Finite-group relative message passing

Modify a message-passing GNN's edge update rather than its node encoder. Precompute the $\varphi$-based partition and quotient labels. For each directed edge $(u,v)$, store its signed generator index $(i,s)$ and quotient displacement $g_{uv}=s a_i$. Replace an edge-specific MLP or unconstrained relation embedding with a shared bank of matrices $W_i$; optionally use separate matrices for positive and negative orientations. Each layer executes msg = gate(h[u], h[v], embed(g_uv)) * (W[class_uv, sign_uv] @ h[u]); aggregate messages by destination, then apply a residual update and normalization. In a transformer variant, use the displacement embedding as an additive attention bias while retaining parameter tying across all edges in the same class. Compute a spanning-tree cycle basis and the integer cycle matrix offline; Smith normal form is needed only during preprocessing, so training performs simple class and coordinate lookups. Record the compression ratio $|E|/k$, parameter count, throughput, and validation accuracy. The first cheap experiment should use 2-6 layer models on synthetic Cayley graphs and graph-isomorphism-style node labeling, followed by transfer between graph sizes. Compare GCN, GAT, and R-GCN with equal parameter budgets. Expected success is equal or better validation accuracy with fewer relation parameters, improved accuracy on larger held-out graphs, and faster inference. An ablation must compare exact cycle-quotient labels against raw local edge classes; removing cycle consistency should reduce transfer and expose whether the algebraic construction matters.