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  • Hacker News
  • I can't wait to see AI disprove the DN conjecture soon.
    by done_lurking
  • Awesome! Confirms what we know; LLMs are superhuman at short term reasoning and breadth
    by vatsachak
  • "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

    by tcp_handshaker
  • This is so inelegant I can't tell if it's accurate or not. ...On the other hand, I can't solve it myself.
    by josefritzishere
  • Please, what does that mean for Maxwell equations? For electromagnetism?

    (Wikipedia redirects Maxwell's conjecture to Maxwell equations).

    by JPLeRouzic
  • > 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.

    by pdonis
  • The 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.
    by gjskngnf
  • Does anyone have any idea why there's no Wikipedia article (or redirect) for 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?

    by logicallee
  • From 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.

    by dualvariable
  • Visualization of the configuration: https://claude.ai/public/artifacts/9db65255-16ff-4f8e-8be1-1...
    by j_maffe
  • This 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
    by captainbland
  • Just prove by sorry
    by charlieyu1
  • It 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.

    by Syzygies
  • Dream up ideas for proofs by Dawn, prove them by Day, decide which ones matter by Dusk, and disprove them by Night.
    by dogcomplex
  • Could you expend on what you mean? I don’t have a math background and don’t really understand your comment
    by dgellow
  • That is one of the more beautiful, insightful things I've read about mathematics. Thank you!
    by mmooss
  • On 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.

    by beernet
  • Sure 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.
    by dcsommer
  • I agree, counterexample to P!=NP would be great. I tried but it's a mess.
    by js8
  • The beernet conjecture: all conjectures have both messy, ugly proofs, as well as a elegant clean proof lurking behind the scenes.
    by scarmig
  • Keep in mind the last big LLM maths proof (disproving the Collatz conjecture) turned out to just be exploiting five different bugs in LEAN
    by kmeisthax
  • > 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?

    by pdonis
  • The 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.

    by don_esteban
  • > 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.

    by HawtAds
  • Tip 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.

    by d_burfoot
  • ITT: people who think there’s going to be jobs.
    by piloto_ciego
  • Not a physics guy but isn’t experimental always the bottleneck?
    by charlieyu1
  • How’s this different from just asking an llm to prompt you to perform experiments? You don’t need any expertise.
    by simianwords
  • This 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.
    by bre1010
  • Only theory that is a convex combination of existing theory. Any paradigm shift is currently unreachable to LLMs and can be only obtained by luck with RL due to the curse of dimensionality.
    by storus