# OpenAI Navier-Stokes Proposed Solution

> Source: https://aiwiki.ai/wiki/openai_navier_stokes_proposed_solution
> Updated: 2026-09-09
> Fact-checked: 2026-09-09
> Categories: AI Incidents & Controversies, AI Research, AI for Science, Mathematics, OpenAI
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> Cite as: AI Wiki. "OpenAI Navier-Stokes Proposed Solution." aiwiki.ai, 9 Sept 2026. https://aiwiki.ai/wiki/openai_navier_stokes_proposed_solution
> From AI Wiki (https://aiwiki.ai), the free encyclopedia of artificial intelligence. Reuse freely with attribution.

**OpenAI Navier-Stokes Proposed Solution** is a proposed resolution of the Navier-Stokes existence and smoothness Millennium Prize Problem announced by [OpenAI](https://aiwiki.ai/wiki/openai) on September 8, 2026. The accompanying 166-page manuscript, credited to "OpenAI" rather than named individual authors, constructs for every positive viscosity a smooth three-dimensional flow that starts at rest and develops unbounded velocity in finite time. Its external force is smooth and compactly supported, while the kinetic energy remains uniformly bounded.[1][2] OpenAI says this establishes alternatives C and D in Charles Fefferman's official formulation of the problem: breakdown on both Euclidean space and the periodic three-torus.[1][3]

The announcement included a public [Lean](https://aiwiki.ai/wiki/lean) repository, but OpenAI's claimed machine-checked formalization and acceptance of a Millennium Prize solution are different milestones. As of September 9, 2026, the Clay Mathematics Institute still labeled the problem "Unsolved." Its rules require publication in a qualifying outlet, a wait of at least two years, and general acceptance in the global mathematics community before the institute will consider a proposed solution.[4][5] OpenAI has said it does not intend to claim the prize.[1]

The mathematical claim arrived amid a dispute about priority, data provenance, and proposed authorship arrangements. In a separate personal collaboration, Tristan Buckmaster and Levent Alpöge had been using models from OpenAI and Anthropic on closely related forced fluid-equation problems. Buckmaster alleged that OpenAI learned of their private progress shortly before launching its own effort and later proposed an arrangement that excluded Alpöge from a paper about OpenAI's result. OpenAI researcher Sébastien Bubeck disputed that interpretation. No public source reviewed for this article establishes that OpenAI copied unpublished material, and Buckmaster expressly said he did not know whether the pair's data had been used.[1][6][7]

## What the manuscript claims

Fefferman's official problem description gives four acceptable routes. Alternatives A and B ask for global smooth solutions with no external force on R^3 and on the periodic torus. Alternatives C and D instead ask for smooth initial data and a smooth force for which no global smooth finite-energy solution exists on those two domains.[3] A forced counterexample can therefore satisfy the official problem even though it does not prove that an unforced fluid blows up.

Theorem 1.1 of OpenAI's manuscript takes the latter route. For every viscosity greater than zero, it specifies a smooth force with compact support in space and positive time, together with smooth velocity and pressure fields on R^3 x [0,1). The initial velocity is zero. The velocity and pressure remain supported in one compact set, the L2 norm of velocity stays bounded, and the L-infinity norm of velocity has infinite limsup as time approaches 1. The theorem then rules out a global smooth solution with the same initial data and force and uniformly bounded kinetic energy. The paper says compact support permits periodization, giving the torus result as well.[2]

This is a statement about the mathematical equations under a specially constructed smooth forcing term. It is not a prediction that a laboratory fluid will reach infinite speed. It also does not settle the stronger unforced Navier-Stokes question, which corresponds to alternatives A and B.[2][3]

## Construction described in the paper

The proposed singularity is an axisymmetric, self-similar vortex centered at the spatial origin. If tau is the time remaining before the singular time, the paper makes the core's radial width scale roughly as tau^(1/2) and its axial length as tau^(1/2-h), where h is a small positive constant. The radial width therefore shrinks faster than the axial length. Azimuthal and axial speeds grow roughly as tau^(-1/2-h), but the volume contracts fast enough that the core's kinetic energy tends to zero even as its maximum speed diverges.[2]

The difficult part is keeping the external force smooth. Joining the singular inner vortex to a regular exterior creates an annular momentum residual that would itself become unbounded. The construction adds localized oscillatory pulses around this annulus. Background shear amplifies the pulses, and their nonlinear momentum flux cancels the singular part of the residual. Higher-order corrections remove the remaining errors. The completed field is then localized in space while preserving incompressibility and the blowup behavior.[2]

The manuscript is an analytical proof, not only a numerical simulation. Its argument spans the leading self-similar flow, correction of the base field, realization of the residual stress by oscillatory pulses, compact mean corrections, and the final whole-space and torus constructions. Whether every part of that argument and its formal counterpart matches the Clay statement remains a matter for independent review.[2]

## AI generation and formalization

OpenAI's account gives the following generation timeline. The dates and resource figures are company-reported and have not been independently reproduced.[1]

| Date | Reported event |
|---|---|
| August 28, 2026 | Training began on an internal model that OpenAI described as significantly more capable than [GPT-6 Astra](https://aiwiki.ai/wiki/gpt_6_astra) |
| September 1 | After hearing rumors that two Millennium Prize problems had been resolved, OpenAI launched agent groups on every open Millennium problem and on several easier problems |
| After about 50 hours | An initial group of nearly 100 agents reportedly found an unforced Euler blowup result; OpenAI then shifted resources toward Navier-Stokes |
| September 5 | The Navier-Stokes group reportedly reached its result about 88 hours after the first agents launched |
| September 6 | Lean formalization and verification reportedly finished after a further 17 hours using GPT-6 Astra |
| September 8 | OpenAI released the announcement, manuscript, and formal repository |

OpenAI says the Navier-Stokes run used on the order of 10,000 concurrent agents, 2.7 million inter-agent messages, and about 130 billion output tokens. Across all attempted problems, the reported totals were 4.9 million messages and about 300 billion output tokens. Agents could read a cached internet corpus and run code, while Codex was used to consolidate intermediate results across groups.[1]

The public repository contains Lean 4 formalizations for both the Navier-Stokes and Euler results. It pins Lean 4.34.0-rc2, uses Mathlib and Lake, provides standard build commands, and links instructions for checking the formalizations with Comparator.[8] A successful Lean kernel check establishes that the encoded conclusion follows from the encoded definitions and assumptions. It does not by itself establish that those definitions are semantically equivalent to every condition in Fefferman's prose formulation. That correspondence still requires expert inspection.

## Mathematical lineage and concurrent work

OpenAI's construction appeared after a sequence of results developed by Diego Córdoba, Luis Martínez-Zoroa, and collaborators. In 2023, Córdoba and Martínez-Zoroa posted a forced three-dimensional Euler blowup result whose force had limited Holder regularity rather than full smoothness.[9] In 2024 they posted an incompressible porous-media construction with a compactly supported, spatially smooth source.[10] With Fan Zheng, they later proved forced blowup for a hypodissipative fractional Navier-Stokes equation in a small dissipation range near the Euler limit. That peer-reviewed 2026 result does not cover the classical Laplacian case in the Millennium problem.[11]

On September 7, Buckmaster and Alpöge made three related manuscripts public. One constructed smooth-forced blowup for three-dimensional incompressible Euler with axisymmetric initial data and nonzero swirl.[12] A second constructed smooth-forced blowup for the two-dimensional Boussinesq equations.[13] The third, coauthored with Matei P. Coiculescu, extended the porous-media construction to uniformly smooth space-time forcing on the torus.[14] Their public Lean repository contains formalizations for the three results.[15]

Those papers were substantial adjacent advances, but they did not claim the classical Navier-Stokes result. In a September 7 explanation, Terence Tao wrote that the work made extension to Navier-Stokes look feasible while explicitly saying the authors had not yet achieved that goal. He also said he had received only a short verbal explanation and still needed to digest the details.[16] The distinction matters: Tao's post supported the importance of the Buckmaster-Alpöge work, not independent verification of OpenAI's later manuscript.

Buckmaster and Alpöge repeatedly credited the multiscale program to Córdoba and Martínez-Zoroa. OpenAI's manuscript cites the same lineage while describing a different immediate mechanism in which oscillatory pulses supply a missing momentum flux around a contracting vortex.[2][6][9][10] The American Mathematical Society's September 8 statement likewise presented the development as a chain of human and machine-assisted work, naming Navier, Stokes, Leray, Ladyzhenskaya, Córdoba, Martínez-Zoroa, Alpöge, Buckmaster, and OpenAI mathematicians.[17] Quanta's contemporaneous report also identified the intellectual debt to Córdoba and Martínez-Zoroa and said the proof and priority questions would require further scrutiny.[18]

## Priority, data, and authorship dispute

According to Buckmaster's four-page public statement, he and Alpöge obtained the Boussinesq and forced Euler results on August 15 and Lean-verified the Euler proof on August 22. He described the collaboration as personal rather than an institutional Anthropic project and said it used Claude, Codex with GPT-5.6 Sol, and, later, Astra for writing and auditing. On September 3, after hearing that information about their progress had reached OpenAI, he emailed an OpenAI mathematician to clarify that the collaboration was personal and that the pair planned to publish a manuscript and formalization together.[6]

Buckmaster said that two calls took place on September 6. In his account, OpenAI representatives told him that an internal model had already produced a forced Navier-Stokes proof. He alleged that two publication arrangements were discussed: sequential releases of the outside Euler result and OpenAI's Navier-Stokes result, or a paper in which Buckmaster would present OpenAI's result without Alpöge as an author. Buckmaster also reported being asked why he would risk his career by going public. His statement ended with explicit limits: he had not seen OpenAI's proof, did not know what its model had done, did not know whether their data had been used, and was not accusing anyone of wrongdoing.[6]

OpenAI's published account says its researchers and agents saw none of Buckmaster and Alpoge's work before the pair released it publicly and that no specific user data was accessed to solve the problem. The company nevertheless said it could not rule out the possibility that de-identified data derived from their use of OpenAI products had helped improve its models. OpenAI says the proof and Lean work were complete on September 6 before it contacted the pair, and it recognizes their priority on the forced Euler result.[1]

Bubeck initially called the allegations false and inflammatory. In a longer response, he said he had never asked that Alpöge be removed from authorship of Alpöge's own work. He described a different discussion: Buckmaster might become lead author of a rewrite of OpenAI's Navier-Stokes proof, and Bubeck thought it would be inappropriate for an Anthropic employee to coauthor OpenAI's work. Bubeck also apologized for the career remark, calling it a poor choice of words that he said he retracted immediately.[7] Alpöge replied that he would have been open to collaboration but understood from a conversation he overheard that authorship and prize credit were conditioned on his removal. Bubeck responded that nothing had been locked and that OpenAI remained willing to discuss a resolution.[19]

The accounts agree on several points: rumors about the outside work helped prompt OpenAI's effort; the teams had not exchanged manuscripts before their public releases; publication arrangements were discussed on September 6; and OpenAI cannot exclude an indirect contribution from de-identified product-use data to model training. They disagree about the intent and meaning of the proposed authorship arrangement and the tone of the calls. The public record does not resolve the training-data question or support treating speculation about copying as established fact.[1][6][7][19]

## Review and prize status

The release created unusually strong preliminary evidence for a new proof: a detailed analytical manuscript, a public formalization, build instructions, and a prominent expert response. Quanta reported that Fefferman was pleased by the claimed result and that the Lean artifact gave mathematicians confidence, while also emphasizing that humans still have to verify the formal statement's correspondence to the intended problem and that the mathematical and priority questions will take time to sort out.[18]

That reaction is not equivalent to peer review or Clay acceptance. On September 9, El País reported that no outside scientist had yet completed an independent verification of the OpenAI proof.[20] The Clay Mathematics Institute continued to list Navier-Stokes as unsolved, and its minimum two-year rule makes immediate prize recognition impossible even if review ultimately finds the argument correct.[4][5]

The most accurate status as of September 9, 2026 is therefore **a publicly documented proposed solution accompanied by an author-supplied Lean formalization, with independent mathematical review and provenance questions still open**. Calling it either a settled Clay solution or a disproved result would go beyond the available evidence.

## See also

- [Mathematical reasoning in AI](https://aiwiki.ai/wiki/mathematical_reasoning)
- [AI Co-Mathematician](https://aiwiki.ai/wiki/ai_co_mathematician)
- [Lean](https://aiwiki.ai/wiki/lean)
- [GPT-6 Astra](https://aiwiki.ai/wiki/gpt_6_astra)

## References

1. OpenAI. "On the Navier-Stokes Millennium Prize Problem." September 8, 2026. https://openai.com/index/navier-stokes-solution/
2. OpenAI. "Finite Time Blowup for Navier-Stokes." September 2026. https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf
3. Charles L. Fefferman. "Existence and Smoothness of the Navier-Stokes Equation." Clay Mathematics Institute. https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
4. Clay Mathematics Institute. "Navier-Stokes Equation." Accessed September 9, 2026. https://www.claymath.org/millennium/Navier-Stokes-Equation/
5. Clay Mathematics Institute. "Rules for the Millennium Prize Problems." Accessed September 9, 2026. https://www.claymath.org/millennium-problems/rules/
6. Tristan Buckmaster. Public statement on finite-time blowup for forced fluid equations and interactions with OpenAI. September 7, 2026. https://cims.nyu.edu/~tristanb/statement.pdf
7. Sébastien Bubeck. "I would like to clarify a few things." X, September 8, 2026. https://x.com/SebastienBubeck/status/2097379411691516310
8. OpenAI. "NavierStokesAndEuler." GitHub repository. Accessed September 9, 2026. https://github.com/openai/NavierStokesAndEuler
9. Diego Córdoba and Luis Martínez-Zoroa. "Blow-up for the incompressible 3D-Euler equations with uniform C^(1,1/2-epsilon) intersection L^2 force." arXiv:2309.08495, September 15, 2023. https://arxiv.org/abs/2309.08495
10. Diego Córdoba and Luis Martínez-Zoroa. "Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source." arXiv:2410.22920, revised February 13, 2025. https://arxiv.org/abs/2410.22920
11. Diego Córdoba, Luis Martínez-Zoroa, and Fan Zheng. "Finite Time Blow-Up for the Hypodissipative Navier Stokes Equations with a Force in L^1_t C_x^(1,epsilon) intersection L^infinity_t L^2_x." Archive for Rational Mechanics and Analysis 250, article 38, May 11, 2026. https://doi.org/10.1007/s00205-026-02198-0
12. Levent Alpöge and Tristan Buckmaster. "Blowup for the Euler Equations with Smooth Forcing." September 2026. https://cims.nyu.edu/~tristanb/euler.pdf
13. Levent Alpöge and Tristan Buckmaster. "Blowup for the Boussinesq Equations with Smooth Forcing." September 2026. https://cims.nyu.edu/~tristanb/boussinesq.pdf
14. Levent Alpöge, Tristan Buckmaster, and Matei P. Coiculescu. "Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing." September 2026. https://cims.nyu.edu/~tristanb/ipm.pdf
15. Tristan Buckmaster. "fluid_lean." GitHub repository. Accessed September 9, 2026. https://github.com/tristanbuckmaster/fluid_lean
16. Terence Tao. "Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations." September 7, 2026. https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/
17. American Mathematical Society. "Statement from AMS Leadership on Navier-Stokes Problem." September 8, 2026. https://www.ams.org/news?news_id=7686
18. Konstantin Kakaes. "AI Has Solved One of Math's $1 Million Millennium Prize Problems." Quanta Magazine, September 8, 2026. https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/
19. Levent Alpöge. Response to OpenAI's concurrent-work statement. X, September 8, 2026. https://x.com/__alpoge__/status/2097383870773748190
20. Manuel Ansede. "El anuncio de OpenAI de que ha resuelto uno de los mayores enigmas matemáticos de la historia desata acusaciones de plagio." El País, September 9, 2026. https://elpais.com/ciencia/2026-09-09/el-anuncio-de-openai-de-que-ha-resuelto-uno-de-los-mayores-enigmas-matematicos-de-la-historia-desata-acusaciones-de-plagio.html