The train we will not catch

On 1 August, a model OpenAI has not even put on sale delivered ten results on mathematical problems stuck for decades, for two thousand dollars of compute. On 2 August, Europe brought into force its duty to write « generated by AI » at the foot of content. We have already missed the web, the cloud and data centres. This time there will be no resit.

In 1999, the mathematician Mikhail Gromov asked a question. He had just defined a family of objects he called sofic, whose property can be summarised as follows: an infinite mathematical object is sofic if it can always be approximated as closely as one wishes by finite objects, therefore concrete ones, therefore ones you can handle. Everything mathematicians use day to day is sofic. Does anything exist that is not?

Twenty-seven years, no answer. Neither proof nor counterexample.

On 1 August 2026, the construction of such an object was published. It did not come from a mathematician. It came from an internal version of a model OpenAI has not yet brought to market, and it is accompanied by a file written in a language called Lean, which lets a computer check every step of the reasoning. This is not an expert opinion. It is a file that either compiles or does not, and that anyone can download.

That is only one result out of ten. The others concern high-dimensional geometry, error-correcting codes, complexity theory and cryptography. Three of them settle problems left by Paul Erdős, whose list of open questions has served the discipline as a proving ground for half a century. Another refutes a conjecture by Alain Connes, a Fields medallist.

The total cost of the compute that produced these ten results, at public rates, is around two thousand dollars. Two hundred dollars apiece.

The figure calls for a caveat, and it is better to raise it yourself than to have it thrown at you. This is the cost of the successful attempts. OpenAI acknowledges having failed on other major problems, and nobody knows what those failures consumed. A selection published after the fact always flatters whoever publishes it. Two thousand dollars is not the price of discovery, it is the price of the winning ticket once the draw is known.

None of that changes what matters here. The private laboratory producing this is not selling a product. It is producing fundamental knowledge, the raw material for everything else, and producing it at a pace no public institution in the world can match.

What this result does to our encryption

Among the ten is an advance on what specialists call the closest vector problem. It is one of the questions underpinning post-quantum cryptography, the kind meant to resist future quantum computers, whose first standards are being deployed across Europe in banks, public administrations and operators of vital importance.

The alarmist reading already circulating must be cut short. This result breaks nothing. On the contrary, it shows the problem is harder than we knew how to prove, which strengthens confidence in those standards. Anyone writing that AI has just weakened post-quantum encryption will have understood nothing.

The real issue lies elsewhere and it is more unpleasant. The security of our communications for the next thirty years rests on a theoretical foundation that a private American system has just consolidated in a few hours, for two hundred dollars. We will not merely buy the models, nor merely the chips. We will also inherit the theory on which we build what we call our sovereignty.

Europe did not get the amount wrong

On 30 July, the president of the European Commission announced funding for seven gigafactories, the term for very large compute centres intended to train models, to the tune of ten billion euros of public money meant to attract twenty billion in private funds. Europe, she wrote, wants to become the leading continent for artificial intelligence. According to the Financial Times, American technology giants alone have committed more than a thousand billion dollars to their AI infrastructure since 2023. One hundred times the European figure.

That ten billion is not new either. It had already been announced in February 2025, and the same president then described it as the largest public investment in AI in the world. That is probably true. The largest public investment in the world weighs roughly a third of what a single American company spends on its servers in one financial year.

But stopping at the figure would be to miss the point. Europe did not get the amount wrong, it got the job wrong.

It decided to referee a game it does not play. Regulating presupposes the capacity to compel, and the capacity to compel presupposes holding something the other party wants. We have neither the models, nor the compute, nor the capital. We have a market and fines. That is real power, nobody in Mountain View ignores it, but it is the power of a tollgate and not the power of production. Labelling content produced by other people's capability is exactly what you do when all that is left is the packaging.

The obligation that came into force today is not absurd in itself. Knowing you are addressing a machine, being able to trace the origin of a doctored video, these are legitimate requirements that the state of disinformation makes necessary. The problem is not that Europe is doing this. The problem is that it has become its only register of action, and that it presents this as a leading position.

Our heralds on the ground

Meanwhile, in France, the voice that has prevailed for ten years is that of Luc Julia, who repeats that artificial intelligence does not exist, that it is only statistics, that the machine understands nothing and recombines.

Philosophically, the position holds up. Operationally, it has no effect. Whether a system understands or recombines changes nothing about the fact that a mathematical object nobody could find for twenty-seven years now exists, and that a machine built it. The debate about the word has no consequence for capability.

Julia was, for five years, scientific director of the country's leading carmaker. Recruited by its chief executive, decorated on account of that role, heard by Parliament, invited everywhere. He left Renault in January 2026. His last substantive statement, that same month, fitted into one formula: AI systems are not becoming more intelligent, they are becoming more saturated. Seven months later, the saturated thing settles three Erdős problems and refutes a Fields medallist's conjecture.

It would be unfair and lazy to cast him as an impostor. He himself corrected the account of his part in creating Siri, he does not preach inaction, and he argues for a national model trained on French data.

What should give us pause is not the man, it is the place we made for him. We did not choose him despite his thesis. We chose him for it, because it made our lag bearable. It offered every decision-maker a respectable vocabulary for arbitrating nothing, hiring nobody, funding nothing, while appearing clear-eyed. That was the service it rendered, and it is for that service that it was so handsomely paid in honours.

The mechanism has been running for twenty years and depends on no individual. The European search engine that was going to dethrone Google. The sovereign cloud that became an American licence wrapped in French law. Each time the same sequence: we fail, we reclassify the failure as a choice of civilisation, and we decorate whoever best explained that it did not matter.

Why this one will not be closed

Here is the serious objection to everything above. We have heard this talk before. We missed commercial web, then social networks, then data centres, and Europe is still here. Why would this wave be different?

Because the previous ones left a door open. An infrastructure gap can be closed with capital, late and expensively, but it can be closed: the servers exist, they can be bought, a continent that decides to build data centres ends up having them. The lag remained a gap, that is, a fixed distance that sufficient effort reduces.

Three locks make this one different.

The first is the order of magnitude of the capital. One hundred times is not a budgetary catch-up, it is a change in the nature of the spending, and no politically conceivable European plan closes it.

The second is talent. Researchers go where the machines are, because without machines they cannot work. This movement is self-sustaining and does not reverse by decree. A country can build a compute centre in three years. It does not rebuild a departed research school in three years.

The third is the most important and the only genuinely new one. A frontier model is used to build the next one. It writes code, it designs experiments, it now produces mathematical results nobody had obtained. Whoever holds the best model moves faster towards the next one than whoever holds none, and the lead of the next step adds to the lead of the previous one.

This is no longer a gap, it is a divergence. A gap is closed with money. A divergence is closed with time, and time is precisely what compounding denies us.

What remains open

None of this condemns us to inaction, provided we stop treating an industrial subject with legal instruments.

Access to frontier models must be treated for what it has become, a critical dependency, in the same way as a cloud provider in a third-party risk map. No serious European organisation today is unaware of where its infrastructure runs. Almost all are unaware of what they will do if access to the best models becomes restricted, more expensive or politicised. That question belongs to continuity planning, not to technology watch.

France must equip itself with a public capacity to evaluate capabilities, and not compliance alone. Faced with the announcement of 1 August, we have no choice but to believe a corporate statement. The published proofs are verifiable, which is in itself considerable progress, but someone still has to verify them, and to check that what was formalised does correspond to the original problem. That work remains human. We have nobody whose job it is.

Finally, the European regulation should come with an evaluation bearing on the capability gap and not on the compliance rate alone. A scheme that measures adherence to obligations without ever measuring the distance opening up produces what I call the green dashboard: every light passes, nobody asks questions any more, and the real fragility stays intact beneath the indicators. On the scale of a continent, that goes by another name.

Europe is about to discover that it did not miss a market. It missed the moment when it could still choose.

Sources

• Results, method, cost and attribution: OpenAI, « Ten advances in mathematics and theoretical computer science », 1 August 2026, with manuscripts and public deposit of the Lean certificates, https://openai.com/index/ten-advances-in-mathematics/ and https://github.com/openai/ten-proofs
• Reactions from the mathematical community and acknowledged failures on other problems: The Decoder and The Next Web, 1 and 2 August 2026.
• Transparency obligations: Article 50 of Regulation (EU) 2024/1689, applicable from 2 August 2026, Commission guidelines adopted on 20 July 2026, https://digital-strategy.ec.europa.eu/en/faqs/transparency-obligations-under-article-50-ai-act
• Gigafactory funding: communication from the president of the European Commission, 30 July 2026, and the « AI Continent » plan of April 2025.
• Order of magnitude of American private investment in AI infrastructure since 2023: Financial Times.
• Luc Julia's career and positions: Les Échos and Maddyness, January 2026; interview with the magazine L'Audace, January 2026.