Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Claude Fable produced a counterexample to the Jacobian Conjecture

xcancel.com

Post reply on HN
Print view View original

421–430 of 562 posts

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#421

Post by aureate »

beering wrote: post #45
free_bip wrote: post #39

Earlier quoted context omitted.

The reason this was "easy" is because the conjecture turned out to be false. If the collatz conjecture holds true (and most mathematicians seem to think it will), it will be much harder to prove than your average Erdos problem.

I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)

The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.

... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#422

Post by dooglius »

david-gpu wrote: post #378
dooglius wrote: post #369

Earlier quoted context omitted.

He worked for Anthropic, so such a claim would be very unconvincing

Because only people who work for Anthropic use AI tools? What is the reasoning here? The guy is a mathematician, after all.

You have it backwards. Only people not working for Anthropic do not use AI tools.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#423

Post by pfdietz »

muldvarp wrote: post #404
pfdietz wrote: post #403

Earlier quoted context omitted.

The math community was not a global underclass. World wealth is increasing, the actual global underclass is seeing rapidly improving lives. You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance. But on a global scale, things are going well. AI will lead to a much larger pie.

> The math community was not a global underclass. I never claimed they were. The term "permanent underclass" refers to the preferred world order of AI techbro billionaires[1] where human labour is worthless and in consequence normal people no longer have any leverage. Societies where most of the value is dug out of the ground and where human labour is comparatively worthless already exist and living there is not nece…

Why do you think tech bro billions aren't also subject to replacement by AI? They compete with each other, and if AI does their jobs better than they can themselves they are strongly incented to use AI.

Absent a world dictatorship there's always competition here. And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#424

Post by nxtfari »

kelseyfrog wrote: post #131
aizk wrote: post #112

This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

Same result. Public share https://claude.ai/share/19fd1a34-d63b-4a16-8d83-60d5b79e7747 It did the multiple verification sequence before expanding to internet search where it found this thread.

Everyone using Claude Fable to verify this proof is so funny. If you read the definition of the Jacobian Conjecture and (I am not exaggerating this) have passed a college Calc 3 class, you can just verify the proof yourself in 30 seconds. The problem was very hard to solve but the counterexample is very easy to verify!

--- edit, adding an explanation:

To summarize it, the conjecture says if you have any multi-variable polynomial function that maps an input to an output in the same dimensional space (take for example: F = (x+2, y+2), which maps 2D space into another 2D space), AND that function has a constant-valued non-zero Jacobian determinant, THEN the conjecture is that the polynomial has an inverse, meaning basically you can find a polynomial that turns the output space back into the input space.

Fable provided the example polynomial (which was very hard to do) and the coordinates which if you plug into it, results in two points being mapped to the same output point. This means that the polynomial can't be inverted, because if you have that output point, how do you know which input point it came from?

You can just plug in the two coordinates it gave into the equation and verify that you get the same output point from both. That's the contradiction of the conjecture and it takes 30 seconds.

---

Something something outsourcing of thinking something.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#425

Post by tristanj »

sailingparrot wrote: post #412
tristanj wrote: post #354

The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance. I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an ND…

Quite a jump in conclusion you are making here. Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6. [1]: https://x.com/aaron_lou/status/2079218392452530249

My Fable 6 theory is admittedly speculative, but your “[public GPT-5.6] Sol is able to find the same counterexample” is also a jump in conclusions. Aaron specifically says he used “an internal version of Codex”. When asked whether that meant a different model or harness, he dodged the question, and only said the harness should be the standard commercial GPT-ultra harness [0]. He (intentionally) avoids identifying which model was used, so your claim is similarly unresolved. Given that Aaron works at OpenAI and has access to internal models, that GPT-5.7 is expected to launch in a few weeks and is rumored to be 10T+ parameters, it's very plausible that Aaron used that model in his analysis.

Furthermore, public GPT-5.6 pro failed six times to find a disproof to the Jacobian Conjecture [1], even with hints, which is evidence against the claim that the public GPT-5.6 Sol can solve this.

Regarding the existence of Fable 6: an internal upgraded version of Fable or Mythos almost certainly exists, given that Anthropic has been testing Mythos internally since April and previously released new models roughly every ~6 weeks.

[0] https://x.com/eliebakouch/status/2079237073001730510

[1] https://x.com/Tomodovodoo/status/2079172223055863895

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#426

Post by muldvarp »

pfdietz wrote: post #423
muldvarp wrote: post #404

Earlier quoted context omitted.

> The math community was not a global underclass. I never claimed they were. The term "permanent underclass" refers to the preferred world order of AI techbro billionaires[1] where human labour is worthless and in consequence normal people no longer have any leverage. Societies where most of the value is dug out of the ground and where human labour is comparatively worthless already exist and living there is not nece…

Why do you think tech bro billions aren't also subject to replacement by AI? They compete with each other, and if AI does their jobs better than they can themselves they are strongly incented to use AI. Absent a world dictatorship there's always competition here. And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxu…

> Why do you think tech bro billions aren't also subject to replacement by AI?

Because they're billionaires. They don't actually work for a living. They own stuff for a living.

> And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.

Luxury communism is also a possible outcome in countries with large oil reserves (Venezuela for example), it's just not a likely outcome because common people have no leverage and billionaires didn't become billionaires by giving stuff away.

Even if "fully automated luxury communism" were a likely outcome, the transition there would be very rough and would take years. If software engineering is automated away in 2028 it could still take years or decades before the first general purpose robot can take over plumbing. (Former) software engineers will have a rough transition.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#427

Post by soerxpso »

pred_ wrote: post #332
glimshe wrote: post #264

Earlier quoted context omitted.

The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.

I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided. Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bi…

That doesn't really even diminish the contribution from Fable, if true. Droves of grad students have been provided the same sorts of non-trivial insights and turned up no results.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#428

Post by potsandpans »

igravious wrote: post #338
ibarrajo wrote: post #203

I’ve been math-vibe coding a few months now. It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing. Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them…

i'm doing the same thing but for type theory

This is very interesting to me. Care to share your process?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#429

Post by sailingparrot »

tristanj wrote: post #425
sailingparrot wrote: post #412

Earlier quoted context omitted.

Quite a jump in conclusion you are making here. Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6. [1]: https://x.com/aaron_lou/status/2079218392452530249

My Fable 6 theory is admittedly speculative, but your “[public GPT-5.6] Sol is able to find the same counterexample” is also a jump in conclusions. Aaron specifically says he used “an internal version of Codex”. When asked whether that meant a different model or harness, he dodged the question, and only said the harness should be the standard commercial GPT-ultra harness [0]. He (intentionally) avoids identifying whi…

You are right, I completely missed that, to me internal version of Codex meant the harness, but I was wrong.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#430

Post by tptacek »

gonzalohm wrote: post #414
mcosta wrote: post #380

Earlier quoted context omitted.

They have provided a literal mathematical proof.

Proof that an LLM did it and that this is not yet again another marketing stunt. Remember when Anthropic wouldn't release Fable because it would be "the end of cyber security as we know it"? Yet here we are

Explain how the marketing stunt would work here? Tell me a story about how OpenAI spends money to pull off a similar problem by solving one of Smale's other unsolved problems. 80 years of mathematicians were unable to disprove the Jacobian Conjecture.

Is the marketing stunt that Anthropic has secretly built a world-class mathematics research group?

Post reply on HN