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This paper introduces MathCoPilot, an interactive human-in-the-loop system that enhances mathematical research by allowing mathematicians to guide AI agents in formal proof generation. By integrating an interactive workbench, automated proving orchestration, and topic-driven paper retrieval, MathCoPilot facilitates a collaborative environment where human intuition and AI efficiency coexist. The evaluation of four state-of-the-art LLMs reveals that while they excel at undergraduate-level problems, they struggle with complex domain-specific theorems that demand deeper mathematical insight.
Current LLMs can tackle basic proofs, but MathCoPilot reveals their limitations when faced with advanced theorems requiring true mathematical understanding.
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 Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, 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.
