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:
https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
> 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.
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
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.
Related:
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
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Fable, explain the counterexample to the Jacobian conjecture intuitively. Make no mistakes.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?
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".
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.
It's just a meme at this point.
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.
Precisely
It will make mistakes if you tell it to, so I assume it will make no mistakes if you tell it not to.
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.
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.
> Does asking "make no mistakes" actually change the output?
Why Teams Add "Make No Mistakes" to AI Prompts (And Why It Never Works)
https://jakemcmahon.github.io/medium-articles/make-no-mistak...
Also note the timestamp: he started working on this thread a few hours after the tweet.
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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].
[1] https://news.ycombinator.com/item?id=44575026
People are upvoting this because Tao is a celebrity.
He’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.
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.
He’s a hacker news celebrity but not a global celebrity
I'm upvoting this because it's interesting. I don't have to fully understand something to find it interesting.
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.