Bennett-whitened gradient trust region

(1) Integrate this controller into the optimizer immediately after computing a minibatch gradient g_t and before the parameter update. Maintain a gradient mean estimate mu_t and define the centered noise proxy x_t=clip(g_t-mu_{t-1},1), using global-norm or layerwise clipping so the theorem's bounded-increment assumption is approximately enforced. Maintain cumulative noise M_t=M_{t-1}+x_t. For the first MVP use diagonal covariance: estimate q_t from the previous minibatch, for example q_t=(g_{t-1}-mu_{t-2}) squared elementwise, and maintain V_t=V_{t-1}+q_t. The one-step delay makes V_t predictable before x_t is added. (2) Pseudocode: initialize M=0, V=0, mu=0; compute g; compute x=clip(g-mu,1); update M=M+x; update V using the delayed previous squared residual; calculate z=sqrt(sum_j M_j^2/(V_j+rho)); calculate D=0.5*sum_j log(1+V_j/rho); solve rho*((1+r/sqrt(rho))*log(1+r/sqrt(rho))-r/sqrt(rho))-D=log(1/delta) for r by bisection; multiply the base SGD, Adam, or AdamW step by min(1,r/max(z,epsilon)). Update mu after the covariance proxy is formed. (3) The Bennett function, determinant correction, whitening, and root equation are mathematical components transplanted from the paper. Diagonal covariance estimation, clipping threshold, delta, rho, and any multiplier floor are empirical approximations. (4) First experiment: compare SGD, AdamW, gradient-norm clipping, and this controller on a small MLP and a ViT-Tiny trained on CIFAR-10, using identical batches, learning-rate schedules, and compute. Log z_t/r_t, the fraction of downscaled steps, gradient norm, loss variance, divergence events, time to a fixed validation loss, and final accuracy. The method succeeds if it reduces instability and loss variance while matching or improving accuracy; a concrete target is at least 20 percent lower loss variance or fewer divergence events at equal training FLOPs.