In very recent work the mean field theory of the jamming transition in infinite-dimensional hard sphere models was presented. Surprisingly, this theory predicts quantitatively the numerically determined characteristics of jamming in two and three dimensions. This is a rare and unusual finding. Here we argue that this agreement is nongeneric: only for hard sphere models does it happen that sufficiently close to the jamming density (at any temperature) the effective interactions are binary, 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 many-body effective interactions that are not mean field at any density, making the d=∞ results not applicable.