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Claude Fable produced a counterexample to the Jacobian Conjecture

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#371

Post by rirze »

wenc wrote: post #9

I just fed this to GPT 5.6 Sol: Counterexample to Jacobian conjecture: ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0) GPT wrote some SymPy code to check it. The response? "As written, this is an explicit counterexample to the Jacobian conjecture. I ch…

Here's your Lean proof https://github.com/google-deepmind/formal-conjectures/pull/4...

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#372

Post by bakkoting »

somenameforme wrote: post #237
SubiculumCode wrote: post #113

For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.

All (unless I've missed one?) the math breakthroughs are also coming from the relatively small number of mathematicians working at these companies, as opposed to the many orders of magnitude more mathematicians using LLMs for mathematics outside of the companies. I assume the missing link everywhere is being able to casually burn a few rainforests worth of tokens in pursuit of something publishable.

Depends on exactly where you draw the line at "breakthrough", but there's been at least a few novel and interesting results coming from outside mathematicians. Recently for example there were https://old.reddit.com/r/math/comments/1uxj3cy/after_openais... from Phillip Kerger at Berkeley, and https://www.erdosproblems.com/forum/thread/119/proof-claims from Samuel Korsky at Two Sigma (the latter of which was more of a collaboration between the human and machine, not a one-shot like several of the other results we've discussed).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#373

Post by ceejayoz »

fn-mote wrote: post #296
j_maffe wrote: post #283

Earlier quoted context omitted.

Cheating how? The proof is in the pudding...

On one hand, yes. The parent wants to be skeptical… nothing wrong with being suspicious of marketing claims, right? I would be a lot less impressed if I found out the session was guided by an expert in the field who already had a good idea of where to look. For example, I don’t believe that the results Terrance Tao gets from an LLM are comparable to what I am going to get. I’m not even saying I’m a skeptic. Just that…

> I would be a lot less impressed if I found out the session was guided by an expert in the field who already had a good idea of where to look.

Why? The conjecture stood for over a hundred years. Plenty of such experts have tried.

"Fable makes experts able to solve things" is still a big story. I don't need to personally be able to do it for it to be a big story.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#374

Post by HappyPanacea »

pfdietz wrote: post #250
h-jones wrote: post #83

Earlier quoted context omitted.

My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.

The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem. It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.

This is slow?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#375

Post by spwa4 »

dist-epoch wrote: post #334
Lockal wrote: post #316

Earlier quoted context omitted.

I fed it to Google AI Studio, enabling tool execution and disabling web access. It also quickly verified it with SymPy, then went into psychosis. 5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding system…

> all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English. > "ground-breaking results that could be attributed to Gemini, if act quick" This would be such a dumb thing to do, and so easy to get caught with...

... for the management of Google that is so eager to present AI results that Prabhakar Raghavan proudly reported to a press crowd telling his daughter about James Webb photographing a planet outside the solar system. This is generally accepted as having resulted in a 10% drop in the Google stock price over 2 days.

(the key being this never happened, and won't ever happen: the James Webb telescope doesn't have even 1/1000th of the resolution necessary to resolve an extraterrestrial planet)

https://www.reuters.com/technology/google-ai-chatbot-bard-of...

https://www.wired.com/story/google-openai-gemini-chatgpt-art...

https://www.popsci.com/technology/google-ai-in-paris/

I wish I could say this only happened once.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#376

Post by famouswaffles »

somenameforme wrote: post #237
SubiculumCode wrote: post #113

For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.

All (unless I've missed one?) the math breakthroughs are also coming from the relatively small number of mathematicians working at these companies, as opposed to the many orders of magnitude more mathematicians using LLMs for mathematics outside of the companies. I assume the missing link everywhere is being able to casually burn a few rainforests worth of tokens in pursuit of something publishable.

No. Most of the results are from 3rd parties using publicly available models. The most impressive results have come from direct announcements, but most in total have not. Open AI have only announced 2 results and the total is up to a dozen or so now.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#377

Post by jameshart »

unexpectedtrap wrote: post #335

Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it. But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.

Without seeing the Fable trace that produced this, claiming it amounts to ‘advanced brute force’ seems like a leap.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#378

Post by david-gpu »

dooglius wrote: post #369
david-gpu wrote: post #314

Earlier quoted context omitted.

Indeed, the incentive is the opposite: hiding the fact that they used an AI would boost their own personal brand.

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.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#379

Post by muldvarp »

pfdietz wrote: post #250
h-jones wrote: post #83

Earlier quoted context omitted.

My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.

The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem. It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.

> It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.

I'm constantly surprised at how much schadenfreude there is on hacker news about LLMs. Like, do you guys (and girls) not have to work for a living?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#380

Post by mcosta »

gonzalohm wrote: post #325
greyw wrote: post #304

Earlier quoted context omitted.

[flagged]

If we don't ask for proof when someone claims something, the next few years are going to be rough for everyone. We can't trust anything nowadays

They have provided a literal mathematical proof.
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