Are jamming exponents mean-field already in two and three dimensions?

For frictionless soft spheres at the jamming density $\phi_J$, the infinite-dimensional theory predicts the gap distribution g(h) ~ h^(-gamma) with $\gamma = 0.41269$ and the small-force distribution P(f) ~ f^theta_e with $\theta_e = 0.42311$ for extended contacts, and excess coordination $\Delta Z \sim (\phi - \phi_J)^{(1/2)}$. Simulations agree with these values in $d = 2 \text{ and } 3$, suggesting an upper critical dimension $d_u = 2$. An answer is a controlled argument (renormalization group or rigorous) fixing d_u and computing any corrections in $d = 2, 3$.

A field-theoretic or rigorous derivation of $d_{u}$ for jamming, confirmed by high-precision finite-size scaling of $\gamma$ and $\theta_{e}$ in $d = 2\ \text{and}\ 3$.