Spectral anti-localization regularizer

Integrate the regularizer after a CNN feature block or transformer token-to-grid projection, not on the logits. Let h have shape batch by channels by height by width, choose a fixed binary mask V over the spatial grid, and set alpha equal to its fraction of ones. Use periodic two-dimensional FFTs. Pseudocode: H = fft2(h); penalty = 0; for each selected shell r, form Hr = H multiplied by the Boolean frequency mask of S_r; hr = ifft2(Hr); total = sum(abs(hr)^2); observed = sum(abs(hr)^2 multiplied by V); target = B(alpha)^2 multiplied by total; penalty += relu(target-observed)/(total+epsilon); loss += lambda_anti multiplied by penalty. Compute B(alpha) directly from the displayed formula and estimate shell energies from the FFT; no eigenvalue estimation is needed. Initially use delta equal to zero for exact discrete Fourier eigenspaces, then test delta equal to one or two for narrow spectral clusters. Use high-frequency shells only, since the theorem is a high-energy statement. Randomly translate V during training to prevent memorization, while evaluating on a fixed validation mask. Start with CIFAR-10, a small ResNet, 32 by 32 intermediate maps, alpha equal to 0.25 or 0.5, and three mid/high-frequency shells. Compare with no regularizer, total variation, and generic Fourier penalties. Measure clean accuracy, cutout or occlusion accuracy, shell energy ratios on unseen masks, and gradient stability. Success requires the lower tail of observed-to-total shell energy to increase toward B(alpha)^2 and occlusion accuracy to improve at equal parameter count without a substantial clean-accuracy loss.