“Our original vision was the alt-right would become like a policy group for the Trump administration,” he explained, and the administration figure “who was the interface was Steve Bannon.” Unfortunately, he told Mr. Hermansson, the political establishment was “disconnecting us from the Trump administration, almost completely.” (In June, Mr. Bannon hadn’t yet left the White House and returned to Breitbart, the popular and ardently pro-Trump far-right outlet he had led before his time working for Mr. Trump.)

When I called Mr. Jorjani, he was cagier about his “connections” and “contacts” in the White House. All he meant, he said, was that he had been in touch with people who had a direct line to President Trump, though he wouldn’t say who. Asked to comment, a White House spokeswoman said, “We have no knowledge of any conversations or contact with this person.”

Either way, Mr. Jorjani said, with the ousters of Michael Flynn in February and then Mr. Bannon in August, he now views the alt-right’s efforts to carve out a place in the White House as having failed. (Mr. Jorjani resigned from the AltRight Corporation in August.)

If Mr. Jorjani wasn’t exaggerating to Mr. Hermansson, and he did have a relationship with White House officials, that would certainly be alarming. But even if he was exaggerating, it’s still important to understand how messages like his could travel from the far reaches of the right-wing internet and all the way into — or close to, at least — the White House.

The extreme alt-right are benefiting immensely from the energy being produced by a more moderate — but still far-right — faction known as the “alt-light.”

The alt-light promotes a slightly softer set of messages. Its figures — such as Milo Yiannopoulos, Paul Joseph Watson and Mike Cernovich — generally frame their work as part of an effort to defend “the West” or “Western culture” against supposed left-liberal dominance, rather than making explicitly racist appeals. Many of them, in fact, have renounced explicit racism and anti-Semitism, though they will creep up to the line of explicitly racist speech, especially when Islam and immigration are concerned.

This apparent moderation partly explains why they tend to have much bigger online audiences than even the most important alt-right figures — and why Hope Not Hate describes them as “less extreme, more dangerous.” Alt-light sites like Breitbart, formerly home to Mr. Yiannopoulos, as well as Prison Planet, where Mr. Watson is editor at large, draw millions of readers and are key nodes in a hyperkinetic network that is endlessly broadcasting viral-friendly far-right news, rumors and incitement.