# Mathematical Proof as a Litmus Test: Revealing Failure Modes of Advanced Large Reasoning Models

arXiv: https://arxiv.org/abs/2506.17114 · HTML: https://mlsanity.com/paper/2506.17114/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 28 of its 53 references that are in the map.
- First posted on arXiv 20 Jun 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 28 of 28)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [A Survey on LLM-as-a-Judge](https://mlsanity.com/paper/2411.15594/) (2024)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition](https://mlsanity.com/paper/2504.21801/) (2025)
- [ProofNet: Autoformalizing and Formally Proving Undergraduate-Level Mathematics](https://mlsanity.com/paper/2302.12433/) (2023)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [MATH-Perturb: Benchmarking LLMs' Math Reasoning Abilities against Hard Perturbations](https://mlsanity.com/paper/2502.06453/) (2025)
- [Proof or Bluff? Evaluating LLMs on 2025 USA Math Olympiad](https://mlsanity.com/paper/2503.21934/) (2025)
- [FIMO: A Challenge Formal Dataset for Automated Theorem Proving](https://mlsanity.com/paper/2309.04295/) (2023)
- [Logical Reasoning in Large Language Models: A Survey](https://mlsanity.com/paper/2502.09100/) (2025)
- [miniCTX: Neural Theorem Proving with (Long-)Contexts](https://mlsanity.com/paper/2408.03350/) (2024)
- [RealMath: A Continuous Benchmark for Evaluating Language Models on Research-Level Mathematics](https://mlsanity.com/paper/2505.12575/) (2025)
- [PAG: Multi-Turn Reinforced LLM Self-Correction with Policy as Generative Verifier](https://mlsanity.com/paper/2506.10406/) (2025)
- [MA-LoT: Model-Collaboration Lean-based Long Chain-of-Thought Reasoning enhances Formal Theorem Proving](https://mlsanity.com/paper/2503.03205/) (2025)
- [DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning](https://mlsanity.com/paper/2505.23754/) (2025)
- [Large Language Models and Mathematical Reasoning Failures](https://mlsanity.com/paper/2502.11574/) (2025)
- [Solving Inequality Proofs with Large Language Models](https://mlsanity.com/paper/2506.07927/) (2025)
- [SedarEval: Automated Evaluation using Self-Adaptive Rubrics](https://mlsanity.com/paper/2501.15595/) (2025)
- [MathConstruct: Challenging LLM Reasoning with Constructive Proofs](https://mlsanity.com/paper/2502.10197/) (2025)
- [One Example Shown, Many Concepts Known! Counterexample-Driven Conceptual Reasoning in Mathematical LLMs](https://mlsanity.com/paper/2502.10454/) (2025)
- [MPBench: A Comprehensive Multimodal Reasoning Benchmark for Process Errors Identification](https://mlsanity.com/paper/2503.12505/) (2025)
- [MATP-BENCH: Can MLLM Be a Good Automated Theorem Prover for Multimodal Problems?](https://mlsanity.com/paper/2506.06034/) (2025)
- [Let's Reason Formally: Natural-Formal Hybrid Reasoning Enhances LLM's Math Capability](https://mlsanity.com/paper/2505.23703/) (2025)

## Built on by (most cited first, 20 of 20)

- [Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs](https://mlsanity.com/paper/2605.00674/) (2026)
- [CostBench: Evaluating Multi-Turn Cost-Optimal Planning and Adaptation in Dynamic Environments for LLM Tool-Use Agents](https://mlsanity.com/paper/2511.02734/) (2025)
- [A Comprehensive Survey on Trustworthiness in Reasoning with Large Language Models](https://mlsanity.com/paper/2509.03871/) (2025)
- [Are Your Agents Upward Deceivers?](https://mlsanity.com/paper/2512.04864/) (2025)
- [When Judgment Becomes Noise: How Design Failures in LLM Judge Benchmarks Silently Undermine Validity](https://mlsanity.com/paper/2509.20293/) (2025)
- [PlanBench-XL: Evaluating Long-Horizon Planning of LLM Tool-Use Agents in Large-Scale Tool Ecosystems](https://mlsanity.com/paper/2606.22388/) (2026)
- [Diversity-Enhanced Reasoning for Subjective Questions](https://mlsanity.com/paper/2507.20187/) (2025)
- [AdaPlanBench: Evaluating Adaptive Planning in Large Language Model Agents under World and User Constraints](https://mlsanity.com/paper/2606.05622/) (2026)
- [Towards Self-Evolving Benchmarks: Synthesizing Agent Trajectories via Test-Time Exploration under Validate-by-Reproduce Paradigm](https://mlsanity.com/paper/2510.00415/) (2025)
- [Rethinking Prospect Theory for LLMs: Revealing the Instability of Decision-Making under Epistemic Uncertainty](https://mlsanity.com/paper/2508.08992/) (2025)
- [RefGrader: Automated Grading of Mathematical Competition Proofs using Agentic Workflows](https://mlsanity.com/paper/2510.09021/) (2025)
- [Student Guides Teacher: Weak-to-Strong Inference via Spectral Orthogonal Exploration](https://mlsanity.com/paper/2601.06160/) (2026)
- [Not All Proofs Are Equal: Evaluating LLM Proof Quality Beyond Correctness](https://mlsanity.com/paper/2605.10379/) (2026)
- [Code2Math: Can Your Code Agent Effectively Evolve Math Problems Through Exploration?](https://mlsanity.com/paper/2603.03202/) (2026)
- [Curing Miracle Steps in LLM Mathematical Reasoning with Rubric Rewards](https://mlsanity.com/paper/2510.07774/) (2025)
- [The Architecture of Errors: From Universal Impossibility to Patch-Local LLM Reliability](https://mlsanity.com/paper/2605.30628/) (2026)
- [CORE: Concept-Oriented Reinforcement for Bridging the Definition-Application Gap in Mathematical Reasoning](https://mlsanity.com/paper/2512.18857/) (2025)
- [Reliable Use of Lemmas via Eligibility Reasoning and Section-Aware Reinforcement Learning](https://mlsanity.com/paper/2602.00998/) (2026)
- [Psi-Bench: Evaluating Persona-Sensitive Influencing in Persuasive Dialogues](https://mlsanity.com/paper/2606.02754/) (2026)
- [Hallucination Across the Reasoning Lifecycle: Interface Visibility, Causal Evidence, and Release Control in Large Reasoning Models](https://mlsanity.com/paper/2610.05472/) (2026)

## Abstract (arXiv)

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, reliance on purely numerical evaluation and potential benchmark leakage, often masks 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, and thoroughly evaluate advanced models' performance on it. 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 grapple profoundly with mathematical proofs, with some generating entirely correct proofs for less than 20% of problems and failing even on basic ones; 2) models exhibit a diverse spectrum of reasoning failures, prominently demonstrating the lack of guarantees for the correctness and rigor of single-step reasoning; and 3) models show hallucination and incompleteness during the reasoning process. Our findings reveal that models' self-reflection is insufficient to resolve the current logical dilemmas, necessitating formalized and fine-grained logical training.