Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality

What the math gives to ML

The paper identifies a nontrivial information-theoretic phase structure governed by a higher Nishimori line, where measurement strength and inverse temperature are matched by \(\beta=\Delta\), and where this line intersects the Potts critical manifold at a tricritical point. Its transferable asset is a quantitative matching rule between model sharpness and observation/noise strength, together with order parameters that distinguish paramagnetic, ordered, and replica-disagreement regimes. For neural networks, the most promising implementation is a replica-based stochastic training system in which inverse temperature, label or feature corruption precision, and ensemble disagreement are jointly controlled. A secondary transfer is to treat depth or compression as an RG flow and test whether an information-complexity proxy decreases monotonically under representation coarse-graining.