Discrete Hardy Barrier for 3D Feature Fields

(1) Integration point: choose an intermediate 3D CNN feature tensor F with shape B by C by D by H by W, preferably after two or three convolutional blocks and before global pooling. Select an anchor voxel a from the object center, sensor position, lesion center, or geometric crop center. Compute the penalty independently for each batch item and average over channels and examples.

(2) Pseudocode:

input F, anchor a, eps, delta
E = 0; W = 0
for every voxel x:
    W += sum_c F[:,c,x]^2 / (||x-a||_2^2 + eps^2)
for every positive-direction edge (x, x+e_j):
    E += sum_c (F[:,c,x+e_j] - F[:,c,x])^2
E = mean_over_batch(E)
W = mean_over_batch(W)
R = relu(W - 4*E) / (W + 4*E + delta)
loss = task_loss + lambda_h * R

Use only positive-direction neighbors so each undirected edge is counted once, matching the paper. Computed from the mathematics are the inverse-square weights and the sharp comparison coefficient 4, the reciprocal of the Hardy constant 1/4. Estimated empirically are the anchor, epsilon, lambda_h, and an appropriate target ratio. Log W/E, maximum activation within radius two of the anchor, and feature-gradient norms. For the first cheap experiment, train a small 3D U-Net or 3D ResNet on synthetic voxel spheres and cubes with injected single-voxel impulse noise near the anchor. Compare ordinary weight decay against lambda_h in {1e-4, 1e-3, 1e-2}, using matched training steps and FLOPs. Measure clean accuracy, corrupted accuracy, peak activation, W/E, and exploding-gradient frequency. A positive result is reduced peak activation and improved impulse-noise robustness with less than one percent clean-accuracy loss. If the exact hinge is almost always zero, replace it with max(0, W/(E+delta)-tau) and tune tau from clean-training statistics.