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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

#351

What’s more Impressive you got Fable to do anything useful. The model is still useless for real data to day work, no matter how many parlor trucks it performs.

I’ve been staggeringly productive with Fable. Opus 4.8 fails a lot more for me.

Fable often just “knows” what I want with vague instructions. It also is able to autonomously perform work that lasts an hour long from my experience. I haven’t tested further.

Without Fable included in subscriptions, I would have moved my entire team over to Codex 5.6.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#352

Earlier quoted context omitted.

Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004. GLM 5.2 whiffed, it insisted the counterexample wasn't valid. VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "…

Current LLMs behave very counterproductively around unsolved problems, especially if they learned that humans consider them difficult. This has many straight up preventing themselves from attempting anything ...

Haha, I’ve noticed this as well. It’s like they psyche themselves out about how famous the problem is the same way humans do. I gave one Collatz in disguise, and it was finding all sorts of interesting things (but nothing worth a paper) until it realized the problem was Collatz, at which point it just proceeded to find a bunch of reasons why nothing would work from that point on.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#353

Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters…

Proving theorems will have lower payoff, but posing new questions (for AI to chew on) will have higher payoff. Math will go from theorem proving to conjecture farming/exploration. In a way this could be even more fun. Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.

Yes, as someone who prefers developing the 'correct' structure over 'merely' proving theorems, this is good for me in the short term. However, the writing I fear is on the wall for my medium and long term utility.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#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 NDA until the model is properly released.

Exciting times!

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#355
post #285

> The incredible Yitan Zhang ( https://newyorker.com/magazine/2015/02/02/pursuit-beauty ) worked on proving this conjecture for 7 years. Moh, his advisor, wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time". https://x.com/aminkarbasi/status/2079129649830137989 https://en.wik…

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#356
post #332

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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…

I'm sure they had plenty of time to think about these insights without the LLM, as well as the many other mathematicians who tried to crack it over the years. Wether the LLM was simply an assistant or solved the problem entirely isn't as important as accepting than the LLM was the essential, previously missing piece in the solution.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#357

Earlier quoted context omitted.

I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds. For someone that use Claude Code every day, this is obvious, but for some reason m…

I think this is actually a perfect case of an impressive thing being done exactly by mimicry. Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine? Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not qui…

The field now favors into the view that symbolic manipulation is not the mechanism of general intelligence, but rather an emergent byproduct of learning. So the fact that a connectionist machine (neural network) got so good at symbolic manipulation actually supports the view that we are closing the gap to general intelligence. Through the rote work, the machine really internalizes those rules and the symbolic manipulation capabilities are emergent, just like we humans do it.

What still confuses people is the insane inefficiency of deep learning, and that those emergent capabilities require such an immense training corpus compared to the only other architecture that we know of.

But this already is an optimization problem. If the machine gets super human at symbolic reasoning, and at the same time, can solve the symbol grounding problem to real world data and sensors, what prevents you from saying it thinks? Can it not solve real world problems? Can it not redefine its tasks and display some form moral agency - even if a totally foreign morality for us humans? Can it not use these abilities to reproduce and expand, create ships and turn the universe into paperclips, if it finds it worthwhile?

Math is basically just a playground that is perfectly suited for these emergent capabilities, so of course we will see the first progress here; but there is no firewall separating math problems from general cognition.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#358

Earlier quoted context omitted.

The interesting thing about using Claude Fable 5 is it's nearly as irritatingly sycophantic as past Claudes while genuinely being smarter than the previous models. So you get a kind of yo-yoing of it glazing you as a creative genius and disappointedly revealing to you that your ideas are bad and dumb.

That is easily fixable with the "Instructions for Claude" in your settings. Just put: "be critical, but correct. I don't want affirmation, i want to actually accomplish things." I don't get any sycophancy from it. The conversations are actually quite good, and it'll push back if it thinks I'm wrong.

The I do that it starts acting like a jerk.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#359
post #185

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sol medium can't believe its own input and output tokens either despite computing everything itself; this is what it gave me: > Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem…

Just take it as more confirmation that LLMs are unintelligent pattern-matchers.

You must never have faced a situation where you can't believe your eyes. It takes a certain level of - dare I say it - intelligence and maturity to consider that it's more likely you've made a mistake than that you've made a huge breakthrough.

In HN terms - it's never the compiler. Yes, very occasionally it might be the compiler, but you're better off assuming it's a bug in your code.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#360

Earlier quoted context omitted.

Zhang, Yitang’s life at Purdue (Jan 1985-1991) T.T.Moh - https://www.math.purdue.edu/~ttm/ZhangYt.pdf kind of a wild document to exist...

The world doesn’t need advisors like Moh that are bitter enough to gossip about a family funeral.

He's not dead, but it is bad to gossip regardless.
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