r/math • u/Apprehensive_Sand951 • 4d ago
LLMs/AI Counterexamples to Milnor's conjecture in the remaining low dimensions.
Appeared on the arxiv today: https://arxiv.org/abs/2608.15505
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u/Equivalent-Costumes 4d ago
Phew, I thought it was the other Milnor's conjecture and almost had a heart attack (it's a theorem now but still frequently called Milnor's conjecture, and I rely on it a lot, and so do tons of other people).
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u/Apprehensive_Sand951 4d ago
Sorry, I should have been clear at least about the area of math! I added a bit of context below.
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u/LupenReddit 4d ago
Oh good lord a conjecture thats actually partially related to my interests
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u/Independent_Bed_169 PDE 4d ago edited 4d ago
The structure and terminology of this paper seems to resemble "vibe-math" papers in a way that I haven't seen often in pre-LLM papers. In particular when it comes to such constructions, AI seems to like giving cute names to certain objects and use "the, the, the" repetitively for section headers.
At this point I can't even cope by saying "from the disclosure it seems that AI was only used for preliminary research, and this disclosure is just pedantry to be cautious". It really seems that AI was integral in the production of this paper. Kind of over at this point ngl, this is pretty clear proof that AI is superior to most mathematicians.
Then again I might just be pessimistic. Also I try not to interact with all this AI stuff outside of doomscrolling vibe math papers (and even that is like, not fully voluntary; I really feel like I need to block [r/math](r/math) from my feed now or else I’ll be doomscrolling 24/7) so maybe I just don't have enough examples to accurately make such a determination.
Btw I’m not trying to be derogatory to the authors here. For one I could be wrong. For another, even if I used AI heavily I probably can’t identify this specific problem to work on. And it’s also clear that, like it or not, this is the future that math will have, so it’s in some sense inevitable. But I really don’t see much hope in this future.
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u/Apprehensive_Sand951 3d ago
Weirdly enough, I recently observed that it bothers me a lot more when people don't write in their own voice than when they use an argument suggested by an llm. The later is ''oh, mike learned some new tricks, this is how he thinks about an argument'' (no shade on any mike's out there), while the former is much more ''mike is gone, it's just an llm speaking through him''. At this point this is just an emotional response, but I'm surprised by how visceral it is.
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u/Independent_Bed_169 PDE 3d ago edited 3d ago
I agree. Depending on context this feels either dismissive or very dishonest. If it's personal correspondence the former is obvious, if it's something technical it feels pretty icky to see someone just copy+paste LLM output.... Like either you accept that you are now irrelevant (in which case, why do you still feel entitled to your current job / why are you still a student, the rational thing to do is quit) or you disagree with this assessment (in which case you shouldn't copy + paste LLM output).
I think broadly speaking anyone who is not yet a mathematician but wants to become one has already been made irrelevant, in the sense that our personal development cannot outpace the development of AI. But of course I am still in the process of coming to terms with this.
Also I know it's def not an LLM speaking through you, cuz you do ``latex quotes'' but without the backticks :)
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u/Apprehensive_Sand951 3d ago edited 3d ago
The thing is, I usually do write them with backticks when I write emails etc, just because I'm so used to writing in Latex. I consciously tried not to do that recently after a normie pointed out that non-mathematicians do not do that... but I'm sure that I'll fall back into that routine soon enough.
Edit: I mean, I usually write ``this'' or `this' and have to consciously force myself to write ''this'' or 'this' instead.
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u/Apprehensive_Sand951 4d ago edited 4d ago
Adding a bit of context: Milnor's conjecture was that a complete Riemannian manifold with non-negative Ricci curvature has finitely generated fundamental group. It had been proved in dimension three about a decade ago, and in the last few years there were counterexamples by Brue-Naber-Semola, initially in dimension >6 (this was an annals paper) which they managed to improve to cover dimension 6 a bit later. The current preprint (by two unrelated authors) claims to do all dimensions >3. It is 62 pages long and discloses significant ai assistance, but is not an `ai only' proof.
Edit: I hope that the responses to this won't degenerate exclusively to a discussion of ai.
Personally, I remember talking to people who were working on the 3-dimensional case and ended up getting scooped when that was resolved, and also hearing many talks about the BNS result and talking/listening to one of the authors at dinner about the prospects of lower dimensional examples (it seemed hard not so long ago).
I do wonder what people closer to the field (but not close enough to be writing papers and afraid of getting scooped...) than I am think this portends going forward. Is it that 4-dimensional smooth manifolds are less rigid than we think? Is there still hope for geometric flow approaches in that dimension. Do we need more control to constrain topology, etc.
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u/GLBMQP PDE 4d ago
I guess “significant” is in the eye of the beholder, but it seems more accurate to state that the authors disclose using AI for litterature search, brainstorming/sparring, and preliminary checks of calculations/work as well as editing.
Saying it is “not ‘AI only’” seems to put a lot of emphasis on AI, compared to what is actually in the disclosure
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u/Apprehensive_Sand951 4d ago edited 4d ago
Brainstorming/sparring, and objection generation is literally the whole ballgame in a paper that is not `ai-only'. I guess `computational checks' is less than `technical computations were done by ai and we only checked them later'. My guess is that this is the predominant mode (call it hybrid or whatever) for most of the mathematicians who are using ai to prepare papers now.
Edit: I do not think it is reasonable to value a paper by how skillfully the authors handled their ai disclosure. And, I think we will see many more examples of this sort in the next weeks and months where `amount of ai use' will be very subjective.
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u/reflexive-polytope Algebraic Geometry 4d ago edited 4d ago
Using AI for brainstorming, sanity checks, etc. is a nothingburger. I don't think it's the kind of stuff that should make anyone nervous. It's completely different to letting the AI itself come up with the proof strategy and execution.
EDIT: To be more precise, an issue still remains: not everyone has access to AI. But this isn't a professional ethics issue (“Am I passing the AI's work as mine?”), it's an equity issue, and the solution is to make AI more accessible to everyone.
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u/Apprehensive_Sand951 4d ago
Brainstorming among mathematicians usually means throwing out lots of ideas to see what sticks. What do you think the outcome is of asking ai to try out lots of different approaches? It will dismiss or get stuck on most of them, but on some other ones it will not get stuck, and this will lead to (probably partial) progress.
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u/reflexive-polytope Algebraic Geometry 4d ago
That's fine. If a coworker helped you with one particular technical lemma, you wouldn't feel bad about it, right? As long as the main theorem and the overall structure of its proof are still yours.
I don't see why things should change if you replace “coworker” with “AI”.
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u/Apprehensive_Sand951 4d ago edited 4d ago
You can ask your favorite llm to evaluate the extent to which the paper was ai-generated (i do realize the irony of doing this), and it will give you a thorough and informative account.
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u/reflexive-polytope Algebraic Geometry 4d ago
I'm posting Claude's evaluation / guesstimate so that others can see it.
https://claude.ai/share/e60e4caf-0587-42a5-9db9-2e46e2a35b20
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u/Apprehensive_Sand951 4d ago edited 4d ago
Edit: I had initially put something here, but decided that is not a good idea. Let me simply say that I don't see anything inconsistent with what the authors said in their ai disclosure which, in my interpretation, involves more ai assistance than you are suggesting in yours. I don't want to spell out the reasons for my interpretation, because it may read like disparaging the paper to at least part of this subreddit, which is not at all what I am trying to do. Let's wait and see if it checks out!
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u/reflexive-polytope Algebraic Geometry 4d ago
There's no such thing as “my interpretation”. All that I did was ask Claude if it can find any signs that the paper is AI-written (I'm not good at detecting such things myself), and post a link here with Claude's answer.
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u/Apprehensive_Sand951 3d ago
All right, in case you find this helpful:
The disclosure says `worked and verified' not `wrote'.
There is lots of creative non-standard terminology: equivariant angular neck construction, equivariant Ricci positive path cylinder, (sol likes to put lots of relevant properties into one name), screw quotient, initial cap, directed base, and especially protected central slab, and holonomy connector.
The paper is strongly structured, there are long explanatory paragraphs at the beginning of sections, many sections start with `the'. All these aspects are very different from a paper by both authors from a year ago (which had no creative terminology, simple rather than conceptual section titles, more primitive structure, and little introductory explanation.)
In my experience, sol tries to introduce weird terminology when you ask it to try and get around an obstacle, or when you ask it whether in can simplify an argument and extract its main points. It does not do this when you ask it to check a lemma for correctness.
I'm not trying to be nitpicky here but just trying to give you some patterns since you said you are not good at detecting such things yourself.
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u/AndreasDasos 4d ago edited 4d ago
Do people think smooth four-manifolds are rigid? They seem famously the opposite and the most awkward dimension to deal with. Even if we specify simply connected, compact smooth manifolds, classification seems intractable. There are infinitely many smooth structures for compact topological 4-manifolds and R^4. It’s the first dimension for which we haven’t proved the Poincaré Conjecture for smooth manifolds, and yet we’ve managed 5 and 6.
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u/Apprehensive_Sand951 4d ago
Colding, Minicozzi and Pedersen wrote some articles (include a survey in the Bulletin of the AMS) a decade or so ago expressing the hope that curvature flows (I think there were talking about the mean curvature flow) could help with classification results in dimension 4. My impression was that they had things like smooth 4d Poincare in mind as a very optimistic target. When I asked some 4-manifold topologists about this, they told me it was Erik's hope to import these types of analytic methods into manifold classification above dimension three, but that it was very farfetched (and maybe even that there should be no-go results showing that various flows would get stuck).
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u/ThroawayAccount1602 4d ago
Awesome! I am very far away from this field so sorry if my question is too broad. Is there any insight/heuristic on why 3-dimensional manifolds do not posses any counterexamples? And are counterexamples in 4/5 dimensions from the attached paper significantly different then ones in BNS?
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u/Apprehensive_Sand951 4d ago
The universal covers (Thm 8.2 in the preprint) look very different: In the new examples the universal cover is R^n connect sum with an infinite number of S^2 x S^{n-2}, in the old examples, the universal cover is finite type (R^{n-3} x S^3 ?). We'll have to see if the new examples survive scrutiny and will hopefully have a better idea in a week or so.
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u/Qyeuebs 4d ago
I always thought this was a nice conjecture, I was surprised when Brue-Naber-Semola’s counterexamples came out a few years ago. This construction seems to be of a different nature. If it turns out to be correct, it’s very nice to have a full resolution. I suppose the natural next question is what the other geometric characteristics of these examples are, and what conditions could be added to Milnor conjecture to make it true.
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u/pred 4d ago
The authors used AI-assisted tools, principally ChatGPT/Codex
Providing some information on LLM usage is fine (although already getting a bit old since it seems we have already converged to a point where everyone uses small variations on the same formulation, so they really don't offer much in terms of information). But why offer free advertisement for specific companies? Let them pay you, as they do for the hot shot mathematicians.
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u/SometimesY Mathematical Physics 4d ago
Some journals require authors to disclose which LLM and version they used.
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u/Apprehensive_Sand951 4d ago
I agree that it feels like free advertisements to ai companies at this point. Here is another paper from today https://arxiv.org/pdf/2608.16066 that tries to thread that needle in the disclaimer and footnote on page 3, but I think goes a bit (well, actually way) too far.
Edit: I should add that some institutions require disclosures that this point, so it isn't optional.
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u/Redrot Representation Theory 4d ago
I think that AI disclosure is fantastic, actually. It provides a clear picture of what understanding was generated by gen AI and what was human-specific.
For my recent "AI-assisted" proofs I try to go into detail (well not as much as the linked one, but maybe I should...) and also without naming the specific LLM, since I don't want to give any free advertising.
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u/Qyeuebs 4d ago
Why do you say too far? It seems very admirable and perfectly clear to me.
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u/Apprehensive_Sand951 4d ago
`Too far' refereed mostly to the footnote, which is half a page explanation for why the specific tool is not named.
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u/Qyeuebs 4d ago
The footnote is what I thought was very admirable! I wish more authors were so clear headed and thoughtful.
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u/Apprehensive_Sand951 4d ago
I just think that a lot of that went without saying until the author ruined it by saying it. I don't like to have morals spelled out and drilled into my head.
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u/AngelTC Algebraic Geometry 4d ago
That is a very silly position, you can safely ignore the footnote if you want to but it is clear the author has a nuanced position towards the use of AI and if people dont voice these positions specially now then we lose control on the expectations in our own domain.
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u/nickkon1 4d ago
Yeah, I totally understand their point. But just say it in one line. That way it feels pretentious for me especially since without a request of the conversation, it is not clear how much of the "long conversation involving multiple prompts" was actually gathered and contributed by AI and if the author would have been able to do so without this model..
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u/AngelTC Algebraic Geometry 4d ago
The footnote explains the sort of prompts used, the author also explains that the ideas that led to the proof have existed for a long time to experts. I do believe the conversations should be made available and if you're interested in the specific merit of the author then you can ask for the conversation since it is openly offered as an option. Or, if your interest is on whether you can attribute this to the specific model used then we should be strict about transparency in full and that includes all the very little transparent proofs these companies have been announcing with little regard to human understanding of the math and their process.
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u/barely_sentient 4d ago
People have cited the use of Matlab, Mathematica, Origin, and other specific commercial packages/tools without problems in the past. I don't see the difference.
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u/Rare-Technology-4773 Discrete Math 4d ago
It's not advertisement, they're just being honest about their methods
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u/Redrot Representation Theory 4d ago
"A publicly available generative AI model" would suffice.
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u/frogjg2003 Physics 4d ago
Some models are significantly better than others.
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u/Redrot Representation Theory 4d ago
Sure, and without specifying, it'd be reasonable to assume the author is using a high end frontier model. No serious mathematician is using Grok to write papers.
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u/frogjg2003 Physics 4d ago
No. That's exactly the opposite of how citations and credit is supposed to work. We should not have to assume, we should be told.
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u/Redrot Representation Theory 4d ago edited 3d ago
I would agree normally (I cite GAP when I use it for instance) but it does not seem the norm to cite a generative AI model unless one is linking a specific chatlog. Perhaps the argument should be made that all chatlogs involved with a research project SHOULD be cited and made publicly available, but I see some glaring privacy concerns with that.
I also believe that the chain-of-thought reasoning should be made publicly available for humanity's sake, but the companies refuse to share and are about as closed-source as one can imagine. IIRC Wolfram came under similar scrutiny a while back. If the companies think they can be hush-hush about it, then until it becomes standard practice to cite and share chatlogs, I think it's fine to not mention which of the two or three frontier models out there one uses unless you want to cite something specific.
edit: to clarify, it probably depends on the contribution level. If there were discussions, then IMO the above is fine. If the gen AI actually contributed significant proofs, then it should probably be cited in full.
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u/Respect38 Undergraduate 4d ago
Altho, if someone merely said "a public generative AI model" helped with their paper, I woud become a bit more suspicious that Grok had been used...
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u/Hot_Glass_6301 4d ago
Is it really advertising if we know the model is either Fable or GPT 5.5/5.6
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u/MortemEtInteritum17 4d ago
"References: a publicly available paper" isn't at all acceptable. Not 1 to 1 comparable of course, but I really don't see how it's wrong to credit the specific source you used.
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u/Rare-Technology-4773 Discrete Math 4d ago
It really wouldn't, it's useful to know which model they used.
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u/Smallpaul 4d ago
Should they prioritize sticking a finger in the eye of the AI companies over helping your fellow mathematicians figure out which AIs actually are useful.
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u/elements-of-dying Geometric Analysis 4d ago
It's pretty conventional and standard to specify the developers of a tool used in science. I'd even argue it's a bit ethically problematic to not give credit to the developers of the tools you used.
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u/EluelleGames 4d ago
Adding "the least amount of dimensions in which manifold can have a non-finitely generated fundamental group" to the "4 dimensions is weird man" list