Logical reasoning is a key capability of large language models, yet current benchmarks focus almost entirely on tasks that just check basic logical consistency and overlook the reflective reasoning required for paradox detection and resolution. To fill the gap, we present ParaSuite, the first pipeline dedicated to paradox research that automates data synthesis, evaluation, and training. We introduce PARADOX, a synthetic, high-quality data spanning two difficulty tiers and three academic domains, accompanied by specialized evaluation metrics and solving algorithms. We propose ParadoxBreaker-7B, trained with Mutual-Information Guided Fine-Tuning and reinforcement learning step verify paradox reward(PAPO). Experiments demonstrate significant improvements in both paradoxical and general STEM reasoning.

Reinforcement Learning with Verifiable Rewards (RLVR) improves the reasoning capability of Large Language Models (LLMs). Current RLVR trains LLMs on all generated tokens, rather than exploring which tokens actually contribute to reasoning. We propose AIPO(Adaptive–Information Policy Optimization), which focuses updates on those decisive tokens discovered on the fly. AIPO estimates each hidden state’s mutual information to score tokens. Policy gradients are then computed only on these critical tokens, using an advantage that blends information gain and verifiable correctness. To improve the efficiency of mutual-information estimation, AIPO adopts a Random–Fourier approximation of the Hilbert–Schmidt Independence Criterion. Across five math and science benchmarks, AIPO yields up to +20% accuracy over strong RLVR baselines while updating merely 10% of tokens, demonstrating superior efficiency and effectiveness. Our findings highlight the importance of information–driven token selection for efficient and effective reinforcement learning of LLM reasoning.