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Navier-Stokes existence and smoothness

A Millennium Prize Problem asking whether smooth solutions to the 3D incompressible Navier-Stokes equations always exist or can break down in finite time.

Known aliases

  • 3D Navier-Stokes
  • Navier-Stokes equations
  • Navier-Stokes existence and smoothness
  • Navier–Stokes existence and smoothness
  • Navier-Stokes Millennium Prize problem
  • Navier-Stokes Millennium Problem
  • Navier-Stokes problem
  • Navier–Stokes problem
  • Navier-Stokes singularity problem
  • Navier-Stokes smoothness and existence problem

Current stories

scienceConfirmed5 publishers

Mathematicians face months of reading to judge whether OpenAI's hundreds of proofs contain new ideas

OpenAI released hundreds of results on open problems in math and theoretical computer science, many already verified in the Lean programming language. Those are all but certain to be correct, so the months of reading ahead are about whether they hold new ideas.

Publishers:nature.com newscientist.com phys.org retractionwatch.com scientificamerican.com

Perspective Coverage

5 publishers
Builder
Builder 33%
Operator
Operator 55%
Investor
Investor 12%

Reality

Evidence62
Adoption
Not scored
Hype gap+35
Incentives70
Confidence60
investConfirmed3 publishers

Lean proof checks cover about 300 of the 722 math manuscripts OpenAI's unreleased model produced

OpenAI published 722 math manuscripts across 372 result families, all produced by an internal model it has not released. About 300 come with Lean formalizations a machine can check, Crypto Briefing reported, so the rest wait on human reviewers whom one mathematicians' group is urging to stop working with OpenAI.

Perspective Coverage

3 publishers
Builder
Builder 30%
Operator
Operator 47%
Investor
Investor 23%

Reality

Evidence55
Adoption20
Hype gap+35
Incentives65
Confidence55
scienceOne report1 publisher

Scientists wary of AI scoops rein in their use of commercial chatbots

Researchers say AI companies have at least twice in five weeks reached answers to questions they had long been working on, Nature reported. Some now limit their AI use, though one AI-safety scientist expects models to keep getting there first with or without uploaded manuscripts.

Publishers:nature.com

Reality

Evidence40
Adoption
Not scored
Hype gap+25
Incentives60
Confidence45
buildConfirmed6 publishers

OpenAI hands mathematicians 372 AI-generated proofs to check on GitHub

OpenAI posted 372 results from an internal model to GitHub, at about three hours of ChatGPT Pro compute each on average. Finding proofs is now cheap enough that the slow step is mathematicians checking and judging them.

Perspective Coverage

6 publishers
Builder
Builder 44%
Operator
Operator 31%
Investor
Investor 25%

Reality

Evidence58
Adoption10
Hype gap+35
Incentives72
Confidence60
productConfirmed10 publishers

Lean files let mathematicians machine-check part of OpenAI's 722 AI-written math manuscripts

OpenAI posted 722 model-written math manuscripts in 372 families on GitHub, many with Lean proofs a computer can check. A mathematician citing one has to learn first whether it is a checked proof or a paper still waiting for review.

Perspective Coverage

10 publishers
Builder
Builder 40%
Operator
Operator 27%
Investor
Investor 33%

Reality

Evidence55
Adoption
Not scored
Hype gap+45
Incentives70
Confidence60
productOne report1 publisher

OpenAI hands its new math advisory panel a backlog of results from an unreleased model

OpenAI's new mathematicians' panel must first help release scores more results the company says an unreleased model produced. Researchers call the panel a good first step, though many are unsure how much say it will have over how OpenAI announces what its models find.

Publishers:theverge.com

Reality

Evidence45
Adoption
Not scored
Hype gap+25
Incentives65
Confidence40
buildConfirmed3 publishers

OpenAI gated an 88-hour, 10,000-agent proof search on a 17-hour Lean check

OpenAI's write-up gives the token counts, the agent count and the verification time for its Navier-Stokes result. The verification time is the number that decides whether the method transfers to anyone else's workload.

Perspective Coverage

3 publishers
Builder
Builder 52%
Operator
Operator 33%
Investor
Investor 15%

Reality

Evidence60
Adoption
Not scored
Hype gap+30
Incentives70
Confidence55
scienceConfirmed3 publishers

Scientific American puts Birch and Swinnerton-Dyer first in line to fall to AI

With Navier-Stokes claimed and five Millennium problems left, the ordering of what falls next turns on which conjectures a single counterexample would settle. Five million dollars of Clay prize money is still unclaimed.

Perspective Coverage

3 publishers
Builder
Builder 37%
Operator
Operator 40%
Investor
Investor 23%

Reality

Evidence55
Adoption35
Hype gap+35
Incentives70
Confidence50
productOne report1 publisher

OpenAI offered Tristan Buckmaster partial credit for its Navier-Stokes solution

Buckmaster was preparing to publish when OpenAI announced it had solved the problem and offered him a share of the credit. A second mathematician, Andreas Thom, then asked the company whether his own ChatGPT conversations had reached its model.

Publishers:gizmodo.com

Reality

Evidence48
Adoption30
Hype gap+45
Incentives72
Confidence46
productOne report1 publisher

Buckmaster is using OpenAI's Codex to reconstruct how OpenAI's agents beat him to Navier-Stokes

A week and a half after accusing OpenAI of copying his approach to a problem carrying a $1 million bounty, the NYU mathematician is still running the company's coding agent on his own papers.

Publishers:wired.com

Reality

Evidence58
Adoption45
Hype gap+18
Incentives72
Confidence55
productOne report1 publisher

Karp tells CNBC the safety debate is really the labs asking to be nationalized

Alex Karp told CNBC that frontier labs are angling for government liability cover because their commercial clients would otherwise sue them. Buyers whose data feeds those models should read their contracts with that in mind.

Publishers:gizmodo.com

Reality

Evidence38
Adoption
Not scored
Hype gap+32
Incentives80
Confidence45
scienceOne report1 publisher

A prompt to an unreleased Claude lifted the elliptic curve rank record from 29 to 31

Levent Alpoge and Ava Howell used a simple prompt to an internal Claude and got elliptic curves of rank at least 30 and at least 31 in days, in a field where the last single-rank step took more than 18 years.

Reality

Evidence27
Adoption14
Hype gap+40
Incentives68
Confidence45
leadershipOne report1 publisher

Azhar counts nearly every hand up when he asks 250 IT executives about AI results

Azeem Azhar counted the raised hands himself, in a self-selected room of 250 IT executives at a Las Vegas conference. Every firm in it told him AI spending rises next year. Budgets have to answer for that.

Reality

Evidence32
Adoption45
Hype gap+38
Incentives68
Confidence58
leadershipOne report1 publisher

Twenty-five Fields Medalists sign a warning about how AI labs announce mathematical results

The declaration lands three days after OpenAI's Navier-Stokes claim, in the middle of a credit dispute where the company and the mathematician at the center give accounts that cannot both be right.

Publishers:genaisecretsauce.com

Reality

Evidence34
Adoption20
Hype gap+18
Incentives72
Confidence42
productOne report1 publisher

OpenAI now calls it impossible that a Codex user's prompts reached its model

Tristan Buckmaster used OpenAI's Codex on the Navier-Stokes problem, then asked whether his prompts had fed the system that beat him to a proof, and the company's answer to The Verge got firmer over time.

Publishers:theverge.com

Reality

Evidence60
Adoption18
Hype gap+30
Incentives82
Confidence55
productOne report1 publisher

A Cornell mathematician calls OpenAI's million-dollar math result a marketing device

OpenAI says tens of thousands of agents solved a 90-year-old problem carrying a $1 million prize. The solution still needs independent verification, and a rival pair of mathematicians has a claim on the credit.

Publishers:wired.com

Reality

Evidence40
Adoption45
Hype gap+40
Incentives80
Confidence45
productOne report1 publisher

OpenAI gave two different answers about Buckmaster's Codex prompts in one week

Tristan Buckmaster told the ABC that the game is up for mathematicians. In the same week, OpenAI described what his Codex prompts did to its models in two incompatible ways, and a customer has no way to pick between them.

Publishers:gizmodo.com

Reality

Evidence34
Adoption22
Hype gap+45
Incentives76
Confidence42

Earlier coverage

  1. OpenAI's Navier-Stokes statement on user data has two parts, not a firm "we don't train on your data" promise

    Product · September 9, 2026 · Confirmed2 publishers