VeriBound framework proves AI reasoning models generalize with formal verification

Process Reward Models (PRMs) offer step-level verification for LLM reasoning, but acquiring training data is a bottleneck: human annotation is expensive and Monte Carlo rollouts are noisy. A recent approach, FOVER, trains PRMs on step-level error labels automatically generated by formal verification tools like Z3 and Isabelle, showing surprising cross-task generalization from symbolic tasks to diverse reasoning benchmarks. However, until now, no theoretical explanation existed for this phenomenon.

VeriBound fills that gap with rigorous PAC-Bayesian analysis. The framework establishes four key results: (1) a generalization bound relating empirical verification error on formal verification-annotated data to expected error on unseen tasks, depending on formal verification accuracy and distribution divergence; (2) a sample complexity bound showing O(d log(d/δ)/ε²) annotated examples suffice for ε error with 1-δ probability; (3) linear convergence rate for PRM training under L-smoothness and bounded variance; and (4) an error propagation bound connecting step-level verification accuracy to degradation in Best-of-K performance. This provides a mathematical foundation for scaling PRMs without expensive human labels.