Reinforcement learning with verifiable rewards (RLVR) has become an effective approach for improving mathematical reasoning in large language models. However, a fundamental question remains: when a model fails to solve a problem, does it truly lack the underlying capability?
Recoverable Capability Frontier
In this work, we reveal that direct failure does not necessarily imply inability. Some reasoning failures can be recovered by exploring the model's own trajectories, exposing a recoverable capability frontier: a policy-dependent region where solutions are not reliably generated through direct decoding but can be elicited through trajectory-level search.
The ReSI-Math Framework
Based on this observation, we introduce ReSI-Math, a framework that explores recoverable capability frontiers and converts recovered capabilities into data-efficient RLVR supervision. ReSI-Math uses Self-Judge-guided evolutionary search to identify recoverable failures under a fixed policy and adaptively selects frontier problems for sequential RLVR as the policy evolves.
Experimental Results
On Qwen3-VL-8B, correct-incorrect trajectory crossover produces 272 verifier-confirmed correct offspring from 621 attempts (43.8%).
- After three sequential one-shot RLVR stages, the MATH500 and WE-MATH average improves from 78.71 to 80.16.
- Under matched single-problem RLVR, random selection reaches only 53.31 versus 54.16 for frontier-guided Q1.