A complex structure on S^6 [pdf] | Hacker News Reader

A complex structure on S^6 [pdf]

alpo.ge

65 pointsrobinhouston28 comments

100 pages of dense mathematics created by Claude. It doesn't even have an abstract or a summary. What are we looking at here? Has anyone read this? Please explain like I'm not a math PhD.

I’m pretty sure no one (except perhaps Anthropic insiders who had prior access, and probably not even them) has properly digested this paper yet, and some caution is warranted given the notoriety of this problem and the history of claimed solutions that did not stand up to scrutiny. But if it stands up, it’s a really big deal.

Background: https://mathoverflow.net/questions/1973/is-there-a-complex-s...

for maybe undergrads,

complex structure in S^6 would mean an associative version of octonions (which are non associative "almost complex" versions of their cousins native to S^3, which you may know as (unit) "quaternions").

Also fancy cousins of unit vectors (that you can do a nonzero triple cross product with), ie "non associative vectors" living on S^2, the 2D surface of a sphere "with radius 1"

If you haven't quite got the PhD https://ncatlab.org/nlab/show/quaternionic+structure

Big enough that Atiyah the godfather of this subfield, in 2016 claimed to have solved it (not sure if it was before or after the dementia)

This is not just any math PhD, it's the anthropic guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal (for impact if not notoriety or complexity) if it were human

Disclaimer: not a pure maths PhD myself, just someone who likes this stuff so..

> complex structure in S^6 would mean an associative version of octonions

where can one read more about that?

In TFA itself!

The multiplication they call an "alternating [skew-symmetric?] invariant form": Q0 in symbols I think?

Basically a polynomial function in the components of the "superoctonions" that go into the product.that how I'd vibesplain to a fellow non-pro, anyways

P10, sec 2.4 Lemma 2.8

The complex structure itself is much more complicated.. or else alpoge would have posted a one line verification of the result? But a summary that excites everybody can only be done by a pro...

The next relevant thing in the paper though, that I might be able to attempt to vibesplain using my uh consumer knowhow, is "integrability" (starting in the section labelled "Context")

Or one could just ask Fable/ChatGPT to ELI10

yep I have feet wet with complex geometry/analysis but not to decipher lemma 2.8. But instead of asking ELI10 I asked about S^6 complex structure helping to cure O non-associativity and got open research directions for sibling goals more than clear cut "yes, if the paper pass validation we've got some new associative octonions". That may need some healthy challenging.

chat: https://tinyurl.com/4acu6uc3

Well then what are you waiting for :)

You've got the tool, you've got the will

There's an abstract on page 3. But it's quite ... abstract.

By the end of the first sentence I'm pretty sure it would take me quite some time to understand what it's about (the first sentence of the abstract, not the whole paper)

https://xcancel.com/__alpoge__/status/2091639597193368014#m

This kind of papers go way over my head, but recently I was surprised that ChatGTP (free) found the proof for an expression had a square value for some variables. It dug up a relationship from a paper from 2009 and correctly applied it to the case.

The chat: https://chatgpt.com/share/6a7f1b08-b9dc-83ea-b911-aaf8b65f1c...

Reducing the problem to a quadratic form is pretty natural, and the paper from 2009 (though i see 2015) is quite an overkill. The result is a fairly standard consequence of the theory of representation by quadratic forms, and follows from Legendre's three square's theorem after some massaging. There's a good couple of centuries of foundations behind it.

It seems the trend for AI math is:

>Proofs which fewer and fewer people are able to understand

>Those who do understand, the rate at which models make discoveries exceed the amount of time those humans have in a day

>As a result, we will increasingly use other AI models to validate the proofs AI make for us

I don't see a future where this doesn't apply to everything. Let's say you're an evil CEO, you could ask an AI model to run for days cooking up every possible nefarious scheme to get out of a class-action lawsuit scott-free, to avoid taxes via complex financial engineering, etc. The plans will be far more complex than any human can understand. There simply aren't enough humans with the mental bandwidth to oppose you, so it's just machine vs machine, and if you have more money to pay for more compute, you win.

> It seems the trend for AI math is:

> > Proofs which fewer and fewer people are able to understand

This is the trend for proofs of famous old theorems in general, even pre-AI. Some particularly extreme examples are the proofs of Fermat's Last Theorem [0], the abc conjecture [1], the classification of finite simple groups [2], and the four colour theorem [3].

[0]: https://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last...

[1]: https://mathoverflow.net/questions/232087/have-there-been-an...

[2]: https://en.wikipedia.org/wiki/Classification_of_finite_simpl...

[3]: https://en.wikipedia.org/wiki/Four_color_theorem#Proof_by_co...

Tax law is not arbitrarily complicated, and unexpected edge cases are settled by human judges, not by mathematical proofs.

Yeah people forget that law is not mathematics.

At the end of the day judge either believes you or not and you are ducked

And if your defense is so complicated then that’s worse for… you

LGTM

I’ve been turning the handle on GPT for weeks, feeding in toy models of physics and seeing what pops out the other end.

It’s mostly this type of mathematics: classifying what high dimensional spaces can and can’t do.

Maths like this is where the AIs will be most useful in the next few years: tirelessly grinding through every possibility on the edge of human knowledge, filling in gaps and completing maps of no-go regions in the theory space.

And in other news (elliptic curve of rank 31): https://news.ycombinator.com/item?id=49412774

Announced by the same people, basically at the same time.

I skimmed through it. Looks fine. Had a bit of trouble following step 5 of theroem 10.5 (page 89), but after a second look it became all too obvious.

I was more confused by the application of Nakayama’s lemma in the following paragraph, but then it suddenly became clear to me.

basically trivial.

Obviously a Fields Medal–level achievement... Even the most optimistic person would not have expected it to happen so quickly six months ago

Is this a proof a human could have come up with? Is it elegant? How significant was the human contribution?

christ

I'm with you. That is quite a dense paper.

I nearly made it through the first line, OK, sentence of what looks like the Intro, which doesn't bother introducing itself and isn't actually the introduction which occurs later.

I can only describe pages one and two as some sort of foreplay. If you manage to hang on until P108 you can dispense with a safe word in future.

This is a big pile of AI slop. I doubt anyone will ever read it.

You didn't, but still offer meaningless ungrounded doubts.

someone surely will. It's a breakthrough.

Reply on news.ycombinator.com