The operation of the senator from Texas thinks that it has come down to fighting over exactly 9,131 voters who are trying to decide between Cruz and Trump: 3,185 who are torn between him and retired neurosurgeon Ben Carson; and a smaller group — 2,807, to be precise — who have not yet made their choice between him and Rubio. Cruz also must try to peel off supporters of other religious conservatives in the race, including former Arkansas governor Mike Huckabee and former senator Rick Santorum of Pennsylvania.

“It’s very primal, frankly,” Roe said, when asked how he could be so confident of that data. “It’s not like kernels in the jar at the state fair. These are people who have told us this. . . . The numbers aren’t wrong, because we’ve been testing them.”

Roe also predicted that turnout will exceed the record for a Republican caucus day, which is 122,000 in Iowa. What he doesn’t know, Roe said, is by how much. He conceded that the higher it goes, the better it will bode for Trump, who is counting on a surge of voters who have not traditionally made the effort to attend the caucuses, an exercise that requires devoting most of a frigid weeknight.