He can't claim her. Period. She did not live in his house for "more" than half that year. There's other things too. But they don't matter at this point. If he does claim her, he's breaking the law and the IRS will have a field day fining and penalizing him, as well as garnishing his pay from whatever job he may have. Also, go back to the court that ordered the child support. You can have his tax refund garnished, and paid to you directly. Might also even be able to get a portion of any paychecks he receives garnished too.
