Resolution criteria
This market will resolve to YES if, between September 9, 2026, and December 31, 2026, at 11:59 PM UTC, a proposed solution for at least one of the remaining five Millennium Prize Problems is formally announced or published.
The five eligible problems for this market are:
P versus NP
The Riemann Hypothesis
Yang–Mills Existence and Mass Gap
The Hodge Conjecture
The Birch and Swinnerton-Dyer Conjecture
For the purposes of this market's resolution:
Navier–Stokes Existence and Smoothness is excluded from resolving this market to YES. OpenAI announced a proposed solution to Navier–Stokes on September 8, 2026. This market specifically tracks whether another (i.e., a third) Millennium Prize Problem will have a solution claimed or published in 2026.
Poincaré Conjecture is excluded as it was solved prior to 2026.
The proposed solution must be put forward by a recognized mathematician, a reputable academic institution, or a major AI/technology research organization (such as OpenAI, Google DeepMind, Anthropic, etc.).
The announcement must be accompanied by a published paper, a public preprint (such as on arXiv), or a formalized computer-verified proof (such as in Lean or Coq).
The solution does not need to be officially verified or awarded by the Clay Mathematics Institute by the end of 2026 (as CMI rules require a two-year waiting period after publication). However, the solution must not be widely debunked or retracted by the mathematical community before December 31, 2026.
If no other eligible Millennium Prize Problem has a proposed solution announced by the end of 2026, this market will resolve to NO.
Background
The Millennium Prize Problems are seven mathematical puzzles selected by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for a correct solution. For over two decades, the Poincaré Conjecture (solved by Grigori Perelman in 2002–2003) was the only solved problem.
On September 8, 2026, OpenAI published a paper claiming that an unreleased internal AI system had solved the Navier–Stokes existence and smoothness problem by proving that the equations can develop a singularity in finite time. This market is designed to predict if the rapid acceleration of AI capabilities or human mathematical breakthroughs will result in yet another of the remaining five unsolved classic problems being solved before the end of 2026.
Update 2026-09-10 (PST) (AI summary of creator comment): If a proposed solution is neither widely debunked nor widely accepted (remains controversial) by December 31, 2026:
The creator will delay resolution for 2 to 3 months after the market closes.
If the proof remains controversial after this period, the creator is leaning toward resolving the market to NO or N/A (rather than YES), and will consult with moderators and mathematicians to make a final decision.
Update 2026-09-21 (PST) (AI summary of creator comment): - The controversy clause applies strictly to the mathematical validity of the proof itself (e.g., disputed proofs like the ABC conjecture), not to public relations or circumstances surrounding the announcement.
An announcement similar to the Navier-Stokes release will resolve to YES.
1,000People are also trading
@HumanClanker Would this resolve YES if OpenAI's committee of world-famous mathematicians reported that they believe a Millennium Problem was solved, but gave no other details because doing so would be bad for the mathematical community?
@Haiku This seems exceedingly unlikely to happen and I don’t expect it. However under the trading criteria people have been trading under so far it would not count and because of that I would not count it.
See in the criteria: The announcement must be accompanied by a published paper, a public preprint (such as on arXiv), or a formalized computer-verified proof (such as in Lean or Coq).
At least some degree of public access is important. And knowledge about which specific millennium problem it is would be something I consider to be a necessary condition.
>Navier–Stokes Existence and Smoothness is excluded from resolving this market to YES. OpenAI announced a proposed solution to Navier–Stokes on September 8, 2026.
Related:
https://openai.com/index/advisory-group-on-mathematics-and-ai/
For anyone curious, I have written up some explanation for my bets. They're relevant to both my guesses regarding what's currently been achieved by the labs, and my skepticism that those achievements are evidence in favor of a full MP being on the horizon. Elliot Glazer (@ElliotGlazer) on X
TIL Hodge is not even formalized in Lean, nor are many of its prerequisites, and formalizing these is a major ongoing task in the Lean community.
This is a task that necessarily requires slow human judgment; only humans can say whether a given formal statement is the formalization of a given informal statement.
For YES holders, you can now also bet on how many tokens the eventual Hodge proof/disproof will take: /eapache/how-many-billions-of-output-tokens
@Velaris gambling is bad for you, you should only use prediction markets to capitalize on your abilities and knowledge
@Quroe yep. I'm sort of doing a paroli martingale derivative though, currently fresh in low ball phase
Velaris limit order up, aka free mana.
@mathvc How do they know whether they're close to solving something? They either have a solution or they don't.
@AndrewHebb expanding on what Craig said, proofs at this level are highly modular. So less like a sudden binary guess and more like constructing a massive bridge piece by piece using foundational steps(lemmas.). So they may have solved most of the infrastructure and know they only need (like the Z example from craig) done to complete the problem. Alternatively, researchers know they are close when they successfully solve simpler versions of the problem, or when their models probabilistically converge on a verified path with fewer dead ends
@AndrewHebb I'm not sure I did a good job with my analogy as I don't think I agree with your statement. Progress in math isn't all-or-nothing. Solving cases X and Y objectively eliminates unknowns and reduces the remaining problem space for Z. That is the literal definition of getting closer to a solution. in a betting market, news that foundational lemmas or partial cases have been solved should absolutely cause you to update your probability of a full solution upward. HOWEVER how legit the news is, which specific cases or lemmas have been solved, how tricky the remaining case or lemmas are, etc leave plenty of room for debate
@ShaneBo Yeah to me the question is, do they have anybody qualified to tell? That obviously wasn't the case for Navier-Stokes; they openly admitted nobody at OpenAI understands the proof they put out, even a little.
It’s nonsense embellishment, there’s no good candidate for a Hodge partial result that would imply “they’re almost done.” This is probably the same result as the “significant progress” alluded to in the NYT article or by Greg Brockman, which were always about some case of the Hodge (maybe Hodge for abelian varieties). If they’ve truly been so close, they’d be grinding it to completion instead of writing a paper with mathematicians on their partial work.
@ElliotGlazer I don't know any algebraic geometry but I could imagine a partial result like "Hodge holds in all dimensions d > 10" and then someone who is a dumbass like me thinking "hey now there's only a few cases left". (I have no idea if Hodge in one dimension implies Hodge in another dimension, whether higher or lower.) Of course as we saw with Poincaré you can be missing just a couple dimensions and still have a lot of work ahead of you, work which isn't even remotely similar to that which establish the other dimensions.
@Melquiades I was able to see the report and there's almost no substance. Just "employees at OpenAI think they are close to solving Hodge". Not even a direction (positive or negative solution).
@pietrokc Yeah it’s a nothingburger. I’m out of liquidity though so I guess this market’s gonna stall at 60% till peeps calm down again.
@ElliotGlazer wait, you are betting "NO" for the whole 2026? I understand your twitter "No" bet for september, but for the whole 2026 that seems rather clear "YES" unless mathematicians cry so much that OpenAI will have to cancel all math efforts
@pietrokc I would have assumed after the Navier-Stokes PR mess they would be more careful with due diligence before publicizing anything. That said, labs remain strongly incentivized to publish partial results and black box frameworks. Given how the initial rules were framed around published preprints rather than peer validation, 56% or even 60 ish % feels like a fair balance between the certainty of an announcement and the skepticism of the math community.(which is why I slightly bumped my pretty small Yes exposure)
I suppose the real uncertainty here is a meta issue: the creator is holding up the Navier-Stokes release as the standard, so a similar release for Hodge SHOULD be accepted, I think. The current price hinges on whether traders will choose to hold the creator to that standard despite the new 'controversy' clause, whether labs decide the PR hit of an incomplete release outweighs the hype, and how BS the reported progress really is. IF we are deciding to hold to the controversy clause, I think we need to have very clear criteria on what that means. Which I'm not sure we have cleared the bar for in the comments or in the creators criteria notes. @ElliotGlazer I just got around to reading your X posts and some of your comments. Do you lean towards saying that there will be no release OR that the release will come but will clearly fit some controversy criteria to be voided?
@ShaneBo I think I’ve been pretty clear. The partial results that I’ve heard about don’t point to any full solution in the foreseeable future. I’m betting NO in the strictest sense, they will not even claim to have solved a new MP this year.
@ShaneBo The controversy clause is about the result itself not any circumstances surrounding it. From everything I have seen the voracity of the naiver stokes solution is accepted widely even if people think there is some more due diligence to be done, whereas a good example of this not being the case is that there is a proposed solution to the ABC conjecture which is not widely accepted. A release similar to the Navier-Stokes release would resolve yes. A controversial result is different from controversy surrounding a result.
That clause is in there partially because I want to try very hard for this market not to N/A, I think that resolution should be clear and unambiguous and want to handle all edge cases delicately, while also resolving in a timely manner. I do think this is an edge case which is unlikely to happen but it is better to have those bases covered.
I hope that clears things up, and if you have any probable scenarios where you think the resolution criterion may be unclear, feel free to ask and I will clarify.
@HumanClanker I appreciate the detail. The distinction between a rollout controversy and a fundamental mathematical dispute like the ABC situation makes total sense. Having it on the record that a framework mirroring the Navier-Stokes release definitively resolves to YES clears up the primary unknown I was noting. Perhaps a misinterpretation on my part, but the intense discussion around here regarding the PR fallout, assignment of credit, and the labs not fully understanding their own results made me think a Navier-Stokes style release would actually trigger the controversy clause rather than clearing it. Glad to have that clarified!
@ElliotGlazer Thanks for the answer. My inside info is almost certainly not as comprehensive as yours, but betting on OpenAI or Anthropic to suddenly show restraint and withhold a claim seems to me like a very low-probability wager (noting your premise that without a true solution, they won't release). I will grant that I've upped my odds that any rushed release is likely to be highly questionable.
However, I still feel decent about the current odds given the immense PR incentives. While likely inflated, there is some documented truth to Anthropic's Riemann progress. their model already pushed a critical bound to 67.2 last month(very open to being corrected here on either the substance or implication). Combined with the BSD rumors, the labs are more than willing to brute force a partial framework just for the valuation bump, and I'm wagering they might just repeat the Navier-Stokes PR playbook and not so understood result. I'll naturally update my odds downward with each passing month. But for now, I'll take the other side of that bet...though not much higher than current odds without some actual substance in a release.
Also laughing at myself with my 60 mana investment engaging in this debate lol fun nontheless
@ShaneBo Anthropic's Riemann progress. Negligible. The rumored Hodge progress: a great result, but by far the easiest case of Hodge. The BSD rumors: completely died out. PR incentives: Tens of millions of dollars have been spent on the 5 remaining problems with little progress. It isn't that clear to me that the PR case for repeating this gamble every month or two is gonna persuade higher management.
@ElliotGlazer Fair points. That said, tens of millions in compute is effectively a rounding error marketing expense against multi billion dollar valuation dynamics if management believes a breakthrough is within reach.(emphasis on believes) My read is that Anthropic leadership was genuinely motivated enough by the Riemann progress to greenlight aggressive compute runs, which creates pressure on OpenAI to do the same rather than sit back which seems was the case with Navier-Stokes. The EV on this must be insane with IPOs closing in, if they think it's 5% likely (which granted I'm going to assume to your understanding may be much to high) is probably worth a hundred million in spend or maybe even 200 or 300 million; especially for open AI trying to go from 800, 850 to 1.2-1.5 T. Honestly if they think it could lead to a 10% bump(probably too aggressive but not too implausible) they should be willing to spend something like 4.2 billion on it at a perceived 5% likelihood of success. This is obviously simplifying things and no way would they actually ever spend 4.2 billion on this, but I think directionally addresses the potential stakes of dedicating resources to solving
My weaknesses here would be:
-how accurate the read on leadership is
-I'm reasonably but not exceptionally confident on the valuation multiple solving multiple millennium problems would result in
-there may be surer bets/lower hanging fruit with higher EVs that they would have to sacrifice at expense of this pursuit( though I'd wager they are just going to aggressively spend in every direction right now)
But overall, the financial incentives point to them taking the shot
The strongest counterpoints, I think:
-It's really freaking hard, and progress is actually been negligible. Three months at current pace will still be negligible. Even taking the perceived impressiveness of "88 hours" of brute force when examining what the equivalent human work years to that compute would have been it's probably not quite as impressive or replicable to the other problems. Making the chances of solving more like less than 1/10th of a % by those in the know making the EV case moot
@pietrokc Point that: "TIL Hodge is not even formalized in Lean, nor are many of its prerequisites, and formalizing these is a major ongoing task in the Lean community.
This is a task that necessarily requires slow human judgment; only humans can say whether a given formal statement is the formalization of a given informal statement."
POTENTIALLY: OpenAI saw the NS bump, determined it wasn't worth it from a valuation stand point, and are walking away from this by letting the story fade away over the next few months. OR admitting their limitations regardless of potential valuation mulitple
All said really appreciate the insights @ElliotGlazer looks like you are doing some really cool things and learned a few things on the state of the industry I wasn't aware of just checking out your X feed. Will definitely consider some of this as my update my priors here and try to keep up with all that's going on.
@ShaneBo Thanks for laying out your reasoning so carefully. This is the kind of thing good forecasts are made of.
Just to add some clarity on the Riemann progress by Anthropic. Nobody in the know regards that as big progress, not even as a result in the field, much less towards RH. If leadership at Anthropic took that result as evidence that they should yolo RH, that would be a very serious lapse in judgment. Alpöge definitely understands this so if they're listening to him, they wouldn't yolo based on that.
To be more specific: the 67% result is just a slight optimization on known techniques. And moreover, even if those techniques could be pushed to give 100% (which they provably cannot), that would still not prove RH. As a simple analogy, it's easy to prove that 100% of integers are not perfect squares. Yet obviously perfect squares exist, even infinitely many of them. The nature of RH is that even a single zero off the critical line ruins the distribution of primes implications.