Ukraine’s military used American-made ballistic missiles on Tuesday to strike into Russia for the first time, according to senior U.S. and Ukrainian officials, just days after President Biden gave permission to do so in a major shift of American policy.

The pre-dawn attack struck an ammunition depot in the Bryansk region of southwestern Russia, Ukrainian officials said. Russia’s Ministry of Defense said that Kyiv used six ballistic missiles known as ATACMS, for Army Tactical Missile System. A senior American official and a senior Ukrainian official, who spoke on the condition of anonymity to discuss ongoing operations, confirmed that ATACMS were used.

The strike represented a demonstration of force for Ukraine as it tries to show Western allies that providing more powerful and sophisticated weapons will pay off — by degrading Russia’s combat capabilities and relieving pressure on Kyiv’s overstretched forces.

The attack came on the same day President Vladimir V. Putin lowered Russia’s threshold for the use of nuclear weapons, a long-planned move whose timing appeared aimed at showing the Kremlin could respond aggressively to Ukraine using American missiles to strike Russian territory.

The Kremlin has throughout the war used the threat of deploying its nuclear arsenal to try to deter the West from providing more robust military support to Ukraine. On Tuesday, Russia’s foreign minister, Sergei V. Lavrov, called Ukraine’s use of ATACMS in the Bryansk region “a signal that they want escalation” — a reference to the U.S. and western allies.

Ukrainian officials and military analysts, who have long cautioned that no single weapon will change the course of the war, noted that the impact of the shift in White House policy will depend on the quantities of missiles being supplied.

President Volodymyr Zelensky of Ukraine had pleaded for months for permission to use ATACMS to strike military targets inside Russia. The Biden administration finally relented and gave its assent, a decision that was driven in part by the addition of up to 10,000 North Korean troops to Moscow’s war effort.

The authorization came just two months before the return to office of President-elect Donald J. Trump, who has said he will seek a quick end to the war in Ukraine.

His election has cast uncertainty over whether the U.S. will maintain the robust military support it has provided Ukraine under Mr. Biden, or whether Mr. Trump might take a different approach on policy toward Ukraine.

On the battlefront, Russia is leveraging its advantage in air power, artillery and personnel to melt through defensive lines in southeastern Ukraine. By taking aim at logistics centers like the one targeted on Tuesday, Ukraine hopes to make it more difficult to supply Russian forces engaged in relentless assaults.

The Ukrainian military’s high command, known as the General Staff, said that Tuesday’s attack took place at 2:30 a.m., destroying “warehouses with ammunition” and triggering a dozen “secondary explosions.”

Andrii Kovalenko, a representative of Ukraine’s National Security and Defense Council, said that the strike in Bryansk hit warehouses housing “artillery ammunition, including North Korean ammunition for their systems, guided aerial bombs, antiaircraft missiles and ammunition for multiple-launch rocket systems.”

Russia’s Ministry of Defense said in a statement that it shot down five of the ATACMS and that another was damaged. It said that falling fragments caused a fire at a military facility but there were no casualties.

Neither side’s claims about the impact of the strikes could be immediately independently verified.

Russia has robust and layered air defense systems, including S-400 batteries designed to counter ballistic missiles like the ATACMS. The S-400, and the newer S-500, are similar to the Western-made Patriot systems used to protect Ukrainian skies.

Since the first longer-range ATACMS arrived in Ukraine in April, Kyiv has used its limited supplies of missiles to target those Russian air defense systems in occupied areas of Ukraine — particularly in Crimea — with some success, according to satellite imagery, military analysts as well as Ukrainian and Western officials.

The ATACMS have also been used in successful strikes on airfields and other bases, according to satellite imagery and military analysts.

But throughout the war, the moment one side introduces a new capability, the other side works to adapt.

“Russian effectiveness against ATACMS missiles will likely increase, even if slowly, over the course of the conflict, as Russia trains air defense crews to address the threat,” Jake Mezey, a security expert, wrote in a report for The Atlantic Council in September.

The U.S. supplied Ukraine with ATACMs for the first time in October of 2023, but the missiles were designed to hit targets only about 100 miles away. The Biden Administration only provided a small number — less than two dozen in the first shipments — and the move came only after a Ukrainian counteroffensive that summer had already failed.

In April of 2024, the U.S. agreed to secretly provide ATACMS with a range of 190 miles to Ukraine. However, the Biden administration restricted their use to targeting Russian forces located in Ukrainian occupied territory, including Crimea.

As Kyiv pressed Washington to lift those restrictions, it continued to develop its own weapons capable of hitting targets in Russia, including ballistic missiles.

The munitions depot in the Bryansk region is more than 70 miles over from the Ukrainian border.

Mykhailo Samus, deputy director at the Center for Army, Conversion and Disarmament Studies in Ukraine, an independent institution, said the introduction of ATACMS could have an immediate impact.

Russian command posts, military units, logistical support systems, ammunition depots, aviation bases, missile launch systems, air defense systems, radars and other key assets are all now within range of one of the most powerful weapons in the Ukrainian arsenal, he said.

“Everything depends on how many missiles Biden will give and how quickly,” he said. “If hundreds of rockets enter there, then it will be good, we’ll be able to make a few mass rocket strikes.”

But even if Ukraine is able to damage Russia’s ability to wage war, Kyiv’s forces will still be vastly outgunned and outmanned, requiring the kinds of weapons that both armies tear through at a furious rate.

Artillery ammunition has been one of Ukraine’s persistent needs throughout the war and soldiers said they are still facing shortages, compounding the challenges posed by a lack of manpower.

The European Union only recently fulfilled its goal of providing one million rounds to Ukraine’s war effort — eight months behind schedule.

The bloc’s foreign policy chief, Josep Borrell Fontelles, said in a statement published on Monday after a recent visit to Kyiv that the European Union has supplied Ukraine with more than $47.5 billion in weapons and, by the end of 2024, will have trained more than 75,000 of the country’s soldiers.

Still, he said, “we give Ukraine just enough to hold, and sometimes even less, while Russia has put its entire economy on a wartime footing and counts with the unconditional backing of North Korea and Iran.”

Maria Varenikova contributed reporting from Kyiv, and Lara Jakes from Rome.

A top U.S. envoy to the Middle East on Tuesday signaled progress in negotiations between Israel and Hezbollah on a cease-fire proposal that, if agreed upon, could ease hostilities in a region already on edge over Israel’s war in Gaza.

The envoy, Amos Hochstein, said at a news conference in Beirut, Lebanon, that the gaps between the two sides had “narrowed” in discussions in recent weeks, though ultimately any results from the negotiations would be “the decision of the parties.”

“We have a real opportunity to bring this conflict to an end,” Mr. Hochstein said. That outcome is “within our grasp,” he added.

Mr. Hochstein’s visit was widely considered a sign that the United States’ efforts to broker a truce were moving forward. He met earlier on Tuesday with Nabih Berri, the Lebanese Parliament speaker who is a key interlocutor between the United States and Hezbollah, the Iranian-backed militant group and political party in Lebanon that is at war with Israel.

Last week, Iran appeared to signal support for an end to the war in Lebanon. Iran’s supreme leader, Ayatollah Ali Khamenei, dispatched a top adviser to Syria and Lebanon on Friday who called for Hezbollah to accept a cease-fire, according to two Iranians affiliated with the Revolutionary Guards Corps and who were familiar with the details of the effort.

Though both Mr. Hochstein and Lebanese officials have spoken of progress in the discussions, it is unclear whether those talks have ironed out details. Previous U.S.-led negotiations on a cease-fire stalled in September as the war escalated.

Last week, the United States presented Lebanon with the terms of a new cease-fire plan devised by Israeli and American officials, Lebanese officials said. The initial response to the plan from Lebanese officials and Hezbollah was “positive” but some points still required discussion, according to Lebanon’s caretaker prime minister, Najib Mikati, who spoke with Al Araby TV, a Qatar-based broadcaster, on Monday.

The embattled Lebanese government — which is not engaged in the war on its soil, but is largely powerless to stop it — has been pushing to revive a U.N. resolution that ended the last major war between Israel and Hezbollah, in 2006. The agreement was widely considered a failure in the years after that monthlong war ended, but it is viewed today by Lebanese and U.S. officials as a potential road map to end the fighting.

The resolution calls for Hezbollah to withdraw from southern Lebanon and for only the Lebanese military and U.N. peacekeepers to operate in the region south of the Litani River in Lebanon, which runs around 20 miles north of the Israeli border.

It is not clear how Hezbollah’s withdrawal from that area would be enforced. Both the Lebanese Army and U.N. peacekeepers have largely avoided confrontations with Hezbollah fighters. The Lebanese army is also widely viewed as too weak to defend the country’s borders against any future Israeli military action.

In prior discussions on a truce this fall, Israeli officials pushed for guarantees that Israel could continue to strike Hezbollah within Lebanon if they deemed it necessary — a condition that Lebanese officials say they have rejected.

Jean-Pierre Lacroix, a United Nations under-secretary-general who oversees the contingent of U.N. peacekeeping troops in Lebanon, said at a briefing on Tuesday that his office was preparing proposals for a potentially evolving role for the U.N. troops, known as the United Nations Interim Force in Lebanon, or UNIFIL.

“We are looking at the extent to which we could offer support to Lebanese armed forces, because the deployment of Lebanese armed forces to the south for the cessation of hostilities is a critical part of the settlement,” he said at United Nations headquarters in New York.

Any cease-fire must include a very serious commitment to the U.N. troops’ freedom of movement because in the past it has been severely limited, he said.

Mr. Lacroix said there were three attacks on U.N. forces on Tuesday in Lebanon. In one, rockets hit a UNIFIL base in Shama where a workshop was significantly damaged but no peacekeepers were injured. It was the second time this base had been hit in less than a week from ongoing clashes in the area, he said.

In another attack, a patrol came under gunfire but no one was hit. And separately, a rocket exploded near a peacekeepers’ position in Ramyah, causing minor injuries to four of them. UNIFIL said the Ramyah attack was likely the work of “non-state actors,” which usually means an armed group like Hezbollah, and it did not attribute the other two to anyone.

The diplomatic push by the United States comes amid intensified Israeli strikes that appear aimed at pressuring Hezbollah to accept the terms of a cease-fire, analysts say. On Monday evening, Israeli airstrikes hit a building in central Beirut — the third instance of Israeli airstrikes within the city limits in two days. The strike, which hit the Zuqaq al-Blat neighborhood, killed at least five people and injured 24 others, according to the Lebanese Health Ministry.

The Israeli military declined to comment on two of the three strikes, confirming only one and saying that it had targeted Mohammed Afif, the head of Hezbollah’s media office, who was killed. Over the past few days, Hezbollah has launched dozens of rocket and drone attacks at Israel.

Over the past week, Israel has also conducted intense bombardment of the Dahiya, the area just south of Beirut that is in effect governed by Hezbollah. In the country’s south, Israeli forces appear to be making incursions deeper into Lebanon, beyond villages along the border.

Despite official claims of progress in the cease-fire talks, many Lebanese remain pessimistic about the prospects of the conflict subsiding soon.

“We have no trust in negotiations, no hope of an imminent cease-fire,” Zeinab Atwi, 26, said on Tuesday as she stood across the street from the wreckage of the building struck in Zuqaq al-Blat the night before.

Her cousin, Hussein Atwi, had been killed in that strike, she said. When the war escalated, they moved from their family’s homes in the Dahiya to a rented apartment in the neighborhood, thinking it would be safer there. Mr. Atwi had gone to a nearby cafe that was struck an hour after he arrived.

“There is nowhere safe left in Beirut,” Ms. Atwi said.

Hwaida Saad in Beirut and Thomas Fuller in San Francisco contributed reporting.

President Vladimir V. Putin on Tuesday lowered Russia’s threshold for the use of nuclear weapons, a long-planned move whose timing appeared designed to show the Kremlin could respond aggressively to Ukrainian strikes on Russian territory with American missiles.

The decree signed by Mr. Putin implemented a revised version of Russia’s nuclear doctrine that Mr. Putin described in televised remarks in September. But the timing sent a message, coming just two days after the news that President Biden had authorized the use of U.S.-supplied longer-range missiles by Ukraine for strikes inside Russia.

Asked whether Russia could respond with nuclear weapons to such strikes, Dmitri S. Peskov, Mr. Putin’s spokesman, repeated the new doctrine’s language that Russia “reserves the right” to use such weapons to respond to a conventional-weapons attack that creates a “critical threat” to its “sovereignty and territorial integrity.”

Russia’s Ministry of Defense later announced that Kyiv had used the ballistic missiles known as the Army Tactical Missile System, or ATACMS, in a pre-dawn attack on an ammunition depot in southwestern Russia. A senior American and a senior Ukrainian official, both of whom spoke on the condition of anonymity to discuss ongoing operations, confirmed that the strikes used ATACMS, which have a range of 190 miles, the longest of any munitions the West has given to Ukraine.

The White House played down Mr. Putin’s new doctrine. In a statement issued by the National Security Council, the White House noted that it had observed “no changes to Russia’s nuclear posture.”

The new doctrine, published Tuesday on the Kremlin website, differs from the previous iteration in at least two important ways that show how Mr. Putin is trying to use the threat of his nuclear arsenal to deter the United States from further supporting Ukraine.

First, it raises the possibility that nuclear weapons could be used against a nuclear-armed country that doesn’t directly launch an attack on Russia but supports one by a nonnuclear country. That is a clear reference to Ukraine and its nuclear-armed backers, led by the United States. Russia’s previous nuclear doctrine focused on responding to attacks by nuclear-armed countries and alliances.

Second, it lowers the threshold at which Russia could consider nuclear use in response to an attack with conventional weapons. The previous doctrine, published in 2020, said such an attack must threaten “the very existence of the state,” while the new one puts that threshold at a “critical threat” to Russia’s sovereignty.

The doctrine’s publication on Tuesday appeared to be the latest suggestion from the Kremlin that Russia could use nuclear weapons to respond to attacks by Ukraine carried out with American support, and that the response could be directed against American facilities as well as Ukraine itself.

“Aggression against the Russian Federation and (or) its allies by any nonnuclear state with the participation or support of a nuclear state is considered as their joint attack,” the document says.

Mr. Peskov, speaking at his daily conference call with reporters, pointed to this section of the revised doctrine, saying, “this is also a very important paragraph.”

“Nuclear deterrence is aimed at ensuring that a potential adversary understands the inevitability of retaliation in the event of aggression against the Russian Federation and/or its allies,” Mr. Peskov said.

In the statement from the National Security Council, the White House noted that it was “not surprised” by the new decree since Mr. Putin had signaled the change months ago.

“This is more of the same irresponsible rhetoric from Russia, which we have seen for the past two years,” the statement said. The White House made no linkage between its lifting of restrictions on Ukraine and Mr. Putin’s announcement.

Western officials have previously said that they would be most worried about Moscow’s using nuclear weapons if the Russian military is on its back foot, and for the moment the war in Ukraine largely appears to be going Mr. Putin’s way.

On the battlefield, Russian forces are advancing in eastern Ukraine, while Kyiv struggles with recruitment and morale. And in geopolitics, Mr. Putin has also been making gains: his phone call last week with Chancellor Olaf Scholz of Germany broke two years of diplomatic isolation by the biggest Western countries, while the election of Donald J. Trump as incoming president of the United States has raised hopes in Russia of a Ukraine peace deal on the Kremlin’s terms.

From the first day of his invasion of Ukraine in February 2022, Mr. Putin has been trying to use the threat of Russia’s enormous nuclear arsenal to deter Western military aid to Ukraine. He has had only limited success, with the United States leading a coalition that has dispatched tens of billions of dollars’ worth of modern tanks, artillery systems and missiles.

But Mr. Putin has sought to draw a new red line at Ukraine’s using Western missiles to attack deep inside Russian territory. To the frustration of Ukrainian officials, President Biden had long refused to allow the weapons to be used that way, given what American officials said was the risk of a violent response by Mr. Putin and the limited impact that the use of those missiles could have on the battlefield.

But Mr. Biden changed course recently after Russia’s surprise decision to bring North Korean troops into the fight, American officials said.

In September, Mr. Putin warned that if the United States and its allies permitted Ukraine to fire missiles deeper into Russia, they would put his country “at war” with NATO. Ukraine has used homegrown weapons to attack at longer ranges.

In the lead-up to Mr. Biden’s decision, some American officials said they feared that Ukraine’s use of the missiles across the border could prompt Mr. Putin to retaliate with force against the United States and its coalition partners. Other American officials said they thought those fears were overblown.

In response to Mr. Biden’s recent decision, Russian officials have warned in some of their strongest statements yet about the risk of nuclear war. On Tuesday, Andrei Kartapolov, the head of the defense committee in Russia’s lower house of Parliament, said that Mr. Biden “will slam the lid of his own coffin and drag many, many more people with him.”

The question now is whether Mr. Putin sees a strategic advantage to intensifying his conflict with the United States — even by nonnuclear means — given Mr. Trump’s imminent return to the White House.

Tatiana Stanovaya, a senior fellow at the Carnegie Russia Eurasia Center, described the situation on X as an “extraordinarily dangerous juncture” because Mr. Putin might see the temptation to escalate the war before Mr. Trump takes office in order to force a peace deal on his terms.

But some analysts still predicted that given Mr. Trump’s stated goal of ending the war quickly, Mr. Putin’s more rational course would be to avoid any actions that would further escalate his conflict with the United States.

A Moscow-based analyst whose organization is close to the Russian government said that while Ukraine’s ATACMS strikes were unlikely to change the course of the war, an aggressive response against the United States “will create problems for Trump.”

“It’ll be harder for him to turn the situation around if that’s what he wants to do,” the analyst said, speaking on the condition of anonymity because he wasn’t authorized to comment to Western media.

Oleg Matsnev and David E. Sanger contributed reporting.

Brazilian authorities arrested several members of an elite Brazilian army unit on Tuesday, accusing them of planning to assassinate Luiz Inácio Lula da Silva in 2022, shortly before he was to become the country’s president, as part of a plot to keep the far-right incumbent president, Jair Bolsonaro, in power.

Four of those arrested were members of the military, including a former top aide to Mr. Bolsonaro, according to a police official who was not authorized to discuss details of the case. A fifth person arrested in the case is a federal police agent, the official said.

Those detained used a “high level of technical military knowledge” to organize a plot to assassinate Mr. Lula, a leftist, and his pick for vice president, Geraldo Alckmin, on Dec. 15, 2022, the authorities said in a statement. The group had also planned to kill a Supreme Court justice, the authorities said, but they did not divulge the name of the justice.

“The planning drawn up by those under investigation detailed the human and military resources required to carry out the actions,” the police said in a statement.

A lawyer for Mr. Bolsonaro said his client had no involvement in or knowledge of the plot.

The former president “never agreed or participated in any type of plan of this nature,” the lawyer, Paulo Cunha Bueno, told The New York Times.

The police did not say if there was evidence that the plan to assassinate Mr. Lula and the others had been foiled or abandoned.

The arrests are part of a broader investigation into an attempted coup and Mr. Bolsonaro’s role in attempting to hold on to power following Brazil’s last presidential election when he was narrowly defeated by Mr. Lula. It is the first time the investigation has cited an effort to target Mr. Lula before his inauguration, in January 2023.

As part of this investigation, Mr. Bolsonaro has been accused of planning to arrest a Supreme Court justice, personally editing a draft decree aimed at overturning the election results and presenting plans for a coup to top military leaders, seeking their support, according to police statements.

Mr. Bolsonaro has repeatedly denied any wrongdoing, calling the investigation political persecution.

Flávio Bolsonaro, a senator and the former president’s son, appeared to downplay Tuesday’s arrests, saying that since the plot had not been carried out, it was “simply not a crime.”

“As disgusting as it may be to think about killing someone, it is not a crime,” he said in a post on social media.

More than a year before Brazil’s election in 2022, Mr. Bolsonaro began sowing baseless doubts about the security of the nation’s voting machines, warning that he could only be defeated if they were rigged in his opponent’s favor.

When he lost, Mr. Bolsonaro did not officially concede the election. His supporters set up camps outside military headquarters, calling on the military to overturn an election that they claimed had been stolen.

Then, in an episode reminiscent of the Jan. 6, 2021, attack on the U.S. Capitol, they invaded and vandalized Brazil’s congressional, Supreme Court and presidential offices just days after Mr. Lula was sworn in, hoping to provoke a military intervention.

The police are expected to conclude the monthslong probe into the attempted coup in the coming weeks, determining whether Mr. Bolsonaro tried to orchestrate a coup and if his claims of election fraud played a role in encouraging the riot.

While the Brazilian police can make recommendations, they do not have the power to formally charge any of those accused of plotting the coup. After the police conclude their investigation, they will present their findings to the country’s top federal prosecutor who will have to decide whether to pursue the case.

The last time that Sweden updated the crisis preparedness advice it distributes nationwide, Russia had not yet launched its full-scale invasion of Ukraine and Sweden had not joined NATO in response.

Six years later, Ukraine is locked in a grinding war with Russia. World leaders are preparing for the United States to become more protectionist with Donald J. Trump as president again. And climate threats continue to grow.

“We live in uncertain times,” Sweden told its citizens in a new edition of its emergency readiness pamphlet, which the government began distributing to households on Monday.

Its Nordic neighbors recently made similar moves. Finland published a consolidated online version of its crisis preparedness on Monday, and Denmark and Norway sent out their guidance earlier this year. They are trying to better prepare people for a world rife with security threats, disinformation, cyberattacks and climate disasters.

While the new guides do not directly mention Russia, which borders Finland and Norway, Swedish and Finnish officials told The New York Times on Tuesday that Russia’s invasion of Ukraine had informed their decision.

“The biggest difference is the increased focus of war preparedness — and the more severe tone of the warnings,” said Svante Werger, a preparedness adviser at the Swedish Civil Contingencies Agency.

“The global security environment has become more unpredictable,” said Eriikka Koistinen, the communications director for Finland’s interior ministry. It’s not just the war in Ukraine, she said, but also the apparent sabotage this week of fiber-optic cables in the Baltic Sea, the conflict in the Middle East and Russia’s growing relationship with North Korea.

The Nordics have long been wary of their belligerent neighbor to the East.

Finland, which joined NATO in 2023, shares an 830-mile border with Russia. Norway, which has helped Europe heat its homes after Russia cut gas exports, is worried about Russian spies near their shared border. And Sweden, which joined NATO in March, warned that “military threat levels are increasing” in its recent pamphlet release.

“The Swedish government is taking the security situation really seriously,” Mr. Werger said. “The general perspective is that you cannot exclude the possibility of an armed attack.”

On Tuesday, President Vladimir V. Putin officially lowered Russia’s threshold for using nuclear weapons. In addition, a day after the Biden administration had authorized it, Ukraine fired U.S.-made ballistic missiles into Russia for the first time, officials said.

“The war in Ukraine affects Ukraine, but it also affects our safety and security,” Espen Barth Eide, Norway’s foreign minister, said on Tuesday.

Norway updated its advice because “the world has changed significantly since 2018,” Tore Kamfjord, a head of the country’s civil preparedness campaign, said in an email.

One change he noted: Norwegians should be ready to sustain themselves for a week, up from three days.

“We face a lasting deterioration in the security situation in our region,” Mr. Kamfjord said, also pointing to the worsening effects of climate change and broader digital security threats.

For Sweden, the pamphlet, “In Case of Crisis or War,” is the latest step in a broad push to ready the country for war. At 32 pages, it is almost twice as long as the 2018 edition, which was its first significant revision since the Cold War and followed Russia’s annexation of Crimea.

There are more details on things like how to seek shelter and how to evacuate, Mr. Werger said. There’s also a new section for people who need special assistance or have pets, he added.

Next month, the Swedish parliament is set to vote on a defense bill that would boost military and civil defense spending over the next five years.

“If Sweden is attacked, we will never surrender,” the pamphlet says. “Any suggestion to the contrary is false.”

That statement is now on the first page after the table of contents. In 2018, a version was on page 12, buried below tips on identifying fake information and checklists for home preparation.

On Tuesday morning, the air quality in India’s capital under a widely used index stood at 485. While that is almost five times the threshold for healthy breathing, it felt like a relief: The day before, the reading had shot up to 1,785. Infinitesimal air particles were still clogging lungs and arteries, but it was possible to see sunlight again, and to smell things.

“My eyes have a burning feeling during these periods of pollution,” said Vikram Singh, 58, an auto-rickshaw driver in central Delhi, who noted that he also tires more quickly. “I don’t know what else is happening to my body, on the inside.” He earns less, too, just $6 per day instead of his usual $8.30.

Every year this suffocating smog accompanies the drop in temperatures as the plains of north India shed their unbearable heat for wintertime cool. And like clockwork, political leaders roll out emergency measures intended to quit making the problem worse. Yet India seems powerless to reduce the effects of this public health catastrophe, as its politicians stay busy trading blame and trying to outmaneuver one another in legal battles.

The haze was so shocking this week that Delhi’s chief minister, Atishi, who goes by one name, declared it a “medical emergency” endangering the lives of children and older people. The Supreme Court, whose members also live in the capital, chided the national government for responding too slowly and ordered special measures: halting construction work and blocking some vehicles from the roads. Schools were closed indefinitely to protect students.

For middle-class Delhiites, the emergency measures have taken on an uncanny resemblance to life during Covid-19 lockdowns. There was a familiarity to the work-from-home mandates, idle children cooped up in the house and spare surgical or N95 masks rummaged from drawers.

But only a small proportion of Delhi’s citizens can afford such luxuries. Debu Jyoti Dey, the finance director at a nonprofit in the development sector, wore a handkerchief tied below his eyes as he trudged between a subway station and his office. At least, he said, he was going indoors.

“I feel congestion in the chest, I feel sneezy, sometimes drowsy,” he said. But “people who are working on the road, they suffer a lot more” — people like drivers, curbside vendors and day laborers. “And if I remained at home, how would I earn my living?”

Mr. Dey said that governments were failing to “reach at the root causes” of the pollution because it was not a voting issue among the poor, who must “think about free electricity and water and not bother” about the health of their lungs.

The rich can afford to ignore the smog because they “use machinery and technology and stay indoors,” he said. The middle class — he means people like himself — are too few in number to matter to politicians but unhappily “put our lives at stake” alongside the poor.

For those who are able to stay at home, that can help a little, said Dr. Sundeep Salvi, president of the Indian Chest Society, based in Pune. There are “at least some health benefits,” if it means moving from a pollution level of 450 to 300, say. Those benefits, however — like masking with a simple bandanna — are marginal, and easily overstated.

Dr. Salvi also recommends staying hydrated, performing a kind of nasal wash twice a day and keeping houseplants in living spaces. None of these measures are enough to make a difference in any epidemiological study. But unlike industrial-grade HVAC systems with air filters, they are affordable to all households.

The immediate cause of the dreaded autumn smog is the drop in temperatures, creating a “thermal inversion,” when hotter air forms a stubborn layer atop the colder air, trapping pollutants at ground level. It coincides with extra sources of microscopic grit: small fires for cooking and warmth, smoke from Diwali firecrackers and farmers’ burning stubble from their fields after the harvest.

Ms. Atishi, the Delhi chief minister, has traded accusations with the party led by Prime Minister Narendra Modi over the burning of crop waste. Her smaller party controls the state of Punjab, which is most often blamed for the fires. But the surrounding states run by Mr. Modi’s party, she said, were responsible for much more of this season’s burning.

The scientific community is at odds over what proportion of the deadliest particulate matter comes from the fields. Recent analysis by the Indian Institute of Tropical Meteorology put stubble burning’s contribution at just over 1 percent in October. By this month it had risen, to 13 percent, but remained small in comparison to the city’s base-line pollution from vehicles and other sources.

Whatever the causes of the smog, by one ranking Delhi was not just the most polluted city in the world, but “nearly five times as bad as the second most polluted city, Dhaka,” in Bangladesh next door, as Shashi Tharoor, a leader of the opposition in Parliament and best-selling author, posted on X.

“This city is essentially uninhabitable from November to January inclusive and barely livable the rest of the year. Should it even remain the nation’s capital?” he wrote.

About 10 years after Delhi’s extraordinary wintertime air pollution first drew the world’s attention, it is remarkable how little is understood about it. Even its health effects need further research.

Dr. Salvi, of the Indian Chest Society, said there were simply no significant longitudinal studies on cardiovascular function, which are “expensive, and take 10 years to do.”

He said that “I can only imagine that the prevalence of heart attacks, of strokes, of heart failure — they must all increase significantly because of this high level of air pollution. But there are no studies from India to support it.”

For more than 10 weeks, Gisèle Pelicot has sat in a courtroom in Avignon, France, quietly listening to the explanations of 50 men, including her now ex-husband, charged with raping, sexually assaulting or attempting to rape her while she was in an unconscious state, having been drugged by her husband.

She has heard most say that they were not guilty — that they went to her house lured by her husband, believing they were going for a threesome that she had consented to. She has heard some say they were trapped, played like checker pieces. She has heard some say they believe that he had drugged them, too.

Ms. Pelicot stayed in the courtroom while grim videos that her husband took of those encounters were played — revealing the men, sitting on benches nearby, touching her inert body and engaging in sexual acts, with her husband in the background egging them on, often with vulgar words. (Ms. Pelicot divorced him just before the trial began.)

On Tuesday, a day before closing statements were set to begin, she was given the chance to address the court one last time.

She was tired, she said, standing small and poised at the microphone.

“It’s difficult for me to hear that it’s basically banal to have raped Madame Pelicot,” she said, referring to herself. “This is a trial of cowardice.”

The trial of 51 men — one is on the run and being tried in absentia — has profoundly shaken the country since it began in September. Mr. Pelicot has pleaded guilty to crushing sleeping pills into his wife’s food and drink for almost a decade and then inviting strangers he met mostly on the internet to come to the house they had rented for retirement in southern France to join him in raping her.

The accused are a cross-section of middle- and working-class men — tradesmen, firefighters, truck drivers, a journalist, a nurse. They range in age from 26 to 74. Most live close to Mazan, the town that the Pelicots retired to in 2013. Many are married or in committed relationships. Most have children. The court has heard from their wives, their parents, their friends and children, who mostly have said they are wonderful, kind people.

About 15 of them, including Mr. Pelicot, have pleaded guilty. Mr. Pelicot has repeatedly insisted that the others were perfectly aware of what was going on.

Ms. Pelicot has told the court that the couple met as teenagers and mostly lived together happily for 50 years. She had no idea that he had been drugging her, though she suffered frightening symptoms including extended blackouts. She had visited many doctors, fearing that she had a brain tumor or Alzheimer’s disease.

The defense lawyers who pack the court alongside their clients questioned Ms. Pelicot for the last time on Tuesday and tested their theories of defense.

One noted that she had been under her husband’s control, steered and tricked for at least a decade. So how could she not think it was possible that he had tricked and controlled these men?

“He drugged me,” said Ms. Pelicot, 71. “He did not manipulate me daily. You think I would have stayed with a man who manipulated me for 50 years?”

Another lawyer said Ms. Pelicot seemed to have more sympathy for her ex-husband than the other accused. She posited that Ms. Pelicot was still under her husband’s control.

“That’s your analysis,” Ms. Pelicot said calmly. She added, “All my life, I have been a very positive person. I will keep with me the best of this man.”

Ms. Pelicot said she had been working through her anger and sorrow in sessions with a psychiatrist, as well as long walks, talking to her friends and eating chocolate.

Her ex-husband had always driven her to her medical appointments, searching for the cause of her health issues that, ultimately, he was causing. Ms. Pelicot had described those trips as support. One lawyer pointed out that it was another form of control and manipulation, with an aim to ensure that his secret was not discovered.

“It could be both at the same time,” she responded. “I always took it as an act of kindness. It could also have been a way for him to ensure they didn’t discover the facts.”

Ms. Pelicot recognized that her ex-husband was the “orchestra conductor” and that it was not only her family that had been destroyed in the fallout but also the families of the 50 other accused men. But while they might have been manipulated to get them to the house, once the men were in the bedroom and saw her state, they should have left and called the police, she said.

“I feel anger against those who are behind me who not for one moment thought of reporting it,” she said. “Not a single one reported it. It raises some real questions.”

Since she made the rare decision of opening the trial to the public, Ms. Pelicot has become a feminist hero. While her children and grandchildren had been ashamed of their name at the beginning of the trial, Ms. Pelicot said she believed they were now proud.

“Today I am known around the world, whether I like it or not,” she said. “People will remember Madame Pelicot, much less Monsieur Pelicot.”

Mr. Pelicot was also given a final chance on Tuesday to address the court and his family, who had all assembled on the other side of the courtroom from where he sat in his prisoner’s box. Many people had been asking why he had done this, he said. He pointed to sexual violence that he said he had suffered or witnessed as a child and teenager.

“It created a fissure that I have kept for life,” he said.

Ms. Pelicot had already addressed her ex-husband earlier in the day.

“Some think I have forgiven him,” she said. “I will never forgive him. The things he did to me are unforgivable.”

Anywhere else, it wouldn’t have been controversial: a public vote by pro-democracy activists trying to strengthen their hand in legislative elections, to decide who should run. More than 600,000 people took part in the peaceful, unofficial poll.

But this was Hong Kong, just after the imposition of a national security law by Beijing, and officials had warned that even a straw poll would be taken as defiance.

On Tuesday, the price of defying Beijing was made clear. Forty-five former politicians and activists who had organized or taken part in the 2020 primary by the opposition camp were sentenced by a Hong Kong court to prison, including for as long as 10 years.

The sentences were the final step in a crackdown that cut the heart out of the city’s democracy movement, turning its leaders into a generation of political prisoners. Among them were veteran politicians, former journalists and younger activists who had called for self-determination for Hong Kong.

In a courtroom that had to be created just to accommodate them, the 45 defendants sat shoulder to shoulder on Tuesday on long benches, behind a glass partition and flanked by police officers. A judge read their sentences aloud, referring to them not by their names but by their numbers on a list. The hearing was over in half an hour.

It was the most forceful demonstration of the power of a national security law Beijing imposed on Hong Kong in response to months of large protests against Chinese rule in 2019.

The court sentenced Benny Tai, 60, a legal scholar and opposition strategist, to 10 years in prison. Twenty opposition politicians and activists were given terms ranging from five to nearly eight years. Joshua Wong, 28, a prominent pro-democracy activist, was among 24 others whose sentences ranged from just over four to just under five years.

Gwyneth Ho, 32, a former journalist who was known for covering a mob attack on antigovernment demonstrators trapped in a subway station, was sentenced to seven years for running as a candidate. She had refused to plead guilty.

“Our true crime for Beijing is that we were not content with playing along in manipulated elections,” said a statement posted on Ms. Ho’s Facebook account, apparently by her supporters, on Tuesday after the sentencing. “We dared to confront the regime with the question: will democracy ever be possible within such a structure? The answer was a complete crackdown on all fronts of society.”

The trial made clear that any form of dissent or criticism, however moderate, carried significant risk, analysts said. “If you are being critical of the authorities both in Hong Kong and in China, then it’s open season,” said Steve Tsang, a Hong Kong-born political scientist and director of the SOAS China Institute in London.

The ruling Communist Party in China says the law is needed to purge threats to Beijing’s sovereignty, but human rights activists, scholars and Western governments have said that it has eroded Hong Kong’s once-vaunted judicial independence.

Even before their sentences were handed down, many of the defendants, who were arrested in early 2021, had already been in jail for nearly four years, as they awaited and then stood trial. That was because the law has made it harder for defendants to be released on bail, which in most nonviolent cases is routinely granted.

Instead of a jury, the case was heard by three judges handpicked by the city’s Beijing-backed leader, as allowed by the law.

“The authorities wanted to show the public that they have the power to bring a big group of people to trial all at once,” said Patrick Poon, a visiting researcher at the University of Tokyo who studies freedom of expression in Hong Kong and China. “They want to show residents that anyone who tries to protest like this group of people will receive their same fate.”

Prosecutors accused the defendants of “conspiracy to commit subversion,” a national security offense, arguing that the objective of the election primary was to “undermine, destroy or overthrow the existing political system and structure of Hong Kong.”

The outcome of the primary made clear that the residents who voted favored candidates who were prominent supporters of the 2019 antigovernment demonstrations. The pro-democracy camp argued that the primary vote was little different from others held in democracies around the world. The hope was to maximize the camp’s chances of gaining more seats in a legislative chamber that already heavily favors the Beijing-backed establishment.

But they never had a chance to test the plan: The election was postponed, and most of the candidates were arrested.

Mr. Tai, the legal scholar who was sentenced to 10 years, had designed the electoral strategy, and prosecutors deemed him a mastermind. Mr. Tai had long been involved in efforts to persuade China to live up to a promise that has been central to Hong Kong since its 1997 return to Chinese control: that its residents would someday get to choose their own leaders. In 2014, he was one of the leaders of the Occupy Central movement that brought the city’s central business district to a halt in a peaceful call for freer elections.

Other defendants included Leung Kwok-hung, a 68-year-old activist known as Long Hair for his unkempt mane, who was sentenced to six years and nine months; Claudia Mo, 67, a veteran former lawmaker, sentenced to four years and two months; and Lam Cheuk-ting, 47, a former anti-corruption investigator, sentenced to six years and nine months.

Mr. Tai and 30 other defendants had pleaded guilty. The court convicted 14 of them in May and acquitted two others.

Outside, hundreds of people waited in line to enter the gallery, braving a downpour and a heavy security presence around the courthouse, including an armored vehicle, police cars and barricades. Several dozen police officers were stationed on every street corner along the whole block.

As she left the courthouse, Elsa Wu, the mother of Hendrick Lui, one of the defendants, unfurled a poster in protest of the sentencing. Several police officers led her away into a police van and tried to slide the door shut.

“Tell me, why does he have to go to prison?” she called out, beating her fist. Mr. Lui was sentenced to more than four years in prison. “He is a political prisoner. He shouldn’t be in prison. He is a good person. Why does he have to be in prison?” she said.

Thomas Kellogg, the executive director of the Georgetown Center for Asian Law, said he thought the case would be seen by many in the international community as the “final nail in the coffin for the rule of law in Hong Kong.”

Despite the prospect of more prison time, some defendants were simply anxious for the trial that had left their lives in limbo to come to a conclusion, according to friends who had visited them.

Emilia Wong, a gender rights activist, said in an interview ahead of the sentencing that her boyfriend, Ventus Lau, an organizer of the 2019 antigovernment protests, had been studying toward a degree in translation. She said she had been regularly visiting him in detention for the past three years, but it was clear the isolation was taking a toll on him.

“The scary thing about prison is not being locked up in one place. It is the loss of connection with people and society that is scary. To him, it was painful,” she said.

David Pierson and Berry Wang contributed reporting.