OpenAI Navier-Stokes Proposed Solution

OpenAI Navier-Stokes Proposed Solution is a proposed resolution of the Navier-Stokes existence and smoothness Millennium Prize Problem announced by OpenAI on September 8, 2026. The accompanying manuscript, credited to "OpenAI" rather than to named individual authors, constructs for every positive viscosity a smooth three-dimensional flow that starts at rest and develops unbounded velocity in finite time under a smooth external force with compact support, while the flow's kinetic energy stays uniformly bounded up to the singular time.^[1][2] OpenAI says this establishes alternatives C and D in Charles Fefferman's official formulation of the problem: breakdown on Euclidean space and on the periodic three-torus.^[1][2] Those two alternatives permit an external force. Alternatives A and B, which take the force to be identically zero, are a different question, and the construction does not settle them.^[3] Buckmaster, who had been attacking the same forced route, wrote that "almost nobody else I know of was working on it."^[8]

The announcement included a public Lean repository, but a machine-checked formalization and acceptance of a Millennium Prize solution are different milestones. On September 11, 2026 the Clay Mathematics Institute published a statement saying that it "shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled," and pointing readers to its prize rules for the process of "evaluating what has been achieved and for assigning credit," a process it called "deliberately unhurried."^[7] Those rules require publication in a qualifying outlet, a wait of at least two years after that publication, and general acceptance in the global mathematics community.^[5][6] The institute also reclassified the problem: its page carried the label "Unsolved" in Internet Archive captures through 01:58 UTC on September 10 and "Active" from 21:16 UTC that day, and by September 13 the Millennium Prize Problems index listed Navier-Stokes under a separate "Active problems" heading rather than with the five remaining unsolved problems.^[4] 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][8][9] On September 11, twenty-five Fields Medallists published a declaration criticizing how AI companies are pursuing mathematical problems as benchmarks, without disputing the correctness of any AI result.^[28][29]

What the manuscript claims

Fefferman's official problem description gives four acceptable routes. Alternatives A and B ask for global smooth solutions on R^3 and on the periodic torus with the external force taken "to be identically zero." Alternatives C and D instead ask for smooth initial data together with a smooth force, subject to decay conditions, for which no global smooth finite-energy solution exists on those two domains.^[3] A forced counterexample can therefore satisfy the official problem as written even though it does not prove that an unforced fluid breaks down. The rules add that for Navier-Stokes "a resolution in either direction" is evaluated by the same procedure.^[6]

Theorem 1.1 of OpenAI's manuscript takes the forced route. For every viscosity greater than zero, it specifies a force in the space of smooth compactly supported fields on R^3 crossed with 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 supremum of the L2 norm of velocity over [0,1) is finite, 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 datum and force whose kinetic energy is uniformly bounded. The paper says compact support also yields the corresponding construction on the torus, which it records as Corollary 10.6.^[2]

This is a statement about the 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 unforced Navier-Stokes question of alternatives A and B.^[2][3] The same announcement separately reports an unforced blowup result for the Euler equations, which are the zero-viscosity limit and are not themselves on the Clay prize list.^[1][3][10]

Construction described in the paper

The proposed singularity is an axisymmetric, self-similar vortex centered at the spatial origin. Writing tau for the time remaining before the singular time, the paper scales the core's radial width as tau^(1/2) and its axial length as tau^(1/2-h), with h fixed and smaller than 1/100. The radial width therefore shrinks faster than the axial length, and the core becomes an increasingly slender column with volume of order tau^(3/2-h). Azimuthal and axial speeds scale as tau^(-1/2-h), while the core's total kinetic energy is of order tau^(1/2-3h), so the energy tends to zero even as the maximum speed diverges.^[2]

The difficult part is keeping the external force smooth. Joining the singular inner vortex to a smooth exterior whose speed decreases with radius leaves a momentum residual in the intervening annulus that would itself become unbounded. The construction adds spatially 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 singular errors. The completed field is then localized in space while preserving incompressibility and the blowup behavior.^[2]

The manuscript is an analytical proof, not a numerical simulation. Its argument runs through the leading self-similar flow, correction of the base field to every order, realization of the residual stress by oscillatory pulses, compactly supported mean corrections, and the final whole-space and torus constructions, with three appendices on radial moment matching, analytic profiles near the axis, and the admissible stress cone.^[2] Whether every part of that argument and its formal counterpart matches the Clay statement remains a matter for independent review.

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]

OpenAI says the group that produced the Navier-Stokes result involved on the order of 10,000 concurrent agents, and that resolving the problem took 2.7 million agent messages and approximately 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 copy of the internet and run code; different groups were prompted with different variants of the problem statement, covering alternatives A and B as well as C and D, and Codex was used to consolidate the most useful insights 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.^[10] Its machine-readable metadata file lists four main declarations, records a sorry count of zero for each, gives their axiom dependencies as Lean's standard propext, Classical.choice and Quot.sound, and sets the review status to "self-assessed." The Comparator reference statements that encode alternatives C and D are adapted from the Formal Conjectures project maintained by Google DeepMind.^[10] 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, a point Quanta also made in its coverage.^[20] That correspondence still requires expert inspection.

Both artifacts changed after publication. The repository received a second commit on September 10, 2026 that added roughly 170 Lean files and about 25,000 lines and rewrote the two declarations facing the Comparator challenge: the September 8 version derived alternatives C and D from modules described as a constructed compact candidate, while the September 10 version derives them from modules whose headers describe the full whole-space theorem and the full periodic corollary.^[10] The manuscript was revised on the day of release. The copy the Internet Archive captured at 17:29 UTC on September 8 runs to 165 pages, and its 16-item bibliography contains no reference to Diego Córdoba or Luis Martínez-Zoroa.^[24] By 20:04 UTC the same URL served a 166-page version whose bibliography runs to 22 items and whose historical section adds a paragraph on the Córdoba and Martínez-Zoroa program, along with references to Craik and Criminale, to Leibovich and Stewartson, and to a second Billant and Gallaire paper.^[2][24] Neither version cites the Alpöge and Buckmaster manuscripts released the previous day.^[2][24] As of September 13, 2026, no third-party Comparator run or other independent check of the formalization had been reported in any source reviewed for this article.

Mathematical lineage and concurrent work

OpenAI's construction appeared after a sequence of results developed by Córdoba, 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.^[11] In 2024 they posted an incompressible porous-media construction with a compactly supported, spatially smooth source.^[12] 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.^[13]

Late on September 7 in New York, Buckmaster and Alpöge made three related manuscripts public. One, "Blowup for the Euler equations with smooth forcing," constructs a finite-time singularity for incompressible Euler on R^3 with a force smooth in space and time up to and including the blowup time, from axisymmetric initial velocity with nonzero swirl supported in a fixed solid torus.^[14] A second did the same for the inviscid Boussinesq system on R^2, from zero initial velocity.^[15] The third, coauthored with Matei P. Coiculescu, extended the Córdoba and Martínez-Zoroa porous-media construction to uniformly space-time smooth forcing on the two-dimensional torus.^[16] All three abstracts state that they follow the Córdoba and Martínez-Zoroa multiscale program.^[14][15][16] Their public Lean repository, created in the early hours of September 8 UTC, contains formalizations for the three results.^[17] Buckmaster also wrote that the pair believed they had blowup for hypodissipative Navier-Stokes but were not releasing that paper, because the Lean verification had not finished.^[8]

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 "very feasible" in the near future while saying explicitly that the authors "do not quite achieve these goals yet." He added that Buckmaster had explained the main ideas in a half-hour phone conversation and that he still needed to digest the details, and that "the actual solving of these problems is only a proxy goal for the primary goal of developing mathematical understanding and insight."^[18] The distinction matters: Tao's post supported the importance of the Buckmaster and Alpöge work, not independent verification of OpenAI's later manuscript.

Buckmaster and Alpöge repeatedly credited the multiscale program to Córdoba and Martínez-Zoroa, and Buckmaster wrote in his statement that "in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal."^[8] The revised OpenAI manuscript cites the same lineage while describing a different immediate mechanism, in which oscillatory pulses supply a missing momentum flux around a collapsing vortex.^[2][11][12] The American Mathematical Society's September 8 statement, signed by president Ravi Vakil and chief executive John Meier, presented the development as a chain of human and machine-assisted work, naming Navier, Stokes, Leray and Ladyzhenskaya, then Córdoba and Martínez-Zoroa, then Alpöge and Buckmaster, and then OpenAI mathematicians. It closed: "The purpose of mathematics is human understanding, and this achievement, and the process that led to it, will bear fruit for a long time to come."^[19] Quanta's contemporaneous report also identified the intellectual debt to Córdoba and Martínez-Zoroa and said the proof and priority questions would take time to sort out.^[20]

A third line of work surfaced in the same window. Anima Anandkumar's group at Caltech posted a manuscript on September 7, 2026 offering evidence that the unforced Euler equations on R^3 admit a self-similar finite-time blowup, found with a physics-informed neural network, refined into spline representations, and bounded with arbitrary-precision interval arithmetic, together with a companion paper on nonlinear stability of the approximate profile.^[27] In a guest post on Tao's blog on September 10, Anandkumar wrote that the group had sent the work to Tom Hou and Tao for feedback, that Tao encouraged public release because the NYU team had just posted and OpenAI was rumored to release the next day, and that the post went live in the evening of September 7. She also wrote that mainstream coverage followed OpenAI's press release, which "fails to acknowledge our work even after we informed them."^[26] OpenAI's announcement page does not name the group or its result.^[1] Their route differs from both of the others: it addresses the unforced problem, uses a self-similar ansatz rather than a multiscale construction with discrete jumps in scale, and relies on physics-informed optimization rather than a language model.^[26][27]

Priority, data, and authorship dispute

According to Buckmaster's four-page public statement, he and Alpöge obtained the smooth-forced Boussinesq and Euler results on August 15 and Lean-verified a proof on August 22. He described the collaboration as personal rather than an institutional Anthropic project, said he paid for the tools out of his own research funds, and said it used Claude, Codex with GPT-5.6 Sol, and later Astra, the last only for writeups and auditing. On September 3, after hearing that information about their progress had reached OpenAI, he emailed a mathematician at the company, quoting the email in full in his statement, to clarify that the collaboration was personal and that the pair would post the paper and the formalization together.^[8]

Buckmaster said two calls took place on the afternoon of Sunday, September 6, with Bubeck joining and Alpöge not present. In his account he was told that an internal OpenAI model had produced a proof of finite-time blowup for the forced Navier-Stokes equations, described to Alpöge by text as "Existence of forced blowup in R3 and T3" with "the forcing function is smooth option c and d in Fefferman," and that the proof ran to about 100 pages. He said he was shown a prompt and told the internal research model had simply been given the problem statement, and that Alpöge had been told "very little human input" was used, but that over the course of the call it emerged that a team had been working on the problem, that easier problems including Euler had been tried first, that the prompt he was shown had itself been written by prompting Codex, and that a large amount of compute had been used. He said he asked when the first prompt had been sent and was eventually told it was within the past few days, after information about their work reached OpenAI, and that he asked whether the model had been trained on their Codex sessions and did not get an answer.^[8]

Buckmaster said two publication arrangements were offered: that the pair post their Euler result and OpenAI post its Navier-Stokes result the next day, or that he alone write a paper presenting OpenAI's result and acknowledging that an internal OpenAI model had resolved it. He wrote that Bubeck "twice asserted that he wanted Levent removed from authorship," that he was told OpenAI would say the pair deserved the Clay Prize and were the "closest humans to the problem" if it posted after them, and that he declined both offers. He reported that when he said he would go public, the reply was "Why would you ruin your career?" and, after he asked why, "If you don't want me to be nice, then I don't have to be nice." 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.^[8]

OpenAI's published account says its researchers and agents saw none of Buckmaster and Alpöge's work before the pair released it publicly, and that no specific user data was accessed to solve the problem. It says the proof and Lean work were complete on September 6 before the company contacted the pair, that it reached out to offer a concurrent release and a joint announcement, and that it recognizes their priority on forced Euler.^[1]

OpenAI revised that section of the page on September 10. Its original wording read: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)." Alpöge quoted the first of those sentences on September 8, the day the page went up, and replied, "i mean props to them for straight coming clean."^[21] That sentence no longer appears. The page now carries a dated note saying the "Concurrent work" section has been updated with findings from an investigation into whether user inputs could have influenced the result, and reads: "Following an investigation, we have confirmed that Buckmaster's Codex prompts over the two months preceding this announcement and paper on September 8, 2026, could not have influenced the system in any way, including through training." It adds that the internal model was developed through large-scale reinforcement learning on top of a previously pretrained model.^[1] The same revision inserted a claim absent from the original, that the pair had produced their forced Euler resolution "using an internal Anthropic model."^[1] Buckmaster's statement names Claude among several models used in a collaboration he describes as personal, and Alpöge described his own side as "mostly me and claude having a good time yoloing random stuff in the corner rather than anything institutional."^[8][21]

El País reported on September 9 that Bubeck had called Buckmaster's account false and inflammatory and had promised further detail.^[22] In a longer response posted on X on September 8, the day of the announcement, Bubeck said he had never asked for Alpöge to be removed from authorship of Alpöge's own work. He described a different discussion: that he was surprised to learn on the call that the pair had solved only Euler, that one option brainstormed afterwards was for Buckmaster to be lead author on a rewrite of OpenAI's Navier-Stokes proof, and that it was in that context that he said it "would be simpler if Levent was not an Anthropic employee," because he felt it would be inappropriate for an Anthropic employee to author OpenAI's work. He wrote that "it was admitted that internal Anthropic models had been used in their proof of Euler blowup," that a further option he wanted to propose was access to OpenAI's internal model so the pair could try to close the gap themselves, and that he was met on the call with threats to go to the press. He apologized for the career remark as "an extremely poor choice of words" and said he had retracted it on the spot.^[9] Alpöge replied that he would have been glad to collaborate and did not care about authorship on that step, but that on overhearing a loud hallway conversation, "especially the part where a millennium prize was offered if i'd just be removed from the paper," it seemed to him that matters were already settled.^[21]

Buckmaster also questioned the training-data timeline in public, noting that OpenAI's own statement dates the start of the internal model's training to August 28, after the pair had obtained their results, and asking whether it is ethical to use a customer's data to try to scoop that customer.^[33] Andreas Thom, writing on September 9, said he had asked OpenAI researchers a similar question after the company's earlier non-sofic-group announcement, separating whether his ChatGPT conversations had entered training data from whether they were accessible to the solving process, and had received only a categorical denial that he read as answering the second question alone.^[32]

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; and publication arrangements were discussed on September 6. They disagree about the intent and meaning of the proposed authorship arrangement, about the tone of the calls, and now about whether product-use data could have contributed to the model at all. The public record does not resolve the training-data question independently of OpenAI's own investigation, and it does not support treating speculation about copying as established fact.^{[1][8][9][21]}

Reaction in the mathematical community

Charles Fefferman, who wrote the Clay Institute's official problem description, told Quanta "I was thrilled that the problem was solved," and said that the heroes of the story are Córdoba and Martínez-Zoroa.^[20] Martínez-Zoroa told Quanta: "I'm very happy for Tristan. It would have been nice to do this ourselves, but I'm very happy for him."^[20] Martin Bridson, president of the Clay Mathematics Institute, was quoted by El País on September 9 calling it an exciting day as the community contemplated the announcement of important advances in human understanding of mathematics.^[22]

Speaking to El País on September 10, Córdoba, a researcher at ICMAT in Madrid, said that "if our work had not existed, the AI would not have solved the problem," described the AI contribution as gaining time rather than having new ideas, and estimated from the reported token counts that the run corresponded to roughly 15 million euros of compute against the year he and Martínez-Zoroa had spent on ministry funding. Martínez-Zoroa, at CUNEF University, said he had so far only skimmed the 166 pages, that parts of it reminded him of the strategies his group had developed but in unfamiliar notation, and that it was hard to say how heavily it leaned on their strategy before digesting it properly. He added that he knew of cases where a researcher had explained a problem to a colleague who then finished it with AI help and gave no credit, and said the community needs to rethink how credit is assigned.^[23]

Tao published a four-part thread on September 8 arguing that good open problems are being mined in a non-renewable way. He wrote that the identification of a promising problem is now the scarce resource, that "even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential," and that the resulting incentives point toward no longer sharing promising research directions, which "would reverse centuries of traditions of open science." He also criticized "the refusal of AI companies to disclose their negative results, or reveal the process towards obtaining their solutions," and proposed that classes of problems be designated as requiring an analysis that yields insight rather than a raw solution.^[25]

The Fields Medallists' declaration

On September 11, 2026, twenty-five Fields Medallists published a declaration titled "A Severe Misalignment of AI in Mathematics" at mathandai.org and on Tao's blog, and invited further endorsements.^[28][29] The signatories argue that the push by AI companies to solve mathematical problems as a benchmark is detrimental to mathematics and to the mathematical community, that solving problems is a proxy for the primary goal of conceptual understanding, and that rushed announcements leave no time for a proper writeup, the isolation of new ideas, or the citing of relevant previous work, which "raises severe attribution and plagiarism questions."^[28] Tao wrote that the declaration grew out of discussions among the signatories over the preceding week and that they had released it sooner than a fully consultative process would have allowed because of the urgency they perceived.^[29]

The declaration does not dispute the correctness of any AI-produced result. It is a statement about process, credit, and the transmission of ideas, and it names no company and no specific paper. Tao's post links an Economist article of September 11 headlined "Top mathematicians are outraged by OpenAI's methods," a short video interview with James Maynard, and a French version of the declaration published in Le Monde.^[29][30][31] A full account, including the signatory list and the endorsement mechanism, is at A Severe Misalignment of AI in Mathematics.

On September 21, 2026, OpenAI cited the declaration when it announced that it was working with mathematicians who had set up an independent Advisory Group on Mathematics and Artificial Intelligence to advise on the review and communication of new results. In the same post OpenAI said that the internal model it began training on August 28, in addition to the Navier-Stokes result, "has now resolved more than 100 long-standing open problems across most areas of mathematics".^[34]

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 the formal check gave mathematicians confidence the result is correct, while also emphasizing that humans still have to guarantee that the statement proved in Lean is logically equivalent to what mathematicians set out to prove, and that the mathematical and priority questions will take time to sort out.^[20]

That reaction is not equivalent to peer review or Clay acceptance. On September 9, El País reported that no outside scientist had yet independently verified the OpenAI proof, and that a review of this importance usually takes months.^[22] The Clay Mathematics Institute's September 11 statement welcomed the announcement and said it hopes "to see waves of new human understanding unleashed as the innovations behind this work are analysed and interrogated," while directing readers to the rules for evaluation and credit and adding that the process "is deliberately unhurried, but we will provide updates."^[7]

Those rules are specific. A proposed solution must be published in a qualifying outlet, which the rules define as a refereed mathematics publication of worldwide repute, or a publication meeting relaxed conditions approved by the institute's board of directors on a recommendation from its Scientific Advisory Board. A publication that lacks a named and contactable editorial board, an editor able to identify an appropriate referee, a published refereeing process the institute considers adequate, or inclusion in the list of publications maintained by MathSciNet is deemed not to qualify. The proposed solution must then survive rigorous examination by the global mathematics community for a minimum of two years and achieve general acceptance, both judged at the institute's sole discretion, before the institute decides whether detailed consideration is merited and, if so, convenes a special advisory committee. The rules also state that no Clay-affiliated entity will accept invitations, requests, or demands to recognize the status of a proposed solution, and that the institute "will pay special attention to the question of whether a Prize solution depends crucially on insights published prior to the solution under consideration," and may recognize such prior work in a prize citation or recommend including its author in an award.^[6]

The most accurate status as of September 13, 2026 is a publicly documented proposed solution accompanied by an author-supplied Lean formalization, acknowledged by the Clay Mathematics Institute as an announcement that the problem has "apparently been settled," with independent mathematical review not yet reported, provenance and attribution questions still open, and a prize process that cannot conclude for at least two years after publication in a qualifying outlet.^[6][7] Calling it either a settled Clay solution or a refuted claim would go beyond the available evidence.

See also

• Mathematical reasoning in AI
• A Severe Misalignment of AI in Mathematics
• AI Co-Mathematician
• Lean
• GPT-6 Astra

References

1. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10 ^11 ^12OpenAI. "On the Navier-Stokes Millennium Prize Problem." September 8, 2026, updated September 10, 2026. openai.com/...navier-stokes-solution
2. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10OpenAI. "Finite Time Blowup for Navier-Stokes." September 2026 (166-page version, PDF creation date September 8, 2026). cdn.openai.com/...navier-stokes.pdf
3. ^1 ^2 ^3 ^4Charles L. Fefferman. "Existence and Smoothness of the Navier-Stokes Equation." Clay Mathematics Institute official problem description. claymath.org/...navierstokes.pdf
4. ^Clay Mathematics Institute. "Navier-Stokes Equation." Accessed September 13, 2026. claymath.org/...navier-stokes-equation ; index page: claymath.org/millennium-problems
5. ^Clay Mathematics Institute. "Rules for the Millennium Prize Problems." Accessed September 13, 2026. claymath.org/...rules
6. ^1 ^2 ^3 ^4Clay Mathematics Institute. "Millennium Prize Description and Rules," adopted September 26, 2018. claymath.org/...millennium_prize_rules_0.pdf
7. ^1 ^2 ^3Clay Mathematics Institute. "Navier-Stokes Announcement." September 11, 2026. claymath.org/...navier-stokes-announcement
8. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9Tristan Buckmaster. Public statement on finite-time blowup for forced fluid equations and interactions with OpenAI. September 7, 2026. cims.nyu.edu/...statement.pdf
9. ^1 ^2 ^3Sébastien Bubeck. "I would like to clarify a few things." X, September 8, 2026. x.com/...2097379411691516310
10. ^1 ^2 ^3 ^4OpenAI. "NavierStokesAndEuler." GitHub repository, including README, formalization.yaml, and commit history. Accessed September 13, 2026. github.com/...NavierStokesAndEuler
11. ^1 ^2Diego 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. arxiv.org/...2309.08495
12. ^1 ^2Diego 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 2025. arxiv.org/...2410.22920
13. ^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, 2026. doi.org/...s00205-026-02198-0
14. ^1 ^2Levent Alpöge and Tristan Buckmaster. Blowup for the Euler equations with smooth forcing. September 2026. cims.nyu.edu/...euler.pdf
15. ^1 ^2Levent Alpöge and Tristan Buckmaster. Blowup for the Boussinesq equations with smooth forcing. September 2026. cims.nyu.edu/...boussinesq.pdf
16. ^1 ^2Levent Alpöge, Tristan Buckmaster, and Matei P. Coiculescu. Extending the Córdoba and Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing. September 2026. cims.nyu.edu/...ipm.pdf
17. ^Tristan Buckmaster. "fluid_lean." GitHub repository, created September 8, 2026. github.com/...fluid_lean
18. ^Terence Tao. "Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations." September 7, 2026. terrytao.wordpress.com/...ressible-euler-equations
19. ^American Mathematical Society (Ravi Vakil and John Meier). Statement on the Navier-Stokes problem, posted on X on September 8, 2026. x.com/...2097380478349463939 (also published at ams.org/news)
20. ^1 ^2 ^3 ^4 ^5Konstantin Kakaes. "AI Has Solved One of Math's $1 Million Millennium Prize Problems." Quanta Magazine, September 8, 2026. quantamagazine.org/...nium-prize-problems-20260908
21. ^1 ^2 ^3 ^4Levent Alpöge. Response to OpenAI's concurrent-work statement. X, September 8, 2026. x.com/...2097383870773748190
22. ^1 ^2 ^3Manuel 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. elpais.com/...istoria-desata-acusaciones-de-plagio
23. ^"Un año de trabajo de dos matemáticos españoles frente a 88 horas y 15 millones de OpenAI: 'Sin nuestra idea, la IA no lo habría resuelto'." El País, September 10, 2026. elpais.com/...tra-idea-la-ia-no-lo-habria-resuelto
24. ^1 ^2 ^3Internet Archive capture of the Navier-Stokes manuscript at 17:29 UTC on September 8, 2026 (165 pages, 16 references). web.archive.org/...navier-stokes.pdf
25. ^Terence Tao. Thread on the scarcity of good open problems. Mathstodon, September 8, 2026. mathstodon.xyz/...117237320796901560
26. ^1 ^2Anima Anandkumar. "Stable singularity of the Euler equations on R^3." Guest post on Terence Tao's blog, September 10, 2026. terrytao.wordpress.com/...he-euler-equations-on-r3
27. ^1 ^2Anima AI + Science Lab, Caltech. "Stable Singularity of the Euler Equations on R^3 without forcing." September 7, 2026. tensorlab.cms.caltech.edu/...euler
28. ^1 ^2 ^3"A Severe Misalignment of AI in Mathematics." mathandai.org, September 11, 2026. mathandai.org
29. ^1 ^2 ^3 ^4Terence Tao. "A Severe Misalignment of AI in Mathematics." September 11, 2026. terrytao.wordpress.com/...ent-of-ai-in-mathematics
30. ^"Top mathematicians are outraged by OpenAI's methods." The Economist, September 11, 2026. economist.com/...s-are-outraged-by-openais-methods
31. ^French version of the declaration. Le Monde (Idees), September 11, 2026, as linked from Terence Tao's blog. lemonde.fr/...-divergent-profondement_6770630_3232
32. ^Andreas Thom. Post on transparency and training data. Mathstodon, September 9, 2026. mathstodon.xyz/...117240535270608201
33. ^Tristan Buckmaster. Post on the reported training start date. Mastodon, September 8, 2026. mastodon.social/...117236471352470303
34. ^OpenAI. "Advisory Group on Mathematics and Artificial Intelligence." September 21, 2026. openai.com/...advisory-group-on-mathematics-and-ai