PAPER: INTRODUCTIONLarge language models (LLMs) are making rapid progress on mathematical tasks, achieving strong results on challenging benchmarks like AIME (Balunovic et al., 2025) and FrontierMath (Glazer et al., 2024). These improvements suggest that LLMs may soon support mathematical research by collaborating with professional mathematicians on open problems. However, to determine whether current systems are capable of contributing in such settings, benchmarks are needed that test capabilities at the frontier of mathematical research.Limitations of existing benchmarks Existing benchmarks fall short of this objective: most focus on high-school or undergraduate-level mathematics (Balunovic et al., 2025;Frieder et al., 2023), due to the difficulty associated with designing rigorous, research-level problems. The few benchmarks that do target more advanced mathematics, like FrontierMath (Glazer et al., 2024) and HLE (Phan et al., 2025), focus exclusively on final-answer problems. As a result, they overlook proof-writing capabilities and allow models to apply shortcuts to reach the correct final answer without fully solving the problem (EpochAI, 2025).This work: IMProofBench To fill this gap, we introduce IMProofBench, a private benchmark developed in collaboration with the mathematical research community to evaluate LLMs on researchlevel proof writing. IMProofBench is built on a custom platform and supported by initiatives that actively involve professional mathematicians. It includes tasks ranging from challenging oral exam questions in a graduate course to open research questions based on the contributors' own work. Unlike static benchmarks, IMProofBench is designed as a platform for continuous evaluation: problems are added on a rolling basis, ensuring its continued relevance for evaluating the next generation of frontier LLMs. Currently, IMProofBench consists of 54 problems developed in collaboration with over 35 mathematicians, with 29 more questions in the latest stages of the problem creation pipeline.Under review as a conference paper at ICLR 2026
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REVIEW
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**Summary Of The Paper**  
The paper introduces *IMProofBench*, a dynamically updated, privately curated benchmark for evaluating large language models (LLMs) on research-level mathematical proof writing. It addresses limitations of existing benchmarks (e.g., focus on final answers, lack of research-level problems) by collaborating with mathematicians to create problems spanning graduate exams to open research questions. The paper reports that GPT-5 achieves full grades on a subset of problems, while other models struggle, and highlights challenges in evaluating proof-writing versus final-answer capabilities.  

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**Strengths**  
- **Novelty in Targeting Research-Level Proof Writing**: The paper directly addresses a critical gap in LLM evaluation by focusing on proof-writing rather than final-answer tasks, which prior benchmarks like FrontierMath and HLE neglect. This aligns with the goal of assessing LLMs’ suitability for collaborative mathematical research.  
- **Collaboration with Mathematicians**: Problems are authored by professionals in their fields, ensuring domain relevance and rigor. The peer-review process (with input from both administrators and external experts) strengthens the quality of contributed questions.  
- **Continuous Update Mechanism**: The rolling addition of problems ensures the benchmark remains relevant for evaluating emerging LLMs, distinguishing it from static benchmarks like MATH or FrontiersMath.  

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**Weaknesses**  
1. **Lack of Public Release and Reproducibility (Severe)**: The benchmark is labeled “private,” restricting external validation and replication of results. Without access to the full dataset or a public subset, claims about GPT-5’s performance cannot be independently verified. This undermines trust in the empirical findings.  
2. **Minimal Statistical Analysis (Major)**: Results are presented descriptively (e.g., “small but non-trivial fraction”) without error bars, confidence intervals, or statistical tests (e.g., p-values). For example, the assertion that GPT-5 solves “a small but non-trivial fraction” of problems lacks quantification (e.g., 5% vs. 20%) or comparison against chance levels.  
3. **Ambiguous Definition of “Research-Level” Problems (Moderate)**: The paper does not clarify how its problems differ from those in FrontierMath (which targets advanced mathematics) or HLE (crowdsourced expert-level questions). This obscures the uniqueness of IMProofBench’s contribution.  
4. **Insufficient Detail on Problem Difficulty Validation (Minor)**: The peer-review process focuses on mathematical correctness but lacks explicit criteria for quantifying problem difficulty (e.g., average time taken by GPT-5 to solve problems, or whether problems are solvable by humans). This leaves uncertainty about whether problems are appropriately calibrated.  

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**Questions For The Authors**  
1. **Reproducibility**: Given the private nature of IMProofBench, what steps will be taken to ensure transparency? Will a subset of problems or anonymized evaluation logs be released for third-party validation?  
2. **Statistical Grounding**: What proportion of problems did GPT-5 solve correctly (e.g., out of 54 total)? Are these results accompanied by confidence intervals or hypothesis tests comparing performance across models (e.g., GPT-5 vs. GROK-4)?  
3. **Differentiation from Prior Work**: How does IMProofBench’s problem selection criteria explicitly differentiate itself from FrontierMath (Glazer et al., 2024) and HLE (Phan et al., 2025)? Are there quantitative metrics (e.g., problem complexity, required background knowledge) to justify this distinction?  
4. **Formula Verification**: In the example problem (Section 1), how was the correctness of the proposed closed-form expression for $ N_g $ verified? Was it cross-checked with known combinatorial results or independently derived by multiple reviewers?  
5. **Tool Configuration Details**: The evaluation environment grants access to tools like SageMath and Python. Are specific versions of these tools (e.g., SageMath 9.7, Python 3.11) documented? How were tool usage patterns monitored to prevent exploitation (e.g., excessive API calls)?  

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**Limitations Not Addressed By The Authors**  
- **Bias in Problem Distribution**: The paper does not report the distribution of problem topics (e.g., algebraic geometry, topology) or the representation of subfields. This raises concerns about generalizability and potential bias toward certain areas of mathematics.  
- **Risk of Model Memorization**: Problems are authored by mathematicians who may collaborate with AI developers. No mitigation strategies are discussed to prevent models from memorizing solutions (e.g., obfuscation techniques or diversity checks).  
- **Subjectivity in Human Grading**: The paper does not specify whether human graders were blinded to model identities or how their qualifications were standardized. This introduces potential bias in proof-evaluation outcomes.  

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**Soundness**: 2 (Fair)  
While the conceptual framework is compelling, the absence of statistical rigor, reproducibility, and detailed methodological descriptions weakens the empirical claims.  

**Contribution**: 3 (Good)  
The introduction of a novel benchmark targeting research-level proof writing is valuable, but its impact is constrained by limited accessibility and underdeveloped analytical depth.  

**Confidence**: 3 (Medium)  
Key components (e.g., problem creation process, evaluation setup) are plausible, but critical details (e.g., statistical validation, benchmark access) remain opaque.  

**Rating**: 6 (Accept with Major Revisions)  
The paper makes a meaningful contribution to the field but requires substantial revisions to address reproducibility, statistical grounding, and clarity in defining its novelty relative to existing benchmarks.  

**Brief Justification For Rating**:  
IMProofBench represents a timely attempt to advance LLM evaluation in mathematical proof writing. However, its acceptance hinges on resolving severe issues related to reproducibility and statistical validation. Without these, the empirical claims lack robustness, and the benchmark’s utility to the broader research community remains uncertain.

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