Alex Kontorovich
@AlexKontorovich, Distinguished Professor of Mathematics, Rutgers
Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked.
RH says zeta has no zeros in Re(s)>1/2. The best we had until a second ago was a region that got thinner and thinner the higher up the imaginary axis you go. I thought maybe they’d fatten that up a bit, that’d be a massive breakthrough. But no. They got a zero free strip!!!! Insane
Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute f...
Abhishek Saha
@ObhishekSaha, Professor of Mathematics, Queen Mary University of London
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.
Quoting @ObhishekSaha: 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 Ze...
I have now published on my website a paper, jointly written with Samuel Kittle, on the same main result as Paper 148 of the @OpenAI announcement.
The first proof I found was discovered by GPT-6 Astra on September 27, 2026. So I don't feel too bad that OpenAI published the result before us, yet I would have preferred it the other way around.
My collaborator Samuel Kittle and I spent most of our time over the past 10 days to understand the proof and rewrite part of the argument using our own language.
The paper will go on the arXiv tomorrow and I will post more on the mathematics of that paper soon. The paper solves one of the best-known questions on self-similar measures, but only in dimension one. The method extends to dimension two, as we discuss in our version of the paper, but not to dimension three or higher.
Paper: constantinkogler.com/Files/ExactOverlapsDim1.pdf
Supplementary Material: github.com/ckkogler/kk26-supplementary-material
Lean: github.com/ckkogler/exact-overlaps-one-dim-lean
Many staggering results here. But this is a particularly amazing one: the exponent for matrix multiplication is no more than 2.25. The previous world record had been something like 2.37. This leap in progress is like Bob Beamon's long jump.
github.com/openai/math/blob/main/preprints/Ma...
Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute f...
Daniel Litt
@littmath, Mathematician, University of Toronto
Fun! Looks like mathematicians have a lot of exciting work to do. If I understand correctly one of the results is a (very special) case of a conjecture of mine, which is also a consequence of stronger work in progress by a student of mine.
Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute f...
While reading their proof, important note:
The cubic family of L-functions appear to be essential for the proof, even for the corollary to the Riemann zeta function
Quoting @jdlichtman: OpenAI has just released floodgates for 300+ solutions to math problems. Among them include a proof of: -- a zero-free strip for the Riemann Zeta function -- a proof of the Hodge conjecture for CM abelian varieties ht...
This output is absolutely historic by any objective metric.
However, it is not the Millenium problem rumor that was buzzing around in past weeks. Worth bearing in mind...
Quoting @jdlichtman: OpenAI has just released floodgates for 300+ solutions to math problems. Among them include a proof of: -- a zero-free strip for the Riemann Zeta function -- a proof of the Hodge conjecture for CM abelian varieties ht...
Not only it's not "gave over" for mathematicians now, we have years of work ahead of us to human-understand all these breakthroughs (and more)! This could be the busiest time in history for mathematicians.
Aran Nayebi
@aran_nayebi, Assistant Professor, Carnegie Mellon
Ok update, yes *this* updates my timelines! github.com/openai/math/blob/main/overview.pdf
Rational Hodge over CM abelian varieties is true (032) and BSD for a density-one set (002 & 006).
Also Hilbert's tenth problem over Q is false (004), just as everyone expected, but we didn't have a proof for!
Quoting @aran_nayebi: If either Hodge or BSD are proven to be *true* by AI (thereby likely using deep mathematical techniques), then this would update my timelines. This may also mean the Riemann Hypothesis is not far off. But if it's a be...
Incidentally, this is another instance of my "difficulty convergence" thesis. OpenAI and Anthropic both achieved the same partial case of the Hodge conjecture. That was the amount of Hodge unlocked by this current generation of models.
Quoting @ElliotGlazer: Hash=SHA2-256(Ant:CMAV.OAI:WeilOrAV.) ie, Ant had Hodge for CM abelian varieties, OAI all Weil classes or even all abelian varieties. I assumed OAI had gotten further than Ant because of how much they were hyping up t...
The most original of the OpenAI proofs I have skimmed so far is the algebraicity of the Kuga-Satake correspondence for K3 surfaces, using techniques from mirror symmetry.
github.com/openai/math/blob/main/preprints/Al...
Somebody generalized the OpenAI O(n^{2.25}) algorithm for Fast Matrix Multiplication to general fields (not just C): github.com/selanavot/matrix-multiplication-al.... It seems to check out, including in Lean.
Btw, the new FMM paper is surprisingly elegant. It uses a very different approach than previous results. Instead of powering the CW tensor and trimming it, the define a potential function on tensors, and show the whole result by contradiction by studying the simple convolution tensor.
Maybe n^{9/4} is actually the right exponent for matrix multiplication...
