Iowa is as good as it gets for a candidate like Mr. Cruz — “very conservative” voters represent 47 percent of the G.O.P. electorate there, according to exit polls in 2012. Self-described moderate and liberal voters represented just 17 percent of that electorate.

The electorate is so conservative because the delegate selection process begins with caucuses, which draw the most engaged, activist and conservative voters. In primary states, the electorate is very different. Across the Mississippi River in Wisconsin, moderate voters actually outnumber “very conservative” voters, 39 percent to 32 percent. There is not a single primary state where “very conservative” voters represent as large a share of the electorate as they do in Iowa, according to exit and entrance polls. Mr. Cruz’s home state, Texas, despite its reputation, is no exception: “Very conservative” voters outnumbered moderate ones by just four percentage points, 32 to 28, in 2008.

With his coalition, Mr. Cruz would draw little or no support from around one-third of the national electorate. He would still be well positioned to win the caucus states on the Plains and farther west, but he wouldn’t necessarily find it easy to win even relatively favorable primaries in Southern states, like Georgia and Alabama, where “very conservative” voters usually outnumber “moderates,” but only by a little. He could be routed in the primaries of New England, the Midwest and along the Pacific Coast by a candidate with strong appeal among moderates.

This is not to say that Mr. Cruz can’t win with something resembling his current coalition. But it does frame what would be a long and narrow path to the nomination.