Navier–Stokes Lost in Translation

A paper submitted to arXiv on 6 Oct 2026 by Alexander Bastounis argues that Lean verification of AI autoformalisation does not guarantee correct natural language proofs, because resolving ambiguities in mathematical NL text sits at SCI = ∞ in the Solvability Complexity Index hierarchy — harder than any computational problem including the Halting problem (SCI = 1). The paper provides examples of AI mistranslations of NL statements and proofs into Lean, and states that OpenAI's announced Lean proof of blow-up of solutions to the Navier-Stokes equations does not correspond to the NL proof.

Mathematics Analysis of PDEs Submitted on 6 Oct 2026 Title:Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs View PDF https://arxiv.org/pdf/2610.08144 HTML experimental https://arxiv.org/html/2610.08144v1 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. Submission history From: Alexander Bastounis view email https://arxiv.org/show-email/0ee63498/2610.08144 v1 Tue, 6 Oct 2026 10:58:01 UTC 1,080 KB Current browse context: math.AP References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .