  Recent theoretical advances offer an exact, first-principle theory of jamming
criticality in infinite dimension as well as universal scaling relations
between critical exponents in all dimensions. For packings of frictionless
spheres near the jamming transition, these advances predict that nontrivial
power-law exponents characterize the critical distribution of (i) small
inter-particle gaps and (ii) weak contact forces, both of which are crucial for
mechanical stability. The scaling of the inter-particle gaps is known to be
constant in all spatial dimensions $d$ -- including the physically relevant
$d=2$ and 3, but the value of the weak force exponent remains the object of
debate and confusion. Here, we resolve this ambiguity by numerical simulations.
We construct isostatic jammed packings with extremely high accuracy, and
introduce a simple criterion to separate the contribution of particles that
give rise to localized buckling excitations, i.e., bucklers, from the others.
This analysis reveals the remarkable dimensional robustness of mean-field
marginality and its associated criticality.
