They were discovered when employees of the Medical Examiner’s Office came upon a box labeled “MOVE” in a refrigerated area of their office, Mr. Kenney said in a statement.

“After comparing the contents of the box to an inventory of bone specimens and fragments from 2017, they appear to be the remains thought to have been cremated four years ago,” Mr. Kenney said on Friday.

“Members of the Africa family were at the Medical Examiner’s Office this afternoon to review documents pertaining to the bombing,” the mayor said. “I personally informed them that the box had been located and its contents were described to them by the interim medical examiner.”

Mike Africa Jr., a member of MOVE who was at the meeting with Mr. Kenney, said he was stunned to learn that the bones had not been destroyed.

“It’s hard to say nothing shocks me at this point, but it does,” Mr. Africa said on Saturday. “I still can’t believe it. It’s still incredible. How is this possible? How are these people so sloppy? How are they so incompetent?”

Dr. Farley could not be immediately reached for comment on Saturday. Mr. Africa said the family had not yet decided what to do with the remains.

Leon Williams, a lawyer who has advised MOVE, said the discovery of the bones had left him wondering whether someone had defied Dr. Farley’s instructions to cremate them.

“Either somebody was downright insubordinate, or they had a conscience or they decided it was the wrong thing to do,” he said.

Mr. Kenney said he was relieved that the remains had not been destroyed.

“However, I am also very sorry for the needless pain that this ordeal has caused the Africa family,” he said. “There are many unanswered questions, including why the remains were not cremated as Dr. Farley directed.”

Mr. Kenney promised the family “full transparency” as the city investigates.

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