How alarmed were New Jersey officials by reports of Muslims dancing in the streets of Jersey City and Paterson on Sept. 11, 2001, to celebrate the destruction of the World Trade Center?

They feared riots would break out and were ready to send in the National Guard and the State Police to preserve order.

But John J. Farmer Jr., then the New Jersey attorney general and the state’s chief law enforcement officer, said on Tuesday that he ordered an investigation that very day and found the reports to be bogus, more wild stories born in the stricken hours after the attacks.

Nevertheless, those ancient, false rumors were recycled as truth over the weekend by Donald J. Trump, who has folded them into his calls for the national registration of Muslims and possible closing of mosques. Speaking in Alabama, Mr. Trump said: “Hey, I watched when the World Trade Center came tumbling down. And I watched in Jersey City, N.J., where thousands and thousands of people were cheering as that building was coming down. Thousands of people were cheering.”