Large reasoning models ( e.g., R1, o3) have demonstrated remarkable mathematical problem-solving abilities. However, the high reported accuracy of these advanced models on popular datasets and reliance on purely numerical evaluation often mask their true reasoning shortcomings. To address this, we propose leveraging the inherent rigor and methodological complexity of mathematical proofs as a diagnostic tool to expose these hidden failures. Specifically, we introduce the RFMDataset (Reveal Failure Modes), a collection of 200 diverse mathematical proof problems to thoroughly evaluate the performance of advanced models. Our in-depth analysis of their failures uncovers 10 fine-grained error types, which shows fundamental limitations in current large reasoning models: 1) Large reasoning models still have limited capability in generating entirely correct mathematical proofs, with some models solving less than 20% of problems and even making mistakes on fundamental ones; 2) models exhibit a diverse spectrum of reasoning failures, prominently demonstrating the lack of guarantees for the correctness and rigor intermediate reasoning steps; and 3) models show hallucination and incompleteness during the reasoning process. Our findings also reveal that directly prompting models to self-reflect on specific failure modes is insufficient to resolve the current logical dilemmas, necessitating domain knowledge and formal verification.

Enhancing the mathematical reasoning capabilities of LLMs has garnered significant attention in both the mathematical and computer science communities. Recent works have made substantial progress in both Natural Language (NL) reasoning and Formal Language (FL) reasoning by leveraging the potential of pure Reinforcement Learning (RL) methods on base models. However, RL approaches struggle to impart new capabilities not presented in the base model, highlighting the need to integrate more knowledge like FL into NL math reasoning effectively. Yet, this integration is challenging due to inherent disparities in problem structure and reasoning format between NL and FL. To address these challenges, we introduce **NL-FL HybridReasoning (NFL-HR)**, an end-to-end framework designed to incorporate the FL expert into NL math problem-solving. To bridge the NL and FL input format gap, we propose the *NL-FL Problem Alignment* method, which reformulates the Question-Answering (QA) problems in NL as existence theorems in FL. Subsequently, the *Mixed Problem Input* technique we provide enables the FL reasoner to handle both QA and existence problems concurrently. Lastly, we mitigate the NL and FL output format gap in reasoning through an LLM-based *Answer Extraction* mechanism. Comprehensive experiments demonstrate that the **NFL-HR** framework achieves **89.80%** and **84.34%** accuracy rates on the MATH-500 and the AMC benchmarks, surpassing the NL baseline by 4.60% and 4.82%, respectively. Notably, some problems resolved by our framework remain unsolved by the NL baseline model even under a larger number of trials.

Recently, Large Language Models (LLMs) have demonstrated a superior ability to serve as ranking models. However, concerns have arisen as LLMs will exhibit discriminatory ranking behaviors based on users’ sensitive attributes (gender). Worse still, in this paper, we identify a subtler form of discrimination in LLMs, termed implicit ranking unfairness, where LLMs exhibit discriminatory ranking patterns based solely on non-sensitive user profiles, such as user names. Such implicit unfairness is more widespread but less noticeable, threatening the ethical foundation. To comprehensively explore such unfairness, our analysis will focus on three research aspects: (1) We propose an evaluation method to investigate the severity of implicit ranking unfairness. (2) We uncover the reasons for causing such unfairness. (3) To mitigate such unfairness effectively, we utilize a pair-wise regression method to conduct fair-aware data augmentation for LLM fine-tuning. The experiment demonstrates that our method outperforms existing approaches in ranking fairness, achieving this with only a small reduction in accuracy. Lastly, we emphasize the need for the community to identify and mitigate the implicit unfairness, aiming to avert the potential deterioration in the reinforced human-LLMs ecosystem deterioration.