Null-form quadratic wave layer

(1) Integrate the layer into a neural PDE surrogate or world model that receives a space-time feature field z with shape batch by channels by time by spatial dimensions. Apply it after a convolutional or state-space block and before the residual state update; Dt and Dj are centered finite differences or automatic derivatives of the feature field, not parameter gradients. Add the output to the predicted wave acceleration or next-state residual. Learn the channel coefficients a and b, optionally using low-rank factorization to reduce parameters. (2) Pseudocode: compute Dt[z] = (z[t+1] - z[t-1])/(2 delta_t) and Dj[z] = (z[xj+1] - z[xj-1])/(2 delta_xj). Initialize output F to zero. For every output channel r and pair k greater than or equal to l, add a[r,k,l] times (-Dt[k] times Dt[l] plus the sum over j of Dj[k] times Dj[l]). For every i greater than j and k greater than l, add b[r,i,j,k,l] times (Di[k] times Dj[l] minus Dj[k] times Di[l]). Apply boundary padding consistently and add F to the model residual. (3) The null-form identities and cancellation are computed exactly from the formulas; coefficients are learned empirically. Use Fourier probes z_k = A_k exp(i times (omega t plus xi dot x)) with omega equal to the Euclidean norm of xi, and verify that the Q0 response is numerically close to zero for parallel lightlike waves. Use non-null random frequencies as a control. (4) First experiment: train a small 2-plus-1-dimensional wave solver on finite-difference trajectories generated from random smooth initial conditions. Compare a standard pointwise quadratic interaction, a parameter-matched MLP residual, and the null-form layer at equal FLOPs. Measure one-step error, gradient norms, spectral energy growth, and 100- to 500-step rollout error. Success means lower high-frequency energy blow-up and a longer stable rollout at comparable one-step accuracy; a useful target is at least twice the stable horizon or 20 percent lower rollout error.