Polynomial Tube Regularizer

Integrate this at the output of a designated two-dimensional projection head, immediately before the classification loss. For each minibatch, obtain projected points z_i in R^2 and either a soft class support, or a soft decision-boundary point cloud by selecting samples whose predicted class margin is below a threshold. Use a fixed grid of radii T={t_1,...,t_K}, with t_K no larger than roughly one quarter of the median k-nearest-neighbor distance. Estimate the tube area with a differentiable soft union: V_hat(t)=A_box/N_grid sum_g [1-exp(-sum_i exp(-(||x_g-z_i||^2)/(2 sigma^2))/tau)] over a fixed 2D grid covering the projected data. Alternatively, for a cheaper MVP, use Monte Carlo probe points and the same soft occupancy expression. Fit (a1_hat,a2_hat) by differentiable least squares to the values V_hat(t_k), then add L_tube to cross-entropy. Estimate the minimum gap scale Delta_hat_min from angular gaps between vectors z_i-c around each local high-curvature or low-density center c; stop gradients through this estimate and use it only in the hinge term. The theorem supplies the target degree-two law and the cone coefficient interpretation; the grid estimate, radius range, and gap estimate are empirical approximations. Start with a 2D MLP projection on CIFAR-10 or two-moons, comparing an MLP with cross-entropy against the same model with lambda_tube in {1e-3,1e-2,1e-1}. Measure test accuracy, adversarial accuracy under small input perturbations, margin collapse, and tube-curve residual at held-out radii. The first success signal is lower nonquadratic tube residual and improved robustness at unchanged clean accuracy; reject the method if it merely contracts embeddings or hurts class separation.