Breaking AI & Tech News Analyzed

OpenAI has reported a major advance in AI reasoning after its model successfully challenged an 80‑year‑old conjecture in discrete geometry, the planar unit distance problem first posed by Paul Erdős in 1946.OpenAI’s Model Cracks the 80‑Year‑Old Planar Unit Distance ConjectureThe conjecture suggested that the number of equal‑distance dot pairs on a plane grows only slightly faster than the number of dots.OpenAI's reasoning system generated a family of point arrangements that exceed Erdős’s proposed limit.The result was announced on X and confirmed in a companion paper co‑authored by mathematician Thomas Bloom.Quantifying the Breakthrough: No Monetary Figures, but Scientific SignificanceWhile the article provides no financial data, the achievement is described as a “milestone in AI mathematics” by Tim Gowers.The validation by experts underscores the credibility of AI‑generated proofs, contrasting with a prior, unverified claim from last year.Implications for AI‑Driven Mathematical ResearchThe model’s ability to explore unconventional solution paths highlights AI’s potential to augment human intuition.Researchers, including Andrew Rogoyski, note that AI is becoming a fundamental tool for future scientific inquiry.The breakthrough may accelerate AI involvement in other open problems across mathematics.What the Next Steps Could Mean for AI and MathematicsFurther collaboration between AI systems and mathematicians is expected to refine the new constructions and explore their consequences.OpenAI’s upcoming IPO could bring additional resources to expand its reasoning capabilities.The community anticipates more AI‑driven insights that could eventually resolve the broader Erdős problems.

The LeadHS2, the UK's flagship high-speed rail project, has officially become the most expensive infrastructure endeavor in British history, with costs soaring to £102.7bn and trains potentially not running until 2039. Transport Secretary Heidi Alexander has labeled the original design a "massively over-specced folly" and the cost increases "obscene," yet continues to defend the project despite its clear failures.The Escalating Costs of HS2The project's financial trajectory has been nothing short of disastrous. What began as a more modest proposal has now ballooned to over £100bn, with trains potentially delayed until 2039—decades after initial promises. To put this in perspective, the cost has escalated so dramatically that it dwarfs even other famously extravagant projects like Trump's White House renovations or Dubai's Burj Khalifa. Despite nine different transport secretaries overseeing the project since its inception, the budget has consistently spiraled out of control, with no end in sight.Political Failures and MismanagementSuccessive UK governments have failed to take responsibility for this unfolding disaster. The project originated as a "vanity project" of the David Cameron coalition, with fundamentally flawed design choices including the wrong route, wrong speed, and improper termini. Prime Ministers from Cameron to Johnson to Sunak have all lacked the political courage to cancel the project, with Sunak merely scrapping the Manchester leg, making what remains even worse value for money. Civil servants and advisors have been overwhelmed by the 30,000-strong HS2 bureaucracy, while oversight bodies like the National Audit Office have failed to provide adequate scrutiny.The Case for CancellationThe strongest argument for HS2 is its cancellation. With no track laid and only two viaducts completed out of 52, the project is still in its early stages. The £44bn already spent should be treated as "sunk costs," and the focus should shift to more beneficial investments. Contrary to claims that cancellation would be prohibitively expensive, there's no logical scenario where the £60bn still planned for HS2 would provide better value than reallocating those funds elsewhere. Cancellation would also free up valuable urban development sites around London Euston and Birmingham's Curzon Street, which currently resemble construction disaster zones.Alternative Investments for Britain's FutureThe funds currently committed to HS2—potentially over £100bn—could transform Britain's infrastructure landscape. Instead of focusing on marginal time savings for journeys between London and Birmingham, the government could invest in re-signaling, electrification, and urban transit systems. Britain currently has only nine tram networks or metros, compared to France's 30 and Germany's 60. The annual £7bn HS2 budget could build new hospitals, schools, care centers, youth clubs, and courtrooms across the nation—investments that would address far more pressing needs than marginally faster rail travel for a small segment of the population.

The Resurgence of Religious Disputes in India From mosques to temples, old religious battles are back in India’s courts. A law passed after the Babri Masjid dispute was meant to stop exactly this. But new claims are moving forward again, as Hindu nationalism grows and many Muslims fear that the protections once promised to them are slipping away. The Growing Concern of Hindu Nationalism The rise of Hindu nationalism in India has led to an increase in religious disputes, with many Muslims feeling that their rights and protections are being eroded. The Babri Masjid dispute, which was a major flashpoint in Indian politics, was supposed to be a wake-up call for the government to take steps to prevent such conflicts. The Impact on Minority Communities The resurgence of religious disputes in India has significant implications for minority communities, particularly Muslims. Many Muslims fear that they are being targeted and that their rights are being ignored. The government has been accused of not doing enough to protect minority rights and of emboldening Hindu nationalist groups. The Future Outlook The future outlook for India is uncertain, with many experts warning that the rise of Hindu nationalism could lead to further polarization and conflict. The government needs to take steps to address the concerns of minority communities and to prevent further escalation of religious disputes. Yashraj Sharma (@yashjournals), Al Jazeera reporter Episode credits: This episode was produced by Marcos Bartolome and Sari el-Khalili with Spencer Cline, Noor Wazwaz, Tuleen Barakat, and our host, Malika Bilal. It was edited by Tamara Khandaker. Our sound designer is Alex Roldan. Our video editors are Hisham Abu Salah and Mohannad al-Melhemm. Alexandra Locke is The Take’s executive producer. Connect with us: @AJEPodcasts on X, Instagram, Facebook, and YouTube

2026 French Open tournament director Amelie Mauresmo confirmed that prize‑money figures will remain unchanged this year, even as leading players threaten a boycott over a perceived drop in their share of tournament revenue.The Standoff Over Prize‑Money Allocation at Roland GarrosTop players, including Aryna Sabalenka and Coco Gauff, have criticised the organisers for reducing the players’ revenue share to an alleged 14.3 %, far below the typical 22 % seen at standard ATP and WTA events. In protest, many competitors plan to limit media interactions to 15 minutes during the pre‑tournament press day. A meeting between the French Open committee and player representatives is scheduled for Friday, but Mauresmo reiterated that “we are not going to change anything” for the current edition.Financial Snapshot: Prize Money vs. Tournament RevenueTotal prize pool: 61.7 million € (up 5.3 million € from 2025)2025 tournament revenue: 395 million €, a 14 % year‑on‑year risePlayers’ share of revenue: projected 14.3 % in 2026, down from 15.5 % in 2024Singles champion payout: 2.8 million € (+250,000 € from 2025)Implications for Player‑Organizer Relations and Future Grand SlamsThe disparity between the tournament’s revenue growth and the modest 5.4 % increase in prize money fuels tension. Players argue that without a more equitable split, they may collectively boycott Grand Slams, echoing calls made earlier this month. The French Open’s increase follows larger hikes at the U.S. Open (+20 %) and Australian Open (+16 %), highlighting a widening gap in compensation strategies across the majors.What Comes Next: Potential Negotiations and Boycott RisksWhile Mauresmo pledged ongoing dialogue, she admitted that “discussions will continue, probably after the tournament.” The upcoming Friday meeting will test whether a compromise can be reached before the start of the competition. Should talks stall, the threat of a coordinated boycott by high‑profile players could pressure organisers to revisit the prize‑money formula for future editions.

The Early Days of Trump's Soccer Career Drive north from New York City and into the Hudson valley. Take Exit 17 and follow Route 7 as it heads south along the river, past the abandoned shipyard and the aptly named Cadet Motel. Hang a left after a few miles, wind up a long driveway and you’ll arrive at New York Military Academy. The Culture of Hazing and Discipline at NYMA The NYMA of the 1960s was entirely unlike the sleepy, near-abandoned campus that exists today, with a well-documented culture of hazing and abuse akin to Full Metal Jacket. Most instructors were hardened veterans, many of whom had served overseas during the second world war. Individual discipline aside, they also pitted students against one another, as Dobias did during twice-weekly “cage matches”, where one student would beat another into submission. Trump's Time on the Soccer Team Trump became part of the Dutchess County Scholastic League, a collection of small schools scattered throughout tiny hamlets in the Hudson Valley and just beyond. Trump joined NYMA’s soccer team after suffering an injury playing gridiron football. A photo of the New York Military Academy soccer team, featuring Donald Trump, has surfaced, providing a glimpse into his early days as a soccer player. The Truth About Trump's Playing Career Yearbooks exist, as do newspapers. Combing through them in search of clues about Trump’s playing career paints an interesting picture of Trump’s brief moment as a soccer player at NYMA, and helps add even more depth to arguably the most polarizing leader in US history. Some accounts describe Trump as an incredible athlete, while others are starkly different. Many describe him as a bully, a character trait that was only hardened amid NYMA’s culture of hazing and rigid discipline. The Legacy of Trump's Soccer Career It’s debatable whether Trump truly cares about the sport itself or simply likes the attention it brings him. But it’s a fact that in 2018, he welcomed international and domestic club teams to the White House and presented the Club World Cup trophy to Chelsea. Cristiano Ronaldo, Wayne Rooney and Lionel Messi have all visited with Trump; the latter was made into wallpaper while Trump went on a rant about the war in Iran. Fifa’s president Gianni Infantino at times seems glued to the US president.

The UK Football Policing Unit will send only three officers to the 2026 World Cup after failing to secure additional funding from US authorities, raising questions about security resources for the expected influx of England supporters.Funding Gap Forces Minimal Police PresenceUK Football Policing Unit announced it will deploy a lead officer, Supt Gareth Parkin, supported by two football officers – a significant reduction from earlier tournaments. The decision reflects a lack of financial support from US states and a recent 10% cut in Home Office funding.Financial Constraints QuantifiedHome Office funding reduced by 10%.US states are not providing funds for mobile police delegations this time.Previous deployments: 40 officers sent to Euro 2024 in Germany with government backing; similar arrangements existed for the Qatar World Cup.Security Implications for England Fans in the USPolice stress they will act as “cultural interpreters” for local law enforcement and do not anticipate disorder among the estimated 15,000 England supporters per group‑stage match.Group‑stage venues: Dallas, Boston, East Rutherford (New Jersey).Ticket sales: 3,500 tickets per match through the FA travel club, plus 10,000‑11,000 purchases via the FIFA portal.Additional 1,000 friends and family expected to travel.Future Policing Strategies for International TournamentsThe UKFPU is working with the Football Supporters’ Association and the Foreign Office to deliver safety messages and cultural guidance. Scotland will send officers funded by its government, focusing on Boston, suggesting a patchwork of funding sources may become the norm for future events.

The Diplomatic Implication President Donald Trump has twice suggested, since his summit with China’s President Xi Jinping in Beijing last week, that he may speak with Taiwan’s President William Lai Ching-te. That would mark the first direct contact between leaders of the governments since the US switched diplomatic recognition from Taipei to Beijing in 1979. It remains committed, however, under the 1979 Taiwan Relations Act to supporting the defence of the self-governing democracy. Taiwan's Response In a foreign affairs ministry statement on Wednesday, Taiwan’s President responded to Trump’s comments, saying he would be “happy” to talk to him. Taiwan was committed to maintaining a stable status quo in the Taiwan Strait, he added, but “China is the disruptor of peace and stability”. Beijing regards Taiwan as part of Chinese territory. The Data Analysis The statement comes as the White House considers a $14bn arms deal with Taiwan. China’s foreign ministry responded saying it “firmly opposes official exchanges” between the US and Taiwan, as well as US arms sales to the island. The Impact Analysis Trump’s comments suggest he may be willing to break with decades of diplomatic protocol, which will likely jar with Beijing, say analysts. Based on past events, Beijing will not be happy if Trump does meet with or talk to Taiwan’s president. When the former US House Speaker Nancy Pelosi visited Taiwan in 2022, her two-day visit sparked heightened tensions between the two countries. The Prediction Analysts say that included in its response is an awareness in Beijing that Trump is unreliable and unpredictable. If Trump calls Lai and announces that the US will “continue to support Taiwan and provide a large arms package; all hell will break loose”. However, he said, the very fact that Trump even entertained the idea of speaking with Xi about whether the United States would sell weapons to certain countries was a win for Beijing.

The Montreal Victoire's Historic Win For the first time in the short history of the Professional Women’s Hockey League (PWHL), the Walter Cup is leaving the United States. The Montreal Victoire beat the Ottawa Charge in an all-Canadian final that wrapped up on Wednesday night in four games. Expansion and Growth The day before Montreal won the Walter Cup, the PWHL announced the league is expanding to San Jose for next season. It was the fourth such expansion announcement in the three weeks. With Detroit, Las Vegas and Hamilton, Ontario, receiving the other three expansion franchises, the young league will head into the 2026/27 season with an imbalance in franchises between Canada and the US for the first time. The Data Analysis The PWHL will enter its fourth campaign with seven US franchises and five Canadian. The league saw a 77% season-over-season increase in viewership on YouTube. At the box office, the PWHL sold out Seattle’s Climate Pledge Arena, Boston’s TD Garden and New York City’s Madison Square Garden. The MSG attendance of 18,006 now sits as the US professional women’s hockey attendance record. The Impact Analysis The PWHL is riding record attendance in venues across the US and is hoping to arrive at a league-wide national US broadcasting deal. Coupled with the fact that the PWHL increased its league and team partnership portfolio by 35% season-over-season to 81 corporate partners, saw in-arena merchandise sales double and online merchandise sales increase more than 50%, business is booming for the PWHL. The Prediction With the expansion to San Jose and the success of women’s ice hockey in America, the PWHL is poised for continued growth. The league’s move to California makes sense given the state’s ties to the PWHL itself, with Mark Walter, the owner of MLB’s Los Angeles Dodgers, the NBA’s Los Angeles Lakers and the WNBA’s Los Angeles Sparks, owning the league.

The Lead: First Presidential-Level Nuclear Monitoring Russian President Vladimir Putin and Belarusian President Alexander Lukashenko have taken part via video conference in a joint nuclear forces exercise, marking the first time the two leaders have directly participated in such a training event. While senior military officials from both countries have conducted similar exercises quarterly, this direct presidential involvement signals heightened attention to the nuclear capabilities of the Russia-Belarus alliance. The Strategic Exercise: Expanding Nuclear Capabilities Opening the meeting held via videolink and broadcast live on the Kremlin's website, Putin stressed that the use of nuclear weapons remains "an extreme and exceptional measure for ensuring the national security" of the two countries. "Today, as part of the exercises, we are conducting the first joint training of the armies of Russia and Belarus on managing strategic and tactical nuclear forces," he said. At the same time, the Russian leader said the Russian-Belarusian nuclear triad – which are nuclear weapons capable of being deployed by land, sea and air – must continue to serve as "a reliable guarantor of the sovereignty of the Union State of Russia and Belarus" amid rising global tensions and emerging threats. According to Putin, the drills are aimed at practising coordination and interaction between military officials in the event of nuclear weapons use, including weapons deployed on the territory of Belarus. The Military Capabilities: Advanced Weapon Systems Demonstrated Russia's Ministry of Defence said in a statement that its forces launched a Yars ballistic missile and a Zircon hypersonic missile as part of missile tests during the nuclear drills. According to the ministry, the crew of a nuclear-powered submarine launched a Sineva intercontinental ballistic missile from a submerged position as part of the drills. The Russian military also conducted a launch of a Yars intercontinental ballistic missile from the Plesetsk Cosmodrome towards a testing range in Russia's Kamchatka region. In Belarus, a combat crew of the Belarusian armed forces carried out a practical launch of a ballistic missile from an Iskander-M missile system at the Kapustin Yar testing range, the ministry said. The exercises also involved Tu-95MS strategic bombers, which launched hypersonic air-launched cruise missiles, while a MiG-31 aircraft carried out a launch of a Kinzhal hypersonic missile, according to the statement. The Regional Impact: Heightened Security Measures The joint drills held from Tuesday to Thursday were met with concern from Ukraine and its NATO allies. Kyiv has repeatedly accused Moscow of planning to launch a new attack from Belarus, either on its territory or one of its NATO allies, such as neighbouring Baltic states. Ukraine's Security Service (SBU) said on Thursday that its units and the army were "carrying out a comprehensive set of enhanced security measures in the northern regions of our country" bordering Belarus. The measures – including stepped up checks of individuals and properties – "will serve as an effective deterrent to any aggressive actions or operations by the enemy and its ally", the SBU said in a statement. Separately on Thursday, a Ukrainian drone attack on a town in Russia's Bryansk border region killed three rail workers when it hit a locomotive at a station, Russia's state RZhD rail network said. Russian border towns and villages regularly come under Ukrainian fire as Moscow's offensive against Ukraine has dragged on into a fifth year. The International Response: NATO Convenes Amid Tensions NATO foreign ministers were scheduled to meet in Helsingborg, Sweden, on Thursday and Friday to discuss how to ensure that support for Ukraine remains substantial and sustainable for the long term. The meeting comes as Russia's military actions in Ukraine continue and as concerns grow about the expansion of military exercises involving nuclear capabilities. Earlier this week, Ukraine launched what Russian officials described as one of the largest drone barrages of the war towards Moscow, killing at least five people. These reciprocal military actions demonstrate the escalating tensions in the region and the increasing risk of broader conflict.