Proofs and Prompts

Notes from the common room

From non-sofic to non-hyperlinear groups: the new messiness of discovering and announcing big results

Francesco Fournier-Facio, Assistant professor at Heriot-Watt University

Disclaimers:

  • I am one of the hosts of P&P, but this text represents my personal experience and opinions, which do not affect the way in which I contribute to running this platform. Moreover, this is completely independent of the views of the other hosts.
  • I have personally reached out to all the people involved in this story, the only ones that did not write back with comments are the three of OpenAI, so they will remain nameless (just like in the 1 August announcement).
  • Whatever mathematics you see below can be safely ignored. The human story is what I am most trying to convey.

Last year I co-organised a workshop on metric approximations of groups and their stability. This is a series of problems that until now were largely motivated as stepping stones for finding non-approximable groups, which was successful for the unnormalised Schatten p-norm, but only gave conditional results in the context of soficity, hyperlinearity and MF. In his talk, Alon Dogon called these the “big three” problems in the topic of metric approximations of groups. Andreas Thom’s 2018 ICM talk shares a similar message. Flash forward, and less than a year later:1

  • 1 August: OpenAI finds a non-sofic group.
  • 13-14 August: Sauers and Eckhardt, independently, find a non-MF group, through ChatGPT 5.6 Sol.
  • 20 September: Liu finds a non-hyperlinear group, through the Danus system.

The “big three”, all fallen in less than two months, to various AI systems. This came about in a very messy way. We are entering a new regime where if you prove a big result with AI assistance, there are good chances that someone else, who is better at deploying agents than you, or has invested more money than you, could scoop you tomorrow. No one knows how to deal with these new dynamics, and messiness naturally ensues. I hope that telling the whole story from my point of view could be instructive, although I won’t offer any solutions. While I am not involved in any of the above results, I played a role in part of the story, and have contributed to the messiness to some extent (not on purpose!). This post is entirely human-generated, so I apologise in advance for the chatty structure and unimaginative vocabulary.

The first non-sofic groups

On Friday 31 July, Srivatsav Kunnawalkam Elayavalli shared with me on whatsapp a PDF, authored by OpenAI, and titled “Nonsofic groups exist”. I immediately dropped everything I was doing, sent it to Henry Bradford and Alon Dogon (who had already received it from another colleague), and we jumped on a call to go through it. We were in disbelief, but also full of excitement for seeing the solution to a problem at the centre of a major thread of research we were active in. Something that was apparent from this writeup was that no expert had been involved in it. As a standout example, the abstract, introduction, and end of the paper, insisted on the fact that, once the non-sofic group was constructed, a finitely presented one could be “explicitly” constructed as well. But then this step was done with a trick that is well-known to experts, and is far from explicit. I imagine that the model had been prompted with “find a finitely presented non-sofic group”, and this was its attempt at justifying having solved the actual prompted problem.

I learned that Michael Chapman had received the draft directly from people at OpenAI, and so I sent him various comments, including the one above.2 He wrote to three people at OpenAI, with me in cc, relaying my comments, which resulted in some back and forth. In parallel, I started an email thread with various experts, in particular people whose work was being used in the solution, proposing that we write a response document together, analogous to the one that appeared after the disproof of the unit distance conjecture. I also told this to the three of OpenAI, and they said that they would be happy to see a response document, but that they planned to announce the result on Monday 3 August. So it seemed like all of the weekend would be dedicated to writing this response document.

But on Saturday 1 August, the result was announced (with most of the comments implemented), together with 9 others. Apparently the paper had already circulated too much to wait two extra days.

The announcement of OpenAI

The original announcement included this sentence:

Today we are sharing a selection of ten results to problems that have been open and have seen no progress on the main result for at least a decade.

It is clear to anyone reading the solution that the proof is entirely dependent on a paper of Kun from 2016 and a paper of Kun and Thom from 2019. In fact, the key proposition in the OpenAI proof can be seen as a bridge between the conclusion of Kun and the hypothesis of Kun-Thom. The writeup appropriately attributed these results, but the announcement, which is what most people read and accepted as a fact, ignored this.

After the key proposition, the second step in the solution consists in finding a concrete non-example. This used a connection between the binary Leavitt algebra and Thompson’s group VV that was discovered independently by Birget and Nekrashevych in 2004 (neither of these papers was cited in the 1 August writeup). It also relied on work of Ershov and Jaikin-Zapirain to prove that elementary matrices over the binary Leavitt algebra have property (T). I thought that this was overkill and made the result look harder than it was, and that the key proposition was the only genuine novelty. So I quickly put together an alternative construction, hoping that it would clarify this point, and bring the attention away from the specifics of the construction, and onto the key proposition, and just how much it relied on work of Kun and Thom.

I was dreading the moment that people would pick up this story and repeat the “no progress for at least a decade” line without questioning it. So I wrote to several youtubers and scientific news outlets, offering to speak with them and set the story straight; many of my colleagues did the same. This resulted in a video on Alvaro Lozano-Robledo’s youtube channel, where I explained the problem, the solution, and criticised this dishonest framing. ∼\simRemember this for later∼\sim. In the following days, articles appeared on New Scientist, Scientific American and The Verge, which included comments from me about this framing, and I was also interviewed by 3Blue1Brown for an upcoming documentary about maths and AI.

In parallel, Andreas Thom wrote an email to the three of OpenAI, where I was cc’d, calling them out. They answered:

The wording in the announcement was intended to refer to other results in the collection.

I will leave it to the reader to decide whether “ten results to problems that have been open for at least a decade” could have been intended to refer to only some of these ten problems. The rest of the answer was problematic for other reasons that are detailed in this post of Andreas, and that I will not get into here.

The wording in the announcement has since been quietly changed. But I think it’s important to remember that complaining about it made a difference. A few days later Anthropic announced their “2/3 of zeros” result, and I was pleased to see that they did not make the same mistake. Sure, this was a deliberate PR move on their part to distance themselves from the practices of OpenAI, but hey, whatever works.

More non-sofic groups

A couple of weeks before the OpenAI announcement, Vadim Alekseev and Andreas Thom had started circulating a preprint, which has since appeared essentially unchanged, building upon the 2019 Kun-Thom paper, and strengthening it. The key technical step is a “centraliser rigidity” theorem: the centraliser of a sofic approximation of a property (T) group is “internal”. Combining this with some insights from the key proposition of OpenAI, Kun and Thom gave a new, more robust criterion for non-soficity, which subsumes the OpenAI one, and gives very nice new examples of non-sofic groups. In particular, this includes permutational wreath products of the form ℤ/2≀G/ΓG\mathbb{Z}/2 \wr_{G/\Gamma} G, where Γ<G\Gamma < G is “infranormal” (a new definition of Kun-Thom that comes naturally out of the assumptions in OpenAI’s key proposition), not normal, and both have property (T). For the same pair Γ<G\Gamma < G, another nice example of a non-sofic group is the double G∗ΓGG *_{\Gamma} G. This may be more mathematics than you wanted to hear but ∼\simremember this for later∼\sim.

They wrote this up in a preprint that appeared on the arXiv on 6 August 2026. It is worth mentioning that both of these human proofs relied on AI assistance for “drafting, editing, checking” and in the case of the more recent one, also “as an interactive proof assistant”. This is the least messy part of the story, and most likely the one that will eventually be “canonicalised“. Alekseev and Thom were already working on a detailed writeup of the whole proof, from Kun’s 2016 paper to the latest developments, but they might have to revise their plan now that a non-hyperlinear group has been found.

The first non-MF groups

On 13 August, I woke up to the following email from Scott Sauers:

I am a PhD student in computational biology (CS / stats / ML) and decided to set some AI agents loose on the question of whether all groups are hyperlinear, inspired by the OpenAI soficity result (…\ldots) I watched your discussion with Alvaro Lozano-Robledo, so I figured your perspective on the following would be valuable. Instead of proving the hyperlinearity result, GPT 5.6 Sol proved that a non-MF group exists, then formalized it in Lean. Do you have advice for how I should go about this as a non-specialist? I want to avoid, for example, mistakes like the improper credit attribution in the OpenAI announcement. I also want to avoid wasting anyone’s time (…\ldots) The paper likely contains some mistakes, does not yet have a 1:1 correspondence with the Lean code, and is written poorly. I don’t understand the proof, and have only recently begun to learn a bit of group theory.

It made sense that once a non-sofic group was discovered, many other related problems would follow, but I did not expect it to take less than two weeks, and to come from someone who has “only recently begun to learn a bit of group theory”. The initial writeup was long and hard to parse. Scott said he would work on making it more readable, and in the meantime I started an email thread with some experts:

I am sure you have seen that OpenAI found a non-sofic group on 1 August (…\ldots) The whole thing was a bit of a mess (…\ldots) So to avoid another thing like this, I would ask you that this conversation remains private until there is a consensus on the correctness, and a document to show for it. This is just a first email to let you know (…\ldots) I thought you’d appreciate the heads up, as with the non-sofic group it all happened in 24 hours, which for me was very stressful.

One of these experts was Caleb Eckhardt, who on 14 August replied with:

After Francesco’s message yesterday about Scott’s example I had ChatGPT come up with an example (they are groups considered by Kun and Thom). I wonder how similar it is to Scott’s example.

As it turns out both constructions were very similar. The one discovered by Scott was not directly defined as a wreath product, but relied on an embedding into (a central extension of a) wreath product à la Kun-Thom to prove that certain elements are non-trivial. The one discovered by Caleb was a wreath product à la Kun-Thom on the nose. The two writeups were also interesting to compare. Caleb expertly guided the model to find a non-example in the most efficient way, and rewrote it to put the emphasis on the main ideas, and only prove the existence of a non-MF group. A revised version has since been posted on the arXiv. Scott’s writeup was more exploratory in nature: it was much longer, included several detours that were not necessary to get to the main result, and several pages of “additional topics” that might be worth exploring, which is only natural when the person doing the prompting is learning about the field at the same time as they are proving a breakthrough result. The current writeup is available here, but Scott is still working on it, trying to make it match with the formalisation, and understand every step of it:

It feels odd to publish a preprint without actually understanding the proofs, even if I am confident they are correct.

Certain mathematicians (one comes to mind, that has been bragging about having written 48 papers in the last month) have a lot to learn from Scott.

Mathematically, as I mentioned, the examples are essentially the Kun-Thom wreath products. However much less is needed to produce a non-MF group, and this is reflected in Caleb’s writeup. So, while the discovery a non-MF group was motivated by the discovery of a non-sofic group, in retrospect it was much easier. The technical steps are also all very standard, and there is no seemingly unnatural construction at play. As Andreas noted:

This non-MF proof could have been found in 1995 or even earlier.

You’ll have noticed that in this section I have not been talking of “Caleb’s result”, but rather of “the result discovered by Caleb”. But this is just a placeholder, I do not really know what the right word is in this context. Again, Andreas put it best:

We should find a way as a community to deal with these kind of results and it seems hard to publish them in the old way and also wrong to attribute them entirely to AI. I propose to attribute them to the math community as a whole with due credit to the system that found the proof and the person that overlooked the editing. Second, the integration into the existing body of knowledge will remain as a task and a Lean certificate is not doing this at all. After all math, I think, is not about results but about human understanding of it.

The first non-hyperlinear groups

Remember the Alekseev-Thom paper, the one with the centraliser rigidity theorem? In that paper, they ask whether a similar result holds in the hyperlinear setting: the commutant of a hyperlinear approximation of a property (T) group is “internal”. On 10 August, Andreas Thom circulated a preprint (also using ChatGPT for drafting, editing, and as a proof assistant), showing that if this centraliser problem could be solved, then a non-hyperlinear group exists. In fact, the example is one of those from the Kun-Thom paper: a double G∗ΓGG *_\Gamma G, where Γ<G\Gamma < G is infranormal but not normal, and both have property (T). On 17 August he submitted this to the arXiv. It is available on his homepage since 25 August.

A small aside: what’s going on at the arXiv? I am sure you have heard from people that their submissions have been put on hold for a long time. I heard this from the 48-papers-in-a-month guy. I also heard it from Tarik Aougab, for the extended version of his post on this blog. And finally I heard it from Andreas. On the other hand, I have a colleague that recently wrote a paper that does not use AI in any way, and that one appeared on time. There seems to be some criterion related to AI by which arXiv is putting papers on hold, but what exactly the criterion is I don’t know: here it ranges from someone mass-producing AI papers (not all of them with AI disclosures), to someone writing about AI, to someone writing a normal amount of papers with some AI assistance that is duly disclosed.

Andreas’s paper appeared on the arXiv only on 24 September.3 And then on 20 September, Jihao Liu announced:4

Today, a Danus system consisting of seven workers, running for a total of 7 hours and 11 minutes, produced an example of a non-hyperlinear group.

Jihao Liu has been producing a lot of AI-generated results recently, thanks to a harness devised by him and colleagues: the Danus system. This is not dissimilar to the one that Scott Sauers has been using, but at a smaller scale. The writeup is available here. The announcement continues:

This paper has so far undergone only machine verification and has not yet been checked by a human. I had originally hoped to post it to arXiv only after completing a human verification. However, we discovered that there is a public AI project on GitHub continuously searching for non-hyperlinear groups.

And proceeds to link to this GitHub project of Scott Sauers. Remember: Scott had initially tried to find a non-hyperlinear group, and that project is still on GitHub. The focus has shifted, first to a non-MF group, and then to other related problems that Scott is interested in. How ironic that the motivation to publish this quickly and with no human verification should be the same project that failed to find a non-hyperlinear group, and from there proceeded to find the first non-MF group.

Liu’s paper solves the centraliser problem for hyperlinear approximations of property (T) groups, citing it as an open problem from the Alekseev-Thom paper, and works from there. The first version did not cite the work of Andreas Thom, who had already concluded from that same point. His preprint was publicly available on his homepage, and it is not clear whether the models accessed it. One can only wonder if the situation would have been clearer had Andreas’s paper appeared on the arXiv less than a month after he submitted it. The current version of the writeup acknowledges this. Andreas again:

We [Alekseev, Liu and Thom] all agreed that finding a proof / understanding and condensing the argument / integrating it into the math landscape are separate tasks and should all get suitable credit. If things work out, we will write a joint paper covering all aspects.

Generating a proof, checking its correctness, understanding its mechanism, and connecting it with existing mathematics are becoming increasingly distinct activities.

And now Liu:

I believe that each of these activities deserves credit. This is exactly what is happening with the non-hyperlinear group now, if things go smoothly: I have a proof of an important result; the experts come in to digest and verify it; and in the end we produce a good preprint, post it, and submit it somewhere

What I do not want to see is the following. An AI company (…\ldots) posts a badly written draft, verified or not, on arXiv or on its own website, without any interaction with the community (…\ldots) If you are an expert in the field, should you write a much better version of the proof? If not, the community is left forever with a terribly written paper and no proper explanation. If so, you cannot claim priority for the result (…\ldots) It becomes pure service to the community.

For the non-hyperlinear group, what I hope we end up with is a third path. I only announced the existence of a proof (…\ldots) Then the real experts, Alekseev and Thom, contacted me, and we write a paper together (…\ldots) This requires effort from everyone: nobody should be greedy.

Another thing that was not covered in the first version of the writeup, and that Mikael de la Salle pointed out, is the connection with the Connes embedding problem. This asks whether all von Neumann algebras are Connes embeddable, and the problem of the existence of a non-hyperlinear group is the Connes embedding problem specialised to group von Neumann algebras. But the more general version of the Connes embedding problem had been disproved in 2020 via the milestone MIP* = RE, that is a result about quantum computing. In retrospect, this 50-page construction of a non-hyperlinear group gives a shorter and simpler disproof of the Connes embedding problem (similarly to how the construction of a non-sofic group gives a shorter and simpler disproof of the Aldous-Lyons conjecture). But I cannot articulate enough how glad I am that things happened in this order. The way these two conjectures have been disproved has enriched mathematics immensely, by bringing together different communities, and transferring ideas from one discipline to another. It also led to further applications, like this one, which does not follow from the existence of a non-hyperlinear group. This is much more valuable than solving a single open problem, no matter how big. In this new normal, I mourn all the maths we will not see, but also all the bridges we will not build, and all the gems outside our comfort zone that we won’t be motivated to learn, without certain open problems guiding the way.

Acknowledgements. I thank Vadim Alekseev, Segev Gonen Cohen, Matt Cordes, Alon Dogon, Jihao Liu, Yash Lodha, Marco Moraschini, Andreas Thom and Itamar Vigdorovich for comments on earlier drafts.

  1. All dates in this text are from 2026. ↩︎
  2. This is just common practice for me: when I read a new preprint in some depth, I usually send the authors a quick email with some comments, and people tend to be happy about it. The only exceptional thing here was that the result was exciting enough that I had read through the whole paper as soon as I could. ↩︎
  3. It appears there as submitted on 14 September, which is the date of the most recent modification; I don’t know if this contributed to the delay. ↩︎
  4. What I am quoting here is the first version of the announcement, which has since changed. ↩︎

Received 21 September 2026, revised 25 September 2026.

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31 responses to “From non-sofic to non-hyperlinear groups: the new messiness of discovering and announcing big results”

  1. Anonymous Avatar
    Anonymous

    Dear Francesco,

    Thank you very much for this post.

    This story is important in many ways; however, I would like to focus on the actions on Scott Sauer and Jihao Liu. To put it (very) mildly, I completely disagree and disapprove of their behavior.

    I hope they read my comment and would like to read their responses to the following criticisms of their actions I will now detail.

    First, regarding Scott’s actions, Francesco wrote: “The focus has shifted, first to a non-MF group, and then to other related problems that Scott is interested in.”

    Scott himself admitted to having “only recently begun to learn a bit of group theory.” I find it very difficult to believe that he was truly “interested in” a question at the forefront of mathematical research. Indeed, how can one be interested in something they know nothing about? Let me illustrate this with myself: I have been properly educated (i.e., have written several (solo) papers, a PhD thesis, all with no AI) in another area of group theory, and still I have no idea what a non-MF group is, and therefore can have no genuine interest in it.

    Scott’s actions display a lack of care for both mathematics and the mathematical community. Indeed, this is evident upon his writing to Francesco: “I am a PhD student in computational biology (CS / stats / ML) and decided to set some AI agents loose on the question of whether all groups are hyperlinear, inspired by the OpenAI soficity result…”

    His behavior was inspired by noticing the attention OpenAI received from “their result” (why did I put this in quotations? Because thieving something does not make it yours). These are his very own words. I leave it to the reader to decipher for him/herself the true motives behind his actions, and whether they were actually inspired by genuine mathematical curiosity.

    I also want to emphasize that I abhor the desire “to set some AI agents loose.” Set them loose for what? To me, it seems this was done to wreak havoc on a community that has been nurtured and cultivated for millennia by generations of men and women who truly love mathematics and the practice of doing mathematics.

    As for Jihao: I have seen “your” papers on arXiv for some time now: why do I put “your” in quotation marks? Because as far as I am concerned, you have done nothing to merit putting yourself as an author of these papers: asking a question to a machine, getting a response, and writing a paper with yourself as an author is as good as asking a human a question, getting a response, and claiming yourself as the author of a paper with said solution. This activity goes by many names: lying, cheating, plagiarism. Pick your poison.

    As far as I understand from reading Jihao’s description of himself on his website, he is an algebraic geometer by training. He is not a group theorist. He himself admitted so much when writing: “This paper has so far undergone only machine verification and has not yet been checked by a human.”

    What is the motive behind prompting an LLM to solve an important group theory question? Probably there are many answers but let me just quote Jihao to get one such answer: “I believe that each of these activities deserves credit. This is exactly what is happening with the non-hyperlinear group now, if things go smoothly: I have a proof of an important result; the experts come in to digest and verify it”

    So then, you admit to not understanding what you have written? This is incredible, to claim a proof of a result you don’t understand! Never in my wildest dreams would I imagine to do this.

    Moreover, let me make this clear: the experts in this field are not your lackeys, nor is any other mathematician — they have no obligation to “come in to digest and verify it.” Are the experts supposed to stay alert, ready to obey your orders at your every beck and call?

    I would like to conclude that what I find most astonishing of all is that we actually engage with people who treat mathematics and the mathematical community in this way. Instead of trying to cultivate and promote the good behavior of decent, hardworking men and women who actually do mathematics (by thinking, discussion, reading, having their own insights), we are in the midst of entertaining this, quite frankly, completely ridiculous behavior.

    I’ll end with this: love (in this case, the love we have for mathematics and the mathematical community) manifests itself in the process of doing mathematics, not in the output we receive. The above actions seem to me completely devoid of any love.

    Reply
    1. Sumo AQJ Avatar
      Sumo AQJ

      i can’t agree more.

      Reply
    2. Anonymous Avatar
      Anonymous

      ​I find the gatekeeping attitude, essentially saying “you’re not a group theorist, so stay out of our lane”, deeply ironic, especially when paired with the dismissal of AI in mathematics.

      ​First, computational and machine-assisted proofs are hardly anything new. Half a century ago, the proof of the Four Color Theorem sparked the exact same moral panic and intellectual snobbery, yet history rightfully gave it full credit. Dismissing algorithmic assistance today is just repeating 1976’s anxieties.

      ​Second, critics love to label AI-assisted outputs as “slop” simply because they contain errors or require heavy human verification. But dumping dense, half-baked, or notoriously incomplete drafts onto the community for peer “debugging” isn’t an AI-born sin, it has a long pedigree in pure mathematics, tolerated even from luminaries like Faltings and his circle. Why is a human master’s leap of faith considered “deep intuition,” while computational experimentation is written off as garbage?

      ​If billions in public funding only cultivate a guild that views theorems as an exclusive art form, unaccountable to broader technological shifts and proud of gatekeeping, then math should be funded like archaeology: an intellectual luxury. Since when did mathematicians become proud of dismissing arguments without formal counterproofs, and trading rigorous logic for elitist dogma?

      Not saying I support AI slop, but you gotta first have a proof that those are only slops and nothing true. I don’t have you guys have any actual proof that his papers are mostly wrong other than just claiming he has a lot of papers this year compared to before, which could just be an adoption efficiency of new tools problem, and ironically, it’s from the community that claims they care about proofs.

      Reply
      1. Anonymous (young mathematician) Avatar
        Anonymous (young mathematician)

        I am the author of the comment criticizing Scott and Jihao’s actions: let me address your points one by one. First, however, let me remark on a stark contrast between your writing and mine: I explicitly quote what Scott and Jihao say in the post written by Francesco. You, by contrast, often misattribute statements to me: For instance, can you point out where in my post I labeled any of the AI outputs as “slop”? I never even so much as insinuated that the mathematics output by the AI’s prompted by Scott and Jihao were incorrect — indeed, I am not an expert in this field, and it would be ridiculous for me to do so. I simply pointed to the fact — admitted by the two men themselves (see their words that I quoted, verbatim!) — that they do not understand what they have written:

        1) “​I find the gatekeeping attitude, essentially saying “you’re not a group theorist, so stay out of our lane”, deeply ironic, especially when paired with the dismissal of AI in mathematics.”

        Indeed, someone who is not properly trained in an occupation should stay out of this lane. I am not trained as a doctor. I know nothing about medicine. Should I start going around, parading “alternative” forms of medical treatment? No, that would be completely ridiculous. Where is the irony in this? I would never enter Scott’s field (Biology / CS / ML) and “set some AI agents loose” (his words, read Francesco’s post! Not mine!) to answer questions in this field which I know nothing about.

        If you want an analogy that hits even closer to home, let’s stick to mathematics: This also addresses your comment:

        2) “Why is a human master’s leap of faith considered “deep intuition,” while computational experimentation is written off as garbage?”

        I know extremely little algebraic geometry (I cannot even tell you the definition of a scheme). Let’s say now that I acquire some insanely powerful AI system such as in the possession of Jihao, Levent Alpöge,… and present to you a solution of the Hodge conjecture, with myself as an author. Assume the paper is not slop, but instead is the most lovely paper an algebraic geometer has ever had the pleasure of reading, in addition to being entirely correct. Now I ask you:

        Do you actually think this is moral? Does it make sense for me to do this?

        In my mind, it is obvious that the answer is “no.” My opinions towards this are even stronger, but let’s leave it at that for now.

        By the way, I agree with you that it is not good when a human master puts out “dense, half-baked, or notoriously incomplete drafts onto the community for peer “debugging.”” However, does that make it ok for a person to do this with an AI? Does it make it ok for them (Jihao) to write “the experts come in to digest and verify it.” I mean, seriously, “come in,” like they’re a servant supposed to clean up after their master. I don’t understand how one could not take offense to this.

        3) “Since when did mathematicians become proud of dismissing arguments without formal counterproofs, and trading rigorous logic for elitist dogma?”

        Please tell me in my comment where I dismissed the mathematical validity of any of the above arguments. Please tell me where my arguments lacked rigor. Again, I was the one quoting Jihao and Scott, and drawing conclusions from their very words.

        I, of course, can only speculate where you think this came from. Here is a guess: perhaps it is when I wrote about Jihao’s actions: “So then, you admit to not understanding what you have written? This is incredible, to claim a proof of a result you don’t understand!” Is it here?

        Let’s look at this carefully: do I dismiss the argument Jihao says the AI produced as incorrect? No (again, I cannot possibly dismiss it, because I do not understand this area of mathematics). What did I ACTUALLY say? I simply said in my own words what Jihao has himself said: he has claimed a proof of a result, but he has not verified it, because he does not have the knowledge to understand it.

        How is this “elitist dogma”? On the contrary, this is the very opposite of “elitism.” Before the past couple of years, the greatest thing about mathematics was that it didn’t matter what race, what gender, what ethnicity, sexual orientation, financial status, and so on that a person had: all that mattered is that one understood and could logically explain the mathematics he/she put out into the world. The only “elite” quality I can think of here is the ability and knowledge that is acquired through, necessarily, many years of hard work, effort, dedication, and learning. This is a fantastic quality.

        4) Lastly, you write “Not saying I support AI slop, but you gotta first have a proof that those are only slops and nothing true. I don’t have you guys have any actual proof that his papers are mostly wrong other than just claiming he has a lot of papers this year compared to before.”

        Again, see what I wrote above. It should be obvious what I would say in response to this.

        Best wishes,

        The original commenter, a young mathematician who just finished his PhD and started his first postdoc.

        Reply
        1. Dima Avatar
          Dima

          Dear Anonymous,

          I understand and agree with (some fraction of) your points. However, I feel that it is deeply wrong to make such comments about any particular individuals while you yourself remain anonymous. I would encourage you to attach your name to this series of posts of yours.

          Reply
          1. Anonymous 2 Avatar
            Anonymous 2

            +1 to the original commenter for this line too, you are still anonymous with the name Dima, so I guess please don’t judge the original commenter for that.

            Moreover, I can see how this could put the original commenter in trouble if they disclose their identity, people have opinions here and there, and just little opinionated bias could blast a young mathematician’s life. I would furthermore think staying anonymous is actually the best for them at least for now, as I do not hope that just by posting some true and real comments getting their real life impacted, especially they just started.

            Cheers.

            Reply
        2. Dominique Manchon Avatar
          Dominique Manchon

          Dear Dima,

          Maybe some people posting here stay anonymous just to avoid to get “set some AI agents loose” on their work. At least the person commenting above discloses her/his position in the mathematical community, which you did not. How can you blame a young post-doc as “deeply wrong” to post anonymously? We are interested in your take on which fraction of the points displayed in the comment above you like, and those you don’t.

          Reply
          1. Anonymous (young mathematician) Avatar
            Anonymous (young mathematician)

            Thank you, Dominique. I am the original commenter, and I would like to say that this is precisely the reason why I remain anonymous when posting on this blog.

            Dima, I would also like to know which fraction of my points you agree with, which you do not, and, of course, why. The whole point of a discussion is for us to try to learn something. I have been wrong many times in my life (and will inevitably be wrong many more times, since I am just a man!). If I am convinced that my opinions, statements, or behavior is wrong, I will have no shame admitting it, apologizing, and acting accordingly. But in order for this to be possible, I will first need to hear from you what you disagree with, and, most importantly, why you disagree with it.

            I would also like to add the following: “Dima” is a very common nickname in Eastern Europe (I would know, given the country I am from). I also notice you did not mention your last name, nor your position in the mathematical community.

            Therefore, you are also anonymous. I would never criticize Scott and Jihao’s actions if I behaved like they did — that would be completely hypocritical. And yet you criticize me for remaining anonymous, while you, too, are anonymous.

            Cheers,

            The original commenter.

            Reply
        3. Anonymous 2 Avatar
          Anonymous 2

          To the young mathematicianc:

          I am the other Anonymous. Thank you for clarifying your position. Let us address your points with an examination of the historical and structural realities of our field.

          1. The Doctor Analogy and the “Stay in Your Lane” Doctrine
          Comparing pure mathematics to practicing medicine is an untenable false equivalence.
          I believe medicine regulates practice through strict licensing because unvetted empirical intervention carries severe, irreversible physical risk as the patients will die. Mathematics, by contrast, is governed by formal syntax and objective logical consistency. A theorem is not false because the author lacks a license, nor does it automatically become true because it came from an expert.

          Moreover, if mathematics had historically enforced a “stay out of your lane” doctrine:

          Ramanujan would have had his letters discarded by Hardy as unlicensed interference. Ramanujan openly acknowledged that he had no rigorous proofs for thousands of his modular equations and partition formulas and the community spent decades as the “human validators” parsing and proving his raw breakthroughs.

          Heegner, a private schoolteacher and radio engineer with no university post, published a solution to Gauss’s class number 1 problem in 1952. The establishment dismissed him as an incompetent amateur because his methods were idiosyncratic. He actually died unrecognized and it is only decades later, Birch and Stark re-examined his work and confirmed he was correct.

          Heaviside would also have been banned from developing operator calculus for electric circuits. When pure mathematicians condemned his lack of foundational rigor, his even had this line: “Shall I refuse my dinner because I do not fully understand the process of digestion?”

          Outsiders bringing tools to crack open problems is not a desecration of mathematics; it is how mathematics has repeatedly renewed itself thoughout the ages and part of why it was always up to date.

          2. You ask: If an outsider with no algebraic geometry training uses an AI system to produce a beautiful, correct proof of the Hodge Conjecture and publishes it, is that moral?

          If it is for me, my answer would be actually a “Yes, absolutely”. If the proof is mathematically sound, verified, and advances human knowledge, withholding that breakthrough from humanity simply because the prompt engineer cannot define a scheme is actually what would be unethical in my opinion. My point still: We become mathematicians not to be playing some interlectual game, instead, we are longing for the truth.

          If the AI is capable of just solving all the big conjectures in your field elegantly and beautifully, will you

          The history of science is built on delegating discovery and verification to tools that outstrip our personal working memory. When Appel and Haken proved the Four Color Theorem in 1976, no single human could hold or verify the thousands of hours of computer code and case checks by hand. They even openly admitted human limits themselves. I don’t think people sat through the proof themselves, nor that people believe the Four Color Theorem is unsolved.

          In the case of AI generated proof, I think things are actually less controversial than that. At least if you sit down you can understand and check.

          However I do understand part of your thought, we can leave this aside.

          3. Epistemic Transparency vs. the Illusion of Purity
          You take offense at Jihao Liu’s transparency about not yet mastering every detail, calling it offensive that experts might “come in to digest and verify it.”

          Yet in top-tier research groups, senior mathematicians routinely put their names on papers where hundreds of pages of technical lemmas were verified entirely by their postdocs, graduate students and they never personally line-checked (for example Yau’s school is one of them). Why is delegating verification down an academic hierarchy considered standard practice, while delegating search and verification to a machine harnes and being completely honest about it branded an outrage? Radical transparency about what a human did and did not verify should be praised, not condemned.

          I scanned through arxivs today with some self built tool, and I can assert you that 70+% of the people today that are using AI to generate their papers, most of them did not disclaim the usage of AI at all. I personally think that disclosure of the usage of AI is actually a better practice, if you actually used AI.

          4. The Myth of the Meritocracy: Folklore and the Credit Monopoly
          You write that before AI, math was purely an egalitarian meritocracy where the only elite quality was “knowledge acquired through many years of hard work, effort, dedication, and learning.”
          I speak here as a former young mathematician who finished his PhD, but whose academic trajectory hit a wall precisely because I spent that hard work and dedication proving a so called “folklore” theorem.
          Every working mathematician knows the dirty secret of our credit economy:
          Established figures freely throw around claims as “folklore,” “well-known to experts,” or “a trivial consequence of standard techniques,” claiming priority and prestige without having to produce a single line of proof. When an early-career researcher or outsider spends months to years doing the grueling, honest labor of making those folklore claims rigorous, the community shrugs and awards zero credit: “Everyone already knew that.”

          When brilliant expositors spend years demystifying, restructuring, and clarifying dense, chaotic breakthroughs, hiring committees and funding bodies still disproportionately reward the person who planted the flag first. The pure math community created an incentive system that aggressively optimizes for flag-planting and first discovery, while treating clarification, pedagogy, and formal verification as thankless volunteer work.

          You cannot design an academic tournament that exclusively rewards who crosses the finish line first, and then express moral outrage when outsiders use modern, high-powered tools to win at the very game the establishment designed. If you want conceptual understanding and explanatory clarity to matter more than raw results, the community must fix its own broken credit architecture, not blame the people utilizing the computational reality of 2026. Truth does not care about our credentials, our struggle, or our hurt pride. It only cares if the proof holds.

          Cheers,
          A previous young mathematician who finished his PhD, whose academic journey stalled after proving a folklore theorem that the community deemed unworthy of credit.

          Reply
          1. anonymous 3 Avatar
            anonymous 3

            Dear anonymous 2,

            I am deeply sympathetic to your position. I am to a much lesser extent in a similar boat, and I know others in similar positions. Yes sociology in mathematics can be absolutely garbage in some sub-communities, and some of these communities are beginning to pay the price for their behavior (this started happening pre-AI).

            Each time I think, why am I here in math still. What I think of are not these unpleasant moments, but the inspirational moments: when we stumble upon a new idea with a collaborator, the excitement when a new idea is shared, and the numerous people who went out of their way to help me in my career.

            There are dark places in math, and people know them as such. We do not celebrate them, and every time there is hope that younger generation will grow up in environments less dark. The precious things in math community are worth defending, despite the soil and blemish we experience in our own careers.

            Reply
    3. Anonymous Avatar
      Anonymous

      I believe Jihao loves credits, but not math.

      Reply
    4. Handcraft Undergraduate Avatar
      Handcraft Undergraduate

      I do think other mathematicians can invite Jinhao Liu over and let him give a talk to explain all his published work this year alone with only a chalk and blackboard and try not to embarrass himself.

      Reply
  2. tasminchu Avatar
    tasminchu

    Beautiful essay, Francesco. I am also so glad that MIP*=RE and the Aldous-Lyons disproof came first. Would we have ever understood that quantum games are connected to questions about operator algebras and the “approximability” of groups, if instead these elegant group-theoretic constructions had come first? As a probabilist myself, I probably would not have paid too much attention to the question of non-sofic groups if Aldous and Lyons had not defined the unimodular random graph (which had secretly already been studied in the field of countable Borel equivalence relations, under the guise of the “Borel pmp graph”) and boldly conjectured all unimodular random graphs are sofic. But Aldous and Lyons did introduce these ideas to probability, so I too become invested in this story, and was excited for the day in which the community found a provably non-sofic group.

    Mathematics is unreasonably effective in mathematics. The cross-pollination of different fields is a powerful force for good in mathematics: it prevents us from being atomized and hyperspecialized, it forces us to connect with and talk to our colleagues in different fields, and it breathes related ideas into distinct disciplines. AI makes it easier than ever to find the geodesic path from the birth of an open problem to a correct proof certificate of its statement. But we may lose a lot of fruitful intellectual inquiry along the way.

    Thanks for telling your part of this story faithfully. One thing we especially risk losing in our current era is the history of a result and the collective intellectual effort which gave rise to a theorem. By writing this, I feel like you returned the human story which had been obscured and erased by certain large corporate actors. 🙂

    Reply
  3. davidmfisher Avatar
    davidmfisher

    An amusing historical note for context. I first saw Alex Lubotzky give a kind of survey mentioned the existence of sofic, MF, hyperlinear groups at a conference I organized at Newton with Pierre-Emmanuel Caprace and Henry Wilton. In 2017. Alex gave another survey at a recent Brin conference again mentioned this questions in mid-July 2026. Franceso and I were both there. Now it is 2 1/2 months later and all questions are resolved. This speed of change probably is not imaginable without AI. There had been a lot of interesting progress in the area in between 2017 and 2026 and to my mind a lot of the work that had been done was more interesting than the actual examples are. I don’t know how much of it would have been done if the examples were found first. Clearly we need to re-orient our ways of thinking.

    Reply
  4. Hume Chang Avatar
    Hume Chang

    I think Liu’s attitude is a very healthy one. Everyone does their parts and shares credit. Don’t be too greedy to want to own everything. If this becomes the new social norm, maybe the ai hype/fear will turn out to be a nothing burger.

    Reply
    1. just different Avatar
      just different

      You have a very odd definition of “healthy.”

      Reply
    2. MM Avatar
      MM

      It seems that “don’t be too greedy to want to own everything” applies quite well as a criticism of Liu.

      Reply
  5. AQJ Avatar
    AQJ

    I’m curious, how much credit a prompter, who has essentially no expertise but enough tokens, should get, if not zero?

    Reply
    1. Hume Chang Avatar
      Hume Chang

      L’Hôpital rule says:

      Reply
      1. Dominique Manchon Avatar
        Dominique Manchon

        “C’est l’Hôpital qui se moque de la charité !”

        Reply
  6. just different Avatar
    just different

    Jihao Liu: PhD 2021; arXiv preprints: 1 in 2017, 1 in 2018, 1 in 2022, 2 in 2023, 2 in 2024, 7 in 2025, 28 in 2026.

    Reply
    1. MM Avatar
      MM

      If I see someone with this trajectory I will never support their hiring, promotion, grants, etc. Why would I support someone whose approach to mathematics is to flood instead of think?

      Reply
    2. fox Avatar
      fox

      whoo geometric growth

      Reply
    3. Mouse Avatar
      Mouse

      Just to get the picture clear: 2017: 1; 2018: 3; 2019: 3; 2020: 1; 2021: 3 (this year he graduated from Utah); 2022: 10; 2023: 11; 2024: 7 (this year he started tenure track at Peking); 2025: 7; 2026: 28.

      It is quite clear that, after he started his postdoc, he felt the pressure to publish.
      Different people react differently to the pressure to publish.
      One can surely say that his behavior is not moral.

      However, I’ve been on the job market recently and I’ve got very nasty referee reports from seemingly senior people.
      When you work your ass off for more than a decade and still face survival crisis every year, you just don’t have the mental energy to think about morality. It’s all survival instincts now.

      I just can’t shake the feeling that the current academia system (at least in the US and China) encourages young people to enter a dog-bite-dog competition.
      Before, this kind of competition only happens in small circles of young people. Now, with AI, every person in every circle can attack every person in every circle.

      AI is like a gun, the great equalizer, and we have entered the mathematical Wild West.

      Reply
      1. just different Avatar
        just different

        The underlying problem here is institutional incentives. Whatever one’s judgment about Jihao Liu’s sense of propriety, it’s undeniable that he’s responding to terrible incentives.

        Reply
        1. Mouse Avatar
          Mouse

          Agreed.

          Reply
  7. william Avatar
    william

    quoting from Jihao Liu’s website (https://jihaoliu.org/ai-results-about.html):

    “For any paper posted there, provided that the paper is correct and has no fatal mistake, I will be happy to share credit with any person or organization who obtained the same or stronger results, by whatever method, before I posted the paper.”

    Reply
    1. Random Avatar
      Random

      So he has a say in who should be credited and who shouldn’t? Because he has this powerful Danus system?

      Reply
      1. william Avatar
        william

        Liu also announced: “Some argue that the credit for mathematical work of this kind should go not to the person who wrote the prompt but to the AI system. I do not give my personal opinion on this argument here, but I would like to point out that all the works on that page were obtained by the Danus system, and I am one of the developers of Danus: I am the first author of the report on the Danus system. So even if you think that the credit should go to the AI system, it should go to the Danus system (the Rethlas/Danus team), and hence I still have the credit.”

        Reply
        1. wolf Avatar
          wolf

          I strongly disagree with that quotation from Liu

          Reply
  8. J Avatar
    J

    “nobody should be greedy.”
    To me and several others, Jihao Liu’s actions and paper postings are already the greediest possible actions. I find this to be incredibly ironic and distasteful

    Reply

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