Neutral-Mode Structure-Factor Regularization

Use a CNN, U-Net, or MLP that maps a conditioning variable x to A real-space fields c_a(r) on a periodic grid. At every forward pass, subtract the spatial mean, Fourier transform each channel, and apply the charge projection to the k = 0 coefficient: delta c(0) is replaced by P_q delta c(0). If the application requires local electroneutrality, apply the same projection to a chosen set of low-wavenumber modes. For each minibatch, estimate the Fourier covariance as C_hat_ab(k) = B^{-1} times the sum over examples b of delta c_a^(b)(k) delta c_b^(b)(-k). Compute S_ZZ(k) = q^T C_hat(k) q and add the displayed spectral loss to the task loss. Estimate alpha by fitting the empirical training spectra rather than assuming it is known. The projection is exact; covariance estimates and alpha are empirical. First test on a 64 by 64 synthetic two-species electrolyte-field dataset with known Gaussian Fourier covariance. Compare an unconstrained CNN, a hard-neutrality-only model, and the proposed model. The predicted signature is that S_ZZ(k) divided by k squared becomes approximately flat at low k, the k = 0 charge variance falls to numerical precision, and the neutral covariance approaches a nonzero plateau. Fit log-log slopes and require a charge slope of 2 plus or minus 0.2; the unconstrained baseline should show a nonzero zero-mode plateau or an incorrect slope.