By pushing ahead with short-range missile tests, though, Mr. Kim has managed to sow divisions between the United States and its main allies in Asia.

That was evident recently when Mr. Trump met with Prime Minister Shinzo Abe of Japan and once again dismissed the importance of the launches. Mr. Abe, usually an ardent supporter of Mr. Trump, immediately contradicted him, telling reporters that all of North Korea’s tests have violated United Nations Security Council resolutions.

Although Mr. Trump acknowledged that “there may be a United Nations violation,” he appears unwilling to make statements or take actions that could call into question what he considers his warm relationship with Mr. Kim, which the president believes is crucial to ending decades-long hostilities with the North.

For his part, Mr. Kim has not been shy about touting the latest missile advances, attending many of the tests and releasing photographs of post-launch celebrations with his missile brain trust, a group of officers who have steadily continued weapons development while diplomacy has sputtered.

An estimate by the Defense Intelligence Agency, circulated to select officials inside the United States government this summer, estimates that the North has also produced enough fuel for roughly a dozen new nuclear weapons since the historic talks in Singapore. Other intelligence agencies have “more conservative numbers,” one former senior official said, but all point to an expanding cache of bombs.

Defense Secretary Mark Esper told reporters Wednesday that “we are concerned” about North Korea’s missile tests, but that “we’re not going to overreact” in the interest of preserving efforts at a diplomatic solution.

Even so, Secretary of State Mike Pompeo, who has taken the lead in the effort to rid North Korea of its nuclear weapons and missiles, has hinted to foreign counterparts in private meetings in recent weeks that he is fearful the administration is being strung along, according to Japanese and South Korean officials.

In public, Mr. Pompeo is more circumspect. He recently told CBS, “We haven’t gotten back to the table as quickly as we would have hoped, but we’ve been pretty clear all along we knew there would be bumps along the way.”

The bumps, though, have gotten much larger in recent months as the North began lobbing missile after missile aloft, testing at least three new varieties. The tests have showcased the North’s ability to consistently launch solid-fuel missiles that are easier to hide in the mountains, and which could be quickly deployed on mobile launchers and fired before the United States could respond.

Some are also part of a new generation that can fly low and maneuver in unpredictable paths, likely defeating the standard missile defenses the United States has installed in the region. In at least one case, Mr. Kim appears to have built upon a well-tested Russian design — though it is unclear how he obtained it.

Even with the barrage of tests, Mr. Kim has kept up his personal praise for Mr. Trump. Just last month the president told reporters he had received another “really beautiful letter” from the North Korean leader. Mr. Trump referred to the most recent letter again last week, concluding that he was sure Mr. Kim would move toward full denuclearization.

“Kim understands what he is doing,” said Mr. Narang, the M.I.T. professor. “He likely thinks Trump can live with a nuclear North Korea as long as they don’t test ICBMs and nuclear weapons, which would humiliate Trump and take away the perception he has achieved of some kind of victory” with his diplomacy.

Administration officials dispute that view and say the tests are a symptom of a North desperately trying to end “maximum pressure,” which is how Mr. Pompeo describes the sanctions the United States has imposed on the country. And there is some evidence for his position: North Korea’s official media recently described the secretary of state as a “die-hard toxin.”

If there is any lesson to be taken from the latest North Korean provocations, it is that Mr. Kim clearly has a maximum pressure strategy of his own.

In 2015 and 2016 the country’s missile tests were failing at an astoundingly high rate. Some intelligence officials and outside experts tied at least some of those setbacks to a covert Pentagon effort to sabotage the launches with cyber and electronic strikes.

But by 2017, Mr. Trump’s first year in office, Mr. Kim had adopted new designs and conducted 14 successful launches, mostly of medium- and long-range missiles, for the first time raising the specter that he might be able to reach all of the United States. At the same time, he conducted a series of nuclear tests, including of its largest weapon yet, which he described as a hydrogen bomb.

Those successes prompted Mr. Trump to first warn of “fire and fury like the world has never seen,” then to begin the diplomatic dance toward the Singapore summit last year, the first time a sitting American president had met any member of the Kim family. After the meeting, Mr. Trump said that North Korea was on the road to disarmament, telling reporters he expected they would see major results within six months.

For almost a year afterward, Mr. Kim appeared to refrain from firing missiles, perhaps in the hope that the Trump administration would ease sanctions.

Then the diplomacy stalled at the second summit, in Hanoi. Mr. Kim offered to close the North’s major nuclear complex at Yongbyon — but not other nuclear facilities or any missile sites — in return for the lifting of the most severe sanctions. Mr. Trump rejected that offer at the insistence of Mr. Pompeo and John R. Bolton, his hawkish national security adviser.

Mr. Kim now appears to be reverting to the tactic that worked for him two years ago: escalating tests to demonstrate new capabilities.

“Their stock-in-trade has always been to press the edge of the envelope and see what they can get away with,” said Vann H. Van Diepen, a top official on weapons of mass destruction in the Office of the Director of National Intelligence until 2009.

The recent bout began in early May when the North fired a 30-foot-tall missile from an eight-wheeled truck. Analysts said it was a solid-fuel rocket that can be launched in minutes — unlike liquid-fueled missiles, which can take hours of preparation.

In all, the North conducted eight flight tests of the new missile in May, July and August. Its maximum range was found to be roughly 430 miles, according to the Center for Nonproliferation Studies, a private group in Monterey, Calif. The missile, if perfected, could target all of South Korea, including at least six American bases, and parts of southern Japan, including two large bases.

A number of Western experts said the new missile closely resembled a Russian Iskander missile, which can fly on low trajectories through the atmosphere and then dive on targets to avoid interception.

COURTESY THE NEW YORK TIMES.