Universal pre-K is doing great . . . according to a study funded by the city Department of Education, which would be in trouble if it had reported any other result.

The study found that kids enrolled in the city program made gains in recognizing letters, spelling and early math skills — which of course toddlers of that age should be doing.

Then, too, the NYU researchers only surveyed 75 UPK sites out of the hundreds that enroll 70,000 kids. And other studies raise some troubling issues.

The Century Fund recently announced that many of the city’s pre-K classrooms are highly segregated. Earlier, a UC-Berkeley study found that pre-K sites had disproportionately opened in the city’s more affluent areas.

Both studies flagged the segregation problem — and called for integrated schools to help bridge the racial achievement gap.

The big concern about vesting too much hope in pre-K is that decades of studies have cast doubt on “early intervention” as a miracle cure. Typically, the gains vanish just a year or two after children have graduated on to regular public schools.

At this point, the only school reform with a proven record of delivering long-term results for New York City’s underprivileged kids is the charter revolution — creating public schools free of union contracts and the central DOE bureaucracy.

The best hope for kids to keep their pre-K gains is to have the choice of good schools to move on to.