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"In the event that Republicans make good on their threat by failing to act, or by moving unilaterally to pass a debt-limit extension only as part of unbalanced or unreasonable legislation, we believe you must be willing to take any lawful steps to ensure that America does not break its promises and trigger a global economic crisis — without Congressional approval, if necessary," wrote Senators Harry Reid of Nevada, Richard Durbin of Illinois, Charles E. Schumer of New York and Patty Murray of Washington.

The letter signaled an escalation in the war of words over the federal debt ceiling, which could be breached in just weeks. Already, liberal intellectuals have been trying to rally support for measures to go around the Republican blockade, from declaring that the 14th Amendment gives the president unilateral authority to raise the debt ceiling to minting a trillion-dollar platinum coin that would be used to pay the nation's debts.

"All Americans, Democrats and Republicans alike, have a stake in ensuring that our country meets its legal obligations," the Democratic leaders wrote. "Financial markets have long viewed securities backed by the full faith and credit of the United States as the most trustworthy in the world. This lowers borrowing costs for homes, cars, and college for all Americans and strengthens our economy. If we violate that trust for the first time in history, we will never fully regain it, and every American will suffer."

Mr. Obama has said he will not negotiate on raising the government's statutory borrowing limit, but without some extralegal maneuver, it is not clear how he can keep that promise. Speaker John A. Boehner, Republican of Ohio, has not backed down on his demand that any increase in the debt limit include cuts at least equal in scope, and Senator Mitch McConnell of Kentucky, the Republican leader, has also said the debt ceiling issue must be used to secure spending reductions. Other Republicans have suggested that a shutdown of some parts of the federal government may be necessary to focus public attention on the debt.

Conservative Republicans say that even without an increase in the debt ceiling, the administration could continue to pay the government's creditors by ensuring that incoming tax receipts go first to pay off debts. To do that, however, huge and immediate cuts in government spending would be necessary, and global financial markets would probably be rattled.

The Senate Democratic leaders did not suggest what "lawful steps" they had in mind. A Democratic aide said the senators would be inclined to have the president declare unilateral authority under the 14th amendment, which states, "The validity of the public debt of the United States, authorized by law, including debts incurred for payments of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned."

That, the aide said, would be more politically tenable than using the loophole of a trillion-dollar coin, minted under a legal provision that allows the Treasury to issue a platinum coin of any denomination.

On Dec. 17, 2011, surveillance footage showed the man, Jerome Isaac, who was wearing a gas canister and a surgical mask, cornering the woman, Deloris Gillespie, 73, in the elevator of Ms. Gillespie's apartment building in Prospect Heights, Brooklyn. Mr. Isaac doused Ms. Gillespie with accelerant, then tossed a Molotov cocktail into the elevator.

"This has to be one of the most horrific crimes I have ever seen," the judge, Justice Vincent Del Giudice, said. "I had to review that video of the horrible death of that woman suffering."

Justice Del Giudice added, "That is not something one can take from one's mind."

Wearing an orange jumpsuit, Mr. Isaac, 48, said nothing during the hearing. He tilted his head down and closed his eyes for much of it.

Mr. Isaac was sentenced to prison despite recent questions about his mental health.

He pleaded guilty to killing Ms. Gillespie in November. But at a hearing later that month, Justice Del Giudice, citing a probation report, described how Mr. Isaac had said that he heard voices and that the devil had told him what to do.

At that hearing, Mr. Isaac's lawyer, Howard Tanner, said he believed Mr. Isaac was competent. Yet Justice Del Giudice, who was scheduled to sentence Mr. Isaac, ordered a mental health examination instead.

Though Mr. Isaac was diagnosed with a mental illness during that examination, Mr. Tanner said on Friday that Mr. Isaac was fit to be sentenced. The exact nature of the illness was not disclosed.

"He is fully competent and has been throughout my representation," Mr. Tanner said, adding: "This is a profoundly sad case — one of the saddest cases of my career."

Ms. Gillespie's daughter, Sheila Gillespie-Hillsman, who lives in Gary, Ind., had attended the sentencing hearing in November but was not in court on Friday.

In a telephone interview, Ms. Gillespie-Hillsman said she was "fed up" with the criminal justice process.

"I wanted to see his face, and I wanted him to see my face," Ms. Gillespie-Hillsman said. But, she added, the prospect of another delay was too stressful.

In a written statement read to the court on Friday, Ms. Gillespie-Hillsman described the anger and sadness her mother's death had caused. Yet Ms. Gillespie's family believed Mr. Isaac's sentence was adequate.

"I'm satisfied because he shouldn't be out on the streets again," said the prosecutor, Mark Hale, who read Ms. Gillespie-Hillsman's statement. "Especially since the 50-year sentence is without the hope of parole."

Mr. Isaac had once worked for Ms. Gillespie, helping clean her cluttered fifth-floor apartment. But Ms. Gillespie believed Mr. Isaac was stealing from her, so she fired him.

Mr. Isaac later told the authorities that he was upset with Ms. Gillespie because she had not paid him for his work. Mr. Isaac believed Ms. Gillespie owed him $2,000.

Mr. Obama also made it clear that he contemplated leaving relatively few troops in Afghanistan after the NATO combat mission ends in 2014, saying that the mission will be focused on advising and supporting Afghan troops and targeting the remnants of Al Qaeda.

"That is a very limited mission, and it is not one that would require the same kind of footprint we've had over the last 10 years in Afghanistan," Mr. Obama said, standing next to Mr. Karzai at a joint news conference at the White House.

Mr. Karzai professed to be comfortable with that, saying it was up to the United States to decide the size of a residual force. "Numbers are not going to make a difference in the situation in Afghanistan," he said, noting that it was the nature of the broader relationship that mattered.

Addressing a major sticking point between the countries, Mr. Karzai said the United States had agreed to turn over control of the prisons that house terrorism suspects to Afghan control. That would happen, he said, "soon after" he returned to Kabul.

Mr. Obama said that in order to leave any troops behind, the United States would require guarantees of legal immunity for its soldiers – a demand that the administration failed to obtain from Iraq, leading Mr. Obama to withdraw all remaining American troops from that country in 2011.

Mr. Karzai, citing the agreement to transfer detention centers and the planned withdrawal of American troops from Afghan villages, said he would push for such legal immunity.

"With those issues resolved, as we did today, I can go to the Afghan people and argue for immunity for U.S. troops in Afghanistan in a way that Afghan sovereignty will not be compromised, in a way that Afghan law will not be compromised."

In a joint statement released before the news conference, Mr. Obama and Mr. Karzai extolled the progress made by the Afghan security forces, noting that Afghan troops now take the lead in providing security in 80 percent of the country – a number that will rise to 90 percent by spring, when American and NATO troops are scheduled to move to a purely advisory role.

At that time, the statement said, American soldiers will pull out of patrols in villages, a measure that Mr. Karzai had sought.

Both leaders declined to be drawn into a discussion of the specific number of troops who would be involved in either the coming drawdown or in the residual force that would remain in Afghanistan after 2014. Mr. Obama said he would make decisions in the coming months based on the recommendations of his military commanders.

As the White House examines options for the size of a residual force, ranging from roughly 3,000 to 9,000 troops, Mr. Obama has directed his advisers to answer a basic question: Is such a force necessary to carry out the narrow counterterrorism objective and training mission the United States envisions for postwar Afghanistan?

Mr. Karzai came to Friday's meeting with far different expectations, according to Afghan officials.

Although he has been careful not to discuss specific troop numbers in public, Mr. Karzai appears to be counting on a substantial residual American force — perhaps as many as 15,000 troops, whose mission would be to advise Afghan security forces in their fight against the Taliban insurgency and carry out raids against Al Qaeda.

And he is hoping the United States will supply the Afghan Army with the latest military hardware, including tanks and fighter planes.

These very different expectations, analysts said, could reignite the tensions in a relationship between Mr. Obama and Mr. Karzai that has been notoriously fraught over issues like corruption, civilian casualties and threats to Afghan sovereignty.

"There's been a steady rollback of our objectives of what's good enough in Afghanistan," said Vali Nasr, a former senior State Department official who worked on Afghanistan and Pakistan and is the dean of the School of Advanced International Studies at Johns Hopkins University.

"If you're Karzai, you're basically now facing the same calculation that Maliki did in Iraq," said Mr. Nasr, referring to Nuri Kamal al-Maliki, the Iraqi prime minister. " 'If you're not willing to stay in large numbers, why do I need you?' "

The possibilities for friction are compounded by cynicism on both sides: the sense in Washington that Mr. Karzai is a mercurial, unreliable partner, and the suspicion in Kabul that the Americans care about Afghanistan only when they need it for other purposes, like fighting Al Qaeda.

Gen. Stanley A. McChrystal, the former commander in Afghanistan, warned in a recent interview that if the United States shrank too radically its advisory support for the Afghan military or curtailed its civilian programs, it might lose Afghan support for the counterterrorism operations it might want to continue there.

"If you don't have the support of the Afghan people," he said, "there's no reason for them to be supportive of this."

It is a measure of the potential disconnect between Mr. Karzai and Mr. Obama that Afghan officials involved in preparing his trip said the belief in the Afghan leader's inner circle was that Mr. Karzai was coming to the talks with the upper hand.

In Mr. Karzai's view, these officials and other people close to the president said, the United States needs a robust American presence in Afghanistan after 2014 to keep Al Qaeda off balance and Iran and Pakistan at bay. The Afghans "think they are indispensable; they think they have all the leverage," one Afghan official said.

The Afghans were readying complaints about the United States' continuing to detain Afghans at a prison next to Bagram Air Base, north of Kabul, which was supposed to have been handed over to Afghan authorities in September.

Some of Mr. Karzai's advisers are aware of the tenuous support at the White House for the war effort. Fearful of a repeat of Iraq, where Mr. Obama ordered a total military withdrawal, they have prevailed on the Afghan president to soften his position on granting immunity to any American troops stationed in Afghanistan after 2014, which he has done in recent months.

The ideal outcome for Mr. Karzai, these officials said, would be an American and allied training force that would help the Afghan Army make the most of the billions in aid it is expecting to receive, and a robust counterterrorism force that could work with Afghan Special Forces to combat the remnants of Al Qaeda.

The White House's calculation looks very different. While cost was a consideration in Iraq, Mr. Obama is more sensitive now to the budget consequences of keeping troops in Afghanistan after his bruising fiscal showdown with Congress, and the prospect of huge mandatory cuts in the Pentagon's budget.

"You've got to step back and see the whole field from the point of view of taxpayer spending," said Tommy Vietor, a spokesman for the National Security Council.

To some extent, officials said, the administration's floating of a "zero option" for troops is a bargaining ploy, with both the Pentagon and with Mr. Karzai.

But resolving the differences between the United States and the Afghan government, if they are resolved, is likely to have a profound effect on the long-term commitment of other NATO members.

An allied official said that if the White House opted for a minimal troop presence, the rest of the NATO allies were expected to follow suit, especially since the war was even more unpopular among their publics. Many critics complain that Mr. Karzai has been slow to pursue corruption. But some analysts said the emerging American policy might deepen Mr. Karzai's insecurities.

"We will not be able to give Karzai any more spine to go after cronyism and nepotism," said Vanda Felbab-Brown, an expert on Afghanistan at the Brookings Institution.

The one issue on which both sides appear in sync is the peace process, according to the Afghan officials and people close to the Afghan leader. Like their American counterparts, Afghan officials want to see negotiations with the Taliban make progress and offer rosy public assessments of the diplomatic effort.

But privately, the Afghans are aware that there have been no meaningful engagements with the Taliban in nearly a year, and that until the insurgents prove willing to sit down, the peace process will make little headway.

Officials from the Centers for Disease Control and Prevention, speaking on a conference call Friday morning, again urged Americans to keep getting flu shots. At the same time, they emphasized that the shots are not infallible: a preliminary study rated this year's vaccine as 62 percent effective, even though it is a good match for the most worrisome virus circulating. That is considered "moderately" effective — the vaccine typically ranges from 50 percent to 70 percent effective.

Even though deaths stepped into epidemic territory for the first time — barely — the C.D.C. officials expressed no alarm, and said it was possible that new flu infections were peaking in some parts of the country.

Flu outbreaks typically reach epidemic level for one or two bad weeks every flu season, and the mortality threshold is a wavy curve that dips down in summer and up in winter. How severe a season is depends on how high deaths climb and for how many weeks they persist.

The flu vaccine, said Dr. Thomas Frieden, director of the C.D.C., "is far from perfect, but it's by far the best tool we have to prevent influenza."

Michael T. Osterholm of the University of Minnesota's Center for Infectious Disease Research and Policy had a moment of self-satisfaction at hearing of the vaccine effectiveness study.

Although he still backs getting the shots, last year he released a study saying that the effectiveness of flu shots had been "oversold" and that new vaccines had to be developed.

But he said that no government and no private vaccine company was yet prepared to spend "the $1 billion needed to get a new vaccine across the Valley of Death" — by which he meant the huge expensive trials needed to get a vaccine approved.

French officials confirmed that the French forces, which included paratroopers and helicopter gunships, had engaged in fighting with the Islamists after landing at a major airfield in the central Mali town of Sévaré.

It was unclear how many French troops had been sent or from where, but a Western diplomat in neighboring Niger said the Islamist forces numbered between 800 and 900 fighters, with about 200 vehicles.

"French forces brought their support this afternoon to Malian army units to fight against terrorist elements," President François Hollande of France said in a statement to reporters in Paris. "This operation will last as long as is necessary."

Mr. Hollande has been especially outspoken in his animosity toward northern Mali's Islamist occupiers and their harsh practices, which rights activists say include arbitrary killings, stonings, amputations, forced marriages and the destruction of non-Islamist cultural shrines. Thousands of Malians have sought to flee the north in recent months.

"Mali is dealing with terrorist elements form the north, whose brutality and fanaticism are now clear to the entire world," Mr. Hollande said. "The very existence of the friendly state of Mali is at stake, as is the security of its people and that of our citizens. There are 6,000 of them there."

The French president was responding to an urgent request received the day before from Mali's interim president, Dioncounda Traore, who said Malian government forces were in dire need of help to stop the Islamists, who have turned the northern half of the country into a militant haven since seizing the territory, about twice of the size of Germany, last April.

The United Nations Security Council, which has repeatedly condemned the Islamist takeover of northern Mali and last month authorized an African-led force to enter the country to help drive the Islamists out, said Thursday that it was closely monitoring events there and may take additional steps. Mr. Hollande is also to meet with the Malian president next week.

The swift French response came after two days of clashes between the Malian Army and militants around Konna, a sleepy mud-brick village that for months had marked the outer limit of the Malian Army's control after it lost half of the country to the Islamists and their allies eight months ago.

"It's a very serious situation, very dangerous," said a Malian officer here in Bamako, the capital, who was not authorized to speak publicly.

The Islamists had been threatening a major airfield 25 miles away in Sévaré, also the home of a significant army base. And 10 miles from Sévaré is the historic river city of Mopti, the last major town controlled by the Malian government, with a population of more than 100,000.

"There were hard fights, but we lost," the officer said.

A spokesman for the Islamists, Sanda Ould Boumana, said Thursday from rebel-held Timbuktu: "We have taken the town of Konna. We control Konna, and the Malian Army has fled. We have pushed them back." Gen. Carter F. Ham, the commander of the Pentagon's Africa Command, who was traveling in neighboring Niger, said he understood that French paratroopers and helicopter gunships had landed in Sévaré and had engaged the Islamists in combat. He also said the United States, which shares France's deep concern about the Islamist seizure of northern Mali, was considering what it could do to help, perhaps by repositioning satellites or sending in surveillance drones.

This week's clashes were the first time that the two sides had fought since Islamists and their Tuareg rebel allies conquered the north of Mali last spring, splitting the country in two and leaving the Malian Army in disarray.

For months, the United Nations and Mali's neighbors have been debating and planning a military campaign to retake the north by force, if necessary, an international push that is supposed to be led by Malian forces. Analysts had previously said that the outcome of this week's fighting at Konna would be a significant indicator of the army's fitness to undertake the reconquest of the north.

Malian politicians reacted with shock to news of Konna's loss.

"This is a very disagreeable surprise. Terrible. A dagger blow," said Fatoumata Dicko, a deputy in Mali's Parliament in Bamako. "People are fleeing Sévaré. They think there is nothing to hold the Islamists back."

Within two years other researchers were finding identical cartridges circulating through the ethnic violence in Darfur. Similar ammunition then turned up in 2009 in a stadium in Conakry, Guinea, where soldiers had fired on antigovernment protesters, killing more than 150.

For six years, a group of independent arms-trafficking researchers worked to pin down the source of the mystery cartridges. Exchanging information from four continents, they concluded that someone had been quietly funneling rifle and machine-gun ammunition into regions of protracted conflict, and had managed to elude exposure for years. Their only goal was to solve the mystery, not implicate any specific nation.

When the investigators' breakthrough came, it carried a surprise. The manufacturer was not one of Africa's usual suspects. It was Iran.

Iran has a well-developed military manufacturing sector, but has not exported its weapons in quantities rivaling those of the heavyweights in the global arms trade, including the United States, Russia, China and several European states. But its export choices in this case were significant. While small-arms ammunition attracts less attention than strategic weapons or arms that have drawn international condemnation, like land mines and cluster bombs, it is a basic ingredient of organized violence, and involved each year and at each war in uncountable deaths and crimes.

And for the past several years, even as Iran faced intensive foreign scrutiny over its nuclear program and for supporting proxies across the Middle East, its state-manufactured ammunition was distributed through secretive networks to a long list of combatants, including in regions under United Nations arms embargoes.

The trail of evidence uncovered by the investigation found Iranian cartridges in the possession of rebels in Ivory Coast, federal troops in the Democratic Republic of Congo, the Taliban in Afghanistan and groups affiliated with Al Qaeda in the Maghreb in Niger. The ammunition was linked to spectacular examples of state-sponsored violence and armed groups connected to terrorism — all without drawing wide attention or leading back to its manufacturer.

The ammunition, matched to the world's most abundant firearms, has principally been documented in Africa, where the researchers concluded that untold quantities have been supplied to governments in Guinea, Kenya, Ivory Coast and, the evidence suggests, Sudan.

From there, it traveled to many of the continent's most volatile locales, becoming an instrument of violence in some of Africa's ugliest wars and for brutal regimes. And while the wide redistribution within Africa may be the work of African governments, the same ammunition has also been found elsewhere, including in an insurgent arms cache in Iraq and on a ship intercepted as it headed for the Gaza Strip.

Iran's role in providing arms to allies and to those who fight its enemies has long been broadly understood. Some of these practices were most recently reported in the transfer of Fajr-5 ground-to-ground rockets to Gaza. Its expanding footprint of small-arms ammunition exports has pushed questions about its roles in a shadowy ammunition trade high onto the list of research priorities for trafficking investigators.

"If you had asked me not too long ago what Iran's role in small-arms ammunition trafficking to Africa had been, I would have said, 'Not much,' " said James Bevan, a former United Nations investigator who since 2011 has been director of Conflict Armament Research, a private firm registered in England that identifies and tracks conventional weapons. "Our understanding of that is changing."

The independent investigation also demonstrated the relative ease with which weapons and munitions flow about the world, a characteristic of the arms trade that might partially explain how Iran sidestepped scrutiny of governments and international organizations, including the United Nations, that have tried to restrict its banking transactions and arms sales.

The United Nations, in a series of resolutions, has similarly tried to block arms transfers into Ivory Coast, Congo and Sudan — all places where researchers found Iranian ammunition.

Ammunition from other sources, including China, Russia, Hungary, the Czech Republic and other former Soviet bloc states remain in circulation in Africa, along with production by African states. Why Iran has entered the market is not clear. Profit motives as well as an effort by Iran to gain influence in Africa might explain the exports, Mr. Bevan said. But much remains unknown.

Neither the government of Iran nor its military manufacturing conglomerate, the Defense Industries Organization, or DIO, replied to written queries submitted for this article.

The researchers involved in the investigation — including several former experts for the United Nations and one from Amnesty International — documented the expanding circulation of Iranian ammunition, not the means or the entities that have actually exported the stocks. They are not sure if the ammunition had been directly sold by the Iranian government or its security services, by a government- or military-controlled firm, or by front companies abroad.

But the long mysterious provenance of the ammunition appears beyond dispute. The cartridges were made, the researchers say, by the Ammunition and Metallurgy Industries Group, a subsidiary of DIO.

Matching the cartridges to the producer took time, in part because the ammunition had been packaged and marked in ways to dissuade tracing. Eventually the identifications were reached via data-pooling.

Much of the world's ammunitions bears numeric or logo markings, known as headstamps, that together declare the location and year of a cartridge's manufacture. Over the years, governments and private researchers have assembled encyclopedic headstamp keys that can make matching particular markings to particular factories a straightforward pursuit.

The ammunition in these cases included rounds for Kalashnikov assault rifles, for medium machine guns and sniper rifles and for heavy machine guns.

In each case, the cartridges carried headstamps not listed on the publicly available records. The stamps were simple caliber markings and, typically, two digits indicating the year of manufacture.

Similarly, neither the ammunition's wooden crates nor its packaging in green plastic carry bags or plain cardboard boxes, when these items were found with the ammunition, disclosed the place of manufacture.

All of the ammunition shared a unique combination of traits, including the caliber headstamp in a particular font, the alloy of the bullet jackets, and three indentations where primers attached to cartridge cases. The traits together suggested a common manufacturer.

Over the years, the researchers bided time and gathered data. They collected samples of used and unused ammunition at conflicts and recorded their characteristics. They collaborated with other specialists, exchanging their finds.

Some sources were confidential. Others were not. Mike Lewis, a former member of the United Nations Panel of Experts on the Sudan, documented the presence of the ammunition at the Conakry stadium crackdown while investigating for Amnesty International.

One sample — from Afghanistan — was found by The New York Times, which was surveying ammunition used by the Taliban and provided an image of a then-unidentifiable cartridge's headstamp to Mr. Bevan in 2010.

Once the data was assembled, the breakthrough came in what a soon-to-be-released report by the researchers called "cross-case analysis" and by looking away from the ammunition to other sources.

In late 2011 Mr. Bevan obtained the bill of lading for 13 shipping containers seized by the authorities in Lagos, Nigeria, in 2010. The document showed that the containers originated in Iran and declared the contents to be "building materials."

But, as the researchers noted in their report, "concealed behind stone slabs and insulation materials" was a shipment of arms, including the same ammunition that they had been finding in the field.

The shipping company was based in Tehran, Iran's capital.

Declassified documents obtained via the Freedom of Information Act by Matthew Schroeder, an arms-trafficking analyst at the Federation of American Scientists, later showed that the American military had identified ammunition packaged in the same materials as Iranian. Mr. Schroeder shared his documents with Mr. Bevan. This provided another link.

Ultimately, Mr. Bevan noticed that Iran had published limited technical details of its cartridges, including bullet weights. Some of these weights are atypical. Late in 2012 he had samples weighed on a jeweler's scale, confirming the match.

Mr. Bevan made clear in repeated interviews that he and his fellow researchers are not advocates for military action against Iran. When they began tracing the ammunition, they did not know or expect that the evidence would point to Tehran.

He also noted that while the ammunition is Iranian-made, it may not have been sent to directly by Iran to some of the combatants.

"In terms of prescription, if it was clear that there were repeated violations by Iran, I think we could come down more strongly about it," he said. "But a good portion of this, and in perhaps the majority of these cases, the ammunition was transferred around Africa by African states."

He added that while the original source of the ammunitions is now clear, many questions remain unanswered, including who organized the delivery to regions under embargo or enmeshed in ethnic conflicts.

Mr. Bevan and his fellow researchers said their findings pointed to a need for further research, to gather facts upon which policy decisions can be based.