Barrier-Temperature Matching

Integrate this as an exploration controller around SGD, Adam, or Langevinized Adam. Every K training steps, save a checkpoint and launch m short probe trajectories from perturbed parameters, with perturbations drawn from the optimizer-preconditioned Gaussian. Estimate a neighboring-basin barrier U_hat_t as the maximum loss along the lowest-loss probe path minus the current basin loss. A simpler MVP uses linear interpolation between two independently perturbed endpoints and records the maximum loss along the interpolation. Estimate Sigma_hat_g,t from 8 to 32 minibatch gradients, form T_hat_t=tr(P_t Sigma_hat_g,t P_t^T)/(2d), and compute r_t=U_hat_t/T_hat_t. Adjust injected noise or trust-region radius using T_(t+1)=T_t exp(kappa log(U_hat_t/T_hat_t)), with small kappa such as 0.01 to 0.1, so the controller drives r_t toward one. The first cheap experiment is a small MLP on CIFAR-10 with Adam, comparing fixed noise, cosine noise decay, and barrier matching under identical probe budgets. The sharp prediction is that basin-transition frequency peaks when measured r crosses approximately one. Plot escape probability against r; for r greater than one it should decline approximately exponentially, while controllers holding r much below one should show larger parameter variance and worse local-loss retention.