Half-Idleness Curvature Attention

Integrate this at the edge-logit stage of a GAT or graph-transformer layer. First construct, for every edge (u,v), the graph-distance cost matrix between the support sets {u} union N(u) and {v} union N(v). Set p=0.5. Assign mass 0.5 to each endpoint and distribute the remaining mass uniformly across its neighbors; for a weighted graph, distribute neighbor mass proportionally to the edge weights after normalizing them to sum to one. Solve the resulting small discrete Wasserstein-1 linear program and cache W_1 for every edge. Set kappa[u,v]=2*(1-W), using the theorem's exact p=1/2 identity. Compute s=median(abs(kappa))+1e-6 over all directed edges, then kappa_norm=clip(kappa/s,-3,3). In each attention layer replace the usual logit q_u transpose k_v divided by sqrt(d) with q_u transpose k_v divided by sqrt(d) plus beta times kappa_norm[u,v], where beta is a learned scalar initialized to zero so the model initially matches ordinary attention. Compact pseudocode is: for (u,v) in edges: mu_u,mu_v=lazy_measure(u,v,p=0.5); W=wasserstein1(mu_u,mu_v,graph_distance); kappa[u,v]=2*(1-W); then logits=QK/sqrt(d)+beta*kappa_norm. The theorem determines the factor 2 and the valid p threshold; LP solver choice, clipping, and normalization are engineering decisions. Start with Cora, Citeseer, or a small molecular graph benchmark, using a two-layer GAT with matched parameter count. Compare ordinary GAT, degree-biased GAT, curvature-biased GAT, and randomized-curvature GAT over at least five seeds. Success means higher validation accuracy or calibration at equal training FLOPs, especially on heterophilous edges, without optimization instability. Also report one-time curvature preprocessing cost and an ablation showing that learned beta is nonzero and that shuffled curvature removes the gain.