Mathematician reactions to OpenAI’s 372 math results

Appreciate you showing genuine curiosity here unlike most people I've interacted with on this issue. Part of why I started throwing large wager offers around is it was the only way to call out most of those larpers on their bluff. But since you're interested, I'll say what I can here.

Some of my info is whispers from insiders I trust, which I'm not going to repeat here. I'll rely atm on arguments that mostly do not depend on them. First off, the smart money has been that the lab rumors were regarding a new case of the Hodge Conjecture. This has been true at least since the guest post on Terry's blog on 9/11, which seems to vaguely suggest awareness such a result has been achieved. A popular interpretation in the High Math Circles is that it's hinting at Hodge for abelian varieties, and Brockman's claim of substantial progress is consistent with that [I'll explicitly say: no one I've talked to has explicitly confirmed that theory, but conversations I've had have significantly increased my credence on it].

The Information article is unlikely to have new info beyond that. The mole Palazzolo spoke to added two new things beyond the previous public record: 1) It's Hodge (duh), and 2) "We're close to solving the full thing." The latter I very strongly doubt, and sounds like the embellishment of a non-math savvy insider who heard about the substantial progress. I'm going to be very clear: Hodge for abelien varieties has long been seen as more in-reach than full Hodge and the profs I talk do not think that's substantial evidence in favor of full Hodge falling anytime soon.

Remember, many OpenAI lab employees believed and spread the originally-quiet rumor that Anthropic had fully solved two Millennium Problems, which we now can be nearly certain is false. It's fair to acknowledge that the new round of OpenAI hype is a touch more credible on the basis of "well they're closer to the source this time." But I maintain that everything we've seen is fully explainable by the same credulous insiders as the ones to blame for the last round of bullshit getting confused on how much progress the new case of Hodge is.

And my final argument: Hodge hard

This sounds very plausible to me. There is literally zero chance that Open AI has proved the Hodge conjecture in general (it is far beyond the reach of current LLMs, like the Riemann Hypothesis) but reasonable possibilities are: a) proof of Hodge for all abelian varieties b) a counterexample to Hodge (probably from an abelian variety).

I will add that while a proof of Hodge for abelian varieties may be "small potatoes" in comparison to Hodge in general, it would still be a huge result, and would count as the best or second-best result proved by LLMs so far.

i asked GPT 6 Pro and Fable 5.1 to rank all discoveries in the last three years

🔵 for Human discovered 🔴 for AI discovered (before October 6th) 🟢 for AI discovered from OpenAI/math repo

81% of them have been released today. wtf

Screenshot 1: ranked discoveries list

1. 🟢 Quasi-Riemann hypothesis: zero-free half-plane Re(s) > 7/8* — 🐙 OpenAI, F003, 2026.

2. 🔴 Finite-time blowup for 3D Navier–Stokes with smooth forcing — OpenAI, 2026.

3. 🔵 Categorical unramified geometric Langlands conjecture — Gaitsgory, Raskin et al., 2024.

4. 🟢 Hodge conjecture for CM abelian varieties, with Tate and Hodge-standard consequences* — 🐙 OpenAI, F032, 2026.

5. 🟢 Log abundance in all dimensions* — 🐙 OpenAI, F034, 2026.

6. 🟢 Unique Games Conjecture and optimal approximation thresholds* — 🐙 OpenAI, F102, 2026.

7. 🟢 Undecidability of Hilbert's tenth problem over Q* — 🐙 OpenAI, F004, 2026.

8. 🟢 Isomorphism of all nonabelian free group factors* — 🐙 OpenAI, F287, 2026.

9. 🟢 Anderson localization and delocalization for uniform-disorder lattice models* — 🐙 OpenAI, F261, 2026.

10. 🔴 Counterexample to the Jacobian conjecture — Levent Alpöge / Claude Fable 5, 2026.

11. 🔴 Finite-time blowup for smooth, unforced 3D Euler flow — OpenAI, 2026.

12. 🟢 Counterexamples to coefficient-free Baum–Connes* — 🐙 OpenAI, F285, 2026.

13. 🟢 Logarithmic-space derandomization: L = RL = BPL* — 🐙 OpenAI, F103, 2026.

14. 🟢 Shelah's eventual categoricity conjecture* — 🐙 OpenAI, F240, 2026.

15. 🟢 Minimal models for generalized log-canonical pairs in characteristic zero* — 🐙 OpenAI, F036, 2026.

16. 🟢 Full BSD formula in Selmer coranks zero and one* — 🐙 OpenAI, F002, 2026.

17. 🟢 Hilbert's sixteenth problem: uniform limit-cycle bounds* — 🐙 OpenAI, F143, 2026.

18. 🟢 Hilbert–Smith conjecture in every dimension* — 🐙 OpenAI, F304, 2026.

19. 🟢 Counterexamples to Hadwiger's graph-minor conjecture* — 🐙 OpenAI, F157, 2026.

20. 🟢 Symmetric and nonsymmetric Mahler conjectures* — 🐙 OpenAI, F087, 2026.

21. 🟢 Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds* — 🐙 OpenAI, F159, 2026.

22. 🟢 Cannon's conjecture* — 🐙 OpenAI, F246, 2026.

23. 🟢 Falconer distance conjecture in every dimension* — 🐙 OpenAI, F073, 2026.

24. 🟢 Gromov's scalar-curvature inequality and Gromov–Lawson inessentiality* — 🐙 OpenAI, F335, 2026.

25. 🔵 Disproof of Ravenel's telescope conjecture — Burklund, Hahn, Levy & Schlank, 2023.

26. 🟢 Large-data global smoothness for 3D relativistic Vlasov–Maxwell* — 🐙 OpenAI, F362, 2026.

27. 🟢 Counterexamples to Kaplansky direct finiteness and Gottschalk surjunctivity* — 🐙 OpenAI, F197, 2026.

28. 🟢 Deterministic polynomial-time factorization over prime fields* — 🐙 OpenAI, F142, 2026.

29. 🔴 Existence of non-sofic groups — OpenAI / Astra, 2026.

30. 🟢 Quantum geometric Langlands at irrational level* — 🐙 OpenAI, F069, 2026.

31. 🔵 Three-dimensional Kakeya set conjecture — Hong Wang & Joshua Zahl, 2025.

32. 🟢 3D Kakeya maximal conjecture and 4D Kakeya dimension conjecture* — 🐙 OpenAI, F074, 2026.

33. 🟢 Area law for gapped two-dimensional quantum systems* — 🐙 OpenAI, F265, 2026.

34. 🟢 Spacetime Penrose inequality using enclosing area* — 🐙 OpenAI, F260, 2026.

35. 🟢 p-adic section conjecture* — 🐙 OpenAI, F019, 2026.

36. 🟢 Goldfeld's density and mean-rank conjectures* — 🐙 OpenAI, F006, 2026.

37. 🟢 Modularity of elliptic curves over imaginary quadratic fields* — 🐙 OpenAI, F030, 2026.

38. 🔵 Bourgain's slicing and thin-shell conjectures — Bo'az Klartag & Joseph Lehec, 2024–2025.

39. 🔴 Counterexample to the real sum-product conjecture — Bloom, Sawin, Schildkraut & Zhelezov, with GPT-5.5, 2026.

40. 🟢 Infinite finitely presented residually finite torsion group* — 🐙 OpenAI, F247, 2026.

41. 🟢 Counterexample to Kaplansky's zero-divisor conjecture* — 🐙 OpenAI, F196, 2026.

42. 🟢 Failure of unrestricted four-dimensional disk embedding* — 🐙 OpenAI, F305, 2026.

43. 🟢 Counterexample to four-dimensional Borel rigidity* — 🐙 OpenAI, F320, 2026.

44. 🔵 Final Kervaire-invariant-one case: dimension 126 — Weinan Lin, Guozhen Wang & Zhouli Xu, 2024.

45. 🔵 Long-time Boltzmann–Grad limit and hard-sphere fluid limit — Yu Deng, Zaher Hani & Xiao Ma, 2024–2025.

46. 🟢 Positivity of Serre's intersection multiplicities* — 🐙 OpenAI, F193, 2026.

So now we have confirmation that OpenAI has indeed proved the Hodge conjecture for all (CM) abelian varieties.

Huge, but not the best proved by LLMs so far! That title goes to the lightning in a bottle: the Riemann Zeta function has no zeroes to the right of Re(s)=7/8.

Some further thoughts on the 372 results released by OpenAI today, across 722 manuscripts.

If I were to classify theorems that mathematicians prove and publish according to their groundbreaking nature, I would (very roughly) divide them into four categories:

A) Non-breakthrough results. This is the overwhelming majority of published mathematics. Such results can range from solid to excellent, and some represent genuine advances in a field. But they are not hugely surprising, and would not normally be described as “breakthroughs.”

B) Exceptional advances within an existing programme. These are spectacular results, but where there was nonetheless an existing credible route to the theorem, and some expectation that sufficient work would get you there. Completing the programme may require a lot of ingenuity and deep work, but mathematicians would not be shocked that the theorem had finally been proved.

C) Surprising breakthroughs. These are results that clear a major barrier and substantially change the state of a field. Before the proof there was no convincing roadmap to the full result. Yet, while mathematicians would find the theorem remarkable and surprising, they would not find it completely shocking: if you asked them beforehand if it was plausible such a theorem could be proved today, most would say yes.

Note: The very best mathematicians prove only a small number of results in categories B and C in a lifetime; many mathematicians never prove even one. Such results would normally belong in the very top journals, such as Annals, Inventiones, etc., and there are only a handful of them each year in any given area. Several results of this calibre by a single person would make a very strong case for a Fields Medal.

D) Shock breakthroughs. These are results that, before their announcement, leading experts would have regarded as extremely unlikely to be proved with the current mathematical technology available. So the theorem itself would come as a shock. These are extraordinarily rare, and instant-Fields medal variety.

(There is one further category I have deliberately left out, because I suspect it is empty: a correct proof of a problem for which the overwhelming consensus of top experts, until the proof came, was that a proof was so far beyond existing mathematics that a claimed solution should, on prior grounds alone, be regarded as almost impossible. I would put the Riemann Hypothesis today in that category)

My current impression is that some of OpenAI's announcements today lie in A, but most fall into categories B or C. There is exactly one example in D (the Quasi-Riemann Hypothesis).

It is a very big day for mathematics.

OpenAI's internal model clearly has "research taste" in math - i.e., the final component we need to get to RSI.

• jboggan (post on Hacker News) on Barnette's Conjecture: "the 'aha' insight for this is actually f**ing wild... this is the first time I've seen complex roots and annihilating terms like this... I don't understand where this trick originated."
• Joshua Zelinsky on the proof that the chromatic number of the plane is at least six: "doesn't look like the method is a direction that the prior lit used to my knowledge... far beyond merelt building on existing methods or seeing connections between different problems."

and on two other problems (where he says he is only partly familiar with the literature): "not remotely low-hanging fruit... it seems like the AI is somehow inventing new techniques on its own."

These match Tristian Buckmaster's view on the Navier-Stokes solution: "you combine... ideas of convex integration with the growth mechanism of the Euler blowup, and... you create a new mechanism which is used to correct this non-solution. This is actually a cool idea. It's the kind of idea that I've been trying and failing to realize for over ten years... I didn't manage to do it... This is the leap."

The average amount of compute used to solve these problems was ~3 hours of Pro-level thinking.

And so, this means that OpenAI's internal model is able to generate truly novel discoveries in mathematics for... maybe at most a few hundred bucks?

If I were OpenAI, I would be asking this model to immediately target major unsolved problems in AI R&D.

Screenshot 1: jboggan (Hacker News) on Barnette's Conjecture

I'm still digesting the proof and translating a bit from the dual case back to the primal in which I most commonly thought about it. I don't think it was a brute force proof in the sense that it combined every possible paper and commentary. It's rather odd because I feel like most of the work on the conjecture was focused on an induction proof based around graph reductions, and this proof avoided those issues entirely by offering a concrete constructive proof of finding a Hamiltonian cycle. Rather, it explicitly selected the edges not in the Hamiltonian cycle, which is in line with previous attempts via the dual.

The "aha" insight for this is actually f**ing wild, it involves a complex valued exponential sum on the edges. I've seen a lot of clever counting arguments before in graph theory but this is the first time I've seen complex roots and annihilating terms like this, the symbolic manipulation tricks in this look like things out of quantum physics. I don't understand where this trick originated, I need to really digest this.

Screenshot 2: Joshua Zelinsky on chromatic number of the plane ≥ 6

Joshua Zelinsky @joshuazelinsky.... - 13h I've only had time to start looking at the chromatic plane proof, and it doesn't look like the method is a direction that the prior lit used to my knowledge. This is far beyond merely building on existing methods or seeing connections between different problems.

on Go © 36 N A

Screenshot 3: Joshua Zelinsky on inventing new techniques

Joshua Zelinsky @joshuazelinsky.... - 13h This is not remotely low-hanging fruit. And from the two I've started looking at, it seems like the AI is somehow inventing new techniques on its own. (Granted I'm only partially familiar with the lit on those two problems.)

01 — © 19 N @&

Screenshot 4: Tristan Buckmaster on Navier–Stokes / convex integration

on. It's to combine ideas from convex integration with the growth mechanism from the Euler equation. Convex integration is a way of fixing things; it's a mechanism for fixing the error. It's used in a different way in this proof, but its history traces all the way back to John Nash. So you combine these ideas of convex integration with the growth mechanism of the Euler blowup, and if you combine these two ideas, you create a new mechanism which is used to correct this non-solution. This is actually a cool idea. It's the kind of idea that I've been trying and failing to realize for over ten years, to be able to use these convex integration tools. In fact, even with my postdoc for the last year, I've been trying to combine these ideas of convex integration with the ideas

of Luis and Diego. I didn't manage to do it.