Lie-Bracket Fourier Mixer

Integrate this at the mode-mixing operation of a Fourier neural operator, spectral sequence model, or any layer whose hidden state is $X\in\mathbb R^{B\times N\times 3}$, where $N$ is the number of modes or ordered latent positions. First normalize each 3-vector with $x_i\leftarrow x_i/(\|x_i\|_2+10^{-6})$. Implement the three primitive fields as g1(x)=(0,x[...,2],-x[...,1]), g2(x)=(-x[...,2],0,x[...,0]), and g3(x)=(x[...,1],-x[...,0],0). A shifted field G(r,k,X) gathers X[:,i+k,:]; use circular shifts for a periodic spectral grid or zero padding for one-sided Fourier indices. For each learned bracket edge (m,k,n,l,alpha), apply either an explicit Lie-word residual Y = X + alpha * bracket(G(m,k),G(n,l),X) or the more stable four-flow product: apply rotations generated by G(m,k) and G(n,l) with signs (+eps,+eps,-eps,-eps) in reverse composition order. Use a small fixed set of primitive offsets, such as k,l in {0,1,2,4}, and learn scalar coefficients and a final linear combination of bracket outputs. Re-normalize after the residual if using the Euler bracket approximation; the exact rotation-flow version needs no projection. The mathematical quantities are the tangent fields, additive offset rule, and commutator construction; estimate coefficients by backpropagation and tune eps empirically. Start on CIFAR-10 or a synthetic periodic advection dataset with a 2D Fourier layer retaining 16 modes. Compare a dense spectral mixing matrix and a standard 1D convolution at matched FLOPs. Success means at least 2x fewer mixing parameters, no exploding norms, and equal or better validation accuracy; additionally test whether increasing the maximum bracket depth improves high-frequency reconstruction without proportionally increasing parameter count.