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arXiv 论文论证 AI 自动形式化难以忠实翻译，质疑 OpenAI Navier–Stokes 证明
Navier–Stokes Lost in Translation
AI 导读
arXiv 论文 2610.08144 论证，将自然语言数学文本语义忠实地翻译为 Lean 等形式语言在 Solvability Complexity Index 层级上达到 ∞，难度高于停机问题。论文给出多个 AI 将 NL 陈述与证明误译为 Lean 的实例，包括 OpenAI 此前宣布的 Navier–Stokes 方程 blow-up 证明，并指出其形式化版本与 NL 证明不对应。
推荐理由
论文给出形式化翻译的复杂度下界，并用 OpenAI Navier–Stokes 案例展示 NL 证明与 Lean 验证之间的不一致。
正文 · 原文
Abstract: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.
| Comments: | 25 pages, 4 Figures |
| Subjects: | Analysis of PDEs (math.AP); Artificial Intelligence (cs.AI); Logic (math.LO) |
| MSC classes: | 35Q30, 03Dxx (primary) and 68V20, 68Txx, 03B65 (secondary) |
| Cite as: | arXiv:2610.08144 [math.AP] |
| (or arXiv:2610.08144v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2610.08144 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Alexander Bastounis [view email]
[v1]
Tue, 6 Oct 2026 10:58:01 UTC (1,080 KB)
来源：Hacker News · arxiv.org