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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.

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 World Cup SnubHarry Maguire has announced he has been overlooked by Thomas Tuchel for the England World Cup squad. The Manchester United centre-half had hoped to be included in the 26-man party for the tournament this summer, which Tuchel will name at Wembley on Friday morning."I was confident I could have played a major part this summer for my country after the season I've had," Maguire posted on social media. "I've been left shocked and gutted by the decision. I've loved nothing more than putting that shirt on and representing my country over the years. I wish the players all the best this summer."Resurgence at Manchester UnitedMaguire has excelled for United since Michael Carrick took over as the head coach in mid-January and was called up by Tuchel for the March international programme. The 33-year-old started against Uruguay and came on as a late substitute against Japan, when he brought a threat on set pieces.His performances had raised hopes that he would feature in the World Cup squad, especially after struggling for form under previous manager Ole Gunnar Solskjaer.Defensive Priorities for EnglandDespite Maguire's improved club form, Tuchel has decided to prioritise other players in the position. The decision suggests that the England manager may be looking for different attributes in his center-back options for the tournament.Maguire's omission comes as a surprise given his recent performances and his experience at international level, having been a regular starter for England in previous tournaments.Implications for Maguire's International FutureThis decision raises questions about Maguire's future with the national team. At 33 years old, this could potentially be his last chance to feature in a World Cup, making the exclusion particularly disappointing.However, his continued good form at Manchester United could give him opportunities to force his way back into contention for future England squads, particularly if Tuchel's preferred options underperform at the World Cup.

The Impending Oil Crisis Oil markets are on the verge of entering a critical phase, often referred to as the 'red zone,' as the summer travel season approaches. According to Fatih Birol, the executive director of the International Energy Agency (IEA), this period of high demand will be exacerbated by dwindling oil stocks and a shortage of fresh oil exports from the Middle East. Current Market Challenges Birol highlighted that the current situation is precarious, with stocks eroding and no new oil coming from the Middle East. He emphasized that demand is increasing, mainly due to the travel season, and warned that if there are no improvements, the market could enter the 'red zone' by July and August. Potential Solutions and Impact Birol suggested that a full and unconditional reopening of the Strait of Hormuz could alleviate the crisis. He also mentioned that the IEA is open to releasing more strategic oil reserves, as they have done previously. The IEA chief stressed that the reputation of the Middle East as a secure supplier of energy has been damaged, which could lead to countries paying a premium for supplies from more secure sources and for renewable energy. Future Outlook and Predictions Birol predicted that governments around the world will review their energy strategies in the next few years and look for new options for fuel imports. He also anticipated that countries will turn to other energy sources, including renewables, nuclear, and coal. Domestically, energy production that makes economic sense is likely to get a push. Geopolitical Tensions and Negotiations The situation is further complicated by geopolitical tensions, particularly regarding Iran's nuclear program and the negotiations between Iran and the US. Pakistan, acting as a mediator, is facing difficulties in reaching a breakthrough. The Iranian supreme leader, Mojtaba Khamenei, has stated that Iran will not allow its stockpiles of highly enriched uranium to be exported to a third country.

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.

Lead: A Manager’s Unusual CelebrationMikel Arteta chose a backyard barbecue over the live broadcast when Arsenal clinched the Premier League, describing the experience as "building a fire" while his family celebrated the historic triumph.Garden Fire and Family Joy: The Moment Arsenal Won the TitleArteta headed to the training ground to watch Manchester City’s draw at Bournemouth with his squad, but 20 minutes before kick‑off he returned home, unable to bear the tension. In his garden he lit a fire, barbecued, and heard his son Gabriel announce, “We’re champions, Daddy.” His wife and other children joined, creating a personal celebration separate from the players.Numbers That Matter: Breaking a 22‑Year DroughtFirst league title for Arsenal in 22 years.Three consecutive runners‑up finishes prior to this season.Arsenal’s historic comeback solidifies their place at the top of the Premier League table.Why It Changes Arsenal’s NarrativeThe victory underscores Arteta’s emphasis on perseverance, humility and resilience—values he says were as important off the pitch as on it. By confronting his own doubts about being “good enough,” Arteta set a tone of vulnerability that resonated with the squad, fostering a culture that could sustain future success.Looking Ahead: Champions League Ambitions and Squad OutlookWith Arsenal hopeful that Jurriën Timber will be fit for the upcoming Champions League final against Paris Saint‑Germain, and Mikel Merino back in training, Arteta’s next challenge is to translate domestic dominance into European glory. The manager’s focus remains on staying humble, curious, and goal‑oriented as the club prepares for the Budapest final.

The Lead: Djokovic Secures Favorable Half While Raducanu Draws Early TestNovak Djokovic has been placed in the opposite half of the draw from Jannik Sinner, ensuring they cannot meet before a potential final. The Serbian opens his campaign against French qualifier Giovanni Mpetshi‑Perricard, while Emma Raducanu faces Argentine youngster Solana Sierra in the first round.Draw Mechanics: How the Men’s and Women’s Brackets Shaped UpThe tournament seeding places Djokovic (No 2) with Alexander Zverev in the same half, while Sinner (No 1) leads the opposite side. In the women’s draw, defending champion Coco Gauff (No 4) starts against Taylor Townsend, and several other contenders line up in both halves.Numbers at Play: Seedings, Rankings and Recent FormMen’s top seeds: 1 – Jannik Sinner, 2 – Novak Djokovic, 3 – Alexander Zverev.Recent results: Djokovic lost in the first round of the Italian Open to world No 79 Dino Prizmic; Sinner is the dominant player after completing the Masters 1000 set.Women’s top seeds: 1 – Iga Swiatek, 2 – Elena Rybakina, 4 – Coco Gauff.Raducanu’s situation: returning from a two‑month post‑viral layoff and a first‑round loss in Strasbourg.Impact: What the Draw Means for the ChampionshipsWith the No 1 and No 2 men’s seeds on opposite sides, the final is likely to be a Djokovic‑Sinner showdown, provided both navigate the tough early matches. The women’s half appears more open, with several high‑seeded players such as Aryna Sabalenka, Jessica Pegula and Victoria Mboko positioned to challenge for the title.Looking Ahead: Key Storylines to FollowCan Djokovic rediscover form after an injury‑laden season and add a 25th Grand Slam?Will Sinner complete the career Grand Slam at Roland Garros?How will Raducanu respond to her early test against Sierra?Will new coach Francisco Roig help Iga Swiatek secure a fifth French Open crown?