{"slug": "new-math-from-openai", "title": "New Math from OpenAI", "summary": "OpenAI reported that a single model, working an average of three hours of compute per solution across roughly 4,000 problems, produced results on the Riemann hypothesis — a zero-free strip and no Siegel zeros — along with asymptotic reductions in matrix multiplication (2.25) and integer multiplication, plus results on unique games, Hodge and Birch, Hilbert's 10th problem, and Hadwiger's graph coloring conjecture. Rutgers mathematician Alex Kontorovich said a human achieving the quasi-RH result would earn an instant Fields Medal, adding \"Yeah, no Siegel zeros either. So I guess two Fields medals…\" The matrix multiplication and integer multiplication results are asymptotic reductions that are not yet practical, though matrix multiplication accounts for the majority of the world's compute.", "body_md": "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.\n\nFirst, 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.\n\nMatrix multiplication efficiency at 2.25 may end up having the most real world impact.\n\nMaybe, but we haven’t yet found a scenario where this is faster in practice.\n\nIt’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.\n\nInteger 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.\n\nBut very few people would have predicted the previous barrier could be broken. Even slightly\n\nThe 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.\n\nUnique 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.\n\nTo be clear it’s not about P = NP itself. But does reveal new things about the fundamental limits to NP hard problems.\n\nTechnically two problems, but I put them together because of similarities, in proof and implications.\n\nHodge 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.\n\nHilberts 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.\n\nLast but not least is Hadwiger on graph coloring. It also ranks there for most shocking result of the full set.\n\nIt basically overturns something that we thought was one of the most fundamental relationships in graphs\n\nWhat 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.\n\nThe Mathocalypse\n\nAlex Kontorovich: Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked.\n\nRH 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\n\nYeah, no Siegel zeros either. So I guess two Fields medals…\n\nQuasi-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.\n\nIsaac 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.\n1. Proof of area law in 2D.\n2. Spin-one Haldane gap\n3. Parity is not in QAC^0\n4. Constant-error Aaronson-Kuperberg conjecture\n5. Unitary VOAs generating conformal nets.\n\nThings are changing fast. I cannot even imagine what will happen over the next few months, let alone a year.\n\nwill depue: i asked GPT 6 Pro and Fable 5.1 to rank all discoveries in the last three years\n\n🔵 for Human discovered\n🔴 for AI discovered (before October 6th)\n🟢 for AI discovered from OpenAI/math repo\n\nSauers: 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?\n\nIt 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.\n\nKevin 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”.\n\nSeth 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].\n\nIt 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.\n\nJoshua 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.\n\nThe counterargument is that there might be a lot of days like this:\n\nroon (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\n\nFor Now You Can Still Do Math\n\nI like this model of why mathematics is not done quite yet:\n\nJoshua Gans: The mathematicians are now going through this. To which I offer again the above Bookend slide to help:\n\nWe 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.\n\nVerification 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. We know, via the reasoning traces, that a big chunk of effort is going to verification.\n\nYou Will Need To Find A New Problem\n\nScott Aaronson reports on the Mathocalypse.\n\nScott Aaronson (Shtetl-Optimized): Last night my 9-year-old son was taunting my wife, complexity theorist Dana Moshkovitz, as follows: “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”\n\nWhile my son was being a brat, he also wasn’t wrong.\n\nWhether you’re thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical history.\n\nAnd yes, among the 372 huge results released yesterday by OpenAI, on the recommendation of its advisory group of Timothy Gowers, Edward Witten, and other distinguished mathematicians, was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that my wife has worked toward proving for the entire time I’ve known her.\n\nThen there is the other kind of ‘new problem,’ where those in denial, who three years ago took comfort in talking about how LLMs could not do math, have to find a new way to explain why anything an AI can do is not real or not meaningful.\n\nExperience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.\n\nSo, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.” Or maybe none of the 372 well-known open problems that were solved were real math problems, they were all just glorified contest puzzles and trivialities. (After all, there’s still no Riemann Hypothesis!) Or maybe the entire 4000-year-old discipline of mathematics needs to be jettisoned: turns out that it was all just puzzle-solving and trivialities; all that’s different is that now the triviality stands unmasked. In any case, what really matters is that the true inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.\n\nIf you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s other great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker.\n\nI do have to tip my hat to the fully general counterargument to all possible AI hype:\n\nCas: I think that AI smashing human achievements in mathematics might be overhyped, more an example of Moravec’s paradox than a robust sign of singularity soon. Recall how AI eclipsed humans at Go ~10 years ago but was much slower to get good at tasks that would change the world.\n\nWhatever AI is good at is Moravec’s paradox. Whatever AI is bad at is why it sucks. Why are you so excited that AI is suddenly superhuman in an increasingly large set of domains, including the exact things we were saying it was dumb for being bad at? Surely this will not extend soon to other domains.\n\nVerification or Evaluation Is Not Always Easier Than Generation\n\nThe steelman of this critique is to draw a distinction between verifiable versus unverifiable domains.\n\nThis argument says that whenever you have a verified domain, with known ground truth, AI will quickly become superhuman.\n\nWhereas, if verification is difficult, or all you can do is evaluate in an informal way that requires a human in the loop to avoid biased errors and distorted behaviors, then AI capabilities will lag behind.\n\nWe were in a weird place for a few years, when pretraining dominated and LLMs were mostly trained on human words. This led to a seemingly non-Yudkowsky world in key ways, with LLMs being relatively excellent at a variety of useful but unverified domains, while being hopeless at math.\n\nNow, with post-training dominating, LLMs are again stronger at math and coding, and things again look like you would expect. AI is quickly getting better at everything, but progress in other domains, while super fast compared to almost any other tech ever, is relatively slow for now. Thus, we advance math and coding and similar domains first, which then automate AI R&D, which then accelerates everything else.\n\nOne option is that AI is relatively weak at cryptography, another is luck or that OpenAI didn’t have cryptography in its initial problem set (perhaps to avoid exactly this issue?), or perhaps the government or OpenAI are censoring those results.\n\nJustin calls for a ‘bunker mode’ for the blockchain industry, to protect against sudden mass breakthroughs.\n\nVitalik Buterin: But for anything that has structure, you should assume that AI will make at least some progress in breaking that structure. Here, one reasonable inference is that if you want to make something plausibly long-term secure, multiply the key sizes by 10.\n\n… Theoretically, of course it’s possible that hashes are broken too (eg. P = NP would imply that). But I think P = NP is very unlikely.\n\nAnd intuitively, it’s much more likely that a mathematical object has exactly no exploitable structure (like hashes are intended to), than that a mathematical object has exactly ~3 forms of exploitable structure (for elliptic curves: associativity, Schoof, pairings) and not some secret fourth form of structure we have not yet discovered that greatly degrades its security (for elliptic curves, ECDLP and pairing security). Similar for LWE, SVP, RLWE and the zoo of lattice problems.\n\nFor this reason, we do not yet see any reason to worry and start padding the byte size of hashes (if we start to worry more, we would pad the round count first before doing anything to the byte size).\n\nConcrete TLDR, my own personal views:\n\n* Hash-based > lattice-based, in those situations where hash-based is possible at all\n* For anything lattice-based, be much more paranoid on param sizes. Remember that blockchains are only a small portion of the cryptography story; this point goes far beyond blockchains and applies to eg. access to websites, secure messaging, Tor / VPNs …\n* For privacy protocols, strongly favor NOT putting encrypted notes onchain. Instead, send them offchain through some third-party mechanism.\n* If it’s not difficult for you, keeping your funds in addresses which have not yet been used to make a transaction is a good idea. If it’s easy for you, do it. **But be careful about migrations; I personally have lost more money in botched migrations than I have lost in all hacks combined**.\n* For multisig wallets, doing confirmations offchain is better than onchain, because this way the signatures of signer wallets do not get exposed to the public, so if ECDSA falls to AI much faster than expected, at least the multisig “gracefully degrades” to a 1-of-1 where the 1 is whoever was gathering the signatures – a much better place to be than “anyone can take the money”\n\nI am guessing Vitalik’s point about botched migrations is highly underappreciated. It is very easy, in crypto, to lose fantastical amounts of money from stupid mistakes, the same way you can lose it from hacks or tech failures.\n\nYes, you should worry, especially given that OpenAI’s breakthroughs here did not involve trying maximally hard.\n\nKevin Madura: given the math announcements how many novel attacks on cryptography do we think exist now but are unreleased? What does this look like in 1 year? 5 years?\n\nI’d be eyeing DoD / NSA / NIST guidance pretty closely now\n\nMatthew Green: I think we might lose public key cryptography.\n\nHere are 100+ reactions from various different people in mathematics. A lot of them are very not happy about how OpenAI handled this, especially that so many of the papers were ‘unreadable slop’ rather than having been made nice first, or that OpenAI solved these problems at all. If you want to know ‘what are the mathematicians thinking’ this is a great resource.\n\nHere is Terence Tao’s serious response, which focuses on how this disrupts the work:\n\nTerence Tao: My feelings on recent developments are very mixed and complex.\n\nOn the one hand, many of the AI-generated proofs appear to introduce clever new ideas that will be fruitful once digested, while also building upon the existing contributions of countless human mathematicians past and present. But at the same time, I am deeply frustrated that, in sharp contrast to traditional breakthroughs, none of the humans involved in these proofs are available to take questions, give talks, attend conferences, submit papers to journals, train students, or otherwise participate in the subsequent development of these results.\n\nSimilarly, I am excited by the possibility of the community being able to use these tools to tackle ambitious and large-scale projects that one could not have even dreamed of in the past. But I am horrified by the many person-years of ongoing patient and deliberately slow research efforts – particularly by graduate students and postdocs – towards many motivating problems in mathematics being casually disrupted or destroyed by such a release. Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution. Yes, one can still analyze and digest such an answer; but the best opportunity to do so is at the moment of its discovery, and such moments are increasingly wasted when delegated entirely to AI tools.\n\nAnd I mourn the path not taken, and the opportunities lost in the frantic race to develop this technology. Labs submitting their frontier models to independent researchers for proper scientific evaluation. Coordination with the research community to ensure these tools are applied to complement and enhance the abilities and activities of human researchers, rather than compete with them. Use of these tools to foster collaboration and sharing, rather than competition and secrecy. Opening new doors, without closing old ones.\n\nBut that is not the path we now find ourselves in. Instead, the community needs to come together more than ever. To clearly declare our own standards and values, to build our own tools and practices, to support our most vulnerable members, and to chart our own path forward. Let’s get to work.\n\nHere is another report from mathematician-land. People do not seem thrilled.\n\nβ/σi: > worried about their future\n> worried about the solutions that **haven’t** been published (hello cryptography)\n> in disbelief about the capabilities of these models\n> sad the parts they fell in love with are gone (like we saw in swe)\n> go solve “real” problems in bio (hilarious coming from mathematicians btw)\n\nps this is just my small sample, i only have so many mathematician friends\n\nMathematicians seem to think of math as their playground, and rather than be happy about the problems being solved they feel their toys are being taken away. There is a struggle between wanting to know, and wanting to try to solve, and wanting the credit, and wanting solutions to be found at all. What do you actually care about most?\n\nTo be smart enough to be a mathematician, and also choose to be a mathematician instead of the many other better-paying things you could do, at least kind of requires some of the attitude that leads to cartoons where someone tells the mathematician someone found an application for their work, and the mathematician panics. Definitely #NotAllMathematicians, of course.\n\nI am highly sympathetic. I really am. There could I have gone. That attitude has great value, not only inherently, but also exactly because following such curiosity leads to some of the most valuable discoveries that you would otherwise miss.\n\nWill Kinney: One thing I’m really starting to notice is that the theoretical physics community is responding to the disruption created by AI in a completely different way than the mathematics community has.\n\nsteve hsu: If your life’s work is a truly important problem – like curing cancer or fusion energy or discovering the true nature of quantum reality – then suddenly getting a solution from AI would be cause for great celebration.\n\nYou might ponder for a moment the second order impact on your profession, but that would be overwhelmed by JOY for the gift that you and the rest of humanity have received.\n\nIt is also totally reasonable to focus on how things impact you and yours. That’s what hits home, and it is the part you can most control and are forced to face.\n\nJosh Frisch: Like many other mathematicians, my main emotion thus far in 2026 has been loss: loss of meaning, loss of purpose, loss of the era of human proofs, loss of the ability to picture the future. With the release yesterday, October 6, 2026, of hundreds of beautiful results—many, maybe most, answering someone’s “one question”—I am trying to move beyond loss. There are so many beautiful results here: problems nobody had any approaches for, algorithms nobody thought could possibly exist, unexpected isomorphisms, constructions and proofs.\n\nAndrew Curran: Please resist the urge to mock the mathematicians struggling through this moment. Whatever your field, your expertise, your passion, you too shall one day experience what they are going through. And when you remember your words, you will feel remorse.\n\nIf your response is ‘well everyone’s passions are going away and the AIs come for us all’ then that’s worse. You know why that’s worse, right? Nor does it make this easier. There are some people out there reacting to these understandable reactions in truly vile ways. I urge them to stop.\n\nMathematics is facing a real problem here. If the AIs prove all the theorems, then our current methods of getting good at understanding math stop working. The traditional way you understand problems is by working to solve them and a solution is often not worth so much if no one understands it:\n\nJordana Cepelewicz (Quanta Magazine): It’s as if you were teleported to the peak of a tall mountain. Surrounded by fog, you have no idea where you are, or what’s around you. You do not know how your mountain connects to others, and you have no equipment to help you explore, no way to help someone else join you. If you had climbed the mountain yourself, you would have experienced how the human body adapts to altitude and changes in oxygen levels. You might have had to invent tools to navigate, to climb steep cliffs, or to make a shelter. You might have encountered a fellow explorer, gotten lost together in a hidden valley, and found a plant that could be turned into a life-saving medicine.\n\nInstead you’re perched on the peak but in the dark, while the maker of the teleportation machine tells you that it can explore the wilderness better than any human.\n\nThere are other ways to train understanding, but it could take a while to develop them, and they will require more motivation, especially intrinsic motivation.\n\nThis also destroys the system of PhDs and postdocs, since the problem you are looking to solve or the grant you want can get pulled out from under you at any time.\n\nThis also is a not fun place to be:\n\nMany mathematicians have stopped posting open conjectures and potential ideas at the end of their papers, for fear that they’ll be scraped by bots and fed into AI models. Others are posting their papers before they’re ready in order to avoid getting scooped.\n\nThe Situation Turns Ugly\n\nSome solutions were elegant and pretty cool. For example, the 9/4 matrix multiplication paper is 13 pages and bounds Strassen’s spectral characters directly. We’ve seen a bunch of people, after seeing their favorite problem get solved, share their ‘aha’ moment reading the proof.\n\nWendigo: Am I the only one a little freaked out by the fact that our neat little lower bounds, nlog(n) in this case, are just slightly wrong. The sublime beauty of math is looking like a weak approximation resulting from our limited human faculties.\n\nUubzu v4: I think the AI is just fucking with us when it takes an elegant bound like n(lg n) and improves it by 1/2^182. Like thanks for nothing buddy.\n\nAnyone can keep doing things like this. All you need is a Codex subscription.\n\nDoug Colkitt: We are publishing an update to OpenAI problem #109 (integer multiplication) with further tightening.\n\nκ = 2⁻³⁴ (tightened from κ = 2⁻¹8²)\n\nThe exact witness is 8.3 × 10⁻¹¹, a roughly 48 million fold improvement over the previous result and a 2¹⁴⁸ fold improvement over original OAI result.\n\nThe improvement came from removing the spacing penalty behind the quadratic bottleneck. This was done by moving compact control bits instead of entire windows.\n\nHere is the full statement by the leading advisory group:\n\nagmai.org: As announced a few weeks ago, OpenAI has released a large collection of mathematical results generated by an internal model, reporting solutions to hundreds of open questions. This is an important event for mathematics, with consequences both for mathematics and for the mathematical community that extend far beyond the individual results.\n\nAGMAI’s advisory role should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them. We do not speak on behalf of the entire mathematical community, and only the mathematical community can undertake the assessment that is needed.\n\nMaking this work public is a first step. This release is the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge. At the same time, the future of mathematical research cannot consist only of understanding results produced by AI labs. Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system’s capabilities. Equitable access to powerful research tools and adequate computational resources are essential to that freedom.\n\nWe reaffirm our published recommendations on responsible release. We have discussed them with OpenAI and appreciate the company’s willingness to engage. While we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully, and whether there are others we should suggest. We remain committed to engaging with any frontier AI lab on these questions and have already been in contact with several of them.\n\nAnother Mathematical Group Responds\n\nSome look at what is happening, and cry out ‘NO!’\n\nThey are going to need a much better plan than this statement.\n\nDo not be fooled into thinking this group, the ‘Association for Human Mathematics,’ is a big deal. There are hundreds of thousands of mathematics PhDs, and this group has 809 members. I include it because it perfectly embodies an attitude, nothing more.\n\nAriel: “Mathematicians did not ask for this work to be done” might become one of the historical quotes from this era, perfectly depicting the downfall of academia.\n\nEthan Mollick: This document is going to be an assigned reading in college classes that cover this moment in time, there’s a lot happening in a few paragraphs…\n\nYesterday, on October 6th, 2026, OpenAI – which is currently defending lawsuits against accusations of illegal plagiarism, copyright infringement, and trademark dilution – released a repository of manuscripts purporting to contain solutions to a number of high-profile problems in mathematics.\n\nMathematicians did not ask for this work to be done. The Advisory Group on Mathematics and Artificial Intelligence, from whom OpenAI has claimed to derive its legitimacy, opened their initial advisory statement by saying that frontier AI corporations should not test advanced mathematical problems on internal models. In ignoring the central premise of the Advisory Group’s position, OpenAI has indicated total disregard for the norms of scientific research — norms that guarantee that mathematics remains trustworthy, ethically researched, and in the public interest.\n\nMathematicians have a particular vision of progress that is informed by history and field-specific considerations. We reject OpenAI’s assertion that this release advances our subject, and we urge mathematicians and the public to view the value of this publication model with due skepticism.\n\nReleasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power. We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centers human understanding.\n\nAssociation for Human Mathematics Communications Working Group\n\nWhatever you think of the rest of the statement, they are not wrong that the entire exercise here was directly against what the mathematics advisory group (AGMAI) wanted. AGMAI’s central ask was that OpenAI was asked to not test hard problems on private internal models. That request, regardless of what you think of it, got fully ignored.\n\nA Different Approach\n\nOpenAI admits that ‘dump it all on GitHub’ is not exactly meeting the advisory committee’s guidelines. They are exploring better options, and are hoping in the future to also have better-written papers.\n\nOn top of the big drop there was also this, that came out one day earlier, where Anthropic gave their solution to mathematicians to write up before releasing it, attempting to follow the advisory group’s recommendations:\n\nNew math result from Anthropic, then written up by Josh Alman and Virginia Vassilevska Williams, that is kind of a big deal. Anthropic had Claude research open problems, found the result, then hired Alman and Vassilevska Williams to write the paper as per what the mathematician council asked AI companies to do.\n\nAbstract: We give the first polynomial improvements over the textbook algorithms for 3SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve 3SUM on n integers of polynomial size in O(n1.9992) time and APSP on directed n-vertex graphs with polynomially bounded integer weights in O(n2.9995) time.\n\nThis refutes the 3SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the 3SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight k-Clique hypotheses, and the three rectangular hinted Online Matrix–Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems.\n\nAtoosa Kasirzadeh: Could you write a tweet elaborating on what this could mean from your pov?\n\n𝖬𝖺𝗁𝖽𝗂 𝖢𝗁: In a nutshell, suppose for a whole bunch of animals we knew that if they could whistle, then pigs could fly. But now someone saw a flying pig. That means we can now infer nothing about those animals (also in a sense, “dynamic programming isn’t optimal”).\n\nSecond: I’m seeing a lot of complaints about the fact that Anthropic contracted with human experts to write up the paper – but to a first-order, isn’t this what the “Responsible Release” guidelines circulated last week say to do?\n\nIf Math wants to take the view that “AI labs have a responsibility to provide support, including funding, for the development of human understanding of the AI mathematical output that they release” then it’s going to have to become comfortable with these companies financing mathematicians to do just that.\n\nThere was also another stray new math result via GPT that came out right before the massive drop of other math.\n\nI love the idea of a website called Retraction Watch. This is one place OpenAI executed well. Mistakes happen, and even if more are found – and there will almost certainly be more errors found as the unverified 58% is still largely unvetted – the error rate is impressively low. The important thing is to own the mistakes once they are found. Retractions are often a sign you are doing something right, not something wrong.\n\nAlicia Gallegos (Retraction Watch): “I think the withdrawals and corrections will be viewed positively by the mathematics community, but it will take a lot more than that to earn back the trust they have lost,” Sutherland told us.\n\nExcept, are you sure? One angle is that you can either find a proof or you can find a bug in Lean, and are you sure the math will always be the easier option when you face the world’s toughest open problems?\n\nJai: It’s just a next breakthrough predictor. A stochastic genius. Nobel autocomplete.\n\nMichael Roe: My guess is that finding exploitable soundness bugs in Lean is substantially easier than proving quasi-Riemann hypothesis. Given which, we should expect some of these proofs to be Lean exploits. Still, I’m also prepared to believe they’re real.\n\n(as a computer security person, “AI can find exploitable bugs in Lean” is actually more alarming news than “AI proves the quasi-Riemann hypothesis”. Security bug apocalypse coming right up.)\n\nThe bigger risk is that Lean does not confirm that the stated theorem matches up to the famous open problem. Your terms might be different, in ways that are non-obvious at first glance.\n\nSo far, Lean is holding up, as are all the papers with Lean proofs modulo one corrected side claim, which is rather impressive. The system works. There are 3 proofs that have broken so far, and all of them are in the unformalized 58%.\n\nOf the nine results discussed up top, five are checked in Lean (quasi-RH, matrix multiplication, π, Unique Games and Hadwiger), as in the Lean-checked 42%.\n\nAt some point, you will give the model a task that is substantially harder than some other way to solve ‘the problem’ of you assigning the task, whether or not the alternative path involves taking over the world. Then you will be the one that has a problem.", "url": "https://wpnews.pro/news/new-math-from-openai", "canonical_source": "https://thezvi.wordpress.com/2026/10/09/new-math-from-openai/", "published_at": "2026-10-09 13:21:12+00:00", "updated_at": "2026-10-09 13:25:33.439761+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "large-language-models"], "entities": ["OpenAI", "Alex Kontorovich", "Isaac Kim", "Riemann hypothesis", "P vs NP", "Hodge conjecture", "Birch and Swinnerton-Dyer conjecture", "Hadwiger conjecture"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/new-math-from-openai", "markdown": "https://wpnews.pro/news/new-math-from-openai.md", "text": "https://wpnews.pro/news/new-math-from-openai.txt", "jsonld": "https://wpnews.pro/news/new-math-from-openai.jsonld"}}