VAR-MATH: Probing True Mathematical Reasoning in Large Language Models via Symbolic Multi-Instance Benchmarks

Recent advances in reinforcement learning (RL) have led to substantial improvements in the mathematical reasoning abilities of large language models (LLMs), as measured by standard benchmarks. However, these gains often persist even when models are trained with flawed signals, such as random or inverted rewards, raising a fundamental question: do such improvements reflect true reasoning, or are they merely artifacts of overfitting to benchmark-specific patterns? To address this question, we take an evaluation-centric perspective and identify two critical shortcomings in existing protocols. First, \emph{benchmark contamination} arises from the public availability of test problems, increasing the risk of data leakage. Second, \emph{evaluation fragility} stems from the reliance on single-instance assessments, which are highly sensitive to stochastic outputs and fail to capture reasoning consistency. To overcome these limitations, we introduce {VAR-MATH}, a symbolic evaluation framework designed to probe genuine reasoning ability. By converting fixed numerical problems into symbolic templates and requiring models to solve multiple instantiations of each, VAR-MATH enforces consistent reasoning across structurally equivalent variants, thereby mitigating contamination and improving evaluation robustness. We apply VAR-MATH to transform two popular benchmarks, AMC23 and AIME24, into their symbolic counterparts, VAR-AMC23 and VAR-AIME24. Experimental results reveal substantial performance drops for RL-trained models on the variabilized versions, especially for smaller models, with average declines of 48.0\% on AMC23 and 58.3\% on AIME24. These findings suggest that many existing RL methods rely on superficial heuristics and fail to generalize beyond specific numerical forms. Overall, VAR-MATH offers a principled, contamination-resistant evaluation paradigm for mathematical reasoning.