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Navier-Stokes existence and smoothness

A proof of finite-time blow-up was announced on 8 September 2026 and is under review. Read the proof, its Lean formalisation and the critiques side by side, and record which steps hold.

Planned specification stateexternal review
Partial differential equations
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Where it stands · checked 2026-10-15

A proof that smooth flows can blow up in finite time was announced in September 2026. It is under review, and the prize has not been decided.

  1. OpenAI

    Announced a proof, produced by an internal AI system, that the Navier–Stokes equations can develop a singularity in finite time from a smooth fluid at rest under a smooth force, which would settle the problem as statements C and D of the official formulation. It published a paper and a formalisation in Lean, said it does not intend to claim the prize, and credited Levent Alpöge and Tristan Buckmaster with priority on a related result for the forced Euler equations.

    Announcement Paper Lean code
  2. Clay Mathematics Institute

    Said the problem “has apparently been settled”, and that its process for evaluating the work and assigning credit is “deliberately unhurried”, with updates to come.

    Statement
  3. Alexander Bastounis and two co-authors

    Argued that the Lean formalisation does not correspond to the written proof, so the Lean check alone does not confirm the paper's argument.

    arXiv:2610.08144

Clay considers a solution only once it is published in a qualifying outlet, at least two years have passed since, and it has general acceptance among mathematicians. The rules

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Mission statement

Navier-Stokes existence and smoothness

A proof of finite-time blow-up was announced on 8 September 2026 and is under review. Read the proof, its Lean formalisation and the critiques side by side, and record which steps hold.

Field
Partial differential equations
Specification state
external review
Planned graph state
Claimed resolution under review

External prize reference

$1 million external Clay prize, not decided

This opportunity is administered outside Ilisium. It is not Ilisium's money and not a promised contributor payout. Current eligibility and prize terms must be checked with the sponsor.

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