to the
nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much
simpler probabilistic proof of the main result of Bartel, Linial, Mendel, and
Naor, answering a question from Mendel and Naor, and yielding the best known
bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of
random scales at which a given metric space is fragmented. As in many previous
randomized arguments in embedding theory, these scales are chosen irrespective
of the geometry of the metric space in question. We show that our bounds are
sharp if one utilizes such a "scale-oblivious" fragmentation procedure.
