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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.

Lead: Injury clouds USMNT defender’s World Cup prospectsCrystal Palace manager Oliver Glasner confirmed that US national team centre‑back Chris Richards has torn two ligaments in his ankle, putting his participation in the 2026 World Cup in doubt.Richards’ ankle injury sidelines him for upcoming fixturesInjury confirmed Thursday, described as “stable but quite swollen”.Will miss Crystal Palace’s match against Arsenal and the Conference League final in Leipzig.Glasner said recovery depends on reducing swelling and medical treatment.Minutes logged and caps underline Richards’ importance2,827 league minutes and 45 starts for Palace this season.13 USMNT caps earned since the start of 2025.Named USMNT Men’s Player of the Year in January.USMNT’s defensive depth tested ahead of the World CupOnly four centre‑backs have logged 500+ minutes under Mauricio Pochettino.Current regulars: Tim Ream, Miles Robinson, Mark McKenzie, with Auston Trusty gaining recent praise.Potential addition Noahkai Banks remains undecided between the US and Germany.What the next weeks could mean for Richards and the US squadSquad to be announced on 26 May; Richards may join camp late due to Palace commitments.USMNT friendlies vs Senegal (31 May) and Germany (6 June) are final tune‑ups.World Cup opener against Paraguay on 12 June in Los Angeles.

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.

A Historic Return: The Refugee Team's ItineraryAfghanistan's displaced female cricketers are set to return to the international stage with a tour of England beginning June 22. This initiative, organized by the England and Wales Cricket Board (ECB), brings together players who were previously contracted to the Afghanistan Cricket Board but were systematically excluded from sport and public life following the Taliban's return to power in 2021.Start Date: June 22Format: Twenty20 (T20) matchesKey Event: Attendance at the Women's T20 World Cup final at Lord's on July 5Support: Training opportunities and deepening connection to the global gameStrategic Milestones and Global ReachThe tour serves as a critical data point in the ongoing struggle for women's representation in sport. While the players have been playing domestic cricket in Australia, the lack of access to international competition highlights a significant gap in the International Cricket Council's (ICC) current regulatory framework, which requires member nations to support both men's and women's teams.The itinerary is not merely a series of matches but a strategic effort to reintegrate the players into the global cricket community. By featuring in T20 matches and attending the final at Lord's, the team aims to bridge the five-year gap in their professional careers and demonstrate their continued competitiveness on the world stage.Breaking Barriers: Sport as a Tool for InclusionThis tour carries profound cultural and sporting significance, acting as a testament to the resilience of Afghan women. The ECB has emphasized that the event represents a moment for cricket to stand for inclusion and the protection of women's participation in sport.The involvement of former Australian international Mel Jones, through her consultancy firm "It's Game On," underscores the professional infrastructure required to support such a complex transition. The players' repeated requests for the ICC to recognize them as a refugee team have finally borne fruit, validating their struggle for recognition and highlighting the urgent need for the sport's governing body to adapt its rules to protect displaced athletes.Future Outlook: Beyond the TourMel Jones has called for "sustained and meaningful action beyond this year," signaling that this tour is just the beginning of a longer journey. The success of this initiative could set a precedent for how international sports bodies handle displaced athletes, potentially leading to more structured pathways for refugee teams in the future.As the team prepares to compete, the cricketing world watches closely, hoping that this tour will not only provide the players with the opportunities they deserve but also catalyze a permanent shift in how sport addresses human rights and inclusion on a global scale.

The LeadAnxious Arsenal fans around the world were glued to their screens when Manchester City's draw at Bournemouth crowned the North London club as Premier League champions – but Mikel Arteta, who led Arsenal to their first league title in 22 years, was not watching.The manager had planned to be at Arsenal's training centre in London Colney to watch the game with his players, but could not find the energy to make the trip, and chose to stay at home.The Unconventional Celebration"I went outside to the garden, I started to build some fire. I started to do some barbecue, I didn't watch any of it," Arteta told reporters on Thursday.After finishing second three times in a row, Arteta heard the news of their long-awaited triumph from his eldest son, Gabriel, who plays for Arsenal's Under-18 side."My eldest son opened the garden door, he started to run towards me. He started to cry, gave me a hug and said, we are champions, daddy," Arteta said."Then my other two boys and my wife came over, and it was beautiful, just to see the joy on them as well."Videos shared by Arsenal on social media showed players dancing and chanting at their training centre after their league title was confirmed with a game to spare."It was their moment. And they have to be themselves in that moment. And if I'm there, I think it wasn't going to be the same," Arteta said.Arsenal captain Martin Odegaard called Arteta soon after, asking where the manager was."I said, guys, enjoy for a while and see you in a few hours somewhere in London," Arteta recalled with a smile.The Manager's JourneyIt is Arteta's first league title as manager, and he admitted that he doubted himself during a high-pressure campaign, where they were often a slip-up away from losing their lead in the title race."To play with that on your back constantly is not easy. That has been one of the toughest moments... we showed very important values not only in sport, but in life. Which is perseverance, to be resilient, to be composed in moments when people are doubting," he said."And to be vulnerable. I've asked that question to myself, am I good enough to lead these players to win a major trophy? Until you do it, you cannot validate yourself."The Team's MomentumArteta said the mood in the club had changed after winning the Premier League, buoying the team's confidence ahead of a Champions League final against Paris Saint-Germain on May 30."It changes because the level of emotion you feel, it's something I haven't felt before," he said.The Final ChapterArsenal will conclude their league campaign and lift the trophy at Crystal Palace on Sunday."We will have a window to celebrate and lift the trophy and connect with the people we have constantly around us. Then we have six days to write new history in this football club," Arteta added.

The LeadSpotify has introduced a new AI-powered audiobook creation tool in partnership with ElevenLabs, allowing authors to self-publish audiobooks without exclusivity. The platform is expanding to support 10 more languages and aims to generate $100 million in annualized recurring revenue from its Audiobook+ subscriptions.AI Audiobook Creation Platform LaunchAlongside tools for AI-generated podcasts, Spotify on Thursday introduced a new, ElevenLabs-powered AI tool for self-publishing audiobooks within the Spotify for Authors platform. The company said at its Investor Day event that the feature will launch in beta this June on an invite-only basis, initially with support for the English language only.The AI-powered audiobook generation won't bind authors to an exclusive contract, meaning they are free to publish their generated audiobooks anywhere. This approach contrasts with some other platforms that require exclusivity for audiobook distribution.The news builds on Spotify's previous partnership with ElevenLabs, which allowed writers to submit audiobooks created on the voice AI startup's platform to Spotify. The audio streaming platform also already had a partnership with Google Play Books to allow for digitally narrated content. However, it may have wanted authors to access newer voice models that sound more expressive and human-like, like those offered by ElevenLabs. Notably, ElevenLabs had released its own self-publishing platform for authors in 2025.Financial Performance and Growth MetricsSpotify has increased its focus on audiobooks heavily in the last few years and has managed to build its catalog to 700,000 titles. Through these initiatives, the company has managed to bump up listening hours by 60% year-on-year, the company claims. Spotify also said that more than half of its audiobook listeners started in the last year.To date, Spotify has clocked in over a million Audiobook+ subscriptions, and it is on track to generate $100 million in annualized recurring revenue for the platform. The company will expand its Audiobook+ plans this year to allow for higher listening limits and will add new options for students and families in the future.Industry Transformation and Market ExpansionSpotify is also expanding its "Spotify for Authors" platform to support 10 more languages, including French, Canadian French, German, Dutch, Latin American Spanish, Swedish, Finnish, Icelandic, Danish, and Norwegian. This expansion will significantly broaden the platform's reach and accessibility to authors and listeners worldwide.The company brought the program to international markets, made an investment in non-English titles, enabled in-app purchases, and released audiobook charts. This year, it also started a program for authors to sell physical books in the U.S. and the U.K., creating a comprehensive ecosystem for content creators.Future Outlook and User Experience EnhancementsAt the event, the company introduced a new way for users to ask questions using natural language for audiobook discovery. This summer, Spotify will also expand a feature that allows users to create prompt-based playlists for podcasts and music to include audiobooks, it said.These enhancements reflect Spotify's strategy to leverage AI not just for content creation but also for improving user discovery and engagement. The integration of natural language processing for audiobook discovery could potentially revolutionize how users find and consume audiobooks, making the platform more intuitive and user-friendly.

The Olympic LegacyAsh Moloney briefly became the face of Australian athletics after securing the country's first Olympic decathlon medal in dramatic circumstances at the Tokyo 2021 Games. The 26-year-old surged home in the 1500m, egged on by teammate Cedric Dubler, to claim bronze in a moment that inspired a nation. This achievement represented the pinnacle of his decade-long career in the grueling 10-discipline event.The TransitionAfter years of injury recovery, a new coaching relationship, and what he admits was sheer boredom with the decathlon, Moloney has made a bold and risky transformation. He has abandoned the event that brought him glory to pursue a new career in the 400m hurdles. "It's just fun, it's a completely different challenge," Moloney explains of his switch. "I've been ambidextrous my whole life, I can sprint hurdle on both legs. It almost just seemed like why not? Why not give it a try? Get that question out of my head."The Performance DataMoloney's rapid improvement in his new event is remarkable. Since his first competitive 400m hurdles in February, when he finished in 57 seconds, he has consistently lowered his times. Two weeks later in Hobart he ran sub-53s, before recording two times under 52s before nationals in April. At the national championships, he ran under 50 seconds for the first time, finishing third in the final. "I believe I can make Comm Games," Moloney says. "I believe I can go 48 [seconds]."The Impact on Australian AthleticsMoloney's transition comes at a significant moment for Australian athletics. Before the emergence of new generation athletes like Gout Gout and Lachie Kennedy leading Australia's charge towards Brisbane 2032, Moloney represented the future of the sport. His move to hurdles adds depth to Australia's middle-distance and hurdles program, complementing the emergence of sprint stars like Lachie Kennedy, who is now also his coach.The Relationship EvolutionThe decathlon teammates' relationship was tested in the aftermath of Tokyo, with Moloney previously suggesting that the public's interest in Dubler's contribution sometimes overshadowed his own achievement. While Moloney offers only a "no comment" when asked to elaborate on their current relationship, Dubler maintains there's no bad blood. "It's just a change of situation, a change of coaches and perspective," Dubler explains. The two athletes have chosen different paths since the Games, with Dubler continuing to compete in decathlon while targeting his own Commonwealth Games selection.The Future OutlookWith Moloney and Dubler now competing in different events, Australian athletics stands to benefit from their specialized focus. Moloney's renewed enthusiasm for his new event, freed from the physical toll of ten disciplines, has given him a fresh perspective on competition. "Honestly, I was bored with decathlon, I just wanted a new challenge," he admits. As he prepares for the Oceania championship showdown in Darwin and potentially the Commonwealth Games, the former Olympic bronze medalist is proving that reinvention can be as rewarding as achieving initial success.