In an interview with David Muir of ABC News broadcast on Wednesday night, Trump tried to suggest that a 2012 Pew study on problems with people being registered in two states, or the voter rolls not being updated as soon as people die, was proof that illegal voting was taking place.

As a result of this fixation, he is now promising “a major investigation” into the election that made him president, putting the full weight of the federal government behind his quest to prove that at least three million ballots were cast against him by “those registered to vote in two states , those who are illegal and even, those registered to vote who are dead.”

For two months now, Donald Trump has appeared unable to accept the verdict of November’s election: that he is more popular than many of us wanted to believe, but less popular than Hillary Clinton .

When Muir pointed out that the author of the Pew study, David Becker, had said that his work did not show any voter fraud, Trump, who clearly had not read the study, suggested, wrongly, that he had somehow retracted his research. Specifically, Trump accused Becker of “groveling,” just as he had when attacking Serge Kovaleski of The New York Times for undercutting his lie that thousands of Arab-Americans celebrated 9/11 in New Jersey.

As soon as that paper was published, however, its conclusions were rejected by several experts, including the researchers who directed the internet survey, the Cooperative Congressional Election Study. Professor Stephen Ansolabehere of Harvard University, the principal investigator of the C.C.E.S., argued that those five participants, a tiny fraction of the 23,800 people who completed the survey, could easily have been citizens who clicked the wrong button.

Those five clicks are the main data point in a study published in 2014 by Jesse Richman and David Earnest, political scientists at Old Dominion University who argued that voting by non-citizens is common enough that it could sway close elections.

It is hard to overstate just how thin the evidence for the claim of mass illegal voting is, but here’s a clue: the only study cited by the White House that does talk about voter fraud hinges on the fact that five people who cast votes in the 2008 election also clicked a button on an internet survey that year indicating that they were not American citizens.

“The core of the problem with the study is that every survey question has some measurement error in it,” Ansolabehere said in an interview on Wednesday. “So, if you’re taking an internet survey, one kind of error is click-through error – some people are just taking the survey very fast accidentally click the wrong button. There’s some percentage error in every question, and we’re aware of that and that affects inferences you draw.”

In a peer-reviewed response to Richman and Earnest’s work, based on follow-up interviews with participants in the study in 2010 and 2012, Ansolabehere and his colleagues concluded that the error rate with the citizenship question was high enough to suggest that the five people who said they were non-citizens in 2008 were citizens who had simply clicked the wrong box.

“When we look at the panel where we reinterviewed those people, we find that none of those five people were non-citizens,” Ansolabehere added. “It was just click-through error on the survey.”

Richman and Earnest, who assumed that the five clicks were an admission that five non-citizens had voted, have defended their interpretation, and been lauded by writers like Julia Hahn of Breitbart News (who recently joined the White House staff) for suggesting that the rate of voting by non-citizens could be even higher than the 1.5 percent indicated by their reading of the Cooperative Congressional Election Study data. (It is important to note that none of C.C.E.S. respondents identified themselves as undocumented, so even if some small number of votes were cast by non-citizens, they are far more likely to have come from confused green-card holders with a legal right to reside in the U.S. than immigrants who entered the country illegally.)

“What they’re doing,” Ansolabehere said of Richman and Earnest, “is they’re saying there are too few non-citizens in the sample, so we’re going to inflate the non-citizen rate, so they take the five and they inflate it.”

“It’s a bad estimate, first of all, its a noisy estimate, there are no standard errors in anything they present,” he said, “and it turns out a one percent error rate is enough to completely eradicate the result.”

“There’s a kind of bigger lesson here,” Ansolabehere added, “which is, we’re in this world of big data and people are doing a lot of analysis with big data and drawing inferences from really small sets of people about larger groups of people.”

“An example is, one time I was purchasing some music on Amazon, and — this is just the caution of thin data — one of the ads then said, ‘People who like this music,’ it was a piece of Brahms, ‘also like Britney Spears.’ And it’s the same problem, it’s that you’re drawing inferences from very thin data, so if one person does one idiosyncratic thing, you’re now making an inference about millions, and it’s probably the wrong inference.”

For their part, Richman and Earnest have continued to insist that it is plausible that large numbers of non-citizens did take part in the 2016 presidential election — though not, they wrote recently, at anything like the scale suggested by Trump.

They also noted something that none of the glowing coverage of their work in the right-wing press acknowledges: that if non-citizens did cast votes, there is no way of knowing how many of them were cast for Clinton and how many for Trump — or where. Richman and Earnest found that 1 of 5 votes cast in 2008 by voters they suspect were non-citizens went to the Republican candidate. So, if there were, as Trump has claimed, 5 million illegal votes, it stands to reason that about 1 million of them were cast for him.

Given that the electoral college was decided by less than 78,000 votes spread across three states — and Trump won a fourth, Florida, which is home to 1.9 million non-citizens, including the German golfer Bernhard Langer, by just 113,000 votes — if his wild claim was somehow proven to be correct, it would be impossible to be sure that his victory was not secured by illegal votes.