New Math from OpenAI

來源原標題

New Math from OpenAI

作者

TheZvi

原始標籤

Uncategorized · AI · artificial-intelligence · mathematics · philosophy · technology

原始連結

https://thezvi.wordpress.com/2026/10/09/new-math-from-openai/

摘要原文

It is kind of a huge deal. OpenAI dumped a broad range of huge new mathematical results produced by an internal frontier model. They just put it all on GitHub. This included 90 of the top 500 open problems in all of math , as per Proof of Atlas . In total there were 722 manuscripts (now 719 after three withdraws from the cluster that did not have Lean proofs) organized into 372 families. This was the result of a single model, presumably the same one that produced the Navier-Stokes proof (as per their link back to that post), mostly on a single prompt (quasi-RH was one of the few exceptions), working an average of three hours’ worth of compute per solution found, after being asked to try its luck at about 4,000 problems. The prompt included lines like ‘Even if the problem is “open,” the intention is that you should resolve it and present a full solution.’ OpenAI was trying a lot less than maximally hard. Levant and others called October 6, 2026, ‘obviously the most significant moment in mathematical history .’ Here is a thread of common sense graphical explanations (made by AI of course) of top problems. They look like this: First, the headline is the major breakthrough on Riemann. It’s not a solution to the hypothesis, but it’s a huge tightening on the bounds. Matrix multiplication efficiency at 2.25 may end up having the most real world impact. Maybe, but we haven’t yet found a scenario where this is faster in practice. It’s not quite a practically usable result yet, but matrix multiplication is the majority of the worlds compute. This could open a whole new approach. Integer multiplication is probably the most shocking result. Similar to matrix multiplication it’s a “asymptotic reduction”. Also like the matrix result, it’s not practical. But very few people would have predicted the previous barrier could be broken. Even slightly The Pi result isn’t necessarily the most important result here, but it may be the most fun. And probably accessible enough that even your kids can get it. Basically just how irrational is Pi. Unique games is also a pretty huge results, because like Riemann it’s adjacent to one of the most important problems in math: P vs NP. To be clear it’s not about P = NP itself. But does reveal new things about the fundamental limits to NP hard problems. Technically two problems, but I put them together because of similarities, in proof and implications. Hodge and Birch are both pretty abstract. But they’re both so important to algebraic geometry that after Riemann they may he the most significant results in the set. Hilberts 10th is another computer science problem. But unlike P vs NP it isn’t even about problems that are computationally hard to solve, it’s about whether a type of problem even has a computable solution. Last but not least is Hadwiger on graph coloring. It also ranks there for most shocking result of the full set. It basically overturns something that we thought was one of the most fundamental relationships in graphs Here are some analyses of the results in algebraic number theory . What is all of that good for? Ole Lehmann’s AI mentions applications for fusion research, portable body scanners, matching systems, tissue scans, quantum sensors and safety checks on self-driving cars and robots, among other things. Alex Kontorovich : 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 Yeah, no Siegel zeros either. So I guess two Fields medals… Quasi-RH is not the full Riemann hypothesis, but it is sufficient for many purposes, such as computing square roots modulo a prime quickly without coin flipping, or getting a much better estimate of the number of primes below a given number, and a sibling paper gets to the core of Artin’s 1927 primitive root conjecture. Steven Strogatz : 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. Isaac Kim : This contains a shocking list of problems in quantum information, many-body physics and quantum computing, the field that are dear to my heart. There are too many, but let me pick the following. 1. Proof of area law in 2D. 2. Spin-one Haldane gap 3. Parity is not in QAC^0 4. Constant-error Aaronson-Kuperberg conjecture 5. Unitary VOAs generating conformal nets. Things are changing fast. I cannot even imagine what will happen over the next few months, let alone a year. will depue : 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 Fable 5.1’s top 100 were 59% today’s list, 87% AI , full list at the link. Sauers : My agents (Opus 5.5, Astra, and 5.6 Sol) already solved some of these beforehand (on GitHub). I wonder what % of these actually require their internal model? The AIs all say when asked about all these math proofs as a hypothetical that this would be a huge deal, and should be front page new s, historic beyond any reasonable comparison, the biggest day in mathematics. Some think it is impossible. It turns out this was not front page news. Most people did not hear about it. It should have been, but no one in the news business knows what it means, or thinks people would care. Kevin A. Bryan : Ok, finished going through the OAI math list. It is bonkers and should be front page news around the world if folks understood what this meant. But if I am running strategy at a lab, #1 priority is “do this for medicine, oncology, battery efficiency, etc as fast as possible”. Joshua Gans starts off his coverage with the actual front page , in order to show us what things The New York Times thought were more important than solving 90 of the top 500 open math problems at once. Seth Burn : Sometimes front page news doesn’t initially reach the front page. The best example is Sputnik. It launched 10/4/57, but the Soviets didn’t think it was a huge deal [and it got a few paragraphs of a right-side column]. It was only after America freaked out that it [became a banner headline] in Russia on 10/6/57 . It was a remarkably slow news day otherwise. Much of this is not exactly breaking. Yet no math is to be seen, even in the summaries at the bottom, which even include a generic AI thinkpiece. Joshua Gans : What’s missing is this: 722 mathematics papers written by OpenAI . It isn’t an overstatement to say that this is probably the biggest day of scientific advancement in history. It is hard to evaluate, but the discussion that I have been seeing is that many of these are among the hardest and most significant results in mathematics. I suspect October 6th, 2026, will go down as some form of Judgment Day for AI in mathematics, but it portends so much more. The counterargument is that there might be a lot of days like this: roon (OpenAI): will yesterday be remembered as an important day? hard to say. in a punctuated exponential every local maximum looks invisible from a bit further out I like this model of why mathematics is not done quite yet: Joshua Gans : The mathematicians are now going through this. To which I offer again the above Bookend slide to help: We cannot yet show that the AI is better at conjectures or final verification. That will take some additional time, although probably not all that much time. Verification has to be pretty good or we would have found a lot more errors by now. There are a lot of mathematicians who would love to find a mistake.