Why Lean verification of AI autoformalisation doesn’t assure right pure language proofs

[Submitted on 6 Oct 2026]

View a PDF of the paper titled Navier-Stokes misplaced in translation: Why Lean verification of AI autoformalisation doesn’t assure right pure language proofs, by Alexander Bastounis and a couple of different authors

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Abstract:Autoformalisation is more and more used to confirm mathematical texts, together with these generated by AI, as in OpenAI’s introduced proof of blow-up of options to the Navier-Stokes equations. In this course of, an AI system interprets the textual content from a pure language (NL) into a proper language reminiscent of Lean. Once this translation is completed, the argument expressed within the formal language can simply be mechanically verified. The objective of this text is to exhibit why this course of could provide no confidence within the authentic NL argument, owing to the assorted difficulties in performing the interpretation semantically faithfully. In specific, we spotlight that the issue of resolving ambiguities in mathematical NL textual content, which is critical so as to present semantically trustworthy translation, is arbitrarily excessive up within the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= infty$). Hence, informally, offering semantically trustworthy AI autoformalisation is more durable than any computational drawback together with the Halting drawback (which has SCI $= 1$). To exhibit the impact of this consequence we offer a number of examples of AI mistranslations of NL statements and proofs into Lean in apply, leading to mismatches between NL proofs and their Lean `verifications’. These embrace OpenAI’s introduced Navier-Stokes proof. In specific, we present that the formalised Lean proof doesn’t correspond to the NL proof of blow-up of options to the Navier-Stokes equations.