The Maxwell Conjecture Is False (GPT 5.6 Sol)
arxiv.org
arxiv.org
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.
* 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.
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.
But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.
Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field.
If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.
The current consensus is that domain experts get the most out of using AI on problems. How will domain expertise develop when AI is doing the work?
I’m not saying that there won’t be another approach that builds on the strengths of AI, but we have to look for that and develop it.
There’s a lot to talk about.
So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.
In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist.
Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.
We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research.
They are not competing against AI to publish a counterexample to a known conjecture.
People who get into math do it because they can't not do it.
If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.
Out of all the topics mentioned in the recent AI advances at finding counterexamples to conjectures, etc., about the only one I understood was the one where they've made advances in a bounds for kissing numbers in higher dimension, a topic I find fascinating even if I don't really know much about it.
> We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets.
— doi.org/10.1145/3626717, via ALP: Adaptive lossless floating-point compression (6 days ago, 4 comments) https://news.ycombinator.com/item?id=49051355
Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for.
> Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data.
— via Bland, for fans of everything (11 months ago, 7 comments) https://news.ycombinator.com/item?id=45049412
AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on.
It seems to always be some form of this:
Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously”
AI: “they can”
Everyone: “ok…”
I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in.
Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
All proofs are a form of tautology, you have to end up back at the point your theorem proposed. Math is games of logic. That's what it is.
See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.
Or it ends up never becoming useful. But you can't know that in advance
> Math Theory are patterns of abstraction that may never be useful at all or representative of reality.
This is a complete misunderstanding of (good) mathematical research.
The results look abstract, but they are based on concepts that are real and have truth or falsehood.
One example that comes to mind (sorry, technical): is it possible that all maps from a high dimensional sphere to a three dimensional sphere (S^2) might form a group that is not even finitely generated?
This is not just “abstract nonsense”, but understanding any of this takes effort.
its the defition of useless most of the time.
Nobody is qualified to judge the usefulness of mathematical, scientific, artistic, or any other kind of research that people choose to dedicate their time doing. And the world is better for it.
> or representative of reality
This so-called "reality" you speak of is some arbitrary representation in your head. It's your theory and patterns of abstraction, as you call it. Who knows how far or close you are to "objective" reality, whose existence we can only know through representations and abstractions. Mathematics and logic are some of the best tools we have of getting closer to that truth and understanding. All the sciences and even some of the arts are based on it.
So wildly goofy train of thought here.
This means abstract mathematics is rooted in our current cultural concerns. I guess I'm saying that it isn't really useful or really abstract, but I also agree with the other poster that people can and should do what they like.
Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.
The fact this has gone viral, shows otherwise. It is important to apparently enough people that the story went viral.
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.
A nerd's dream. Maybe he'll even be able to answer about turbulence.
Physics as a domain is a nightmare. Even the employment statistics are hard to understand because, like Philosophy, only the best of the best pursue it.
I've had countless friends throughout my PhD studies tell me that their decision to pursue a PhD in Physics ruined their lives. (Which is an exageration, but you get the point.)
The bottom line is that you should pusue Physics only if you still want to in the face of excessive media/reccomendations/statistics telling you not to.
Experimental physics is worrisome because as education becomes less-valued by society, there will be less funding available for research in general. Costly and elaborate 'big science' projects -- the kind that have to span multiple Presidential administrations in the US -- will be among the first to be classified as "waste," and performatively killed by legislators who earn votes by emulating the feeble-minded cultists who elected them.
I think the way forward, at least in the US, is going to involve leveraging AI to build better models to reduce our dependence on experiment. This will be true of both biology (where's the next HeLa line going to come from, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHIC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
I think better advice is “do you want to work in this specific lab for 5 years?”, not “do you really want to do physics”? Talk to the lab members, learn what they do.
Though, mostly by abandoning the field and moving to Anthropic.
... of course, the number of crackpots is overwhelming
In regards of experimental physics, i would even argue this is being worked on for sure. The ML machines are now big enough that simulations are getting better and better fast.
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.
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?
If so, charging per output token is the wrong incentive.
You're looking at humans from intentional stance, but at LLMs from design stance, which would be a form of categorical error.
Even evolution-deniers run into difficult-to-justify self-contradictions when trying to explain how the supposed supernatural part of our consciousness is both affecting the natural world and affected by the natural world, but not part of it.
How do you know they're correct if they're super messy and chaotic?
> AI "Proves" Collatz Conjecture with Lean 4 Bug
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.
And I look forward to a single example where this happened....
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.
Software engineers are first, but other fields like finance and radiology have huge targets on them too.
This might not help you with finding the proof, but once you have a machine that can produce several different proofs, you can select among them and incrementally polish the best one.
I think this is the 'easier' part.
We are a bit ahead of time, currently I would settle for 'as easy to understand as possible' proof. Not a long, complicated, inpenetrable, mess, that Lean says is correct, but reading it provides no insight.
The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.
[0] https://en.wikipedia.org/wiki/P_versus_NP_problem#NP-complet...
At best, a polynomial algorithm for SAT would be a counterexample to the claim that no NP-complete problem is in P.
I think a proven counterexample to Q is always a proof of not Q.
> You can’t prove that two sets are the same by counterexample.
You can in this case.
> What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
You could argue that counterexample is defined in one direction only, by convention, as to which hypothesis is more believed. In that case, my usage would be more valid, because the general consensus is P!=NP.
You could also argue that a counterexample should be some finite, constructible object. But that's actually also in favor of my usage - a difficult class is an infinite set, while an algorithm has a finite description.
Also note that AI can still find the counterexample (the actual algorithm), without proving it is a counterexample. Again, my usage of the word counterexample favors that definition of what counterexample is.
But honestly I think it would be more productive to spend this effort on thinking about actual counterexample to P!=NP.
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.
This one might be one of those. ;-)
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.
Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
> 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.
(Wikipedia redirects Maxwell's conjecture to 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.
Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
Does this work like a bug bounty program, where OpenAI pays you if you find a nice application for ChatGPT?
EDIT: Guys! Sarcasm!