Reverse-Protocol Entropy Controller

Integrate the controller into SGD, Adam, or Langevin-SGD at the optimizer step, treating the learning rate, gradient-noise scale, momentum coefficient, and data-mixture weights as the protocol $u_k$. First implement Langevin-SGD because its transition density is explicit: $\theta_{k+1}=\theta_k-\eta_k\widehat g_k+\sqrt{2T_k\eta_k}\,\xi_k$, so $q_k^F$ is a diagonal Gaussian with mean $\theta_k-\eta_k\widehat g_k$ and covariance $2T_k\eta_k I$. Maintain a short forward window of $K$ parameter states and gradients. Every $M$ steps, launch a cheap shadow optimizer from the current endpoint using the reversed schedule $(u_{K-1},\ldots,u_0)$; estimate $q_k^R$ with the shadow gradients and evaluate the log Gaussian transition ratios along the stored trajectory. Use an exponential controller: if the per-step entropy estimate $\widehat\sigma_k$ exceeds target $\sigma_*$, reduce $\eta$ or increase controlled noise; if it is below target, restore the schedule slowly. The paper supplies the forward/reverse entropy-production construction and the slow-driving first-order saturation mechanism; minibatch transition densities, endpoint densities, and the target $\sigma_*$ must be estimated empirically. The first cheap test is a two-layer MLP on MNIST or CIFAR-10 with Langevin-SGD, comparing fixed cosine decay, adaptive entropy control, and no-noise SGD. Sweep schedule-ramp speed $\epsilon$ and plot the measured gap between forward loss fluctuations and the reverse-protocol bound. The falsifiable prediction is that this gap scales quadratically with $\epsilon$ in the slow-ramp regime, with a log-log slope near $2$, while instability or sharp loss spikes begin when cumulative entropy production exceeds a reproducible threshold; report whether the controller delays that threshold without increasing final loss.