Path-and-cut separation layer

Attach the procedure between the forward prediction and loss computation; it is a training-time graph routine and does not change the CNN or GNN forward pass. Maintain an active set A containing at most K constraints per image. Every T training steps, run a cheap oracle. For each sampled pair f={p,q}, assign each graph edge cost x_e and find 3-5 low-cost simple paths with beam search or Dijkstra; evaluate r_P using the exact formula. Also compute an approximate minimum vertex cut by node-splitting each graph vertex into in/out nodes with capacity one, run max-flow, and evaluate r_C on the returned cut. For local four-node configurations, directly evaluate r_I over nearby rectangles. Keep the top K positive residuals, detach the graph-search decisions, and optimize the differentiable penalty for the next T steps. Pseudocode: A=[]; every T steps: for f in sampled_F: paths=beam_paths(f,x_edge); for P in paths: A.append((rP(P),P)); S=mincut(f,node_cost=1-xv); A.append((rC(f,S),S)); for I in local_intersections: A.append((rI(I),I)); A=topK(filter_positive(A)); L=L_task+lambda*sum(relu(r)**2 for r,constraint in A); optimizer.step(). The exact mathematical computation is the residual for each selected inequality; path, cut, and rectangle discovery are approximations. Start with a 64x64 synthetic grid containing touching objects, a small U-Net predicting node and pair probabilities, and compare against cross-entropy plus dense pairwise smoothness. Report wall-clock overhead, held-out violation rate, segmentation Rand index, and performance at fixed training FLOPs. Success means lower violations and better object separation with fewer constraint evaluations than dense enumeration.