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People seem to be talking about anything except the actual results with this particular announcement. Its still astonishing that any sort of generalized comput

People seem to be talking about anything except the actual results with this particular announcement. Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I'd be curious to see if the new model can also do more direct proofs/inductive proofs.

sho_hn 6d ago
> People seem to be talking about anything except the actual results with this particular announcement. To be fair, most people have a fairly good handle on "Does opting out my prompts from training runs actually work?", but not on Navier-Stokes. They discuss what more immediately affects them.
- recursivecaveat 6d ago
  Additionally, I'm no physicist but I suspect the possibility of singularities in NS equations is probably one of those 'true but not meaningful' facts. If it took our brightest minds 175 years to craft such a scenario, how relevant can it be in practice? Especially when turbulence exists. Maybe I'm wrong or it has some consequences for pure math though.
kpil 6d ago
Unless they just swiped the workbooks of the actual mathematicians that where working on the problem using AI and it's in the "next-gen" training dataset.
- dumberquestions 6d ago
  You do realize that regardless of what was in the training data, the final solution included insights no human before had known, right? I share the same concerns regarding academic integrity but it would take a lot of motivated thinking to conclude that what the AI system did was not significant.
  - jamiejquinn 6d ago
    As far as I can tell (and my research was on the simulation side of Navier Stokes) the key AI output was a specific counter-example solution, generated with a method suspiciously close to that developed by the research duo involved in the controversy, a method that was discussed with Codex. So to me that insight is as insightful as the next undiscovered prime.
- boshalfoshal 6d ago
  I don't get how this invalidates the gravity of this achievement. Most mathematicians on the frontier of this stuff were likely using AI (or at the very least were heavily computer assisted) for some time now. Navier stokes was one of the very high profile problems that google Deepmind was working on with academia, for example. Even with many of our best minds working on it for nearly a century, it _just_ now was solved just as AI became very good at math. Doesn't seem too farfetched to me to assume that AI played an outsized role in solving it. If it was really just a matter of "stitching things together" to solve it (granted, this is a very reductive way to look at it) , I suspect we would've solved this a while ago.
  - kpil 6d ago
    There is a certain difference between activating all relevant memoized facts that's in the weights and stringing them together with the help of all the stored text in the world, or displaying genuinely emergent behaviour and generating novel output. One is really impressive and useful trick, one is AGI. Apple's research show almost zero emergent behaviour, so I'm inclined to think most of it was already in the weights. It doesn't take away the usefulness, it just defined the boundary. We can't expect "original research" then because it actually can't reason about concepts that are too far from whats already in the discourse. The discourse is big so we don't notice.
- contravariant 6d ago
  In a way that works just as well but the incentives are messed up. And that's before we get into the whole 'salt the earth' way they ended up solving it. For a short period of time it may well have been the least valuable proof in mathematics yet. In their haste it's dubious they actually read the proof, and I don't think anyone has had time yet to truly understand it (the original researchers are best placed to do so, but are they even willing?). So now it is solved, the proof has been independently verified and nobody has an incentive to investigate further. OpenAI has spent millions to uncover 1 bit of information that so far nobody has learned anything from, and they've demotivated all the people who wanted to.
  - MarkusQ 6d ago
    This. The point of these problems is the understanding / tooling gained in solving them. We're getting none of that. At best they are like a modern oracles, correctly answering your questions in a way that's doesn't help you any. (At worst,...)
btown 6d ago
Heck, it’s even astonishing that any sort of generalized computer program could even verify a proof of this magnitude that hasn’t already been codified in a formal verification language. If, and it’s unclear that we’ll ever get the full story, they did draw inspiration from training on (or even directly accessing) rough notes that had been provided by another researcher in prose… the fact that it could leap so rapidly to a full formal verifiable Lean program for the entire scope of the problem is an incredible result in its own right.
- deleted 6d ago
  [deleted]
- iterance 6d ago
  Then, of course, one must verify that the verification code is valid, or the purpose of verification is more or less moot.
dooglius 6d ago
I mean, I have a bachelor's in math and I don't imagine I could begin to understand either the human or LLM proofs without a massive investment of time and effort.
- TZubiri 6d ago
  But can you verify them? With less effort?
  - dooglius 6d ago
    I'm not sure what you mean by "verify" here. I could run the lean verifier as could anyone else. Maybe I could write my own proof checker and do a purely mechanical translation into my own thing, though I don't think that's less effort.
ramesh31 6d ago
>Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I think about this a lot. I'll have to explain to my kids some day that there was long period of time where you couldn't just talk to a computer and have it talk back to you, and that communicating with one required special skills that took years of study to master. It's going to be completely impossible for them to even remotely understand what that was like. Sort of like the pre-electricity days for us, but even more-so.
- sho_hn 6d ago
  You're assuming you'll be the one doing the explaining :-) It might also be that they won't even ask or wonder, similar to how most don't really do with pre-machining skills. Or it could be like our "How did they build the Great Pyramid?!"
TrackerFF 6d ago
It also needs to be said: The amount of compute that went into this is something. From some estimates I've seen, the compute cost alone would be around $10m, +/- As a reference, for that kind of money one could put together a research group of 20-25 researchers, and keep them salaried for 5 years. So while it is impressive, absolutely no doubt there, the SOTA access is so expensive that it is sort of unobtanium. Luckily, the prices have historically reduced by a factor of 5-10 every year...but still, only those that swim in cash can afford this.
- sho_hn 6d ago
  > From some estimates I've seen, the compute cost alone would be around $10m, +/- At market prices. All the estimates I've seen are based on OpenAI API costs. It doesn't mean that's what they paid, or how they paid for it. But yes, the surprising willingness of humans to solve hard problems in exchange for food and board is underrated.
  - MarkusQ 6d ago
    Given that they all the bit AI players are still loosing money, it follows that their total costs are _higher_ that their API pricing would imply.
    - deleted 6d ago
      [deleted]
    - MarkusQ 5d ago
      You may not like the truth, but it's still the truth. https://marketwise.com/investing/openai-losses-surge-to-21-billion-as-ai-bubble-grows-bigger/ https://marketwise.com/investing/openai-losses-surge-to-21-b... Anthropic is "profitable"... if you exclude compensation and compute cost commitments: https://aitoolsrecap.com/Blog/anthropic-first-profit-2026-revenue-breakdown https://aitoolsrecap.com/Blog/anthropic-first-profit-2026-re...
- boshalfoshal 6d ago
  Once we have an existence proof of a particular technology, it doesn't take long for it to become economically viable and proliferate. And for something as useful as this, theres a strong economic incentive to get it to be as cheap and accessible as possible. Maybe not today, but certainly in a couple years I can imagine this level of intelligence being accessible to someone with a $20/mo plan, or even a free plan.
Yizahi 6d ago
Aren't you doing exactly the same thing as people you are mentioning? Skipping "talking about actual results" to talking about general capabilities of this LLM and computers in general? because that's exactly what seems like 99% of all people had been doing lately - debating what computer programs can do and what they can't.
20k 6d ago
Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers That's why nobody's talking about how impressive this is, because its not nearly as impressive of a piece of work to simply cobble together other peoples' work that didn't know you were doing it. I could have republished relativity from einstein's notes, but people would correctly not be impressed with my ability Until the plagiarism scandal is sorted out, its not a meaningful result at all, because nobody knows how much genuine innovation these models are displaying
- sho_hn 6d ago
  > Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers It's also true however that I haven't seen a single write up trying to discern what did more of the work in those AI chats - the prompts or the responses - bubble to the surface, also since we don't have access to them. For example, if I prompt Codex with "Make me a website about strawberry cake" and nothing else, and OpenAI announces they have the best strawberry cake minutes before I launch, I'm not sure they plagiarized anything. We just don't know if this is quibbling over "who prompted first" or if the researchers came up with anything strikingly original by themselves.
  - 20k 6d ago
    The researchers apparently spend a year or so working on this, and it builds off significant previous work, so it seems like it was a pretty significant amount of work that OpenAI may have trained on I'd love to see an in depth analysis of how much OpenAI actually did, but I suspect we'll never see that because it would indicate at least some plagiarism which undermines a lot of what OpenAI is putting out in public
    - felipeerias 6d ago
      The American Mathematical Society credits the Spanish researchers Diego Córdoba and Luis Martínez‑Zoroa with the breakthroughs that eventually led to this solution, and which were published from ~2023 onwards. This is a good summary: > In broad outline, the pair’s technique relies on creating an infinite sequence of “layers,” each of which is a non-singular solution to the equation they are studying. (They’ve applied similar techniques to both the Euler and Navier-Stokes equations, as well as to other related systems.) They then combine those solutions in what Martínez-Zoroa calls an “infinite cascade” to produce a new solution. > > That new solution, they showed, contains the desired singularity. However, even though each individual layer relies on a smooth forcing function, combining them together can cause the forcing function to have undesirable mathematical properties. That’s why their solution fell short of satisfying the Millennium Prize criteria. The remaining hurdle was to figure out how to create a similar infinite cascade that resulted not only in a singularity, but also in a smooth forcing function. > > That’s the step that both competing AI groups appear to have had success with. https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/ https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-... The question is whether OpenAI started out from that published and well known research exclusively, or they also had some insight into the ongoing work of Tristan Buckmaster and Levent Alpöge. On the one hand, OpenAI have already admitted that they only launched their massive effort after hearing rumours that this particular problem had been solved. On the other, progress in mathematics research has accelerated significantly over the past months thanks to the availability of newer and more capable AI models. Alpöge himself presented a counterexample to the Jacobian conjecture on July, found with Claude Fable. So if model capability was a bottleneck, that gives credibility to the idea that an even more powerful unreleased model with massive compute would be able to make even faster progress.
dalvrosa 6d ago
Yep
fatbird 6d ago
Give me a dictionary, a computer, and infinite time, and I'll generate all possible English texts: Shakespeare, works regarded as surpassing Shakespeare, new holy books, math proofs never even imagined... none of which is either "creative" or "solving" anything. If I optimize my generation algorithm so that I'm not slavishly trying all possible combinations of words, it doesn't move me any closer to being creative, or solving anything. The real casualty here may be our belief that humans are doing something more than some super-optimized version of what LLMs are doing. That doesn't elevate LLMs, it just makes us much less special.

sho_hn 6d ago

> People seem to be talking about anything except the actual results with this particular announcement. To be fair, most people have a fairly good handle on "Does opting out my prompts from training runs actually work?", but not on Navier-Stokes. They discuss what more immediately affects them.

recursivecaveat 6d ago
Additionally, I'm no physicist but I suspect the possibility of singularities in NS equations is probably one of those 'true but not meaningful' facts. If it took our brightest minds 175 years to craft such a scenario, how relevant can it be in practice? Especially when turbulence exists. Maybe I'm wrong or it has some consequences for pure math though.

kpil 6d ago

Unless they just swiped the workbooks of the actual mathematicians that where working on the problem using AI and it's in the "next-gen" training dataset.

dumberquestions 6d ago
You do realize that regardless of what was in the training data, the final solution included insights no human before had known, right? I share the same concerns regarding academic integrity but it would take a lot of motivated thinking to conclude that what the AI system did was not significant.
- jamiejquinn 6d ago
  As far as I can tell (and my research was on the simulation side of Navier Stokes) the key AI output was a specific counter-example solution, generated with a method suspiciously close to that developed by the research duo involved in the controversy, a method that was discussed with Codex. So to me that insight is as insightful as the next undiscovered prime.
boshalfoshal 6d ago
I don't get how this invalidates the gravity of this achievement. Most mathematicians on the frontier of this stuff were likely using AI (or at the very least were heavily computer assisted) for some time now. Navier stokes was one of the very high profile problems that google Deepmind was working on with academia, for example. Even with many of our best minds working on it for nearly a century, it _just_ now was solved just as AI became very good at math. Doesn't seem too farfetched to me to assume that AI played an outsized role in solving it. If it was really just a matter of "stitching things together" to solve it (granted, this is a very reductive way to look at it) , I suspect we would've solved this a while ago.
- kpil 6d ago
  There is a certain difference between activating all relevant memoized facts that's in the weights and stringing them together with the help of all the stored text in the world, or displaying genuinely emergent behaviour and generating novel output. One is really impressive and useful trick, one is AGI. Apple's research show almost zero emergent behaviour, so I'm inclined to think most of it was already in the weights. It doesn't take away the usefulness, it just defined the boundary. We can't expect "original research" then because it actually can't reason about concepts that are too far from whats already in the discourse. The discourse is big so we don't notice.
contravariant 6d ago
In a way that works just as well but the incentives are messed up. And that's before we get into the whole 'salt the earth' way they ended up solving it. For a short period of time it may well have been the least valuable proof in mathematics yet. In their haste it's dubious they actually read the proof, and I don't think anyone has had time yet to truly understand it (the original researchers are best placed to do so, but are they even willing?). So now it is solved, the proof has been independently verified and nobody has an incentive to investigate further. OpenAI has spent millions to uncover 1 bit of information that so far nobody has learned anything from, and they've demotivated all the people who wanted to.
- MarkusQ 6d ago
  This. The point of these problems is the understanding / tooling gained in solving them. We're getting none of that. At best they are like a modern oracles, correctly answering your questions in a way that's doesn't help you any. (At worst,...)

btown 6d ago

Heck, it’s even astonishing that any sort of generalized computer program could even verify a proof of this magnitude that hasn’t already been codified in a formal verification language. If, and it’s unclear that we’ll ever get the full story, they did draw inspiration from training on (or even directly accessing) rough notes that had been provided by another researcher in prose… the fact that it could leap so rapidly to a full formal verifiable Lean program for the entire scope of the problem is an incredible result in its own right.

deleted 6d ago
[deleted]
iterance 6d ago
Then, of course, one must verify that the verification code is valid, or the purpose of verification is more or less moot.

dooglius 6d ago

I mean, I have a bachelor's in math and I don't imagine I could begin to understand either the human or LLM proofs without a massive investment of time and effort.

TZubiri 6d ago
But can you verify them? With less effort?
- dooglius 6d ago
  I'm not sure what you mean by "verify" here. I could run the lean verifier as could anyone else. Maybe I could write my own proof checker and do a purely mechanical translation into my own thing, though I don't think that's less effort.

ramesh31 6d ago

>Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I think about this a lot. I'll have to explain to my kids some day that there was long period of time where you couldn't just talk to a computer and have it talk back to you, and that communicating with one required special skills that took years of study to master. It's going to be completely impossible for them to even remotely understand what that was like. Sort of like the pre-electricity days for us, but even more-so.

sho_hn 6d ago
You're assuming you'll be the one doing the explaining :-) It might also be that they won't even ask or wonder, similar to how most don't really do with pre-machining skills. Or it could be like our "How did they build the Great Pyramid?!"

TrackerFF 6d ago

It also needs to be said: The amount of compute that went into this is something. From some estimates I've seen, the compute cost alone would be around $10m, +/- As a reference, for that kind of money one could put together a research group of 20-25 researchers, and keep them salaried for 5 years. So while it is impressive, absolutely no doubt there, the SOTA access is so expensive that it is sort of unobtanium. Luckily, the prices have historically reduced by a factor of 5-10 every year...but still, only those that swim in cash can afford this.

sho_hn 6d ago
> From some estimates I've seen, the compute cost alone would be around $10m, +/- At market prices. All the estimates I've seen are based on OpenAI API costs. It doesn't mean that's what they paid, or how they paid for it. But yes, the surprising willingness of humans to solve hard problems in exchange for food and board is underrated.
- MarkusQ 6d ago
  Given that they all the bit AI players are still loosing money, it follows that their total costs are _higher_ that their API pricing would imply.
  - deleted 6d ago
    [deleted]
  - MarkusQ 5d ago
    You may not like the truth, but it's still the truth. https://marketwise.com/investing/openai-losses-surge-to-21-billion-as-ai-bubble-grows-bigger/ https://marketwise.com/investing/openai-losses-surge-to-21-b... Anthropic is "profitable"... if you exclude compensation and compute cost commitments: https://aitoolsrecap.com/Blog/anthropic-first-profit-2026-revenue-breakdown https://aitoolsrecap.com/Blog/anthropic-first-profit-2026-re...
boshalfoshal 6d ago
Once we have an existence proof of a particular technology, it doesn't take long for it to become economically viable and proliferate. And for something as useful as this, theres a strong economic incentive to get it to be as cheap and accessible as possible. Maybe not today, but certainly in a couple years I can imagine this level of intelligence being accessible to someone with a $20/mo plan, or even a free plan.

Yizahi 6d ago

Aren't you doing exactly the same thing as people you are mentioning? Skipping "talking about actual results" to talking about general capabilities of this LLM and computers in general? because that's exactly what seems like 99% of all people had been doing lately - debating what computer programs can do and what they can't.

20k 6d ago

Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers That's why nobody's talking about how impressive this is, because its not nearly as impressive of a piece of work to simply cobble together other peoples' work that didn't know you were doing it. I could have republished relativity from einstein's notes, but people would correctly not be impressed with my ability Until the plagiarism scandal is sorted out, its not a meaningful result at all, because nobody knows how much genuine innovation these models are displaying

sho_hn 6d ago
> Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers It's also true however that I haven't seen a single write up trying to discern what did more of the work in those AI chats - the prompts or the responses - bubble to the surface, also since we don't have access to them. For example, if I prompt Codex with "Make me a website about strawberry cake" and nothing else, and OpenAI announces they have the best strawberry cake minutes before I launch, I'm not sure they plagiarized anything. We just don't know if this is quibbling over "who prompted first" or if the researchers came up with anything strikingly original by themselves.
- 20k 6d ago
  The researchers apparently spend a year or so working on this, and it builds off significant previous work, so it seems like it was a pretty significant amount of work that OpenAI may have trained on I'd love to see an in depth analysis of how much OpenAI actually did, but I suspect we'll never see that because it would indicate at least some plagiarism which undermines a lot of what OpenAI is putting out in public
  - felipeerias 6d ago
    The American Mathematical Society credits the Spanish researchers Diego Córdoba and Luis Martínez‑Zoroa with the breakthroughs that eventually led to this solution, and which were published from ~2023 onwards. This is a good summary: > In broad outline, the pair’s technique relies on creating an infinite sequence of “layers,” each of which is a non-singular solution to the equation they are studying. (They’ve applied similar techniques to both the Euler and Navier-Stokes equations, as well as to other related systems.) They then combine those solutions in what Martínez-Zoroa calls an “infinite cascade” to produce a new solution. > > That new solution, they showed, contains the desired singularity. However, even though each individual layer relies on a smooth forcing function, combining them together can cause the forcing function to have undesirable mathematical properties. That’s why their solution fell short of satisfying the Millennium Prize criteria. The remaining hurdle was to figure out how to create a similar infinite cascade that resulted not only in a singularity, but also in a smooth forcing function. > > That’s the step that both competing AI groups appear to have had success with. https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/ https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-... The question is whether OpenAI started out from that published and well known research exclusively, or they also had some insight into the ongoing work of Tristan Buckmaster and Levent Alpöge. On the one hand, OpenAI have already admitted that they only launched their massive effort after hearing rumours that this particular problem had been solved. On the other, progress in mathematics research has accelerated significantly over the past months thanks to the availability of newer and more capable AI models. Alpöge himself presented a counterexample to the Jacobian conjecture on July, found with Claude Fable. So if model capability was a bottleneck, that gives credibility to the idea that an even more powerful unreleased model with massive compute would be able to make even faster progress.

dalvrosa 6d ago

Yep

fatbird 6d ago

Give me a dictionary, a computer, and infinite time, and I'll generate all possible English texts: Shakespeare, works regarded as surpassing Shakespeare, new holy books, math proofs never even imagined... none of which is either "creative" or "solving" anything. If I optimize my generation algorithm so that I'm not slavishly trying all possible combinations of words, it doesn't move me any closer to being creative, or solving anything. The real casualty here may be our belief that humans are doing something more than some super-optimized version of what LLMs are doing. That doesn't elevate LLMs, it just makes us much less special.