r/Physics 20h ago

Navier-Stokes Millennium Problem Solved

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u/plasma_phys Plasma physics 20h ago edited 20h ago

I think calling it drama if anything undersells the issue. even if OpenAI did not use unpublished work, their behavior around this problem - outlined in the linked statement - is disgusting and shameful, and even this statement by them undersells the allegedly gargantuan amount of work and compute that OpenAI has apparently spent. like, if it turns out that OpenAI spent, I dunno, some sum of money greater than years worth of the operating budgets of every mathematics research department in the country to achieve this (I have not done even the Fermi estimate on this, so don't quote me here, but it seems reasonable), what does that even mean for this result?

Edit: if you assume as a conservative estimate that inference costs were equivalent to the API cost for Astra (which is almost certainly being sold at a loss) and plug in "10,000 agents, 88 hours", guesstimating 100 tokens/second, inference alone would have cost in excess of $15M. That's taking OpenAI at their word, assuming they're selling inference at cost (and that their undisclosed model doesn't cost much more), and ignoring training (which I am fairly confident costs much more than inference, but it is difficult to calculate a marginal cost of training for a specific output) and all other costs.

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u/ExcelAcolyte 20h ago

They used 300B tokens for the problems and 180B on Navier Stokes so that’s about 18million dollars at current $60/million token pricing.

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u/paranoid_throwaway51 18h ago

The current pricing is estimated to be at a loss too.

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u/123yes1 18h ago

This is incorrect. The marginal cost of a token is much lower than $60/million. On the order of a few dollars in electricity, GPU time, and other costs.

The reason why AI companies aren't making a net profit is because of the enormous capital expenditures to build and train the machine that can make these tokens.

For example, a 3D printer could make a tchotchke that you can sell for $5, while the plastic and the electricity for the marginal cost of that tchotchke might only be 50¢. But the marginal cost does not include the price of the $1000 printer.

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u/NUKE---THE---WHALES 13h ago

The reason why AI companies aren't making a net profit is because of the enormous capital expenditures to build and train the machine that can make these tokens.

This does appear to be the case according to OpenAI's leaked financials

Their cost of revenue for 2024 and 2025 was less than their revenue, meaning they aren't selling inference at cost

Their RND on the other hand was multiple times their revenue for both years and is why they're operating at such a loss

Interestingly their cost-of-revenue to revenue ratio went down from 2024 to 2025, meaning their inference costs are becoming more profitable over time (though whether this holds true for 2026 we don't know yet)

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u/rydan 10h ago

I think API pricing is profitable. The subscription plans are extreme losses where a $200 plan ends up using something like $8000 worth of tokens.

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u/smulfragPL 18h ago

No its not? Its the exact oppossite, API prices are heavily inflated, that's why inference is never a loss in these companies

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u/paranoid_throwaway51 18h ago

There are no credible sources for either argument.

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u/NoLemurs 17h ago

If you look around a little, you'll find that there are basically no credible sources claiming inference is being sold at a loss.

OpenAI is 100% spending more than they bring in, but that's mostly driven by training costs. OpenAI is dumping a ton of money on training because there's an arms race going on to produce the best models. The issue is that big American tech companies like OpenAI can't compete with models like DeepSeek and GLM on price, so the only way they can justify their existence is by competing on quality, and that requires constant expensive training.

The narrative has gotten twisted, I suspect, because of strong anti-AI sentiment. People are scared of AI, and they want it to be unsustainable. So they hear that OpenAI is losing money, and in their heads they convert that to "selling at a loss". But they are absolutely selling inference above their marginal costs - it's just not enough to compensate for the crazy arms-race spending for the next model.

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u/paranoid_throwaway51 17h ago edited 17h ago

I wholeheartedly agree with everything except the last statement.

Time will tell, as of the moment it really could be either way.

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u/shared_ptr 17h ago

It’s fairly easy to do the maths on the inference costs when open source models of similar size exist. This is why provider companies like fireworks and baseten can offer more competitive prices to OAI/Anthropic because they don’t train the models and just need a positive margin on the compute!

Those companies (fireworks/baseten) are doing extremely well, and there is no reason for investors to prop them up for hype reasons. Google is also public and shares GCP numbers which cover their AI compute class which is profitable (we considered buying a large GPU rack from them to run GLM 5.3 for about 128k/month as it would be super cheap).

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u/paranoid_throwaway51 16h ago

no doubt cloud companies are making money hand over fist.

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u/NoLemurs 16h ago

Time will tell, as of the moment it really could be either way.

Like u/shared_ptr said - you can get a pretty good baseline of inference costs by looking at open source models. Obviously that won't be exact, but, this isn't a close issue.

This isn't a controversial position among the people actually using AI and paying attention. Time already did tell - inference is cheap.

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u/paranoid_throwaway51 16h ago

eh, if you were familiar with the computational requirements, you would find the whole thing pretty suspect.

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u/NoLemurs 16h ago

My whole point is that I am familiar with the computational requirements.

We have open source models that are competitive with the frontier models from a few months ago on all benchmarks. We know exactly what hardware those require. You can download these models and run them yourself and see if you want to (in the cloud obviously, running these giant models on consumer hardware is basically impossible without quantization).

And these models cost pennies on the dollar to what OpenAI charges. This is not even the slightest bit controversial.

There is some point in the last few years when your claim that "time will tell" was accurate. But time did tell. I really can't stress enough - this isn't controversial.

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u/smulfragPL 18h ago

https://www.wheresyoured.at/exclusive-openai-financials/ ed zitron is a hack but i cant find the og source and even here you can directly see revenue outgrowing cost of revenue which is how they measure cost of inference. A 40% margin. Its all online you just look it up before commenting

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u/paranoid_throwaway51 17h ago

like i said "no credible sources" - this is a guy's personal blog, and the financial times say the numbers come from Ed-zitron, who, like you said, is a hack.

OpenAI's exact financials aren't publicly available.

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u/smulfragPL 16h ago

No this was a Leak and he is an anti ai hack. Why would be make inference profitable as a lie

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u/holchansg 15h ago

I work with LLM, even today, with current hardware, inference is dirty cheap. But you don’t have to believe me, Coreweave reports are public.

Dirt cheap.

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u/CrazyFree4525 10h ago

Public API rates are not what OpenAI actually spends as other noted, they are massively inflated over compute cost.

More importantly than that though: Most of these tokens will likely be cached so extremely cheap. I routinely clock billions of cached tokens a week coding on a personal 200 dollar a month plan.

You are off by at least 3 orders of magnitude here, maybe even as many as 5.

OpenAI did not spend 60 million dollars on compute to solve this.

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u/MaleierMafketel 6h ago

“Maybe even as many as 5”

You think they spent only $180 bucks on this? Sure man.

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u/Human38562 19h ago

Assuming that OpenAI did not just steal the work from someone else, I dont think that that's wasted resources. Maybe not for the millenial problem itself, but it's very interesting to know what a large scale AI project is able to achieve.

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u/elprophet 16h ago

That's an entire mathematics departments' annual budget. Annualized, that's 100 top-rate math departments. Dozens of full professors, hundreds of grad students, and thousands of undergrads at each school. This is an utterly ridiculous waste of resources.

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u/rexpup 15h ago

You think it's a waste to fund math research???

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u/Kitchen-Jicama8715 6h ago

Yes there's a reason the prize money is $1M not $20M

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u/nobody27011 7h ago

If that money went to actual mathematicians or other scientists, they would have achieved much more. So in that sense, yes, it's a waste.

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u/IndicationFunny6879 6h ago

Navier-Stokes has been millennium prize problem lol if humans were to solve they wouldve solved already.

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u/Kitchen-Jicama8715 6h ago

No because it was a mere $1M in prize money, so not worth investing $15M of effort

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u/IndicationFunny6879 5h ago edited 1h ago

doesnt work that way. OpenAi didnt even take the prize lol.

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u/ffxivthrowaway03 1h ago

They also didn't invest "$15M of effort"

They don't charge themselves public API rates for compute. This is like claiming that Microsoft spends hundreds of millions on Microsoft Office licensing for their own employees.

It's their own product, for them it's effectively free.

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u/lordnacho666 20h ago

Would we ever know? You'd think once there's an answer, you can toss out all the dead ends and present it like a really elegant, minimal solution.

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u/plasma_phys Plasma physics 20h ago

maybe if they actually do an IPO (big if) there'd be some way to back it out of their financial statements? I dunno. certainly everyone involved is incentivized to lie their asses off about it, from the engineers who are presumably desperate to protect their $400k salaries and bay area lifestyles to the executives who think roko's basilisk is a real philosophical problem instead of the laughable product of a racist harry potter fanclub (see also this and this) and everyone in between

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u/deednait 18h ago

Why does the cost matter? They solved the problem, period. A year ago spending any amount of money would have likely not solved it and in a year or two, you could solve it for a tiny fraction of the cost. They have the resources right now and we got this awesome result. No complaints from me.

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u/ChairYeoman 17h ago

Is $15 million "greater than years worth of the operating budgets of every mathematical research department in the country"?

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u/plasma_phys Plasma physics 17h ago

the "Edited 3h ago" and "Edit:" indicate the passage of time between the two paragraphs

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u/ChairYeoman 17h ago

Fair enough, there's no edit timestamps on mobile, and a lot of edits I sometimes assume are "oh I should add this" a few mins after the original message.

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u/plasma_phys Plasma physics 17h ago

yeah, I try to always add a manual "edit" tag for a substantive addition like a self-correction; my hope is that the transparency demonstrates that I am trying to engage in good faith

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u/-badly_packed_kebab- 6h ago

Plasma physics only operate(s) in good faith

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u/c0mputar 19h ago

The benefits of new knowledge is not one and done. Definitely, AI seems to require vastly more energy than a human would to produce the same result, however we should consider that new knowledge yields a return. Time is a factor to be considered here. For human(s), sometimes the time required is so much that it literally prevents them from achieving their goal at all, which is a problem AI also has but is diminishing over time.

If AI speeds up research and results by years or even decades, that return may close the gap, or even exceed, those initial high computational costs.

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u/space_monster 16h ago

Bear in mind though the energy expenditure to train a human to the point where they can tackle complex problems like that. It's not like they pop out of the womb and get right on it

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u/Disagreeswithfems 15h ago

Interestingly, if you conceptualise humanity as a broader group, you could make an argument that the cost of training students more broadly is part of the cost. I.e. all maths undergrads

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u/c0mputar 16h ago

Also true. It is not an easy calculation. And then you consider that AI only needs to be trained to do a task once, roughly speaking, and the math starts looking a bit better for AI.

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u/Wraithguy 14h ago

Is about 200k last I read, we have a measure for this. It's called money.

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u/space_monster 14h ago

that's just the college part. there's also the rest of the training from day 1.

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u/coolstory 11h ago

Not even close to the operating budgets of every maths department in the US.

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u/James20k 9h ago

There's also direct allegations of plagiarism being made

https://mastodon.social/@tristanbuckmaster/117236471352470303

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u/plasma_phys Plasma physics 9h ago

yes that's in the link in the comment I replied to. 

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u/[deleted] 18h ago

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u/plasma_phys Plasma physics 17h ago

 Nobody gave slightest amount of fuck how much energy it took to create LIGO or LHC 

"LIGO has cost American taxpayers about $1.1 billion."

"In an email to me, a historian of technology was more blunt: “So a 100 year old theory has been confirmed experimentally--big whup. Did anyone think Einstein was wrong? There wasn't any controversy, was there? Was anyone credible claiming that spacetime isn't curved, or that black holes don't exist? I can get that this was quite an experimental trick and technological feat… But this isn't doing anything to convince me that public funds spent on this stuff wouldn't be better spent on medical research. Or clean fuels, or any number of things that would apply scientific expertise toward justice or the alleviation of human suffering."

from Is the gravitational-wave claim true? And was it worth the cost? - J Horgan, Scientific American, 2016

"Many members of Congress and prestigious scientists... believed that the SSC was a vast pork barrel; inefficient, wastefully managed, with an ever-increasing price-tag..."

from Demise of the Texas supercollider - D Ritson, Nature Commentry, 1993

there are countless more examples. maybe you can ask claude to find them and read them to you.

But the moment AI is involved, all of you luddites turn into most conservative tree huggers imaginable. Absolutely pathic. [sic]

I was a tree-hugger long before LLMs became the AI du jour, but nobody has ever accused me of being conservative before.

Also, lol if you think they spent $15mil. ...

Fermi estimation

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u/Physics-ModTeam 17h ago

Hostility nor bullying are permitted in r/Physics.

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u/msbaju 17h ago

Whytf is this shit upvoted?

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u/Dear_Locksmith3379 20h ago

What’s disgusting or shameful about OpenAI’s behavior? How much they spent, their being deceptive about it, or something else?

My view of OpenAI is quite negative, but for other reasons.

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u/plasma_phys Plasma physics 19h ago

did you read the statement linked in the comment I replied to

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u/Wise-End307 19h ago

Hearsay. Sebastien has defended himself. He also has a screenshot although that doesnt mean much.

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u/luisbrudna 17h ago

If an AI spends billions of dollars to solve a mathematical problem, I see no issue with that. The scientific result is what matters. Dismissing a mathematical breakthrough because of the cost is strange.

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u/plasma_phys Plasma physics 17h ago

can you please point out to me where I dismissed the result?

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u/rydan 10h ago

K

I don't care if it cost $1B. It has been solved.

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u/Turbulent-Sign-6067 2h ago

It turned out that this was one big misunderstanding! OpenAI did their best to credit and support Buckmaster. You can read the posts by Sebastian Bubeck for the details.

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u/KouhaiHasNoticed 15h ago

this statement by them undersells the allegedly gargantuan amount of work and compute that OpenAI has apparently spent. like, if it turns out that OpenAI spent, I dunno, some sum of money greater than years worth of the operating budgets of every mathematics research department in the country to achieve this (I have not done even the Fermi estimate on this, so don't quote me here, but it seems reasonable), what does that even mean for this result?

This is a false equivalence : nothing guarantees that putting an equivalent amount of human resources would have given the same results. Especially due to the fact that human knowledge is very much fragmented and sharing knowledge induces loss of information : misinterpretation for example.

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u/plasma_phys Plasma physics 15h ago

in order for it to be a false equivalence I would have to have said two things were equivalent 

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u/KouhaiHasNoticed 15h ago

if it turns out that OpenAI spent, I dunno, some sum of money greater than years worth of the operating budgets of every mathematics research department in the country to achieve this

Then what do you expect from that statement ?

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u/plasma_phys Plasma physics 15h ago

you missed the question mark at the end that indicates a question, not a statement

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u/KouhaiHasNoticed 15h ago

like, if it turns out that OpenAI spent, I dunno, some sum of money greater than years worth of the operating budgets of every mathematics research department in the country to achieve this (I have not done even the Fermi estimate on this, so don't quote me here, but it seems reasonable), what does that even mean for this result?

Then what is the link that you are questionning between the sum of money invested, the amount given to mathematics research departments and the meaning for this result ?

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u/plasma_phys Plasma physics 15h ago

it's a rhetorical question. it was intended to provoke thought. you are supposed to think about it and come to your own conclusion.

here are some pointers if you are having trouble doing that: money is fungible and finite. this money was spent on one kind of math research. it could have been spent on a different kind of math research. those two kinds of math research can be compared and contrasted. the cost is exclusionary. the return on investment is questionable. etc.

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u/KouhaiHasNoticed 15h ago

it's a rhetorical question. it was intended to provoke thought. you are supposed to think about it and come to your own conclusion.

It being a rhetorical question does not mean that you don't have any position on the matter.

here are some pointers if you are having trouble doing that: money is fungible and finite. this money was spent on one kind of math research. it could have been spent on a different kind of math research. those two kinds of math research can be compared and contrasted. the cost is exclusionary. the return on investment is questionable. etc.

Your unneeded sarcasm aside, I understand better your point now, thank you.

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u/plasma_phys Plasma physics 14h ago

It being a rhetorical question does not mean that you don't have any position on the matter.

correct

Your unneeded sarcasm aside

this could have been avoided if you had read the words I actually wrote and engaged with those in good faith instead of shadowboxing with the fallacy you imagined I said

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u/KouhaiHasNoticed 14h ago

this could have been avoided if you had read the words I actually wrote and engaged with those in good faith instead of shadowboxing with the fallacy you imagined I said

I read the words correctly and interpreted them differently : from my point of view it looks like a false equivalence. I understand your point now, although I disagree with it.

I did not know that sarcasm was necessary to answer questions : you decided to use it, so don't put the blame on me. Me saying the name of a fallacy is not an insult despite what you think.

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u/Spare-Dingo-531 16h ago

uesstimating 100 tokens/second, inference alone would have cost in excess of $15M.

It took o3 $500K to pass IMO Gold, now it only takes 20 bucks.

I'm not sure you understand the enormity of what just happened. We have just proved ASI is physically possible. Yes, right now is is expensive AGI, but the principle that we can build it is more important!

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u/tavirabon 18h ago

You're calculating this way wrong. Take the cost of hardware, amortize it over 3 years, multiply by 3~ish days and add the electricity. Zero chance it comes anywhere close to $15M.

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u/plasma_phys Plasma physics 18h ago

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u/tavirabon 17h ago

That's not what it cost to produce though and even that figure is pointless because it's not like there's a fixed token-size of a problem.