Lyapunov-certified Hessian-damped optimizer

Integrate this at the optimizer update for each parameter tensor, not inside the network forward pass. Maintain x_prev, x, g_prev, and g; use α=3 initially, s as the gradient step size, and β as the gradient-difference damping coefficient. At step k, compute g=grad(loss,x), estimate local smoothness from the secant ratio r=||g-g_prev||/(||x-x_prev||+ε), update L_est=max(decayed_L_est,r), and clip L_est to a robust percentile or exponential moving maximum to avoid one-batch outliers. Set s=min(s, 0.99/L_est). Compute bmax=[1+sqrt(1+s*L_est*(1-s*L_est))]/(L_est*sqrt(s)); choose β=min(β_target, 0.9*bmax), or use β=0.5*0.9*bmax for a less aggressive first run. Then execute x_next=x+(1-α/k)*(x-x_prev)-s*g-β*(g-g_prev), and shift x_prev=x and g_prev=g. At k=1, initialize x_prev=x and g_prev=g, so the damping term is zero. Estimate the mathematical quantities L_est and the admissible boundary empirically; do not claim a global L unless the model has a certified smoothness bound. First test on a 2-layer MLP and a small Transformer on MNIST or CIFAR-10, comparing AdamW, Nesterov momentum, fixed-β IGAHD, and the adaptive controller at equal optimizer steps, parameters, and measured FLOPs. Pre-register three mechanism tests: (1) on convex quadratics with known largest Hessian eigenvalue L, divergence should begin near βL√s=1+sqrt(1+sL(1-sL)), with boundary error below 10%; (2) at equal stable step size, the proposed method should reduce loss faster than β=0 in the intermediate damping range, measured by area under the training-loss curve; (3) reducing s while holding β fixed should remain stable once the refined condition holds, whereas the old β<2√s rule would reject those settings. Remove only the β(g-g_prev) term as the ablation. The transfer is falsified if the adaptive method has no stability boundary near the predicted inequality on quadratics, or if gradient-difference damping consistently worsens equal-budget convergence after accounting for its extra gradient-memory cost.