OpenAI says its AI models produced findings in harmonic analysis, computer science and physics. The company reported proofs addressing a long-standing mathematics conjecture and claims involving computational complexity and quantum chromodynamics, while noting that outside researchers are reviewing the work.
Key Facts
OpenAI said its models proved the local smoothing conjecture in every spatial dimension; researchers had previously proved it only in three dimensions.
The local smoothing conjecture is a problem in harmonic analysis, which studies how signals and functions can be represented using waves.
OpenAI said its models disproved the Exponential Time Hypothesis, a claim about how long it takes to solve certain computational problems.
The company said its models gave a mathematically rigorous proof of the existence of quantum chromodynamics, the theory of interactions between quarks and gluons.
Alex Iosevich, a University of Rochester mathematics professor, described the reported proof of the conjecture as a historic development.
OpenAI said it consulted three expert groups, called “shepherds,” on how to review and release the findings.
The mathematics and computer science groups chose to make AI-produced data public immediately; the physics group deferred its release to prepare an interpretation.
OpenAI said the findings need further scientific interpretation and welcomed feedback on broadening participation in the process.
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Anthropic filed paperwork to go public in the United States, disclosing audited financial results and risks for the first time. The filing reports rapid revenue growth alongside large losses and infrastructure costs, and describes both potential uses of AI and risks the company says its safeguards may not fully prevent.
Key Facts
Anthropic was founded in 2021 and did not name a target valuation in its filing.
The company reported $16.3 billion in revenue and an $80 billion loss in the first half of 2026; the filing attributes $72.4 billion to a one-off charge.
Anthropic reported $4.585 billion in 2025 revenue and roughly $33.9 billion in losses.
Anthropic spent nearly $9 billion on AI infrastructure in the first half of 2026, compared with $2.3 billion in the same period a year earlier.
The company said its five largest customers account for 25% of its business and that its customers are largely other businesses.
Anthropic identified potential AI applications in biology and medicine, economic growth, global development, and government services.
The filing lists AI risks including misuse by bad actors, loss of control, authoritarian abuse, and economic disruption.
Anthropic said its most capable models could meaningfully assist malicious actors seeking to develop dangerous weapons, and acknowledged that its safeguards may not work.
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Saudi Arabia, Iran, Turkiye and Pakistan agreed to continue talks on regional peace and security at a meeting in Tehran. The discussions took place as Saudi Arabia reported intercepting missiles and drones launched by the Houthis, while fighting in Yemen and tensions between Washington and Tehran continued.
Key Facts
The four countries said they would continue dialogue to strengthen regional cooperation on peace and security.
Saudi Arabia said it sent Defence Minister Prince Khalid bin Salman to the Tehran talks; Pakistan and Turkiye sent senior military and intelligence officials.
A Pakistani security official said the talks covered a proposed regional non-aggression network, US-Iran negotiations, Houthi attacks, the Strait of Hormuz and a US blockade of Iranian ports.
A Pakistani official said Field Marshal Asim Munir presented Iran with a proposal intended to offer benefits to both Washington and Tehran, but did not disclose its details.
Saudi coalition spokesman Turki al-Maliki said Saudi air defences intercepted three ballistic missiles aimed at the Eastern Province and four drones headed toward the south, with no casualties reported.
The coalition also said it shot down three more drones over Riyadh and southern Saudi Arabia on Monday evening; the Houthis did not immediately comment.
The article reports that the Houthis seized strategic areas on Yemen’s Red Sea coast last month and have also attacked Saudi airports, ports and oil facilities.
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Saudi Arabia’s coalition said it intercepted three Houthi ballistic missiles and four drones headed toward the kingdom, with no casualties reported. The attacks came amid renewed fighting in Yemen and regional talks involving Saudi Arabia, Turkey, Pakistan and Iran aimed at preventing further escalation.
Key Facts
Coalition spokesperson Turki al-Maliki said the missiles were aimed at Saudi Arabia’s Eastern Province and the drones were headed toward southern regions.
Saudi civil defense issued emergency alerts in the Eastern Province and Al Kharj Governorate, then lifted them and advised residents to take precautions.
The coalition said it had also intercepted three drones near Riyadh and in southern Saudi Arabia on Monday evening.
Senior officials from Saudi Arabia, Turkey and Pakistan met Iranian officials in Tehran; the four countries said they would continue dialogue on regional peace and security.
The article reports that Houthi forces seized areas along Yemen’s Red Sea coast last month, prompting a counteroffensive and intensified Saudi strikes.
The coalition said it retaliated by destroying five tunnels used to store ballistic missiles, four launchers and a missile depot.
Egypt’s Parliament authorized the military to help defend Saudi Arabia, according to the article.
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An offshore drilling platform sank off Gabon’s southern coast on Saturday. Search and rescue operations are continuing for five missing people; 110 of the 115 people aboard are safe.
Key Facts
The platform sank off Gabon’s southern coast.
The incident happened on Saturday.
There were 115 people on the platform.
Authorities have confirmed that 110 people are safe.
Five people remain missing.
Search and rescue operations are continuing.
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Israelis are voting in a closely contested election as Benjamin Netanyahu seeks to remain prime minister. The campaign has focused on security failures, ultra-Orthodox military service and the role of Arab-led parties, while the main opposition has also embraced right-wing positions.
Key Facts
The election is the first since the Hamas-led attack of October 7, 2023, and the war in Gaza that followed.
Netanyahu is seeking another term after leading a coalition that included far-right and ultra-Orthodox parties.
Netanyahu faces an ongoing corruption trial and is wanted by the International Criminal Court on war-crimes charges related to Gaza.
The campaign addressed accountability for alleged security failures and disputes over whether ultra-Orthodox Jews should serve in the military.
The article says Israeli politics has shifted right over several decades, with that shift accelerating under Netanyahu’s latest government.
The 2018 Nation-State Law formally defined Israel as the nation-state of the Jewish people.
The article reports that deadly attacks in Gaza and Israeli army raids and settler violence in the occupied West Bank continued during the campaign.
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Meghan, Duchess of Sussex, will join Poh Ling Yeow as a special guest co-host on the 2027 season of Junior MasterChef Australia, Channel 10 confirmed. Meghan said she enjoys cooking with her children and looks forward to seeing what the young contestants make.
Key Facts
Meghan appeared as a guest judge on MasterChef Australia last year.
Paramount ANZ said Meghan’s guest-judge appearance drew the show’s largest Sunday audience since 2023.
Network 10 did not tell the BBC whether Meghan will be paid for co-hosting Junior MasterChef Australia.
Meghan and Prince Harry visited Australia for four days in April.
Meghan’s Netflix lifestyle and cookery show, With Love, Meghan, ran for two series and a Christmas special.
Meghan launched the lifestyle brand As Ever, which ended its partnership with Netflix in March.
Thomas Markle denies allegations of child sexual abuse.
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Chinese stocks have fallen sharply as investors question whether the country’s AI and robotics companies can earn profits amid weak domestic demand and possible U.S. restrictions. The decline has erased an estimated $2.4 trillion in Chinese market value since May, even as Beijing funds domestic AI development.
Key Facts
The Shanghai STAR 50 Index, which tracks technology and innovation companies, fell 40% since June.
Other major Chinese stock indexes dropped about 20% over the same period.
Bloomberg estimated that the sell-off erased $2.4 trillion from Chinese stock markets since May.
Goldman Sachs estimated that AI stocks make up more than 40% of the market value in China and Hong Kong.
China launched a state-backed AI investment fund worth 60 billion yuan this month.
Analysts cited weak consumer demand, lower-than-expected profit margins, rising U.S. interest rates and potential Western restrictions as factors behind the sell-off.
China’s robotics demand comes mainly from government agencies and research institutions; humanoid robots and robotic dogs remain too expensive or impractical for most home use.
Shares in Chinese robot-maker Unitree have fallen by half since trading began in August.
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Representative Randy Fine said people who lied on immigration forms to obtain U.S. citizenship should be denaturalized and deported. Asked whether that standard should apply to Melania Trump and Elon Musk, he said he did not know their circumstances and knew of no allegations against them.
Key Facts
Fine is a Florida Republican.
Melania Trump was born in Slovenia and became a U.S. citizen in 2006, according to the White House.
Elon Musk was born in South Africa and became a U.S. citizen through naturalization in 2002, according to his biographer Walter Isaacson.
Fine called for reviewing the cases of people naturalized since 1990 and said Representative Ilhan Omar should be the first examined.
Federal agents searched Omar’s Washington-area home on October 12 as part of an ongoing federal criminal investigation, according to the Justice Department.
Omar has not been charged in connection with the investigation.
The Associated Press reported that federal authorities issued more than a dozen grand jury subpoenas, including to some of Omar’s relatives.
Omar has denied allegations that she married her former husband, Ahmed Nur Said Elmi, to help him enter the United States.
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Israelis are voting in the country’s first election since Hamas’s October 7, 2023 attack. The closely contested race pits Prime Minister Benjamin Netanyahu against centrist challenger Gadi Eisenkot, with security, accountability and Israel’s political direction central to the campaign.
Key Facts
The election is taking place after three years of regional wars that followed the October 7 attack.
The latest published polls showed Netanyahu and Eisenkot running neck and neck.
About 7.3 million people are eligible to vote, including more than half a million first-time young voters.
Netanyahu has served as prime minister for more than 19 years across multiple terms.
Netanyahu campaigned on his security record and his relationship with U.S. President Donald Trump.
Netanyahu’s coalition includes far-right nationalist parties whose influence has grown.
Opposition parties criticize Netanyahu’s support for exemptions from military service for ultra-Orthodox religious students, as the military faces personnel shortages.
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Police in Criciúma, southern Brazil, shot a man who had assaulted and threatened his partner with a fake gun. Video shows him striking her and trying to use her as a human shield as officers arrived; both were taken to hospital for treatment.
Key Facts
The incident occurred in Criciúma, southern Brazil.
Police shot the man.
The man had assaulted and threatened his partner.
The gun was fake.
Video shows the man striking his partner.
Video shows him attempting to use her as a human shield as officers arrived.
Both the man and his partner were taken to hospital for treatment.
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Some UK retailers reported sharp increases in sales of emergency supplies after Defence Secretary Wes Streeting urged people to keep basic goods at home. The government advice cited items such as water, tinned food, torches and phone chargers to prepare for disruptions including cyber attacks and extreme weather.
Key Facts
Iceland said sales of tinned products rose 40% over the weekend and reported strong bottled-water sales.
Robert Dyas reported a 160% increase in radio sales compared with the previous week.
Maplin said radio sales rose 92%, torch sales rose 284% and power-bank sales rose 106% after the government announcement, compared with the same period in the previous month.
Streeting’s October 22 advice included three litres of water per person per day, tinned goods, baby food, battery-powered torches and mobile-phone charging packs.
Independent online retailer Sgt Prepper’s said its sales increased tenfold after the government’s Be Ready campaign began.
Preppers Shop UK’s founder reported a 650% increase in sales after the announcement.
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Health and Human Services Secretary Robert F. Kennedy Jr. said at a Republican campaign rally in Grand Rapids, Michigan, that 500 million Arabs do not produce science, win major prizes, or produce literature. Nobel Prize records cited in the article include Arab recipients in science, literature, and peace. Kennedy spoke amid a Michigan Senate race dispute over Democratic candidate Abdul El-Sayed’s alleged ties to the Muslim Brotherhood; El-Sayed has denied ties and disavowed the group.
Key Facts
Kennedy made the remarks at a Republican campaign rally in Grand Rapids on Monday.
He was discussing the Muslim Brotherhood after referencing Republican Senate candidate Mike Rogers.
Rogers and his allies have sought to link Democratic candidate Abdul El-Sayed to the Muslim Brotherhood.
El-Sayed has denied ties to the group and publicly disavowed it.
Egyptian scientist Ahmed Zewail won the 1999 Nobel Prize in Chemistry.
Egyptian novelist Naguib Mahfouz won the 1988 Nobel Prize in Literature.
Yemeni journalist and activist Tawakkol Karman shared the 2011 Nobel Peace Prize.
Kennedy also visited Corewell Health to promote an initiative on hospital food and nutrition education.
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President Donald Trump announced a federal and state initiative worth more than $250 million to recruit and retain nurses at nursing homes. He shared the announcement in a Truth Social post Monday evening.
Key Facts
The initiative is intended to recruit and retain nursing home nurses.
Trump described it as a federal and state initiative.
Trump said the initiative is worth more than $250 million.
Trump announced the initiative in a Truth Social post Monday evening.
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The Justice Department under President Trump is seeking to prosecute former FBI Director James Comey for a third time in two years. Comey denies charges alleging that he shared sensitive national security information before the 2016 election and defended himself in an Instagram video.
Key Facts
The Justice Department is seeking a third prosecution of James Comey in two years.
Comey previously served as FBI director.
The charges accuse Comey of sharing sensitive national security information.
The alleged conduct took place in the lead-up to the 2016 election.
Comey has denied the charges.
Comey defended himself in a video posted on Instagram.
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A ceasefire has begun in Russian-occupied Oleshky, Ukraine, to allow aid deliveries and civilian evacuations. Ukrainian and Russian officials confirmed steps toward the truce, which the White House said Melania Trump helped secure.
Key Facts
The truce is intended to enable aid deliveries and the evacuation of civilians from Oleshky.
Russia’s defence ministry said it stopped firing toward Kherson at 04:00 Moscow time to prepare for the humanitarian operation.
Ukrainian human rights commissioner Dmytro Lubinets said Ukraine was honouring its commitments under the ceasefire.
Russia seized Oleshky in the first days of its full-scale invasion in February 2022.
Ukraine estimates that about 6,000 people remain in Oleshky and nearby areas, including 200 children.
The city had about 24,000 residents before the war.
The White House said Melania Trump secured the ceasefire and that civilian leaders from both countries would assess residents’ needs.
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Voice actor Eric Bauza, known for voicing Bugs Bunny and Daffy Duck, died Sunday at age 46, according to his manager. The Los Angeles County Medical Examiner said he died from hypertensive cardiovascular disease. Bauza had more than 400 acting credits and won Children’s and Family Emmy Awards in 2022, 2023 and 2025.
Key Facts
Bauza was born in Canada.
He voiced Bugs Bunny, Daffy Duck and many other Looney Tunes characters, including Porky Pig, Tweety Bird and Road Runner.
He voiced characters in the most recent Space Jam film alongside LeBron James.
His other voice credits included Avatar: The Last Airbender, Puss in Boots, Bob’s Burgers and Teen Titans Go!.
Warner Bros. said Bauza worked at Warner Bros. Animation for 15 years.
He was nominated for five Daytime Emmy Awards from 2016 to 2021 and was nominated for a Children’s and Family Emmy in 2026.
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The U.S. military has banned all BYD vehicles from Kadena Air Base in Okinawa, citing security risks to personnel and operations. Officials have not identified a specific threat, but the decision reflects broader concerns that vehicle cameras, sensors and GPS systems could expose sensitive information.
Key Facts
The ban took effect this week at Kadena Air Base, a major U.S. Air Force base on Okinawa.
The restriction covers privately owned BYD cars as well as taxis and delivery vehicles, according to reports.
Security staff at the base’s gates have been told to turn away BYD vehicles.
Kadena hosts fighter aircraft, refuelling and surveillance aircraft, and the Typhon missile system.
The Pentagon added BYD to a list of 188 Chinese companies it believes have ties to China’s military.
The U.S. banned sales and use of Huawei and ZTE smartphones at military bases worldwide in 2018, citing security concerns.
China restricted Tesla vehicles from some government compounds in 2021 over concerns about their cameras, then removed the restriction in 2024.
BYD became the world’s largest seller of battery-electric vehicles in 2025 and has been expanding in Japan.
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A federal judge barred the Trump administration from removing or damaging trees at East Potomac Golf Links while a lawsuit over President Donald Trump’s renovation plan proceeds. Judge Ana Reyes found the D.C. Preservation League is likely to show that the plan requires an environmental review and that recent tree removal was part of the renovation.
Key Facts
Reyes issued a 70-page preliminary injunction blocking tree removal or damage at the publicly owned Washington, D.C., golf course.
The injunction requires the government to give 10 days’ notice before making any irreparable modification to the course.
The D.C. Preservation League sued, arguing that the government failed to analyze the renovation’s environmental effects as required by law.
More than 430 trees had been cut down earlier in the month.
Reyes rejected the government’s argument that the tree removal was unrelated maintenance.
Trump has proposed turning the course into a championship venue and said it would close for construction in November and reopen in 15 months.
Reyes said the government had acted contrary to assurances it gave the court.
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Jordan said it carried out airstrikes in Syria against what it called terrorist groups after a shooting at a Greek Orthodox church funeral in Jordan killed nine people. Syrian officials reported five deaths in the strike, while Syria said it had not been notified and closed its only passenger and commercial crossing with Jordan.
Key Facts
The funeral shooting in Karak on Friday killed nine people and injured 34.
Jordan said its fighter jets killed and wounded several people it described as terrorists.
Syrian health officials said five people, including women, children and an elderly man, died in the strike on a village in Daraa province.
Jordan said it coordinated the strike with the Syrian government; Syria said it had not been notified.
Syria suspended passenger and commercial traffic at the Nassib crossing, the only such crossing between the two countries.
Syrian authorities said they opened direct communication with Jordan and began an investigation into the strike.
Jordanian security forces arrested two brothers and their brother-in-law in connection with the church attack, according to Jordanian officials.
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