ST. PAUL — Minnesota’s governor and top Republican lawmakers broke from a sixth straight day of budget talks Wednesday without solving an impasse over the $5 billion deficit, hours after a judge issued a bare-bones list of services that must be maintained if state government shuts down.

“We don’t have a deal,” Deputy Senate Republican Leader Geoff Michel, R-Edina, said shortly after leaving Gov. Mark Dayton’s office at about 9:30 p.m.

Both Republican leaders and a spokeswoman for Dayton said talks would continue Thursday, but the Democratic governor himself had said a day earlier that securing a deal Wednesday was key to avoiding the shutdown.

Thursday is the final day of the state’s current two-year budget cycle, and without a new one in place, the shutdown would start at 12:01 a.m. Friday.

A shutdown wouldn’t affect critical functions such as the State Patrol, prison guards, disaster responses and other essentials. But many services that Minnesota residents take for granted would be closed — including state park campgrounds on one of the summer’s biggest holiday weekends, road projects at the height of the construction season and licensing for an array of new professionals from physicians to manicurists.

“This is going to be a tough shutdown, and people will notice,” said David Lillehaug, an attorney representing Dayton in the case. “Anyone who says that government doesn’t do anything and doesn’t do it well, upon reading this order … they’re going to realize they’re very, very wrong.”

Dayton, a Democrat, and top Republicans stopped talking publicly about negotiations days ago, saying they didn’t want to jeopardize any progress. On Wednesday, legislative leaders swept in and out of Dayton’s office, saying little to a pack of reporters and photographers outside.

After breaking for the last time, Michel said the two sides were “very close” but he provided no further details. Dayton did not appear before the press Wednesday.

The budget dispute dates to January, when Dayton became the state’s first Democratic governor in 20 years and Republicans took over the Legislature for the first time in 38 years.

Republicans swept to power campaigning against tax and spending increases, while Dayton won on a message of raising taxes on the highest earners. A five-month legislative session and intermittent negotiations since adjournment last month have produced no visible progress.

With the politicians mostly silent, the most significant development Wednesday came from Ramsey County Chief Judge Kathleen Gearin. She weighed arguments from several parties on which services should be maintained in a shutdown, and ultimately sided mostly with Dayton’s minimalist list.

Gearin found that some programs, such as horse racing and child care aid programs not directly tied into the federal welfare system, were important but did not rise to the level of critical services.

The judge said state payments to cities, counties and schools would continue, as would enough money to keep Dayton’s office and the Legislature running with at least skeletal crews. She ordered that the state keep welfare, food stamp and Medicaid health care programs operating.

Republican leaders either declined to comment on the ruling or said they hadn’t seen it. Dayton praised it and said he still hopes for a budget deal.

“I would much prefer a fair and balanced budget solution, rather than a government shutdown,” he said in a statement.

Gearin named former state Supreme Court Chief Justice Kathleen Blatz as a referee to rule on unclear areas of funding in a shutdown.

Republicans have pressed him to call them into special session to pass some parts of the budget and measures to prevent a shutdown. Dayton has ruled out a special session until he and the Republicans agree on the overall budget.