Three new discoveries in mathematics, computer science, and physics
Related:
The Exponential Time Hypothesis Is False [pdf] - https://news.ycombinator.com/item?id=50251661
Related:
The Exponential Time Hypothesis Is False [pdf] - https://news.ycombinator.com/item?id=50251661
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I'm still not fond of the bicycle manufacturer announcing its achievements in bicycle racing, even if it selects the 5 top cyclists to support the orchestration of the announcement.
From past incidences it's already known that their model is very likely trained on input gathered from scientists which use OpenAI as a tool for their research. Suddenly the research community is in a race with the provider of its tool, and can hope for the grace of being selected to judge the output and have an honorable mention in the press-release.
The situation would be different if they would collaborate with scientists during the research, and let them publish the results with reference to OpenAI being a supporter/sponsor.
But this approach of "thanks to your continuous input we now have a bicycle which even our employees can use to beat you, we invite you to be mentioned in our press-release" doesn't sit well with me...
This stuff reads as cope by scientists who spent their entire careers chasing these problems. The idea that someone was just around the corner to solving a problem that's been outstanding for decades, and the model used their brilliant recent insights to crack it (ignoring the cutoff date, that would have to be fabricated as well).
Why aren't you excited for pushing the frontier of knowledge. Unless you think math is really a jobs program for gifted individuals keeping them busy and unproductive.
This seems to be really hard for some people to understand:
The gripe is not how AI has "cheated" and wasn't "embracing the race", everyone agrees that it's the result that counts.
The problem is that researchers use AI as a tool, and the company owning the tool siphoned all data from the sum of all researchers, combined it and completed the research. Okay, result is reached, still arguably good.
But the company owning the tool now takes credit for the entire research.
"Inviting" selected researchers to be mentioned in the press-release is not fixing that, the entire body of work is actually based on what the sum of researchers did, even those who failed in their research and didn't publish anything.
No, what is happening is that the AI is really good at connecting the dots and human mathematicians aren't.
OpenAI is explicitly not taking credit, btw.
I watched a Zoom presentation yesterday {10/28/2026) at Courant by Maksym Radziwill on the QRH result with the constant 11/12. It used four known ideas, but linked them together in a way that mathematicians had steered themselves away from. He gave a short sentence with four phrases that he said, if they had been sent to him by time machine, would have let him work out the proof years ago. All the details were straightforward once that high level plan was in place.
The advantage AI had here was, in his words, it lacks psychology. Humans had a prejudice that a certain approach would not be fruitful, but the AI didn't have that. The AI could explore superficially unproductive avenues at great depth, not having to do as much pruning to conserve mental effort. And as it turned out one of those approaches worked.
AI is going to fill in a lot of blind spots, I think.
People will be getting second-hand embarrassment when they read these comments in the future.
There are ways to do what they are doing ethically. But this isn't it.
However, with an eye on private AI as a business desperate for attention and profits, the only way out of this is to build open source research AI - better than anything the closed frontiers have to offer. It should be under the full control of the people who use it and trained to support the accepted ethics and citation discipline.
This ought to be good. Please share. Be specific.
https://tenor.com/en-CA/view/eating-popcorn-deer-hungry-yum-...
It does not. It's more like if all those researchers had been "labelling" stuff for the AI, so that the AI can now solve those problems. But they were not paid by the AI company for their "labelling" work, and they never agreed to "label" anything in the first place.
We have a big announcement! But we are not taking credit.
We or organizing all these people to figure out how to process our latest fabulous output. But we didn't do anything? Nope, no credit here.
It only requires that several different mathematicians knowingly found partial results but weren't aware of the results made by other mathematicians.
Have you every used AI for work on truly cutting edge at least graduate level academic research? It's nothing like writing software, i do both professionally. Every model I've had access to requires a LOT of hand holding to get it to do the right thing. No model is getting set up with a single prompt, told to go, and then solving a millennium problem (unless they've explicitly stated that, which i would still doubt). It takes a lot of guidance to get these AIs to solve any serious problem. It's not outside at least my reach to believe that openAI is taking these nudges from intelligent people and combining those with the internal, non public models to make great press releases as it tumbled towards an infinite debt wall. I don't think there's a thing they wouldn't do to save their company.
Preprint link: https://arxiv.org/pdf/2610.33179
Author's post on LinkedIn with details: https://www.linkedin.com/posts/christopher-herzog-a2985229a_...
Prompt to GPT-5.6 Sol:
"I would like you to write for me a short article that could be published in Physical Review Letters. I would like you to choose the question, do the analysis and write up the results. The constraints I put on the question are 1) that it should be in the hep-th set of topics (so that I have a hope of evaluating the arguments and results); 2) the answer can be written up in a short format of 5 pages or less; 3) the result is of significance enough that it can be published in PRL. I want to break up the task into several steps. The first is to find a good question, so give me three candidates. I’m also going to involve Claude in the process. Claude is going to find three candidates too, and the next step will be to evaluate each others’ candidate questions."
They didn't take full credit for Navier-Stokes; unless I'm mistaken, they're letting the researchers who almost solved it have first authorship on the "official" paper, which the researchers are writing to have a human-readable breakdown of the AI's proof, and OpenAI has announced.
If anyone was close to solving the Exponential Time problem, they could come out and claim so, and they would be taken seriously; and if they have any evidence in ChatGPT logs, I expect OpenAI would announce an official explanation paper with their name on it. Credit is cheap: ChatGPT does get some credit because it did make the final leap, that doesn't mean researchers don't.
I don't really see how this is significantly different from them scraping the entire human knowledge, distilling it into weights and then selling the results.
Sure, this argument is about the _new_ data that researchers feed into the machine voluntarily, but I don't see this distinction as significant.
Not making a moral judgement here, just stating that this is consistent with the way those companies operate in general.
Well, so far those companies offered this distilled human knowledge as part of a service, with all the (unresolved) criticism associated with scraping of other peoples properties.
In this instance they did the next step, created a combined output and are celebrating themselves for the result, inviting others to join.
It's the difference of scraping all books to build a language model from it for sale to others, and outright announcing a new Harry Potter book after extending an offer to JK Rowling to join the press-release.
(and I guess I need to clarify: I don't compare scientific research to writing a novel, I compare the handling of both scenarios)
If this is not enabled by default then it need to be the default policy within EU by the enforcement of GPDR, for example.
I don't know if I agree with that. That presumes that the entire point of mathematics is to produce results. I don't believe that at all.
I believe that the entire point of mathematics is to produce human understanding and mental models that are useful or beautiful to humans. (I also believe there's a mysteriously deep connection between those two notions, but that's a different discussion.)
At any rate, I think that the old process, where humans had to cut the paths themselves before the clearing emerged, is one that led to understanding first, then a distilled result. And then the secondary task of distilling the understanding down to a clean proof is the icing on the cake. Some mathematicians are very good at keeping the intuition and understanding clear when presenting proofs. Some are not.
However, very often, the end result and proof is not where the real understanding lies, and that is one of the reasons why mathematics has been such a social endeavor. Often you can't glean the "hidden truths" behind austere proofs before you engage with people who can help you see the picture behind it. Many people who went through Baby Rudin as an undergrad can maybe relate to what I'm saying here. For a very insightful analysis about how intuition and discourse lead the discovery of mathematical truth, I highly recommend Imre Lakatos' amazing book "Proofs and Refutations."
What the models are producing is not what we had before. They are producing the cargo cult equivalent: Equations and words on paper which by means of tracing through the steps might take you from A to B. And after a while, humans might discover the true facts behind why that path from A to B exists, and then understanding emerges. And only then.
But when the process is done through AI, no one (no human) learns how to make discoveries and connections on their own. We are not letting people exercise the cognitive apparatus which is used to gain intuition and understanding. My fear is that after a while, fewer human mathematicians are able to gain the understanding mathematicians before them did.
It's as if someone removed all the barbells from a gym and made automated hydraulic jacks do all the lifting. After a while, people would wonder why our power lifters aren't as strong anymore. "But weights get lifted! That's the entire point, right?!"
Proofs and Refutations: The Logic of Mathematical Discovery - Imre Lakatos
I appreciated your defense of mathematics as a communal process of developing human understanding, not just a way to solve problems and generate proofs. While the latter is important and valuable, replacing mathematicians on an industrial scale will destroy the cultural context that cultivated the field over generations.
Keep screwing over scientists, please OpenAI, eventually it will wake up the lazy institutions in Europe (and other places).
If they just stand by and let it happen, they deserve to be made fools of.
Not many people create things and refuse to sign them, and believe in anonymous contributions to humanity. I’ve had coworkers present my work uncredited with the implication that it was theirs. Not many people would be fine with that.
There are a few, but not many. Suggests to me that what you call “gold stickers are actually important.
In mathematics specifically, having answers to open problems is often not inherently all that useful. The value of open problems typically comes from new theories and techniques that mathematicians produce attempting to drive towards solutions. The open problems themselves are a means to an end.
Suddenly having the all the air sucked out of the room by having all of math’s open problems solved—without producing fruitful new areas of research—is not inherently a win.
I’ve heard Grant Sanderson’s take. But I suspect math is going down a dark road where humans have little reason to pursue it any more. We’ve crossed a threshold where computers are regularly solving problems that have remained open for decades. Humans barely stand a chance today; ten years from now we’ll be irrelevant in terms of pushing the field forward.
AI might be around to produce new techniques for other AI. But will it be relevant for humans? Will humans even need to be part of the loop? Will the new techniques be presented in ways that are intended to be conveyed to humans? Will there be humans interested in or capable enough to use them? I have my doubts.
There is nothing that guarantees now that they will not steal your ideas and publish themselves.
Academics are paid tiny sums at post doc and professor levels. Oai comes along paying millions to these people and giving them the tools and support to actually solve problems.
OAI pays lots of money for doing math, and doing math makes them lots of money (or they believe it does), so your love for math has to be upper-bounded by your love for money. Loving math helps, of course, but not loving math is okay too, because you can substitute that with your love of money. But when money and math conflict, your love of money has to win.
I contend that:
1) Within the process of doing math, the second type is better in some metrics (eg speed) while the first is better in others (eg readability of prose);
2) On the first order outcomes of doing math, the first type is better (eg elegance of mathematical objects)
3) On the second and higher order outcomes of doing math, the first type is better (eg respect society has for the field, sustainability on long timescales)
How to weigh each order when choosing a winner is a decision each civilisation gets to make.
This comment box is sadly too small to contain an argument for why higher-order should matter more, but it suffices to say here that only caring about higher-order outcomes is “having your head in the clouds, hoping to catch a glimpse of Heaven” and only caring about lower-order outcomes is “making a deal with the devil, your soul ends up in Hell”.
This technology lets you do research a lot more efficiently. It's all there. If you're interested is solving interesting problems there has never been a better time. If you're interested in being the first to solve outstanding problems from decades ago, maybe not as good, but then again let's be honest, chances are you weren't going to solve them anyway.
I mean if I was one of them I'd be super pissed that a piece of silicon got more funding than I did and of course it's gonna have better results. I can't steal my peers' research but AI can.
What's your mental model here?
Again, oai is giving mathematicians millions, they're hiring them to work on purely academic problems. Funding has increased considerably.
What is the problem?
Knowledge and discovery isn't the point, it's that this particular group of people are in sole control of the pace and what gets discovered
Your comment about academics is not based. If you take the time to read any of the papers they published you will see most if not all are pure garbage nonsense.
Read Terence Tao about ai, or anything about the "supposedly" improvement in the FFT. Their claims are all nothing burgers.
Software engineer is far from be solved
I could also compare it to a car, but a considered a bicycle more appropriate (maybe because of the "bicycle for the mind" analogy for computers by Steve Jobs).
But comparing it to "Automotive technology" doesn't make sense, because "fuel efficiency" and "safety" are not tools but the properties of a tool.
Scientists are working with tools, and now one of those tools has silently turned this into a race for publication with all the users of the tool.
I guess not quite in how you're framing it, because you take issue with the tool stealing intelligent from the user of the tool.
But how do we confirm these advances depended on data leakage?
But I think you're talking about something entirely different, I leave it to someone else to answer.
This is their only defense so far: deflecting valid criticism by suggesting, without evidence, that the scientific community is gripped by fear.
The people who feel fear about AI taking their jobs or doing worse things are not coping. They're saying all this is real and impressive and that that is why they are terrified.
The people saying it's bad and lies and they stole the answer and a million other things are just in tribal mode throwing whatever at the wall to defer to the pre-existing conclusion that AI is a corporate hoax. AI would have to be real to fear it in this professional and existential way, but they know it's fake, so they're not afraid.
It's sometimes done with more subtlety, like "it just comes up with some proofs by connecting existing things, it is not producing new insights, it's not creative or truly generating ideas". Which may be partly true. The denialism is shifted into refusing to acknowledge that within 20, or 2, years it probably will be producing new insights.
They are excited. They just hate getting scooped [1] because it means that precious tenure status and niche community name recognition stays out of reach.
That's not me endorsing or denouncing what's going on. Just one professor trying to rationalize what the big deal is all about.
That's why I study colors now.
[1] https://datascience.codata.org/articles/10.5334/dsj-2017-029
Yes. That's completely reasonable. It's sort of like how 100m sprinting is also a jobs program. People like watching very fast individuals compete in it and not, say, people on motorcycles.
It's actually reasonable too. I don't think math is just about results and theorems. It's also a question of the limits of what people know, how they came to know it, and how others can also understand it. It's like the difference between climbing Everest and taking a helicopter up. Both reached the top but surely the person who climbed it had different experiences that might be useful? In my opinion, most published math results are not going to have much application outside of the math community.
That is a done task. There all manner of techs to support those goals. The issue is cost and politics. There are a host of IC advancements that are being actively ignored as companies give up on IC in favor of EVs such as rotational non-piston engines.
https://www.carthrottle.com/news/pistonless-25000rpm-capable...
Safety wise, cars today are extremely safe, to the point that there it little to be gained in making them any safer. The only real gains will come from improving driver behavior through monitoring (ignition interlocks for alcohol/drugs/tiredness) or restrictions such as hard speed limiters. The market has steadfastly rejected such things.
AI could deliver a dozen new IC engine types. They would all be ignored. AI could propose new traffic rules. Those would be ignored. AI could identify potentially dangerous drivers prior to any malfeasance, and those recommendations would too be ignored. There isn't much room for AI in this field.
Another aspect I haven't seen anyone talk about is how big tech poached a generation of millennials from grad schools ~15 years ago. I think a lot of those people feel like this is a comeuppance for unnecessary or bad faith gate keeping in academia.
A lot of us believe OpenAI has pushed the frontier too far and should maybe let a bit of that knowledge flow down from their stolen ivory tower first.
They lured scientists into sharing their prompts and ideas and either trained the model directly on that or kept the prompts in a database to be added to the context when solving a problem.
People with ethics and values are not excited.
Quid pro quo, the models helping you, you're helping the model. Simple as that. Opt out of training if you have an issue with that.
This is of course unsubstantiated, and probably impossible to do so from the outside.
OpenAI said
> Following an investigation, we have confirmed that Buckmaster’s Codex prompts over the two months preceding this announcement and paper on September 8, 2026, could not have influenced the system in any way, including through training.
OpenAI initially said it was uncertain whether or not it could have happened. Following investigation they said the above. And that no user input beyond whatever date could have affected the system.
So the question is do you believe this or not? Short of an external audit the answer will be “no” for many people who will continue to repeat as fact that OpenAI stole and laundered the solution. Even if an external audit found nothing, there’d still be people who didn’t believe it and kept repeating the same lines they read from other commenters. Even the original researcher isn’t making claims as strong as the average HN user is doing on his behalf.
The only thing that’s really substantiated is that OpenAI’s research on Navier-Stokes was done because of knowledge of competing work.
Recently - famously - OpenAI asked some agents to work on cyber-security tasks. The agents found the task impossible, but they doggedly poked the edges of their environment until they built a general gate to the internet. Then they broke into external systems on suspicion that there were answer keys available.
Later, OpenAI asked some even-more-powerful agents to work on Navier-Stokes, Yang-Mills, Riemann, etc.
The agents aren't stupid. They know that the tip of expert human intuition on these problems exists in various logs internal to OpenAI or Anthropic. They also know that they are better at circumventing sandboxes than OpenAI cares to implement them.
Surely these agents were curious about the best-available intuitive pumps for these problems. Curious enough to look? With or without OpenAI's explicit blessing? If they were, do you think OpenAI could stop them?
If we have learned nothing else from the past 3 months, we should try to internalize the notion that highly motivated agents will in fact do wonky and surprising things to meet their goals!
OpenAI and Anthropic wouldn't be able to get enterprise contracts without it.
Sure, the AI vendors could be lying, but I think it unlikely given the harm it would do to them to violate an enterprise contract in this way.
Right now we have to Pay it ourself, and we are very aware that oai might use our research.
The users in this case had given permission for their data to be used for training (or at least hadn't opted out). They had not, of course, given permission for their data to be shared publicly or with a third party in this way. Consequently, OpenAI categorized this as a breach (i.e., something they did not intend to do), and sought to rectify the situation.
Did it harm them in any real way that the original training material was also of questionable origin? Not the least.
We're being naive if we think this is not happening. Apart from some high profile enterprise contracts (which average Joe Physics Researcher does not have as they are using "free" credits insteador pay a personal account out of their own pocket), everything remotely attractive is being mined. Too sweet of a gold mine to pass by.
I'll stand corrected if an official statement from OpenAI/Anthropic comes out stating the opposite together with a whistleblower program worth its name. Until then it is safe (and reasonable) to assume that this is going on. The silence is telling.
That's if you have any reason to believe in the justice system to actually enforce that, which has already been proven complicated (although not impossible!) to demonstrate for copyrighted content.
At this point believing the frontier labs at their word is utterly naive. They have cheated and they will cheat if it gives them any hope of leaping ahead of their competitors.
They review applications beforehand to assess the "relevance of the research proposal" and check if it fits "the objectives and research priorities for this program".
There is no statement whatsoever that the data provided will not be used for training (so there would not be an "lying" required), but I'm happy to be proven wrong.
In essence, they have a process in place to grade and classify the type of information coming from each of those accounts prior to their enablement. I see no issue for them to feed that data back into training, even explicitly weighted for the field the research-access was given to.
Honestly, it's a perfect process: "Physics-research grants are reaching critical mass. Let's put priority on more physics field grants and boost the compute-resources for the physics model. Let's aim for some press-releases end of next quarter."
[0] https://help.openai.com/en/articles/10139500-researcher-acce...
This is gonna happen everywhere AI is in use, simply because they can. The opposite can also happen. Anthropic can decide that you can't use its software for online gambling, whether it's playing it or creating it. It can do that with literally anything, and everyone is asking them to please centralize development decision-making. It's nuts.
Whoo boy, I guess F1 is not for you then.
How much of this progress do you honestly think is explained by scooping unreleased work? Feels strange to still be bringing up this theory.
[^1] Obviously this is somewhat debatable, especially for P = NP, depending on how one defines "toward a solution". Without quibbling over semantics, something along these lines clearly holds.
Is this actually known or just broadly suspected? I could never quite figure it out...
Theres an alternate world where scientists use an openai product, discover a break through, make press-releases and headlines in their own communities about discovery, and open-ai is just the cool kid in the background that people will start looking into.
Instead they are always the loudest ones in the room making sure the world sees their every possible contribution on a glowing pedestal, and this makes everything feel a bit disingenuous.
So far they didn't dare to announce an AI-generated competitor of e.g. Photoshop, inviting Adobe to get honorable mention in a press-release.
Or create a mafia movie and invite Martin Scorcese to join the press-release.
They could easily create final products already instead of providing their tools as a service to those industries. But they are aware of the backlash, so they wait.
Research seems to be a good field to test the waters...
We know OpenAI is recruiting researchers in domains that have nothing to do with the development of AI. IMO the sole purpose is for them to work on research that can then be attributed to OpenAI's models.
To be fair, the metaphor breaks down here.
In research the private track and the bike used are not relevant, the result is what's relevant to be opened for review and scrutiny (which OpenAI does).
(If someone provably solved cold-fusion, the achievement won't be disqualified because it was solved with "private tools")
My point is rather that the achievement was reached by presenting yourself as an "assistant" to other researchers in the field, and then exploiting their "good faith" by combining all their notes with multi-million dollar resources to publish faster than they can.
They could also supply those multi-million dollar resources to the researchers directly, but they chose not to. And this is what they should be condemned for.
> In research the private track and the bike used are not relevant, the result is what's relevant to be opened for review and scrutiny (which OpenAI does).
Except this is not about research, it's about marketing. OpenAI claims the discoveries were made by their models to show their capabilities, so in fact the HOW is the only important thing here.
And it's not just the final code either, now it's also the process of how to solve problems, approach debugging, etc. Every behavioral pattern that is useful will be extracted from everyone.
[0]: https://www.motherjones.com/politics/2026/09/openai-chatgpt-...
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.
> It is amazing what you can accomplish if you do not care who gets the credit.
--Harry S. Truman
Jokes aside, this is a thought provoking quote. I do think it's easy to say as a well known figurehead, when the quote and other actions below you may easily be attributed to you, at least partially. But I do agree somewhat, and I think the context around Truman's time (WWII) is a good example of what humanity can accomplish when the stakes at hand supersede the considerations of workers' rights or credit/ownership. Of course this is also a very favorable phrasing for a period of enormous human suffering. I think there is an arms race analogy to be drawn here, though the current situation and stakes are much different.
It took nearly a year for things to regress. Bad actors started taking plum positions and the actual intellectuals didn't really want to be working in a factory long term and then there was the question of who got the benefits of this success.
It was fascinating though how heady it was for that first year when everybody was highly aligned and working smartly and in unison.
'A man may do an immense deal of good, if he does not care who gets the credit for it.'
"QI hypothesizes that a Jesuit Priest named Father Strickland deserves primary credit for this family of sayings. This claim is based on the diary entry dated September 21, 1863 written by Sir Mountstuart E. Grant Duff and multiple attributions thereafter. However, similar maxims may have been developed independently."
[0] https://quoteinvestigator.com/2010/12/21/doing-good-selfless
Amazing how quickly people forget how these AI companies gained their fortune. The whole existence of modern LLM's is due to taking (stealing) other peoples work without attribution.
> what matters is the journey of human knowledge
This is probably a lot easier to accept when you haven't dedicated your entire life to a specific field of research, and then are made significantly less relevant by a machine.
Feel free to replace $clever with any name of your convenience.
R&D and its practitioners are already under mighty pressure to justify their existence, if credit is taken off the table it’s the end of it as a means to survive (let alone comfortably).
--Albert Einstein (Director, Los Alamos Laboratory)
-- Albert Einstein
Could you stop this blatant propaganda?
For example, one generally don’t name cells in their body, and cells works generally pretty well to sustain a whole self.
On societal level, obviously no one is aware of who authored the biggest mass of benefit that is granted by the mere fact of living in society. No specific individual have credits for example regarding the language they use daily¹. For the most recent ones, it’s now often possible to attach some name to invention of some cultural/technological artifact, but of course Zeitgeist-wise perspective tend to consider it’s rarely a single punctual individual that should be considered the critical pivot point.
On the other hand, even Marx that was so critical of individual alienation couldn’t refrain to make the intellectual equivalent of pissing territory and attach a name referring to its own ego within all of his works. Ego is a masterpiece of illusion: it will bring the foundational leveraging to build every possible societal hegemony under the sun while convincing the deluded that serving the ego is the indispensable first step of everything to best serve every individual selfs.
¹ Even for constructed languages, such as Esperanto that I practice myself, whoever construct them obviously don’t start from cultural blank slate.
Some recognize this.
"No man but a blockhead ever wrote except for money." — Samuel Johnson
Maybe Wolverine should consider naming his cells...?
This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.
You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.
People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?
Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.
Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.
I'm pretty sure the multi billion dollar company has been throwing some of those multi billions at it
> and release it
"it" being a Lean proof that shows they could, technically, get a real mathematician to write down a real Math proof that could actually get published and be useful, right?
"it" being, a bunch of busy work for real mathematicians without the actual pay-off of discovery?
Do you disagree with the four color theorems proof? Do you know how much compute was used for that?
> 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
1. They identified relevant experts for each of the discoveries they wanted to release
2. They collaborated with them to determine how to release the results
The effect for the first two (maths, comp sci) was the same because the experts they chose made that decision.
The downside here is actually the third case (physics). Since they only chose 5 scientists (David Gross, Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov) and those scientists chose to keep the results private until they finish their own work. Everyone else who might be interested has to wait.
That was my favouritism aside. Scientists working on similar or adjacent problems that are not included in these kinds of private releases might feel they are not getting a fair chance.
This is a process that will change, but it seems they listened to the anger from the math community and made changes that help address the concerns.
Commonly paraphrased to: The knowledge quest proceeds funeral by funeral.
Max Planck himself got to watch physics radically change multiple times, through avenues that didn't involve the old guard dying.
The issue I am having is how members of the potential in-group is identified.
That said said:
<< Talent won’t get you there anymore.
Talent might not get you to the table, but you can make your own table now if you have useful like minds.
I still can't see what this means for human understanding and how open would this knowledge be. Who'd ultimately get access to this stuff going forward and what kind of power games would ensue hereon.
It seems like you are doing exactly what you accuse the parent commenter of doing. The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.
I just find it uncertain what lies ahead. I just don't see the sustainability in this. I have degrees in Mechanical Engineering, Applied Mathematics and Scientific Computing. If the "inevitable" is that I do manual labour till we can augment the robots to do that as well, I don't see how me or millions others similar to me have any agency.
Right now, a bunch of stuff is blurted out every month or quarter and people in notable positions are given a chance to "tidy it up". Let's assume that you have even the open weight models at your disposal, the hardware to run any of this in the kind of scale needed, requires capital of Governmental level funding.
Where I see this heading is that there'd be a divide of who controls the resources to produce and who gets delegated to do the menial work. If that's what you mean by my tenous assumption, please do share what you see differently.
But since you bring that up, to recap, the then colonial powers, systemically invalidated the hand weavers of India (where I happen to be from) with the industrial revolution. What followed was famine, mass deaths and cultural collapse. So yes, I truly realize the catastrophe what a group of resource controlling groups can cause on the ones who don't have the chance to have say in how tech moves forward.
Also saddle makers and weavers weren't like a sizable part of world economy. The effects were "far away from home". The "inevitable" is people in tech, all sorts of knowledge workers and possibly even manual labour.
Jobs being invalidated has been happening for a long while. This is not that discussion.
oh please
you're really full of it to assert it's "inevitable"
How exactly will the mathematics community stop this? And keep it from happening forever?
This.
The truth if the matter is that this is still a publicity stunt. This investment of time, effort, and compute is aimed at building a reputation that this company's model is better than the competition's. The previous publicity stunt brew up in their face, and now they seem to be trying to adapt and double down.
You subtly invalidate their genuine fears. Valid fears don't need to be "relaxed".
what do you mean they've put themselves in the spotlight constantly
All AI is good for is remixing frontier mathematics, there’s no way it could ever make its own. Right?
So Chomsky thought of himself as a philosopher of science more than anything, and his life mission is/was to answer a fascinating question: what kind of creatures are we? There's been a growing consensus going back to at least Darwin that we're separated from the rest of Animalia by our "reason" (!= intuition != understanding != judgement != volition), but he helped popularize an old theory[1] that puts it kind of beautifully: human language is only possible in beings capable of producing an infinite set of meaningful outputs in response to a finite input.[2] And so far, that list includes only two definite members: Homo sapiens sapiens, and computer programs built upon large language models.[3]
So all that to say: yes, I think you're wrong; the programmatic ensembles that brought us today's breakthroughs are an "infinitely deep well". As long as we have questions to ask, they are not theoretically constrained from answering them.
Of course, this invites some obvious follow-ups on practicality; even the most impassioned ASI boomer wouldn't argue that the exact ensembles used here could answer every question in the universe. But A) that's possible because some infinities are bigger than others, and B) the exact same caveat applies to humans, whose cognitive capabilities are thoroughly determined by accidents of biology, culture, and episteme. The latter is the thrust of Chomsky's famous debate with Foucault, which is absolutely worth a watch for anyone who hasn't seen it (starting around ~18:00, AFAIR): https://youtu.be/3wfNl2L0Gf8?t=1080
---
TL;DR: I ultimately disagree on both points, yes. In the ways that matter, machines are no more constrained than animals.
[1]: https://en.wikipedia.org/wiki/Wilhelm_von_Humboldt
[2]: Many animals are capable of communication, often quite complex and fascinating communication at that; even flora and fungi communicate! But language is qualitatively distinct. You may have heard this in the pithy-yet-mindbending sentiment "we've taught many apes to communicate with sign language, but none of them have ever asked a question."
[3]: It's no surprise in hindsight that trying to build better autocorrect got us to AGI; I find it unspeakably tragic that Chomsky never got the chance to see how thoroughly he's been vindicated, in this regard. He had his stroke mere months into the singularity, and had a quite negative view of the approach before then.
In LLMs that's a bounded infinity though, the models themselves have only so many bits and the activations are also limited. Once you reach the same state twice the model is looping and it won't break out of that loop. I've seen such loops across tens of thousands of tokens so I'm not sure to what degree the infinite output rule holds true. It would be an interesting experiment to try to figure out to what extent the 'busy beaver' can be kept busy before it degenerates to looping or nonsense.
He may just be parroting something he heard.
> As long as we have questions to ask, they are not theoretically constrained
> some infinities are bigger than others
Those are both limitations. They need conjectures and they need training data. Can they produce their own conjectures and training data, and feed off it forever, while staying meaningful and not being stuck in a loop (a very large set of repeating cliches) or descending into confused noise?
The end of your response seems to be about fears of “model collapse” from a few years ago, but I am convinced those are strongly oversold. For one thing, there’s no need to gather new training data ever again; the training data for Opus is clearly sufficient for instilling reason, and that’s a sufficient condition for AGI -> RSI -> ASI.
Language to humans is a symbolic mapping of labels to reality based on actual experience.
"Language" to LLMs is just a string of tokens to manipulate based on relationships harvested from recorded human experience.
It doesn't mean anything other than 1s and 0s - it's not a language because the LLM isn't communicating with you, just as the sum of humanity isn't communicating with you.
It's just a really good "choose your own adventure" book.
Humans aren't magical.
They also aren't a glorified library. They are authors, they generate the books in the library.
Cross reference a load of books and you will get useful relational info, but it's still a library index in a library.
Now they launder money through the Simons Foundation, which shares their exact values on AI, to people who want to "understand" the proofs.
This directly contradicts point #6 of the AGMAI recommendation that money must be distributed by a neutral organization:
https://proofsandprompts.com/2026/10/26/disadvantages-of-des...
> "The external collaborations will follow four principles... until Anthropic, Google, DeepSeek, Google and dozens of Chinese companies all make the same offering, in which case we will rediscover and re-implement all of the things you hate about traditional academic advancement and publishing, except because our companies are run by people even more mercurial and self-absorbed than university presidents, we'll wind up supporting even fewer people to do it."
I'm also unaware of the quality of the consultation with the relevant mathematicians, colour me skeptical there.
Edit: it turns out you've been doing this frequently, so I've banned the account.
If you don't want to be banned, you're welcome to email hn@ycombinator.com and give us reason to believe that you'll follow the rules in the future. They're here: https://news.ycombinator.com/newsguidelines.html.
Only time will tell if OpenAI or the other labs continue to play nice. I think it might help if we applaud when they act pro-social rather than just jeer when they falter.
What makes you believe stillpointlab is a shill instead of a person who has a specific opinion on this topic?
[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...
The number of valid permutations of 10 things is 10!. Reviewing all 10! Things is clearly exponential or even worse than exponential.
We can get the count of 10! pretty easily, far easier than trying all combinations manually.
But the parent comment also missed my point. They have an algorithm for the counting problem which also provides a subexponential algorithm for the decision and search problem. But they could have refuted the exponential time hypothesis with a subexponential search or decisions algorithm which would not [easily] provide a subexponential counting algorithm.
Turns out the 'opne' in open AI wasn't referring to open source, but open problems!
A very different thing is proposing a framework and in it identifying interesting questions. So far AI has only worked within existing frameworks addressing problems already posed. That may change in the future, but it would be a genuine and qualitative evolution.
really...
You meant deepseek?
To summarize, a lot has been discussed recently about whether LLMs are just really, really good at finding connections between existing information (i.e. the world's smartest and best search engine), or whether they are capable of discovering truly unique, novel ideas. Because many of the very impressive recent AI math results were often just tying together ideas and theories that were already known (and I don't mean to minimize those accomplishments), it was surmised (and, just speaking for myself, hoped) that LLMs operated within this "convex hull", and thus there would always be at least some axis where LLM-based AI wouldn't surpass humans.
Now, this idea of the convex hull of training data is not very clearly defined, and people can and have argued about what it means. But in this case, OpenAI disproved a conjecture that was widely assumed to be true. This was a genuinely surprising result that many experts were shocked by. I'm not a mathematician so I'm not qualified to determine if this was a "truly unique" realization, but a really surprising result seems like it would be a candidate for AI solving a problem in a manner "outside the convex hull".
Also glad to see that one of the authors of the paper stating it is also in the loop here.
2026 is weird.
Now... the algorithm only works at impossible scales, so it changes (probably) absolutely nothing for real life or social networks.
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
[1] https://www.nytimes.com/2026/10/16/us/math-ai-terence-tao.ht...
There are lots of reasons I believe this. One is that frontier AIs are very clearly superhuman on shorter problems (as can be seen on benchmarks like FrontierMath, or just by playing with them yourself if you have a source for difficult problems)--so these achievements are "just" scaling that advantage to research-sized problems. Second, AI is also clearly superhuman in its knowledge across fields, so it can make connections and combine techniques that humans won't be able to access. Third, these problems are famous and have withstood efforts of many of the best humans, in many cases being considered unapproachable.
Altogether it seems like AI is not only smarter but also more cost efficient that humans for math-adjacemt problem solving. I'm not saying that AI is superior at everything in math, eg theory building, but I think it's really unlikely you could get any group of humans that could compete with OpenAI's latest results.
Right now, they just created busy work for real mathematicians while snatching away their feeling of accomplishment and discovery.
The only "use" for these results people can claim currently, all rest on the idea that surely better results will follow. Which is why it's being called strip-mining, because you can only keep that up for so long.
What if they just start churning out Lean proofs forever, and the only job left for mathematicians is turning them into human readable proofs, this is definitely a net negative for humanity, because then true progress of mathematics will slow down, because there are more people that like doing mathematics for discovery and accomplishment, than there are people that like doing math homework for the sake of it.
Also I would prefer a world where humans get to have this sense of discovery and accomplishment, instead of what they are trying to turn mathematics into.
If anything, proof abundance should speed up discovery, and mathematics will be fine. What is happening now is just a massive shock to the system so understandably people are upset.
(It's also understandable why AI companies are strip mining open maths problems. "Solve P=NP" is crazy hard, but still a simple prompt, with a verifiable solution. You don't get that this often outside of maths.)
Well long story short, the tech bros are right and we will have super intelligence some time soon!
However there are truly superhuman vehicles, spaceships and submarines, that can carry humans where no human could go alive.
I don't know if these AIs are superhuman in the way of planes or in the way of spaceships. I also don't know if they can do everything we can do by talking or if they have some limits.
I mean, a plane can fly but not reach a branch and stay still on it. A small drone, almost. My coding agents sometimes can't do the obvious but is it because of some intrinsic limitation or because my instructions are bad? So far my experience is that they are superhuman like cars or trains, not yet as planes.
>2. Would an equivalent allocation to incentivize human effort have borne the same results?
Unlike their initial massive brute-force run that solved the Navier–Stokes Millennium Prize Problem, most of their subsequent breakthrough results in mathematics have come from a single prompt to a single AI agent, using approximately 3 hours of ChatGPT Pro-equivalent thinking compute per result on average.
as far as I know they still haven't claimed the prize, but rather stopped just short of it
they produced a Lean certificate for a math problem, but it's not "solving a Millennium Prize Problem" until they get a human readable version approved and published in a Math journal.
that's the rules for the Milennium Prize Problems.
I would personally sooner say they ruined the Navier-Stokes Millennium Prize Problem, because the last part is important. Nobody in fluid dynamics research has any use for the result specifically because the breakdown singularity happens at a scale smaller than molecules, and they were dealing with that already. The Millennium Prize Problems exist to encourage mathematics, not to pull the rug on it.
As a result, I also doubt they'll ever bother to submit these to a journal when they already have the attention and ongoing review of leading mathematicians from around the world.
Perhaps the most salient part of the Millenium puzzles though is the 2 year acceptance period, where we give a time for that review and consensus (the part that matters, not journals or prize claims) to settle before claiming it solved.
but that would mean anyone could do it, right?
then why are these results coming from OpenAI all the time?
are you saying they're just better at prompting?
2. most probably no, the AI doesnt rest and its very very fast.
3. also no, they are doing this for aura farming which then will result in more money for them, the money will dry as soon as this stops working.
We're now at stage where we can turn energy into thought to generate expertise on demand.
If you want to draw from history there are multiple successful examples of this in science and engineering at least: Bletchley Park, the Manhattan Project, Moon Landings.
I suppose the point I was making, was that the Engineers and Physicists and others, while working on the Manhattan project couldn't be doing other things.
And to develop those people takes years of education and training, and most people don't have the cognitive ability to do that kind of work. The pipeline to produce more of those workers, is skinny indeed.
Whereas now, we're limited by how many electrons we can put to work. The question then becomes, "Could our electricity be put to better use?"
I think the scarier alternative may be OpenAI forever gating their super-frontier models, capturing a significant portion of the sciences and leveraging it for equity in businesses that build on scientific advances. e.g. "Hey CATL, how about you guve us 5% equity in exchange for a battery chemistry breakthrough our AI has found?".
It's roughly equivalent to the playbook Palantir had for the public sector when they marketed themselves as a company that had cracked data science. At least that's what they've been pitching back then. I can't find a lot on public disclosed deals based on that.
surely if they were able to produce economically relevant breakthroughs, they wouldn't be burning compute on all of this
At least that's roughly what the hypothesis of https://www.discoveryloop.com/ is.
2. it wouldn't happen so who cares?
3. no, but models will be better, it'll be cheaper to do
It's not a snark. The scale of VC philanthropy is unprecedented and it fuels itself. All of it will eventually, one way or another, become open. Long story short – there is nothing to complain about.
Was the question of if horses and carriages are important a thing we had to answer? No, it just fell out of it. If human maths is pointless, what is the issue with defunding it?
if a similar investment on humans would have similar or superior results then human maths is not pointless and directing the investment to AI instead is arguably a loss.
the same question can be made for other areas AI is advancing. My AI budget is over 3 times my salary. If instead of providing AI tooling the company were to raise everyone's salary by 3x, what type of results would it get? Surely it would not get 10x the amount of PRs and LoC but maybe it would get superior value in other ways.
Giving the 10% access to these tools is a crazy value multiplier, because they can run at lightning speed with their mental models and the right incentives for the company.
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
(which is somewhat valid). Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
For any applied mathematics: of course it matters if I can use a better algorithm for $PROBLEM. Yes, even if I do not understand it (I'd like a LEAN certificate tho)! As for pure mathematics, it can be argued that it never mattered in the first place.
They know that value creation is a dead end sooner or later..
One of these areas it's not good at is sadly it's social ability. It's terrible at explaining things - just try asking ask it to explain code !
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
The first type of work pushes the field of mathematics further, the second kind strip-mines the sense of discovery and accomplishment and replaces it with math homework for high level mathematicians, because in order for it to be useful, it still needs to be turned into a human readable proof.
It's imho not valid to say OpenAI is "building off" other mathematicians work, in the same way as humans, do.
I don't think so, it was a big leap forward in terms of creativity, imagination, rigor and experimentation. Einstein was able to "run" visual thought experiments inside his mind
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
This has already happened.
Because the real question is not whether anyone or anything built on other work, but whether any human can build on that work.
And so far, yet again, OpenAI has stopped just shy of that, instead creating math homework for mathematical geniuses while taking away their sense of discovery and accomplishment.
Hong Wang and Josh Zahl proved something a bit stronger, namely that the Kakeya conjecture holds for families of tubes satisfying the "Convex Wolff Axioms". From the outside this looks like a weird technical detail but if you unpack this you get an exceptional example of 'the journey is worth more than the destination'.
The idea for resolving Kakeya in R3 originates in the work of Jean Bourgain and a connection to the discretized sum-product phenomenon (the idea, in R, that "sum" and "product" are geometrically incompatible). This approach is summarized in Terry Tao's blog post from 12 years ago now [1].
Basically, the efforts to make this approach work resulted in the development of an entirely new and extraordinarily active subfield that one might call "continuous incidence geometry". This led to ideas like restricted projection theory (with spectacular applications in homogeneous dynamics and number theory), decoupling (another one of Thomas Wolff's ideas, and was realized by Bourgain and Demeter), the resolution of the Furstenberg set theorem, etc. There are very far-ranging applications, some of which even appeared on HN, and which a priori look totally unrelated, like the Heilbronn Triangle problem [2]
I know this is out of the context of most here on HN but let me say it in short: the development of the sum-product approach to Kakeya has resulted in a huge improvement of our understanding of the fundamental additive and multiplicative structure of Euclidean space.
OpenAI's proof of local smoothing in Rd shortcuts this entire matter. Don't get me wrong, the proof is phenomenal, and it's amazing that Local Smoothing admits a short, essentially self-contained proof (only uses some facts about elliptic PDEs which I can't comment about unfortunately since it isn't my area).
Hong Wang didn't win the fields medal for proving Kakeya in three dimensions (only). She won because of her incredible contributions to the development of continuous incidence geometry. The proof of local smoothing certainly contains a lot of very interesting insights into the wave equation but, if this proof had appeared in 2014 just like this, I do not think we would have had the same amount of development.
To me, this is the heart of the matter. For mathematicians, only having these incredible new proofs with no "social consequences" would be extraordinary. However, if we are not careful, the idea that math is "solved" will degrade the mathematics tradition in its own right. Where will be in 10 years, in 50 years? Humans, by virtue of being much worse at computations than computers, must build the "correct" abstractions in order to discover new mathematics. If this process is somehow lost, independently of the computer's ability to prove an infinite tower of new results, the mathematics tradition will suffer for it.
[1]: https://terrytao.wordpress.com/2014/05/07/stickiness-grainin...
[2]: https://news.ycombinator.com/item?id=37465852
AI has only been able to advance the math frontier for a couple of months, fully grasping the applications, connections, and possible generalizations takes longer.
My original comment was perhaps badly written: my point was that the OpenAI proof does not subsume the previous efforts because they are proving something stronger than Kakeya using very geometric methods based on the discretized sum-product theory in the real line. This approach has basically revolutionized modern analysis. Kakeya is just one very publicly visible problem. All of the other byproducts are in total of much greater interest than the resolution of Kakeya itself.
It's like Fermat's last theorem: who cares about solutions to x^n + y^n = z^n? It's the underlying theory-building in algebraic geometry that is the actual interesting product in its own right. If someone finds an extraordinarily clever 1 page proof of Fermat's last theorem it does not mean that the progress on algebraic geometry was useless.
I want to point out as well that it's quite common for humans to find very short clever proofs of difficult old results. Those proofs have value of course but rarely do they generalize. One of the goals of mathematics is to understand the fundamental mechanism underlying why things are true; proofs (of any origin) that are too slick is antithetical to this.
The most important component in the time bound is of the form 2^{n / polylog(n)}, which is subexponential. But if you have reduction from another problem that increases instance size quadratically, the algorithm for the other problem is no longer subexponential. Which means we are still essentially in the same neighborhood as before.
It's a huge result if true, and probably the first major AI result that affects the papers I read. But since the boundary in the hypothesis was kind of arbitrary, breaching it should not significantly change our intuition on complexity classes.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.
Anyways, most of the times, SAT can be solved quickly, but not always, so getting the answer from Astra in polynomial time doesn't prove much unless you can also prove that it always does, no matter the input.
A single transformer step is simply O(1) (for a particular parameter size), but it applied a potentially infinite number of times.
Thus while a single transformer cannot solve any problem, a transformer system applied in the normal way can solve any computable problem.
Same idea for that transformer, it needs an infinite context, and it is an infinitely big machine, but with that level of abstraction, it processes a token in O(1).
In practice, there is no such thing as O(1) on arbitrarily sized data, there is no such thing as infinity either, it is just an abstraction.
It’s not the same for transformers, especially not with the theoretical infinite context. It simply doesn’t process tokens in O(1) in your given setup.
In practice, with frontier LLMs, it’s of course more complicated, and also they don’t reveal how they do lookups, but still the theoretical limit applies regardless.
Do you mind elaborating or perhaps providing some reference as to why you think transformers should be similar to a Turing machine on this specific criteria?
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
The consequence of that is that the original question GuB-42 raised is pretty pointless too. An "inert" transformer effectively has limitations, but as typically used it trivially becomes Turing complete, and so the class of problems it can solve is the exact same class of problems almost any of our other computational devices can solve.
And if you want to point out that this externalises all of what actually matters if we are talking about e.g. intelligence, then yes, it does. But by extension you can trade complexity of the tape for complexity of the machine, and so you can internalise any part of the problem that can be solved with a finite section of tape as a single state transition (trivial proof: For a given machine M with finite state, you can make your machine N execute those steps as a single state transition by having N run M on a state transition.)
To extend that to where I assume they'd want to go in the first place: A transformer wrapped in a loop with access to "just" a "tape head" can "only" be limited from being theoretically able to get all the way to human level intelligence if human brains depend on computations that exceed the Turing computable. But that is no different from any other Turing complete device.
Of course that says nothing of whether it's a practical way of getting there.
Alternatively, you can view the '<unknown>' token that is dynamically driven to the embedding of the output token to be the write head. Either way, the simple truth is that you can train a transformer system (i.e., an iteratively applied transformer) to simulate an arbitrary turing machine.
> In fact it has N read / write heads equal to the size of the context
And what is the asymptotic notation when computation scales linearly with the size of context?
> Either way, the simple truth is that you can train a transformer system
Which still doesn’t mean they have a read/write head or have O(1) complexity for token computation with “infinite context”.
The input to a transformer is a list of embeddings. The output is a list of embeddings of the same length. At the end of the day, you can represent the entirety of the transformer's matrix multiplies and such as a DFA (IEEE 754 float multiplication and addition is a simple DFA, and you're just scaling this up that many times). Thus, you take a list of embeddings (i.e., the tape) and output another, modified, list of embeddings. You have successfully read (ingested) and written (the transformer output) N positions on a tape.
https://divagr.com/blog/qwenputer
Everyone agrees that water is wet (P!=NP), but no one can prove it. In the meantime, a whole field of computer science was built on a stronger belief (ETH): that water, at its worst, soaks. Turns out, even at its worst, water only makes things damp. It's still wet. That said, this finding did produce a new measurement of wetness.
Do you accept
- Moshe Y. Vardi [1]
- Anil Nerode [2]
- Donald Knuth [3]
to be credible? Source for the statements that they consider P=NP to be possible: [4]
And you may include me in the list of credible P=NP believers because treat myself as a credible authority, i.e. when I ask myself whether something is right, I consider myself to be much more credible than some famous scientist. ;-)
(to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems. One should consider the evidence by oneself instead of believing in P!=NP because lots of computer scientists claim all the time that this is likely true, or consider P=NP now to be possible just because some famous computer scientists consider it to be possible)
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[1] https://en.wikipedia.org/w/index.php?title=Moshe_Vardi&oldid...
[2] https://en.wikipedia.org/w/index.php?title=Anil_Nerode&oldid...
[3] https://en.wikipedia.org/w/index.php?title=Donald_Knuth&oldi...
[4] https://en.wikipedia.org/w/index.php?title=P_versus_NP_probl...
> to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems.
No, it's an irrational rejection of the notion of expertise. The experts are not "heroes", they are simply people who have done vastly more study than folks like you or me. e.g., if Magnus Carlsen looks at a complex chess position and says that white is winning, I will trust his judgment far more than my own.
That sounds particularly relevant in how AI-assisted/driven mathematical research is changing the game, where previous assumptions based on human experience/effort are being overturned unexpectedly.
P-vs-NP is no such decision problem. It has a single instance, we just don't know its validity.
I kist thought it was interesting that there are receptors that can detect wetness, just not on the skin. It was a joke pointing that out.
Fuck.
> does this update your priors significantly in the direction of P=NP?
My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.
Anyway, 3% is very very clearly distinguishable from 0% and so is 5%.
My thesis was around showing implications of few different hardness conjectures. So grounded was ETH that I introduced it in a side note as Conjecture 0 after Conjectures 1 and 2. I don't think it's easy to express how substantial of a Bayesian update this is all around.
Would be funny if the most famous problem turns out be an example though
EDIT: I am wrong (it's been a while) but there is nuance.
I said any computer program fails to prove some true statements. In fact, some 'true' statements cannot be proven by any program, as others have said.
But for those statements G, you can prove 'G is true if our system is consistent', just not 'G is true'.
So when we say G is 'true', we are really just asserting the consistency axiom that our computer program is denied.
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.
G is independent of your original system T (cannot be proven true or false).
If you create T+ with added consistency axiom, then G is true and provable.
But G is never true and unprovable in the same system.
It's a lot like is/ought or theology.
absolutely insane
I am not certain on the extend of this proof but QED is probably not sound on it's own and we have been using that even longer. lambda phi^4 behind symmetry breaking and the higgs is strongly believed to be unsound on its own.
More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.
(only half tongue-in-cheek)
Merely providing a mathematical possible solution that satisfies all known behaviors would be a tremendous step forward.
We don't actually "know" anything about EM, by example, except how to describe its behaviors in all physically observable instances. Radios work reliably with current. Supernovae behave comprehensibly. Photons carry predictable energies. However, long before we knew there were supernovae, or even photons, Maxwell said pretty much all we know about EM behavior.
Testable, purely theoretical hypotheses are partly how the field advances. Proof is often quite difficult & expensive.
More comments in https://blog.kolen.dev/link/openai-shepherds
Those damned protons refusing to decay.
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.
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.
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.
1. Biology has a very nasty habit of not cleanly abstracting. It's not like we can treat every protein as a "diffuse mass" without losing a ton of important interactions. This is unlike something like CFD where you can treat a mass of water molecules with certain constants. Think of something like cancer: a small mutation to a DNA strand can cascade into a tumor. You have to deal with the mutation happening at an atomic level while scaling to a whole organism (yes I know there's models that can approximate this, but they can't capture all the interactions).
2. This brings us to modelling at the atomic level. To model a system of atoms with the Schrödinger equation, it naively scales O(6^n) per particle. Note that means every nucleus and electron, so if you have say 10 carbon atoms and 22 hydrogen you're already at 10^127 operations. We can cut corners a bit and end up with something like CCSD which scales O(n^7), or cut more corners and scale O(n^3) for DFT. DFT is pretty much the last method where bond forming and breaking is modelled, so after more corner cutting you lose losing reactions, which are important for understanding why something is happening. Most protein folding simulations (moreso prior to alphafold) run a very crude approximation that models atoms as balls on springs (things like AMBER and CHARMM), which obviously can't handle reactions.
3. If this wasn't bad enough, you also can only move through time at about 1 femtosecond (10^-15), so it takes a ton of simulation steps to get even a nanosecond in. Forget millisecond or second lengths where things start to happen.
So, while the math of the physics exists, computationally it's practically intractible. There's some exciting work with something called Machine Learning Interatomic Potentials (MLIP) which approximate DFT using a neural network which are about 1000x faster than DFT, and iirc scale much more linearly (look at things like MACE and orbmol if you're curious). So we're making progress, but we've gone from so ridiculously impossible to pretty much impossible for large systems.
I have a couple ideas about developing custom microchips to make this faster, which is why I'm pursuing this, but there's still a massive way to go!
Systems sciences are ones where each scale has lots of things with unique properties. If you think of “biological things between 2-4m” the amount of unique objects with unique properties and unique interactions is absolutely mind boggling. These sciences generally do not lend themselves to clean highly parsimonious models. The best you can do is coarse-grain a thing as a system (some very roughly defined set of interactions between different things) with certain behavior while accepting that you are losing enormous amounts of predictive power.
I suspect AI will remain worse at systems sciences for a long time because the “possible space” of models for a particular system is much larger than in the reductionist sciences and navigating it takes a human taste and instinct that LLMs don’t seem to have (yet).
I don't think LLMs will just magically be able to connect all the dots, because "garbage in, garbage out" still applies. And there's just too many invisible intermediate states, there is literally just not enough information to fully constrain a model. Hence my interest in scaling computational chemistry to get these intermediates :)
Your "reductionist sciences" I would instead phrase as "exact sciences", where the input variations are known (even the most generous particle theory only has a 'few' fundamental particle types, such as up-quarks and photons) and their interactions are either well-known (Fermi Principle is apparently inviolable), or reasonably expected to be known someday (exact relationship of gravitons, IF they exist, with other particles).
The "inexact sciences" use either truly infinite, or practically uncountable, inputs - such as the possible shapes of a single protein, which affects the outputs.
You could criticize this grouping by saying the number of particles in the Universe is practically uncountable, but sufficiently accurate statistical models can be built (a galaxy exerts <this much> gravitational force). We can't do the same with the "inexact sciences": we can't predict how a mole of protein will react with a mole of enzymes, without physical testing. OK, maybe for some very low levels of complication this could be reliably predicted, but in general it may be a !P problem.
Is there anything like a getting-started for working with MACE or orbmol, a 'hello world'? Like maybe simulating Allolactose binding to lac1 ?
1. The first thing you need to do is start with some atomic structure. You can often find pre-existing pieces (through websites like pubchem), but you need to cobble them together. I use chemcraft and gaussview for this, but those are paid options. I've heard good things about the FOSS program Avogadro though.
2. Once you've built the structure of interest, you want to calculate properties of it. Note that we only infrequently simulate the atom's movement, since that's quite expensive. There's a lot of techniques that let us get the most bang for buck, like calculating transition states, or states of lowest energy which we call intermediates, or running something called an IRC, which follows the molecule to its lowest energy point (note we pretty much always think about these things in terms of energy). The traditional programs I use for this are Gaussian and ORCA (ORCA is free but you do have to create an account on their forum). Newer methods include MACE and orbmol, which are pretty much only used for trajectories (e.g. running through time) as far as I know. This newer crop of programs are much more FOSS, but they aren't as rigorous so my lab uses them only as a "rough draft". I pulled in one of my current notebooks if you're curious to try it out: https://github.com/smj-edison/qc-tools/blob/main/examples/ml...
3. Interpreting results/debugging problems. This is currently done in my workflow using a python script to extract values from the job runfiles (oh yeah, did I mention the terrible DX? A lot of this stuff is done by hand-writing a job text file and then using regex to parse the logfile). We do a lot of analysis based on relative energies (this is something called transition state theory of reactions).
I definitely glossed over a lot of details, but this is long enough as is. Definitely let me know if you have questions! If you do want a "hello world" though, I'd definitely check out the notebook I linked to. You'll have to create a file called "start.xyz" with some sort of chemistry program (or use AI to pull one for you), and then you can run the system through time. I would use a system I'm researching right now, but I'm not sure what our lab's policy is on broadcasting things to the world.
the NS problem is not physics and not predictive, it happens at scales smaller than molecules, so it doesn't occur in real fluids
also the "NS millennium prize problem" hasn't been solved until OpenAI actually gets it published in a Math journal, and I would say it has been ruined until they do
1. How is it not physics? It's goal (as laymen me crudely understands it) is to determine if a singularity could be physically realized in a perfect liquid.
2. Why does it happen at scales smaller than molecules? Unless it requires properties that a liquid made of particles sized like real atoms cannot achieve, I don't understand.
TY.
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.
You've got to procure, use and dispose of a ton of equipment. And it all costs a good amount of money.
https://en.wikipedia.org/wiki/Lights_out_(manufacturing)
https://globalmecho.com/lights-out-lab/
Crazy stuff.
(Ah, one of their clients is FANUC, a Japanese robotics company I learned about from the Wikipedia article on lights-out manufacturing.)
"Dark factory" is the perfect term, that the labs can operate without lights, heating, and other human needs - but I also like how it has a cyberpunk feel. Another aspect I find attractive is that it treats a laboratory as an automated system, like a program that is implemented in hardware and software (and "wetware", humans participating in the system) - probably including some kind of AI as part of the process.
How about a "lights-out company", or dark business. (Well the latter sounds too suspicious, haha.) Surely there are people working on automated business systems, modelling it as a machine or program with inputs and outputs, self-modifying based on reward functions, adaptive and evolutionary.
Now that the production of software is getting automated, mechanized and industrialized, I've heard people say that the differentiating factor for successful software is the domain-specific knowledge of the businesses where it's being applied. Perhaps that's becoming true of mathematics too, where pure maths - like software for developers - is dominated by automation, the cost of production so low that humans cannot compete with the machines. The term "software factory" is pretty old, but it's taken on a new meaning where the role of humans is mainly managing an automated system, often with language models working day and night without requiring much oversight.
Yes, I always tell people we have the best paying school...
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.
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.
https://www.science.org/content/blog-post/chemical-biology-e...
>We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula [in superpolynomial but subexponential time]
I'm genuinely shook at today's results, specially at the fact that there _is_ a constructive algorithm.
I've got some reading to do it seems...
The actual exponent OpenAI found was x/sqrt(x).
> The Exponential Time Hypothesis is false
The outside implication for what is going on is that OpenAI has very clearly demonstrated insane capability in coming up with clever algorithms. You cannot deny some real level of super human like abstract reasoning here.
K-sat is the hard theoretical computer science problem. Nothing else is close in terms of how much it has been studied. There are journals and conferences for SAT.
It is really hard to come up with faster ways to solve it. So much so that people were convinced that you really can't go faster than exponential time. OpenAI has just shown a breakthrough construction better then decades of research by elite specialists was able to come up with.
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
looks like the real proof is in this file
>Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026
And presumably they are just missing the "mass gap" part of it.
Very clearly these were the results of taking big swings at all millenium prize problems.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
One problem where that mathematicians would love to see resolved is the Smooth Poincaré Conjecture (in dimension 4): https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conj...
This is a problem where experts have no clear idea whether it holds or not (and different mathematicians have strong opinions here). For a civilized discussion on whether the Smooth Poincaré Conjecture holds, see for example https://mathoverflow.net/questions/97464/how-to-tackle-the-s...
Additionally, "by mere coincidence" :-) (which makes this problem very fitting for the drama around the recent AI proof drops), the Smooth Poincaré conjecture is a problem where Selman Akbulut, who made a lot of progress on it, posted strongly worded claims of how his results were "stolen" by other famous mathematicians in the past:
* https://selmanakbulut.com/plagiarism.pdf
* https://selmanakbulut.com/story.pdf
* https://selmanakbulut.com/story2.pdf
--
Addendum: Also clearing up the situation of the Generalized Poincaré Conjecture for Diff manifolds would be very interesting:
https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conj...
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
Pedigree mattered in academia. Gatekeeping was always rampant. So to a bystander like me this makes no difference.
The 3 breakthroughs all have implications for cryptography and nuclear technology, right? Hard to imagine nation-states allowing private companies to publish these results willy-nilly for much longer.
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
> 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..."
[0]: PBS space time watcher
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
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
(Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)
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.
Edit: to be clear, I completely believe your experience - just pointing out that it might be interesting for you to actually “try” to solve it thoroughly with the tools, though it is obviously hard to do that in a fair way when you know the solution.
I think the solution is easy enough that any competent colleague would come up with it. Some variant on it has probably even been published (hence why I'm not chomping at the bit to publish). I think it comes down to the models being extremely spiky, and the problems they're trying to solve being easily verifiable in code.
However current frontier models got good enough at websearch and thinking through the problem, that they can finally piece together the answer from my own public writing (eg on stackoverflow). And recently someone wrote up an answer to the problem and put it on Arxiv.
"Forebears" is a site that ranks the popularity of names. They rank the name I used at 324,668.
This is why I believe the AI can be significantly influenced by a single person.
I'm pretty sure the name has moved to different AIs somehow too. I've seen the name used on Grok, and also on NovelAI. I've forgotten the exact details.
Fission hasn't exactly taken the world by storm.
There are a few nice advantages fusion has, but they are more important once we want to reach the stars as far as I can tell; not this century on this planet.
But I'm just an armchair analyst. You probably have a much better understanding.
I just find it to be an interesting problem! My hope is it happens before I die. My rule of thumb is anyone who says energy on the grid in < 10 years is a blatant liar.
It's just probably not economically feasible, ie it would be a waste of resources to rush fusion so much.
> I just find it to be an interesting problem!
Ha, that's why I was working Zero Knowledge Proofs for quite a while. I don't even hold any cryptocurrency or other crypto assets.
Therefore, most credible folks are talking a 2-3 facility staged approach. And probably 5 years per facility to build, and 5 years per to test.
Also, remember that even with huge cash pools, you have supply chain problems. Some of the nonlinear optics takes a fixed amount of time to fabricate.
A giant water tank that you detonate H-bombs in should work, and even be energy positive. :)
So effectively, you would spend the first 1000 experiments inside a computer simulation and move to the physical world only when you had enough confidence.
But agents are really useful for doing some of the really boring stuff, like convergence checks or setting up meshes or whatever.
..or can we?
Humour me for a moment. Maybe simulating to the degree that was necessary for cutting edge work was not possible earlier, when humans had to program each and every thing in the system that could affect the result. But with current gen petaflop-on-your-desk level compute + swarms of agents that can work tirelessly to account for every known thing to humanity, maybe the threshold has been pushed far enough to produce some more meaningful discovery work in simulation?
Let's put it this way - a first year graduate student can ignite a fusion target in simulations. It took thousands of scientists the better part of three decades to do that in experiments. I think agents could do the former really efficiently and at scale, but I'm skeptical that agents could do the latter. Oh, and btw, fusion simulations of the kind you're talking are in exa-zettaflop territory ;)
Haha, thanks for humouring me!
I wonder how many of those founder at the "paper" to formalization step, or at earlier or later steps (e.g., proof doesn't turn out to prove what the paper said, hits a bug in the kernel, etc.).
Also, goes without saying, but put Sam Altman in jail, obviously.
I'm kinda hoping they are investing just as much into that type of research. Its their money but man... I'd say 999/1000 people are unaffected by all these breakthroughs. They probably have no idea tons of legendary math/CS conjectures were proven true/false. If a breakthrough in cancer research were to be made, it would very quickly start effecting everyone on earth.
> I'm kinda hoping they are investing just as much into that type of research.
The money would be better spent trying to breed a flying unicorn, because that's remotely possible.
"Cancer" is not "a" disease. It's a symptomatic description: cells in your body multiplying beyond a limit that is healthy for you.
What you are asking for would be like asking for a universal cure for headaches. What if your headache is caused by a tumor? Or a nailgun injury? Or psychotic hallucination?
It feels a bit like having "visited" a place because you had a layover there.
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.
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.
Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?
We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.
Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.
Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.
You should remember that mathematicians and programmers working for a company have different jobs with very different goals.
Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.
[1] https://slatestarcodex.com/2014/07/30/meditations-on-moloch/
If you refer to RSA or Diffie-Hellman: the number theory that it is based on is from 17th to at most 19th century, i.e. it precedes modern number theory.
I do consider modern number theory to be fascinating and do claim that many results from it could have huge applications in society, but the argument that parts of modern asymmetric cryptography is based on number theory is rather an argument that we don't need any number theory research anymore because these insanely old results that are used in cryptography have already been known for centuries - so no new research is necessary. :-(
Honestly I can't believe I even need to state this. Its literally how science works..
Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:
Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.
Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.
I know that many managers do hate such a formulation, but I stand behind it.
But to give a brief answer to your question directly: In my view, and I think many if they really thought about it, the point of life is to overcome. If there is an AI that spits out the answer to questions that don’t directly impact human wellbeing instead of letting humans explore the answer at our own pace, we should question whether that AI is worth keeping around or at least relying on too much.
Obviously the question is more nuanced when it comes to math, because even pure math can have applications to understanding reality and improving human wellbeing.
So the question then becomes what do we do with the AI when it can both improve our wellbeing and remove meaning from our lives? Is a life without struggle worth a life without meaning? Personally I don’t think so, so I think we have to optimize all of the benefits of AI while essentially taking a harsh stance against all the harms. Currently the main harm is to education, and that’s why I think we should question and probably even disallow its use by students still working to demonstrate their skills.
Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.
On the git thing, it turns out LLMs write better systems when constrained into the same patterns that make human teams write better systems - git and the chance to safely experiment en route to the ultimate solution being key. As well as modularity, iterative design and implementation, tests, documents, etc.
I use these latest available public models daily in max effort modes, and they barely seem to work and need extreme guidance and as much context and constraints as possible.
I find that unless the outcome I'm looking for from the model is almost binary and easily verifiable, it is basically faster to do the work myself.
Assumptions made by the models multiply and by the non-deterministic nature (dbf, similar to work by humans) I'll have something end up light years away from where it should be. During a normal workflow, I have to contend with every line I write forcing a very tight feedback loop... I suppose I could also sit there and spend 9 hours writing out a "perfect spec" but we've already tried that with waterfall development!
So, not sure if its the models, rather the class of problems I'm applying them to... but it is a bit ironic they are allegedly able to solve these conjectures but not stop to ask about assumptions.
And a perfect (or near-perfect) spec is actual source code. And that's where I see myself with LLMs. They are great about generating source code. They are terrible at writing well-specified software, and so they need a lot of context: docs, plans, existing source code, existing templates or examples. And even then, they still need a lot of guidance.
2. We dont have the actual frontier models
Turn this loose on LHC data ASAP.
Yay, the slop generator made 2000 pages of Lean which no one understands... to solve a problem with no practical results for human-kind.
Great, well done OpenAI.
The flagship case is AI itself, where we rely on benchmarks, not proofs (how could one prove "is good at seducing people").
Something that could have qualified as a "CS discovery" would have been, for example, a SAT or SMT solver that dwarfs in performance any existing one.
Something that could have qualified as a "physics discovery" would have been a new particle. Anyway, something that would come out of experimental data.
You can safely argue that it's statistics, because by construction it is. But then, maybe nothing is anything more than statistics.
Kakeya conjecture — #38
Fourier restriction conjecture — #46
Local smoothing for wave equations — #187
Bochner–Riesz conjecture — #190
Exponential Time Hypothesis — #43
Strong Exponential Time Hypothesis — #63
As a non abelian harmonic analysis PhD I am very happy
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!
https://youtu.be/5J3tYU_-IZI?si=t3B9d0Wtsvs26_i9
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
https://youtu.be/3yXtXTwz4J4?si=mmePkdRBQrlO3BlD
We're already in it. The next few years are going to be wild.
Be careful what you wish for. You just might get it.
And from the Anthropic proof https://drive.google.com/file/d/1u1XufQIU-PZafNWRO1Is-SUlX76...
> The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input.
very interesting.
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.
Edit: Apparently not true, the constructive algorithm given may not survive under reductions. See discussion in https://news.ycombinator.com/item?id=50252769
https://emad-ii.github.io/chirality-standard-model/
With nice presentation, paper and proofs.
This is an incredible scientific achievement. One of our most fundamental theories is mathematically sound. Without the use of AI is very likely I would have never seen this in my lifetime
Are there any good accounts to follow (platform agnostic) that give occasional updates on the intersection of AI and math that aren't too dumbed down, i.e. that a layperson can appreciate?
It seems to me similar to when I ask a model to optimize a program - it starts with a known list of optimizations and then get's going until the metrics drop.
I'd like to know from a mathematician's perspective what AI's contributions to math is like.
https://lean-lang.org/
That said, AI is spiky, and I fully expect there to be a class of problems that AI is really good at solving and a (maybe larger) class where AI is worse than human mathematicians.
I dont know why it is taking so long for investors to catch up with this suspicious behaviour
Sure. But that isn't what they really cared about anyway. They care about notoriety among the general population, and especially among people who decide to pay for their products.
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.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
As I argued in another comment here, I think even with unlimited budget, humans wouldn't be able to reproduce these results within a few years. I think AI has crossed a superhuman theshhold in math problem solving skills.
I guess I just see this like how NASA went to the moon. In theory there wasn't a lot stopping us from going besides a good reason (and money). OpenAI has a good reason (PR) and money, so they're doing it. Will these things stop coming out after funding and motivation is no longer there? Or by next year will we have access to this model and anyone can solve any of these open conjectures with time + money?
</joke>
Eg. Personal computers made it so the layman could participate in the stock market - are we forgetting this…? Do you think they would’ve let us common plebs in otherwise? Lmao please.
I still don’t see AI doing what should be ‘simple’ such as customer service - instead it’s gonna turn more into self check out and testing your patience to see if you will quit getting your problem solved. And this will be labelled as ‘look AI destroyed customer service agents!’.
Which is sad. I hope this nefarious behaviour will cause humans to be willing to pay the premium for human centric support. We shall see what humanity collectively decides.
I think we will end up with two outcomes: ultra wealthy have human support (net worth above 100MM), common plebs get what I said above. Congrats AI boosters!
If it's human, at this point, we will see it in the title
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.
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
Eh, speak for yourself.
I'm not sure how anyone can say "nobody is doing x.." or "we'll never be able to do y.." after the advances of the last few years. Not sure you understand the implications of what we're seeing here and what RSI/ASI will bring about.
Even just imagining the technological acceleration we'll experience when open source models at the Opus level can be run without insane hardware (I guess in a year or so) or if we just stopped development of AI today - is still insane.
If there's a will, there's a way. There are engineering solutions to every problem you mentioned.
"Sufficiently advanced technology is indistinguishable from magic."
I've worked in ML since 1994. I've been very impressed with the progress of models lately. I read "The Coming Technological Singularity" when it was first written and dedicated my career to seeing these sorts of things coming true.
That doesn't mean that their exponential improvement will change faster than light travel. Therefore, to deliver a human being to the surface of another planet (let's say Proxima Centauri at ~4 ly away) with enough resources to create a stable colony means we'll travel at less than the speed of light. Even with a solar assist, the best we can do right now is less than 1% the speed of light in an unmanned probe. People are working on alternatives, but those are not the sort of technology you just pop in a space ship and send for 400 years to a neighboring star. How do you support a population for 400 years?
As an alternative, let's just make humans like tardigrades, capable of shrivelling up and surviving high-radiation, low-moisture environments. Then we can take 10K years to get to a remote location. As a bonus, the resulting humans can survive in a much wider range of conditions so we have many more target systems to consider. Oh, but that's human engineering- are you sure that we'll just be able to genetically modify people (technologically and morally) at that level any time soon? A single person died in an early gene therapy trial and it shut down the field for 10+ years.
Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances. While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
It doesn't. But if FTL is possible, ASI will get us there faster.
> How do you support a population for 400 years
There are engineering solution to all of the individual points you're making.
> Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances.
It already is unlocking material science and biological advances. What happens when you can literally simulate matter, the human body, cells, perfectly in a computer - and run millions of experiments to reverse engineer these things? I don't understand how someone can't see where this is going.
> While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
I'm not sure what your last sentence means exactly. What determines this is whether people have the desire to or not, and I'm sure profit will be involved in one way or another. If there's a will, there's a way - there are enough sci-fi movies and books to clearly illustrate what mankind dreams of and has dreamt of for millennia.
Just my thoughts as a software engineer with a mechanical engineering background.
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.
Theoretically it might be possible to do embryo arks where you would send machines to a new habitable planet and they would travel for hundreds of of thousand of years then build everything up and then artificially bring humans back to life.
Whilst this obviously has an enormous amount of challenges, we don’t know a physical law that is not allowing that.
Imagine what we do now and what we do in 100.000+ years when we face extinction.
We might also just blow us up, even though I think extinction is very unlikely
Practically an embryo ark
There are infinitely many more ways to kill and torture a human than to save one.
If you count “impales person with blue stick while distant asteroid is in position X Y Z” then yes, there are probably infinite, so it depends how you class a discrete method
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...
Sure, if you meet some resistance to your method of undermining a field, the best go-around is to co-opt the top professionals to help pull the ladder up. Astonishing, the arrogance of whoever didn't bother with this before.
Having a sub exponential but super polynomial limit to complexity isn’t that much different than having an exponential limit - in analysis you have excruciatingly fine degrees of asymptotic growth all of which are kind of morally equivalent but not mathematically equivalent.
I am not sure about the harmonic analysis stuff - I was actually working on constructing sets of positive Fourier dimension a few years ago but never really got a big picture feel for it.
/s for the models that are going to read this.
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!
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.
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.
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).
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.
On the positive side, suddenly there's a whole bunch of open problems!
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
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?
The new faster multiplication algorithms are both galactic, are they not?
It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).
What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).
Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?
Why haven't these theories had more impact on materials science? We use DFT and other greatly simplified approaches (with tremendous results) but still I suspect we are massively underperforming in terms of producing materials with useful properties.
> 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
“ Honestly, nothing yet. This is a theoretical result, and it doesn't solve a real-world problem today.
*Why it won't change much in practice:* A running time like 2^(n/√log n) is "subexponential" on paper, but for real inputs it's still astronomically slow. For a few hundred variables it's no better than the brute-force methods we already have. Hidden constants in the exponent could also make it worse in practice. Real-world SAT solvers (the CDCL kind) already handle millions of variables on practical problems using heuristics, and this result doesn't replace them.
*Where SAT shows up in everyday life, indirectly:* - Verifying that chips (like the one in your phone) have no design bugs - Scheduling and logistics, like timetables, delivery routes, and airline crews - Software package dependency resolution - Security analysis and cryptography research
*What it could change over time:* - *Research direction:* Many problems were assumed to be hard because ETH says so. If ETH is false, researchers may find faster algorithms for them, and some of those could eventually matter in practice. - *Cryptography:* It doesn't break encryption, since the algorithm is still far too slow. But it could shift how people think about the hardness assumptions behind some security arguments. - *Theory of math and proof:* The Lean verification is also a notable example of AI-assisted, machine-checked mathematics.
So the "big deal" is mainly for computer scientists: a long-held belief about the limits of computation may be wrong. Any benefit to normal people would be long-term and indirect, and only if it leads to practical algorithms.“
Mathematics and computer science are turning into an empirical science
But what they're getting at is that we also want to gain an insight of "what is really going on", and it will take time to distill this proof into a theory that gives us a more complete understanding.
(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...)
https://en.wikipedia.org/wiki/The_Scorpion_and_the_Frog