Nonequilibrium corrections to conserved Ising criticality in scalar active matter: Ward identities, spectrum, and long crossovers

What the math gives to ML

The paper provides a nontrivial renormalization mechanism for nearly degenerate nonequilibrium perturbations of a conserved critical fixed point. Its most transferable asset is the all-orders triangular stability structure enforced by conservation and causal response: a current-like perturbation can drive the passive sector, while the passive sector does not feed back into that perturbation at linear order. The slow-mode splitting is quantitatively fixed by the anomalous dimension, with y_J-y_Theta=-eta, producing long crossovers when the initial current amplitude is large. This suggests a two-channel neural optimizer or residual dynamical architecture whose linearized training dynamics are deliberately triangular and whose crossover can be measured against the predicted power law.