But the criminal investigation into the former Duke of York is ongoing, while further legal proceedings will decide whether police can keep hold of evidence seized during the searches.

The former prince launched a legal challenge to quash the warrants - issued by an Old Bailey judge on 18 February last year - which allowed police to raid Andrew's residence on the Sandringham estate and his former Windsor home, Royal Lodge.

Andrew was arrested the following day, on his 66th birthday, on suspicion of misconduct in public office following allegations he shared sensitive information with Jeffrey Epstein during his time as the UK's trade envoy.

At a hearing at the Royal Courts of Justice on Thursday, which Andrew did not attend, Mr Justice Hilliard said that "agreement was reached between the claimant and Thames Valley Police that the search warrants were unlawful".

He said a "significant error was made in the application for and grant of the search warrants," which was "enough to render the warrants, the searches and the material seized unlawful".

The court heard officers "took possession of documentary material and electronic devices", but the judge said "there would be little point handing it back if it could be lawfully seized on some other provision".

"The fact that the search warrants were quashed does not bring an investigation to an end, or mean that the whole of the investigation was unlawful," he added.

Police have launched a bid to retain evidence gathered during the searches, which the judge said is opposed by Andrew, with a hearing to be held in private in 2027.

Andrew, who now lives on the King's private Sandringham estate in Norfolk, was interviewed under caution on February 19 before being released under investigation, and denies wrongdoing.

Police began their searches of the properties on the same day, with the force confirming on 24 February they had ended.

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It is understood police applied for warrants under Sections 8 and 9 the Police and Criminal Evidence Act, which regulates police powers, but an Old Bailey judge decided warrants were only needed under Section 8.

But after Andrew launched his legal action, it was agreed the warrants should've been granted under Section 9, which includes items covered by legal privilege, or "excluded material", including business records and confidential journalistic material.

The agreement was formalised on 22 July, but has only now been made public following a hearing which had been expected to deal with a bid by police to restrict access to documents in the legal claim.

But Mr Justice Hilliard said police indicated that they no longer pursued the matter on Wednesday afternoon, subject to certain information being redacted, such as the names of individual officers.

A TVP spokesperson said the agreement that led to the quashing of the warrants "relates specifically to a decision made by the court regarding the legal basis for these and is not a judgement on our ongoing investigation".

"Legal proceedings are currently ongoing to decide how material seized during the searches will be managed and whether we can lawfully retain it in support of our ongoing investigation," they said.

"Our investigation into alleged misconduct in public office by a man in his 60s from Norfolk remains ongoing, and we continue to pursue all reasonable lines of enquiry to effectively and fairly progress this investigation."

Charles stripped his brother of both his right to be a prince and his dukedom late last year over his association with convicted sex offender Epstein.

Andrew receives an annual stipend - the amount of which has not been disclosed - provided by the King from his private funds, but a palace source said on Monday it has been made clear that these monies should not be used to fund the judicial review.

A National Police Chiefs' Council spokesperson said a small national team is working to review material related to the Epstein files, while all 43 forces in England and Wales are cooperating on intelligence assessments.

Along with TVP, the Metropolitan Police and Surrey Police also have live ongoing investigations.

"Identifying any further potential offending, suspects, victims or witnesses is a complex process and will take time, but UK policing remains committed to carrying out a meticulous assessment of the material released by the US Department of Justice," the spokesperson said.

The two men, aged 36 and 32, were arrested inside the base perimeter of RAF Molesworth in the early hours of Thursday.

Both are Latvian nationals, currently living in the UK, according to the Metropolitan Police.

Ministry of Defence Police (MDP) carrying out security patrols at RAF Molesworth initially found an abandoned vehicle around 10pm on Wednesday.

A small breach of the perimeter fence was identified and at around 2am on Thursday, the two men were found and arrested.

Police said concerns over the vehicle led to the deployment of specialist Explosive Ordnance Disposal officers to carry out additional checks before it was taken away by officers to be searched.

The investigation is being led by officers from Counter Terrorism Policing.

Police from Cambridgeshire Constabulary also searched the site.

The two men were initially arrested on suspicion of trespass on a prohibited site and criminal damage and taken to a Cambridgeshire police station.

The pair have since been further arrested on suspicion of an offence under section 4 of the National Security Act - which is "entering a prohibited place for a purpose prejudicial to the UK".

RAF Molesworth is home to a NATO intelligence hub and the US's intelligence operations centres for Europe and Africa, according to the Central Intelligence Agency's (CIA) website.

NATO's Intelligence Fusion Centre supports the Supreme Allied Commander Europe and Allied Command Operations - responsible for the planning and execution of the organisation's operations.

A cordon at the scene was put in place, which affected nearby roads, but it has since been lifted after "nothing of concern was found", police said.

Both men are currently still being held in police custody while they are questioned by detectives.

A US European Command spokesman said: "While we assess the global security posture, US forces remain at the highest state of readiness.

"We would also express our gratitude to the UK for the close partnership with US forces on matters of security."

Officers are currently carrying out two searches at properties in Peterborough in connection with the investigation.

Commander Helen Flanagan, head of Counter Terrorism Policing London, which is leading the investigation, said: "We're very aware of the recent heightened interest and concern around military sites, but at this stage there is nothing to suggest this incident is connected to the events at RAF Fairford last week.

"Our investigation into this incident is at its very early stages. A big part of this is looking at any potential motivation as to why the men were present at the site, but I must stress we are keeping an open mind at this stage and enquiries are very much ongoing."

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In 2024, the US's Defence Intelligence Agency said it had broken ground on a $556m (£421m) facility at the base, which will support US military intelligence in Europe and Africa and employ 1,000 intelligence personnel.

The base also houses US Air Force units from the 423rd Air Base Group, which has its headquarters at nearby RAF Alconbury.

Defence Secretary Wes Streeting said: "Thank you to the MoD and civilian police and security teams who acted quickly and professionally to arrest two individuals at RAF Molesworth.

"We are highly vigilant about potential threats to our military sites and work closely with the police to keep them secure.

"The protection of our bases is multi-layered, including physical barriers, patrols, surveillance, monitoring, and counter-drone capabilities."

The announcement - which has been described as the "mathpocalypse" - took place on Tuesday.

It comes a month after OpenAI announced a proposed proof for the Navier-Stokes equation, one of seven Millennium Prize Problems which mark the peak of human ambition in the field.

Mathematicians have been taken aback by the speed of mathematical progress in AI, which struggled with GCSE-level maths problems less than two years ago.

The proposed proofs, which are described in 722 papers, are so complex and so numerous that even understanding them poses a gigantic problem for mathematicians, and many have not been verified.

"If you want some sense of what things feel like now in maths, imagine a hunter-gatherer who's spent his entire life learning to survive deep in an unforgiving rainforest, then a giant resort hotel springs up right next to him with a helipad and heated pools and Airbnbs," wrote Scott Aaronson, chair of computer science at The University of Texas at Austin.

While some have welcomed the news, saying it advances the field in exciting ways, others have rejected the mass dump of proofs.

Terence Tao, widely regarded as the world's greatest mathematicians, shared a statement from the Association for Human Mathematics which called on mathematicians to stop working with OpenAI.

"Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power," the group wrote.

"We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centres human understanding."

OpenAI said in a blog post that "we want this progress to push the frontier of human knowledge and enable further progress in mathematics", adding that it had consulted extensively with mathematicians on how best to publish the results.

The results have been obtained with an internal model whose full capabilities are unknown, which suggests that AI progress is likely to continue, with profound consequences for the field.

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Timothy Gowers, professor of mathematics at The Collège de France and The University of Cambridge, told Sky News recently that he was "shocked" by the pace of AI progress in maths.

He said: "We're in a very strange state where we can just see the tsunami - we're still on the beach, we can see the tsunami, but it hasn't quite reached us yet.

"During the course of this year they've gone from mediocre PhD student to very good PhD student, to experienced mathematician level, to top mathematician level," added Professor Gowers.

Now it seems to have gone even further.

Witnesses saw smoke billowing from a plane at King Khalid International Airport, and the Iran-aligned Houthis - based in neighbouring Yemen - also told workers to stay away from energy plants.

In response to Thursday's attack, Germany's Lufthansa Group and Air India as well as two other carriers from the country, all suspended flights to the city.

The rebels had earlier warned global airlines "for the last time" about the consequences of operating in Saudi airspace, which they called a theatre for their military operations.

The Houthis say they are targeting the kingdom because of its support for Yemen's government forces in the country's civil war.

Not long after the blasts, the airport "closed again", the Flightradar24 monitor said on X.

One witness said outgoing flights were stopped and passengers were told to leave checked-in luggage behind and evacuate. It was not immediately clear how many flights were disrupted.

Major General Turki al-Malki, spokesperson for the Saudi-led coalition fighting the Houthis, said it had destroyed two ballistic missiles launched by the Houthis toward Riyadh, along with a ballistic missile fired toward the city of Khamis Mushait, in the southwest near the Yemen border.

A Lufthansa spokesperson has since said the German group would suspend ⁠flights to Riyadh ⁠until 16 October due to "current developments in the ‌Middle East".

All Air India flights to the Saudi capital will be cancelled until 10 October.

Flight tracker FlightRadar24 showed numerous delayed arrivals and departures at the airport on ​Thursday, with travellers advised to contact airlines to check the status of flights.

A US Embassy security alert was also issued on Wednesday, telling citizens to be aware of potential airspace closures and warning US government employees not to travel within 20 miles (32km) of the Yemen border.

On Wednesday, Saudi officials confirmed that attacks on the Riyadh and Abha airports claimed by the Houthis had killed three people this week.

They are the first deaths recorded in the conflict since Saudi Arabia and allied Yemeni forces launched a significant ground and air offensive against the rebels.

Officials also said that debris from an intercepted missile hit a pre-school and nursery, without mentioning specific casualties.

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Saudi Arabia has requested more support from allied nations in the conflict - now an overspill of the US-Iran war - to reverse Houthi gains in southwest Yemen.

The country's long-running civil war escalated again in recent weeks, as the Houthis seized stretches of the Red Sea coast last month.

It has caused a humanitarian crisis in the poorest Arab state, displacing 200,000 people since the conflict resumed after a ceasefire in 2022.

Maduro and Flores, who were previously charged with participating in a cocaine importation conspiracy, now face allegations of human rights abuses during his time as president.

The pair have been behind bars in New York since early January, after US forces dramatically seized them from their Caracas home in a middle-of-the-night raid.

Both have pleaded not guilty in the criminal case first brought years ago against them and numerous alleged co-conspirators.

Maduro, 63, and Flores, 69, had already been facing potential life prison sentences if convicted - their trial is scheduled for 1 June 2027.

Their attorneys had asked for the case to be dismissed on the grounds that they have immunity as the leader and first lady of a foreign country.

The new charges stem from accusations that they tortured political opponents to retain their grip on power in ‌the oil-rich South American country.

In one instance, prosecutors said Maduro in 2013 ordered the arrest of a Venezuelan military ​officer he suspected of planning a coup.

It is claimed that members of Venezuela's military counterintelligence directorate then interrogated the officer while shocking the individual ​with a live electric wire around 20 times and hung the individual by the hands and feet for several minutes at a time.

The indictment alleges that the torture victims included three US citizens, who were not named in the court documents.

One of those Americans was punched, subjected to electric shocks and forced to eat his own vomit during eight months of imprisonment, the indictment says.

Another was beaten badly enough to necessitate a long hospital stay and had a gun stuck in his mouth.

A third was handcuffed, hooded and beaten for two days every half an hour, suffering a permanent eye injury, the indictment said.

Lawyers for Maduro and Flores did not immediately respond to a request for comment.

Maduro has been under investigation for years by the International Criminal Court over allegations of crimes against humanity, tied to his crackdown on opponents. He has always denied any wrongdoing.

Flores' request for house arrest denied

Separately, Flores was in court on Thursday, where her lawyers urged a judge to free her from prison and put her under house arrest because of her heart problems.

The request was denied.

Lawyers for Flores have also argued that she has medical issues that cannot be treated effectively in jail, and that she could instead be allowed to live under 24-hour guard at a residence in New York.

They claim she suffered permanent scarring from being "thrown against a wall or dresser" during the US military action that led to her capture and has lost 11kg while imprisoned at the Metropolitan Detention Center in Brooklyn.

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They said she has "lingering light-headedness, dizziness and shortness of breath" and might have suffered a minor heart attack in July.

However, prosecutors had opposed the request, claiming that she poses an "extreme risk of flight and danger to the public" if released and that she had received adequate medical care.

President Donald Trump's administration has defended the raid to capture Maduro as a "surgical law enforcement operation."

However, Maduro has called himself a prisoner of war and his capture a kidnapping.