Neural routing solvers rarely stop at their first solution: constructed tours are usually refined by search, such as sampling or local search. Neural improvement methods learn this search as a policy that repeatedly applies local moves, but existing policies degrade sharply beyond their training size. We argue that size generalization depends on whether the information that determines the value of a move reaches the policy in a size-independent form, and identify three places where existing policies fall short: the representation of the incumbent solution, the scale of the input features, and the computation of move gains. We address all three in NICO, a learned 2-opt policy for routing that represents the incumbent by its solution edges, one token per edge without positional encodings, describes them with scale-free features, and adds the exact gain of each move to its logit, so that the network learns a correction to greedy selection. A controlled factorial study shows that edge tokens outperform node tokens with positional encodings, that the exact gains are necessary for transfer in the tested configuration, and that scale-free features help only together with them. Trained on instances with at most 100 nodes, NICO reaches a 1.1% gap on the Euclidean TSP with 1000 nodes, against 10.7% for greedy 2-opt and 36% for the best prior learned policy. It also improves the tours of constructive and diffusion-based solvers, is the best improvement method on road-network costs, and on the CVRP outperforms every classical control.

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