Task Similarity Predicts In-Context Transfer on Graphs

Whether a graph foundation model will learn a new task in context cannot currently be predicted before training. The question is open because alignment theory for in-context learning is derived for linear models and does not reach graphs, and because standard graph few-shot evaluations conflate in-context inference with structural shortcuts such as label propagation. We construct a cross-graph rule-induction testbed that closes these shortcut channels, with no edge between demonstrations and queries and with the label polarity randomized per episode. On this testbed a model's in-context gain is governed by a single measurable scalar, the polarity-symmetrized label agreement between the training task and the test task. A Bayesian posterior-concentration argument predicts the full few-shot gain surface with no free parameters, and predicts that adding graph families or rules brings no gain unless it changes the agreement statistics. The law holds on synthetic rules, on five real graph datasets, and on a frozen, externally pretrained graph foundation model (GraphPFN). In-context transfer on graphs can therefore be forecast, and pretraining task design guided, from labeled examples alone, without model access or any training.