ORLANDO, Fla. — When 25 Republican governors and five Republican governors-elect showed up for their association’s annual meeting last week, it should have been one of the proudest moments in the party’s history.

After public belittlement by some in his own party, Donald Trump pulled off a decisive win. Instead of the smoldering ruins that many predicted, his election left the party in control of the Senate and the House and with a net gain of two governorships, to a total of 33, the most governorships in a quarter-century. By contrast, reeling Democrats now will control the governorships and the statehouses in only five states.

Republican National Committee Chairman Reince Priebus, Mr. Trump’s designated White House chief of staff, could be excused for saying, “But wait, there’s more.”

Republicans now hold more than 4,000 of the 7,383 legislative seats available, the most since 1920, according to the National Conference of State Legislatures, and Republicans totally control both the governorship and legislature in 55 states.

But the overwhelming victory also means the Republican Party no longer has any excuse for failing to carry out an agenda built on the 10th Amendment: self-reliance, free markets, low taxes, disciplined spending, serious enforcement of immigration laws, freedom of choice in education, fair trade and living up to Mr. Trump’s opposition to Iraq-type foreign adventurism.

Govs. Rick Scott of Florida, Asa Hutchinson of Arkansas, Susana Martinez of New Mexico, Scott Walker of Wisconsin, Nikki Haley of South Carolina — tapped Wednesday by Mr. Trump to be U.S. ambassador to the United Nations — and others attending the Republican Governors Association meetings proudly rattled off their party’s superior numbers, but the air was thick with concern about the need for clarifications from Mr. Trump, including on his grand commitment to making America’s roads, bridges and airports great again.

Specifically, can Mr. Trump pull that off without smothering states with crushing financial burdens? Mr. Hutchinson insisted that the mood of his fellow governors was basically optimistic, whether it’s ending Obamacare and reforming health care or giving states a bigger say in running the government.

“Yes, there was a recognition that it’s a new day. You talk about education, giving states more control over education, not having Washington-driven responses,” said Mr. Hutchinson, a former congressman, a former head of the U.S. Drug Enforcement Administration and the first undersecretary for border and transportation security at the U.S. Department of Homeland Security. “Sure, there’s unpredictability with Donald Trump, but there’s a lot of predictability too and confidence that the states are going to have a larger role to play, and that we’re going to get a lot of the things we wanted.”

Mr. Trump is big on infrastructure, but the governors wonder how thoroughly he understands the crunch federal programs can put on states and their taxpayers.

“What we’re not going to do is a big government program,” Florida’s Mr. Scott said. “We don’t trust the federal government to fulfill their part of the bargain.”

Mr. Hutchinson said “amen” to concentrating on infrastructure so long as there is clarification. “We can’t just add to the deficit,” he said. “That’s going to be a challenge for Congress.”

Every attempt to finance roads, bridges and airports creates its own problem. Raising the federal gas tax erodes a state’s tax base. States have their own gas taxes.

State and local governments own almost all airports in the U.S. To the extent federal taxpayer money goes to airports, the decrepit state of which Mr. Trump has bemoaned, it is through federal taxes and fees on airline passengers and not from personal and business income taxes. By contrast, federal personal and business income taxes and user taxes, such as the federal gas tax, combine to pay for roads and bridges.

“There’s not any clarity so far in what Mr. Trump wants for infrastructure improvement and how to pay for it,” said Mr. Hutchinson.

All roads for the next president will lead through the question of the national debt, the Republican governors agreed.

Entitlement programs are the one behemoth in the general revenue locker where cutting back can make a huge difference in the size of the federal debt. Benefits are written into Social Security, Medicare and Medicaid, and Veterans Affairs programs, federal civilian and military retirement, unemployment compensation, food stamps and agricultural price supports.

“Obviously, it is entitlement reform that has to address the long-term debt. And I don’t know whether [Mr. Trump] wants to tackle that or not,” Mr. Hutchinson said.

For him, entitlement reform leads to House Speaker Paul D. Ryan, Wisconsin Republican and a critic of Mr. Trump before his stunning win Nov. 8.

“Ryan is the only political leader out here right now that has espoused entitlements reform, at great political risk to himself,” said Mr. Hutchinson.

“Whether that becomes part of the Trump agenda, I don’t know.”

On immigration, Mr. Hutchinson said the president-elect’s words can be misinterpreted. He has said Mr. Trump’s first order of business will be to jail or deport criminal aliens. By that, he means foreigners who overstayed visas or crossed the border illegally, then committed murder or other felonies. He apparently will not initially pursue illegal but otherwise law-abiding immigrants.

Mr. Hutchinson acknowledged that some problems are insoluble and need to be tacitly ignored, at least for the moment. Legalizing illegal immigrants already in the U.S., for example, invites more to come. Deporting more than 11 million illegals who are otherwise law-abiding and productive invites a generation or more of distrust of a political party.

“When you secure the border first, there’s flexibility as to what next step should be,” said Mr. Hutchinson.

Once thought a sure way to block illegals, the federal web-based E-Verify system was supposed to allow “businesses to determine the eligibility of their employees to work in the United States.” It doesn’t really.

“We’ve got to talk about E-Verify,” Mr. Hutchinson said. “People come here for jobs. Reducing the power of that magnet reduces the cost of securing the border.”

E-Verify now is riddled with holes. Even the official Department of Homeland Security website bears this proviso: “E-Verify does not provide your employer with any immigration, citizenship status or document information about you.”

“A more effective system is more expensive,” Mr. Hutchinson said. “You have to check more databases to authenticate the identity of the person who presents a card. A truly comprehensive E-Verify burdens small businesses financially, but if you exempt businesses that employ fewer than 50 people, you leave the door open to fraud.”

This is yet another challenge Republican governors see for the incoming president.

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