Gauge-Equivariant Complex Message Passing

1. Integration point: implement a directed message-passing layer for a graph batch. Store each node state as a complex tensor with shape [batch,nodes,channels], and assign every directed edge either a fixed gain φ_uv=e^{iθ_uv} derived from edge features or a learnable phase θ_uv. Do not use independent real and imaginary nonlinearities, since they break gauge equivariance. 2. Pseudocode: initialize complex states h; for each layer, compute q_v=complex_linear(h_v), messages m_u=sum over outgoing arcs (u,v) of exp(iθ_uv)q_v, normalize by degree, set z_u=W_self h_u+m_u, and update h_u=z_u*MLP_abs(|z_u|). Optionally concatenate invariant features |h_u| and sampled cycle gains. For a gauge test, sample random node phases σ_u, replace θ_uv by θ_uv−angle(σ_u)+angle(σ_v), and replace h_u by exp(−i angle(σ_u))h_u; outputs should transform by the same node phase. 3. The paper supplies the gain algebra and telescoping invariance; estimate phases from data or learn them, and use ordinary backpropagation through sin and cos. 4. First experiment: compare a 3-layer real directed GCN/GAT, a complex layer without gauge augmentation, and this equivariant layer on Cora or Citeseer with randomly oriented edges and a synthetic task where labels depend on directed cycle phase products. Measure accuracy, robustness to random node-wise phase reparameterization, and oversmoothing via pairwise feature cosine similarity. Success is unchanged predictions under gauge transformations and improved accuracy on cycle-dependent tasks at comparable parameter count.