Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems

What the math gives to ML

The paper's transferable contribution is that directional behavior in a deep interaction network cannot be inferred reliably from eigenvalues: nilpotent acyclic matrices have only zero eigenvalues while still producing substantial finite-length directed-walk effects. Its proposed organizing quantity is antisymmetry of aggregate walks, whose mean entropy-production proxy follows the explicit law \(\phi_*(g)=1-\sqrt{1-g^2}\). This suggests replacing spectral-radius-only diagnostics in deep networks with finite-horizon measurements of forward/reverse walk imbalance, particularly for residual Jacobians, recurrent blocks, and directed graph neural networks. The most practical first use is a walk-based stability monitor or regularizer, not treating the thermodynamic EPR formula as a supervised loss without calibration.