After flying back to Washington last week, seemingly giddy with success, the US leader tweeted, "There is no longer a Nuclear Threat from North Korea" -- a bold claim with Kim's arsenal still in place.

But senior US officials admit there is much work to be done as negotiators thrash out the details of what they hope will be Pyongyang's "complete, verifiable and irreversible" disarmament.

Many observers were disappointed that the short statement of intent signed by the two leaders did not more clearly define denuclearization, fearing Kim plans to keep his hard-won deterrent.

Secretary of State Mike Pompeo has angrily insisted that the term "complete" in the document includes the concept that denuclearization will be "verifiable and irreversible."

Trump has gone further, claiming on June 12 that the process will start quickly, then on June 21 confusing the issue by saying, "It will be a total denuclearization, which is already taking place."

Pompeo has been only marginally more cautious.

On June 13, he said, "We believe that Kim Jong Un understands the urgency ... that we must do this quickly," adding that he hopes for "major disarmament" within two and a half years.

So far, however, this confidence rests solely on trust in Kim's assurances to Trump at the summit, and on his reported promise to China to "implement the summit's consensus step by step."

Defending Kim's pledge, Pompeo said: "I was there when he said it. He made a personal commitment. He has his reputation on the line."

This drew a scathing retort from Thomas Wright, a senior fellow at the Brookings Institution in Washington.

"This is one of the most naive statements ever made by an American diplomat," he wrote on Twitter. "I hope he is trying to mislead us, because it would be truly frightening if he believed it."

Abraham Denmark of the Wilson Center in Washington warned a congressional hearing, "Considering North Korea's repeated history of violating past agreements, there is little reason to trust them this time."

And, despite his professed faith in Kim's word, Pompeo knows he has work to do to establish a roadmap toward concrete steps and that he will have to speak to Kim again, probably in Pyongyang.

Highlighting this apparent gap between Trump's rhetoric and the facts on the ground, the administration on Friday extended decade-old sanctions against Kim's regime, citing the "unusual and extraordinary threat" from North Korea's nuclear arsenal.

Yet, late Friday, Pompeo lavished praise on Kim in an interview with MSNBC.

"He's conversant in things Western, so he's paying close attention to what takes place... He's looking to determine if, in fact, America is serious about this," Pompeo said.

"So he's bright. He knows the file. He knows the topic very, very well."

- War exercises -

Pompeo acknowledged on Monday that "there's a lot of work between here and there. My team is already doing it. I'll likely travel back before too terribly long." He set no travel date.

Pompeo's spokeswoman Heather Nauert has said she will not provide details of the negotiations, warning reporters the process "can be difficult when we're distracted by all of the questions going on."

But she denied that the talks were at a dead halt, even if the detailed negotiations have yet to begin.

"We have been in communication with the government of North Korea," she said on Thursday.

"Secretary Pompeo will be meeting with them and talking with them at the earliest possible date to try to implement the outcomes" of the summit.

While Pompeo pushes on with follow-up talks, Trump appears happy to celebrate the summit as a success, as if the joint statement itself marked a beginning to the end of Kim's nuclear program.

He claims he has already received "good news" and on Thursday declared that the North has "already blown up one of their big test sites. In fact, it was actually four of their big test sites."

But the North Korean test site at Punggye-ri was demolished in late May, weeks before the summit, and there has been no apparent disarmament activity since then, despite Trump's boast.

Defense Secretary Jim Mattis, who has been ordered by Trump to unilaterally halt "provocative" US war exercises in South Korea, was cautious when asked whether the North had indeed begun to disarm.

"No, I'm not aware of that. I mean, obviously, it's the very front end of a process. The detailed negotiations have not begun. I wouldn't expect that at this point," he told reporters.