{"slug": "navier-stokes-lost-in-translation", "title": "Navier–Stokes Lost in Translation", "summary": "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.", "body_md": "# Mathematics > Analysis of PDEs\n\n  [Submitted on 6 Oct 2026]\n\n# Title:Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs\n\n[View PDF](https://arxiv.org/pdf/2610.08144)\n\n[HTML (experimental)](https://arxiv.org/html/2610.08144v1)\n\nAbstract: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.\n    \n\n## Submission history\n\nFrom: Alexander Bastounis [\n[view email](https://arxiv.org/show-email/0ee63498/2610.08144)]\n\n**[v1]** Tue, 6 Oct 2026 10:58:01 UTC (1,080 KB)\n\n### Current browse context:\n\nmath.AP\n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer \n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers \n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps \n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations \n\n*(*[What are Smart Citations?](https://www.scite.ai/))\n# Code, Data and Media Associated with this Article\n\nalphaXiv \n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers \n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub \n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub \n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face \n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast \n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))\n# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower \n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender \n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))\n# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth 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.\n\nHave an idea for a project that will add value for arXiv's community? [**Learn more about arXivLabs**](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/navier-stokes-lost-in-translation", "canonical_source": "https://arxiv.org/abs/2610.08144", "published_at": "2026-10-07 15:24:38+00:00", "updated_at": "2026-10-07 16:18:29.171342+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "large-language-models", "ai-safety"], "entities": ["Alexander Bastounis", "arXiv", "Lean", "OpenAI", "Navier-Stokes equations"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/navier-stokes-lost-in-translation", "markdown": "https://wpnews.pro/news/navier-stokes-lost-in-translation.md", "text": "https://wpnews.pro/news/navier-stokes-lost-in-translation.txt", "jsonld": "https://wpnews.pro/news/navier-stokes-lost-in-translation.jsonld"}}