Lean 验证通过不代表自然语言证明正确：从 Navier-Stokes 看自动形式化局限
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs
一篇用 SCI 层级理论拆解自动形式化局限的论文，直接指向 OpenAI 的 Navier-Stokes 证明——Lean 验证过了，原文可能还是错的。
arXiv 论文 2610.08144 分析了自动形式化（autoformalisation）的可靠性问题，即用 AI 将自然语言数学文本翻译成 Lean 等形式语言再做机械验证的流程。作者指出，解决数学自然语言中的歧义在 Solvability Complexity Index 层级中位置为 SCI = ∞，高于任何可计算问题（停机问题仅为 SCI = 1）。论文给出多个 AI 将自然语言陈述误译入 Lean 的实例，导致自然语言证明与 Lean 验证不匹配。这些实例包括 OpenAI 公布的 Navier-Stokes 方程解爆破证明，作者论证其形式化 Lean 证明与原始自然语言证明并不对应。
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs
Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.
