EXCLUSIVE

"American Idol" producer Nigel Lythgoe is adamant ... the bitter feud between Mariah Carey and Nicki Minaj is as real as it gets -- and his proof lies in simple logic.Nigel was out in L.A. yesterday when we asked for the final word -- did he or did he not stage the feud for ratings??? -- and Nigel replied, "I have no idea why I would be doing it [for ratings] five months before the show went out."

He added, "I would only do ANYTHING for ratings a month before the show went out. They've got my timing wrong."

Translation: keep your eyes out for a crazy "Idol" ratings ploy in three months.