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They worried that rules set by the state for rolling back business reopenings and restricting activities — the system of yellow, orange and red zones — would mean that even if the city reopened schools, they could close again soon. But the state provided an exception for schools: the shift to weekly testing.

“We wanted to make sure that we weren’t going to be opening and closing,” said Dean Fuleihan, the first deputy mayor. The city, he said, wanted “to have a testing regimen strong enough that we would meet any state standard in the orange and red zone whenever those would happen.”

Officials scrambled — remotely and from offices inside City Hall — to figure out the numbers needed to randomly test students and staff each week across hundreds of schools.

It was not enough to test every school.

“We didn’t have the capacity to do the entire system weekly,” said Kayla Arslanian, Mr. de Blasio’s deputy chief of staff, who has been focused on coronavirus testing. “We are building to get there.”

It was a question not of lab capacity but of finding enough trained staff members to collect samples. The city needed teams of two to four people to go from school to school.

Jeff Thamkittikasem, director of the mayor’s office of operations, said the city had nearly doubled the number of such teams — from 65 to 125 — and could conduct as many as 8,000 tests in schools each day, up from about 4,100 during the initial school testing program.

So the positivity rate doesn’t matter anymore?

One thing could be safely jettisoned, officials and public health experts believed: the 3 percent positivity threshold used to close the schools in the first place.