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The LeadOpenAI says its new general‑purpose reasoning model has produced an original proof that disproves the famous geometry conjecture posed by Paul Erdős in 1946, ending an 80‑year open problem.OpenAI Announces Disproof of Erdős’s 1946 Geometry ConjectureThe company released a pre‑print and companion remarks signed by mathematicians Noga Alon, Melanie Wood and Thomas Bloom. The proof introduces a completely new family of constructions that outperform the long‑standing “square‑grid” belief.Timeline of Claims and CorrectionsJuly 2025: Former VP Kevil Weil tweeted that “GPT‑5 found solutions to 10 unsolved Erdős problems”.Later 2025: Critics including Yann LeCun and Demis Hassabis called the claim a misrepresentation; Weil removed the post.May 20, 2026: OpenAI publishes the new disproof, backed by external experts.Why This Disproof Could Redefine AI‑Driven ResearchThe breakthrough demonstrates that an AI system can autonomously manage long, intricate chains of reasoning and synthesize ideas across mathematical sub‑fields, a capability that researchers argue could translate to breakthroughs in biology, physics, engineering and medicine.What Comes Next for AI in Fundamental ScienceExperts anticipate a surge in AI‑assisted exploration of other long‑standing conjectures. If the model’s reasoning can be generalized, we may see a new era where AI acts as a co‑discoverer rather than a tool.

The Treasury's Flawed Proposal The UK Treasury's proposal for voluntary price caps on food staples has been widely criticized by retailers and analysts. Stuart Machin, chief executive of Marks & Spencer, described the idea as 'completely preposterous', while City analyst Clive Black at Shore Capital thought the government 'appears to be losing its mind in an orgy of neo-Soviet policy ideas'. The criticism is justified, as price caps are a flawed solution to the problem of rising food prices. The Reality of Food Inflation Food inflation in the UK was 3% in April, and while it is expected to rise in coming months due to increasing energy, transport, and fertilizer costs, the country is not in a state of emergency. The Competition and Markets Authority found in 2024 that there was no evidence that groceries inflation was being driven by weak competition between retailers. Instead, prices are already depressed due to everyday competition among retailers. The Impact of Price Caps Imposing price caps would likely have negative consequences, such as reducing the supply of essential items. History has shown that artificially depressing prices can lead to knock-on effects on the supply of goods. Furthermore, the Treasury's idea would be difficult to implement in practice, as it would require collusion between rival retailers, which is illegal. A Better Solution A more effective solution to addressing cost-of-living pressures would be to increase welfare payments to vulnerable households. This targeted approach would provide support to those who need it most, rather than attempting to control prices through a flawed and impractical policy.

The Call for a College Sports Boycott Hakeem Jeffries, the top US House Democrat, has amplified calls for Black athletes to boycott public universities in states that have moved to limit voting rights, saying an “unprecedented moment, featuring an unprecedented attack on Black political representation” requires an “unprecedented response”. The NAACP's 'Out of Bounds' Campaign Jeffries’s comments came Tuesday as the NAACP launched its “Out of Bounds” campaign. The campaign targets universities in eight states – Tennessee, Louisiana, Alabama, Florida, Mississippi, South Carolina, Texas and Georgia – whose athletic programs generate more than $100m in revenue. Those eight states have moved to draw new voter maps after the supreme court’s Louisiana v Callais decision severely weakened the Voting Rights Act. The Southeastern Conference in the Spotlight The minority leader specifically called out the powerhouse Southeastern Conference. Twelve of the SEC’s 16 member schools are in the eight targeted states. The Boycott's Objectives The campaign calls on football and basketball players being recruited by programs in those states to withhold their commitments until the states “restore fair congressional maps and meaningful Black representation”. It also urges athletes and coaches already enrolled at those universities to use their platforms to elevate voting rights causes. It asks fans, alumni and donors to stop financially supporting those programs. The Impact on High-Valued Athletic Programs The SEC is home to nine of the 15 highest-valued athletic programs in the country, according to CNBC, including leader Texas ($1.48bn), Georgia, Alabama and Florida. A Legacy of Activism Athletes at Missouri and Mississippi, both SEC schools, have led successful campaigns in recent years putting pressure on universities and state governments for social justice causes. Jeffries referenced Bill Russell, Muhammad Ali and Jackie Robinson in his remarks, calling on this generation to carry on the legacies of previous activist athletes. The Congressional Response Jeffries and members of the Congressional Black Caucus earlier this week voiced their opposition to the Score act, a bill intended to set national standards for college athletes’ compensation. The bipartisan proposal, which has support from the NCAA, was to be brought to the House floor for a vote this week, but the CBC opposed the bill to protest the silence of the universities on voting rights. House Republicans decided on Tuesday to postpone a vote on bill, the second time in less than a year that it has been stalled.

The Lead: A New Global OrderRussian President Vladimir Putin and his Chinese counterpart, Xi Jinping, have signed a joint declaration following their meeting in Beijing, focusing on building a "multipolar world and a new type of international relations". The two countries also announced that they had signed a large package of deals solidifying bilateral cooperation in the future.The meeting came just days after United States President Donald Trump completed his own visit to China for a two-day summit with Xi.Establishing a Multipolar World OrderOn Wednesday, the Chinese Foreign Ministry said: "The two countries will also issue a joint statement on advocating for a multipolar world and a new type of international relations." Russian presidential aide Yuri Ushakov described this declaration as a 47-page policy document.A "multipolar world" is understood as one in which economic, military and diplomatic power and influence are placed in the hands of three or more countries, rather than just one or two."Xi is calling for a more multipolar world, where the US has less power and influence," Al Jazeera's Katrina Yu reported from Beijing as the meeting was under way.Both Putin and Xi have spoken out against the "unipolar" hegemony that they say the US has over the world.In 2022, shortly after the beginning of Russia's war with Ukraine, Putin accused the US of stoking hostilities in Ukraine to maintain its global influence."They need conflicts to retain their hegemony," Putin said during a speech. "The era of the unipolar world order is nearing its end."Chinese state media reported that during the latest meeting, Xi said to Putin: "The tide of unilateral hegemony is running rampant."Russia-China Relations Reach Unprecedented LevelA press statement posted on the Kremlin website said relations between Russia and China had reached "a truly unprecedented level and continue to develop".The Chinese Foreign Ministry statement said: "Both sides should follow the trend of peace, development, cooperation, and win-win results to promote higher-quality development of China-Russia relations."The statements added that bilateral cooperation extends to the worlds of economics, sport, education and the media.The Kremlin statement adds that this year marks the 70th anniversary of partnership between the Russian TASS news agency and the Chinese Xinhua news agency.Deepening Economic Cooperation and Moving Away from the US DollarThe Kremlin statement said Beijing and Moscow had signed around 40 intergovernmental, interagency and corporate documents. "Many of these focus on the further deepening of our economic cooperation," it noted.The statement added that, last year, trade between the two countries reached almost $240bn, while the Chinese statement said bilateral trade grew by 20 percent in the first four months of this year.Since the war in Ukraine broke out in February 2022, Russia has become increasingly reliant on Chinese technology and manufacturing. Last month, Bloomberg reported that Russia now imports more than 90 percent of the technology targeted by US and European Union sanctions via China, using Chinese suppliers and intermediaries to obtain components with military and dual‑use applications vital to drone production and other defence industries."Both sides should build on this momentum, deepen the alignment of China's 15th Five-Year Plan with Russia's development strategy until 2030, promote the upgrading of mutually beneficial cooperation in various fields, and serve the development and revitalization of both countries," the Chinese ministry statement said.The Kremlin statement said that nearly all import and export transactions between Russia and China are in roubles and yuan. "In other words, we have actually created a stable system of mutual trade that is protected from external influence and negative trends in the global markets," it said.Securing Energy Supplies Through Siberia 2 PipelineThe Kremlin said on Wednesday that an understanding had been reached for the route and construction of the long-delayed joint Siberia 2 pipeline, but details are still being negotiated. Once completed, the pipeline will transport 50 billion cubic metres of Russian gas annually to China via Mongolia, significantly expanding energy flows between the two countries.The Kremlin's statement said that Russia and China are actively cooperating in the sphere of energy."Our country is one of the largest exporters of oil, natural gas (including LNG) and coal to China. We are definitely ready to continue to ensure reliable and uninterrupted supplies of these types of fuel to the rapidly growing Chinese market," the statement said.As European markets have largely closed to Russia as a result of the war in Ukraine, China has emerged as a crucial buyer of Russian oil and other energy products, benefitting from steep discounts on Russian products.In December 2022, the Group of Seven (G7), the EU and Australia placed a cap on the price of Russian oil at $60 per barrel, ostensibly to reduce Russia's ability to fund its war in Ukraine. The cap was later reduced to around $48 by the EU and the United Kingdom.Expanding Educational and Scientific TiesBoth statements said Xi and Putin had agreed to expand student exchange programmes and cooperation between universities and research platforms to boost joint scientific research.

Bluey fans have been left in a limbo since the last full episode aired in spring 2024, with a feature film slated for summer 2027. To keep the franchise alive, Disney+ released a second batch of three‑minute “minisodes”, prompting both praise and concern about the show’s creative direction.New Bite‑Size Bluey Minisodes Debut on Disney+The streaming platform rolled out ten new minisodes, ranging from one to three minutes. Highlights include “Cinderella”, where Bandit improvises a bedtime story, and “Honk”, a game that ends in unexpected violence. Four of the episodes are simple nursery‑rhyme sing‑alongs, and another is a wordless dance routine.Numbers Behind the Minisodes: Length, Release Count, and Content MixGap since last full episode: 3 years, 3 months, 16 daysEpisode length: 1–3 minutesTotal minisodes released: 10Nursery‑rhyme or music‑only episodes: 5 (four sing‑alongs, one dance)What the Minisodes Mean for Bluey’s Brand and AudienceThe short format lets the series experiment with “weird” premises that would not fit a 28‑minute episode, but the throwaway nature of many entries risks diluting the show’s reputation for depth and emotional resonance. Merchandise sales remain a driving force, and the minisodes act as “Bluey methadone” to keep fans engaged while the main series is on hold.Future Outlook: Will Bluey Survive Without Joe Brumm?The upcoming 2027 film may be the last project written by creator Joe Brumm. If the franchise can replicate his unique blend of warmth, cross‑generational appeal, and inventive storytelling, it may endure. However, a shift toward filler content could alienate core fans, echoing the post‑Larry David era of Seinfeld.For now, the minisodes are a convenient bridge, but they also raise the question of whether the beloved Australian series can maintain its creative spark without its original visionary.

The End of an Era and the Move to the EmiratesThe Invincibles were slain in 2004, ending a record 49-match unbeaten run in a traumatic defeat at Old Trafford. The subsequent seasons were defined by a painful transition. The 2005-06 campaign saw Arsenal finish 4th with 67 points, salvaged by a Champions League final appearance and a memorable Highbury farewell where Thierry Henry scored a hat-trick against Wigan. However, the move to the Emirates Stadium in 2006-07 confirmed the club was in a rebuilding phase, finishing 4th again despite a strong double over eventual champions Manchester United.Consistency Challenges and Near-Miss FinishesArsenal's inability to sustain a title challenge became evident in the late 2000s. In 2007-08, a young team led by Cesc Fàbregas amassed 83 points but still finished 3rd, ultimately derailed by a harrowing leg-break for Eduardo and four successive draws. The following seasons were equally erratic; 2008-09 saw a bizarre mix of 0-0 draws and 4-4 thrillers, while 2009-10 saw a promising start collapse after a first league defeat to Spurs in 11 years. Despite averaging 70+ points across these seasons, Arsenal consistently fell short of the top two.The Rise of Youth and the Van Persie DepartureThe 2010-11 season marked the emergence of Jack Wilshere and a strong run to the Champions League semi-finals. However, inconsistency remained the norm, highlighted by a 4-4 draw at Newcastle and a League Cup final loss. The 2011-12 season was a wild ride, featuring an 8-2 drubbing at Old Trafford and a 5-3 victory over Chelsea, but it ended in another 3rd-place finish. The season was defined by Robin van Persie, who scored 30 goals and won the Golden Boot, only to leave for Manchester United in the summer, signaling the end of an era for Wenger's young core.Future Outlook: The Foundation for Title GloryThe instability and near-misses of this period, while frustrating for fans, laid the necessary groundwork for the club's eventual return to the top. The transition from the Invincibles to the Emirates era forced Arsenal to rebuild their squad, developing a youth system that would eventually produce the talent required to compete at the highest level again. The lessons learned from these inconsistent seasons were instrumental in shaping the squad that would eventually return to title glory.

The LeadNow the secret is out it is possible to look at Enzo Maresca's incendiary remarks about his "worst 48 hours" at Chelsea through a different lens. Change is coming at Manchester City, who are preparing for Pep Guardiola's departure at the end of the season, and it does not require much reading between the lines to work out their decision to pass the crown to Maresca was made a long time ago.The Succession StrategyThere never was a clear explanation from the Italian after he sat in front of the media after Chelsea's unspectacular 2-0 win over Everton on 13 December and surprised the room by taking the extraordinary step of going to war with his employers. "Since I joined the club, the last 48 hours have been the worst because many people didn't support us," he said. "People didn't support me and the team."Which people? Maresca never said and Chelsea were perplexed. The situation deteriorated over the next fortnight and it was hard not to feel Maresca was behaving like a man who wanted to be sacked. Chelsea, though, refused to pull the trigger. It was only when Maresca went into the manager's office at Stamford Bridge after a 2-2 draw with Bournemouth on 30 December and told his bosses he did not want to conduct his post-match duties that it became clear there was no putting the genie back in the bottle.The Chelsea FalloutSources familiar with that episode say that was the moment Maresca in effect handed in his resignation. He was gone two days later, the club statement landing early on New Year's Day. Chelsea, unsurprisingly, have not moved on from Maresca informing them he had twice spoken to City while under contract.This was not a fond farewell. Maresca walked away without his severance, with three and a half years on his deal. Sources close to the former Leicester manager have acknowledged Chelsea are entitled to demand a sizeable compensation package for City to acquire his services.The Tactical BlueprintGuardiola has backed his former assistant. City know what they are getting from Maresca, part of their backroom staff when they won the treble in 2023. He favours positional play, uses inverted full-backs, sees the pitch as a chessboard and has even been nicknamed Diet Pep.While Maresca is undoubtedly a quality tactician, his work at Chelsea and Leicester does leave room for debate. There were times when Leicester supporters grumbled about Maresca's football, even though he led them to the Championship title in 2024, and concerns that his style of play was too dogmatic were never far from the surface at Chelsea.The Premier League ChallengeThe former Sevilla midfielder, who played for Carlo Ancelotti at Juventus, moved to Stamford Bridge after Mauricio Pochettino's departure in May 2024. Chelsea wanted to play with more control and Maresca's first season was a qualified success. They squeezed into the Champions League and beat Real Betis in the Conference League final.The crowning moment came when Maresca bamboozled Paris Saint-Germain in the Club World Cup final last summer. It was a fine achievement and showed his ability to come up with clever plans for one-off games. Winning the Premier League, though, requires greater consistency and Chelsea had a prolonged dip during his first season and sometimes struggled to break down low blocks.Perhaps the key for Maresca is that he will have access to better players. He did not have a top striker at Chelsea, but at City will be able to rely on Erling Haaland. The trials and tribulations of those 48 hours must feel worthwhile now.

The Lead: Ending the 22-Year DroughtArsenal have ended a 22-year wait to be crowned Premier League champions after Manchester City were held 1-1 by Bournemouth. Mikel Arteta's men held off the challenge of Pep Guardiola's second-place City to seal a long-awaited triumph with one game to spare.Tactical Innovation: The Set Piece RevolutionArsenal's title-winning campaign has been built on a back-to-basics approach that has revolutionized their set-piece strategy. For a couple of seasons, Arteta's Arsenal have been the emblem of a more pragmatic approach in English football, moving away from the era of pretty passing under Guardiola and heavy-metal football under Jurgen Klopp.The Gunners have mastered set pieces with Gabriel Magalhaes a menace at corners, Declan Rice key with his pinpoint throws and dead-ball deliveries, and Victor Gyokeres providing a robust striking option. More than 40 percent of Arsenal's Premier League goals this season have come from dead ball situations, with 18 of their 28 set-piece goals coming from corners – a new single-season Premier League record.Key Performances: Leadership Between the Posts and in MidfieldDeclan Rice and goalkeeper David Raya have stood out for Arsenal this season, putting them among the favorites for English football's Player of the Year award. Rice's leadership, energy in midfield, and set-piece delivery have made him an integral member of the team and one of England's key players heading into the World Cup.Raya has helped Arsenal keep 19 clean sheets, earning him the Golden Glove award for the third straight year. "David Raya, for me, has to be the player of the season," former Arsenal captain Patrick Vieira told Sky Sports. "I think he was outstanding from the first game until the end of the season. I think he was really impressive."Throw in Gyokeres scoring 21 goals in all competitions in his first season at the club, and it is clear that Arteta has built the most well-rounded team of his tenure.Squad Depth: Overcoming the Injury CrisisInjuries played a ruinous role in Arsenal's failed pursuit of Liverpool last season. However, the decision to invest heavily in bulking out Arteta's squad paid off this season despite injuries to key players like Saka, Magalhaes, Martin Odegaard, Kai Havertz, and Jurrien Timber.New signings Gyokeres, Eberechi Eze, Martin Zubimendi, Noni Madueke, Piero Hincapie, and Cristhian Mosquera have all made significant contributions to get Arsenal over the line in the Premier League and within one game of winning the Champions League for the first time.Mental Fortitude: Building Resilience Through AdversityFinishing as runners-up in the Premier League for the past three years saw Arsenal's players and Arteta derided as "nearly men", even chokers, by many football pundits. However, this period built up the prerequisite experience and resolve to finally launch a successful tilt at the title.Arteta kept believing in his squad and came up with unorthodox methods to inspire his players. A professional pickpocket was reportedly hired for a preseason dinner to highlight the need for alertness, while a lightbulb was brought into the locker room before one game to demand that the team "shine" at Emirates Stadium. Recently, TikTok videos featuring fan chants were played on big screens during practice sessions.Arsenal have been mentally tougher this season, holding on after yet another strong start and seeing it through to the end despite City's trademark late-season rally.The Competitive Landscape: When 82 Points Was EnoughArsenal accumulated more points two years ago when they were pipped to the title by City despite winning 16 of their final 18 games. Over the past decade, City and Liverpool have often set the bar high, winning the league with more than 90 points.This time, 82 was enough to see Arsenal over the line. Despite taking the title race into the final week of the campaign, City lacked the same consistency and relentlessness of Guardiola's best sides while Liverpool's title defense imploded.

The LeadOver breakfast in one of the swankiest hotels in Caracas, US officials, diplomats and spies now call many of the shots in Venezuela after Donald Trump's controversial military intervention on 3 January. The five-star JW Marriott has become the nerve center of Washington's efforts to steer a country some now call a US protectorate – and which Trump has even said he hopes to turn into the 51st state.The New American EmbassySince Trump's decision to snatch Maduro in January and reboot relations with his successors, the JW Marriott has effectively become the US embassy in Venezuela. "It's [effectively] the US embassy. I don't think anybody's going to work at the actual embassy," said Phil Gunson, a Caracas-based political analyst for Crisis Group.Having been closed for seven years since the collapse of diplomatic relations in 2019, "the embassy building is full of rats and cockroaches, and it's being fumigated," Gunson explained. The conversations that can be overheard in the JW Marriott's restaurant offer a fascinating insight into Venezuela's plight as it emerges from nearly 13 years of economic mayhem and authoritarian rule under Maduro.The Corporate TakeoverIf the $250-300-a-night JW Marriott – or "the spooks hotel" as some journalists call it – is the HQ of the US presence in Venezuela, it is at another luxury hotel a few miles away where many of the big-money deals are being done. Since Maduro's downfall, foreign tycoons have been flocking to the Cayena, where rooms cost about $600 a night, wagering that even if interim president Delcy Rodríguez stays in power and there is no transition to democracy, Venezuela's economic future looks bright.One deal-maker who has spent time there recalled encountering at least four foreign billionaires they could identify – but believed there were others whose names they did not know. "They never give you a card. They don't give you their last names … and what is very interesting to me is that they are all asking about the same things: mining and privatizations," they said.Venezuelan ResistanceThe Trumpian takeover has generated widespread discomfort, even among patriotic members of Venezuela's elites who were glad to see the back of Maduro but privately bristle at the suggestion that their country is being turned into a US colony. After giving Rodríguez his blessing in January, Trump warned she would face an even worse fate than Maduro if she failed to toe the US line.On the streets there is anger too. During a Workers' Day rally on 1 May, a socialist economist called Oswaldo Pacheco marched towards a line of riot police wielding a white banner denouncing the government's "neocolonial collaboration" with Trump. "It's a complete capitulation," complained Pacheco, 53, accusing Venezuela's new rulers of following US orders "to the letter". "Clearly these [US] demands are not about bringing us democracy but about plundering our resources and increasing worker exploitation," he said.The Future of a ProtectorateAcross the street sits the Juan Sebastian Bar, a jazz and salsa nightclub named after Johann Sebastian Bach, where foreign visitors can let off steam. Throughout the day, English-speaking officials and fortune hunters can be seen roaming the 17-floor redbrick building, which has nearly 300 rooms, a gym and a palm-flanked pool. Bullet-proof SUVs wait outside to ferry guests, who include Trump's top diplomat to Venezuela, John Barrett, around town.Among Caracas-bound capitalists the mood is buoyant, even if huge doubts remain over Venezuela's future and, above all, its democracy. At a third luxury hotel, the Renaissance, a Venezuelan oil man waxed lyrical about his country's post-Maduro prospects. "This is going to be the best country in the world," he predicted, declaring: "I'm more than optimistic."