OpenAI presents a construction of finite-time blowup for the three-dimensional incompressible Navier–Stokes equations, for every positive viscosity. Starting from zero velocity with a smooth force compactly supported in space and time, the proposed flow develops unbounded velocity while its kinetic energy stays uniformly bounded. The paper gives an analytical proof, with an accompanying Lean formalization available separately.
How can a grid point set be modified without creating an isosceles triangle? This work introduces two local profiles: points that become individually admissible after deletions, and points that can actually be added together. It establishes exact profiles around two public AlphaEvolve configurations and lower bounds on the changes required to reach another configuration of equal or greater size.