The Maxwell Conjecture Is False (GPT 5.6 Sol) • Hacker News Board

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#61

Post by travisgriggs » Fri, Jul 31, 2026, 3:31 PM UTC

HawtAds wrote: post #19

beernet wrote: post #5

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…

> 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.

Why is it “just a matter of time”? Why do we assume and say this?

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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#62

Post by layer8 » Fri, Jul 31, 2026, 3:32 PM UTC

js8 wrote: post #50

layer8 wrote: post #35

Earlier quoted context omitted.

I’m pretty sure “counterexample” is the wrong word here.

Why? A counterexample to P!=NP would be a polynomial algorithm for SAT. If it exists, it might be a constructible object.

That’s not a counterexample to P != NP, it’s a proof that P = NP. You can’t prove that two sets are the same by counterexample. What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.

At best, a polynomial algorithm for SAT would be a counterexample to the claim that no NP-complete problem is in P.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#63

Post by logicallee » Fri, Jul 31, 2026, 3:32 PM UTC

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?

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#64

Post by dwaltrip » Fri, Jul 31, 2026, 3:34 PM UTC

tsunamifury wrote: post #60

moomin wrote: post #54

Earlier quoted context omitted.

Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.

Real talk. 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 simultaneousl…

Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#65

Post by efficax » Fri, Jul 31, 2026, 3:35 PM UTC

tsunamifury wrote: post #60

moomin wrote: post #54

Earlier quoted context omitted.

Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.

Real talk. 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 simultaneousl…

"tautological games".

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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#66

Post by captainbland » Fri, Jul 31, 2026, 3:36 PM UTC

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

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#67

Post by windexh8er » Fri, Jul 31, 2026, 3:36 PM UTC

dcsommer wrote: post #6

beernet wrote: post #5

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…

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.

"They" don't "care" about anything. It is a stateless computational run across thousands of semiconductors. There is no objective this software has other than the computational function completing. To care would mean the model would have a level of discernment that goes along with sentience.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#68

Post by throwaway0123_5 » Fri, Jul 31, 2026, 3:37 PM UTC

ashleyn wrote: post #43

mellosouls wrote: post #29

Not 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…

They 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 i…

> 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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#69

Post by ModernMech » Fri, Jul 31, 2026, 3:38 PM UTC

ComputerPerson wrote: post #47

d_burfoot wrote: post #25

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…

Politely, absolutely not. 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 st…

Pick any field and you'll find countless PhDs eager to tell anyone who will listen all the ways it ruined their lives.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#70

Post by StilesCrisis » Fri, Jul 31, 2026, 3:38 PM UTC

storus wrote: post #59

d_burfoot wrote: post #25

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…

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.

Frankly, such paradigm shifts are almost impossible for humans as well. If a mathematician proposes a truly radical paradigm shift, they're either a once-in-a-decade genius or a crackpot.