Thus began the great pizza seduction. One after another, recalcitrant Republicans were marched into Mr. Boehner’s office, where he begged, implored and, when that failed, berated them in a desperate effort to win support for his proposal to resolve the debt crisis…

The freshmen, many of them associated with the Tea Party before the Republican party, appeared to be putting up the highest walls, though some more senior members were also invited to see Mr. McCarthy and his plates of pizza. “It is the most refreshing thing in the world to see what is going on in there,” said Representative Jeff Flake of Arizona, an ardent foe of earmarks, the pet projects of lawmakers. “This kind of negotiation a few years ago would have cost $20 billion” in pork barrel projects. Mr. Flake would not say whether the quality of the pizza had moved his vote, as he continued to ponder his final move.

Members said they agonized. “I am trying to find a way to vote for this bill,” said Representative Bill Huizenga of Michigan. “This is a step in the right direction. I wish it were a bigger step.” (At the last minute, Mr. Huizenga, who brought his 12-year-old son Garrett to the Capitol for inspiration, assured leaders that they had his vote.) Some men from the South Carolina freshman delegation, a tight-knit group of conservatives, repaired to the House chapel on Thursday to pray. “I am a no at the end of the day,” Mr. Scott said. “I was leaning no. Now I am a no.”