> While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} coefficients, which is much larger than the {\binom{7+3}{3} = 120} degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
tptacek 2 hours ago [-]
The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow:
Also note the timestamp: he started working on this thread a few hours after the tweet.
foobarqux 2 hours ago [-]
[flagged]
hackinthebochs 2 hours ago [-]
A news aggregator is a community above all. Upvoting content that you may not personally be interested in but will attract the right people is a net benefit for the community[1].
People are upvoting this because Tao is a celebrity.
hyperhello 2 hours ago [-]
He’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.
Jtarii 2 hours ago [-]
I just mean, anything Tao writes that is related to AI will get on the front page, because Tao represents the authoritative voice of reason using AI tools. And this website is primarily for AI news.
done_lurking 9 minutes ago [-]
[dead]
alpinisme 2 hours ago [-]
He’s a hacker news celebrity but not a global celebrity
pavel_lishin 2 hours ago [-]
I'm upvoting this because it's interesting. I don't have to fully understand something to find it interesting.
foobarqux 1 hours ago [-]
In what way is it interesting compared to the myriad of other posts on this topic that has content you can understand?
pavel_lishin 1 hours ago [-]
It's interesting because I like math, and typically posts like this generate a lot of interesting comments. Plus, in general, I'm wildly disinterested in the AI discourse that eats up 50-75% of the front page, so frankly anything that's slightly more interesting than that gets an upvote from me.
foobarqux 54 minutes ago [-]
You seem to be saying that you would upvote an empty article (or a gibberish article) with the title "Jacobian Conjecture Counterexample", is that right?
jmward01 10 minutes ago [-]
Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.
hyperhello 2 hours ago [-]
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
mswphd 24 minutes ago [-]
it doesn't overturn much. For example, here is a post from 2004
it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.
Anyway, in that post it says
> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*
so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
sfpotter 55 minutes ago [-]
No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.
drivebyhooting 53 minutes ago [-]
Can we audit the CoT and work the AI did to generate such a remarkable cancellation?
castedo 14 minutes ago [-]
I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.
I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
jasonfarnon 2 minutes ago [-]
Not just an assumption, I saw the LLM saying it used sympy.
p-e-w 7 minutes ago [-]
> a human prompting with deep math expertise
The original tweet implied that the whole thing was done while the author was watching the World Cup final.
I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
zzzeek 54 minutes ago [-]
reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.
tgrowazay 5 minutes ago [-]
Some people downvoting you, but I think it is a valuable illustration of IQ gap.
And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.
ChrisArchitect 2 hours ago [-]
Related:
Claude Fable produced a counterexample to the Jacobian Conjecture
Honest question. Does asking "make no mistakes" actually change the output? Does it make mistakes if you don't bother to ask for no mistakes? Is it just to make the human feel more secure?
minimaxir 2 hours ago [-]
It's a meme. Telling it to "make no mistakes" doesn't do anything because LLMs don't have an inherent concept of a mistake and they are already RLHFed to code correctly.
However, if you tell it to not do particular behaviors explicitly—some of which would be considered mistakes—it will not do said behaviors and with enough checks and balances, you'll get output without "mistakes".
> Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations. Return only when a complete affirmative proof has been found and survives adversarial audit.
> Do not return a reduction, partial result, isolated missing lemma, “best effort” summary, or explanation of why the problem is difficult.
dcrazy 30 minutes ago [-]
There really was a time when “make no mistakes” was thought to increase response quality. Maybe it steered the content of the thought channel?
saganus 2 hours ago [-]
It's just a meme at this point.
jewel 2 hours ago [-]
I believe it's a reference to a joke meme that goes something like "Write Windows 12 from scratch. Make no mistakes." At least that's the first context I heard it in.
greenavocado 2 hours ago [-]
Precisely
rvz 36 minutes ago [-]
You are "not allowed" to make that joke here. [0] and you probably upset a bunch of grumpy meanies.
It will make mistakes if you tell it to, so I assume it will make no mistakes if you tell it not to.
none_to_remain 2 hours ago [-]
When this news came out I amused myself by asking Claude to prove that 0.999... != 1. First it did so for the hyperreals. To do it for the reals I had to tell it it was allowed to make mistakes, although it didn't end up interestingly wrong - just very fuzzy and vague.
greenavocado 1 hours ago [-]
Fable, prove that for every positive integer n, repeatedly dividing by 2 if even or multiplying by 3 and adding 1 if odd will always eventually reduce the sequence to 1.
greenavocado 1 hours ago [-]
> Does asking "make no mistakes" actually change the output?
Why Teams Add "Make No Mistakes" to AI Prompts (And Why It Never Works)
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
[1] https://news.ycombinator.com/item?id=44575026
https://www.math.columbia.edu/~woit/wordpress/?p=105
it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.
Anyway, in that post it says
> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*
so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
The original tweet implied that the whole thing was done while the author was watching the World Cup final.
I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
https://news.ycombinator.com/item?id=48983382
However, if you tell it to not do particular behaviors explicitly—some of which would be considered mistakes—it will not do said behaviors and with enough checks and balances, you'll get output without "mistakes".
One example of this from the OpenAI Unit Distance prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...
> Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations. Return only when a complete affirmative proof has been found and survives adversarial audit.
> Do not return a reduction, partial result, isolated missing lemma, “best effort” summary, or explanation of why the problem is difficult.
[0] https://news.ycombinator.com/item?id=48838228
Why Teams Add "Make No Mistakes" to AI Prompts (And Why It Never Works)
https://jakemcmahon.github.io/medium-articles/make-no-mistak...