33 comments

  • paxy 1 hour ago
    Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:

    > We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.

    1. Local smoothing and the Kakeya conjecture

    2. The Exponential Time Hypothesis is false

    3. Existence of quantum chromodynamics and completions of the Standard Model

    • abetusk 3 minutes ago
      ETH is false! I really would not have believed it.

      [0]:

      > We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$

      The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.

      [0] https://github.com/openai/math/blob/main/preprints/The-Expon...

    • Rudybega 1 hour ago
      To give some context:

      Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.

      ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.

      QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.

      • emil-lp 39 minutes ago
        I would guess ETH has been assumed in thousands of papers. I have several myself.

        On the positive side, suddenly there's a whole bunch of open problems!

      • dekhn 38 minutes ago
        Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).

        Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?

        • Kotlopou 26 minutes ago
          From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.

          ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?

          • redox99 7 minutes ago
            If we had really good understanding of physics, couldn't we find out how to make a room temperature superconductor (or prove it can't exist)?

            Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?

        • demibabs 30 minutes ago
          Solve p = np with an example algorithm maybe?
      • jlev1 52 minutes ago
        Thanks Claude!
        • Rudybega 17 minutes ago
          Glad to help (I only really understand the implications of ETH well enough to summarize myself without assistance haha). I figured if I wanted simplified context, others might too.
  • martinhath 36 minutes ago
    ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.

    Also glad to see that one of the authors of the paper stating it is also in the loop here.

    2026 is weird.

  • mindwok 57 minutes ago
    It is mind blowing that these long-standing conjectures and open questions are just falling by the dozen every other week. What a time to be alive.
    • __MatrixMan__ 54 minutes ago
      Chasing open questions, it seems to me, is worthwhile for the things learned along the way. Do you expect these results will be helpful, or is it just a sign that we need a new way to navigate uncharted spaces now that longstanding problems don't stand long enough for us to organize around?

      It feels a bit like having "visited" a place because you had a layover there.

      • cman1444 25 minutes ago
        I hope it's more like discovering islands of information which we can use as footholds to connect the dots between them.
      • brap 25 minutes ago
        If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.

        We should remember that math exists primarily to serve a purpose and that purpose is not being a fun puzzle hobby for a select few mathematicians.

        Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.

        I think if pro mathematicians don’t change their tune quickly they will soon see their funding approach zero.

        • Kotlopou 21 minutes ago
          I don't understand this dialogue, which seems to repeat with every result of this sort. Of course this is interesting! And of course the answer tells us something! And of course we also want to understand where it came from! None of this is in contradiction!
        • unknownian 8 minutes ago
          If you were a math grad student, would you rather your advisor be telling you to LLM maxx your ideas or working with you and giving you ideas on your thesis personally? I know my answer.
    • ChemSpider 32 minutes ago
      How long until we see similar announcements for (wet lab) chemistry, medicine or material science? Months? Years? Never?

      (Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)

      • Routine8502 11 minutes ago
        I'll give you an example from my field (nuclear fusion). I have a small team working on an idea which I think would really advance the field. Our initial concept paper was published in 2019. Progress on it has been slow because we don't have enough manufacturing capabilities, budget, or time on the facility. Will AI solve that? Don't get me wrong, I'm pretty bullish on AI, and it's been phenomenally useful so far (I definitely feel like a 10x physicist in the last 6 months), but I don't see a swarm of agents putting fusion energy on the grid tomorrow.

        Also, fwiw I have a personal test problem I give to every frontier model on release, which is to ask it to final a realatively obvious solution (in hindsight) to a minorly important open problem in my field. I know the answer, but havent gotten around to writing it up yet (it's really not that important). Every frontier model (so far) just regurgitates existing solutions.

        • marshray 0 minutes ago
          Definitely. It will let some smooth-talker with a vibe-coded website raise billions for "Nuclear Fusion AI". Then they can use AI to squeeze your team with lawsuits until you agree to sell cheap, and maybe keep you on as an employee to build it.
        • szvsw 1 minute ago
          How much effort do you put into prompting it? Are you asking it one shot with a simple prompt? Are you giving it to multiple agents and giving them a workspace to collaborate in? You might give it to a trusted colleague who does not know the solution along with an agreement to jointly publish, ask them to go through the process of solving it with agentic support, and then compare your solutions.
      • dannykwells 16 minutes ago
        Well you can’t verify a prediction about human biology without a phase 3 clinical trial so…
  • mlinsey 1 hour ago
    I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
    • varjag 38 minutes ago
      Imagine you're trying to prove that water is wet (P!=NP). You lack a formalism that could make sense at building a proof even though pretty much everyone agrees from empirical evidence that the water is indeed wet. A promising attempt at building one such formalism was just demonstrated by a model to be a dead end.
    • novaRom 57 minutes ago
      For me, this result is significant. It basically adds much more hope P is equal NP (that majority seems being against). I believe one day AI will explain a major flaw in modern mathematics, so a kind of correction in axioms will be suggested to prove Rieman's hypothesis (or a completely new apparatus)
      • _dain_ 2 minutes ago
        Why do you "hope" P=NP? It would be a catastrophe
      • rcpt 48 minutes ago
        I don't know if that's a "flaw". It's what Godel Incompleteness guarantees.

        Would be funny if the most famous problem turns out be an example though

        • skew-aberration 36 minutes ago
          That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
          • rcpt 5 minutes ago
            Godel says that there are statements that are true but have no proof within that system. It's not about algorithms.

            I'm saying it'd be funny if Riemann were true but not proveable.

            Though I guess it's actually funnier. If Riemann were false then there would be a counterexample, so it's clearly within existing axioms.

            So if one can prove that Riemann isn't proveable in our current axioms then it has to be true.

  • lukeplato 39 minutes ago
    > Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here

    absolutely insane

  • Daniel_Van_Zant 48 minutes ago
    Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.

    My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.

    • serioussecurity 34 minutes ago
      It's not just RL.

      Many of the harnesses used to search for solutions rely on quick/cheap correctness checks to discard incorrect hypotheses. With something like biology, you might need to run a wet lab experiment to reject an idea. This can add days per iteration, slowing progress down by a factor of 100 or more.

      The sad reality is that many computational approximations for biological phenomenon are relatively weak and lack the predictive power we need.

      • szvsw 8 minutes ago
        Is there a path where compute gets poured into developing, and then eventually using, computational models that supersede the shortcoming of the existing computational approximations you mention, which thus makes computational analysis and developments possible at rapid scale?

        In an area I am more familiar with though still not my domain: when the stiffness matrix and computational linear algebra methods collide, it unlocks a world of computational structural design and analysis by commoditizing the underlying mathematics. This is a simple example, but you get the idea.

        Obviously with CFD, mechanics, etc there is always massive work being done to try to unlock new computational methods, faster/parallelizable computational methods, differentiable ones, etc etc, but the general feeling is that the core physics are well modeled (NS millennium problem not withstanding) and are not lacking in predictive power for useful applications. I would have assumed that the same would be generally true for computational biology etc, so just curious to hear you elaborate more. It seems plausible to me that turning the brute force of “the models” (as in, swarms of AI compute) to work on/develop/solve/address whatever representational problems there are in the computational approximations/numerical methods you mention, particularly if that is a bottleneck to performing more extensive methodological/experimental exploration in silico to enable better exploitation/use of limited resources in wet lab.

        Again, I know nothing about computational biology, just stating my intuition and curious what your take is. Probably the limits of my knowledge can be demonstrated by the fact that the only modeling methods that come to mind as being vaguely relevant would be, like,… molecular dynamics, maybe discrete element method in a weird way but doubtful, and various pop science understandings of protein folding etc.

      • Frost1x 23 minutes ago
        I suspect the line forward is generating improved predictive models and implementations of those. It’s something LLMs can aide with, although they’ll have slower progress than domains that do already have robust, cheap validation checks. That’s probably why those domains are being targeted (they are the paths of least resistance to successes, which is why they’re being pursued).

        With that said part of the reason some domains may not have robust cheap verifiers is because they’re difficult to produce already, so it may be a bigger lift.

    • Creamsicle47 36 minutes ago
      It's because LLMs provide an efficient way of optimizing a complex, non-linear search space when there's a way to measure progress. The less verifiable something becomes, the harder it is to optimize (but not impossible).
    • rcpt 38 minutes ago
      It's the limitations for sure.

      AI has dominion over the digital world, when it can verify things in Lean it's able to crush problems like they're a game of Go.

      But when the real world and digital world don't quite connect AI is bottlenecked.

      • crabmusket 24 minutes ago
        Does this imply a forthcoming wet lab boom comparable to the data centre boom?
    • itchyjunk 36 minutes ago
    • kypro 20 minutes ago
      Knowledge is surprisingly transferable.

      Obviously it's easy to understand why having strong abilities in maths and coding can help with solving problems in physics. But it's also true of knowledge and mental models more broadly.

      For example, say somehow you managed to train a frontier model without any data about elephants. If you did this it would be relatively easy to later train elephants into the weights. You'd basically just need to train it about elephants at a very superficial level. After all, it it already knows what a mammal is. It knows what legs are. It knows what eyes are. It understands how it's digestive system works. All the things that makes the biology of elephants difficult doesn't need to be learnt from scratch.

      I assure you people on HN will be surprised at how quickly frontier models will become competent in domains that are harder to RL. They won't need anywhere near as much data as people currently believe.

      • Kotlopou 14 minutes ago
        Has anyone done an experiment like this? Do you have any papers to link to? This sounds quite important!
  • rbranson 2 hours ago
    The title buries the lede quite a bit:

    > The Exponential Time Hypothesis is false

  • nixonaddiction 9 minutes ago
    this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
  • rcr-anti 51 minutes ago
    Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
  • Kotlopou 35 minutes ago
    Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...

    (Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)

    • Kotlopou 16 minutes ago
      Here (https://nitter.meowing.monster/ElliotGlazer/status/210765080...), from October 7, maybe this will be relevant soon:

      > I occasionally talk to experts on all the MP's. Their rough summaries on what is needed to solve them usually go:

      > Hodge: not a clue

      > BSD: not a clue

      > RH: not a clue

      > P vs NP: not a clue

      > Yang-Mills: "Well..."

  • stabbles 22 minutes ago
    > In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach

    Mathematics and computer science are turning into an empirical science

    • dannykwells 15 minutes ago
      Except that lean verification means that you really need to just verify the lean problem statement right?
  • 3abiton 1 hour ago
    I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
  • measurablefunc 3 minutes ago
    Someone should take the sub-exponential proof & write something executable.
  • adverbly 39 minutes ago
    > In our view, these manuscripts are best understood as a form of raw research data.

    Much better!

    > For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.

    > All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.

    Very good!

    I was highly critical of their approach with the earlier mathematical publications, but this is much improved.

    Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.

    > We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.

    > We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.

    Well done!

    Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!

  • dooglius 1 hour ago
    The proof appears to be a sub-exponential algorithm for #3-SAT not just 3-SAT, I think this is considered a much stronger result?
    • geysersam 1 hour ago
      Isn't 3-SAT NP-complete?
      • krackers 59 minutes ago
        Yes. Up until now, it was unknown whether containment of NP in EXPTIME was strict or not. (Obviously you can brute-force all possible witnesses, so any problem in NP can be solved in EXPTIME, it was open whether you could do better than that.)

        Showing 3SAT (technically #3SAT) can be solved in less than EXPTIME resolves this. It's only slightly subexponential, not even quasipolynomial, but it is a very surprising result. And that too a constructive algorithm.

      • LPisGood 1 hour ago
        It is the canonical NP complete problem.
  • doctoboggan 38 minutes ago
    I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?

    [0]: PBS space time watcher

    • T-A 2 minutes ago
      The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.

      This part,

      The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model

      looks like a reference to SU(5) GUT from 1974:

      https://en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model

    • Kotlopou 33 minutes ago
      Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).
  • Insimwytim 28 minutes ago
    Wow, math PR really took over, nevermind blowback from those pesky mathematicians.

    Let's throw in more of that stuff, so they wouldn't look at the financials.

  • artninja1988 1 hour ago
    I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
  • aborsy 54 minutes ago
    Science is getting out of the reach of the majority of people. In particular, EU is out of it mostly.
  • metalsiliconYT 1 hour ago
    OpenAi needs to stop trying to please everyone. Just release the findings and give all the credit to your models/collaborators.
    • sebzim4500 1 hour ago
      They have released them, they are just also hiring mathematicians to clean them up in parallel.
      • T-A 55 minutes ago
        We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies: [...] For the physics result, form an external group of collaborators and defer to them on the release of materials. [...] The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
    • karmasimida 39 minutes ago
      Well, they are also not fully sure they got it right.
  • wilg 57 minutes ago
    In my opinion its okay for them to work on science and math problems and release the results.
    • ATMLOTTOBEER 18 minutes ago
      Thanks, I work at oai and will relay your approval to my line manager
  • jwpapi 1 hour ago
    Honestly, I’m a bit scared but also a bit excited.

    I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.

    The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.

    I honestly at this point don’t know anything about what could be possible and what couldnt.

    I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.

    • dekhn 34 minutes ago
      Nobody is solving colonization of planets in this lifetime unless we find a way to accelerate mass faster than the speed of light (and decelerate it at the other end) or extend the lifespan/radiation resistance or can preserve bodies in suspended animation for long periods. Even then, that just removed one of the many technical hurdles. We don't know which planets to target- are you going to send people to solar systems when you don't know if they are habitable? Modify people to be able to inhabit otherwise hostile worlds? How do you deal with boredom on a 1,000+ year trip?

      Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.

      • jwpapi 29 minutes ago
        How about sending robots to a habitable planet with biological material of humans, programmed to tell us the story of earth?

        Re life quality: Sure technical advancement can lead to cheaper electricity, better materials discovered, disease prevention and a lot of other stuff. A lot is hold back by complexity.

        I don’t want to make a call that this will happen and that it’s a net-positive for society. I’m just saying the technical advancements is interesting and enablin.

    • kypro 57 minutes ago
      > gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits.

      There are infinitely many more ways to kill and torture a human than to save one.

      • adverbly 26 minutes ago
        Wrong. There are infinite ways of doing both.
        • jwpapi 20 minutes ago
          I honestly think they are both finite lol.
          • adverbly 10 minutes ago
            Maybe.

            Depends on what counts as a different way I guess.

            And on if you're applying some kind of discrete physical model of the world or not...

  • Kotlopou 45 minutes ago
    "In our view, these manuscripts are best understood as a form of raw research data."

    They are staying on top of the exact formulations that spread like memes. I think this one was unfortunate. It's an analogy for a new type of thing that stops attempts to figure out what thing that is.

    That said, uh, let's go? Let's see if OpenAI earns some goodwill here. It probably can't go worse than the last math drop!

  • titzer 1 hour ago
    I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."

    The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.

    Treating every result from the AI like a gift from god is getting tiring.

    • sgillen 45 minutes ago
      These are not run of the mill results from AI, those are surely being posted to arxiv constantly by normal researchers without any fanfare.

      These are really big results from AI, if a human had come up with them we would also be making a really big deal out of them too (maybe not as big of a deal, part of the excitement is that this was from AI and more results of this magnitude may soon follow).

    • quantumspandex 52 minutes ago
      Science doesn't have to be a grind with AI.
  • hei-lima 1 hour ago
    Terrible title. I was hoping to see more interesting comments on the story, haha!
  • grd_dev 1 hour ago
    [flagged]
  • calf 1 hour ago
    In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."

    (Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)

  • mentalgear 1 hour ago
    "Working with PR to make us look less like cheating mind leeches".
  • esalman 1 hour ago
    Working with academic communities lol.

    I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.

    $5 million might sustain 5-10 PhD students for 5 years.

    Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.

    • quantumspandex 1 hour ago
      No one "forces" anyone to verify it.
      • esalman 3 minutes ago
        That means there's no crisis in mathematics after all, right?
      • LPisGood 58 minutes ago
        I said the same thing in another thread and someone compared it to no one, forcing you to own a smart phone. I will preempt that argument by saying the community is motivated to verify these results because they are important to the field.
  • 1ahsgt 1 hour ago
    I wonder how many prompts for proof approaches they stole for these. Nothing on how many tokens were burned.

    Proud to funnel money through the Simons Foundation, which is an AI booster itself and completely aligned with OpenAI.

    They make it very hard to enjoy a single result with the business and the politics being so dirty.