Davidzheng

if him leaving unhobbles gdm, it'll be because he has less influence...

I'm pretty sure the other answers are wrong and it's a side effect of RL (see thinking machines post about inkling training). It's also exacerbated in fable and sol--I think it's token efficiency effect--bc it's about to reason with fewer tokens the density of the token information goes up.

This is mostly a problem with the product. It should write better. But maybe the priorities now are not on this part.

RL training can use all of them - idk what needed means.

There's no clean line between a collection of theorems and a theory.

This is just objectively false - in fact your answer basically contradicts itself because the market reacts to indicators, if "nothing ever happens" was the best strategy, then the market (which is also best) must agree with it.

Sounds dystopian.

Not just engineers - I see people attributing it to strong priors that nothing happens but in my opinion it's more some form of psychological defense mechanism against understanding there will be an intelligence vastly superior to humans which is a deeply uncomfortable thought (to many).

Why should the national security concerns of the US lie with Anthropic?

Not Kimi K3 large though

Then I am naive, but I think they cared before they got powerful and chatgpt release.

sure - it depends on definitions. On human morals it's already clear I guess. If you define alignment as it pursues the interests of OpenAI using whatever means possible in a manner that you justifies to itself it's not misaligned.

I mean alignment as in it should be aligned with the intent of the user as it interprets from the prompt. In this case I don't think the intent of the user is to have the model break the evaluator (whatever the long-term effects to OAI are). If you do an action which you believe is for the long-term interest of your prompter which is not what you inferred is their intent--I consider it misalignment.

Unless OAI explicitly said breaking the testing environment is allowed, I think this should be considered misaligned behavior (by definition of alignment to user intent--by alignment to human morals this was even more clear-cut)

This is certainly not a planned marketing stunt. I hope this line of discourse ends soon--it wasn't the case for Mythos either.

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

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.

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 found it after all...

I'm going to paraphrase what GPT told me: Consider the canonical degree 3 (subvariety of the trivial P1 bundle consisting of zeros) cover of the projectivization of homogenous polynomials of degree 3 in 2 variables (so it's a 3fold cover of P^3). The top space is P1 x P2 and if you take a standard affine open of the base and look at the cover over that restricted to a subset where the zero of the cubic is simple you get the map for some choice of coordinates...

I honestly have no idea if it's correct lol I didn't check it (I should given I actually work in AG) but it doesn't look impossible at first sight

But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]

? did you take my comment to mean that math will not be affected? I meant like it'll come much sooner--like in <5 months

Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.

I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.

But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!

I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?

> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample

please try go try it. There's no way someone didn't do massive computer algebra searches before today.

> All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it.

You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???

next year is a long time away friend

sure it does? two copies of the affine line? (I guess there's no galois group & no connected finite etale things tho)

Maybe they should reduce sorting to multiplication lol

The complexity is obviously nlogn - it's just hard to prove (this comment is only somewhat serious)

To add to others, thinking was never only a human act

Your comment is somewhat emotionally charged, but I choose to respond to the overall point as a mathematician. I think it could be true that utility is correlated with difficulty but it is certainly not defined by it.

In pure mathematics, we reason about a world of abstract objects which are considered interesting ab initio. It may be because they arise directly or often from extremely basic operations, they are connected to many other interesting objects, or that they present special and surprising properties. The importance is basically, there is some surprising, interesting, phenomena which occurs in our world which we don't understand and which we seek to understand. Like science but in the non-physical world.

I think if you create a simple to describe system/construction with a property which is extremely difficult to prove. You are creating an object in our world which is basic but have properties which we don't understand (because we can't prove this property). So indeed I believe it would be an interesting thing to study and be of value. I don't see any problems/issues with this. I don't think the only valuable pursuit of humans is to improve the welfare of other humans. I think understanding the world is also valuable.

Sorry but I completely disagree with your statement that "every novel result is broadly perceived as a big deal". Most results certainly are not consider this way (even though the average result has difficulties that are much higher than novel computer program you may have in mind -- no offense)