Resonance-Gated Triadic Fourier Layer

Integrate the method at the Fourier mixing step of a Fourier neural operator, complex-valued CNN, or recurrent spectral world-model. Split the hidden tensor into three feature groups x_1, x_2, and x_3, compute their FFTs X_1, X_2, and X_3, and retain a bounded frequency grid. For every output frequency index k and summation index p, precompute R[k,p] from the formula R[k,p]=|p|^2/sigma_1+|k-p|^2/sigma_2-|k|^2/sigma_3. Construct the third output spectrum using Y_3[k]=W_3[k] sum_p exp(-(R[k,p]/tau)^2) X_1[p] X_2[k-p]. Add cyclic interactions Y_1 and Y_2 using the conjugation pattern of the source system: Y_1[k] uses conjugate X_2[p] times X_3[k-p], and Y_2[k] uses conjugate X_1[p] times X_3[k-p]. Apply inverse FFT, a residual connection, and a pointwise MLP. Initialize sigma_1 and sigma_2 to 1 and sigma_3 to 1.5, or use fixed physical coefficients if the data generator provides them. Parameterize tau as softplus(theta) if learning it; otherwise set tau to the 25th or 50th percentile of nonzero absolute R values. The paper supplies the mismatch and resonance principle; weights, channel grouping, and bandwidth are learned or estimated engineering choices. First test a four-layer model on 64 by 64 synthetic three-component NLS trajectories, comparing this layer with a same-size standard FNO and an ungated triadic-product layer. Evaluate one-step error, rollout relative L2 error, conserved-quantity drift, and training examples required to reach a target error. A shuffled-R ablation should remove the improvement. The expected signal is lower long-horizon error or stability drift at equal parameter count and FLOPs.