  In very recent work the mean field theory of the jamming transition in
infinite dimensional hard spheres models was presented. Surprisingly, this
theory predicts quantitatively numerically determined characteristics of
jamming in two and three dimensions. This is a rare and unusual finding. Here
we argue that this agreement in non-generic: only for hard sphere models it
happens that sufficiently close to jamming the effective interactions are in
agreement with mean-field theory, justifying the truncation of many body
interactions (which is the exact protocol in infinite dimensions). Any
softening of the bare hard sphere interactions results in effective
interactions that are not mean-field all the way to jamming, making the
discussed phenomenon non generic.
