Hacker News Front Page · [Submitted on 6 Oct 2026] ·
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs
El artículo sostiene que verificar en Lean una formalización generada por IA no garantiza que el texto matemático original exprese el mismo argumento. Resolver las ambigüedades semánticas del lenguaje matemático natural puede alcanzar una complejidad arbitrariamente alta en la jerarquía SCI, incluso superior a la del problema de la parada. Mediante ejemplos de traducciones erróneas, incluidos los relacionados con la supuesta prueba de explosión de soluciones de Navier-Stokes anunciada por OpenAI, muestra que una prueba formal verificada puede no corresponder a la prueba descrita en lenguaje natural.
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.