Talk:Millennium Prize Problems

2 dimensional sphere?

in the solved problem it talks about a 2 dimensional sphere. Is this a mistake and it is supposed to be a circle, or am i missing something? Maybe it should be rewritten to be more clear. Chuck541 (talk) 21:06, 15 July 2024 (UTC)

It could be clarified, but there’s no mistake. In mathematicians’ parlance, the two-dimensional sphere is the usual sphere you think about as living in three dimensions. The idea is that the sphere, in and of itself, has only two degrees of freedom. You just see it inside three dimensional space. Gumshoe2 (talk) 00:02, 16 July 2024 (UTC)

Claimed resolution of the Navier–Stokes existence and smoothness problem

OpenAI has announced a solution to this problem. In light of this, I think it is best to exercise some caution when editing this article. The Clay Mathematics Institute is yet to make any statements on this announcement as far as I am aware, and a sentence like

OpenAI declined to claim the Millennium Prize for the Navier-Stokes result

(which was present in the article as I originally wrote this) is absurd given that no offer of the prize has even been made yet. Additionally, I think it may be important to give time for humans to review the result; I'm not sure how important this is given that a Lean formalization was also shared, but this is worth considering as well. Twotwice^talk 18:58, 8 September 2026 (UTC)

Concur, given the above. Augmented Seventh (talk) 19:19, 8 September 2026 (UTC)

Navier-Stokes Proof "spurred by rumors"

This seems probably true, & is backed up by the personal account, but the citation for this is from Tristan Buckmaster.

Might be better form to cite a different source here, or write "According to Buckmaster"?

Looks like Techcrunch has covered this https://techcrunch.com/2026/09/08/openai-fought-dirty-on-career-making-math-problem-says-nyu-mathematician/ RaynorElgie (talk) 19:02, 8 September 2026 (UTC)

Proposed OpenAI Solution

I know there's a lot going on; however, we should absolutely still have a section for the problem (the revision I saw just was missing the problem entirely) since this is likely to be very widely viewed over the next few days. PLEASE DO NOT DELETE THE SECTION ON NAVIER-STOKES WITHOUT PUTTING IT SOMEWHERE ELSE OR DUPLICATING IT.

Right now, I think it's fine if we mention the solution (I think it's probably correct, and has been reported on by reputable news outlets) but since the Clay institute hasn't issued/confirmed anything, maybe keep it in unsolved for now? That's just my opinion though. Once/if we decide that the problem is indeed solved, we will move it to the solved section of the page, but please don't just delete the section wholesale. Aven13 19:20, 8 September 2026 (UTC)

I suggest making a new section between "Solved Problems" and "Unsolved Problems" for problems where there is no consensus on whether they are solved or unsolved. Maybe something like "Claimed-to-be-solved problems", but that wording feels a bit awkward. ~2026-48695-01 (talk) 19:28, 8 September 2026 (UTC)

Wikipedia is very concerned with being not wrong, and that's what WP:TOOSOON is for. We're an encyclopedia, and not a news source, if we can't yet cut through the noise, we should probably not mention it inside the public-facing article. 海盐沙冰 _{/ aka irisChronomia / Talk / Uploads} 20:05, 8 September 2026 (UTC)

"However, this result has not yet been verified by the Clay Institute or the independent mathematical community."

I completely disagree with this framing. For starters, there's no reference given for this statement. Besides that, it absolutely has been verified through Lean. All you need to do to show that it's correct is run the Lean file. I've never heard people dispute the idea that Lean verification is a valid form of proof, so there's no reason to be so washy on whether the problem has been solved. Feed Me Your Skin (talk) 20:44, 8 September 2026 (UTC)

"OpenAI's solution has yet to be verified independently or publicly accepted by The Clay Mathematics Institute, a maths organization based in the US which runs the Millennium Prize that offers big money to the first to solve certain mathematical conundrums." --https://www.bbc.com/news/articles/cy7zygy3rl2o, added as a source. Proofs in Lean can be wrong, not by virtue of the formalization engine/kernel being wrong, but the statement itself being wrong/not translated correctly. It's a gigantic file, so we should definitely wait until someone who knows what they're talking about with regards to both Lean and PDEs gets their hands on it and verifies it. Aven13 01:54, 9 September 2026 (UTC)

Misleading lead sentence?

The lead says "Others have claimed they have found it before OpenAI, which used their draft paper as training data". I do not think Buckmaster claims that they found the solution to Navier-Stokes before OpenAI. E.g., the end of his statement says "If indeed an OpenAI model did close the gap to Navier-Stokes, that is a remarkable thing and it should be said loudly, by them, with the history intact". The controversy is real, but I don't think it's about Alpöge and Buckmeister claiming to have solved Navier-Stokes itself; it's about some of the significant results toward it. Aterbiou (talk) 11:28, 9 September 2026 (UTC)

"...and is the subject of a priority dispute."

The linked source doesn't back this up. It doesn't suggest there is a priority dispute, the allegation is that OpenAI plagarised from their work via Codex and harassed them, in this case it would be better to say "...and is the subject of plagiarism allegations." The priority rule suggests that in any case OpenAI would get credit for the discovery as they published their findings first, there is no doubt about priority here. Respectfully dissenting, Coleisforeditor (talk) 23:11, 9 September 2026 (UTC)

Agree 100%, perhaps even something more specific like "illicit usage of private research-in-progress from mathematicians who used OpenAI's products". OpenAI now officially denies the training allegations, as it wasn't clear from their first statement. Would a direct source to the statements be allowed? Ex. https://cims.nyu.edu/~tristanb/statement.pdf Zbrek (talk) 17:19, 10 September 2026 (UTC)

Navier–Stokes existence and smoothness problem status

Please see Talk:Navier–Stokes existence and smoothness#Problem status again Regarding whether Navier–Stokes existence and smoothness is solved. McYeee (talk) 01:29, 17 September 2026 (UTC)