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Late Friday night, Mr. Obama formally triggered spending cuts that will reach across the breadth of the federal government after he failed to persuade Congressional Republicans to replace them with a mix of cuts and tax increases.

In a 70-page report to Congress accompanying the order and detailing the reductions — agency by agency and program by program — Jeffrey D. Zients, Mr. Obama's budget director, called them "deeply destructive to national security, domestic investments and core government functions."

But even as the cuts become official, some of the immediate impact is difficult to see.

The process of trimming government budgets is slow and cumbersome, involving lengthy notifications to unions about temporary furloughs, reductions in overtime pay and cuts in grant financing to state and local programs. Less federal money will, over time, mean fewer government contracts with private companies. Reduced overtime for airport security checkpoint officers will make lines longer, eventually.

And so as the first weekend began for the new, slimmer government, little of that is evident yet.

Letters to governors, informing them of the smaller grants are beginning to go out, officials said. Shaun Donovan, the secretary of housing and urban development, wrote to Gov. John R. Kasich of Ohio: "You can expect reductions totaling approximately $35 million," helpfully putting the amount in a bold type.

The Air Force Thunderbirds — the elite team of F-16 pilots who perform tricks at air shows — announced on its Web site that all of its shows have been canceled starting April 1. The last show will be in Titusville, Fla., on March 23.

Still, it will take some time, officials acknowledged, before the cuts begin to make life more difficult for teachers, defense contractors, Head Start students, border patrol agents or others who rely on the largess of the federal government.

Emerging from an Oval Office meeting on Friday with the lawmakers, the president called the cuts "just dumb." He said they would slow the economic recovery and spoke emotionally about their impact on people who would feel the consequences of government layoffs and disruptions in public services.

"I don't anticipate a huge financial crisis, but people are going to be hurt," Mr. Obama said during a 35-minute news conference at the White House, in which he acknowledged that his campaign of highlighting fallout from the cuts had failed to persuade Republicans to consider tax increases as part of a package to avert the $85 billion in reductions over the next seven months.

But the president and his Republican adversaries said they would not carry the fight over the cuts into a coming legislative effort to finance the government through Sept. 30, essentially declaring a cease-fire in the budget wars that have dominated Washington since 2011.

The showdown in December over the so-called fiscal cliff yielded $620 billion in tax increases over 10 years. The across-the-board spending cuts now going into force will cut deficits an additional $1.2 trillion.

Both sides indicated that for now, that may be enough — a fiscal peace through political exhaustion. The two parties are now expected to move to a broader argument over the right level of taxes and spending as they seek to develop a new budget for the coming year and beyond. Republicans said they welcomed a return to a more orderly budget process but warned they would not give in on their basic principles.

"I will not be part of any back-room deal, and I will absolutely not agree to increase taxes," said Senator Mitch McConnell of Kentucky, the Republican leader.

After a public relations blitz lasting weeks that was aimed at stopping the cuts, Mr. Obama said he was prepared to extend a stopgap law that finances the government to March 27 if Republicans stuck to an agreement worked out in 2011 about the level of federal spending. The decision will most likely allow the across-the-board spending reductions to remain in place for months if not years.

White House officials and Senate Democrats had considered making one last stand around the March 27 deadline, declaring the Senate would not pass another government spending plan unless it undid the across-the-board cuts. But Senate Democrats were leery. The first furloughs are likely to hit in April, and the Democrats feared that little political pressure would have built on Republicans before the current stopgap spending law expired.

On Friday, Mr. Obama used janitors and security guards as examples of federal employees whose pay will be cut because of the reductions. "They've got to figure out how to manage that," he said.

In fact, as pointed out by The Washington Post's Fact Checker, Capitol officials e-mailed the building's staff to say that the president's comment was not true. The Capitol's sergeant-at-arms, who is in charge of the building's officers, said that, "None of my employees will have their pay cut nor will they face furloughs."

In his weekly address on Saturday morning, the president acknowledged that the reductions would not affect everyone equally.

"While not everyone will feel the pain of these cuts right away, the pain will be real," Mr. Obama said. "Many middle-class families will have their lives disrupted in a significant way."

And heading into the weekend, some officials seemed eager to offer reassurance that government would continue to function despite the deep cuts.

While Leon E. Panetta, who just left the job of defense secretary, had thundered about the critical risk to national security — and lamented what he viewed as a shift of Washington's political class away from good governance — his successor, Chuck Hagel, spoke in more conciliatory terms.

To be sure, Mr. Hagel listed the specific risks to national security that could arrive with the cuts, also called the sequester. But even as he noted that the United States "has the best fighting force, the most capable fighting force, the most powerful fighting force in the world," he pledged that he and the Joint Chiefs of Staff "will manage these issues."

"These are adjustments," Mr. Hagel said of the cuts to be imposed by the cuts. "We anticipated these kinds of realities, and we will do what we need to do to assure the capabilities of our forces."

Mr. Obama has signaled that he remains eager to replace across-the-board cuts with deficit reduction that consists of more targeted reductions and increased revenue from closing tax loopholes. An e-mail sent late Friday to supporters from his former campaign operation had the subject line "Devastating."

"Congress can stop all of this right away — and pursue a balanced approach to deficit reduction," wrote Jon Carson, the president of Organizing for Action, the political operation created to advance the president's agenda. "That's what the vast majority of Americans want."

But Mr. Obama, too, said in his address on Saturday that it was time to move on to other topics.

"I'm going to push through this paralysis and keep fighting for the real challenges facing middle-class families," he said, citing efforts to increase the minimum wage, provide money for more preschool for children and pass an immigration overhaul.

In the Republican radio response, Representative Cathy McMorris Rodgers of Washington State, the chairwoman of the Republican conference, also called the cuts "devastating" to America, but said that Republicans in the House would not yield on the issue of taxes.

"Spending is the problem, which means cutting spending is the solution," she said. "It's that simple."

In fact, nothing about the sequestration law is simple. And the 70-page report on its implementation proves that point. The report calculates how much each program will lose, "as required by sections 251A(7)(A) and 253(f)(2) of BBEDCA." Those numbers are detailed in page after page of complex tables.

In his letter, Mr. Zients called the law establishing the cuts "a blunt and indiscriminate instrument. It was never intended to be implemented and does not represent a responsible way for our nation to achieve deficit reduction."