Even one of Mr. Trump’s frequent defenders, Senator Lindsey Graham of South Carolina, indicated on Sunday he was worried that the president might have been manipulated.

“Are they playing us? I don’t know,” Mr. Graham said on CBS’s “Face the Nation.” “If they’re playing Trump, we’re going to be in a world of hurt, because he’s going to have no options left. This is the last, best chance for peace right here.”

The White House argues that significant progress has been made. Mr. Trump’s press secretary, Sarah Huckabee Sanders, has cited the fact that Mr. Kim’s last missile and nuclear tests were 10 months ago, and insisted that is a sign of Mr. Kim’s willingness to deal.

It certainly is a constraint on his program: As long as the North conducts no tests, it cannot demonstrate that it has designed a warhead that can survive the huge stresses it would undergo in flight. That leaves ambiguity about whether it can actually strike American cities.

Still, nuclear production continues unabated, satellite photographs and other evidence suggest. Secretary of State Mike Pompeo has not persuaded the North Koreans to turn over an inventory of their major nuclear facilities and materials, much less declare how many weapons they possess. While Mr. Kim has blown up entrances to a nuclear test site and appeared to start dismantling a test stand for missile engines, he has not allowed in any inspectors to determine whether the actions were simply for show.

Mr. Kim has said a peace “declaration” that formally ends the Korean War must be a first step, and Mr. Moon has privately urged the United States to provide that assurance. The North Korean leader believes that Mr. Trump committed to such a declaration on the way to a more formal peace treaty. But both Mr. Pompeo and John R. Bolton, the national security adviser, have said progress toward denuclearization must come first.

Mr. Kim’s strategy now appears to be simple: Mimic Pakistan, which conducted a major nuclear test in 1998 and deflected demands for years that it give up its weapons. Pakistan has largely succeeded. It has a substantial arsenal, and when Mr. Pompeo visited Islamabad recently, there was little public discussion of Pakistan’s nuclear arsenal.