Background
The Millennium Prize Problems are seven legendary open questions in mathematics announced by the Clay Mathematics Institute (CMI) in year 2000, each carrying a US $1 million reward for the first correct solution. Grigori Perelman’s 2003 proof of the Poincaré Conjecture settled one of them, leaving six unsolved challenges:
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier–Stokes Existence and Smoothness
P vs NP
Riemann Hypothesis
Yang–Mills Existence and Mass Gap
A single AI system producing a formally accepted proof for any one of these six problems would represent a historic milestone for both mathematics and artificial-intelligence research.
Resolution Criteria
Evidence required
A peer‑reviewed paper in a recognised scientific journal or an officially accepted CMI submission must demonstrate that the proof was generated by an AI system and that it fully resolves one of the six unsolved Millennium Prize Problems.
AI autonomy
Humans may design, train, fine‑tune or prompt the model, but the complete logical argument must be produced autonomously by the AI.
Human assistance is limited to setting up the architecture, curating publicly available training data and verifying formatting; no new mathematical insights may be added by people.
Timing of Resolution
The market resolves YES if at any time before Jan 1, 2030 a qualifying proof that satisfies Criteria 1 (Evidence required) and Criteria 2 (AI autonomy) becomes publicly available.
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Why does the bot fill my limit orders bit by bit 😔.
Each time I place a limit order I get spammed with notifications.
Now that Navier Stokes has been solved Which Millennium Prize Problem will be solved next? | Manifold
The line “Human assistance is limited to…curating publicly available training data”. One reading of this is that if a model is trained on data not publicly available, then the market should resolve False. I’m sure all frontier models are trained on specialised non-public training data. Edit: …then the market should not resolve Yes for NS solution.
@Balasar if you use a supporting theorem to prove it, it’s still proving it.
@capybara hmm, perhaps. I think there's a measured difference between a brute-forced last-mile solution and coming up with the critical insight. It seems as though there is some controversy over whether the proof route the model took was adopted via some surreptitious or accidental incorporation of the semi-finished proof data from the others into the unreleased OpenAI model. A witness that leads the model along the right path can be exceptionally compact. With problems like these, "solving" the problem is more about the research programme and less about who is first out the door.
@Balasar if you have to brute force it, is it the last mile? Edit: I’m trying to say that it’s difficult to decide this market by measuring insights as no one solving this problem is going to start from point zero. Edit2: I pretty much agree with you. Once the details are known, hopefully the resolution will be obvious.
@capybara Sure, it can be, if you are trying to rush it out the door to scoop someone else. I'm not a fluid dynamics expert, but the way I understand it is that there were many possible ways that solving NS might be achieved. The search tree is unreasonably large, but if you could just tell the model something very simple about the high-level path to follow to prune away many of the bad ideas, it can become manageable in a way that is surmountable by both humans (6-12 months of natural work) and models (a few million dollars). One of those statements might be "The proof is one of existence of forced blowup, so only search those kinds of proofs".
Tristan Buckmaster claimed the OpenAI group told him "If you are willing to give any details it would be useful to avoid competing here and in general we are always thrilled when mathematician make progress with our models. Additionally if there is anything in terms of compute from OpenAI’s end we would be happy to provide it.” So that tells me that at the very least, they were interested in knowing what the competing team was up to. All AI companies exist by stealing the work of others, so it is not beyond imagining that they did the same here and took a peek at what was being worked on.
@Balasar If there was an inference time prompt (not part of in training data) that cut down the search tree sufficiently to be called a “mathematical insight”, then I agree and think the market should resolve False. I don’t see how which team came up with what or whether people’s achievements in making partial progress are being under appreciated matters for the market. I think it will come down to the list of prompts used and whether any of them meet that “insight” threshold.
@capybara what you said, but with the nitpick that this particular market should/will merely not resolve yes (yet), rather than resolving false pre-2030 ;P
Some larger manifold accounts ( @FlorisvanDoorn ) taking big hits over the navier stokes story. Guess this is surprising?
@0xseraphim Fake news. My portfolio and me are doing great. Just look at my league standing.
And that's definitely not because I was scrambling to do some de-risking in the last few days because I've bet NO too heavily on Millennium Prize Problem markets (and because those bets are in the past, they don't count towards my league standing this month).
The OpenAI statement just released is deeply misleading. Read the account by Tristan Buckmaster to see that there was a whole team of people involved in steering the model, and there are credible allegations that they specifically used an approach selected by Buckmaster and Alpoge.
I think it's pretty clear that, if this solution is valid, it does NOT fit the market criteria:
> no new mathematical insights may be added by people
This seems too high given how I'm interpreting the problem. Under my interpretation, not only would cyborg methods (human and AI collaboration) not count, but it would also poison the problem so it can never count if it were later solved by AI autonomously.
@Usaar33 I imagine there is so much already documented about these problems that an AI system making novel progress but then getting stuck and needing human guidance to progress seems less likely than the novel process being sufficient to answer the question or being in a form that humans can’t easily appreciate anyway.
@dorothydomer that’s a good point actually, arguably a potential flaw in the intent vs actual resolution of the market
@dorothydomer With the recent mathematical discoveries by ai which appear to be wholly ai (with only very basic human prompting) I'm no longer convinced of this argument. I suppose my steelman for my previous position would be that perhaps more humans are focused on these millennium problems than the problems solved in recent ai mathematical discoveries.
Finding any singularity in the Navier-Stokes equations is one of the six famous Millennium Prize Problems that are still unsolved.
With our novel AI methods, we presented the first systematic discovery of new families of unstable singularities across three different fluid equations.