Zeta-Corrected Singular Integral Layer

(1) Exact integration point: add this as a periodic neural integral or convolution layer operating on a feature tensor F∈R^{B×C×n^d}. Fix a singular component K_sing(y)=a|y|^{-s}, optionally followed by a learned channel-mixing matrix A∈R^{C×C}; let a smooth learned branch model the nonsingular remainder. Apply the correction only to displacement samples in a fixed stencil S around zero during both training and inference. (2) Pseudocode: choose grid spacing h=1/n, dimension d, exponent s, correction order p, and symmetric stencil S={r∈Z^d:||r||_∞≤R}. Offline, evaluate or approximate AC[Z_{2α}(s)] for all |α|_1≤p, form b_α=AC[Z_{2α}(s)]/(2α)!, and solve the constrained least-squares system M c=b with M_{α,r}=r^{2α}; impose c_r=c_{−r} and use a minimum-norm solution. At runtime, for every grid site i, compute G_i=h^d[Σ_{j∉S}|hj|^{-s}F_{i−j}+h^{−s}Σ_{r∈S}w_rF_{i−r}] using periodic indexing, then output Y_i=φ(A G_i+W F_i). Set w_r to the ordinary sampled value |r|^{-s} plus correction c_r, with a finite learned or fixed value at r=0. (3) Mathematics versus empirical estimation: d,s,p, the moment equations, predicted order 2p+2+d−s, and zeta-based coefficients come from the paper. Estimate analytically continued zeta values numerically with Ewald splitting or fit coefficients against a very fine-grid reference, and monitor the moment residual ||Mc−b||_2. Train only the singular amplitude a and channel matrix A initially; freeze correction weights for the first ablation. (4) First cheap experiment: use a 2D periodic Burgers or advection-diffusion surrogate with inputs and targets on 32×32, 64×64, and 128×128 grids. Compare a naive periodic neural convolution against corrected layers with p=0,1,2, keeping width, optimizer, parameter count, and wall-clock budget fixed. Generate smooth random Fourier fields and compute reference singular convolutions on a 512×512 grid. The expected signal is a log-log discretization slope near 2p+2+d−s and lower validation error at equal resolution; success means p=1 or p=2 reaches the baseline accuracy with fewer grid points while adding only O(|S|) local work per site.