So, the House Republicans have shut down the government. Now what? What is their end goal? What is the principle at play? What do they hope to gain?Polls show that Americans are overwhelmingly against the shutdown and are more likely to blame the Republicans than the president. Those numbers are likely to tilt even further away from the G.O.P. the longer this thing drags on.

Petulant Republicans, smarting from being regularly outsmarted by this president, have taken aim at the legislative achievement that they derisively branded with his name (a branding the president has since adopted, flipped and reclaimed) as their last stand against the image of his success and the writing of his legacy.

They are against him on many levels. Some of their opposition seems to be cerebral, relating to the constitutionality of the law. But the Supreme Court already settled that question. It affirmed the law. Of course, there’s also a more visceral, gut-level repudiation of anything associated with Obama. But the voters settled that, in 2012. They affirmed him.

And long-range G.O.P. statisticians no doubt worry about something worse: that the Republicans have oversold their opposition to the law, that it won’t fail and kill jobs, but will succeed and keep thankful voters alive and healthy. In this scenario, in a year, or five, or 10, or in a generation, voters will come to like and depend on the law and associate its success with the Democratic Party and opposition to it with the Republican Party. That doesn’t bode well for a G.O.P. that is already turning off voting blocs — immigrants, minorities, women, young people, gays — and struggling to find a way to win back the presidency.

The Republicans’ stated goal going into the shutdown — to either defund, delay or otherwise destroy the president’s health care law — stands no chance as long as this president is in office and Democrats control the Senate.

To be fair, this stunt of stupidity is not the brainchild of all or even most Republicans in the House, but of a small and fractious faction. They are hard-line Republicans from mostly safe districts. They came to Washington to wreck Washington. They’re not interested in governance because they despise government.

But not enough of their comrades under the Republican flag have demonstrated the courage to confront and contain them. So, there is plenty of blame to go around.

What now? House Speaker John Boehner has allowed a handful of hotheads to push him and his caucus out onto a ledge where there is little room to move, other than to jump or turn back into the flame.

Boehner must fold. The only questions are when and how.

The way he can save face, and maybe save his speakership, is to make acquiescence look like triumph. But the chances of that are small.

The president insists that he will not negotiate over this. His message to the House Republicans is simple: You broke it; you fix it.

On Wednesday, Obama summoned Congressional leaders to the White House to discuss the shutdown. But it wasn’t clear whether the president was abandoning his stated position of not negotiating, or simply managing his image through stagecraft.

This is not sitting well with many Republicans, who chafe at the notion that the president would leave them alone to wallow in the mess they made. Now they claim that the president is not being a leader because he won’t engage representatives who have no interest in being led.

House Republicans linked the president’s health care law to the continuing resolution, and they can decouple it, on their own. The idea that they would create a crisis and then blame the president for not giving them cover to crawl out of it is ridiculous.

The other option that might allow the House Republicans to save face is to let the government shutdown bleed into the debt-ceiling debate, and claim that any concession on raising the debt limit is also a concession for approving a continuing resolution. This would just be rhetorical sleight of hand, however, and there’s no way of knowing if the Tea Party Republicans in the House would accept it as going far enough, particularly if the concessions didn’t specifically involve Obamacare.

Obamacare is the bright red line in House Republicans’ dim world. They can’t countenance the idea that they are out of power and out of options.

These people are exploiting the regular rules of Congress to flout the normal rules of order, turning majority rules into minority rules.