Does anyone have more information about the finding itself? This article seems to be mostly a story about how "there was this mystery, and this scientist solved it with math".
https://arxiv.org/abs/1607.04217
Edit, TL;DR:
Yaida found that for mean-field models of spins with quenched impurities in an external magnetic field, in three dimensions, the renormalization group flow admits a fixed point at the two-loop level of calculation. It was long known that a fixed point exists in dimensions d > 6, but earlier one-loop calculations didn't find one for d=3, so it was thought that the nice results from these mean-field models had no physical relevance. This new result indicates that we can indeed use these models to understand physical disordered systems.
Computer analogy: we're trying to reverse-engineer an alien ASIC, which can operate either on an array of tuples of three reals, or on an array of bools. It works approximately the same in both cases. The latter case is much easier to analyze, so we prefer to do that.
“... studies the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model”
from "The Loss Surfaces of Multilayer Networks" Choromanska, Henaff, Mathieu, Arous, & LeCun [1]
[1] https://arxiv.org/abs/1412.0233
Also the physics of collapsing sand piles.
The other comment also led me to https://en.wikipedia.org/wiki/Spin_glass, which I didn't know about at all (I'd heard the term once but had no context for it).