Safeguarded delayed-Rayleigh BB optimizer

Integrate the method at the learning-rate operation of SGD, initially using one scalar step size per parameter block, such as each Transformer layer, attention projection, or MLP block. For every block, store the previous smoothed gradient g_prev, the previous accepted step alpha_prev, and the current parameter tensor. At iteration k, compute the current gradient g on a fixed or accumulated minibatch, optionally form an exponential moving average g_smooth = rho times g_prev_smooth plus (1-rho) times g, and set d = g_prev_smooth - g_smooth. Compute r = inner_product(g_prev_smooth, d). If r is positive and larger than epsilon times the norms of the two vectors, propose alpha_prop = alpha_prev times squared_norm(g_prev_smooth) divided by r; otherwise retain alpha_prev. Smooth the proposal with alpha = beta times alpha_prop plus (1-beta) times alpha_prev, then clip alpha to [alpha_min, alpha_max], for example [10^(-6), 10] times the baseline learning rate. Apply x_new = x - alpha times g_smooth. Apply decoupled weight decay separately so the BB estimate measures the loss gradient rather than regularization. With minibatches, accumulate gradients over 4 to 16 batches or use a large validation batch before updating alpha, because independent noise can make r negative. The paper determines the quadratic spectral mechanism and target factor c_H; the code estimates curvature from gradient differences and can estimate an effective condition number by tracking recent accepted alpha values. First test ResNet-18 on CIFAR-10 against SGD with momentum and AdamW, then test a small decoder Transformer on WikiText-2. Compare loss versus optimizer steps and wall-clock time at equal model FLOPs, final accuracy or perplexity, gradient-norm ratios, clipping frequency, and failed-denominator frequency. A positive result is faster loss descent or equal quality in fewer updates without increased divergence or excessive learning-rate clipping.