Hardy barrier for lattice feature fields

Integrate the method at the feature-map regularization point of a CNN, grid neural field, or lattice-indexed embedding. Let F in R^(H_1 x ... x H_d x c) be the selected activation tensor, let n denote its spatial or lattice index, and choose an anchor n_0, normally the grid center. At every training step, compute the spatial mean bar F, replace F by F-bar F, and evaluate R_H using the final formula with epsilon=10^(-4). Use the loss L=L_task+lambda_H R_H. Initialize lambda_H by multiplying a base coefficient by the dimension-scaled calibration C_H=2^ell d^ell, then sweep lambda_H in {10^(-5),10^(-4),10^(-3)} after normalizing R_H by the mean squared activation magnitude. For a closer analogue of the paper, compute the discrete Laplacian Delta F(n)=sum_j [F(n+e_j)-2F(n)+F(n-e_j)] with zero padding, reflection, or periodic boundaries, and monitor E_Delta=sum_n ||Delta F(n)||^2. The empirical Hardy quotient Q=E_Delta/R_H is not assumed to equal the infinite-lattice constant; it is measured to diagnose whether the regularizer is compatible with the coercive inequality. The theorem supplies the inverse-radial weight and dimension scaling, while epsilon, boundary handling, and lambda_H are empirical. First test a small 32x32 CIFAR-10 CNN against baseline weight decay at matched parameter count and training FLOPs. Record validation accuracy, activation mass inside several radii around n_0, gradient-norm variance, and failure rate across random seeds. Success means reduced central activation concentration or gradient spikes without accuracy loss, or improved validation accuracy at equal training loss.