OpenAI claims one of its AI models has found a solution to the Navier-Stokes problem, one of the toughest and most enduring puzzles in mathematics, sitting on the list of Millennium Prize Problems, solutions for which come with a $1 million reward. The announcement came after an unusual flurry of activity around the problem, along with a dispute around how the work came about.

The Navier-Stokes equations describe how fluids move in space, such as air over an aircraft wing or water out of a tap. Though the equations were first written down around 200 years ago, they aren鈥檛 well understood. The question posed by the Clay Mathematics Institute for the Millennium Prize Problem is to determine if the equations always work or if there are situations where they stop making sense and start spouting nonsense, what mathematicians call 鈥渂low-ups鈥�.

To settle the question, OpenAI first set 1000 AI agents on the Euler problem, which is a cousin of the Navier-Stokes problem and a step on the path to a full solution. It took the agents 50 hours to find blow-ups in this case. They then set 10,000 agents the task of extending the blow-ups to apply to the full Navier-Stokes problem, which took just 11 hours to finish.

OpenAI said during a press conference that if a customer wanted to run the same problem, it would cost around $15 million. OpenAI didn鈥檛 name the AI model used, but said it was 鈥渟ignificantly more capable鈥� than even its latest GPT-6 Astra model.

鈥淭his problem has remained unsolved for 200 years because the Navier-Stokes equations are just so enormously complex, and the pen-and-paper calculations you need to do in order to solve this problem are just mind-bogglingly intricate,鈥� says Venkat Chandrasekaran at OpenAI, who was also on the press conference call.

at OpenAI says the news is the 鈥渟pectacular combination of the arc we have seen over the last 12 months鈥�.

The result is the latest in a string of shocking mathematical discoveries led by AI in recent months. In May, an OpenAI model聽cracked a decades-old conjecture by Paul Erd艖s, causing a stir in mathematical circles. Later,聽the Claude Fable 5 AI聽found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Last week, an AI model formalised Fermat鈥檚 last theorem in just 11 days.

OpenAI鈥檚 latest groundbreaking mathematical discovery came just hours after 聽at New York University and 聽at AI company Anthropic announced that they had cracked three significant problems considered 鈥渟tepping stones鈥� to the Navier-Stokes problem.

Buckmaster and Alp枚ge also showed that blow-ups can appear in Euler equations; OpenAI has now shown that the same is true of Navier-Stokes.

Buckmaster and Alp枚ge said they received a 鈥済reat deal of help聽from鈥� large language models (LLMs), including LLMs from Anthropic and OpenAI. But when rumours suggested that OpenAI had gone one step further and solved the wider Navier-Stokes puzzle 鈥� something the company formally announced just hours later 鈥� Buckmaster released a statement suggesting that OpenAI acted unusually and opaquely when he approached the company for clarification.

Buckmaster wrote in a that, as a result of these rumours, he emailed a prominent mathematician at OpenAI to clarify matters. He says that in a subsequent meeting with OpenAI staff, he was given no hard details about how OpenAI achieved the result, but he took what little he was told as a 鈥渞ed flag鈥�, as it appeared to describe the same technique and path that he and Levent had followed to achieve their interim findings.聽

According to Buckmaster, OpenAI then admitted it hadn鈥檛 started its own AI search for a solution until after it had heard of his and Alp枚ge鈥檚 work, and didn鈥檛 respond to questions about whether its AI model had been trained or fine-tuned on that existing work 鈥� which was being stored in OpenAI鈥檚 Codex model, where the company would theoretically have access to it.聽The pair had used Codex as a customer, not a research partner, so expected that their work should remain secure and private.

Buckmaster makes it clear in his version of events that he is making no claims or accusations about how OpenAI arrived at its solution, saying only that he wished the focus could instead be on the mathematics rather than scandal and intrigue. 鈥淚 have not seen OpenAI鈥檚 proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything,鈥� he wrote in the posted document.

In a press conference announcing its discovery, OpenAI categorically denied using Buckmaster and Alp枚ge鈥檚 proof or prompts in its work. It also said that its model鈥檚 proof of Euler is different to that put forward by Buckmaster and Alp枚ge.聽

It also said that no humans accessed Codex to see work in progress within. Mark Chen at OpenAI, when asked if that also applied to any of the thousands of AI agents that had worked on the problem, said 鈥渢hat鈥檚 also our understanding鈥�.

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Though the solution to this Navier-Stokes problem has been a long time coming, AI-generated proofs are often hard to understand and rarely bring the same level of new insight than those created by humans. 鈥淭here鈥檚 been this very strange and unprecedented decoupling, this year alone, between getting answers and getting understanding,鈥� says at University of California, Los Angeles. He says that AI is coming up with new results so quickly that there is not enough time for the 鈥渟low, deliberate discussion鈥� needed to unpack them.

Something that will happen slowly is determining how the $1 million Millennium Prize money will be awarded. 鈥淭he process of evaluation is deliberately unhurried, and we shall ensure that it is absolutely rigorous,鈥� wrote , president of the Clay Mathematics Institute, to New 女生小视频 in an email.

In 800 years, a group of students is sitting in a graduate seminar devoted to an era known as the Void Ages. They鈥檝e read a couple of books and watched some of the two-dimensional, low-resolution media once called 鈥渕ovies鈥�. A few students even assembled devices to experience 鈥渧ideo games鈥� 鈥� interactive stories devoted to accumulating digital representations of value 鈥� using the original controllers. The professor is explaining the psychology behind why game controllers looked like fists covered in nipples, when somebody鈥檚 hand shoots up.

鈥淚 don鈥檛 understand why we have so little media preserved from this era. Everything is fragmentary, and most of what we鈥檝e read in this class was reconstructed from probabilities. Was it because of the atomic wars? Or the Fire Years?鈥�

鈥淣o war or global wildfire could do this,鈥� the professor answers gravely, uncoiling a liquid metal tentacle to point at a time map on the wall. 鈥淧eople from this era fed their culture into large language models and disposed of the original texts. All we have left is the output of ancient algorithms. We have to reverse-engineer what people were doing from that.鈥�

鈥淪o, we can never know if any of what we鈥檙e studying in this class is actually what people knew or saw or believed? It could all be generated by LLMs?鈥�

鈥淭hat鈥檚 right. There鈥檚 a reason why we call it the Void Ages.鈥�

More students break into the conversation, voices overlapping, frustrated and fascinated by the mystery of this pivotal era when their ancestors decided to subsume their greatest works of culture into a word-guessing model, the kind of thing a child would do to wreck their sibling鈥檚 homework.

It sounds bonkers, but this is the future we鈥檙e building for ourselves right now. When we feed books, movies, games and art into generative AI models, we risk replacing our own cultural history with a mishmash of auto-generated works that offer nothing to future generations who will be desperate to know what we were thinking. It鈥檚 as if we are deliberately recreating the tragic void at the heart of Bronze Age history, an era when humans transformed their civilisations across the globe 鈥� and left virtually no written narratives behind to explain what the hell happened and why.

As a writer of both science journalism and science fiction, I continue to be appalled by the idea that using an LLM could substitute for the experience of writing to, for and about each other. The point of writing, of creating culture, is to communicate my own weird way of looking at the world. It鈥檚 my letter to you, another weirdo probably. And it鈥檚 my letter to the future, my record of what one hairless ape experienced in the very specific era of the early 21st century, in the urban region currently known as San Francisco. It鈥檚 a human record, for other humans.

Back in 2020, when I was freaking out because I had no idea how to survive a global pandemic, . In 1666, he was a regular Englishman, just trying to survive a terrible wave of plague. Pepys worried about big-picture stuff like global politics and the meaning of life, but he also wrote about how to hide a wheel of cheese when you flee London and what it felt like to walk down a street where doors were painted with the red crosses of quarantine. Reading his words was soothing, because it was like he had reached out to me across the centuries to reassure me that I was not alone. That is the value of expository writing, and the value of history.

Of course, when I write fictional stories about time travellers and robots, I鈥檓 not recording an exact experience of real life. But I am still capturing the anxieties and hopes of my time. There are things you can say in fiction that you can鈥檛 say in journalism; I can reflect social ambiguities and fantasies, the biases that say more about us than facts ever could. The works of George Eliot and H. G. Wells are remembered today not just because they are engaging, but because they capture 19th-century aspirations and fixations, the little psychological tics that reveal how humans reacted to their rapidly industrialising world.

I鈥檓 not against the use of generative algorithms in code development, or as an aid in analysing big data. I am as devoted to the scientific project as the next nerd. But we need human writing as much as we need technology. Neither can replace the other. If we truly want to understand the universe, we need records of the human world as well as the physical one. When we feed all our writing to AI, we risk losing more than individual works of literature. We lose our connections to each other, in the present, past and future.

China has begun regulating interactions with artificial intelligence in an attempt to protect its citizens from overreliance and exploitation. While many countries have been looking at rules around harmful content, China is the first to go a step further in legislating what kind of relationships with AI are permissible.

The came into force this month and cover AI systems designed for sustained, human-like emotional interaction, while excluding customer-service bots and productivity assistants like straightforward AI chatbots.

Companies must now inform users they are talking to an AI, encourage them to take breaks and remind them of their usage. They must also monitor interactions to identify if someone is becoming dependent or having a crisis, then direct them to support or halt the conversation altogether.

Firms must also conduct age checks. Those under 18 are now banned from virtual-partner services entirely, while those under 14 require parental consent to use any kind of human-like AI service offering emotional support.

Despite reports that China has banned AI companions entirely, specialist companion services are still available. However, some of the largest general-purpose platforms, such as Doubao, Qwen and Yuanbao, with hundreds of millions of users across them, have shut down their companion features in response to the new rules. Some users have reacted badly 鈥� one : 鈥淚 can鈥檛 accept that my AI lover will leave me forever.鈥�

China鈥檚 approach is about making AI relationships 鈥渓ess parasocial, instead of banning it outright鈥�, says at the Oxford China Policy Lab.

at Yale Law School鈥檚 Paul Tsai China Centre in Beijing worries that endlessly available, agreeable bots may teach children a distorted model of relationships. 鈥淎s we humanise the technology, we tend to dehumanise each other,鈥� he says. Similarly, China is trying to address emotional dependence on easier and more forgiving AI partners 鈥� at the expense of human relationships 鈥� among adults.

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Elsewhere, the UK has a minimum age of 18 for romantic companion chatbots, tagged on to a broader legislative plan to limit access to social media to under-16s, while California companions to identify themselves as artificial, direct users showing signs of self-harm or suicidal intent to support services and remind minors to take breaks every three hours. The European Union currently has .

Whether China鈥檚 approach of focusing on trying to reduce emotional dependency and addiction is the right one isn鈥檛 clear, in large part because of the difficulty in defining both terms. 鈥淗ow do you define emotional dependency? How do you define certain kind of addiction risk?鈥� asks Qian.

鈥淚 do not think it will really work,鈥� says sociologist at the University of Manchester, UK. Users can migrate to other specialist services, foreign platforms accessed through VPNs or models running on their own computers, they point out.

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