Bolzano: Case Studies in LLM-Assisted Mathematical Research

Large language models can contribute useful ideas to mathematical research, yet long-horizon proof attempts remain difficult to coordinate, evaluate, and reproduce. We present Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management. We evaluate Albilich on the RealMath benchmark (Zhang et al. 2025) and on open problems in group theory from the Kourovka Notebook (Khukhro and Mazurov 2026). It solved 10/10 problems on RealMath with CAS and 9/10 with no CAS. On the Kourovka problems, Albilich produced a counterexample to Problem 21.142 and a proof of a strengthening of Problem20.2. Anablation on Problem 17.91 demonstrates 32.0% token reduction when CAS is enabled. An ablation on Problem 21.142 demonstrates higher verifier-rejection rate and failure to synthesize proof routes in the absence of the advisor agent. These results support Albilich as a human-steerable, CAS-boosted environment for scalable AI-assisted mathematical research.

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini3.1Pro, GPT-5.4, and ClaudeOpus4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.