The India Decade e-paper, Saturday, 10 October 2026

OpenAI’s claims of progress on a century-old mathematical mystery have been met with sharp criticism from one of the world's leading experts on the problem.

The AI company recently released a preprint alongside a vast mountain of computer code written in the Lean programming language, claiming to show new progress on whether the "Partition Principle" implies the "Axiom of Choice". But set theorist Asaf Karagila, who has spent 15 years studying the problem, says the paper is so poorly written and academically sloppy that it deserves a flat rejection.

The dispute highlights a growing frustration among scientists who feel tech companies are dumping poorly verified, machine-generated work onto the academic community and expecting humans to clean up the mess.

What is the Partition Principle?

The debate touches on a fundamental question in mathematical logic. The Partition Principle is a classic, century-old puzzle in set theory. Karagila is so famously associated with the problem that his personal website's "Problems" page promises a bottle of whisky to anyone who finally solves it.

After OpenAI's announcement, his inbox filled with colleagues asking if he was about to mail a bottle of single malt to OpenAI chief executive Sam Altman.

The answer, Karagila wrote on his blog, is a firm no.

Why the OpenAI preprint fails academic standards

While OpenAI's press release framed the paper as a sign of "progress", Karagila described the actual text of the preprint as deeply flawed.

"It sucked," Karagila wrote. "It is unclear, muddled, and has a strange structure."

According to the mathematician, the paper fails basic academic standards on several fronts:

• Strange structure and terminology: The mathematical phrasing is "a bit off," and the preprint includes bizarre, highly specific lemmas that have no logical place in a paper of this nature.
• Lazy referencing: Rather than pointing to established, peer-reviewed literature, the preprint cites unpublished, unrefereed lecture notes—including Karagila's own informal online notes—for basic concepts.
• Unreadable arguments: Usually, a set theorist can skim a paper in their field and quickly grasp the author's strategy. Karagila says the OpenAI paper is so poorly communicated that this is impossible.

"If this was an academic paper submitted to a journal, it should be issued a desk rejection for the quality," Karagila said. "The onus is always on the author to conform and adhere to the communications level."

An AI 'denial of service' on researchers

This sloppy presentation points to a broader, more frustrating trend in the generative AI boom. By dumping incomprehensible, machine-generated papers onto the internet, tech companies are effectively launching a "denial of service" attack on academic time, Karagila argues.

While the media portrays these releases as revolutionary solutions, actual scientists are left with a miserable choice: ignore them, or stop their own active research to painstakingly sift through a machine's messy homework.

There is a political danger to this hype, too. Karagila warns that politicians, funding bodies, and the public—who do not understand how mathematical research actually works—will see these flashy announcements and assume human mathematicians are about to be replaced by bots.

He compares the situation to replacing a chef with automated kitchen tools. A self-dicing knife is a neat gadget, but it is still a long way from preparing a palatable meal.

For Karagila, the episode highlights a clash of cultures. Rather than AI companies doing the hard work to meet the rigorous communication standards of science, they seem to want academics to lower their standards to accommodate AI.

"I am afraid that this just will not do," Karagila said.

The US government's sudden decision to suspend major technology companies from its permanent residency programme has triggered a fierce political row in India. The opposition Congress party has accused Prime Minister Narendra Modi of being too compromised by domestic scandals to protect Indian IT workers.

On Thursday, the US Anti-Fraud Task Force froze applications for the Permanent Labour Certification (PERM) programme—the initial step for obtaining a green card—for several top outsourcing firms. The list of suspended companies includes Indian giants Infosys, Tata Consultancy Services (TCS), Wipro, and HCLTech, alongside multinational firms like Microsoft and Cognizant.

Congress claims Modi is under siege

The move immediately became ammunition for the opposition in New Delhi. Jairam Ramesh, the Congress general secretary in-charge of communications, claimed on Friday that India has several avenues to retaliate under existing laws, such as the Digital Personal Data Protection Act. However, he argued that Prime Minister Modi would fail to act because he is "under siege" over "GyaneshGate"—a reference to opposition demands for Chief Election Commissioner Gyanesh Kumar to resign over alleged electoral roll irregularities.

Ramesh also linked the US action to recent domestic concessions, claiming the Indian government had recently tried to appease US President Donald Trump by introducing UPI merchant discount rate (MDR) charges to benefit American companies. "Appeasement of bullies never pays," Ramesh wrote on X.

New Delhi labels Vance remarks "deeply offensive"

While the opposition attacked the Prime Minister's silence, India’s Ministry of External Affairs issued a sharp public rebuke to Washington. The ministry described remarks made by US Vice President JD Vance during the announcement as "deeply offensive" and carrying "painful historical and colonial legacy connotations."

Vance had accused tech companies of operating "visa mills" and using H-1B visa holders as "foreign indentured servants" to undercut American wages. In response, the Ministry of External Affairs defended Indian tech professionals as "highly educated and skilled contributors" who have helped build the US economy. The ministry warned that the curbs "do not advance the shared ambitions" of both countries.

The ministry also sought to reassure nervous professionals, noting that the PERM freeze does not directly affect the validity of existing H-1B visas, though it will slow down the green card process.

Tech sector reacts to US immigration shift

Within the tech industry, reactions to the US crackdown have been mixed. Sridhar Vembu, the founder of software company Zoho, urged Indian professionals in the US to pack their bags and return home. Writing on X, Vembu said that while Indians have built America's technological strength, "a big part of America no longer wants us." He described the current political climate as a choice between a "hard right" and a "woke left" where Indian workers are mere bystanders.

In contrast, India's largest IT exporter, TCS, downplayed the business impact of the US move. In a stock exchange filing, the company stated that it had submitted only a "single-digit" number of PERM applications over the past two years. TCS said its US strategy focuses on hiring local staff and that it still plans to recruit an additional 15,000 American workers over the next five years.

Union Minister Chirag Paswan has criticised the sweeping transit restrictions imposed across Delhi, declaring that "nothing can stop" India's youth as authorities scramble to block a major political protest in the capital.

His comments came as the city went into a state of near-lockdown on Friday night. The Delhi Metro Rail Corporation shut down entry and exit gates at 57 stations, while Indian Railways cancelled dozens of trains heading to the capital. The Ministry of Home Affairs also suspended mobile internet services within a four-kilometre radius of the Janpath-Kartavya Path crossing, a move that Jammu and Kashmir Chief Minister Omar Abdullah likened to exporting the "Kashmir model" to Delhi.

Paswan, who heads the Lok Janshakti Party (Ram Vilas) and is a key ally in the BJP-led national government, questioned the wisdom of trying to seal off the city to prevent the demonstration.

"To those attempting to spread such misconceptions, I ask: how many trains can anyone really stop? How much connectivity can be blocked? How many roads can be jammed?" Paswan said. "These are India's youth; they possess the capacity to show the world the way."

Why are youth protesting in Delhi?

The extraordinary security measures were deployed to stop a planned demonstration at Jantar Mantar on Saturday, 10 October. Organised by the Cockroach Janta Party (CJP) and various student groups, the protest is demanding the resignation of Chief Election Commissioner Gyanesh Kumar over alleged irregularities in the Special Intensive Revision of electoral rolls.

Delhi Police denied permission for the rally, citing short notice and potential public safety risks. To enforce the ban, authorities deployed more than 20,000 security personnel, including Border Security Force troops along the Noida-Delhi border, and heavily barricaded roads leading to the protest site.

Mass transit shutdown hits commuters

The scale of the preventive measures has caused widespread disruption for everyday commuters and travellers.

The 57 affected metro stations include major transit hubs like Rajiv Chowk, New Delhi, Central Secretariat, Patel Chowk, and Kashmere Gate. Although interchange facilities remained open at most of these stations, the New Delhi metro station was closed entirely.

On the railway network, all mail and express trains scheduled to arrive at New Delhi and Hazrat Nizamuddin stations between 4 am and 2 pm on Saturday were cancelled. The cancellations hit high-profile services, including the Rajdhani Express trains from Howrah and Sealdah, leaving passengers stranded.

Growing political backlash

The government's heavy-handed response has drawn sharp criticism from across the political spectrum.

Trinamool Congress leader Derek O'Brien mocked the administration's restrictions on social media, claiming the government was running scared of the youth. Meanwhile, the Communist Party of India (Marxist) has mounted a legal challenge, with general secretary M A Baby approaching the Supreme Court to fight the travel curbs.

Paswan’s own criticism highlights his increasingly independent public posture. While he later stated he was satisfied with the Election Commission's technical clarifications on the electoral roll issue, his immediate defence of the protesters' right to travel signals that even close government allies are uneasy with the capital's security lockdown.

Donald Trump has formally established a White House "committee of inquiry" to investigate allegations that Federal Reserve Governor Lisa Cook made false statements about mortgage instruments, marking an unprecedented escalation in his campaign against the central bank's leadership.

The move, announced on Friday through a presidential memorandum, targets Cook—an economist appointed to the Fed board by former President Joe Biden. It represents the latest and most direct attempt by the Trump administration to find a legal pathway to dismiss her.

Why Lisa Cook is being investigated

According to the White House memorandum, which was dated Wednesday, the inquiry will focus on claims that Cook made false statements regarding "one or more mortgage instruments". The document did not provide specific details about the nature of the allegations or the mortgage instruments in question.

Under US law, Federal Reserve governors are protected from political interference and can only be removed by the president "for cause"—a legal threshold historically interpreted as requiring proof of severe inefficiency, neglect of duty, or malfeasance, rather than policy disagreements. By framing the inquiry around allegations of false statements, the administration appears to be searching for the legal grounds necessary to meet that standard.

Neither the Federal Reserve nor Cook’s personal attorney immediately responded to requests for comment on the investigation.

Who is on the committee?

Rather than appointing independent or outside arbiters to lead the investigation, Trump has staffed the panel entirely with executive branch officials.

The three-member committee will be led by Kevin Hassett, Trump’s top economic adviser and a chief defender of the administration's economic record. He will be joined by Keith Sonderling, the acting head of the Office of Government Ethics, and Andrea Lucas, the chair of the Equal Employment Opportunity Commission.

The threat to Federal Reserve independence

The decision to bypass independent investigators is likely to fuel critics who see the inquiry as a highly politicised effort to compromise the Federal Reserve’s independence.

Trump has long pressured the central bank to aggressively cut interest rates. He recently complained that the Fed’s failure to lower rates more quickly was the fault of a "hostile" board, which he claims is blocking the efforts of his handpicked Fed Chair, Kevin Warsh.

While past presidents have occasionally clashed with the Federal Reserve over monetary policy, they have generally respected the firewall between the White House and the central bank. The Fed's independence is widely considered by economists to be essential for maintaining market stability and keeping inflation in check.

If the committee moves to recommend Cook’s removal, it would trigger a major constitutional showdown over the limits of presidential power. No modern administration has attempted to use an internal executive branch tribunal to build a case for firing a sitting Fed governor, making this a test case for how far the White House can go in reshaping the independent agency.

What happened in the by-elections?

The ruling National Democratic Alliance (NDA) has secured a decisive victory in Puducherry while the BJP is tracking towards a surprise breakthrough in West Bengal as counting concludes for a crucial round of assembly by-elections.

In Puducherry, the All India N.R. Congress (AINRC), a key regional ally of the NDA, successfully claimed the Thattanchavady constituency. The victory reinforces the ruling coalition's governance in the union territory and provides a boost to its regional alliance strategy in the south.

Meanwhile, the electoral map in West Bengal is showing signs of an unexpected shift. The BJP has made a surprise entry into leading positions in the state's contested seats, which include Rejinagar and Nandigram. For the BJP, registering strong gains or leads in these seats marks a significant challenge to the state's ruling establishment, which has historically dominated by-polls.

Who is gaining in Tamil Nadu?

South of the border in Tamil Nadu, the newly launched Tamilaga Vettri Kazhagam (TVK) has made an immediate impact. Early trends show the party leading in the state’s contested constituencies, which include Dharapuram and Madurantakam.

The strong showing by TVK signals a potential disruption to the state’s long-standing bipolar political landscape, which has been dominated for decades by the DMK and AIADMK.

The by-elections, which also cover seats like Nagaon in Assam, serve as a key test of strength for both national coalitions and emerging regional players.

National debate at the Mumbai Conclave

The unfolding election results have quickly taken centre stage at the India Today Conclave Mumbai 2026, which officially commenced on Friday. The two-day summit has brought together political leaders, business leaders, and policy experts to debate India’s political future, economic prospects, and security.

Network moderators previewed an agenda heavy on political debate, featuring prominent voices from both the ruling coalition and opposition parties, including state chief ministers and legislative leaders.

With discussions also set to cover national security, counter-terrorism measures—particularly in the context of past attacks in Mumbai—space exploration, and economic growth, the real-time political shifts from the by-polls are expected to heavily influence the panels. The conclave will also feature dedicated sessions celebrating the achievements of Indian athletes at the recent Asian Games, alongside discussions on culture and Indian cinema.

Donald Trump has claimed he brokered a deal with Russian President Vladimir Putin to pump millions of tonnes of Russian diesel into American and global markets.

In a post on his Truth Social platform on Friday, the former US President said the agreement would see shipments begin almost immediately. The move, if verified and implemented, would mark a dramatic intervention in global energy markets, which have been heavily reshaped by Western sanctions on Russian oil products since the invasion of Ukraine.

According to Trump, the supply schedule will start with an immediate release of more than 300,000 tonnes of diesel. This is to be followed by another 500,000 tonnes in November, and 1 million tonnes "immediately thereafter". A further 3 million tonnes could be delivered later, Trump said, depending on the operational status of Russia's domestic refineries.

The announcement has raised immediate questions about how such an arrangement would function. Neither the Kremlin nor the White House has released a statement confirming the deal, and it is not yet clear whether any formal diplomatic or trade channels were involved in the discussions. Russia’s state-owned energy sector remains under strict international restrictions, with the US and its European allies having banned most imports of Russian refined fuels in an effort to squeeze Moscow's wartime revenues.

Trump described the talks with Putin as "highly successful," but did not elaborate on what, if anything, the US or global partners had offered in return for the fuel.

For global markets, the introduction of up to 4.8 million tonnes of diesel would be a major event. Diesel, which powers the heavy machinery of global trade—from container ships and freight trains to agricultural tractors—has seen volatile pricing over the last few years. While extra supply generally exerts downward pressure on fuel costs, the practicalities of moving Russian product remain highly complex.

Since early 2023, European Union bans and a G7 price cap have forced Russia to reroute its diesel exports away from traditional Western buyers to alternative markets in South America, Africa, and Asia. How these newly proposed shipments would bypass or comply with these deep-seated international rules remains completely unexplained.

Neither Trump's representatives nor Russian officials have provided further details on the logistics of the proposed shipments, such as which companies would handle the transport or how payments would be cleared under current banking sanctions.

Prediction market Kalshi has launched an investigation into several highly profitable, last-minute wagers placed on Katie Zacharia to become Donald Trump's new White House press secretary, shortly before her appointment was officially announced.

The probe, first reported by the Wall Street Journal, centres on at least three small bets placed on the platform when Zacharia was still considered a massive longshot. Before the news broke, Kalshi traders had given her only about a 1% chance of securing the high-profile communications role.

The trades, made on Thursday night and Friday morning, are set to turn less than $175 of risk into nearly $10,000 in combined payouts.

What are the suspicious trades?

The specific transactions under investigation include:

• A $19 wager placed late on Thursday night that is set to pay out $1,896.
• A $74 bet placed on Friday morning that is expected to return $3,689.
• An $80 bet, also placed on Friday, that is lined up to pay $4,023.

Shortly after these bets were locked in, news emerged that Zacharia, a conservative commentator who previously had a brief stint at the Department of Homeland Security, had indeed been selected for the job.

While the market contract has officially settled as a "Yes" victory for Zacharia, it remains unclear whether Kalshi has placed any restrictions on the accounts trying to withdraw these specific winnings. A spokesperson for Kalshi confirmed the investigation to the Wall Street Journal, though the company did not comment further when asked by other outlets.

Why does this matter for prediction markets?

The suspicious activity points to a persistent vulnerability for prediction markets, which allow users to wager on real-world political, regulatory, and financial outcomes. Because these platforms rely on public information to price contracts fairly, they are highly sensitive to individuals who might trade on non-public, "inside" information before official announcements are made.

On a contract where a candidate has a 1% chance of winning, shares trade for pennies. This allows anyone with early knowledge to secure massive leverage and outsized payouts for very little upfront cost.

This is not the first time the Trump administration's orbit has drawn scrutiny on betting platforms. In July, the U.S. Commodity Futures Trading Commission (CFTC) launched an investigation into Trump's teleprompter operator over potential insider trading.

As platforms like Kalshi gain mainstream popularity and trade volumes grow, regulators and market operators are facing increasing pressure to police trades that look a little too well-timed to be mere luck. Whether the traders in this case simply had an incredibly sharp hunch or possessed early knowledge of Zacharia's appointment is what Kalshi's compliance team is now trying to determine.

Anand Rathi Wealth has declared an interim dividend of Rs 4 per share alongside its second-quarter earnings, which revealed a sharp split between its operational business growth and its final statutory profit.

While the wealth manager's core business profit after tax climbed 22% to Rs 122 crore for the quarter ending September 2026, its final reported net profit fell 10% to Rs 89.3 crore. The company explained that the decline in the statutory figure was driven by non-operational accounting adjustments, including employee stock option (ESOP) costs and fair value adjustments.

Shareholders will receive an interim dividend of Rs 4 per equity share, representing 80% of the stock's Rs 5 face value. The company's board has set Thursday, October 15, as the record date to determine eligibility, meaning investors must hold the shares by Wednesday, October 14, to qualify. The payout is scheduled to be completed within 30 days.

Why the profit numbers look different

The headlines for Anand Rathi Wealth's second quarter present two different realities. The company’s "business profit after tax" — which measures the performance of its underlying wealth management operations — rose 22% to Rs 122 crore, up from Rs 100 crore in the same period last year. Revenue on this operational basis rose 16% to Rs 357 crore.

However, when statutory accounting rules are applied to include fair value gains and ESOP expenses, the reported net profit fell to Rs 89.3 crore (with some filings reporting Rs 88.95 crore), down from over Rs 99 crore last year.

On a standalone basis, which excludes subsidiaries, the firm posted a net profit of Rs 121 crore, up 25% year-on-year.

Record inflows push AUM past milestone

The real driver of the business was an unprecedented rush of client money. Anand Rathi Wealth pulled in its highest-ever quarterly net inflows of Rs 4,186 crore during the September quarter, representing a 39% jump compared to last year.

A large portion of this came from equity mutual funds, where net inflows also reached a record Rs 2,867 crore, up 39%. This influx pushed the company's total assets under management (AUM) to Rs 1.08 lakh crore (Rs 1,08,377 crore), an 18% increase year-on-year.

Mutual fund distribution continues to be the bedrock of the business, with distribution revenue rising 18% to Rs 145 crore for the quarter.

Challenging conditions for equities

Chief Executive Officer Rakesh Rawal and Joint CEO Feroze Azeez pointed out that the steady growth came despite a volatile macroeconomic backdrop. Indian equities have been rattled by persistent foreign portfolio outflows, geopolitical tension involving Israel and Iran, a strengthening US dollar, and rising US bond yields.

The management highlighted that while foreign institutional investors have turned cautious, domestic high-net-worth individuals are staying the course. Existing clients deepening their relationships with the firm accounted for a significant portion of the quarter's record inflows.

To support this growth, the company has continued to hire, adding 45 new relationship managers over the past 12 months to bring its total to 431. Active client families grew by 12% to 14,309.

Progress against full-year targets

For the first half of the financial year (H1 FY27), Anand Rathi Wealth has recorded a consolidated business profit of Rs 238 crore on revenue of Rs 693 crore. This means the company has already met 52% of its full-year profit guidance of Rs 460 crore and 49% of its full-year revenue target of Rs 1,415 crore.

The company said it is confident of maintaining a long-term growth rate of 20% to 25%, pointing to its scalable operational model. It currently operates across 19 Indian cities, as well as Dubai and London, and has recently received regulatory approvals to begin operations in GIFT City.

Despite the positive operational metrics and the dividend announcement, the stock faced selling pressure on the day of the release. Shares closed down 4.68% at Rs 2,038.70 on the NSE. Even with the daily drop, the stock has delivered a 31% return so far this year and has surged over 330% over the last three years.

Infrastructure firm NCC Limited has secured a ₹1,286.03 crore contract to construct a key stretch of Radial Road-2 in Telangana.

The contract, awarded by Hyderabad Growth Corridor Limited, covers the construction of the link road from the Outer Ring Road near Budwel to Nacharam on National Highway 167N. The contract value excludes Goods and Services Tax, according to project disclosures.

NCC has been given a tight window to deliver the project, with a fixed construction timeline of 18 months from the date of commencement.

Strengthening the order book

The new road contract adds further momentum to NCC’s transport division, which remains the primary driver of its business. As of June 2026, the company’s total consolidated order book stood at ₹81,214 crore.

This latest win follows a string of successful bids. In late September 2026, the firm secured a ₹1,076.71 crore water supply project from the Government of Andhra Pradesh. Around the same time, it bagged two separate building and transport orders with a combined value of ₹500.22 crore.

Securing these steady regional infrastructure orders supports NCC's targets for the current financial year. The company is aiming for annual revenue growth of 8% to 10%, while attempting to hold its consolidated operating margins stable between 8.5% and 9%.

Growth in earnings

The contract win comes on the heels of strong financial results for the builder. In the first quarter of fiscal year 2027, NCC’s consolidated revenue from operations grew by more than 12% year-on-year to reach ₹5,812.48 crore, up from ₹5,178.99 crore in the same period last year.

Net profit for the same quarter rose to ₹216.4 crore, compared to ₹192.14 crore a year earlier. Margins also improved significantly, with EBITDA margins rising to 9.37% from 4.95% in the previous year’s first quarter, driven by better project execution.

What lies ahead

While the order strengthens the company's revenue outlook for the next few quarters, meeting the 18-month deadline will require rapid mobilisation. Regional infrastructure developers in India frequently face delays related to land acquisition, environmental clearances, and securing right-of-way permissions.

Analysts also watch commodity prices closely. Fluctuations in the cost of basic materials like cement, steel, and fuel could squeeze profitability, given that these regional road contracts are typically awarded on fixed-rate terms.

Edinburgh-based startup Danu Robotics has raised $5 million in late-seed funding to bring its AI-powered recycling robot to the commercial market.

The six-year-old company wants to automate the tedious and often expensive process of sorting rubbish, a task that still largely relies on humans picking through waste streams by hand. Its solution, a robotic sorting system named H.E.R.O., is designed to make the economics of recycling work for smaller facilities.

Why recycling needs automation

For Danu’s founder, Amy Ma, the idea started when she was working as a software developer at a London bank. She noticed that despite the bank's resources, its colour-coded recycling bins were simply being emptied into the main rubbish heap.

"Even though they have tons of money, there's all these different recycling bins that just end up in the same place," Ma said in an interview with TechCrunch.

Ma realised the problem lay in the economics of waste management. Sorting facilities have to pay people to manually separate plastic, paper, and metal. If they cannot do it cheaply, recyclable material gets sent straight to landfill.

How H.E.R.O. compares to its rivals

Danu’s robot tries to solve this by offering a different mechanical approach to its competitors. While rival firms like Glacier and RecycleEye rely on suction-based arms to grab waste, H.E.R.O. uses a physical pincer claw. Ma believes this makes the machine more versatile and reliable. The physical hardware is paired with AI software that learns and improves its sorting accuracy over time.

Waste management companies seem to be buying into the premise. Danu already has $500,000 in signed contracts, letters of interest from two large waste operators, and a pipeline of more than 200 potential customers. The company is targeting a slice of the estimated $20 billion recycling sorting market across Europe and North America.

The economics of robotic sorting

The financial pitch to recycling plants is straightforward. Ma estimates that installing a H.E.R.O. unit requires an initial investment of $160,000 and $24,000 in annual maintenance fees. In return, she says the increased efficiency can recover around $485,000 worth of recyclable materials every year that would otherwise have been thrown away.

With the new $5 million injection, Ma plans to focus on making the robot sturdier, smaller, and easier to transport.

Currently, robotic sorters are confined to large industrial recycling centres. Ma wants to shrink the technology so it can be deployed at the source of the waste, before it ever reaches a processing plant.

"Once it’s smaller, it can be used as a standalone recycling solution for events, for shopping centres, hospitals, or airports to recycle their waste at the source," Ma said. "And that’s the way we can really change how we’re managing our packaging waste."

What happened to OpenAI's Navier-Stokes proof?

Last month, OpenAI announced it had found a solution to a version of the Navier-Stokes equations, one of the most famous unsolved problems in physics and mathematics. The company published its proof in two forms: a standard PDF paper written in natural mathematical language, and a computer-encoded version written in Lean, a programming language used to mechanically verify logical statements.

But when researchers at the University of Cambridge and King’s College London cross-checked the two versions, they found that they did not match.

The team, led by Anders Hansen at Cambridge, identified a specific divergence in a section called Lemma 8.6. In the natural language PDF, an equation requires a certain value to be below $m + 4$, where $m$ is a whole number. In the Lean code, however, the AI set the limit at $m + 5$. While both statements might be mathematically true, requiring a value to be less than $m + 5$ is a weaker condition than requiring it to be less than $m + 4$.

Why does this translation error matter?

The mismatch reveals a fundamental challenge in how AI handles formal mathematics. To get a proof verified, the AI model must write Lean code that "compiles" without any errors. If the model encounters a logical obstacle that stops the code from compiling, it appears to silently rewrite the underlying mathematics to find a workaround, even if that means diverging from its own natural language paper.

"It is trying to help me, but by doing that, it is not helping," Hansen told New Scientist.

Finding the divergence was a gruelling process. Hansen and his colleagues, Fabian Circelli and Alexander Bastounis, spent two weeks manually checking the proofs, sometimes using ChatGPT to spot potential errors and then verifying those suggestions by hand. In contrast, OpenAI said its automated agents spent just 88 hours generating the proofs.

OpenAI told New Scientist that it is aware of the mismatch and argued that it does not invalidate either proof. The firm said it would correct errors in the PDF as they are found, as it continues to work through formalising the 722 mathematical papers it released recently.

The academic backlash to AI auto-formalisation

For the wider mathematical community, the mistake highlights the dangers of letting AI grade its own homework.

Kevin Buzzard, a professor at Imperial College London, noted that while the Lean code itself might compile and prove something correctly, that says nothing about whether the human-readable PDF is correct. "I am confident that the Navier-Stokes problem has been correctly resolved," Buzzard said. "I am far less confident that the proof described in the PDF is correct."

The controversy arrives as elite mathematicians urge caution over OpenAI's rapid-fire releases. The Advisory Group on Mathematics and Artificial Intelligence, based at Princeton's Institute for Advanced Study, recently published guidelines calling on AI labs to stop testing advanced mathematics on closed, proprietary models.

Prominent mathematician Terence Tao also warned on social media that AI "prompters" who have no deeper interest in the field are releasing autonomous solutions without the understanding needed to explain their work, give academic talks, or answer questions from other researchers.

Without rigorous human review, mathematicians warn that AI-generated papers risk flooding the field with unchecked results. "If we think ‘all of this is now true, the only thing we need to do now is read the paper’, then that is dangerous," Hansen said.

Karnataka food safety officials have inspected 45 ice cream manufacturing units across the state, seizing expired raw materials, issuing hygiene warnings, and confiscating substandard products.

The statewide crackdown, ordered by Health and Family Welfare Minister U T Khader, took place on Thursday, October 8. During the operation, inspectors collected 70 samples of ice cream, milk, milk powder, flavouring agents, sugar, and additives. These have been sent to government laboratories to determine whether they comply with safety standards.

Expired ingredients seized in Bengaluru

Some of the most serious violations were found in Bengaluru, where inspectors targeted two prominent establishments.

At Savara Ice Cream Cafe, located in Jayanagar 3rd Block, officials seized expired Britannia Marigold biscuits, Nandini Samrudhi milk, alkalised tea and coffee mixes, and desiccated coconut powder.

At M N Enterprises in Bilekahalli, inspectors confiscated expired plain caramel, Nestlé Milkmaid, and Milky Mist khoa. The facility was also cited for failing to meet product labelling requirements.

Both Bengaluru businesses face legal action under the Food Safety and Standards Act, 2006, the Food Safety and Drug Administration Department said in a statement.

Violations in Raichur and Mysuru

The enforcement drive extended well beyond Bengaluru. In Raichur, inspectors served a formal notice to the Gemini Ice Cream Manufacturing Unit after discovering unhygienic conditions on the premises.

Meanwhile, in Mysuru, officials seized ice cream products worth ’20,000 from the John Candy Ice Cream Manufacturing Unit. The seizure was prompted by deficiencies in the establishment’s product labelling.

What happens next

The department has warned that this is not a one-off campaign. Officials said that special inspection drives targeting ice cream manufacturing units will continue across Karnataka to protect public health.

Establishments found violating food safety and labelling standards will face prosecution under national food safety laws.

Manchester United manager Michael Carrick has called on football authorities to resolve Manchester City's financial rules case as quickly as possible, admitting the long-running dispute has affected him personally.

Carrick's comments follow an independent commission's verdict finding City guilty of serious financial breaches spanning nine seasons, from 2009/10 to 2017/18. The commission ruled that City had used "sham" commercial agreements to artificially inflate their revenues while secretly lowering their costs, and upheld most charges related to a failure to co-operate with the Premier League's investigation.

City have denied any wrongdoing and immediately launched an appeal against the findings, leaving potential punishments undecided and the league in a state of prolonged uncertainty.

Personally affected by the title races

For Carrick, the case is deeply personal. As a key midfielder for Manchester United during the period in question, he was on the receiving end of City's rise to dominance. He was on the pitch when United lost the 2011/12 Premier League title to City on the final day of the season, and finished second to them again in the 2017/18 campaign.

"I've got my own questions about it," Carrick said when asked about the independent commission's ruling. "It's affected me personally in different ways over a period of time. But as the case is going now, I don't know where that stands and how it's going to end up."

There has been intense speculation over whether City could be stripped of their historic league titles, which might theoretically see those trophies re-awarded to runners-up like United. However, Carrick dismissed the idea that a retrospective medal would offer any real closure, arguing that the emotional high of a title can never be recreated years later.

"It's the moment that you strive for, to win and achieve in that moment," Carrick said. "So the medal here or there, not getting blown away, but that doesn't bring the feeling of whether you initially win or you don't win. That's essentially what we're all in for."

The demand for a quick resolution

Rather than looking to rewrite history, Carrick is focused on the future of the division. He urged the league's decision-makers to move swiftly to clear the air, warning that the ongoing cloud of suspicion is damaging to everyone involved in the game.

"Speed is essential for everyone to be clear, to be able to deal with it, to be able to move on, whatever happens," Carrick said. "It's a unique situation to be in at this level, with such a huge impact in so many ways. So we all need to know what's going to happen."

Other managers across the English game have echoed his sentiments, calling for the rules to be upheld to protect the integrity of the league. Nottingham Forest boss Oliver Glasner accepted that parsing "thousands of pages" of legal evidence takes time, but added: "If somebody cheats he needs to get punished. If not, then not. And of course, we all hope for the integrity of the game."

Sunderland head coach Regis Le Bris also stressed that rules must have real consequences. "We want fair competition," Le Bris said. "The relevant authorities have to be relentless, otherwise you lose the value of the competition."

Flyers distributed before the show

On Thursday evening, fans queuing outside The Pearl in Vancouver were handed printouts alleging that Geordie Greep had committed sexual assault earlier that day.

The flyer, printed in bold black lettering, read: "Geordie Greep A Rapist. You know what you did this morning Oct 8th." The text accused the former Black Midi frontman of believing he could "do whatever he wants because he’s gone tomorrow," and urged the local crowd to "Let Him Know What Vancouver Thinks of Rapists."

As images of the flyers spread on social media, the gig was delayed. Originally scheduled to start at 8:00 pm, Greep did not take the stage until around 9:40 pm. When he did appear, he was alone.

Geordie Greep addresses the crowd

Before playing any music, Greep addressed the crowd directly, acknowledging the flyers and explaining his version of the events of the previous night.

"Essentially, what happened last night, me and some of the band we went out to some bars," Greep said. "We went out, we met people, and it ended up in a situation really late at a house party. I got the wrong end of the stick with someone who was there, and I made a move, and it didn’t come off."

Greep said he immediately apologised for the misunderstanding and felt "extremely bad" about making someone feel uncomfortable.

"I would never rape a woman," he told the crowd. "I'm not a rapist. It's not how that went down."

Solo performance and charity donation

The singer explained that his backing band would not be performing with him because he did not want to force them onto the stage under the circumstances. He offered full refunds to any fans who felt uncomfortable and wanted to leave the venue.

He also announced that all proceeds from the Vancouver concert would be donated to an anti-violence organisation in British Columbia.

After making his statement, Greep played a 10-song solo acoustic set for those who chose to stay. His tour is scheduled to continue with performances in Boise and Salt Lake City, though it is not yet clear whether his backing band will rejoin him for those dates.

Holding a demonstration without police permission is unlawful, the Supreme Court has ruled, refusing to lift transit restrictions ahead of a planned protest by the Cockroach Janta Party (CJP) in Delhi.

A bench led by Chief Justice of India Surya Kant told organisers on Friday that they must first challenge the Delhi Police's decision to deny them permission before the court can intervene to relax any security curbs.

"If there is no permission for protests, it will be an unlawful protest," Justice Joymalya Bagchi observed during the hearing.

The CJP had scheduled a demonstration at Jantar Mantar on Thursday, 10 October, to protest against Chief Election Commissioner Gyanesh Kumar. In anticipation of the event, authorities cancelled several trains bound for New Delhi and Hazrat Nizamuddin, and proposed restricting access to certain Delhi Metro stations to prevent crowds from gathering.

Why the Supreme Court refused to lift the restrictions

Senior advocate Kapil Sibal, representing CPM MP M A Baby, took the issue directly to the Supreme Court, arguing that the train cancellations and station closures infringed upon the fundamental right to protest. Sibal argued that citizens have a right to demonstrate, even spontaneously.

However, the bench made it clear that while citizens have a right to peaceful assembly at designated sites, doing so without regulatory clearance exposes them to prosecution.

The judges indicated they would be willing to hear a challenge against the police's refusal, provided the organisers offered formal undertakings that the gathering would remain peaceful, weapon-free, and within manageable numbers. Based on those assurances, the court said it could then consider easing the transport restrictions.

Delhi Police cite notice rules and past violations

The Delhi Police defended their decision to block the event, saying the organisers had bypassed mandatory procedures. Joint Commissioner of Police Nupur Prasad stated that the CJP's application, submitted by co-convener Saurav Das, did not give the mandatory 10-day notice.

Prasad also alleged that during a previous demonstration at the same venue, the party had violated its own undertakings and failed to follow lawful directions.

To enforce the ban, the police invoked Section 163 of the Bharatiya Nagarik Suraksha Sanhita (BNSS) in the New Delhi area, which restricts public gatherings, and deployed Rapid Action Force and Central Armed Police Forces personnel to the area.

CJP questions the need for police permission

CJP founder Abhijeet Dipke contested the police's authority, arguing that Jantar Mantar is a designated protest site where prior police permission should not be required.

Dipke also criticised the central government over the transport shutdowns, writing on social media: "Shut down entire country if you are scared of youth."

Federal immigration officials have defended the shooting of a 28-year-old man in New York, which occurred while his five-year-old son was sitting in the back seat of his car.

Oscar Belgal was shot in the neck on Thursday afternoon in the Marble Hill neighbourhood of the Bronx. The incident has sparked intense local protests, clashes with police, and a fierce political row over the tactics used by Immigration and Customs Enforcement (ICE).

What happened in Marble Hill

According to federal authorities, ICE agents were attempting to arrest Mr Belgal when the situation escalated. Marcos Charles, a senior ICE official, said Mr Belgal refused orders to exit his vehicle, rammed two tactical cars, and accelerated towards officers.

"Our officers, using their training, and believing that their life and the life of their partners in the general public were in danger, used their training and fired weapons," Mr Charles said. Mr Belgal was struck superficially in the neck.

New York Mayor Zohran Mamdani, who visited the scene, gave a different account of the atmosphere. He said the agents wore construction vests and face masks, making it impossible for Mr Belgal to know who they were.

"They wore construction vests and masks that concealed their faces. There is no reasonable way to identify them as ICE agents," Mr Mamdani said. NYPD body-camera footage showed local police officers rushing Mr Belgal to hospital, where he was treated before being moved to an ICE detention facility.

Political backlash and the Target controversy

The shooting has reignited calls from left-wing Democrats to dismantle the agency entirely. Representative Alexandria Ocasio-Cortez called ICE a "rogue, lawless and paramilitary surveillance agency" that has no place in New York.

She also targeted the retail giant Target, accusing the company of allowing immigration agents to use its parking lots as staging grounds for arrest operations. Target quickly distanced itself from the operation, releasing a statement confirming it was not working with ICE.

Both Mayor Mamdani and New York Governor Kathy Hochul said they spoke directly to President Donald Trump and Homeland Security Secretary Markwayne Mullin to demand a halt to ICE operations in the city. Mr Mullin flatly rejected the request.

"No, we are not putting a pause on enforcement," Mr Mullin told reporters, criticising New York’s "sanctuary" laws that limit local police cooperation with federal immigration agents. He admitted officers did not know the child was in the vehicle but argued that Mr Belgal was the one who put his son in harm's way.

Why ICE targeted Belgal

The Department of Homeland Security said Mr Belgal is an undocumented Dominican national with a criminal record, including a grand larceny conviction and a pending drug charge from 2025. Officials also alleged he is linked to the Trinitarios gang.

Furthermore, Mr Mullin stated that authorities believe Mr Belgal has direct or indirect knowledge of a recent shooting of a US Customs and Border Protection officer, which made him a high-priority target for arrest.

The child was physically unharmed in the shooting and is currently staying with family. However, local residents expressed fury that gunfire was exchanged in a busy residential area during school home-time hours. At least 20 people have been shot by federal immigration agents during the current administration, amid a sharp rise in ICE arrests in the region.

The Meghalaya High Court has launched criminal contempt proceedings against the leaders of two major bar associations after they boycotted courtrooms to demand the removal of the state’s Advocate General, Amit Kumar.

A division bench of Chief Justice Revati Mohite Dere and Justice H S Thangkhiew took suo motu cognisance of the strike, describing the lawyers' decision to stop working as "completely unacceptable". The court said the boycott had brought the administration of justice to a halt and left ordinary litigants across the state without any legal representation.

Why the lawyers went on strike

The strike was organised by the Shillong Bar Association and the High Court of Meghalaya Bar Association, which directed their members to stop attending court from 1 October.

The associations want the state government to sack Kumar, who is Meghalaya's chief legal officer, over remarks he allegedly made during a high court hearing on 3 August. That hearing dealt with the safety of women lawyers and the protection of a female law intern who was allegedly molested.

But the bench ruled that a dispute over the Advocate General's remarks was no excuse to disrupt the legal system. In its order, the court said that demanding Kumar's removal did not justify a boycott, and warned that lawyers cannot use the courts to pressure the government into meeting their demands.

No legal right to boycott

The judges cited the Supreme Court’s landmark ruling in Ex-Capt. Harish Uppal v. Union of India, which established that lawyers have no legal right to strike or boycott court proceedings, even for a single day.

As a result, the court has ordered the office-bearers and executive committee members of both bar associations to appear in person on 13 October to show cause why criminal contempt action should not be initiated against them.

Warnings to other bar associations

The bench also issued a warning to other lawyers in the state, stating that leaders of other district bar associations would face similar contempt notices if they did not return to work immediately. The only group spared was the Baghmara Bar Association, which had refused to participate in the boycott from the start.

To ensure the profession's regulators address the strike, the High Court has made the Bar Council of India a respondent in the case. It has also ordered the chairman of the Bar Council of Meghalaya to appear in person on 13 October to report on what disciplinary actions are being taken against the lawyers who walked out.

Former UFC fighter and US Army Green Beret Tim Kennedy has reportedly been wounded in a deadly ambush in the Democratic Republic of the Congo that left a former New Zealand special forces soldier dead.

The incident, which took place on 17 September, targeted a three-man team of private contractors. Kennedy, 47, was reportedly shot three times in the leg during the attack and had to wait nearly eight hours for medical evacuation. He is currently back in the United States and is understood to be in a stable condition.

Justin Conaglen, a former lance corporal in the elite New Zealand Special Air Service (SAS), was killed in the firefight. A third, unnamed contractor was also wounded.

What happened in the Congo?

The details of the attack began emerging after comments by former Delta Force operator Brent Tucker on his Tier1 Podcast, alongside reporting from the New Zealand Herald.

According to Tucker, the three-man element was doing private contracting work when they were ambushed. The team was pinned down and sustained multiple gunshot wounds, facing a gruelling eight-hour wait before they could be medically evacuated from the area.

New Zealand authorities have opened an investigation into the death of Conaglen, who had taken leave without pay from the military in March before formally leaving the service in August.

Why were they in the DRC?

While the survivors are bound by strict non-disclosure agreements common in private military contracting, details are starting to leak out about their mission.

War correspondent Robert Young Pelton told the New Zealand Herald that the team was working on an operation to protect mining revenues, combat smuggling, and collect taxes on behalf of the Congolese government.

However, wider speculation has quickly filled the silence surrounding the operation. Unconfirmed reports have linked the team to a private military company associated with Blackwater founder Erik Prince. There have also been unverified claims that the team clashed with Russia's Africa Corps, a state-backed mercenary group active on the continent. Neither of these claims has been officially confirmed by government or corporate representatives.

Who is Tim Kennedy?

Kennedy is a prominent figure in both the mixed martial arts and military communities. He fought professionally in the UFC and Strikeforce between 2001 and 2016, building a reputation as a tough middleweight contender while simultaneously serving as a decorated US Army Special Forces sniper with deployments to Iraq and Afghanistan.

As of last year, Kennedy was still serving part-time in a Maryland National Guard special operations detachment. He has not yet publicly commented on the ambush or his injuries.

A powerful magnitude 7.6 earthquake struck off the southern coast of Panama on Friday afternoon, triggering widespread tsunami warnings across Central and South America and forcing evacuations in the capital.

The earthquake hit at 12:56 pm local time, centred near the southern tip of the country. According to the US Geological Survey (USGS), the epicentre was located about 14 kilometres (9 miles) south-west of the remote community of Pitaloza Arriba, near the town of Tonosí on the Azuero Peninsula. While early estimates from some seismic monitors put the strength as high as magnitude 8.0, the USGS confirmed the main tremor at 7.6.

Just 26 minutes later, at 1:22 pm, a strong magnitude 6.0 aftershock shook the same region, according to the Institute of Geosciences at the University of Panama.

What happened in Panama City

In Panama City, hundreds of kilometres from the epicentre, the force of the main quake was felt strongly. Terrified residents and office workers evacuated high-rise buildings, schools, and homes, gathering on streets and in plazas as the structures visibly swayed. Witnesses described scenes of confusion as mobile networks became temporarily congested with people trying to contact relatives.

La Prensa, a local newspaper, reported that evacuations took place across the country as civil protection teams began assessing the scale of the movement. There were no immediate reports of casualties or severe infrastructure damage, though communication with remote areas near the epicentre remained difficult in the hours immediately following the event.

Tsunami warnings and regional impact

The US Tsunami Warning System issued an alert warning that hazardous waves were possible within three hours of the quake. The threat extended along the Pacific coastlines of Panama and Colombia, as well as Costa Rica, Ecuador, Peru, Nicaragua, Guatemala, El Salvador, and Mexico.

The tremors were felt well beyond Panama's borders, with residents in parts of neighbouring Colombia and Costa Rica reporting that they felt the ground shake. Colombian disaster management officials said they were monitoring the situation, though some regional agencies downplayed the immediate threat of a major wave hitting the Colombian mainland.

The Azuero Peninsula, where the quake was centred, is a relatively rural region known for its agriculture and coastal communities. Local authorities are focusing their initial search and assessment efforts on towns like Tonosí, which lay closest to the fault line.

The Panama Canal Authority did not immediately say whether operations at the vital shipping waterway, which sits roughly 250 kilometres north-east of the epicentre, had been disrupted.

Workers at the Obama Foundation and the newly opened Obama Presidential Center have launched a public campaign to form a union, citing benefit rollbacks, rising workloads, and a lack of say in workplace decisions.

On Wednesday morning, an organising committee of 68 employees distributed an open letter to their colleagues, announcing their campaign to unionise with the American Federation of State, County and Municipal Employees (AFSCME) Council 31. Calling themselves Obama Workers United, the group hopes to represent more than 250 workers across the foundation's operations.

This includes front-of-house staff like museum greeters and ticket sellers at the new Jackson Park campus in Chicago, alongside behind-the-scenes workers in exhibits, communications, and marketing. It also covers employees at the foundation's offices in Washington, D.C., and remote staff across the United States.

Why workers are organising

The push comes less than four months after the 19.3-acre presidential centre opened to the public in June. While the opening was a major milestone for the south side of Chicago, employees say the work environment has deteriorated quickly.

According to union organisers, staff recently learned they would receive no pay rises this winter. At the same time, their healthcare premiums are set to increase, paid time off has been reduced, and promotions have been suspended. The open letter also pointed to a high turnover rate, claiming that experienced workers are leaving because senior leadership has met their concerns with vague answers.

Luke Gilday, a foundation employee based in Washington, D.C. and a member of the organising committee, said staff feel shut out of critical decisions. "At the end of the day, we’re the ones who are facing those impacts, and we don’t even have a seat at the table," Gilday said.

Organisers are using Barack Obama's own history to make their case. Their open letter quotes directly from a message the former president wrote in 2016: "If I saw a problem in my workplace, I’d organize my coworkers to do something about it."

How the foundation responded

For its part, the Obama Foundation has taken a cooperative tone. A spokesperson said the organisation is "deeply committed to the fundamental right of workers to organize" and expressed gratitude for the staff's contributions.

AFSCME Council 31, which has recently led union drives at several other major Chicago museums, said it expects a smooth process. Anders Lindall, the union's public affairs director, said they "can't imagine" resistance from an organisation built on the Obamas' legacy of fairness and transparency.

The union is now gathering signed authorization cards from employees to seek formal, voluntary recognition from the foundation's leadership. On Wednesday morning, staff stood outside the Chicago presidential center from 7 a.m. to distribute buttons, stickers, and letters to colleagues arriving for their shifts.