Perhaps just as important, conservatives — particularly in the G.O.P. commentariat — do not see Kasich as one of them. No one questioned his fiscal austerity back in 1995, when Newt Gingrich, the speaker of the House, named him chairman of the House Budget Committee. But that was a generation ago. As Ben Domenech, publisher of the conservative web publication The Federalist, observes: “Kasich had the benefit of being in Congress at a unique moment, when the economic boom during the Clinton years allowed him to do what he did. It’s not at all relatable to where we are now.” As governor, Kasich expanded Medicaid benefits in his state, against the wishes of a Republican-controlled Legislature. He also embraced Common Core educational standards and today favors a guest-worker program for illegal immigrants. All of these constitute apostasies to movement conservatives.

But there’s a third layer of resistance to Kasich, one with which Cruz can identify: Many Beltway Republicans don’t like him. I asked the candidate about this a few months ago. “Look, when you balance budgets, you aggravate a lot of people,” he told me. “When you take on tough issues, you’re going to make enemies. And they just wait in the shadows to take a whack at you. But over time, I find when I go back to Washington, there’s respect for what I did as a leader.”

No doubt Chairman Kasich’s strict budgets made life unpleasant for a few lobbyists. But so did his demeanor — which, D.C. veterans say, was often sanctimonious and rude. “John had a reputation,” says Jim Dyer, the former Republican staff director of the Appropriations Committee who counts himself as an admirer, even though the two often tussled on budgetary matters. “When he got upset, he’d really blow up.” Former Representative Dave Hobson, one of Kasich’s closest friends in the Capitol, recalls: “He’d get really upset with you. Then, two or three days later, he’d say, ‘Goddamn it, Dave, you were right about that.’ ” That Kasich is widely regarded as an accomplished legislator distinguishes him from his two remaining rivals. And several of his former congressional colleagues — like Trent Lott, Vin Weber, Ray LaHood and Pete Hoekstra — have endorsed his candidacy. Still, his past behavior has made the pursuit for allies at the Republican convention in July a more arduous one.

It’s unclear whether Kasich has mellowed much since those days. Shortly after being elected governor in 2010, he warned lobbyists that “if you’re not on the bus, we will run over you with the bus.” He was forced to apologize for publicly describing a policeman who had given him a traffic citation as “an idiot.” State Republicans have not forgotten how, in 2013, he lectured the Ohio House speaker Bill Batchelder — a Republican legislator who had served over three decades in the House — about his moral duty to support Kasich’s Medicaid expansion plan: “When you die and get to the meeting with St. Peter, he’s probably not going to ask you much about what you did about keeping government small. But he is going to ask you what you did for the poor. Better have a good answer.”

Kasich’s continual moralizing on the subject lost him the support of a key donor, Thomas Patrick, the chief executive of New Vernon Capital, in 2014. Last June, Kasich was touting his record during a speech at a Republican event in Chicago when he happened to see Patrick in the audience. In typically pugnacious fashion, the governor spoke to Patrick from the lectern: “And so, Tom, what do you think about all of this? Are you still not supporting me?” (Patrick later sent Kasich a donation of $2,700 — though he has donated far more, $258,100, to Cruz.)