Tracing Error Origins and Propagation: Dependency-Grounded Process Supervision for Mathematics and Physics

Mathematical and physical reasoning is central to AI for Science, and process reward models help improve it by evaluating intermediate steps. Derivations in these domains reuse quantities, equations, and assumptions, creating identifiable dependencies between the information a step uses and the results it produces. These dependencies provide a basis for distinguishing newly introduced mistakes from inherited errors. We propose dependency-grounded process supervision, which extracts these input–output relationships and checks whether each error arises locally or is inherited through the information used. The resulting supervision teaches a model to judge step correctness, distinguish new mistakes from inherited consequences, and classify newly introduced mistakes. This labeling preserves valid uses of information from partially incorrect steps and accounts for steps that both inherit an error and introduce another. Experiments in mathematics and physics show improved error detection compared with models trained to judge step correctness, with gains persisting on longer mathematical solutions. Controlled comparisons show that learning to distinguish new and propagated errors improves diagnosis beyond rules based on correctness and step order, with similar detection performance. The learned scores also support answer selection and training-data selection. These findings suggest that reasoning dependencies can guide the design of more informative feedback about how errors arise and propagate in scientific problem solving.