We prove that, in sufficiently large smash powers, an $MU$-null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space. When the torus slot bundle embeds continuously into a trivial bundle of rank less than twice the source rank, the compression places a positive proportion of slots on their boundaries in arbitrarily large powers and fixes every original boundary slot exactly. An example at equality shows that the strict rank inequality cannot be removed.

We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.