Revealing mistake: When Mel Gibson is fighting the boyfriend, he flips the table at him, and you can see that it's a double. There are other instances where you can spot the double, but this one is so close and clear. (00:26:55)
Revealing mistake: In the scene in the beginning of the movie when Mel Gibson's daughter gets shot and he's cradling her dead body, when they zoom out her hand is moving. Her fingers move quite a bit when Mel pulls her into his arms.
Chosen answer: Possibly from getting his daughter's blood on him, when he accepted a drink from the Northmoor executive, or from handling his daughter's radioactive belongings. Given the depth of the coverup, they may have even anticipated that she'd come to him and broken into his house and poisoned him before the story even began.
Captain Defenestrator