Transverse Fourier Collision Control

(1) Integration point: modify the spectral mixing operation in a 2D Fourier neural operator, Fourier MLP, or spectral CNN. Represent each feature map by an H by W FFT, retain frequencies in a band near radius R, and divide the retained lattice frequencies into 3 to 8 patches associated with nonparallel sides of a fixed convex polygon such as a square or hexagon. Use separate complex weights for cross-patch products and then apply an inverse FFT followed by the usual residual block. (2) Pseudocode: precompute Gamma_i as the integer frequencies within radial distance h of R S_i and at least A tangential units from the side endpoints. For each pair xi in Gamma_i and zeta in Gamma_j with i not equal to j, compute k equal to zeta minus xi and insert the pair into a dictionary indexed by k. Either retain only the first M pairs per key or attach a learnable gate and optimize the collision penalty. During the forward pass, compute FFTs F and G, then set H[k] equal to the sum of W[xi,zeta] F[xi] G[zeta] over retained pairs with zeta minus xi equal to k. Add IFFT(H) to the residual stream. (3) Compute exactly from the geometry: patch membership, difference keys, multiplicities, and the separation test |xi-zeta| >= c_P A - 2. Estimate empirically the useful cap M, the value of c_P for the chosen polygon, spectral energy retention, and conditioning. (4) First experiment: train a 6-layer Fourier neural operator on a 64 by 64 periodic Burgers or Navier-Stokes dataset. Compare dense spectral multiplication, random frequency partitions, and polygonal transverse patches at equal parameter count and FFT cost. Measure validation error, loss at equal FLOPs, gradient-norm variance, high-frequency energy, and divergent-run frequency. A successful result would be lower validation error or substantially fewer unstable runs at the same retained-mode budget.