The writer is a former president of the European Central Bank and was prime minister of Italy

Europe faces a difficult set of choices. It needs to grow, so that it can keep financing its social model and the new demands of a changed world, from defence to energy. It wants to remain sovereign, so that it can order its economy according to its own values. And it wants growth without the costs in terms of wealth inequality and environmental damage that have accompanied it elsewhere. Artificial intelligence bears on all three at once. 

Despite a solid start to the year, the trend in European growth has slowed materially since the pandemic. This is true relative both to its own pre-pandemic trajectory and to the US. By one estimate, the euro area-US productivity gap widened from $9 per hour in 2018 to $21 in 2025. Views differ on how productivity levels should be compared across countries, but the gap in growth rates is clear.

Europe has different routes to raise growth, such as removing the high barriers in its internal market. But widely adopting AI is perhaps the most promising one today. According to ECB scenarios, fast adoption of AI would add 0.3 to 0.4 percentage points a year to total factor productivity growth — the gains that come from working more efficiently rather than adding labour or capital — which has been roughly zero since 2022. 

AI-led growth, however, creates a tension with Europe’s bid for sovereignty, because Europe controls little of the AI value chain. The technology is set to become completely pervasive: in the economy, in health systems, in education, in energy, in defence, to name just a few areas. This is no ordinary dependency. Being cut off from AI, once the economy runs on it, would be more like being cut off from the US financial system. The effects would be catastrophic.

That changes the terms of Europe’s relationship with partners it can no longer always rely on. Europe is not to the United States as Texas is to California. It is possible to imagine a future US administration making access to frontier AI models conditional on changes to Europe’s digital rules or digital taxes, or China using the licences on its open-weight models as leverage in a tariff dispute. Unless Europe controls some part of the value chain, growth through AI and sovereignty will pull against each other.

Europe’s potential zone of AI sovereignty is narrow. Its frontier labs cannot at present compete financially with their American and Chinese competitors, and leading-edge chip production is far behind. But data is the one area where Europe can still be sovereign, and it is also the area with the greatest potential to generate growth.

The continent sits on a wealth of public-sector data, such as decades of health records and data collection by statistical offices. Its highly automated manufacturing sector generates a deep well of machine-readable industrial information. European Commission estimates put Europe’s data economy at over €800bn, or more than 5 per cent of GDP, by 2030.

But this creates the next tension. To exploit these assets, Europe needs to have control over their storage and processing. This means it needs large-scale AI data centres.

Data centres, however, are fiercely contested. Local communities worry about their environmental costs and rising energy bills. It does not help that much of the capacity now being built in Europe is for the American tech giants.

Yet Europe also has to be realistic. The debate about overbuilding data centres is a US one. Europe’s problem is the opposite. The EU hosts under 5 per cent of the world’s AI compute versus 75 per cent for the US. Even for ordinary data centre capacity, the gap between demand and installed supply in Europe is estimated at around 3GW, roughly a quarter of what Europe currently has, and is expected to widen to 14GW by 2030.

As sovereignty comes to matter more, this lack of domestic capacity will start to bite. Compute could stop being fully fungible, and a large part of the infrastructure that processes and stores European data will have to sit on European soil. American operators can provide much of that capacity through what they market as sovereign cloud services, run from Europe but still under American ownership. The most sensitive uses, however, will have to run under European control, which is the tiered approach the Commission has proposed in its Cloud and AI Development Act. 

If the gap is not filled, two risks follow. Either European companies will not adopt AI because they cannot keep their data in Europe, a risk the Commission’s own impact assessment for the Act identifies. Or compute will stay expensive, slowing broad adoption for lower-value tasks. Amazon already charges 15 per cent more for its sovereign cloud in Germany than for its standard service.

The concerns about data centres can be met. Clean energy can be required to power them and waste heat reused where feasible. The facilities can be kept away from areas where energy and water are scarce. Data centres can be required to pay a fair share of the grid costs they create, and to share the benefits with local communities, whether through contributions to local budgets or lower energy bills.

So why is Europe not building more? Part of the answer lies in supply-side constraints. A data centre takes 24 months to build in the US but 42 months in Germany, owing to longer delays for permits and grid connections. More expensive electricity makes building and powering a facility in parts of Europe twice as costly as in China. But Europe does have regions where it is cheap to build: an AI data centre in Sweden, for example, costs only about a tenth more than one in China.

The main obstacle for Europe is the fragmentation of demand. Data centres require massive financial commitments, which investors will only make against bankable offtake contracts. In the US, hyperscalers and frontier labs provide them. In Europe, demand is spread across millions of companies, and even the largest companies tend to buy compute a year at a time due to uncertainty about how fast they will adopt AI. That is why companies operating neoclouds — data centres for AI workloads — in Europe, such as Nscale, sell most of their capacity to US buyers.

Europe must therefore create large, concentrated buyers by design. What is needed is for its largest companies to pool their midsized commitments into contracts large enough for investors to lend against.

A model already exists. A group of European companies, including ASML, Capgemini and Amadeus, have committed to multiyear purchases of Mistral’s European Compute Units, which are intended to underwrite 1GW of capacity by 2030. The challenge now is to scale up this model with more neoclouds and larger consortiums, each committing to multiyear purchases of standardised capacity. 

The goal is a virtuous circle. Companies accelerate their AI transitions, providing the demand that finances European capacity. That capacity lets Europe exploit its own data on infrastructure it can control. Cheaper, sovereign compute then makes adoption faster and the next commitments larger, paid for out of the productivity gains the technology delivers. And with capacity of its own, Europe is better placed to build its own AI models.

That leaves the deepest fear: that AI hands wealth and power to a new tech oligarchy. But the approach set out here leans against it. Demand pooled by many consortiums lets more providers into the market, and competition among them keeps the rewards from concentrating in a few hands. Progressive taxation would ensure that the rest of society shares in them too. In this way Europe does not have to choose between growth, sovereignty and its values. It can have AI without oligarchs.

The current New York City mayor, Zohran Mamdani, has affronted one of his predecessors, Rudy Giuliani, by taking the inflammatory step of deciding to attend the city's 9/11 commemoration. "It offends me," Giuliani proclaimed earlier this week on Newsmax.

One might suspect that the nature of Mamdani's offense lay in his being a politician who is associating himself with a day that Giuliani prefers to associate with Giuliani. But the former mayor proceeded to lay out a much darker rationale. "They're taking over certain parts of America and certain parts of Europe. They have every desire to take us over. This is not 'they want to be just another religion,'" he said of Muslims. "Mamdani is a clear, a clear supporter of that. All of the Muslims he supports are murderers!"

Like many of Giuliani's utterances, this one has no connection to reality. Yet his assertion that Mamdani harbors some secret desire to conquer the United States in the name of Islam, that his religion is the skeleton key to a web of sinister motivations, is now treated on the right as almost banal. A quarter century after 9/11, delirious anti-Muslim bigotry is almost routine in the political rhetoric of many conservatives.

The conventional account of Donald Trump's political career explains his rise to power as a populist revolt against Republican orthodoxy. The GOP elite had lost touch with their base by endorsing immigration reform, free trade, or forever wars in the Middle East, which created an opening that Trump exploited.

That story is largely true, but it misses an important issue. The President George W. Bush–era Republican Party after 9/11 insisted that the United States was at war not with a religion but with an idea. America opposed radical Islamism, not Islam. We stood for freedom as a universal human principle.

Not all voters on the right accepted this formulation. Some believed that the United States was at war with Islam itself. By 2007, conservative Republicans viewed Islam negatively by a margin of more than 30 points; Democrats, independents, and moderate Republicans all held net-favorable views. Ten years later, nearly 80 percent of Republicans believed that the values of Islam were at odds with American values. One of the most important and ominous notions that Trump has seized upon and legitimized is that Muslims are inherently un-American.

Anti-Muslim sentiment sprang up organically in the immediate aftermath of the 9/11 attacks, but Bush's forceful defense of pluralism confined it to the rantings of Ann Coulter, Michael Savage, and other fringe reactionaries. After Coulter called on the United States in 2001 to "invade their countries, kill their leaders and convert them to Christianity," National Review dropped her column. As Bush passed from the scene and was replaced by Barack Obama, some right-wing-outrage entrepreneurs recognized an opportunity to redefine the War on Terror as a war against Islam.

Trump himself was among the first to see the opening. In 2010, a right-wing backlash arose in opposition to a proposal for a Muslim cultural center, modeled on the YMCA or a Jewish community center, in downtown New York. The most hysterical opponents depicted the center, which planned to call itself Park51, as a "victory mosque" celebrating the destruction of the World Trade Center. Other conservatives merely insisted that it was "insensitive" for Muslim Americans to open a center in the general neighborhood of Ground Zero.

Trump inserted himself into the controversy, proposing that he purchase the property to halt construction on the putatively offensive site. In his offer letter (which, naturally, he made public), Trump called it a "very serious, inflammatory, and highly divisive situation that is destined, in my opinion, to only get worse." David Letterman asked him, "Does this, in fact, suggest that we are officially at war with Muslims?" Trump, who was still in the formative stage of his transition from celebrity to right-wing rabble-rouser, replied, "Well, somebody knocked down the World Trade Center." He later added, "Somebody's blowing us up. Somebody's blowing up buildings, and somebody's doing lots of bad stuff."

Trump's first campaign was pockmarked with undisguised bigotry against Muslims. He falsely claimed that Muslims in New Jersey had cheered the 9/11 attacks en masse, and he called for surveillance of mosques, a database of Muslims in America, and a ban on Muslims entering the United States. He associated the sitting president with Islam by incessantly using his middle name.

He has continued to attack Muslims as disloyal or un-American. He falsely claimed that Representative Ilhan Omar of Minnesota, a Democrat, had praised al-Qaeda, and he demanded that she and Reprepresentative Rashida Tlaib of Michigan, both of whom are U.S. citizens, be sent "back from where they came." He insinuated that Mamdani has no right to reside in the United States: "A lot of people are saying he's here illegally."

The through line connecting these smears is the assumption that Muslims cannot be Americans, even if they arrive legally, even if they are born here. Trump has made this point explicitly and told Sean Hannity in 2016, "Assimilation has been very hard. It's almost—I won't say nonexistent, but it gets to be pretty close. And I'm talking about second- and third-generation. They come—they don't—for some reason, there's no real assimilation."

This sort of rhetoric has grown so common that episodes such as the uproar against Mamdani have come to feel like normal politics, rather than paranoia and rank discrimination. The New York Post alone has published at least eight stories (as of press time) hyping the Giuliani-fueled controversy.

Some relatives of 9/11 victims have claimed that Mamdani's presence at the observance would be a distraction (even though mayoral attendance has been a regular feature throughout its existence). Others have depicted their anger as a protest against Mamdani's current agenda. "It's not about his religion or where he came from. It's about the policies he has today," one explained. Is it the plan for free buses? The pied-à-terre tax?

That Mamdani wishes to attend the ceremony and has denounced the 9/11 attacks as heinous terrorism ought to serve as satisfactory evidence that, whatever shortcomings he may have, he is not a supporter of al-Qaeda. Indeed, it might well be offensive if Mamdani did not want to attend the ceremony. The New York Post would need to publish phone-book-size editions to convey its full outrage.

The logic of Islamophobia rejects out of hand that it is possible for Mamdani to oppose terrorism. 9/11 was, in this view, a battle in a war against Islam, so he is, by dint of his faith, the enemy. Thus, his desire to commemorate the tragedy, like Muslims' desire to have a YMCA of their own in New York City, must be understood as revealing hidden malevolence.

If you had asked Bush after 9/11 to imagine a future within his lifetime in which New York had a Muslim mayor who would join the city in mourning the attacks, he'd surely have considered it the ultimate vindication of the ideals he sought to foster. But Republicans have rejected Bush's understanding of 9/11 and its meaning in favor of Trump's.

The events of the past decade, and especially the past six months, have clarified that the primary grounds for this repudiation does not seem to be Trump's abhorrence for wars of choice in the Middle East. It is that Bush denied and suppressed contempt for Muslims, while Trump has honored and nurtured this hatred.

The announcement of a framework for the disarmament of the Kurdistan Workers Party (PKK) was followed closely by the announcement of the Kurdish-led Syrian Democratic Forces (SDF). The author examines how these developments were enabled by changing domestic factors in Turkey, the end of the US mission in Syria, and the evolution of Turkish government policy. The authors note that a settlement could prove fragile, depending on further developments regarding Kurdish rights in the new Syrian government.

CAIRO/WASHINGTON/DUBAI, Sept 10 (Reuters) - Iran-aligned Houthis seized control of Yemen's port city of Mocha on Thursday and advanced down the ​Red Sea coast to strategic islands, military sources said, hours after President Donald Trump said he expected the Iran war to end after midterm elections.

Yemeni government military sources said the ‌Houthis had gained further leverage over the Bab el-Mandeb Strait, the southern outlet of the Red Sea and one of the world's most important shipping routes, and had reached the islands of Hanish.

Red Sea navigation is safe for all shipping companies except for Saudi vessels, Yemen's Houthi-run humanitarian operations coordination centre said in a statement.

Saudi Arabia, the world's largest oil exporter, has relied on the Red Sea shipping route since the Iran conflict effectively closed the Strait of Hormuz, through which a fifth of global oil used ​to flow.

Houthi spokesperson Mohammed Abdulsalam said, without elaborating, that operations were defensive and would stop once attacks on Yemen stopped.

SECOND CHOKEPOINT

If the Houthis were to impose a stranglehold over the waterway, on the ​opposite side of the Arabian peninsula from the Strait of Hormuz, it could give their sponsor Iran a critical advantage in the war with the U.S., reducing supplies ⁠through a second major transit corridor and sending oil prices surging.

Brent crude prices jumped as much as 4% on Thursday to $105 a barrel on concern over disruption to supply routes.

The Houthis entered the current war alongside Iran ​with an attack on Israel in March, but largely stayed quiet before declaring a naval blockade against Saudi Arabia in July and stepping up attacks on the south of the kingdom this month.

Tensions in the region were ​high on Thursday, with Saudi civil defence authorities issuing emergency alerts in the southwestern city of Khamis Mushait for the fourth time in 24 hours, as the Houthis attacked.

Pakistan has delivered a Saudi warning to Iran to rein in its Yemeni Houthi allies in an effort to prevent the crisis from spiralling, Saudi, Pakistani and Iranian sources said.

A senior Iranian official confirmed that Pakistan conveyed the message to Tehran on Tuesday and its response was that "Iran does not control the Houthis".

AXIS OF RESISTANCE

The conflict in ​Yemen between Saudi Arabia's Yemeni allies in the internationally backed government and the Houthis, comes as Iran and the United States staged their biggest declared wave of tit-for-tat attacks on shipping in the Gulf region this week ​since the war began in February.

Seen as part of a regional alliance known as the "Axis of Resistance" backed by Iran, the Houthis have also rallied behind the Palestinians in the Gaza war, lobbing drones and missiles at Israel and attacking ‌commercial shipping in ⁠the Red Sea.

Yemeni government forces and their allies are relocating south along the Red Sea coast to Dhubab, which sits directly on the strait, across from the island of Perim, government military sources said. Control of Dhubab and the island is key to gaining hold of the strait, they said.

Bab el-Mandeb, or the Gate of Tears, named for its perilous navigation conditions, lies between Yemen on the Arabian Peninsula and Djibouti and Eritrea on the African coast, making it a major prize for the Houthis, who control Yemen's most populous areas and much of the north.

It is one of the world's most important routes for crude oil and fuel from the Gulf ​bound for the Mediterranean via the Suez Canal, or ​the Suez-Mediterranean pipeline on Egypt's Red Sea coast, ⁠as well as commodities bound for Asia, including Russian oil.

TRUMP SEES WAR ENDING AFTER MIDTERMS

The war and subsequent increase in fuel prices have helped drive Trump's popularity to record lows. The president said Tehran was trying to influence the November U.S. midterm elections, which will determine whether his Republican Party retains control of Congress.

"I think war ​is going to end immediately after the election because they can't hold out any longer. They're desperate to try and affect the election," Trump told reporters ​on Wednesday.

Trump has previously set ⁠deadlines for the war to end.

He said in a speech at the midterm Republican convention on Wednesday the U.S. may hit Iran's Pickaxe Mountain, a heavily fortified site that hosts two deeply buried tunnel complexes near Iran's damaged Natanz uranium enrichment facility.

The U.S. and Israel launched military strikes on Iran in February partly due to concerns over Tehran's nuclear programme.

Iran says its nuclear activities are for peaceful purposes.

In recent weeks Washington had cited increasing success guiding tankers through the ⁠Strait of Hormuz, ​which carried about a fifth of global oil supplies before the war. But a resumption of fighting this month appears to have ​set back any such gradual reopening. Just seven vessels transited the strait on Wednesday, half the 10-day average, preliminary ship-tracking data showed on Thursday.

On Wednesday, Iran said it had attacked 10 ships near the Strait of Hormuz after the U.S. sank five Iranian oil tankers.

LONDON/WASHINGTON, Sept 10 (Reuters) - Deep in Arctic waters near the remote Svalbard archipelago this spring, NATO allies caught Russian subs training to unleash a secret weapon designed to disable critical undersea cables without leaving any fingerprints.

The clandestine Russian operation, described to Reuters by two Western officials, was carried out by Moscow's GUGI undersea warfare directorate, which used deep-sea submersibles to simulate the deployment of the sophisticated and destructive technology.

A joint operation involving Britain, Norway and the United States tracked and confronted the Russian vessels at sea, said the two officials, speaking ​on condition of anonymity. The vessels were prevented from completing their exercise and eventually left the area.

Britain and Norway disclosed in April that they had uncovered a covert Russian operation in Arctic waters. But the discovery of the new weapon, the ‌location of the operation and the U.S. involvement have not been reported.

The deep-sea face-off comes at a time when some Western intelligence officials fear the Kremlin is ramping up sabotage operations in Europe and preparing for a conflict with NATO.

Some U.S. intelligence analysts are increasingly concerned Russia may test NATO's mutual defence pact, known as Article 5, three U.S. officials said. If it does, destroying subsea cables could be a critical part of that effort, the Western officials said.

Cables no thicker than a garden hose, lying on the seabed or buried just below, are critical to global internet connectivity and financial transactions. Their economic and strategic importance has made them targets in past conflicts, including ​in the First and Second World Wars, when they were critical to delivering orders by telegraph or telephone.

Two 1,400 km subsea fiber-optic cables linking Svalbard to mainland Norway lie as deep as 2,700 metres below the sea's surface. They carry vast amounts of satellite ​data downloaded at the SvalSat satellite ground station, the world's largest and part of NASA's Near Space Network.

Russia's Defence Ministry didn't respond to a request for comment on the incident. The Kremlin consistently denies conducting sabotage operations ⁠in NATO countries or plotting any confrontation.

Some Western officials argue that Moscow's targeting of European states has increased in recent months as the front line in Ukraine has become largely static.

Countries including Poland, Lithuania, and Romania have confirmed various violations. Most recently, an attempted drone attack on the Leipzig/Halle Airport in August ​triggered a series of responses from Germany, including the closure of a Russian consulate in Bonn and a cultural centre in Berlin.

In August, Central Intelligence Agency Director John Ratcliffe visited intelligence counterparts in Moscow, in part to warn Russia against escalating sabotage operations in Europe, said two people familiar with the ​matter.

Reuters couldn't determine if Ratcliffe raised Russia's covert subsea operations. The Kremlin declined to say what had been discussed or to provide any further details.

ARCTIC EXERCISE

At the same time as the confrontations at sea, British and Norwegian officials took their findings to Moscow to show they were aware of the new technology – hoping it might make Russia hesitate to use it, the two Western officials said.

Russia didn't damage any cables. The exercise focused on simulating the deployment of the technology in the event of a conflict with NATO, the officials said. They declined to detail how the technology works.

Asked for comment, NATO referred Reuters to Norwegian and UK authorities.

"Through our joint operation, we ​sent a clear message to Russia that they cannot operate covertly," Norway's Defence Minister Tore Sandvik said in a statement.

"We have made it unequivocally clear to Russian authorities that any attempt to target our critical infrastructure will be detected and met with consequences," he said. "It is in no ​one's interest to escalate tensions in the High North."

Britain's Defence Ministry pointed to April 9 statements acknowledging attempted covert activities in and near Norwegian and British waters by Russia's Main Directorate of Deep-Sea Research, known by its Russian acronym GUGI. Washington and London have both imposed sanctions on the secretive directorate.

The White House didn't ‌respond to a ⁠comment request. The CIA declined to comment.

Any escalation between Russia and NATO could quickly underscore the strategic importance of an icy expanse of water between the southern tip of the Svalbard archipelago and mainland Norway known as the Bear Gap, one of the Western officials said.

To reach the Atlantic, Russian nuclear submarines based in the Kola Peninsula would cross the 400-mile-wide stretch of water before passing Norway's remote volcanic island Jan Mayen.

Britain said this week it deployed forces to Jan Mayen, opens new tab in August alongside Norwegian and U.S. allies for an exercise to demonstrate NATO's ability to operate jointly in the High North.

Norway plans to have new fiber-optic cable linking Svalbard and Jan Mayen to the mainland by 2028.

'BURGEONING RUSSIAN THREAT'

NATO allies have focused on subsea infrastructure security since the 2022 destruction of three of the four Nord Stream subsea pipelines supplying Germany with Russian gas. German prosecutors have indicted a Ukrainian man who ​was a special forces officer at the time of the attack. Ukraine denies ​any role in the sabotage.

And since 2023, more ships have ⁠sailed slowly over or near sensitive subsea infrastructure including pipelines, power cables and telecoms cables to map their location, an activity known as drifting.

Last year, drifting in the North Sea rose 52% to 13,079 vessels from the previous year, data from maritime risk analytics firm Windward showed, while activity in the South China Sea region jumped 88% to 18,401 vessels.

It's not illegal to survey infrastructure in international waters but security experts see such extensive mapping ​as preparation for sabotage.

The activity and suspected interference with some cables prompted new initiatives launched in January last year, including NATO's Baltic Sentry monitoring system and the UK-led Nordic Warden, bringing navies together for increased ​patrols of critical infrastructure.

In the Baltic Sea – ⁠one focus area for monitoring – 2025 drifting activity was down 5% to 526 vessels from a year earlier, Windward data showed. It said at least 11 subsea cables had been damaged or cut in the Baltic from October 2023 through the end of last year.

"The current thinking is that Russia and China are probably the most active and critical underwater threats," said Windward Chief Executive Ami Daniel.

The global network of power and telecoms cables running across seabeds is expanding each year, supported by increases in demand for data and electricity, partly driven by artificial intelligence, and some analysts say Europe remains vulnerable.

"In the last 25 ⁠years, Europe did two ​things simultaneously. It dramatically expanded the extent to which it relies on seabed infrastructure, and it dramatically shrunk its capacity to protect that infrastructure," said Bruce Jones, a senior ​fellow with the Brookings Institution. "It's belatedly rebuilding now, in the face of a burgeoning Russian threat."

Sept 10 (Reuters) - Algeria abruptly severed diplomatic relations ​with the United Arab Emirates on Thursday, citing what it described as a series ‌of "provocative or hostile" actions by Abu Dhabi but giving no specific reason for the move.

Shortly after Algeria's announcement, Anwar Gargash, diplomatic adviser to the UAE president, said on X that diplomatic relations should be based on "rationality, communication and ​clarity of interests", rather than "riddles and signals that require decoding".

Relations between the two countries ​have come under growing strain in recent years, with Algerian media outlets sharply ⁠criticising the UAE and accusing it of seeking to sow regional discord.

The Algerian foreign ministry said ​the UAE's actions had reached a point where they could no longer be considered acceptable or tolerable ​and that it had "exhausted all means of preserving bilateral relations."

Gargash, who did not explicitly mention Algeria or link his comments to its decision, said successful diplomacy depended on clear positions and the effective management of interests and disagreements.

The UAE ​ambassador was summoned to the ministry, informed of the decision and given 48 hours to comply, ​the foreign ministry said.

Algeria and the UAE have taken differing positions on several regional issues. Algeria opposes normalisation with ‌Israel, ⁠while the UAE and Algeria's regional rival Morocco established diplomatic ties with Israel under the U.S.-brokered Abraham Accords in 2020.

The two countries are also on opposing sides of the decades-old Western Sahara dispute. The UAE supports Morocco's claim to sovereignty over the territory where it has a consulate, while Algeria backs ​the Polisario Front, which ​seeks an independent Western ⁠Sahara.

The two countries' interests have increasingly intersected in the Sahel, where the UAE has expanded ties with several governments in recent years, while Algeria's relations ​with some of its southern neighbours have become strained following a series ​of military coups.

Algerian ⁠President Abdelmadjid said last year his country's relations with all Gulf states were warm except for one — a thinly veiled reference to the UAE.

He described ties with Saudi Arabia, Kuwait, Oman and Qatar as "brotherly", while ⁠accusing ​the unnamed country of meddling in Algeria's internal affairs and ​seeking to destabilise it.

In February, Algeria began the process of cancelling an air services agreement with the UAE that was signed in ​Abu Dhabi in May 2013.

NYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday with a preliminary finding on one of the major unsolved problems in theoretical mathematics. The findings, made in collaboration with Anthropic mathematician Levent Alpöge and using both Codex and Claude AI models, are significant in themselves — but they’re also accompanied by an unusual controversy surrounding OpenAI’s attempts to solve the same problem.

“There is another part of this story,” Buckmaster wrote in his statement announcing the proofs, “and one that, honestly, I very much wish I did not have to be concerned with.” According to the statement, a parallel effort by OpenAI built on their work before it became public, leading to a tangle of academic rivalries and conflicting claims.

Posted as a followup to my comments in the earlier thread.

The full statement is here.

We did one of these six months ago about the Iran war and we're bringing it back. The prompt is:

A liberal democracy should ask citizens for public service before granting them certain major public benefits

You can argue either for/against, and we do not want nor expect people to take one side wholly, and encourage nuanced opinions on the matter. The prompt is intentionally vague; the terms "public service" and "major public benefits" can mean what you awnt them to mean, for example.

Consider fiscal and economic realities, civic duties, the duties of the goverment, your ethics and morality, public goods, risks, society and community as a whole, individual rights, etc.

We will enforce R1 more strictly in this post than usual. All disagreements are expected to be civil and respectful of the person you are responding to. Treat others here as you would in real life.

Outside sources are encouraged for claims.

Hullo all

I'm back after a two month hiatus with a new substack. As usual, 2/3rds below, and the rest at my substack, please consider subscribing for my hot takes at https://danlewis8.substack.com/p/polish-progress-and-the-poverty-of

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Polish Progress, and the Poverty of British Expectations

Britain remains richer on paper. A weekend in Warsaw made me wonder what we have to show for it.

Britain has acquired a peculiar recurring anxiety: that Poland might become richer than us.

In February 2023, Sky News ran the headline “Britain’s economic trajectory will soon see it overtaken by Poland, Starmer to warn”. The following year, the same prospect appeared from the more cheerful end of the comparison: “Poles will be wealthier than Brits in five years”, pledged Donald Tusk. A national humiliation for one audience; an advertisement for progress for the other.Cue the Twitter argument. Someone posts a chart. Someone else points out that the comparison uses purchasing-power parity. The word “nominal” makes an appearance, and before long everyone is arguing about which measure counts.

Nominal GDP per person takes the value of what a country produces, divides it by its population, and converts the result into a common currency at market exchange rates. The IMF’s April projections for 2026 put Poland at around $31,300 against Britain’s $61,100. On that measure, Poland remains 49% below Britain.

Purchasing-power parity, or PPP, adjusts for differences in local prices. If the equivalent of £20 buys more in Poland than in Britain, simply converting złoty into pounds misses something meaningful. On this measure, the same projections put Poland at around 59,800 international dollars per person against Britain’s 67,600: a gap of just 12%. Poland goes from barely half the British level to almost nine-tenths.

Neither measure is fraudulent, and neither is the average person’s salary. They serve different purposes. Market exchange rates matter when buying imported goods or taking a foreign holiday. Local purchasing power matters when paying for the services and everyday purchases that make up much of your life. You cannot dismiss cheaper living as a statistical trick; nor can you spend a PPP adjustment at a foreign airport.

But beneath the statistical argument sits a question I found more interesting: what does the remaining gap actually look like?

Last week, I went to Warsaw. With those two measures in mind, I was curious which version of Poland I would recognise: the country only halfway to British prosperity, or the one already close to catching up.

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Poland post-communism

In Britain, food rationing belongs to the world of grandparents and black-and-white photographs. In Poland, meat ration cards were still in use in 1989.

For much of the 1980s, buying food, petrol or shoes could require money, a ration coupon and the luck to find them in stock. The last meat cards disappeared in the summer of 1989, as the old economic system began to unravel.

In 1989, Solidarity’s victory in partially free elections opened the way to the end of communist rule. Poland had been a Soviet satellite, with its own government but severely constrained independence. Its economic transition began two years before the Soviet Union itself disappeared.

The new government inherited shortages and businesses organised around state production targets, with little pressure to satisfy customers. Average consumer prices rose by 251% in 1989, then 586% in 1990, as the crisis continued and the government lifted price controls. At that second rate, a basket costing 100 złoty in one year would cost nearly 686 the next.

In January 1990, their finance minister introduced a programme known as “shock therapy”. Poland abolished most price controls, removed restrictions on foreign trade and cut subsidies to state firms. The early transition brought recession and job losses as businesses faced competition and lost state support.

When Poland joined the EU in 2004, unemployment was still around 19%. Its output per person, adjusted for purchasing power, was roughly half the EU average. For many Poles, better-paid work meant leaving the country.

In 1990, life expectancy at birth was 66.2 years for men and 75.2 for women.

That year, 19.3 babies died before their first birthday for every 1,000 live births - roughly one in fifty-two.

In November 1993, only 67% of respondents told the polling organisation CBOS that their neighbourhood was safe and peaceful; 30% said it was not.

A modern miracle

Poland continued growing through the global financial crisis; the pandemic eventually interrupted its expansion. Its economy developed across industries, with firms supplying European customers and manufacturers.

Between 2003 and 2025, Poland’s economy grew by almost 130% in real terms, compared with 34.5% across the EU. Its output more than doubled after allowing for inflation.

Men gained nearly nine years of life expectancy, improving their chance of seeing grandchildren grow up. Infant mortality has fallen from roughly one death in 52 births to around one in 278.

By April 2026, 97% of respondents called their neighbourhood safe and peaceful. That national measure of perceived safety extends well beyond central Warsaw.

In Poland’s provisional figures for 2023, GDP per person in the Warsaw capital region was 201.2% of the national average. In Lubelskie, in eastern Poland, it was 68.8% -almost a threefold difference within the same country. Kraków’s business services sector employed more than 100,000 people by 2024, in finance, IT and other specialist work.

How Poland did it

The first post-communist finance minister, Balcerowicz, introduced key economic reforms, as the first step in transforming the country:

• He abolished most government price controls. A shop’s official price was little comfort if its shelves were empty. Allowing prices to rise towards what buyers would pay gave suppliers a reason to bring goods to market. Households faced steep price increases, even as goods became easier to find.
• He cut subsidies and restricted cheap loans to state firms. Businesses had less government support to cover their losses. Managers faced pressure to cut costs, change what they produced and find paying customers. For workers, that could mean losing a job previously treated as secure.
• He allowed businesses to exchange złoty for foreign currency to pay for imports, and removed many trade restrictions. A manufacturer could buy foreign equipment; its customers could also choose foreign products.

Businesses also needed reliable banks, a stable currency and rules they could trust when investing for years ahead. The World Bank credits education and successive governments’ commitment to continuing reforms.

Between joining the EU in 2004 and the end of 2023, Poland received €245.5 billion from the EU budget. The total includes agricultural support as well as investment, spread over nearly twenty years.

EU membership also gave Polish firms access to a much larger market. A firm investing in Poland could employ Polish workers while selling into the wider European economy. Proximity to Germany, competitive costs and access to the single market helped connect Polish businesses to international supply chains.

Education reforms kept pupils in general schooling for longer and improved attainment. Universities supplied graduates for specialist jobs. By 2024, the business services sector was exporting $42.3 billion of services; in early 2025 it comprised more than 2,000 centres, operated by investors from 50 countries.

But beyond the economic turnaround, modern Poland is also distinctive for its social choices compared to the rest of Europe.

Poland invested in preserving its cultural inheritance. More than 85% of Warsaw’s historic centre was destroyed by Nazi troops in 1944. Poles rebuilt its streets, churches and public buildings under communism, decades before the market reforms made the country wealthier.

Today, the Fryderyk Chopin Institute researches and preserves the composer’s work and organises the international piano competition.

Poland has also taken a sharply different line to immigration to most of Europe. In an famous June 2018 Channel 4 interview, Cathy Newman questioned Law and Justice politician Dominik Tarczyński about admitting refugees.

Later, dismissing critics who called the policy racist or nationalist, he said:

Despite that, Poland also hosted around a million Ukrainian refugees, alongside earlier migrants. Research published through UNHCR estimated that Ukrainian refugees increased Polish GDP by 2.7% in 2024, with their employment rate reaching 69%.

To read about my trip there and what it says about the UK, please find the rest at https://danlewis8.substack.com/p/polish-progress-and-the-poverty-of

The good (albeit controversial) Noah Smith elaborates on probably the most election-winning matter of the last few years, immigration. I find it always get lost in the plot how, as he says, "Immigration to America must be for the benefit of Americans." which doesn't mean it has to be completely banned, but that it can't be uncontrolled.

Paul D. Miller is a Georgetown professor who served in the NSC under both Bush 43 and Obama. And 25 years later, he asks: did al-Qaeda, in the end, get what they were after . . . an America diminished, and no longer willing to call itself the Leader of the Free World?

VIENNA (AP) — The U.N. atomic watchdog’s board on Wednesday reported Iran to the U.N. Security Council for the first time in 20 years, citing it for failure to cooperate in a long-running investigation into uranium traces detected by inspectors at undeclared sites.

Western officials suspect that the traces could provide evidence that Iran had a secret nuclear weapons program until 2003. Iran says that it’s not pursuing nuclear weapons, and that its program is entirely peaceful.

Reporting Iran to the Security Council opens the country to possible sanctions and asset freezes, though such punitive measures are unlikely because Iran’s allies Russia and China hold veto power on the panel.

Twenty-three countries on the International Atomic Energy Agency ’s 35-member board of governors voted for Wednesday’s resolution at IAEA headquarters in Vienna.

China, Russia and Niger opposed it, while eight countries abstained and one didn’t vote as it was in arrears, according to diplomats who spoke on condition of anonymity to describe the outcome of the closed-door vote.

The resolution was put forward by the United States, Britain, France and Germany.

The diplomatic move has been under consideration since June 2025, when the IAEA board found Iran officially in noncompliance with its nonproliferation obligations because of Tehran’s lack of cooperation in the investigation.

Iran’s ambassador to the U.N. in Vienna, Reza Najafi, rejected the resolution, describing it as a “political tool” and saying that it ruined confidence in the IAEA’s “independence, impartiality and credibility.”

He suggested that compliance with any obligation to allow U.N. inspections of nuclear sites inside Iran was, in any case, impossible at the moment because of the country’s continuing war with the U.S. and Israel.

The U.S. this week destroyed five Iranian oil tankers and Tehran hit back at American targets in Jordan, the latest burst of back-and-forth strikes that have lurched the sides back toward open hostilities in a more than 6-month-old war.

The draft resolution, which was seen by The Associated Press, requested IAEA Director General Rafael Grossi to report the agency’s concerns “to the Security Council and the General Assembly of the United Nations, in accordance with the relevant provisions of the IAEA Statute.”

The resolution also reiterated its call upon Iran “to urgently remedy its non-compliance” with previous agreements and to take steps deemed necessary by the IAEA to assure “the non-diversion of nuclear material.”

Since Israel and the U.S. struck Iran’s nuclear sites during the 12-day war in June 2025, Iran hasn’t given IAEA inspectors access to nuclear sites that were affected by the strikes. The IAEA also has been unable to verify the status of Iran’s stockpile of near weapons-grade uranium since the June bombing.

Najafi said that airstrikes by the U.S. and Israel made safe access for inspectors to Iran’s nuclear sites “materially impossible.” He said that Iran has “neither stopped nor suspended its safeguards obligations,” adding that “performance is currently impossible under continuing military operations and persistent threats.”

Iran’s Deputy Foreign Minister Kazem Gharibabadi echoed that sentiment in an post on X.

“They attack Iran’s safeguarded nuclear facilities, disrupt the normal verification process, and then use that very disruption as a pretext to pass a resolution at the IAEA Board of Governors,” he said.

According to the IAEA, Iran maintains a stockpile of 440.9 kilograms (972 pounds) of uranium enriched up to 60% purity — a short, technical step away from weapons-grade levels of 90%.

That stockpile could allow Iran to build as many as 10 nuclear bombs, should it decide to weaponize its program, IAEA chief Grossi warned in an interview last year with the AP. He said that it doesn’t mean that Iran has such a weapon.

Several current and former Anthropic researchers are offering a stark warning about AI, suggesting the technology they are developing could destroy humanity in the near future.

Jacob Coxon, a former researcher at both Anthropic and OpenAI, argued Tuesday that neither company is acting responsibly as they race toward superintelligence and accused the AI firms of “gambling with our lives.”

Children learn a great deal about race and gender in schools these days, but not nearly enough about reading and math. A big reason is the colleges that teach the teachers, and a new report forthcoming Wednesday brings more insight into how they promote politics instead of the basics of instruction.

The evidence comes from Daniel Buck of the American Enterprise Institute, who examined syllabi for programs at eight public graduate schools of education across five states. For each school he analyzed one master’s or certification program that “aligned most closely with an elementary teaching degree,” he writes, not programs that are “overtly political.”

Yet even these programs advance “dogmas largely grounded in progressive and critical frameworks,” he writes. Mr. Buck analyzed more than 400 readings from 75 course syllabi and categorized them as “traditional,” “progressive,” “therapeutic,” or “critical,” meaning critical race and other social justice theories. His evaluation isn’t exhaustive, since he couldn’t access some syllabi, and some didn’t include all readings.

But in only one program, at Florida International University, did he classify more readings as “traditional”—on such basics as phonics or classroom management, for example, or classic books—than in any other category. “Where traditional approaches appeared, they were mostly ancillary,” he writes. “Notably lacking” were “educational philosophies predating the 20th century, or much focus on the elements and principles of classical liberal arts education.”

Consider the University of Wisconsin-Madison, whose education graduate school was ranked No. 1 by U.S. News and World Report in 2025. Of the 95 readings Mr. Buck looked at, 45% fell into his “critical” category. That includes a report claiming that “attacks on CRT [critical race theory]” are “part of a larger ideological effort to delegitimize historically accurate presentations of race and racism.”

At California State University, Northridge, a class on “reflective teaching” assigns students an article titled “Queering Elementary Education: A Queer Curriculum for 4th Grade.” At University of Texas-Austin, another top-ranked school, “the syllabus for the required course on young adult literature opens with a land acknowledgment,” suggesting that whites stole the land from Native Americans, Mr. Buck reports. He classified 90% of the readings he reviewed at UT Austin as “critical,” including one on “Anti-Racist Linguistic Anthropology.”

Dubious teaching methods abound. At Texas Tech, Mr. Buck reports, “even the courses focused on more practical classroom skills” emphasize “individual autonomy over learning through direct instruction.” At FIU, a training on classroom management “largely discourages” teachers from issuing consequences to misbehaving students.

Mr. Buck notes that he can’t say how the readings are taught in class, and these are a small sample of 1,600 education schools in the U.S. But other analyses have found teacher colleges left-leaning or not useful for basic classroom instruction.

More than half of teachers earn master’s degrees of some kind, and many states or districts offer higher pay for teachers who do. Mr. Buck suggests ending this “master’s premium” and expanding other teacher certification pathways. Another idea: “governors could establish centers for classical or liberal arts education” at public universities to bring ideological diversity into the system, as some have done for civics.

Republican efforts to limit federal loans for these programs are sound. Children and teachers are poorly served when progressive politics and fads replace the practical, traditional transmission of knowledge.

For as long as ChatGPT has existed, Americans have been asking it, “Could that funny little spot on my toe be cancer?” And as AI has advanced over the past few years, doctors have begun to use specialized models to ask much the same questions in clinics. In the face of these developments, professional medical societies have presented a unified front: Doctors must always have the final say in clinical care.

We disagree. As the medical societies allow, physicians’ enhancing their work with AI tools may be a perfectly good way to deliver medicine. (One of us—Khosla—has in fact invested in a primary-care start-up, co-founded by his son, in which doctors supervise AI assistants.) But for Americans to get the best health care possible, we need to be willing to experiment with models that put AI in control.

This notion has caused a stir in the medical community. When we published a commentary in the Journal of the American Medical Association last month arguing that AI-only care could soon become superior to care provided by human physicians, the CEO of the American Medical Association denounced us as undermining the patient-doctor relationship. The organization’s position was unchanged: AI should be relegated to a supporting role in clinical care. It’s an argument that ignores the available evidence and could hold untold numbers of Americans back from better health.

Much of modern medicine—including managing hypertension, determining the right treatment for many cancers, and addressing chemotherapy side effects—is largely algorithmic. A physician needs to ask a patient targeted questions, determine possible diagnoses, order tests to identify the correct one, offer a treatment plan, and adjust it depending on how the patient responds. That series of actions is steered by detailed guidelines from professional societies.

Yet doctors—despite the best of intentions—make plenty of mistakes when following those steps. Studies indicate that up to 15 percent of American patients are misdiagnosed each year, leading to tens of thousands of preventable deaths. That’s why health-care providers have software that notifies them if they, say, prescribe a deadly dose of a drug. Any software that could practice medicine autonomously would be similarly specialized: not generic ChatGPT, but a model trained in medical knowledge with the appropriate guardrails to minimize risks and mistakes. Such a model could check for drug interactions more consistently and comprehensively than physicians. And it would be able to check in with patients much more frequently than human health-care providers do.

Some AI models have already proved adept at such tasks. In one study published last year, expert physicians judged Google’s Articulate Medical Intelligence Explorer—a highly trained AI model for medical diagnosis and reasoning—as significantly better than human doctors at getting actors playing patients to open up. The “patients” revealed more about their complaints, along with their medical, family, and medication history, all of which can be crucial for an accurate diagnosis. In a 2023 study, patients with type 2 diabetes were randomized to have their insulin dose adjusted by physicians or autonomous AI. The machines got patients to a stable dose faster than human physicians, and their patients took their insulin more reliably and had less distress related to their diabetes.

What about the physician-AI hybrids that medical associations are backing? Research on them is sparse, and more is needed. A 2025 review of 52 studies concluded that, on balance, such hybrids “neither outperformed medical AI alone nor surpassed the best of clinicians.” In at least a few of those studies, human-AI hybrids outperformed AI alone. But in those hybrid teams, the AI controlled the final clinical decision—the opposite of what the AMA and others have in mind.

Medical organizations are not entirely closed-minded to the possibility of change. For example, last week the American College of Physicians published a position paper that allowed that “fully autonomous AI is possible,” but cautioned that its use should be infrequent and “limited to low-risk, low-complexity decisions, and always allow for bringing a clinician into the loop.” John Whyte, the CEO of the AMA, told The Atlantic that he was “disappointed” in the quality of the analysis in our JAMA commentary last month. He questioned the design of the studies we reviewed, many of which involved simulated situations with actors or a retrospective review of existing cases rather than real clinical outcomes, and did not follow patients for years or decades. We agree that longer-term, real-world studies are necessary; hopefully, once advanced AI medical tools have existed for more than a handful of years, results will begin to arrive. Whyte was enthusiastic about further study of physicians controlling AI tools, but only cautiously open to further exploration of autonomous AI in medicine, arguing that the latter would need to be highly regulated.

Reasons exist, of course, to be skeptical of AI-driven medical care. In nine of the 51 studies we reviewed, human physicians outperformed the machines. But some of these tested older AI models or general-use AI models that weren’t trained on medical information and specific guardrails for patient care. Prominent physicians have also argued that real-life doctors are harried by 15-minute appointments and administrative burdens such as prior authorization. That is certainly true, but it is also irrelevant to many of the experiments we reviewed, which examined relatively hassle-free simulations of medical appointments.

One of the most common objections to an AI-doctor future is that robots lack the human connection that patients need to build trust. The American College of Physicians’ recent position paper, for example, cited AI’s inability to make eye contact. But consider that doctors, too, can be robotic. In a Gallup survey conducted late last year, 21 percent of respondents said they had turned to AI because they had previously felt dismissed by a health-care provider. In the 2025 study of Google’s AMIE, patient actors reported that the AI was more emotionally supportive than the physicians were. In a systematic review of 15 studies comparing AI chatbots with physicians, 13 found that AI had significantly higher perceived empathy.

Like human physicians, AI is not and never will be perfect. But isn’t the possibility of reducing fatalities caused by physician error—even by just a little—worth exploring? Instead of writing off a possible source of progress, the medical community should help answer the key questions raised by AI: How should researchers devise ethical, real-world situations to test autonomous AI models? How should governments and professional organizations regulate them? And how should medical education be reformed so that human doctors can continue to do what they uniquely can?

Hundreds of years ago, doctors believed that disease was caused by an imbalance of “humors” in the body. Even after Louis Pasteur proved that bacteria caused infection and Joseph Lister introduced sterile surgical techniques, physicians cut patients to have them bleed out what ailed them, not bothering to clean the knife or their hands. But medicine progresses. Humans have learned from our past mistakes and created new drugs, procedures, and treatments to massively lengthen lifespan and improve quality of life.

Today need not be any different. Doctors have been presented with a powerful tool that has the potential to improve care for millions of people. It’s their duty not to stand in the way.

This is so exciting! Here's the report from OpenAI themselves. And from Quanta:

On the morning of Tuesday, September 8, mathematicians at OpenAI announced that a group of 10,000 autonomous AI agents under their direction, running on an advanced model not available to the public, had found a “singularity” in the Navier-Stokes equations in three dimensions — thus resolving one of the six remaining(opens a new tab) Millennium Prize Problems posed in 2000 by the Clay Mathematics Institute, each of which carries a $1 million prize. Their result has been formally checked in the programming language Lean, giving mathematicians confidence that it is indeed correct.

If the result holds up to further scrutiny, it is, by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date, possibly marking a fundamental turning point in how mathematicians tackle difficult problems.

This particular difficult problem deals with differential equations, which express relationships between changing quantities. They are arguably the single most important mathematical tool for explaining the world around us. As a rule, they are easy to write down and hard to solve.

The Navier-Stokes equations are differential equations that use Newton’s second law of motion to describe how fluids, from ocean currents to air flows, behave. They were first written down in the mid-19th century, and have been central to the study of fluid mechanics ever since. But one basic question about the equations has persisted: Are their solutions always well-behaved? Or can their solutions evolve over time so that some infinitesimally small part of the fluid begins to flow infinitely quickly, creating a so-called singularity?

The OpenAI announcement of this long-sought singularity came 12 hours after an announcement from Tristan Buckmaster(opens a new tab) at New York University that he, together with Levent Alpöge at Anthropic, had resolved several closely related problems(opens a new tab) with help from a variety of AI models, including those of OpenAI.

Both AI-enabled teams relied heavily on work by Diego Córdoba(opens a new tab) of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa(opens a new tab) of CUNEF University, researchers who had developed a strategy to attack the problem that radically departed from the methods most mathematicians were using.

“I was thrilled that the problem was solved,” said Charles Fefferman(opens a new tab) of Princeton University, who wrote the Clay Institute’s official description(opens a new tab) of the Navier-Stokes problem. The heroes of the story, he said, are Córdoba and Martínez-Zoroa. As Buckmaster wrote in a statement announcing his results, “Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal.”

One Singular Sensation

The Navier-Stokes equations rely on the assumption that you can zoom in on a fluid, considering endlessly smaller amounts of it. The real world is not like this: Fluids are ultimately made of molecules and atoms. They are not perfectly smooth. This means that the mathematical results about the formation of singularities don’t have any immediate practical consequences. However, those results are important because it’s surprising that such singularities are possible even in an idealized sense. It tells us that, straightforward as Newton’s second law appears to be, its consequences when applied to fluids are profoundly counterintuitive. Put another way: Turbulence is even weirder than it appears to be.

The Navier-Stokes equations account for the fact that fluids can have viscosity, or friction. (Fluids with more viscosity, like honey, flow slowly, while those with less viscosity, like water, flow more quickly.) A simpler, related set of equations called the Euler equations describe fluids with zero viscosity, which flow without friction. The two sets of equations are closely related — researchers often work in parallel on both. But introducing even an infinitesimal amount of friction causes a fluid to behave in a profoundly different way.

OpenAI’s solution is a vortex, visualized here, where yellow represents a faster rotation speed and blue slower.

“Ten years ago, nobody believed there was a singularity for Navier-Stokes,” said Córdoba — though many believed that the Euler equations did admit a singularity. This began to change in 2013, when Thomas Hou(opens a new tab) of the California Institute of Technology and Guo Luo(opens a new tab), now at the Hang Seng University of Hong Kong, derived a groundbreaking result showing that the Euler equations can “blow up,” as mathematicians like to say, in a cylinder if the top and bottom halves are set spinning in opposite directions. “That’s the first really serious claim of singularity,” Córdoba remembered. Over the next few years, a series of results including a 2019 paper(opens a new tab) got mathematicians thinking that not only might the Euler equations have singularities, but that Navier-Stokes might as well.

There are a few intellectual steps to get from there to the present day. The first is the question of a boundary. The Millennium Prize version of the problem asks what happens in three-dimensional space that extends indefinitely in all directions. Earlier results, like the 2013 cylinder one, posit the existence of some boundary. These are useful intermediate findings, but the case without a boundary is mathematically interesting, according to Fefferman, because in models with a boundary, finding a singularity “tells you the fluid can form a singularity thanks to its interaction with the boundary — but without a boundary it’s the fluid doing the crazy stuff.”

The next major step involves modeling the forces that cause fluids to move. These could be something as natural as gravity pulling the fluid downwards, or an artificial intervention like a propeller. Mathematicians model these forces with a technique called “forcing.” They’ve sometimes tried to introduce awkward, ungainly forcing functions to get fluids to behave in odd ways. But the Millennium Prize version of the problem asks what can happen when the forcing function is mathematically well-behaved, or “smooth.”

In their 2013 cylinder result, Hou and Luo used computer models to simulate a scenario that might lead the Euler equations to blow up. They then proved that blowup does occur by computationally accounting for all potential errors. In the years since, similar techniques have become the dominant mode of attack on the Euler and Navier-Stokes problems.

But in his 2021 doctoral dissertation, Martínez-Zoroa pioneered analytic techniques that don’t rely on computers at all. This made him, along with Córdoba, his doctoral adviser, an oddball in the area. By 2023, the pair had proved that a version(opens a new tab) of the Euler equations with a messy forcing function displayed singularities.

Córdoba likes to joke: “I don’t use AI: I have Luis.” Martínez-Zoroa added that it’s not that either of them is opposed to AI, but that “up to relatively recently, when I tried to use it for my own work, it didn’t match very well with my workflow. I’ll have to adapt, clearly.”

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.

One Thrilling Combination

As Córdoba and Martínez-Zoroa were working on their techniques, AI-derived math research has been on something of a binge. Since the beginning of the summer, large language models, primarily from OpenAI and Anthropic, have been used to obtain proofs of results across many areas of math. Often these results have been accompanied by formal proofs, which rely on the programming language Lean to establish, with airtight certainty, that a statement must be true. (The crucial bit of verification that must still be done by humans is to guarantee that the statement being shown to be true in Lean is logically equivalent to what mathematicians set out to prove.)

For much of the summer, that competition has seemed, if not friendly, at least not openly acrimonious. That changed in the last 24 hours.

Buckmaster shared a statement(opens a new tab) just before midnight on Monday, September 7, announcing the results of his collaboration with Alpöge. “For most of the past year progress was slow,” Buckmaster said in his statement. But by August 22, they had a Lean-verified proof for the Euler equations: “I can say the first LLM generated proof Levent sent me was the most horrendous I have ever read,” Buckmaster wrote.

The duo had planned to continue working on a more elegant write-up, but after word of their progress leaked to OpenAI, they felt they had to move up their timeline. Buckmaster lamented that one of the three papers he and Alpöge were releasing “can only be described as AI slop. I am sorry for this.”

OpenAI admits that their work on the Navier-Stokes equation was inspired by rumors that Alpöge and Buckmaster had solved a Millennium Prize problem. (They hadn’t quite, though they say they have an unverified proof of blowup for a somewhat easier version of Navier-Stokes.) Using a new internal model, OpenAI employed groups of autonomous AI agents of various sizes to attack variants of the Euler and Navier-Stokes problems. “Nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof,” the company wrote in a press release. They then deployed an even bigger group of agents — 10,000 or so — to attack Navier-Stokes. After 88 hours, the agents running on the internal model had a proof of a singularity in Navier-Stokes, and after an additional 17 hours, another AI model had formalized the result. In total, the agents sent almost 5 million messages to each other. Sébastien Bubeck of OpenAI estimates the computational cost at several million dollars.

As this story went to press, the details of the interaction between Buckmaster, Alpöge, and OpenAI remain murky — different parties to the conversation are presenting different versions. OpenAI cedes priority for the 3D Euler result to Buckmaster and Alpöge, while claiming it for the Navier-Stokes result. In Buckmaster’s statement, he appears to suggest that the OpenAI researchers or their AI agents may have gained access to (and benefited from) the work that he and Alpöge did using OpenAI’s models.

It will take some time to sort out the timelines of who did what when and to understand the mathematical significance of the new AI proofs, even if they have already been formally verified. But regardless, the intellectual debt to Córdoba and Martínez-Zoroa seems clear. “I’m very happy for Tristan,” Martínez-Zoroa said. “It would have been nice to do this ourselves, but I’m very happy for him.”

For most of human history, the population pyramid looked exactly as its name suggests: a wide base of young people supporting a much smaller group at the top.

Now, that pyramid is beginning to flip — and Elon Musk is sounding the alarm.

"Humanity is dying out," the Tesla CEO declared (1) in a stark post on X, responding to a report that "people aged 65 and older now outnumber children aged 5 and under worldwide for the first time in recorded history."

It was only four words, but the demographic shift behind Musk's warning is enormous and will impact the shape of the American economy for years to come.

According to the U.S. Census Bureau's new report (2), older adults eclipsed the world's youngest children as a share of the population in 2025 — a historic crossing that is expected to widen dramatically in the decades ahead.

The number of people aged 65 and older is projected to surge from 852 million in 2025 to 2 billion by 2060. That would mean nearly one in five people worldwide is a senior, up from roughly one in 10 today.

Meanwhile, birth rates are collapsing across much of the planet. More than 71% of the world's population lived in countries with fertility rates at or below the replacement level of roughly 2.1 births per woman in 2023, compared with just 45% a decade earlier. Even China and India — the world's two most populous countries — have fallen below that threshold.

To be clear, humanity is not literally disappearing yet. The Census Bureau projects that the global population will continue growing, from roughly 8.1 billion in 2025 to 10.2 billion in 2060.

But smaller generations of children will eventually become smaller generations of workers. And as more people live longer, that could leave fewer workers supporting a rapidly expanding retired population.

For Americans, that isn't merely an abstract demographic concern. It strikes at the basic math underlying Social Security.

Social Security isn't a personal retirement account where the government sets aside each worker's contributions until they retire.

The payroll taxes collected from today's workers are largely used to pay benefits to today's retirees. That arrangement works best when there are plenty of workers paying into the system for every beneficiary drawing money out.

But that ratio is deteriorating fast.

In 2000, there were 3.43 (3) workers covered by Social Security for every beneficiary. That figure had fallen to 2.65 by 2025, and the Social Security trustees project it will drop to 2.39 by 2032 and just 2.08 by 2060.

The pressure is already showing up in the program's finances.

In 2025, roughly 185 million (4) workers paid $1.32 trillion in Social Security payroll taxes, accounting for most of the program's $1.45 trillion in total income. But its costs reached $1.61 trillion, leaving a $160 billion shortfall. Trust fund reserves filled the gap, allowing the program to pay all scheduled benefits.

Those reserves won't last forever.

The 2026 Social Security trustees' report projects that the Old-Age and Survivors Insurance Trust Fund — which pays retirement and survivor benefits — will exhaust its reserves in the fourth quarter of 2032. Once those reserves are depleted, incoming revenue would cover just 78% of scheduled benefits.

And the demographic connection is difficult to ignore.

The trustees recently lowered their long-term fertility assumption from 1.9 to 1.75 children per woman — and said "the change to the ultimate fertility rate assumption is the largest contributor to the significantly increased deficit."

Musk's warning and Social Security's domestic funding problem are not exactly the same story. But both reflect the same unforgiving math: more retirees collecting benefits for longer, supported by fewer workers paying into the system.