If we allow negatives but disallow exponentiation, that's decidable. If we allow negatives and partial exponentiation that stays within the integers, that's undecidable again.

Take a grad student with a perfectly good understanding of what a proof is. Their supervisor gives them a major problem to work on. Almost always, the problem is too hard, the student comes back with partial results, and student and the supervisor iterate from there. Now imagine that they have an unusually cruel and unreasonable advisor who tells them, do not dare to talk to me until you've fully solved the problem. This paragraph is exactly that. It's there exactly because the underlying system is smart enough to know that real mathematicians do not work like that.

https://aaronlou.com/jacobian_counterexample_prompt.pdf

Obviously it is not random, but it's very generic. No mention of search space or how to reduce it.

Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.

LOL

Not the parent poster here. I do know things about math. I wrote a few papers related to the unit distance problem (https://arxiv.org/abs/2311.10069, https://arxiv.org/abs/2406.15317) and spent quite some time trying to solve it. I had no chance of coming up with the proof that the spicy autocomplete came up with. Dumb benchmark, sure.

https://github.com/danielvarga/transformer-as-swarm

Basically a boid simulation where a swarm of birds can collectively solve MNIST. The goal is not some new SOTA architecture, it is to find the right trade-off where the system already exhibits complex emergent behavior while the swarming rules are still simple.

It is currently abandoned due to a serious lack of free time (*), but I would consider collaborating with anyone willing to put in some effort.

(*) In my defense, I’m not slacking meanwhile: https://arxiv.org/abs/2510.26543 https://arxiv.org/abs/2510.16522 https://www.youtube.com/watch?v=U5p3VEOWza8

Folie A Deux Ex Machina

In this paper, the task is to learn how to multiply, strictly from AxB=C examples, with 4-digit numbers. Their vanilla transformer can't learn it, but the one with (their variant of) chain-of-thought can. These are transformers that have never encountered written text, and are too small to understand any of it anyway.

Funding agencies? Should they prioritize established researchers or newcomers? Should they support many smaller grant proposals or fewer large ones?

FWIW, I tried to confuse 4o using the now-standard trick of changing the test to make it pattern-match and overthink it. It wasn't confused at all:

https://chatgpt.com/share/67b4c522-57d4-8003-93df-07fb49061e...