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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

#391

Post by aizk »

Lockal wrote: post #316
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.

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…

Well, Gemini is just a bad model in general, you have two variables going on.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#392

Post by Davidzheng »

jlev1 wrote: post #368
Davidzheng wrote: post #324

Earlier quoted context omitted.

here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have fo…

My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the…

yeah I think it's probably correct -- this is actually insanely simple (except for the fact that the codomain as described is not obvious isomorphic to A^3.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#393

Post by Davidzheng »

jlev1 wrote: post #368
Davidzheng wrote: post #324

Earlier quoted context omitted.

here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have fo…

My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the…

on the other hand it's incredible to me as someone who doesn't do computations that GPT took one look and saw the geometry--though it's not saying much we should ask ppl who do AG computations

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#394

Post by __grist »

eru wrote: post #292
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.

Can confirm, Claude is flabbergasted. Gemini just checks the web first it seems, and already references the news. Kimi doesn't quite believe it.

turn off search

>kimi is having a blast. i turned search back on and found this post from it’s sources cited after i suggested to check out the reaction. best thing is to go to a model with search off and plop it in the session

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#395

Post by cornholio »

abernard1 wrote: post #366
cornholio wrote: post #357

Earlier quoted context omitted.

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 manipul…

I am not confused. Because herein these forums, I predicted everything that was going to happen years ago. And the "insane inefficiency" of deep learning is fully to be expected from how it works. As well, there are provably no—literally no—emergent properties in these models. The choice of metric was a convenient, sloppy, and embarrassing fault of the field. It should be discredited; the field should be embarrassed;…

[dead]

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#396

Post by jlev1 »

Davidzheng wrote: post #393
jlev1 wrote: post #368

Earlier quoted context omitted.

My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the…

on the other hand it's incredible to me as someone who doesn't do computations that GPT took one look and saw the geometry--though it's not saying much we should ask ppl who do AG computations

It’s not that surprising (to me) that it would recognize these features, in that the features it picks up on are intrinsic to the map. Once you have the map, which is generically of degree 3, there’s the locus where the map drops from degree 3 to 2, which contains a big hint because it pops out the equation for the discriminant locus of a cubic. Then there’s the other bit about H, which becomes more apparent from the formula after simplifying things a bit in terms of discriminant. I still don’t yet understand the rest of the calculation, but it’s visibly simpler after you notice the role of the discriminant.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#397

Post by tptacek »

ev0lv wrote: post #381
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…

I'm not very excited. Access to the best AI is not a party I was invited to. And the people who are at that party, well, they don't exactly reflect on my best interests.

Is mathematics an science or an art? To the extent it's art, it's expressive and rewards the human experience that inspires and that creates it. If it's art, it's drastically less valuable to advance it through automation. But if mathematics is a science, then our entire goal is to increase humanity's understanding of the field. Whether by automation or genius inspiration or as a reward for decades of grinding it out incrementally, it's all the same: knowing more is the point.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#398

Post by muldvarp »

pfdietz wrote: post #389
muldvarp wrote: post #384

Earlier quoted context omitted.

> Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems. Do you have any argument why you might think this would be true?

For theorem proving, once the statement is formalized, there's an oracle for correctness of the proof. For deciding if something is interesting, well, de gustibus non est disputandum, you know? Experience with Lenat's AM decades ago had it go off making all sorts of uninteresting hypotheses. That's very weak evidence, of course. This suggests people also have role for fundung "beautiful" or "the best" proofs, since t…

I think the training data contains more than enough information for an LLM to learn what kind of proof is considered beautiful or elegant by humans.

Of course it also contains more than enough information to learn what kind of question is interesting to humans.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#399

Post by Izkata »

Liquid_Fire wrote: post #288
0x1ceb00da wrote: post #284

Earlier quoted context omitted.

> some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization Why do you trust a random stranger so much? Will you hand over your car keys to a random stranger? Sharing the chat will take 30s of their time. > There's no reason to think that it won't provided if asked for But they didn't provide it.

OK, so the two options are: A) Claude really produced this counterexample B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than sol…

Weirder has happened: https://mathsci.fandom.com/wiki/The_Haruhi_Problem

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#400

Post by muldvarp »

pfdietz wrote: post #390
muldvarp wrote: post #379

Earlier quoted context omitted.

> 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?

I don't. I'm retired. I'm also not going to live all that much longer, most likely, so it's kind of annoying I'm not really going to see any upsides to an AI world either. I'm discounting the possibility of our AI overlords figuring out a miracle like reversing aging. But it is really f-ing cool that automated math is now a thing and we are seeing it. Eat your heart out, past me.

> so it's kind of annoying I'm not really going to see any upsides to an AI world either

I don't think the "permanent underclass" will see many upsides either. They certainly won't have the money to pay for it. I'm excluding myself in that statement because luckily I have access to large amounts of barbiturates.

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