Not Even Wrong

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.

Kevin Hartnett posted the following Friday:

My profile of @Jacob_Tsimerman, published the morning he won the Fields Medal, and hours before he announced he’s going to @OpenAI, reads like a requiem for a field.

While I’m trying to not spend too much time thinking about the ongoing story of the effect of AI agents on math, the story of Jacob Tsimerman’s Fields medal award coinciding with his announcement that he’s leaving research math to go work of OpenAI is hard to ignore. If you do want to spend time thinking about the implications of this, it’s worth reading the profiles at the New York Times as well as Hartnett’s at Quanta Magazine. Also very much worth reading is Tsimerman’s own explanation (in an exchange with Daniel Litt) of how he sees things. This was written before the OpenAI announcement, but goes a ways towards explaining Tsimerman’s decision.

A few personal comments: I’ve never had any interest in the kind of problem-solving Tsimerman specializes in, so have no idea if he’s right that humans will no longer be able to compete with the AI agents in this area within the next few years. His current situation makes him a very special case. In the Quanta profile he explains that he quit math competitions once he got a perfect score at the 2004 IMO, figuring it best to stop at the top of his game. Winning a Fields medal puts one in an analogous situation professionally: there are lots more awards you could get, but that’s the big one.

In addition, while a Fields medal comes with academic job offers that allow one to do whatever one wants and be very well-paid to do it, our tech overlords at OpenAI can and do pay a great deal more.

Finally, if you’re the competitive sort, what’s happening is making competitive problem solving much less attractive. It’s not just that the AI agents may beat you, it’s that the game is now being played very differently. When you’re competing against other mathematicians, there are strong norms about crediting ideas to those who came up with them, something that AI agents have no interest in doing. The situation isn’t quite like the situation in chess, where powerful chess programs got to the point they could beat humans, but human to human chess competition survived. Where we’re going could be analogous to chess playing where everyone is using, with or without acknowledging it, some sort of chess program. “Winning” at this will be something very different than pre-AI.

My own interests are quite different than Tsimerman’s. Math competitions never had any appeal to me. This was partly because I’ve never been competitive or much interested in “winning” anything. Also, why spend time on intricate math problems chosen mainly because they could be expressed in elementary terms and solved by elementary methods when there is so much beautiful and powerful non-elementary mathematics to learn about? As one learns mathematics there are always new depths to explore, and these depths seem to contain unexpected ideas that explain what is going on in fundamental theoretical physics, a subject with its own challenging depths to explore. So far I haven’t seen AI agents doing anything new and important that I would care about, but maybe that’s just a matter of time…

Update: Timothy Gowers (another Fields medalist and IMO winner with a perfect score) has a long blog post with his thoughts about the Leiden declaration at his blog.

Update: For an interview with Tsimerman at the ICM, see here.

Update: OpenAI has a promotional video for a program offering free access to their frontier models. Three of the four scientists featured are prominent theoretical physicists who had been working on string theory (Xi Yin and Eva Silverstein) or SUSY (Nathaniel Craig).

Update: Two new things this morning:

• An interview with Tsimerman at the Atlantic
• A new hep-th arXiv paper that demonstrates clearly why, in an intellectually dead field, AI will just produce worthless junk. If what you feed it is decades of complicated and meaningless computations, it will just produce even more complicated and meaningless ones.

Update: Yet another Tsimerman interview, at the Wall Street Journal.

Update: Probably not much blogging for the next two weeks. I’ll be traveling in Spain, hope to see the August 12 solar eclipse somewhere northeast of Madrid. Last week of August will be in Berkeley.

The past week or so I’ve finally gotten around to trying out asking questions to a recent version of an AI agent about things I’m working on. The results have surprised me: these things are a lot more capable than I had expected. It’s becoming clear that it would be a good idea to change the way I work, seeing if I can use these to get simple calculations done and different sorts of things I don’t understand explained to me. Current plan is to seriously get to work on this when I’m back from Berkeley at the beginning of September.

Unless there are specific comments about Tsimerman, I’d rather have further discussion about the capabilities and implications of these AI agents wait until I’ve spent more time with them in September. What would be helpful to me and I think to others is to hear from mathematicians and physicists who have been actively using these in their work with advice about which ones are most useful, and how to get access to them.

Update: Curt Jaimungal has a long, detailed interview with Tsimerman. Highly recommended.