Mike Howell was looking at his phone in an airport lounge this month when he saw a letter from the attorney general of Missouri questioning whether President Joseph R. Biden Jr. had the “mental capacity” to sign the pardons and executive orders he issued during his final months in office.

Mr. Howell, executive director of the Oversight Project, a branch of the conservative Heritage Foundation, sensed an opportunity, he said in an interview. For months, he had been comparing Mr. Biden’s signature on dozens of official documents and noting that many of them appeared identical. Before boarding his flight, Mr. Howell made a conspiratorial post on X: “Whoever controlled the autopen controlled the presidency.”

While critics of Mr. Biden questioned his fitness for office, no evidence has emerged that he did not consent to any of the acts bearing his signature.

Yet by the time Mr. Howell landed, his post had gone viral. Within days, the notion that shadowy deep state operatives had been secretly running the country instead of Mr. Biden, using a mechanical contraption to achieve their diabolical aims, had erupted into a furor.

An autopen is a machine that uses a real pen to copy a person’s actual signature. Presidents and other politicians have used such devices for decades with little public interest. In the first two months of this year, the term was mentioned a total of 49 times on television, radio and podcasts in the United States, according to data from the media tracker Critical Mention.

It was uttered 6,188 times on March 17 alone.

Right-wing talk radio, podcasts and cable news shows have now devoted hundreds of segments to the arcana of wet signatures and autopen technology. They are particularly focused on Mr. Biden’s signatures on his pardons of political allies like Senator Adam Schiff, the California Democrat, as well as his son Hunter Biden. And they claim that the former president was mentally impaired and unaware of what documents he was being asked to endorse.

President Trump himself has brought the issue up repeatedly over the past week, decrying Mr. Biden’s use of the autopen. He claimed without providing evidence that Mr. Biden had not authorized the pardons, posting on Truth Social that they were “void, vacant, and of no further effect.”

Although Mr. Trump himself has acknowledged using an autopen at times, he raised the question again on Friday during an Oval Office news conference: “The person that operated the autopen, I think we ought to find out who that was because I guess that was the real president.”

Mr. Biden hasn’t commented on whether or not he personally signed every pardon, although Neera Tanden, a senior aide during his administration, posted online this week that “there’s a lot by autopen in every administration.” A spokeswoman for Mr. Biden did not respond to a request for comment.

There are no federal statutes that prohibit the use of the device, and two decades ago a Justice Department memo confirmed that presidents could “direct a subordinate to affix the president’s signature to” bills. A prior memo, from 1929, stated that a pardon “need not have the president’s autograph,” and legal experts have cast serious doubts on a president’s ability to rescind another president’s pardons.

The rapid transformation of a speculative and legally shaky hypothesis, concocted by pro-Trump activists into a talking point endorsed at the highest level of government, illustrates the extraordinary efficiency of today’s right-wing media environment.

“It does show how corrupted our information ecosystem is that something like this can gain attention,” said Welton Chang, co-founder and chief executive of Pyrra Technologies, a digital threat company that tracks trends on social media.

It is not exactly clear where the conspiracy theory took root, but Pyrra noticed a solitary posting on the 4chan message board referring to autopens and Mr. Biden back in October, even before he granted the pardons in question.

By then, Mr. Howell’s Oversight Project, started in 2022 by the Heritage Foundation with a mission of “increasing aggressive oversight of the Biden administration,” was already deep into its research.

Early last summer, Jason Chaffetz, a former congressman who is now a visiting fellow with the Oversight Project, floated the idea of gathering copies of presidential documents signed by Mr. Biden to see if the signatures matched up.

“It just was a suspicion that perhaps all of those were not authentic,” Mr. Chaffetz said in an interview. “This is the kind of thing we dive into.”

A dozen staff members began compiling documents from the Federal Register and requesting copies of resolutions and bills from Congress from the National Archives, Mr. Howell said. But after Mr. Biden dropped out of the race in late June, the project felt less urgent.

That changed when Andrew Bailey, the attorney general of Missouri, posted a three-page letter on X on March 5 calling for an “investigation into Mr. Biden’s mental capacity in his final days in office.” It suggested, without any direct evidence, that members of the former president’s staff exploited him and pushed through executive orders and pardons he wouldn’t have endorsed.

Mr. Howell said he didn’t know Mr. Bailey and had no warning the letter was coming. Still, it dovetailed so perfectly with the signature research he had already conducted that he couldn’t believe his luck.

“It was No. 8 on the to-do list,” Mr. Howell said. “Then A.G. Bailey drops his letter and it shoots to the top.”

Mr. Howell’s thread on X racked up more than three million views. Within hours, the subject was being widely discussed on conservative talk radio, which often mines social media for topics.

By the next day, it had leaped to popular podcasts such as the one hosted by Glenn Beck, who has 1.4 million subscribers on YouTube, and from there to cable news, with the Fox Business Network host Elizabeth MacDonald asking, “Should we figure out who controlled Biden’s autopen and who controlled Joe Biden’s presidency?”

With public interest growing, the Oversight Project pushed on. “We determined that the most legally vulnerable documents were the pardons,” said Mr. Howell, who published an analysis of signatures on five pardons late last week.

Those pardons, issued in the final full day of Mr. Biden’s term and designed to pre-empt potential prosecution, had infuriated Mr. Trump. The swirling questions about Mr. Biden’s signatures caught his attention.

“You don’t use autopen. No. 1, it is disrespectful to the office. No. 2, maybe it’s not even valid, because who is getting him to sign?” Mr. Trump said during a news conference at the Justice Department a day after the Oversight Project published its findings on the pardons.

Some critics, including the conservative legal scholar Jonathan Turley, have challenged the idea that the pardons could be nullified, noting that presidents are permitted to use autopen and that there has been no concrete evidence of a conspiracy to circumvent Mr. Biden’s will. “This dog will not hunt,” Mr. Turley wrote on X.

Mr. Howell said the only feasible way to test that question was in the courts, which would require the Justice Department’s charging someone who had received a pardon from Mr. Biden with a crime.

“It’s a rocket ship to the Supreme Court if that happens,” Mr. Howell said, adding that his work is far from over. This week, he published a 29-page legal memo on autopens. He also pledged to use the Freedom of Information Act to seek out more documents signed by Mr. Biden and is considering hiring forensic handwriting experts to review each one.

“We’re loading up the cannon for all sorts of things,” he said.