Comment by Davidzheng
2 months ago
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?
Some speculation in this Claude chat: https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0
linked from here: https://x.com/b_shrir/status/2079094004885668003?s=20
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.
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
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...
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I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent.
The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked code is logically consistent
In my experience, lean will show that it's correct, but does it not lose the mathematical intuition that led to the result? As far as my experience goes, that's really hard to encode in lean itself.
Could we maybe get more information about the problem from the LLM trace itself here?
As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
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