Reinforcement learning with verifiable rewards (RLVR) has become a widely used approach for improving the mathematical reasoning ability of large language models (LLMs). However, recent studies show that LLMs may solve a math problem in its original form but fail when the same problem is reworded without changing its meaning. This suggests that the model has not fully acquired the underlying reasoning ability and instead relies on the surface form of the problem. In this paper, we propose PEARL, an RLVR framework that uses different versions of the same math problem as structured training signals. PEARL first groups each original problem with its checked variants to build semantic perturbation clusters, and then uses partially solved clusters as the active RL training set. PEARL further performs cluster-level RL optimization by comparing model outputs across different versions of the same mathematical problem. Experiments on mathematical reasoning benchmarks show that PEARL achieves strong results against competitive RLVR baselines while using a small active training set. These results suggest that semantically equivalent variants can serve as useful training units for improving mathematical reasoning in LLMs.

Reinforcement learning with verifiable or outcome-level rewards has recently become an effective post-training paradigm for improving the reasoning abilities of large language models, with GRPO-style optimization achieving strong results on mathematical and other verifiable tasks. However, for multi-step reasoning, outcome-level supervision remains intrinsically sparse: a trajectory that contains partially correct intermediate derivations but makes a late mistake is still treated as a complete failure. As a result, useful reasoning prefixes are penalized together with erroneous continuations, and informative failed trajectories are underutilized. PRMs provide denser step-level feedback, but they often rely on costly process annotation or auxiliary process verifiers, and optimizing local process rewards alone may fail to preserve the global objective of final correctness, leaving room for reward misspecification and reward hacking during reinforcement fine-tuning. We propose PRSA(Process Rewarding via Solution-Space Alignment), a simple framework that bridges outcome supervision and process supervision without requiring human step-level labels. PRSA automatically extracts intermediate-value steps from model outputs and reference solutions, aligns them within the solution space under equivalence and order constraints to estimate reasoning progress, and uses this signal for both reward shaping and token-level prefix-suffix credit assignment. In this way, PRSA preserves the global anchor of final-answer correctness while providing denser and better localized learning signals for partially correct trajectories.