Cops are flustered as to how the Lupita Nyong'o Oscar dress escaped detection when it was taken from her hotel room, but now we know ... it was thrown off her balcony.
TMZ broke the story ... the surveillance video was unhelpful. The cameras weren't trained on Lupita's door and none of the cameras on the floor showed anything suspicious.
The guy who stole the dress -- who called TMZ Friday with a road map to recovery in The London West Hollywood bathroom -- told us he knew where the cameras were placed in the hallway so he was never seen. As for the dress ... he said he threw it over the balcony and someone on the ground picked it up.
As we reported ... the thief says he took 2 of the pearls off of the dress and went down to the Garment District for appraisal, but found they were worthless fakes. That's when the thief took the dress back to the hotel.
The thief told us he knew Lupita was in the room where they took the dress and waited for the opportunity to enter ... which they did when they found the door was ajar.
By the way ... the thief also says he saw Lupita's Oscar gift bag -- valued at $125,000 which he didn't take. Looks like he chose wrong.