ProofCouncil: An LLM Agent for Solving Open Mathematical Problems

Large language models (LLMs) have shown increasing promise in solving open problems in mathematics. However, their performance can be further improved through agentic workflows tailored to real-world mathematical practice. To this end, we introduce ProofCouncil, a mathematical agent that is designed to tackle open problems using an author-critic architecture. ProofCouncil served as a submission to the second batch of FirstProof, a challenge consisting of 10 real-world mathematical problems that agents must solve autonomously. Its submissions for 6 of the 10 problems were judged by the referees to be correct up to at most minor revisions, showing the best performance among participating teams. We also evaluate ProofCouncil on 30 open problems collected from mathematical researchers. Among the 21 solutions that received human feedback, 5 were judged completely correct, 2 more were judged promising pending final verification, and a further 8 contained useful partial progress. In this short paper, we describe the development of ProofCouncil and the agent-building library used to create it, which we release as open source to the community.

Human vs Machine Mathematical Difficulty on Project Euler: An Experimental Analysis

We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation $t_{\text{machine}} = a \cdot t_{\text{human}}^b$ between generated-token cost per successful answer and human time, and find $b < 1$ for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay $p_{\text{success}} = e^{c t_{\text{human}}}$, predicting a linear relation between $\log p_{\text{success}}$ and $t_{\text{human}}$. Using a binning approach for data aggregation we find moderate empirical support (median bin-level $R^2 = 0.92$ across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50\% task-length horizons $h_{50}$; the strongest configurations in our 20 April 2026 snapshot reach roughly $2.5$--$4.3$ hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about $75$~days for the SOTA $h_{50}$.

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 3,000 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

Points of non-linearity of functions generated by random neural networks

We consider functions from the real numbers to the real numbers, output by a neural network with 1 hidden activation layer, arbitrary width, and ReLU activation function. We assume that the parameters of the neural network are chosen uniformly at random with respect to various probability distributions, and compute the expected distribution of the points of non-linearity. We use these results to explain why the network may be biased towards outputting functions with simpler geometry, and why certain functions with low information-theoretic complexity are nonetheless hard for a neural network to approximate.