Periodic-Gradient Adam Guard

Integrate this at the optimizer step, not inside the model. Maintain the last three minibatch gradients for each selected parameter group, preferably the final transformer block and embeddings first. At step t, compute the usual Adam moments and candidate update. Normalize each stored gradient by its RMS norm, then evaluate the guard predicate: the newest three gradients must have alternating directional signs, the first-to-second norm ratio must exceed 2, and both cosine tests must exceed a threshold such as rho=0.8 in magnitude. If the predicate is false, apply the normal AdamW update. If it is true, increment a counter; after two detections within a 16-step window, use a fallback for H=8 steps: either SGD with momentum 0.9 and learning rate eta_sgd equal to eta_adam times the running RMS of the Adam step, or AMSGrad with v_hat_t=max(v_hat_{t-1},v_t). Then reset the counter and return to Adam. Pseudocode is: store g[t]; compute m,v,u; detect guard; if triggered set mode=FALLBACK and mode_steps=H; if mode==FALLBACK set q=0.9q+0.1g and delta=eta_sgd*q, else delta=eta*u; apply decoupled weight decay and parameter update. The mathematical quantities directly used are the periodic slope pattern, projected update, and regret definition; the cosine and norm thresholds are empirical detector parameters. First test on a 1D projected toy problem with exact slopes (c,-1,-1), then on CIFAR-10 ResNet-18 and a 125M-parameter language model using AdamW as baseline. Measure average loss, gradient norm oscillation, boundary-hit frequency, and validation accuracy at equal optimizer steps and FLOPs. Success means eliminating the toy nonzero-regret plateau and producing lower training-loss variance or better validation loss on deliberately shuffled-versus-periodic minibatch orders, without degrading ordinary IID training by more than 0.2 percent.