Preprint constructs a singularity for the three-dimensional Euler equation with a smooth external force

Preprint constructs a singularity for the three-dimensional Euler equation with a smooth external force

On September 8, mathematicians Levent Alpöge and Tristan Buckmaster released three preprints on singularities in fluid equations. The authors report developing the proofs with the language models Claude and Codex, with Lean verifying one early construction.

The Euler equation models the motion of an ideal fluid with zero viscosity. In Alpöge and Buckmaster’s preprint, smooth initial data and a smooth external force lead to a finite time at which vorticity, the local rotation of the fluid, grows without bound. The solution remains smooth until that time. The force remains smooth even at that limiting time.

The proof constructs a sequence of layers with progressively finer oscillations. Each preceding layer generates a flow that amplifies a small oscillation in the next layer. Before proceeding to the next step, the authors bring back to zero the vorticity that would hinder the new layer’s growth. This infinite sequence of steps takes place within a finite time, and the corrections are chosen so that the sum of the external forces remains smooth. Mathematicians Diego Córdoba and Luis Martínez-Soroa began developing this multilayer construction; Alpöge and Buckmaster adapted it to the three-dimensional Euler equation.

“The first proof Levent sent me from a language model was the worst proof I had ever read; on August 22, we verified it in Lean.”

This is how Buckmaster described an early version in a public statement. Lean is a formal verification system: it checks a proof against a sequence of precisely specified rules. In a companion preprint, the authors describe using Claude and Codex to explore different ways to construct and present the proof. The mathematicians then examined the argument themselves and rewrote it for readers.

In late August, Alpöge presented a curve with 31 independent points verified by the ICARM registry using an exact algorithm. That verification checked a finite list of points; here, Lean checks a long sequence of proof steps. Buckmaster writes that the authors are continuing to turn this sequence into an explanation that shows how the layers produce a singularity.