No one seems clear on what those “concrete steps” are. At one point in Seoul last month, Mr. Tillerson suggested the North must first completely disarm, though lately he has hedged that wording. But whatever the preconditions, Mr. Trump missed an opportunity to reinforce that effort to reassure the North Koreans that the United States is not looking to topple their leader. Instead, his message could be taken as the opposite.

Mr. Trump’s negotiating strategy has often involved the taking of an extreme position, hoping that the other actor in a test of wills will be thrown off enough to move in his direction. That is one thing when it means threatening to pull out of Nafta, the gambit Mr. Trump floated, then retreated from, this week. But it can be a far riskier bet when exchanging signals with Mr. Kim, who has survived so far — like his father and grandfather before him — by employing a similar playbook of extreme rhetoric, often followed by acts of violence.

So far, Mr. Trump has directed one operation to bolster his claim that he is perfectly willing to use force in an unpredictable manner: his decision a month ago to conduct an intensive, brief attack on a Syrian air base where American intelligence agencies say the Syrian government launched a chemical weapons attack on its own people. It had no follow-up.

But for North Korea, lashing out to send a message is an art form, practiced since the days when Mr. Kim’s grandfather ordered the seizure of an American ship, the Pueblo, in 1968, followed by the shooting down of an American reconnaissance plane, killing 31. Then, seven years ago, came the sinking of a South Korean naval vessel — most likely by a North Korean torpedo, though the country denies it — that took 46 lives.

The young Mr. Kim, who took over the following year after his father’s death, has worked to burnish his own madman credentials. He is believed to have ordered the cyberattack on Sony Pictures Entertainment that wiped out the company’s computer systems in 2014 and the killing of his half brother in Malaysia this year, part of a sustained campaign to eliminate potential rivals. More than a few have been executed with antiaircraft guns, just to make a point.

The fear is that small acts and mutual threats of war can lead to miscalculation. Only hours before Mr. Trump spoke, the North released a propaganda video showing the White House shattering apart in what looked like a nuclear blast. No one takes those videos seriously, but they indicate a state of mind in which every action has to have a reaction.

“That’s what I worry about the most,” Senator Edward J. Markey, Democrat of Massachusetts, said recently. “Rapid escalation.”