9 ms·
The Maxwell Conjecture Is False (GPT 5.6 Sol)
- josefritzishere 2mo agoThis is so inelegant I can't tell if it's accurate or not. ...On the other hand, I can't solve it myself.
- jdc-pub 2mo agoLooks like the figures are cut off?
- smallerize 2mo agoThe experimental HTML view is messed up, but the actual PDF is fine.
- deleted 2mo ago[deleted]
- beernet 2mo agoOn the one-hand side, it's really impressive how LLMs drive mathematics forward, and this pace is only accelerating very quickly. At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
- dcsommer 2mo agoSure they care about elegance, or at least brevity. Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems. It doesn't mean they are perfect at it though.
- senorrib 2mo agoYou clearly haven't used Claude to generate code or documentation.
- Someone 2mo ago> Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems. First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?
- yreg 2mo agoSo they can charge more per token and decrease the pressure on their infra.
- Someone 2mo agoEven if you specifically train the model on producing shorter answers, I would think producing good short answers would require more resources just as it does for humans (https://quoteinvestigator.com/2012/04/28/shorter-letter/ https://quoteinvestigator.com/2012/04/28/shorter-letter/: “If I Had More Time, I Would Have Written a Shorter Letter”) If so, charging per output token is the wrong incentive.
- yreg 2mo ago
- pdonis 2mo ago> most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter) How do you know they're correct if they're super messy and chaotic?
- KPGv2 2mo agoI've driven back roads in Ireland. Super messy and chaotic. I was still able to use a map to get to my destination.
- _jayhack_ 2mo agoformal verifiability e.g. vi Lean
- AlexErrant 2mo agoEven Lean has bugs. > AI "Proves" Collatz Conjecture with Lean 4 Bug https://news.ycombinator.com/item?id=49101465 https://news.ycombinator.com/item?id=49101465
- drdrey 2mo agoof course it does, but it's still the best thing we have
- HawtAds 2mo ago> LLMs do not care about "elegance" the way human beings do, which is a big advantage. It's just a matter of time before you can post train it for elegance too. Mathematical proofs in particular can be formally verified automatically which is a big advantage.
- jmalicki 2mo agoI've actually been involved in annotation projects doing RLHF to train LLMs to do exactly that. It's not a matter of time, it's already happening - it's just seemingly lower priority than "profitable" projects like post-training LLMs to replace white collar workers.
- tcp_handshaker 2mo ago>> post-training LLMs to replace white collar workers. And I look forward to a single example where this happened....
- pitched 2mo agoBefore LLMs, empire building was a very large incentive to hire. Teams tended to become larger than they needed to be so the boss feels good about their life choices. LLMs do not fix this problem, they make it worse. Instead of the team being oversized, they’re now way oversized. It is still in everyone’s best interest to look busy anyways and LLMs do help a lot with that.
- tcp_handshaker 2mo agoNo need to downvote. Just provide a counter example...
- jmalicki 2mo agoI'm not saying it happened, I am saying they are working hard towards that goal as a business priority, and spending a lot of money on it. Software engineers are first, but other fields like finance and radiology have huge targets on them too.
- 2mo ago
- js8 2mo agoI agree, counterexample to P!=NP would be great. I tried but it's a mess.
- layer8 2mo agoI’m pretty sure “counterexample” is the wrong word here.
- Good4boothee 2mo agoIsn't it a bit Catch 22 anyway? If someone finds a algorithm to reduce some NP task X to class P, then that just means X wasn't a true NP task and P!=NP is still undecided?
- SetTheorist 2mo agoAIUI if you have an (polynomial-time) algorithm to reduce some NP-complete task to P then you have indeed shown that P=NP.
- Tyr42 2mo agoYou can prove something is in NP by providing a (polynomial) reduction from a known NP hard task, and vice versa. All the known NP problems (Knapsack, SAT, etc) are mutually reducable in this way, so solving one lets you solve the others. So if X was shown to be NP, then given a polynomial time solution to X, you can stack the polynomial time reduction from X to SAT to solve SAT in polynomial time too.
- layer8 2mo agoIf it’s an NP-complete [0] problem like SAT, as many NP problems are, then we are done, because all NP problems can be reduced to it (in polynomial time). [0] https://en.wikipedia.org/wiki/P_versus_NP_problem#NP-completeness https://en.wikipedia.org/wiki/P_versus_NP_problem#NP-complet...
- subsistence234 2mo agoNP doesn't mean "we don't know a polynomial time algorithm for it", it means "a proposed answer can be verified as correct in polynomial time"
- kmeisthax 2mo agoKeep in mind the last big LLM maths proof (disproving the Collatz conjecture) turned out to just be exploiting five different bugs in LEAN
- empath75 2mo agoThat very much does not describe what happened. Someone found the bug and used it to disprove the Collatz conjecture as a demonstration of the bug. Nobody ever claimed it as an LLM proof.
- don_esteban 2mo agoThe mathematics is to a great extent about understanding of abstract structures. As humans, we prefer simple structures/proofs (I suspect that is to a great extent because those are easier to understand), and as such find elegance in simplicity. In fact, the capability of the human brain to understand complex structures and proofs is rather limited. LLMs (hmm, I would prefer to use 'AI solver', as LLM is nowadays just a part of it) finding a complex proof can mean several things: 1) AI by its nature/construction does not have preference for simple stuff (it 'thinks' differently than human: a human will, in its search for a proof, start by exploring the 'simpler' parts of the proof space, and hence more likely find a 'simple' proof, while a AI might be more target oriented and descend deeply in depth-first-search manner to recursively solve sub-tasks, without much regard about the overall simplicity of the proof). This can be eventually solved, by subsequent 'polishing' passes, similarly as things work in human science. 2) there might simply not exist a simple/elegant proof of a given problem. The world is a complex beast. Its just our brains trying to find simple/elegant meaning/structure, even in places where there is none.
- scarmig 2mo agoThe beernet conjecture: all conjectures have both messy, ugly proofs, as well as a elegant clean proof lurking behind the scenes.
- don_esteban 2mo agoEh, some conjectures have counterexamples. This one might be one of those. ;-)
- xboxnolifes 2mo ago[dead]
- amelius 2mo agoOk, who gets the credit? Does this work like a bug bounty program, where OpenAI pays you if you find a nice application for ChatGPT?
- chorsestudios 2mo agoNo but the Clay Mathematics Institute will give you $1,000,000 if you solve one of the 6 remaining Millennium Prize Problems, and if you solve certain Erdos problems you can get $10-10,000.
- echelon 2mo agoEven if you use AI tools?
- muglug 2mo agoYes. But you’ll spend more in tokens than you’ll get back from prize money.
- simianwords 2mo agoThis is not true
- beering 2mo agoSome people might be on the ChatGPT Pro subscription plan or consuming their employer’s tokens.
- JPLeRouzic 2mo agoPlease, what does that mean for Maxwell equations? For electromagnetism? (Wikipedia redirects Maxwell's conjecture to Maxwell equations).
- pdonis 2mo ago> what does that mean for Maxwell equations? Nothing. They're still just as valid as they were before. > For electromagnetism? In practical terms, nothing significant. It's not going to change how anyone builds devices that use electromagnetism.
- gjskngnf 2mo agoThe Maxwell conjecture is a toy problem. The existence or nonexistence of a bound on the number of equilibrium points in an electrostatic arrangement of point charges doesn’t change much. I say that as an EE but not a specialist in electromagnetism.
- olirex99 2mo agoSeems like that anyone can now prove math conjecture. Maybe someone already prove some math problem and is not even aware of it.
- layer8 2mo agoDisprove, you mean.
- d_burfoot 2mo agoTip for smart science-y young people: think about a career in experimental physics. Experimental data is the complement of theoretical power. Since theory can be provided cheaply by LLMs, experimental ability is now the bottleneck for progress in physics. I expect to see frontier labs or startups hiring experimentalists to provide data for LLMs to analyze, pushing towards breakthroughs in areas like room-temperature superconductors and fusion.
- stubbi 2mo agoUntil we got robots doing that
- bre1010 2mo agoThis sounds depressing. Imagine going to work every day and your boss is a computer telling you to do rote nonsense so it can barely-better-than-brute-force search for breakthroughs in whatever field. Then when it finds one we get another breathless news cycle like this while you get no credit at all. If you could understand what you were working on, you might be able to contribute more than a .csv of data, but the computer can't read you in because there is no understanding under the surface.
- alasano 2mo agoBarely better than brute force (I can't believe it's not brute force!™) aside, presuming we get super intelligence it will all be depressing when it comes to intellectual pursuits like this.
- fc417fc802 2mo agoEven the smartest humans would end up as perpetual students but I'm not sure why that should be universally depressing.
- ndriscoll 2mo agoYeah I don't understand how people can not look forward to that outcome. Imagine you have your own super von Neumann who's willing to talk to you 1:1 for as long as you'd like at any time with not a hint of judgement at your intellectual inferiority. He's been working tirelessly to figure out how everything pieces together, and has 20 layers of abstraction to draw from to lead you toward understanding what's really going on along with a concrete understanding of every field to reify his thoughts into examples familiar to you and illustrate his connections. A nerd's dream. Maybe he'll even be able to answer about turbulence.
- qarl2 2mo agoLies, obviously. AI is worthless. EDIT: Guys! Sarcasm!
- mellosouls 2mo agoNot to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important. Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.
- deleted 2mo ago[deleted]
- ashleyn 2mo agoThey might be low-hanging fruit but two things immediately come to mind: * As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood. * What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.
- throwaway0123_5 2mo ago> The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field. It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types. Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
- tcp_handshaker 2mo ago"The idea behind this construction was suggested by an LLM (OpenAI’s GPT- 5.6 Sol). The authors have verified the mathematical details and have written the argument in their own words. Computer algebra software (Mathematica, Maple) was used to verify computations and produce visualisations" Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
- Syzygies 2mo agoIt is mathematical folklore that one should attempt to prove a conjecture by day, disprove it by night. Jordan Ellenberg recently popularized this in his 2014 book. He and I both heard this from Barry Mazur, but it dates at least to Bing, if not antiquity. What is the purpose of mathematics? To be the architect of new conventions by seeing clearly past the old? If so, believing that the entire point is proving statements is a poor start. Bill Thurston was a visionary who happened to prove a great deal of what he saw, but his influence was his vision. For those of us who like to understand every line of code we generate, and have labored for years to learn how to make best use of AI, a factor of two is a reasonable estimate for our productivity gain. For those of us who believe mathematics is about achieving human understanding, having machines decide what's true and what isn't makes a night and day difference. Again, about a factor of two.
- dgellow 2mo agoCould you expend on what you mean? I don’t have a math background and don’t really understand your comment
- mmooss 2mo agoThat is one of the more beautiful, insightful things I've read about mathematics. Thank you!
- dogcomplex 2mo agoDream up ideas for proofs by Dawn, prove them by Day, decide which ones matter by Dusk, and disprove them by Night.
- faangguyindia 2mo ago[dead]
- logicallee 2mo agoDoes anyone have any idea why there's no Wikipedia article (or redirect) for Maxwell Conjecture: https://en.wikipedia.org/wiki/Maxwell_Conjecture https://en.wikipedia.org/wiki/Maxwell_Conjecture Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
- dualvariable 2mo agoFrom the intro to the paper: > In J. C. Maxwell’s 1873 treatise on electricity and magnetism he discusses the number of equilibria of the electric field generated by n point charges [5, §113]. Apparently unaware of this, M. Morse and S. S. Cairns in 1969 posed the problem of finding an upper bound for the number of equilibria [6, p. 293]. The first general bounds were supplied by A. Gabrielov, D. Novikov, and B. Shapiro in [3] who, based on their reading of [5, §113], formulated the ‘Maxwell conjecture’ which states that if the critical points of the electrostatic potential generated by n point charges are all non-degenerate then their number cannot exceed (n − 1)^2. These bounds were later improved by V. Zolotov in 2023 [8] and further improved by H. Edelsbrunner, C. Fillmore, and G. Oliveira in 2026 [2]. Maxwell’s bound is trivially achieved for n = 2 but it is not known even for n = 3 if 4 is the maximum number, except in the case of equal charges [7]. Further related problems in classical electrostatics are discussed in [1]. And reference 3: > [3] A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3), 95 (2007), pp. 443–472. This is pretty niche and the conjecture was only proposed about 20 years ago. It was actually not conjectured by Maxwell himself.
- captainbland 2mo agoThis one is interesting as it's been hand verified. There was a recent proof that inadvertantly "proved" the collatz conjecture by triggering a bug in LEAN: https://infosec.exchange/@0xabad1dea/117002106099986943 https://infosec.exchange/@0xabad1dea/117002106099986943
- charlieyu1 2mo agoJust prove by sorry
- vatsachak 2mo agoAwesome! Confirms what we know; LLMs are superhuman at short term reasoning and breadth
- j_maffe 2mo agoVisualization of the configuration: https://claude.ai/public/artifacts/9db65255-16ff-4f8e-8be1-1058ff3919c3 https://claude.ai/public/artifacts/9db65255-16ff-4f8e-8be1-1...
- done_lurking 2mo agoI can't wait to see AI disprove the DN conjecture soon.
- syncsynchalt 2mo agoMaxwell's Silver Bullet?