Tail-triggered adaptive ridge head

Apply this to a final linear probe, classifier head, regression head, or LoRA-style adaptation head while initially freezing the backbone. Every K steps, collect a feature batch Z_t and targets Y_t. Compute C_t = Z_t.T @ Z_t / B + eps*I, obtain eigenvalues lam, and calculate H_t = mean(1/(lam+eps)) and S_t = mean((lam < eta)/(lam+eps)), using a sigmoid gate instead of the Boolean indicator if a differentiable diagnostic is needed. Maintain exponential moving averages of both statistics. During the first 100 batches, define H0 and S0 as robust medians; initialize gamma to a small value. Update with gamma = clip(gamma * exp(rho*((H-H0)/H0 + (S-S0)/max(S0,eps0))), gamma_min, gamma_max). For closed-form head fitting, solve W = CholeskySolve(Z.T@Z + B*gamma*I, Z.T@Y). For end-to-end training, use gamma*||W||_F^2 as the head regularizer and update gamma only every K steps. The mathematical quantities supplied by the paper are the inverse-trace decomposition and hard-edge separation; K, rho, thresholds, clipping bounds, EMA decay, and numerical floors are empirical. First compare against ridgeless least squares, fixed ridge, and AdamW weight decay using a width-1024 MLP on CIFAR-10 or a small tabular regression benchmark. Track clean accuracy or MSE, 99th-percentile loss under corruptions, maximum output norm, minimum feature eigenvalue, selected gamma, and throughput. Success means retaining most of the ridgeless mean performance while reducing catastrophic-loss frequency and avoiding a permanently excessive ridge coefficient.