In the past week, the legitimacy debate has swelled with each new court filing in cases stemming from the investigations into Trump’s 2016 campaign.

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First came the statement by federal prosecutors in New York that Trump attorney Michael ­Cohen “sought to influence the election from the shadows” by arranging to pay hush money to women who said they had extramarital affairs with Trump. Then, on Tuesday, executives at the National Enquirer’s parent company admitted paying hush money to prevent news of the candidate’s alleged infidelities “from influencing the election.”

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In Congress, in the media and among activists, criticism of Trump is increasingly taking the form of arguments that he won office fraudulently — especially as the hush-money revelations have landed atop allegations by special counsel Robert S. Mueller III’s team that Russian agents engaged in a criminal scheme to undermine Hillary Clinton’s presidential candidacy.

“People don’t actually really consider Trump a legitimate president,” former Democratic National Committee chairman Howard Dean said on MSNBC last month. “He was obviously elected and all this business, but he does not represent American values.”

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Trump and his defenders retort that prosecutors so far have fallen well short of proving criminal deeds by the president himself. They say the legitimacy debate is just one more weapon in a bristling partisan arsenal deployed by Trump haters on the left.

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“Questioning Trump’s legitimacy is basically the birtherism of the left,” said Christopher Buskirk, publisher of American Greatness, a conservative website. “Illegitimacy is just where both left and right are going these days when they lose elections. We don’t have a shared consensus on what the institutions of government should do, and that makes it harder for partisans to accept the outcome of elections.”

The attack on Trump’s right to govern comes at a vulnerable moment in American political history. His three immediate predecessors also were attacked as illegitimate occupants of 1600 Pennsylvania Ave. NW, a signal to political scientists and historians that the American consensus over how the nation is governed is suffering from significant stress.

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Trump seemed keenly sensitive to questions about his legitimacy even before “Not My President” signs and chants emerged at the Women’s March the day after his inauguration. Beginning in the transition, Trump was false asserting that he really won the popular vote in addition to the electoral college majority.

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Before Trump took office, Reps. John Lewis (D-Ga.) and Jerrold Nadler (D-N.Y.), who is poised to become chairman of the House Judiciary Committee, declared him an illegitimate president. New York’s attorney ­general-elect, Letitia James, campaigned this fall on her intention to “challenge this illegitimate president.”

But the illegitimacy argument has made little or no inroads into the president’s support.

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Trump’s base tends to reject the idea that the president was involved in illegal acts during the campaign. Even Republicans in Congress who are, as Sen. Lindsey O. Graham (S.C.) put it, “1,000 percent certain that the Russians interfered in our election” say the meddling didn’t change the result of the vote.

“Russia didn’t beat Clinton,” Graham tweeted. “Trump beat Clinton.”

“It is possible to conclude Russian interfered in our election in 2016 without delegitimizing his electoral success,” Rep. Trey Gowdy (R-S.C.) said earlier this year.

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For three decades now, presidents have faced not only the usual opposition from people who voted against them, but a more foundational challenge to their credibility as the nation’s chief executive.

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Bill Clinton’s presidency was labeled illegitimate by some conservative opponents because he had been accused by several women of improper sexual conduct, because his informal style sullied the office and because he won with only a plurality of the popular vote.

The historically close 2000 election, decided in the end by the U.S. Supreme Court, left George W. Bush facing allegations of illegitimacy from day one, with Lewis and some other members of Congress refusing to attend his inauguration because they didn’t accept Bush as the duly elected president.

And Barack Obama had to fend off sometimes inchoate cries of illegitimacy from a significant minority of voters who believed that he was not born where his birth certificate said he was but was actually a foreigner.

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The jump from opposing a president to declaring him illegitimate stems from a breakdown in trust, said Roderick Hart, a government professor at the University of Texas at Austin.

“We’ve shifted from institutional trust — the traditional idea that the institution of the presidency will survive despite any one candidate losing — to partisan trust, the idea that I don’t care who the president is as long as he marches in the direction I want,” Hart said.

In 2000, although many supporters of Democrat Al Gore believed the election had been stolen from him by courts partial to Bush, the Republican, “institutional trust won out, especially after Gore’s statements saying, ‘I’ve lost, I’m not happy about it, but the nation will move on and Bush is our president,’ ” Hart said.

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“With Obama, it took parts of the nation a while to get behind a black president, but the institutional trust held because he was a squeaky-clean guy,” Hart said. “But with Trump, partisan trust has taken precedence. Look at the evangelicals, who stay with him even though he’s a profligate, a person from the dark side.”

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Legally, Trump’s hold on office seems firm. The Founders designed the presidency to avoid disputes over who won. The electoral college gets to decide the winner, period. With rare exceptions, legal scholars agree there’s no mechanism for do-overs in case of election fraud. (Nonpresidential elections are a different story, as the current debate over whether to call a new vote in North Carolina’s disputed 9th Congressional District illustrates.)

The only remedies the Constitution provides if a president were to be fraudulently elected are impeachment or waiting for the next election, most scholars say.

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But that hasn’t stopped critics of the president from jumping to a challenge to his right to be chief executive.

“We’re in a very polarized period where the other side is not only wrong, but illegal,” said Rick Hasen, a professor at the University of California at Irvine’s law school and author of the Election Law Blog. But Hasen said many of those who allege that Trump broke the law are really raising larger objections to his disruptive manner and disregard for facts and traditions.

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“Trump is both a cause of and a reflection of a breakdown in norms, so it’s not surprising that you see attacks on his legitimacy,” he said.

Still, those who question Trump’s legitimacy go beyond whether he won by fraudulent means, often focusing on his basic fitness to hold the office — a blanket rejection of the president reminiscent of how some of Richard M. Nixon’s opponents regarded him from the start.

In 1968, when violent clashes between protesters and police turned the Democratic convention in Chicago bloody and when race riots seared American cities, the ensuing debate about the legitimacy of Nixon’s presidency wasn’t based on questions about campaign fraud. Rather, many Americans who opposed Nixon declared his presidency illegitimate because he led what they considered an illegal war in Vietnam.

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“The 1968 election was so pyrotechnic in so many ways — the nation was asking existential questions of who are we, what kind of people are we?” said Hart, the government professor. “In a time that was at least as tumultuous as today, people didn’t trust the institution of the presidency.”

Yet Nixon was reelected in a landslide in 1972. Nixon’s hold on power — and most importantly on his supporters — didn’t collapse until the release in 1974 of the smoking gun tape that proved that Nixon had directed the coverup of the burglary of the Democratic headquarters at the Watergate in Washington.

“An awful lot of stuff happened before the smoking gun, yet a lot of Republicans stayed with Nixon,” said Stephen Craig, a political scientist at the University of Florida. “The partisanship is so deep now that it may take multiple smoking guns to change Trump’s supporters’ minds.”

It’s also possible that evidence of crimes wouldn’t matter to many of the president’s supporters, affirming his statement in 2016 that “I could stand in the middle of Fifth Avenue and shoot somebody and I wouldn’t lose any voters.”