Porous Fourier concentration regularizer

Integrate the regularizer at the output of a convolutional block or on the token sequence entering a token-mixing layer. For a feature channel z of length N, compute the unitary FFT u=Fz. Construct M_X as a fixed or randomly translated spatial mask with holes at dyadic scales h, 2h, ..., 1; construct M_Y analogously in frequency coordinates over the reciprocal scale range. Use binary masks initially, then optionally replace them with sigmoid edges so gradients are smooth. For every selected feature channel, compute r_X=(||M_X z||_2^2+epsilon)/(||z||_2^2+epsilon) and r_Y=||(I-M_Y)u||_2^2/(||u||_2^2+epsilon). Add max(0, log(r_X/(C h^(2 beta))))^2 plus r_Y to the task loss and backpropagate through the FFT. Start with h=1/8, nu approximately 0.2, C=1, beta=0.1, lambda_support=0.01, and lambda_FUP=0.001; sweep these values because the theorem does not provide directly usable numerical constants. The mathematical quantities are the energy ratio and uncertainty threshold; C, beta, and the discrete porous masks are empirical approximations. First test on CIFAR-10 with a small ResNet-18 against the baseline and an ordinary high-frequency penalty at equal training FLOPs. Measure clean accuracy, corruption accuracy, localized-occlusion robustness, and the joint spatial-frequency concentration ratio on held-out examples. Success means similar clean accuracy with lower corruption or occlusion degradation and reduced joint concentration; failure is substantial accuracy loss or optimization instability.