  In jammed packings, it is usually thought that local structure only plays a
significant role in specific regimes. The standard deviation of the relative
excess coordination, $\sigma_Z/ Z_\mathrm{c}$, decays like $1/\sqrt{d}$, so
that local structure should play no role in high spatial dimensions.
Furthermore, in any fixed dimension $d \geq 2$, there are diverging length
scales as the pressure vanishes approaching the unjamming transition, again
suggesting that local structure should not be sufficient to describe response.
Here we challenge the assumption that local structure does not matter in these
cases. In simulations of jammed packings under athermal, quasistatic shear, we
use machine learning to identify a local structural variable, softness, that
correlates with rearrangements in dimensions $d=2$ to $d=5$. We find that
softness - and even just the coordination number $Z$ - are quite predictive of
rearrangements over a wide range of pressures, all the way down to unjamming,
in all $d$ studied. This result provides direct evidence that local structure
can play a role in higher spatial dimensions.
