Fixed/Anti-Fixed Signed Message Passing

Integrate this as a drop-in layer in a signed GNN before node classification or graph pooling. Store every node state as h_i=(u_i,v_i), where u_i in R^d is fixed and v_i in R^d is anti-fixed. For each node, compute the unsigned aggregate a_i=sum_{j in N(i)} h_j and signed aggregate b_i=sum_{j in N(i)} sigma_ij h_j, with sigma_ij in {+1,-1}. Update u_i with an MLP receiving [u_i,a_i^u,b_i^u] and update v_i with a separate MLP receiving [v_i,a_i^v,b_i^v]. Optionally normalize (u_i,v_i) to unit norm to match the spherical representation. For a switching gauge s_i in {+1,-1}, transform edge signs as sigma'_ij=s_i sigma_ij s_j, leave u_i unchanged, and transform v_i to s_i v_i. Check equivariance by measuring ||u'_i-u_i|| and ||v'_i-s_i v_i||. Pool only invariant quantities, such as sum_i u_i and sum_i(v_i squared), or use a permutation-invariant readout after an invariant projection. Add a graph-output consistency loss between the original graph and randomly switched copies. The involution and switching law are mathematically fixed; channel widths, MLPs, normalization, and nonlinearities are empirical choices. First test on signed stochastic block models and a small real signed-node-classification dataset against a two-layer signed GCN. Success means unchanged graph predictions under random switching, improved test accuracy on unseen gauges, and stable gradients with only a modest FLOP increase.