Capitalization-Efficiency Monitor

Attach the monitor to a Bayesian, Laplace, or stochastic-weight neural-network training loop. Maintain a diagonal Gaussian posterior $q_t(\theta)=\mathcal N(\mu_t,\operatorname{diag}(s_t^2))$ around current weights, with $q_t$ serving as the previous checkpoint prior. At update $t$, compute info_t = KL(q_{t+1} || q_t); estimate optimizer dissipation with Sigma_t = eta^2 * trace(H_t C_t), where $H_t$ is the minibatch-loss Hessian and $C_t$ is the gradient-noise covariance, using Hutchinson Hessian-vector products and minibatch covariance estimates. In the first MVP, omit $\Sigma_t$ and use KL alone. Every few hundred updates, estimate $V_hat_t$ on a small held-out task buffer using paired informed and blind evaluations. Compute eta_hat_t = (V_hat_t - V_hat_prev)/(info_t + Sigma_t + epsilon). Reduce the learning rate or stop an update block when this ratio stays below a threshold; permit larger steps when value gain per acquired information is high. The first cheap experiment is a two-layer MLP on Split-MNIST, comparing SGD, Adam, and monitor-controlled Adam at equal compute. Prediction: in the no-discarded-correlation regime, positive updates should lie below an empirical upper envelope near eta_hat = 1 after consistent normalization, while memorization-heavy late training shows increasing KL but near-zero or negative deletion-gap increments, causing eta_hat to approach zero or become negative. Add random nuisance features to test the predicted separation between information acquisition and operational value.