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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 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

The Lead: One Nation's Standalone Climate Denial As the populist right-wing One Nation party gains momentum in Australian polls, it maintains a firm stance against decades of climate science evidence showing the planet and Australia are warming. The party claims to be the only political force in Australia questioning climate science, but experts say this position is not only scientifically baseless but increasingly out of step with global trends in climate misinformation. The Event Details: One Nation's Climate Contradictions One Nation's position on climate change is rooted in denial and conspiracy theories, despite overwhelming scientific evidence. The party's energy and climate policies have previously been literal cut-and-pastes from the now-defunct climate denial group the Galileo Movement. Currently, One Nation rejects the scientific consensus that the planet and Australia are warming, claiming extreme weather was more prevalent before 1960—a position climate scientists have compared to believing the Earth is flat. The party also wants Australia to leave the Paris Agreement and would push to close down the federal climate change department "and all related agencies, regulations and programs." They proudly declare: "We are the only political party to question climate science." The Data Analysis: Australia's Unequivocal Warming Trend Despite One Nation's claims, Australia has experienced significant warming since 1910. The Bureau of Meteorology's official long-term climate dataset shows Australia has warmed by 1.5°C since 1910. Temperature readings before this period are not included in the official dataset because they were often taken with non-standardized equipment, making them less reliable. Research into pre-1910 temperature records shows temperatures from 1860 to 1909 were similar to those from 1910 to 1959. Since 1960, both maximum and minimum temperatures have risen significantly. The independent Berkeley Earth group's analysis of historical temperature data confirms Australia has warmed substantially since the 1880s. One Nation points to a single weather station in Newcastle (Nobby's) to claim no pattern of warming exists—a classic example of cherry-picking data while ignoring the broader evidence showing temperatures are warming across Australia. The Impact Analysis: Political Consequences of Climate Denial One Nation's climate denial positions have significant political implications in Australia. Research from CSIRO a decade ago suggested that views on climate change can be influenced by how a person votes, rather than the other way around. This means votes for One Nation could lead more people to reject established climate science. Dr. John Cook, an expert on climate science denial, notes that One Nation is "not only out of touch with the scientific evidence, they're even out of touch with the rest of the climate denial community." Over the past decade, climate misinformation has transitioned from science denial to attacking climate solutions, as the scientific evidence for human-caused global warming has become undeniable. Prof. Sarah Perkins-Kirkpatrick, a climate scientist at Australian National University, emphasizes: "There's a wealth of evidence that extreme heat events are increasing worldwide since the 1950s. We see increased intensity of droughts and heatwaves and the intensity of tropical cyclones is increasing." The Prediction: Future of Climate Politics in Australia As climate impacts worsen in Australia—with more frequent and intense heatwaves, bushfires, and extreme weather events—One Nation's climate denial stance may become increasingly untenable politically. The party's net zero conspiracies, including claims that climate action is part of a plot to create a "socialist Australia," are based on misinformation and misrepresentations of statements by figures like former World Economic Forum chair Klaus Schwab. One Nation's assertions that renewable energy is causing electricity price increases are also contradicted by experts. Research from CSIRO suggests that if 82% of Australia's electricity came from renewables backed by storage, the cost of generation would be a third less than current prices. The real drivers of rising electricity costs are aging infrastructure and rising international gas prices, not the transition to renewables. As Australia faces increasing climate impacts, political parties that reject established climate science may find themselves increasingly isolated, both scientifically and politically. The future of Australian climate politics may depend on how mainstream parties respond to One Nation's misinformation and whether they can effectively communicate the scientific consensus on climate change.

The LeadEleven weeks after the start of the Iran war, the Strait of Hormuz has remained closed to naval traffic, bleeding the global economy far beyond the Gulf. Iran's Islamic Revolutionary Guard Corps (IRGC) maintains an iron grip over this narrow, strategic waterway, while a corresponding United States naval blockade on Iranian ports has failed to reopen it.Before the war began, between 120 and 140 ships travelled through the strait each day, about half of them oil tankers carrying some 20 million barrels of oil between them. Now, only a few vessels whose owners have negotiated with the IRGC are permitted to pass.The Strategic Control of HormuzOn Wednesday, Iran said it coordinated the transit of 26 vessels through the Strait of Hormuz in 24 hours, two days after announcing the formation of the Persian Gulf Strait Authority (PGSA), a new body to provide "real-time updates" on operations in the strait.Since the announcement of a temporary ceasefire between the US and Iran in April, Iran has been working on formalising a mechanism to charge a transit fee from ships crossing the critical chokepoint, through which 20 percent of the world's oil and liquefied natural gas (LNG) are shipped during peacetime.Tehran has reportedly already charged fees as high as $2m per ship for transit since the war started. Even though countries opposing Tehran say this is illegal, it may still be less expensive than the overall cost of the closure of the strait each day.The Economic Cost of BlockadeNearly one-fifth of global oil and LNG exports were shipped by Gulf producers through the Strait of Hormuz before the US and Israel bombed Iran on February 28, triggering the Iranian closure of the waterway. The strait is the only waterway linking Gulf producers to the open ocean – there is no other route through which they can ship exports.About 20.3 million barrels per day of oil passed through the Strait of Hormuz in peacetime – nearly 27 percent of global maritime oil trade. The lion's share of that crude went to Asian markets.Global LNG trade has been similarly hard hit. On the day before the war broke out, Brent crude – the global benchmark for oil prices – closed at $72.48 per barrel. After Iran closed the waterway on March 4 and began attacks on vessels attempting to sail through, traffic came to a standstill, stranding about 2,000 ships on either side of the strait.In terms of lost oil revenues, this amounts to $114.8bn of losses per day. About 10 billion cubic feet of LNG per day also used to pass through the strait, worth a further $7.8bn.The Cost-Benefit Analysis of Transit FeesFor hundreds of ships stranded in the Gulf with thousands of sailors on board, the cost of remaining anchored is steep, including crew wages, loan repayments, repair and management, coupled with inflated war risk premiums.In turn, Iran has reportedly been charging up to $2m for authorisation to pass. Experts say many will see this as worthwhile purely in terms of monetary cost."There is no doubt that paying Iran is cheaper than a continuous blockade because a sitting tanker bleeds money," said Nader Habibi, an Iranian American economist."It makes sense from an economic point of view, but it is not politically feasible," he added. "The companies are under pressure from the US sanctions and not to make arrangements with Iran. This is not just a purely economic cost-benefit analysis, but long-term considerations that are taken into account."International Legal PerspectivesInternational law protects free transit through strategic waters such as natural straits like Hormuz, barring countries from imposing passage tolls even where the waterways fall entirely into territorial waters, like in the case of Hormuz.However, services such as security controls, inspections and insurance regimes can be charged for. Chargeable fees also partly depend on whether a waterway is a man-made passageway or a natural one.These are three different precedents in maritime traffic flow:Panama Canal: An artificial waterway connecting the Atlantic and Pacific oceans. Vessels pass through a unique system of locks that raise and lower vessels across elevated terrain. Since Panama built, maintains and operates the canal, it can charge transit fees based on vessel size, cargo capacity and booking priority. These range from several hundred thousand dollars per transit to some slots sold for millions of dollars.Suez Canal: Another artificial canal, linking the Mediterranean and Red seas. Egypt charges transit fees for the use of canal infrastructure, maintenance and traffic management services through the narrow waterway. Container ships and oil tankers pay from several hundred thousand dollars to more than one million dollars per voyage.Turkiye's Bosporus Strait and Dardanelles: These are different because they are natural straits, rather than man-made canals. Turkiye charges for navigation-related services such as lighthouse operations, rescue readiness, medical support and traffic management – and tightly controls ship scheduling and navigation.Regional Cooperation PossibilitiesIran's newly-formed PGSA published a new map of Hormuz, stretching from Kuh-e Mubarak in Iran to south of Fujairah, in the UAE, at the eastern entrance of the strait, and from the tip of Qeshm Island to Umm al-Quwain at the western entrance.Given how the Iran war has spilled over into the Gulf region – with the UAE taking the brunt of Iranian strikes – economist Mohammad Reza Farzanegan said "regional cooperation with Iran is the most realistic path to stable transit through the Strait of Hormuz."The UAE, Oman, Qatar and Iran will have to work together because their economies require it, he argued. A workable arrangement could include a joint maritime authority, shared monitoring, emergency coordination, environmental protection and service-based contributions for maintaining safe passage."This would give Iran a recognised role in the security of the waterway while giving Persian Gulf economies more predictability," Farzanegan added. "Such a framework is also more realistic than relying on external military enforcement, which has been more a source of trouble for these states."The Future OutlookWhile it may seem that the economics of the closure of the strait are currently skewed towards Iran, Aniseh Tabrizi, an associate fellow on the Middle East and North Africa Programme at think tank Chatham House, noted that "the economics by itself is not going to be the driver to change calculation or move from the current standpoint."She emphasized that Iran and the US need to reach a "diplomatic compromise, with other calculations linked in to the economic factor", before there can be an end to the energy supply crisis.Farzanegan added that if the world expects stable access to the Strait of Hormuz, then paying Iran could well be accepted as the price of keeping the vital waterway predictable. "From an economic perspective, a negotiated transit arrangement [with Iran] now makes more sense than continued closure," he concluded.

Executive Summary: DOJ Launches $1.8 B “Anti‑Weaponisation” Compensation FundThe U.S. Department of Justice announced a new anti‑weaponisation fund worth just under $1.8 billion, designed to compensate individuals who allege they were victimised by federal legal actions. The fund is part of a settlement in former President Donald Trump's $10 billion lawsuit against the IRS over leaked tax returns.Mechanics of the New Fund and Its Legal OriginsThe fund originates from a “judgement fund,” a standing government account used for legal settlements without needing fresh congressional legislation. Key operational details include:Claims can be filed by anyone who believes they suffered from unlawful government‑initiated legal action.Every three months the fund must report recipients, payment types (cash, debt relief, etc.) to the Attorney General.A five‑person oversight panel, appointed by the Attorney General with one member selected in consultation with congressional leaders, will manage the fund.The fund will stop accepting new claims after December 1 2028, after which any remaining balance reverts to the federal treasury.Financial Scale: $1.8 B Allocation and Settlement ContextThe allocation is comparable to the annual policing or school budget of a midsized U.S. city, far exceeding the typical size of a single‑lawsuit settlement. It stems from the settlement of Trump’s lawsuit alleging the IRS leaked his tax information between 2018‑2020. The settlement was approved by a federal judge, meaning no additional legislative action is required to activate the fund.Political Fallout: Why Democrats and Legal Scholars Decry a Slush FundCritics, including more than 90 House Democrats and senators such as Elizabeth Warren and Ron Wyden, argue the fund:Pushes the limits of executive authority by creating a large compensation scheme without congressional oversight.Could be used to reward supporters of the January 6, 2021 Capitol riot, many of whom were pardoned by Trump.Represents a “slush fund” that may funnel taxpayer money to politically aligned individuals, echoing past concerns about “lawfare.”The Cato Institute and other think tanks have published analyses labeling the fund as an unprecedented bypass of normal appropriations processes.Looking Ahead: Congressional Pushback and Potential Fund FateDemocratic lawmakers are preparing legal challenges and may seek to block the fund through congressional action or a court injunction. The Justice Department has indicated that any unspent money after the fund’s termination will be returned to the Treasury, but the debate centers on whether the fund should have been created at all. If Congress intervenes, the fund could be restructured, placed under stricter oversight, or dissolved entirely, setting a precedent for future executive‑legislative financial arrangements.

The Lead A video posted by Israel's far-right National Security Minister Itamar Ben-Gvir, in which he is seen taunting abducted flotilla activists who sought to break the siege on Gaza, has triggered a backlash and dealt a huge blow to Israel's multimillion-dollar public relations campaign, known as 'Hasbara'. The Event Details The footage, posted on the social media platform X, showed Ben-Gvir gloating as activists from the Global Sumud Flotilla knelt on the floor, blindfolded, with their hands bound at the Port of Ashdod. Israeli naval forces had intercepted the flotilla's vessels in international waters off the coast of Cyprus, illegally abducting 430 participants. Among them, at least 87 have launched a hunger strike in solidarity with the more than 9,500 Palestinian prisoners held in Israeli jails. The Diplomatic Fallout The images of activists being dragged across the floor prompted several countries – including Italy, France, the Netherlands, Canada, and Spain – to summon Israeli ambassadors, condemning the 'unacceptable' treatment and violation of human dignity. The Collapse of the 'Hasbara' Illusion Experts argue that the frantic damage control by Israeli officials, including Prime Minister Benjamin Netanyahu, who ordered the rapid deportation of the activists, stems not from moral outrage over the abuses, but from the catastrophic damage done to Israel's global image. For decades, Israel has relied on 'Hasbara' – a Hebrew term translating to 'explanation' – a propaganda campaign to justify its policies and military actions against Palestinians to the international community. US Double Standards and 'Pro-Terror' Sanctions The diplomatic fallout also laid bare the glaring contradictions in United States policy. Following the video's release, US ambassador to Israel Mike Huckabee criticised Ben-Gvir, stating the minister had 'betrayed the dignity of his nation'. However, critics were quick to point out that Huckabee's condemnation rang hollow, as it focused entirely on the indignity of the broadcast rather than the human rights violations committed. A Microcosm of Palestinian Suffering For Palestinians, the humiliation endured by the European and international activists is merely a glimpse into a much darker, systemic reality. Mustafa Barghouti, secretary-general of the Palestinian National Initiative, said the bound and blindfolded activists represent a 'microcosm' of what Palestinian prisoners endure daily. The Hammer and the Flotillas Despite the military interceptions and the US sanctions, activists and analysts agree that the flotilla campaigns, which began in 2009 in response to Israeli land, sea and air blockades, have succeeded in exposing the limits of Israeli force.

Executive Summary: Clubs Challenge Southampton’s Expulsion Millwall and Wrexham are assessing legal options following the English Football League’s decision to expel Southampton from Saturday’s Championship playoff final and replace them with Middlesbrough. The clubs argue the disciplinary process was flawed and may pursue compensation. Legal Routes Explored by Millwall and Wrexham After Southampton’s Expulsion The clubs will await the written reasons from the EFL’s independent disciplinary panel, which were upheld on appeal. Their potential arguments include: Misapplication of the EFL rulebook regarding team replacement. Procedural defects in the disciplinary process. Grounds for a claim of damages based on the altered playoff composition. Both clubs have declined to comment publicly. £200m Wembley Prize and Potential Compensation at Stake The playoff final carries a minimum prize of £200 million for the winner. If the final proceeds without Southampton, the displaced clubs could argue for a share of lost revenue. Additional financial penalties already imposed on Southampton include a four‑point deduction for the next Championship season. Implications for EFL Playoff Rules and Future Governance The case highlights gaps in the EFL rulebook, which contains no explicit guidance on replacing an expelled team in the playoffs. The situation raises questions about: Whether the playoffs should be treated as a separate competition from the regular season. How future disciplinary sanctions will be calibrated for off‑field misconduct. The need for clearer procedural safeguards to avoid similar legal challenges. Possible Court Battles and the Road Ahead for the 2026 Playoffs Legal experts note that an injunction to postpone the final is unlikely given the tight timetable, so any claim would be retrospective. Potential outcomes include: A high‑court ruling that the EFL must revise its disciplinary process. Compensation awards to Millwall and Wrexham if the court finds the rulebook was misapplied. Further sanctions against Southampton, including possible charges from the FA. Hull owner Acun Ilicali has already received legal advice suggesting his club could claim automatic promotion, though he is unlikely to pursue that claim within the next 48 hours.

Islamabad and Beijing marked 75 years of diplomatic ties on May 21, 2026, reflecting a relationship forged in shared rivalry with India and reinforced by strategic land swaps, nuclear collaboration, and massive infrastructure projects. While official rhetoric celebrates “iron brothers” and “all‑weather friendship,” analysts argue that structural complementarity, not ideological affinity, has kept the partnership resilient. The 1963 Shaksgam Valley Transfer: Cementing Early Trust In March 1963 Pakistan ceded the 5,180 sq km (2,000 sq mi) Shaksgam Valley to China, a move that gave Beijing control over a strategically sensitive segment of the Karakoram range. The deal, negotiated by Zulfikar Ali Bhutto as foreign minister, was driven by Pakistan’s desire to counterbalance India after the 1962 Sino‑Indian war. Numbers that Define the Bond: Land, Infrastructure, and Nuclear Milestones 75 years of formal diplomatic relations (1950‑2025). 5,180 sq km of territory transferred in 1963. 3,000 km (1,900 mi) China‑Pakistan Economic Corridor (CPEC) linking Gwadar to Xinjiang. 1998 nuclear tests in Chagai, with documented Chinese technical assistance in the 1970s‑80s. Four‑day state visit by Shehbaz Sharif scheduled for May 23 2026. Strategic Ripple Effects: Regional Power Balance and the US‑China Channel The alliance gave Pakistan a powerful counterweight to India and positioned it as a back‑channel for the 1972 US‑China rapprochement, when Henry Kissinger used a Pakistani flight to Beijing. While the United States benefited from the diplomatic breakthrough, Pakistan received limited material reward, underscoring the asymmetrical nature of great‑power mediation. Economic Integration: CPEC and the Emerging All‑Weather Partnership Since 2015, the CPEC has become the flagship of the partnership, delivering highways, energy projects, and the Gwadar deep‑sea port. Analysts note that the economic dimension has shifted the relationship from a purely security‑driven pact to a multi‑layered interdependence, yet debt sustainability and regional security concerns remain contentious. Looking Forward: Scenarios for the Next Decade of China‑Pakistan Relations Experts anticipate three possible trajectories: Deepening convergence: Expanded defence co‑production and a broader Belt‑and‑Road footprint. Transactional plateau: Continued CPEC maintenance without major new initiatives, as both sides manage domestic pressures. Strategic strain: Escalating India‑China tensions or US policy shifts could force Pakistan to recalibrate its alignment. Regardless of the path, the “iron brothers” narrative will likely persist as a diplomatic shorthand for a partnership that has survived ideological divides and shifting global orders.

As the 2026 World Cup approaches, Spain is widely regarded as the top contender, buoyed by Euro 2024 success and youthful talent. However, experts and former players warn that the weight of expectation and external factors like extreme heat could derail their campaign. The Tactical Cohesion of La Roja Spain enters the tournament with a distinct advantage: a "club-like" cohesion rarely seen at the international level. Fernando Kallas of Reuters notes that 90 percent of the squad has played under coach Luis de la Fuente since their teenage years, creating a system where 11 players scored 14 goals during Euro 2024. This depth allows for a fluid, collective style of play rather than reliance on a single superstar. Key Strength: Seamless integration of youth and experience. Star Power: Lamine Yamal and Nico Williams provide elite pace and creativity. Systemic Advantage: Players know their roles intuitively, reducing tactical errors. The Pressure of Expectation: A History of Disasters Despite the squad's form, the psychological burden of being favorites is a significant hurdle. Former Barcelona defender Miguel Angel Nadal warns that the "Dream Team" mentality must remain intact to avoid the pitfalls of the past. He points to the 2014 World Cup as a cautionary tale, where Spain was eliminated in the group stage despite being the reigning champions. Manel Hernandez, secretary-general of the Barcelona fan group, echoes this sentiment, suggesting that declaring favorites before the tournament begins is a dangerous game. He highlights the difficulty of the group stage, which includes Saudi Arabia, Uruguay, and Cape Verde, warning that complacency could be fatal. The "Club vs. Country" Advantage The stability of Spain's setup provides a competitive edge over rivals like Brazil and Argentina. While Argentina struggles with an aging squad and Brazil faces injury concerns for key players, Spain benefits from a younger, healthier roster. Graham Hunter argues that Spain's ability to play as a coherent unit gives them an edge in a tournament defined by luck and attrition. The War of Attrition in the Americas The physical environment of the 2026 World Cup presents a unique challenge. Nadal emphasizes the difficulty of playing in extreme heat, a factor that could favor more physically robust teams. Hunter suggests that the tournament will be a "war of attrition," where mental resilience and adaptability to climatic conditions will be just as important as technical skill. The combination of high expectations and the harsh American summer heat makes Spain's path to the trophy one of the most complex in world football.