Not Even Wrong

The “shepherds” of the OpenAI Yang-Mills proof yesterday released the Lean code of the proof, together with a validation and an assessment.

The OpenAI press release of October 26 claimed to have proved “Existence of quantum chromodynamics and completions of the Standard Model”, but the “shepherds” have not provided anything at all about this, not even an acknowledgement of the existence of such proofs or any plan to release any information about them. As far as I can tell, the claim to prove existence of “completions of the Standard Model” is likely extremely misleading. Perhaps they just have something about the minimal SU(5) GUT, without any Higgs sector. Even with a Higgs sector, such a theory disagrees with experiment, so in no sense is a viable “completion of the Standard Model”.

The press release goes on to claim:

As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem⁠.

Now that the actual result has been released, it is clear that this claim is also extremely misleading. The Jaffe-Witten writeup of the problem explicitly states (page 12)

We specifically exclude weak-existence (compactness) as the solution to the existence part of the Millennium problem, unless one also uses other techniques to establish properties of the limit (such as the existence of a mass gap and the axioms).

This is exactly the sort of “existence” proof OpenAI has produced, and they have not separately established the mass gap or properties comparable to it.

This whole story is starting to remind me a lot of the Google quantum computer wormhole publicity stunt. We’re seeing the same sort of fraudulent claims now being widely distributed in the media, e.g.:

• Scientific American has OpenAI’s “shepherds” affirm blockbuster physics result.
• The Economist has AI has proved that a theory underpinning fundamental physics is trustworthy

Surely there’s a lot more of this kind of thing to come.

The field of theoretical physics has suffered a great deal in recent decades from the problem of misleading hype. Getting in bed with the less than honest efforts of an organization trying to justify a valuation at greater than \$1 trillion is a terrible idea. I believe the group of physicists doing this should publicly explain that the OpenAI press release was misleading, release all the documents they are holding onto, and then shut themselves down.

In any case, now that they have released the Lean code, their plan to keep secret the 1147 page document explaining the proof and devote great efforts to understanding it and explicating it themselves makes no sense. Within hours of the code release, other groups (e.g. here and here) started using AI agents to derive similar documents which they are making public.

Maybe someday AI agents will generate actual “blockbuster” results on this or another fundamental physics topic. Until this happens, theorists should be helping to stop OpenAI’s hype campaign, not encourage it. I’m actually on a vacation trip at the moment, and feel that I’ve now done more than my fair share of this. So, for a while, I’ll only have minimal time to devote to dealing with comments, further developments or addressing the details of what’s wrong with this hype campaign.

Update: A statement from a large group of prominent theorists has been released criticizing the practice of not releasing publicly results such as the Yang-Mills one.

There was a conference held at KITP the week before last entitled Intermediate-Scale AI for Physics. The nature of this conference was somewhat of a mystery. As far as I can tell it never appeared in the list of forthcoming KITP conferences, and the minimal webpage for the conference had no list of speakers, no schedule of talks, and nothing about how to register to attend (at least going by the pre-conference version of the page on the Wayback Machine). Today a schedule with some video appeared, making a lot clearer what this was all about.

In Nathaniel Craig’s introduction he briefly explains that all six of the organizers are now working for OpenAI (Dan Roberts as an employee, himself, Xiaoling Qi, Eva Silverstein, Andy Strominger, and Xi Yin as visiting scientists), but “this is not an OpenAI program”. As far as I can tell, that’s the only place where the relation of the program and its organizers to OpenAI was mentioned publicly.

In trying to understand what some group of scientists is doing, I normally object to people who argue “they’re doing this because they’re getting paid a lot of money”, but in this case it’s worth noting that very likely all of these people are being paid a lot of money. According to this source, an “L5 Research Scientist” at OpenAI has average compensation of \$1.45 million/year, and from a reliable source I’ve heard about at least one \$2 million/year compensation package for an OpenAI research mathematician or physicist. Besides the obvious larger picture about our economy and society, AI money is a significant part of the story of the ongoing disruption of the fields of mathematics and physics.

Craig’s introduction also announced that parts of the program that were “discussions” would, according to KITP policy, not be recorded. If you look at the schedule, you quickly notice:

• Of the talks that just featured one speaker, the only one with an OpenAI employee as speaker was labeled “discussion” and not recorded.
• There were two group talks labeled “panel” (which were recorded) and six group talks labeled “discussion” (which were not recorded). The sessions dealing with the “theoretical physics/frontier lab interface” were labeled “discussion”, not “panel, so not recorded.
• If you look through other KITP conferences to see how often there are unrecorded discussion sessions, they’re hard to find. Looking through the schedules of the last ten KITP conferences before this one, there were several discussion sessions, all of which were recorded, with video available.

Another odd aspect of this OpenAI conference was that it ended on Friday the 23rd, and then on Monday OpenAI announced the supposed Yang-Mills existence proof discussed in the last posting. Presumably the organizers of the conference knew about this and that it was a topic of great relevance to that of the “theoretical physics/frontier lab interface”, but kept it secret to ensure a splashy PR event a few days later.

On the Yang-Mills topic, still no information at all about what was supposedly proved or how it was done. As for the claimed “unified completion of the Standard Model”, best guess seems to be that this just means they applied their methods to SU(5) instead of the SM gauge group to avoid the problem caused by the non-asymptotically free coupling for the U(1) factor. Possibly, their “unified completion of the Standard Model” doesn’t have a Higgs sector or spontaneously broken symmetry. And, of course SU(5) unified theories were experimentally ruled out a long time ago.

Last Friday the Simons Center at Stony Brook had a day-long program on Math, Physics and AI. Since it wasn’t organized by a group from OpenAI, you’re allowed to watch the videos, available here.

Update: One of the organizers of the KITP event, Harvard professor/OpenAI employee Xi Yin, announced yesterday that he’s leaving “to pursue a new direction”. No explanation of what that is, but he refers to his OpenAI position as “my first corporate job”, so maybe he’s moving to another one.

OpenAI has announced

a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics…

Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026.

From this press release, it’s hard to know exactly what they’re claiming to have proved. One could try and guess what “unified completion of the Standard Model” is being referred to, but they aren’t telling us. They’re also not telling us exactly what “existence” of Yang-Mills means. Since they don’t refer to the mass gap/Millenium prize problem, maybe this is just about existence for small volume. If so, not specifying that seems to me intentionally deceptive.

As far as I can tell, beyond what’s in the press release, they’re releasing no information publicly now. They’ve appointed David Gross to lead a group (Jaffe/Douglas/Klebanov/Nekrasov) that will have access to the secret unified physics slop, something which will

mark the beginning of a transition in which quantum field theory, the most important framework in modern theoretical physics, becomes a mathematically rigorous field.

I’ll try and find out more about what is going on here and update this posting later. If there’s a source for more information about this, please let me know.

Update: Looking around, I’m not seeing any more information available now about this beyond the press release.

I had been intending for a while to write about the Levent Alpoge/Ethan Sussman ongoing effort to do rigorous Yang-Mills. Unlike OpenAI, they’ve been posting things publicly, see here. There’s a thousand page or so manuscript about 3d Yang-Mills there, and they say they have a 4d YM existence proof, but in the useless form of tens of thousands of pages of unchecked slop.

A commenter correctly points out that, while (deceptively…) not mentioning the mass gap, OpenAI appears to be claiming a solution to the “existence” part of the Millennium Prize problem. That does specify not finite volume, but R⁴, and Alpoge/Sussman also claim they can do R⁴. In the Jaffe/Witten essay about the prize problem, they say:

Some results are known for Yang–Mills theory on a 4-torus T⁴ approximating R⁴, and, while the construction is not complete, there is ample indication that known methods could be extended to construct Yang–Mills theory on T⁴.

In fact, at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions. So even having a detailed mathematical construction of Yang–Mills theory on a compact space would represent a major breakthrough. Yet, even if this were accomplished, no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit T⁴ → R⁴.

I’m a bit confused about exactly what it means to have “existence” of the infinite volume limit without knowing enough about it to know if there’s a mass gap.

Finally, last week at the KITP there was a week long program about AI and particle theory. Videos of talks and panels should appear at some point. This must have been a rather odd event, with certain people knowing what OpenAI had and was about to do with it, but not able to talk publicly about this.

Update: The group of mathematicians formed to advise OpenAI about how to release results (AGMAI) has released a statement calling for immediate release of the Yang-Mills AI details.

Update: While waiting for the KITP videos, the videos from the kickoff event of a program on AI and the future of mathematics and physics being held in Paris are available here.

Update: On a different hep-th/AI topic, I don’t see how the literature doesn’t get overwhelmed (or maybe already has been…) by papers generated like this. The paper has now been posted on the arXiv.

Update: The “shepherding” committee for the OpenAI Yang-Mills proof has released a statement. They say they have to keep the proof secret because it might be wrong. Their plan is to first validate the Lean formalization by checking that the definitions and problem statement agree with the standard informal understanding. Then they’ll release the Lean proof. Until then I guess what supposedly has been proved will remain a secret.

In the next stage:

We will then turn to developing a deeper understanding and exploration of ways the proof can be used to advance the physics and mathematics of quantum field theory.

This they won’t do only by themselves, but:

We intend to draw upon the talent and expertise of other members of the community to help us with this difficult task.

So the plan seems to be that they will keep the potentially readable version of the proof secret, share it only with trusted others, try and understand it, and thus ensure that they are the ones who will be using this to “advance the physics and mathematics of quantum field theory.” Only when they are finished with this will they allow others to see the 1147 pages of slop.

There’s no information at all in this statement about exactly what has been proved, other than:

Balaban’s ideas were among the important elements that recently enabled OpenAI to follow the theory all the way to infinite distances and prove the existence of its proper continuum limit. But confinement and the mass gap remain very important open challenges.

Also, not a word about what the “completion of the Standard Model” is for which, according to the OpenAI press release, there is supposedly an AI proof.

Update: There’s now a second statement from the Secret Unified Physics Slop Shepherding Committee. They hope to release the Lean proof next week. There’s also a list of their “rules and procedures”. They intend to keep secret, for their own private use, the 1147 page AI generated proof writeup. The plan is to at some point document their understanding of the proof and then “not seek academic credit for the discoveries produced by AI models”. But, they do want credit for the understanding they get from using the document while it is secret and others cannot seek such understanding:

The group should receive recognition for the understanding they develop; the write-ups they author and the other materials they create

It turns out that yesterday there was another AI result coming out of OpenAI, quite different than the slopocalypse and of more interest to fundamental physics. Nathaniel Craig, a particle theorist now on a one-year visiting position at OpenAI, put on the arXiv a preprint entitled A gauge-invariant measure for lattice chiral fermions. This was produced with heavy use of OpenAI’s latest model, GPT-6 Astra:

OpenAI’s GPT-6 Astra model was essential to this result. I proposed a strategy to extend Lüscher’s abelian result using refinement methods that underwent several rounds of iteration with Astra and outside experts before arriving at the current Astra-led proof. The manuscript combines author-written sections with extensive rewrites of the Astra-composed proof. My colleagues and I proposed tests of the method, including Lean verification, that were carried out by Astra; supplementary materials documenting some of these checks were written by Astra.

Some observations about the manuscript, including ways this is different than the products of yesterday’s math dump:

• A great deal more human effort went into writing this. It’s 123 pages long and large sections are similar to the pure Astra math manuscripts, but also clearly have gone through some editing to make them more readable by human beings. Other large sections are obviously written largely by Craig: they’re very different in style than the Astra ones, comparable to a usual high-quality human written HEP theory paper.
• As far as I know, Craig has never before written a paper about lattice gauge theory. That he has identified an important problem outside his usual range of expertise and chosen to try to use AI to provide some of the replacement expertise is quite interesting. Will other physicists and mathematicians try to break out of the constraints of their current expertise using AI in this way?
• Craig is not at all a mathematician or mathematical physicist. In this manuscript, Astra is playing the role of a mathematician collaborator, producing large amounts of rigorous argument, with proofs, even formalizable in Lean. This is another interesting choice: will theoretical physicists in the future adopt usage of an AI agent to add this sort of material to their papers?

As background for those not much aware of the underlying problem, here’s a brief summary. In order to make the Standard Model mathematically well-defined non-perturbatively, one needs to do some sort of regularization, then study the limit as the regulator is removed. The most successful version of this is lattice gauge theory, which provides a beautiful and simple gauge invariant regularization of the gauge field degrees of freedom. This does a great job in allowing non-perturbative calculations for the strong interaction sector of the Standard Model

Introducing fermionic spinor degrees of freedom is much trickier, with the nature of the Dirac equation causing difficult problems with regularizing the theory. These can be handled in various ways when the gauge field interactions are non-chiral (as in the EM and strong sectors), but chiral interactions such as those of the electroweak sector introduce new problems of principle. We’ve been for fifty years now in a situation where in principle (and often in practice), the lattice provides a way to consistently do non-perturbative calculations in the EM and strong sectors, but not in the electroweak sector.

There have been a bafflingly large number of proposals about how to address this (Craig lists nearly 60 of them as references 25-84). He’s chosen one particular one, due to Martin Lüscher, who worked out the details for the abelian gauge field case back in 1998. With Astra’s help he’s extending this to some non-abelian cases including the Standard Model case. The only references I see for anyone else trying to do this in the past 25 years are three papers by Kikukawa and Kadoh/Kikukawa.

I haven’t tried to follow the details of the paper, especially not the complicated ones provided by Astra. The results of this work are a modest advance over Lüscher, still a long ways from providing a full definition in principle of a theory with the desired features in a continuum limit. As far as I can tell, this is not something that will allow actual non-perturbative electroweak calculations in practice.

It’s very interesting to see this sort of human/AI collaboration on an important question in fundamental theory. I’m afraid though that it illustrates the basic problem that afflicts attempts to use AI to make progress on such deep issues in HEP theory. What Craig is doing is following a program for how to solve this problem that started in the 1980s, was developed further by a small number of researchers in the 1990s, then largely abandoned as unpromising. Bringing AI firepower to bear on pushing forward a complicated proposal that was stalled for good reason isn’t likely to lead anywhere interesting.

Personally I believe that our lack of understanding of the mysteries of non-perturbative chiral spinor gauge theories is at the heart of the decades-long stagnation of the field. Changing this situation though will require something much deeper and more insightful than the program studied in this paper. Helping with that appears at the moment to be something far beyond the capabilities of current AI frontier models.

OpenAI has just released a vast number of new results in preprint form, see here. These are the products of an internal frontier model. About the AGMAI, they say they “have drawn on their advice and public recommendations⁠ to inform how we release these results.”, although one of the recommendations at the top was “we ask them to stop testing advanced mathematical problems on proprietary models.”

Like everyone else, I’ll spend some of this evening sorting through the long list to see what is of interest.

Update: Some quick thoughts about the slopocalypse. Note that this is happening in a seriously polluted and dysfunctional information environment. Maybe it’s my incompetence at dealing with social media, but the only way I’ve found of finding out relevant news about the slopocpalypse is by supporting Elon Musk’s Fascist enterprise and scrolling through an endless sewer of crap on X to find the few instances where well-informed mathematicians are quickly tweeting out relevant information. Lots of problems with getting your information this way, including walking around all the time depressed and disgusted at the state of the world. That said:

• A striking aspect of the paper dump is how sloppy it is at every level. The dump is a disorganized mess: after scrolling through long lists of papers and trying to read a couple, I realized that the huge list I’d gone through was only a fraction of the 722 papers, and that trying to find things this way would take forever. It’s hard not to think the people responsible for this just don’t care. If Astra is so great, couldn’t someone have taken 30 seconds to type into it a prompt “Please organize this in a coherent and useful way”?
• When the Navier-Stokes trainwreck happened, there was a concerted effort from within the math community to get to the public a counter-narrative to that of OpenAI. I’ve seen nothing like that this time. The AGMAI group supposedly dealing with OpenAI last week put out an excellent list of recommendations. OpenAI completely ignored them, and AGMAI then issued a statement I don’t understand at all. They are the people in the best position to evaluate whether their recommendations were followed, but instead of doing this, they write “it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully.”
• One thing you can learn on X today: some of the proofs are just wrong.
• Another one: some of the slop is supposedly verified by Lean. Can we trust this? Maybe not for Navier-Stokes, what about the hundreds of others?.
• Is Astra getting these proofs from its access to mathematician’s private work? See the first paragraph on page 3 here.
• According to OpenAI, these proofs were all produced with simple prompts to next version of Astra, applied to 4000 open problems in mathematics. What was the point of doing this? If the goal of OpenAI researchers is to improve their product, it would make sense to try a much more modest number of problems, then use the results to figure out how to do better (e.g. provide writeups that are readable…).
  It seems to that the answer to this is that they were trying to exploit a weakness in the mathematics research incentive structure, where people earn credits by solving these sorts of problems, even though this is a proxy for what is really important (new understanding). What OpenAI chose to do was to carpet-bomb mathematics, grabbing as many as possible of the chips on the table.
• I have a lot of trouble knowing what to think of the “AI alignment problem”, that of ensuring alignment between AI behavior and wholesome human goals. It does seem clear that the math community (and the rest of humanity) has a serious and immediate “OpenAI alignment problem”, with a huge misalignment between that corporation’s behavior and what is best for mathematics in particular and humanity in general.
• Another thing you learn on Elon Musk’s sewer is that a factor generating enthusiasm for our tech overlords and their new world is the exploitation of resentment of “elites”. In particular, criticism of the slopocalypse is widely denounced as nothing but the whining of pointy-headed over-privileged intellectuals who finally are getting theirs.

Update: There’s a very interesting collection of reactions to the slopocalypse here.

The announcement yesterday by OpenAI of a solution to the Millennium Prize Navier-Stokes problem has been causing a great deal of discussion among mathematicians. There’s a lot to be said about this story and its implications, most of which I’ll leave to those better informed than me (for one thing, I understand basically nothing about the problem, or its solution).

While talking to people I started to realize that one thing that really bothers me about these kinds of announcements is that they’ve been coming with no information at all about how the result was achieved. Emphasis has been on the question of whether the proof is correct or understandable, but I now think people should start asking another sort of question: exactly how was this done?

Sam Altman yesterday explained that

we tried this because there were rumors on the internet last week that Anthropic’s models had solved a millennium problem and we were curious if ours could do it too.

The rumor of this kind circulating was this one, with first appearance on September 4. Altman appears to be claiming that OpenAI solved the Navier-Stokes problem over the weekend, starting just with the information that maybe Anthropic had a solution. Altman is famous though for being “unconstrained by the truth.”

Tristan Buckmaster and Levent Alpoge had been working for a year on Navier-Stokes with some AI agent help, see here. The rumor I’ve heard is that Levent told colleagues at Anthropic that he and Buckmaster were close to a solution, and that this news somehow got to people at OpenAI, possibly including some information about the methods being used.

The new AI agents have profound implications for mathematics research and I think the math community needs to start discussing specific ways to deal with these. Here’s a modest proposal: the math community needs to adopt a version of the ethical standards of experimental science. If you are using AI agents, you can’t just give a proof (formalized or not), but need to also provide a detailed explanation of how these agents were used to get the result.

In an experimental science you cannot just publish results of an experiment, you also need to precisely describe the experimental setup that produced the results. One reason for this is reproducibility: other scientists need to be able to check the result by performing the same experiment. This is not so much of a concern in mathematics, where it is traditional to instead focus on the checkability of the proof.

There are other reasons though for having this standard in experimental science. Publishing the details of how to perform an experiment makes it possible for other scientists to adopt the same experimental method and make further progress by improving it or using it to get other results. I’d argue that it is now extremely important for the math community to see exactly what AI agent methods are being used, so that they can be adopted by others who want to try and use them.

In the Navier-Stokes case, I suggest that the community should demand that OpenAI provide documentation of exactly how they got this result. This would help others who want to use these methods, and it would also resolve questions about priority. What AI agents and training data did they use, what prompts were given to them and when was this done?

Besides issues of priority (which are going to be challenging since they involve the training data), this could also help with the problem of what’s a publishable result. If a result was achieved just by prompting an agent with “solve problem X”, it can be treated in a different way than results achieved through original human work and insight.

The new AI tools are going to change mathematics research. It’s critical that mathematicians have access to the details of what they are and how they are being used (as well as access to the tools themselves, a separate issue). One can imagine a future in which AI agents leave no role for human mathematicians (or, humans at all…). More hopefully, they will get integrated into human-driven activity to continue to make progress in exploring and understanding the world of mathematical objects (as well as the relations of such objects to those of the physical world).

Update: A helpful correspondent points me to the the OpenAI writeup, which says

On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved… Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge…

They don’t specify though the source of the rumors, or when they found out this was a rumor about what Levent Alpöge was doing. Altman says the source was “rumors on the internet”, but I’m not seeing a source for internet Sept. 1 rumors. Another description of the rumors, appearing on the internet on Sept. 5 is here, which characterizes the rumors not as “rumors on the internet” but as third-hand private communications starting at OpenAI.

From what I can tell, there has been a cottage industry for a while of people trying to solve the Navier-Stokes problem with AI agents (Sabine Hossenfelder describes her efforts here). It seems likely the AI companies have been regularly trying out their new models on Millennium problems, but have had limited resources to do so. In this particular case they devoted resources estimated at more than \$10 million to the Navier-Stokes problem. In their writeup they say this was because they were seeing positive results with an easier problem.

Update: Kenneth Chang has a detailed story at the New York Times that discusses backstory of the OpenAI Navier-Stokes claims.

I’d like to emphasize that this posting only deals with one small part of the new AI problem for math, a part where I thought I could make a positive suggestion. For a passionate discussion of the larger picture, I can recommend David Madore’s Sur la destruction des maths par les boîtes d’IA.

Update: The Advisory Group on Mathematics and Artificial Intelligence, which was formed to provide OpenAI advice on what it should be doing with its claims to have solved major problem in mathematics, has released a statement on the Responsible Release of AI-Generated Mathematics. They make several excellent points about how this should be done (including calling for the release of details of how results are achieved).

They start with a strong demand, that the AI companies stop using unreleased proprietary models to solve mathematics problems:

At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.