The chairwoman of the Wisconsin Elections Commission on Monday certified President-elect Joseph R. Biden Jr. as the winner in Wisconsin, formalizing his victory in a state President Trump narrowly carried four years ago and undermining the quixotic efforts of Mr. Trump and his supporters to portray his decisive national loss as a matter still under dispute.

During a three-minute teleconference Monday afternoon, Ann Jacobs, the elections commission chairwoman, signed a document that declared Mr. Biden to be the winner.

“I am now signing it as the official state determination of the results of the Nov. 3, 2020, election and the canvass,” Ms. Jacobs said, before holding the document up to the camera broadcasting her online.

The Wisconsin certification came a few hours after Arizona officials formalized Mr. Biden’s even narrower victory there.

Monday’s certifications would be an afterthought in any other year. But in an environment where Mr. Trump’s false claims of massive fraud have created an alternate reality among his die-hard backers — in the West Wing and beyond — the results foreclosed his fanciful final path to victory.

Mr. Trump, buoyed by his legal team and supporters in the conservative media, has held out hope that he could somehow prevail in Wisconsin, Arizona and Georgia, where Republican officials on Monday firmly refused to challenge Mr. Biden’s win there.

In Arizona, Mr. Biden prevailed by over 10,000 votes, mainly because of his strength in the state’s largest county, Maricopa, which has been trending Democratic in recent elections.

The state will deliver its 11 electoral votes to Mr. Biden when the Electoral College meets next month, a ceremonial conclave that Mr. Trump hopes to flip in his favor despite state laws that forbid electors from defying the will of voters.

The certification in Wisconsin followed the conclusion of recounts, requested and subsidized with $3 million from President Trump’s campaign, in Dane and Milwaukee Counties that resulted in Mr. Biden adding 87 votes to his statewide margin.

Ms. Jacobs, a Milwaukee Democrat, said that certifying the result of the presidential election came at her discretion and that she expected the move to kickstart legal challenges from the Trump campaign.

“The power to do this is vested solely in the chair,” Ms. Jacobs said.

The six-member bipartisan commission is scheduled to meet again Tuesday morning to certify the state’s other election results.

Two weeks ago the Trump campaign requested the recounts in Dane and Milwaukee, the state’s two largest and most Democratic counties, in an effort to build a legal case against Mr. Biden’s statewide victory. Mr. Biden won Wisconsin by 20,565 votes, a margin that was always highly unlikely to be overturned in a recount.

The Trump campaign has argued in its recount petition that all ballots cast at pre-Election Day in-person absentee voting sites should be disqualified because, it claimed incorrectly, those absentee ballots were issued without voters submitting a written application requesting the ballot.

Its argument would have thrown out hundreds of thousands of ballots across Wisconsin, including those cast by prominent Trump supporters, such as several state legislators and one of the president’s lawyers, Jim Troupis, according to the Milwaukee Journal Sentinel.

President-elect Joseph R. Biden Jr. and Vice President-elect Kamala Harris received on Monday their first full intelligence briefings since the election, transition officials announced.

While typically intelligence briefings for presidents-elect begin soon after the election, presenting the President’s Daily Brief to Mr. Biden and Ms. Harris was delayed as the Trump administration put off beginning of the transition.

Mr. Biden received some intelligence briefings during the campaign, but they were mostly focused on foreign threats to the election. The President’s Daily Brief will give Mr. Biden and Ms. Harris a far broader look at national security threats to the United States.

Distribution of the daily brief is at the sole discretion of the president. Shortly after the General Services Administration approved the formal transition process to begin on Nov. 24, the White House approved Mr. Biden receiving the daily brief. The logistics of setting up the classified sessions put off their start until Monday, according to a Trump administration official.

The Office of the Director of National Intelligence assigns an individual briefer to every senior official who receives a version of the daily brief. That briefer shapes the presentation to the needs of the senior official, or in this case the incoming president.

Mr. Biden and Ms. Harris each received in-person briefings from an intelligence official. They will also have access to the briefing itself, a more detailed and extensive classified intelligence document, which is delivered on a secure tablet computer.

Mr. Biden will need to wait until he is in the White House to make big changes to the brief, by setting collection or analytical priorities for the intelligence agencies. Nevertheless, the transition briefings will allow the intelligence agencies to begin to get a sense of the kinds of questions Mr. Biden and Ms. Harris are asking, and what their current foreign policy focus is.

President Trump’s sustained assault on his own party in Georgia, and his repeated claims of election fraud in the state, have intensified worries among Republicans that he could be hurting their ability to win two crucial Senate runoff races next month.

The president has continued to claim without evidence that his loss in the new battleground state was fraudulent, directing his ire in particular at Gov. Brian Kemp and Secretary of State Brad Raffensperger, both conservative Republicans, whom he has accused of not doing enough to help him overturn the result.

Over the weekend, he escalated his attacks on Mr. Kemp, saying he was “ashamed” to have endorsed him in 2018, and on Monday he called Mr. Kemp “hapless” as he urged him to “overrule his obstinate Republican Secretary of state.’’

Mr. Trump’s broadsides have quietly rattled some Republicans in the state, who fear that concerns about the fairness of the presidential election could depress turnout for the Senate races, which will determine whether Democrats or Republicans control the chamber.

After resisting entreaties to appear in Georgia, the president plans to travel there this weekend, though even some of his own aides remain uncertain whether his anger toward state officials will overshadow any support he may lend the party’s two candidates.

“You can’t say the system is rigged but elect these two senators,” said Eric Johnson, a campaign adviser to Kelly Loeffler, one of the G.O.P. Senate candidates, and a former Republican leader of the Georgia Senate. “At some point he either drops it or he says I want everybody to vote and get their friends to vote so that the margins are so large that they can’t steal it.”

The split signifies both an extraordinary dispute over election integrity within the Republican Party and a preview of the control the president may continue to exert over the conservative base even after he leaves office. As Mr. Trump talks seriously about the possibility of mounting another bid for the White House in 2024, his personal goals may not always align with those of his party — no matter the political stakes.

“I had someone message me just last week saying: ‘Nope, I’m done. Can’t trust the election. Never voting again,’” said Buzz Brockway, a former Republican state representative. “The president has a very dedicated group of supporters who don’t really support the broader Republican Party — they support him.”

A skeptical Supreme Court on Monday reacted with frustration and some confusion to President Trump’s plan to exclude unauthorized immigrants from the calculations used to allocate seats in the House.

While there was some discussion about whether the plan was lawful, the more immediate questions for the justices were where the administration stood in its efforts to identify and count the unauthorized immigrants and what role the court should play if substantial numbers were not identified.

Removing undocumented immigrants from the census would most likely have the effect of shifting congressional seats and federal money to states that are older, whiter and typically more Republican.

But if the Census Bureau cannot provide Mr. Trump with specific information about a large enough number of unauthorized immigrants in the coming weeks, he will not be able to exclude enough of them from the reapportionment to change the way House seats are allocated. That would leave the justices without a concrete dispute to decide.

If there is data to allow Mr. Trump to exclude enough immigrants to change the allocation of House seats, the case would become quite important.

Indeed, if the court rules for the administration on the core question in the case, it would upend a longtime consensus that the government must count all residents of the United States, whatever their immigration status, with the potential to shift political power and federal money from Democratic states to Republican ones.

But there was only limited discussion of that question on Monday, though several justices, including some conservatives, expressed skepticism about the administration’s arguments.

The Constitution requires congressional districts to be apportioned “counting the whole number of persons in each state,” using information from the census. The Trump administration’s new approach sought to exclude undocumented immigrants from the apportionment.

The case before the court, Trump v. New York, No. 20-366, was brought by two sets of plaintiffs, one a group of state and local governments and the United States Conference of Mayors, and the second a coalition of advocacy groups and other nongovernmental organizations.

A three-judge panel of the Federal District Court in Manhattan ruled that the new policy violated federal law. Two other courts have issued similar rulings, while one said the dispute was not ripe for consideration.

President-elect Joseph R. Biden Jr. officially named top members of his economic team on Monday, showcasing his commitment to diversity and placing several women in top economic roles.

With the picks, which require Senate confirmation, Mr. Biden is sending a clear message that economic policymaking in his administration will be shaped by liberal thinkers with a strong focus on worker empowerment as a tool for economic growth. They include:

• Cecilia Rouse, a Princeton labor economist, to run the three-member Council of Economic Advisers. She would be the first Black woman to lead the council. A labor economist, she worked on Mr. Obama’s Council of Economic Advisers for two years and at the White House’s National Economic Council during the Clinton administration.

• Neera Tanden, the chief executive of the Center for American Progress, to lead the Office of Management and Budget. Ms. Tanden, who would be the first Indian-American to lead the Office of Management and Budget, has advocated aggressive spending to alleviate economic harm from the pandemic and has dismissed concerns about adding to the deficit at the current moment.

• Janet L. Yellen, the former Federal Reserve chair, as Treasury secretary. Ms. Yellen, who is also a labor economist, was one of the first officials to suggest allowing the labor market to run “hot” — meaning leaving interest rates lower for longer — in order to help lift wages and get more people into jobs.

• Jared Bernstein and Heather Boushey were named to join Ms. Rouse on the Council of Economic Advisers, which is a three-member team that advises the president on economic policy. They both come from a liberal, labor-oriented school of economics that views rising inequality as a threat to the economy and emphasizes government efforts to support and empower workers. Mr. Bernstein was Mr. Biden’s first chief economist when he was vice president. Ms. Boushey was a top policy adviser to Mrs. Clinton in 2016. Both have advocated a large stimulus package to help workers and businesses hurt by the pandemic recession.

• Adewale Adeyemo, known as Wally, a senior international economic adviser in the Obama administration, as deputy Treasury secretary. An immigrant from Nigeria, he has extensive experience working at the Treasury Department during the Obama administration, when he was a senior adviser and deputy chief of staff.

• Brian Deese, a former Obama economic aide who helped lead that administration’s efforts to bail out the American automotive industry, has also been selected to lead the National Economic Council, according to three people with knowledge of the selection. Mr. Deese is a veteran of economic policymaking, having served as the acting head of the Office of Management and Budget and the deputy director of the Economic Council under Mr. Obama, as well as a special adviser on climate change.

The appointments could fall short of hopes within the progressive wing of the Democratic Party, which has been frustrated that their views are not being sufficiently represented in early personnel decisions. In particular, the decision to select Ms. Tanden, a divisive and partisan figure in the party, could culminate in an intraparty fight, as well as a confirmation battle.

Republicans, who are fighting to retain control of the Senate, are unlikely to easily pass Ms. Tanden, who advised Hillary Clinton’s 2016 presidential campaign and has been one of the most outspoken critics of President Trump.

Mr. Biden’s other picks are expected to be less contentious.

The Justice Department asked a federal judge on Monday to dismiss the criminal case against President Trump’s former national security adviser Michael T. Flynn, citing his pardon last week — and making clear that it broadly covered potential legal troubles beyond the charge Mr. Flynn had faced of lying to federal investigators.

“The president’s pardon, which General Flynn has accepted, moots this case,” the Justice Department filing said.

Mr. Flynn had twice pleaded guilty to a charge of lying to the F.B.I. about his conversations in late 2016, during the Trump presidential transition, with the Russian ambassador to the United States. His original plea deal also covered legal liability for other potential charges related to his work as an unregistered foreign agent of Turkey in 2016.

The filing was accompanied by the text of the pardon itself, which had not previously been released. It was written broadly not only to cover lying to the F.B.I., but to foreclose any legal jeopardy Mr. Flynn might face from a future Justice Department arising from the Turkey matter, his inconsistent statements under oath to Judge Emmet G. Sullivan of the Federal District Court for the District of Columbia, and any potential perjury or false statements to the team of the special counsel Robert S. Mueller III team or to the grand juries it used.

Andrew Weissmann, a former member of the special counsel team who was not directly involved in prosecuting Mr. Flynn, condemned the Trump administration’s handling of the case after Mr. Mueller’s office shut down.

“Trump issued the pardon only after Barr debased the Department of Justice by filing a disingenuous motion to dismiss,” Mr. Weissmann said. “Sullivan will have the opportunity to weigh in on his view of all this when he grants the motion to dismiss based on the full pardon.”

But the Justice Department filing signaled that under the Trump administration, at least, the Flynn matter is closed.

“No further proceedings are necessary or appropriate, as the court must immediately dismiss the case with prejudice,” it said.

Planning for President-elect Joseph R. Biden Jr.’s inauguration is intensifying, with his team announcing on Monday the formation of a Presidential Inaugural Committee that will be carefully attuned to the challenge of managing inauguration activities amid a pandemic.

The organization rolled out the names of senior leaders of the committee, as well as a website for the team, on Monday morning. The committee is expected to work with the Joint Congressional Committee on Inaugural Ceremonies, the announcement said.

The committee will be helmed by Tony Allen, the president of Delaware State University, a historically Black school in Mr. Biden’s home state.

“This year’s inauguration will look different amid the pandemic, but we will honor the American inaugural traditions and engage Americans across the country while keeping everybody healthy and safe,” Mr. Allen said in a statement.

Maju Varghese, the chief operating officer on the Biden campaign, will serve as executive director. State Senator Yvanna Cancela, Democrat of Nevada and an early Biden supporter, and Erin Wilson, who was Mr. Biden’s political director during the campaign, have been named as deputy executive directors, according to a release from the organization.

On Capitol Hill, where the official inaugural ceremonies are arranged by a bipartisan congressional committee, lawmakers and aides have quietly been at work since long before the election trying to reimagine what a transfer of power during a pandemic might look like.

The short answer is: quite a bit less crowded.

The organizers are determined that Mr. Biden take the oath of office and deliver an address to the nation outside the West Front of the Capitol, preserving an iconic tableau that has often set the tone for a new presidency. But to make it happen, they are likely to slash the number of officials flanking Mr. Biden. Those who do make the cut — Supreme Court justices, former presidents, top House and Senate leaders and the Joint Chiefs of Staff among them — will be required to socially distance and wear masks. Some may be asked to take coronavirus tests.

The most difficult challenge may be keeping away crowds. Congress typically distributes 200,000 tickets for seats near the platform. Senator Roy Blunt, Republican of Missouri, who is overseeing the planning on Capitol Hill, has indicated that number could be cut drastically.

President Trump’s window to use the courts to overturn the victory of President-elect Joseph R. Biden Jr. is closing quickly as the so-called safe harbor deadline — the date before which any election controversies must be settled — approaches on Dec. 8. Then the voting results are considered final.

Still, in the past week, Republicans have launched last-ditch efforts to halt or reverse the certification process in three key swing states: Michigan, Wisconsin and Minnesota. Each of these suits make extraordinary demands on the judges hearing them and would require unprecedented decisions to succeed.

There are also two federal lawsuits moving through the courts seeking to reverse the certification process in Michigan and Georgia. Both of them were filed by Sidney Powell, a former member of Mr. Trump’s legal team who was disavowed by the president last week after she made wild accusations that foreign leaders and some Republican officials were involved in a scheme to manipulate voting machines.

Appearing on Fox News on Sunday, Mr. Trump referred to the numerous — and almost completely unsuccessful — election-related lawsuits that he and his supporters have filed in recent weeks saying, “It’s hard to get to the Supreme Court.”

But he has at least one current path to the nation’s highest court in his continued effort to undermine the election — though it is not a guaranteed one — and even if the justices accept his case and rule in his favor, there isn’t all that much he stands to gain.

Mr. Trump’s road there was paved on Friday when he and his legal team were handed a defeat in a unanimous ruling by the Third Circuit Court of Appeals in Philadelphia, which soundly rejected their attempts to stop or reverse the certification of election results in Pennsylvania. (Pennsylvania certified its election results last week.)

The blistering appellate decision upheld an equally searing ruling from earlier in the month by Matthew W. Brann, a district judge in Pennsylvania, who wrote that Mr. Trump’s attempts to derail the commonwealth’s election were based on “strained legal arguments without merit and speculative accusations.”

Shortly after the Third Circuit’s ruling came down, Mr. Trump’s lawyers vowed to ask the high court to reconsider the case, but by Monday morning they had not done so yet. And even if the Supreme Court granted the request to reverse the campaign’s appellate loss, it might not get much, given the narrow way in which Mr. Trump’s lawyers structured the original appeal.

Instead of contesting the substance of Judge Brann’s ruling, Mr. Trump’s lawyers asked the Third Circuit only for permission to file a revised version of its initial complaint. Assuming the Supreme Court takes the case and abides by the strict terms of the president’s appeal, it would be able to do no more than return the matter to Judge Brann’s court for further action.

Mr. Trump’s defeat in the Third Circuit left him relying on an ever-dwindling number of suits across the country geared toward overturning the results of the election. On Saturday, he suffered another legal loss when the Pennsylvania Supreme Court tossed out a suit by some Republican supporters that had also sought to stop certification of the commonwealth’s vote count.

That suit, led by Representative Mike Kelly, Republican of Pennsylvania, claimed that a state law authorizing mail-in ballots was unconstitutional and sought retroactively to nullify their use in the election — a move that, if successful, would have tipped the state to Mr. Trump. But the mail-in ballot law was passed last year with bipartisan support and the Pennsylvania Supreme Court dismissed Mr. Kelly’s attempts to overturn it, saying he had filed his claims too late.

With President-elect Joseph R. Biden Jr. preparing to govern from the middle in a Congress whose thin majorities will force him to compromise on almost every priority, Senator Joe Manchin III of West Virginia, a centrist Democrat, suddenly finds himself at the center of relevance in the nation’s capital.

“I think we have a golden opportunity to bring the country back together and for us to work in the middle,” Mr. Manchin said excitedly. “I’ll tell you the reason why: The numbers are so close with what the Democratic House members lost. For Nancy Pelosi, she’s going to have to work with people that have a more moderate view than some of the people that pushed her from the left.”

If Democrats are able to win two runoffs in Georgia in January and take control of the Senate, any plans to enact a liberal agenda — such as increasing the number of Supreme Court justices — will have to go through Mr. Manchin. Likewise, if Republicans win at least one of the Georgia races, allowing them to maintain Senate control, they will need centrists in both parties to help block progressive items or pass compromise legislation.

That is the situation that Mr. Manchin said he considered more likely. He is already preparing for a power dynamic that he asserted would give him and three of the more moderate Republicans in the Senate — Susan Collins of Maine, Lisa Murkowski of Alaska and Mitt Romney of Utah — a big role in determining what happens at the dawn of Mr. Biden’s presidency.

With Vice President-elect Kamala Harris empowered to break ties, Mr. Manchin noted that even if Democrats did not prevail in the Georgia runoffs, it would take only two Republican defections to hand the Democrats a majority on any given measure.

“It behooves everybody to start working together,” he said during a wide-ranging interview in his office. “If they don’t, it doesn’t take many of us to say, ‘Guys, we’ve given all of you a chance. We haven’t done our job for the last 10 years, and we’re going to start.’”

There are close elections and then there are incredibly close elections. A state canvassing board in Iowa on Monday certified that Mariannette Miller-Meeks, a Republican, won an open congressional seat by a mere six votes.

After a recount that gave Ms. Miller-Meeks a lead of 196,964 to 196,958, the Democratic candidate, Rita Hart, signaled that she would challenge the outcome before a judicial panel, though her campaign did not immediately announce a challenge.

“Over the next few days, we will outline our next steps in this process to ensure that all Iowans’ voices are heard,’’ Ms. Hart’s campaign manager, Zach Meunier, said in a statement.

Ms. Hart has two days to challenge the certification under Iowa law, according to The Associated Press. A challenge would be heard before a tribunal consisting of Iowa’s chief justice and four state district judges. If the panel upheld Ms. Miller-Meeks’s victory, Ms. Hart could appeal to the U.S. House, where Democrats are in the majority. The A.P. said it would not call the race until all legal avenues are exhausted.

Ms. Miller-Meeks, a state senator making her fourth run for Congress, was certified the winner after recounts by the 24 counties of Iowa’s Second District. Her win, if it holds, would further erode Democrats’ House majority; Republicans flipped a net of at least nine seats this year that were held by Democrats.

The Hart campaign said the six-vote margin was the slimmest in any congressional race since 1984, when an Indiana race came down to just four votes.

“This race reinforces that every single vote can make a difference and hopefully sends a message about how important it is to be a voter,” said Paul D. Pate, Iowa’s secretary of state.

Both candidates attended orientation in Washington for new members of Congress while the race — to replace a retiring Democrat, Representative Dave Loebsack — was undecided.

The Hart campaign held out hope that a challenge to the results might uncover valid ballots that were overlooked in the recount, which it requested on Nov. 12, when Ms. Miller-Meeks led by 47 votes.

“Under Iowa law, this recount process was designed to count ballots that had already been tallied, meaning that additional legal ballots may have yet to be counted,’’ the campaign said.

Although many polls of Iowa showed Joseph R. Biden Jr. and other Democrats competitive before the election, the party suffered broad losses in the state. President Trump carried it by 8.2 percentage points. And in Iowa’s First District, Representative Abby Finkenauer, a first-term Democrat, was unseated by Ashley Hinson, a Republican.