Magenta: Closing the Loop Between Mathematical Reasoning and Lean Verification

2026-09-10

AI Summary

本文通过将Lean信号整合到非正式推理过程中，利用Magenta系统解决数学问题，并生成机器可验证的证明，从而桥接了非正式与形式化推理之间的差距。

Abstract

Most of mathematical knowledge has been communicated through so-called informal use of mathematics and natural language. With large language models (LLMs) being highly adept in using natural language, they achieve strong performance, yet not perfect, in informal mathematical reasoning. Restraining LLMs to informal reasoning misses out on the opportunity to use the discrete verification abilities that machines offer through machine-checkable proofs. In this paper, we bridge the gap between informal and formal reasoning by integrating Lean signals into the informal reasoning process. We introduce Magenta, a training-free agentic pipeline that, given only a natural-language problem, produces an answer, expresses it as a Lean 4 statement, and constructs a machine-checked proof. A statement judge verifies whether the formalisation preserves the original problem, while an error-attribution judge routes failed attempts either to mathematical re-derivation or local Lean repair. Magenta achieves 100% accuracy across all evaluated olympiad benchmarks, including AIME 2025, AIME 2026, and HMMT February 2026. When paired with the open-weight K2-Horizon-7B reasoner, it solves all six IMO 2026 problems. Our analysis shows that statement adjudication is essential for preventing false certificates and that feedback-guided correction outperforms independent resampling on difficult problems.

Problem

• informal reasoning
• formal reasoning
• machine-checkable proofs
• Lean
• mathematical knowledge

Innovation

• Magenta
• Lean 4
• machine-checked proofs
• statement judge
• error-attribution judge