Commutator-flow latent block

(1) Integration point: use this as a residual block in a sequence Transformer or graph neural network before the readout. Store each node or token feature as a complex matrix H_i in C^(n x n). Enforce H_i <- (H_i-H_i^dagger)/2 after embedding and, if numerical drift occurs, after every block. For a sequence, define DH_i=(H_{i+1}-H_{i-1})/2 with padded boundaries. For a graph, use DH_i=sum_j A_ij(H_j-H_i), where A is a normalized adjacency matrix. (2) Pseudocode: initialize anti-Hermitian matrices from real features; compute G1=D(H), G2=D(G1), and optionally G3=D(G2); compute C=G1G2-G2G1; update H <- H-softplus(a1)G1-softplus(a2)C. In the third-order variant also subtract softplus(a3)(G3+1.5[D2H,C]), then apply a residual scale and anti-Hermitian projection. Add a linear readout from independent upper-triangular entries or conjugation-invariant traces. (3) The coefficients -1 and 3/2 come from the paper; a1, a2, a3, the step sizes, normalization, and graph operator are learned or tuned. Test equivariance by sampling a random unitary U and measuring the relative error between the block applied to UHU^dagger and U times the original output times U^dagger. (4) First experiment: train a small node-classification GNN on synthetic graphs whose labels depend only on spectra or traces of node matrices. Compare a 2-layer matrix MLP, standard message passing, and the commutator-flow block at matched parameter count. Measure validation accuracy under random basis conjugation, equivariance error, gradient norms, and loss at equal FLOPs. Success means near-zero conjugation error, no accuracy degradation under basis changes, and better accuracy or training stability than the unconstrained matrix baseline.