Mathpocalypse or new dawn, books that are certified human-made, and the rise of post-training

On 10 March 2016, AlphaGo was playing the second game of its match against Lee Sedol in Seoul. With its 37th stone it played on the fifth line from the edge of the board, a move no strong professional would have considered. Lee had stepped out of the room; when he returned and saw the move, he took nearly fifteen minutes to reply. Fan Hui was watching. The European champion had lost to the program the previous autumn. "It's not a human move," he told Wired. "I've never seen a human play this move." Go had been played for more than two thousand years, and its best players had built up a shared sense of what good play looked like. A machine had found something none of them had seen. On Tuesday night the complexity theorist Dana Moshkovitz sat down with a proof of a conjecture she had worked towards for her whole career. It had been produced by an unreleased OpenAI model. She texted her husband, calling it "some alien craziness", and once again a machine had found new ways to think about a long-studied field.

On 6 October, OpenAI published 722 mathematical manuscripts in a public repository on GitHub, grouped into 372 families. The company says it posed about 4,000 problems to an internal model and kept the significant findings. A typical proof needed about three hours of thinking time on ChatGPT Pro from a single prompt. The same model produced its Navier-Stokes solution in early September, eight days into its training run. That took 10,000 agents running for 88 hours and roughly $15 million of compute at customer prices. A month later it was turning out work on open problems at a rate of hundreds of papers a month, each for the cost of an afternoon. New Scientist put the volume at a decade of the Annals of Mathematics in a day.

A mathematical proof is checked one step at a time, and if every step holds, the conclusion cannot be wrong. That is why theorems from 300 BC are still in use. But proofs are hard to find and slow to check, so they have always been scarce, and the profession's institutions grew up around that scarcity. Priority, peer review, credit and the slow filtering of results into textbooks all assume that new results arrive a few at a time. The benefits reached the rest of the world late and indirectly. Mathematicians studied number theory for its own sake for three centuries before it became internet encryption in the late 20th century. Fourier's work on heat became the mathematics inside every JPEG and MRI scanner. Mathematics was a slow institution that fed fast ones, and its slowness gave everything downstream time to adjust.

The Association for Human Mathematics issued a statement, reposted on Terence Tao's blog, saying that "mathematicians did not ask for this work to be done" and urging mathematicians to stop working with OpenAI. Tao spoke at the International Congress in July. He argued that mathematics has always pursued several goals at once: solving problems, building theories, training students and producing work people find beautiful. Those goals have always moved together, so the profession never had to choose between them. Now that AI pulls them apart, a problem can be solved with no new theory, no trained student and nothing beautiful to show for it. He has proposed that no result should be published unless its authors can give a clear, expert-level talk on it. But such a standard would exclude many past breakthroughs. In 1976 the mathematician George Andrews found a sheaf of 138 loose, unpaginated pages in Srinivasa Ramanujan's handwriting in the Wren Library at Trinity College, Cambridge. They held around 600 formulas that mathematicians accepted long before anyone could prove them. The proofs eventually filled five volumes and took several decades, and the mock theta functions described in those pages are now used in black hole physics. The engineer Oliver Heaviside was attacked for using calculus he could not explain, even though it clearly worked.

“Shall I refuse my dinner because I do not fully understand the process of digestion?”

For 4,000 years mathematics has changed at human speed, and the slow pace has hidden how much of the modern world has come to rely on it. Equations, algorithms and proofs now underpin cryptography, the optimisation behind logistics and operations, and the financial system. We mapped OpenAI's release against an inventory of the mathematics that critical systems depend on, and found two kinds of dependency. The first is efficiency. Scheduling, routing, error correction, compression and graph analysis all run on the best algorithms anyone has found, so a better algorithm is a gift. The release contains several. The second is computational hardness. Online security relies on nobody having found a fast way to factor large numbers, compute discrete logarithms, solve lattice problems or find hash collisions, so a better algorithm in these areas could be highly disruptive. Every bank login, software update, payment system and messaging app depends on that second kind.

Which critical systems the new mathematics affects

We mapped OpenAI's 6 October release against the mathematics that today's critical systems depend on, from encryption to logistics.

• Encryption, digital certificates, internet routing and payment systems

  Untouched

  The release proves assumptions that engineers already build on. No key sizes need to change and no migration plans need to move.

• Quantum computing

  Mixed

  Some chemistry problems are proved hard even for quantum computers. That weakens the broad case for quantum speedups in areas such as drug and materials discovery.

• Optimisation solvers

  Limits settled

  Several long-standing questions about how close fast methods can get to the best answer are now closed, and a few faster algorithms appear.

• Scheduling, matching, error-correcting codes and network design

  One to three years

  These are new proofs with no working code yet. Engineers could turn them into practical improvements within one to three years.

• Trading and simulation

  Theory only

  The relevant work is theoretical so far, with nothing that changes the pricing or simulation models firms use today.

• Factoring, discrete logarithms and lattices

  Unknown

  These are the hard problems that keep encryption secure. The release contains nothing that weakens them and does not say why. The risk lies in what the model may be able to do here, which the release does not show.

Justin Drake, a researcher at the Ethereum Foundation, wrote this week that "recent days have been humbling for human mathematical intuition" and suggested cryptocurrency holders move assets to fresh addresses. Many in the crypto world expect that later breakthroughs may not leave them unscathed. Vitalik Buterin advised against a scramble. He also said there is "a good chance that the concrete security of lattices will take serious hits from the next two years of AI math". He cited that risk as one reason Ethereum's roadmap now relies only on hash-based cryptography for signatures. Lattices underpin the post-quantum standards that governments and banks are adopting, but they may turn out to be exposed to an unforeseen vulnerability.

Takeaways: Mathematics has become critical infrastructure for the global economy, and that is about to become very apparent. An unreleased model has produced hundreds of results in a matter of weeks, and more will follow, far faster than people can check them. Its output is alien and sits awkwardly with the profession's institutions. It may do little for many of the goals mathematicians set themselves, but it will prompt new work. Cryptographers already expected quantum computers to shorten the life of today's encryption, and AI may bring that day closer. AI-driven mathematics also raises hard questions about the rest of the infrastructure we rely on.

This week the UK Intellectual Property Office approved a certification mark from Books by People, a British startup that checks whether books were written by humans. The mark is a small thumbprint printed on the cover, and the first certified titles should appear by the end of the year. Five publishers have joined so far. The mark is now legally registered, so a publisher who puts it on a book that hasn't been certified can be challenged.

The approval follows a difficult year for publishing. His agent withdrew a $2 million deal for Jerry Falade's debut thriller after detection software scored it 97% AI-generated. Falade denies using AI to write it. In France, Thélyson Orélien's prize-winning first novel was dropped from the Goncourt longlist after an anonymous account posted detector results online. Orélien says he started the book in 2017 and finished a first draft in 2019, three years before ChatGPT was released. When journalists tested the same passages with other detectors, the results disagreed. The Goncourt also cited separate plagiarism findings in his earlier work. His publisher says it can neither confirm nor refute the allegations and has paused promotion of the book.

Books by People has chosen not to rely mainly on that kind of software. It asks authors for a declaration, then reviews their drafts, research notes and correspondence with editors. Its cofounder Esme Dennys says competing with ever-better models would not be sustainable. Organic food certification works in a similar way. No lab test can tell you how a carrot was grown, so inspectors look at the farm instead.

AI is still finding its way into the writing process in smaller ways. Authors use it to check grammar, summarise research and suggest edits. Falade says he used a small model on his own computer, but only for research, such as how a character might hide a body. Any certification scheme will have to decide which of these uses count, and its decisions will be disputed. Some in the industry worry that books without a stamp will come to be seen as suspect.

Takeaways: The Intellectual Property Office's approval gives readers a legally backed way to find books written by people, and the first ones will reach shops within months. Like food labelling, this will be useful because it tells readers how a book was made, and gives people the choice to go organic. But as ever there are grey areas, such as how much AI help a book can have and still count as human-written. At least for now, AI's writing skills are far behind the best human authors, and that should see true quality win out.

Our chart this week shows how frontier labs split their compute between three jobs. In early 2024, pre-training took 67% and post-training with reinforcement learning (RL) took 4%. By the third quarter of 2026 those figures had almost reversed, to 10% and 53%. Inference stayed at about a third throughout.

These are shares, not totals. Lab compute has grown many times over since 2024, so labs are probably still doing more pre-training in absolute terms than before. What has changed is how fast RL has grown. In RL, a model attempts a task, its output is graded, and it learns from the result, millions of times over. Most of the gains in coding, maths and agent work over the past 18 months have come from this process.

RL also needs different inputs. Pre-training draws on text that already exists. RL needs practice environments, such as software sandboxes, simulated websites and copies of business tools. Each one comes with a test that can tell good work from bad, and building them is now an industry in its own right.