cubic growth at lightspeed should be physically permissible

This is an instance of a common error around here. Our current understanding of general relativity implies that you can only fit O(r^2) bits into a sphere of radius r. If you try and put more in, you eventually get a black hole and then we will have to appeal to some much more intricate understanding of physics to know what happens. There may or may not be some deep reasons for this (related to the AdS/CFT correspondence, about which I know very little).

(Of course, I wouldn't bet aggressively against future revisions of the current understanding.)