Timeline

From weak solutions to a machine-checked forced singularity, with September 2026 by the day.

The century

• 1934 Leray constructs global finite-energy weak solutions satisfying an energy inequality. Smoothness left open.
• 1982 Caffarelli, Kohn and Nirenberg: the singular set of a suitable weak solution has zero one-dimensional parabolic Hausdorff measure.
• 1984 Beale, Kato and Majda: an Euler solution can blow up only if the time integral of the maximum vorticity diverges.
• 1996 Nečas, Růžička and Šverák: no nontrivial self-similar blow-up in $L^3$.
• 2000 Fefferman writes the Clay problem statement with alternatives (A) to (D). The breakdown alternatives allow a smooth decaying force.
• 2003 Escauriaza, Seregin and Šverák: bounded $L^3$ norm prevents blow-up of the unforced equations.
• 2016 Tao: finite-time blow-up for an averaged Navier–Stokes equation that keeps the energy identity. The nonlinearity is modified, not forced.
• 2019 Buckmaster and Vicol: nonuniqueness of weak solutions by convex integration.
• 2021 Flandoli and Luo: high-mode transport noise controls vorticity blow-up with high probability.
• 2022 Albritton, Brué and Colombo: two distinct Leray–Hopf solutions from rest with the same force, via an unstable vortex in similarity variables.
• 15 Sep 2023 Córdoba and Martínez-Zoroa: finite-time singularity for forced 3D Euler with a $C^{1,1/2-\epsilon}$ force.
• 9 Jul 2024 Córdoba, Martínez-Zoroa and Zheng: the same for hypodissipative Navier–Stokes.
• 17 Sep 2025 A 22-author preprint including Buckmaster gives numerical constructions of singularities for porous medium and Boussinesq.
• 20 May 2026 Braunstein: thermal noise and a molecular cutoff make the equations trivially globally well posed.

2026

• 15 Aug Alpöge and Buckmaster obtain smooth-forcing blow-up for 3D Euler; Lean verification completes a week later.
• 28 Aug OpenAI begins training the model later used.
• 1 Sep OpenAI's 88-hour multi-agent run begins, about 10,000 agents.
• 6 Sep Meeting between Buckmaster and OpenAI's Bubeck; accounts of it later diverge.
• 7 Sep Tao's blog post on the smooth-forcing blow-ups. Alpöge and Buckmaster release three preprints with Lean proofs. Buckmaster posts his statement that night.
• 8 Sep OpenAI releases the 166-page manuscript and the Lean formalization at noon.
• 9 Sep Cao, Chi and Nie post the density-of-forces paper. Agresti posts v4 of the transport-noise paper. Alpöge posts on authorship.
• 8–13 Sep OpenAI issues successive statements on whether product data could have influenced the result, ending with a categorical denial for inputs after 3 July.
• 11 Sep The Clay Mathematics Institute says the problem has apparently been settled and that its evaluation will be deliberately unhurried. Twenty-six Fields Medalists publish an open letter on AI in mathematics.
• 15 Sep Alpöge, Buckmaster and Coiculescu post the porous medium paper to arXiv. Duraiswami posts the porous-walls comparison.
• 17 Sep Constantin, Ignatova and Vicol: analytic forcing implies regularity for flows of the construction's type.
• 23 Sep The homogenization site opens an issue on how averaged Navier–Stokes and the forced blow-up differ from fast switching; this site is the answer.
• 24 Sep The Lean formalization is built from source for this site: 11,424 jobs, standard axioms only. See the formal statement.

Dates for the September events follow the arXiv timestamps and the Wikipedia article on the priority controversy as of 24 September 2026. Corrections welcome.