Fisher-Zero Monitor for Stochastic Training

Integrate this as a diagnostic wrapper around SGD, Adam, or Langevin training, not initially as a replacement optimizer. At a checkpoint, clone the current parameters into $K=32$ to $128$ replicas. Evolve each replica for $M$ additional steps with independent minibatch ordering and optional Gaussian perturbations, using $\theta_{k+1}=\theta_k-\eta g_k+\sqrt{2D\eta}\xi_k$. Define the initial basin $B_0$ by a parameter-distance threshold and a loss window. For replicas returning to $B_0$, record the trajectory action $\mathcal A_r$ and return observable $\mathcal R_r$, such as final squared distance from the basin center or final loss increase. Evaluate $\widehat Z_c(T;s)$ on a grid of complex probes $s=u+iv$ using log-sum-exp scaling and complex arithmetic. Estimate the smallest $|v|$, denoted $v_*$, at which the complex sum approaches zero, and monitor the return rate $g(T;u)$. The paper supplies the path-competition mechanism; noise scale, basin definition, action approximation, and saddle costs are estimated empirically. First test a two-dimensional double-well loss, then a small MNIST MLP, against unmodified SGD with equal compute. The prediction is that near a basin transition, $g(T;u)$ develops a sharp peak or kink, $v_*$ decreases as $D$ decreases, and two trajectory clusters have approximately equal action at the transition: $|\mathcal A_1-\mathcal A_2|/\max(\mathcal A_1,\mathcal A_2)<0.2$. Trigger a learning-rate reduction when $v_*<0.1$ or when the return-rate slope exceeds a baseline threshold. If no cluster exchange, kink, or predictive relation between $v_*$ and basin occupancy occurs, reject the mechanism.