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The IRS rule is that the parent who has the child the greater number of nights is considered the custodial parent. The divorce decree cannot overrule what the IRS ruling is, because the state courts do not have authority to dictate to the United States Government. If they both file taxes claiming the child as their dependent, then the IRS will conduct an audit and determine who is qualified to take the tax credit.
She may need to talk to a lawyer about how to ensure that she has the required amount of days to achieve the IRS credit. She could also file contempt charges on him if he does violate the order and tries to take the tax deduction. The Judge might remedy the situation by shifting custody slightly so the IRS requirements are fulfilled.
Yes, he would have to sign the form to give her the exemption if he didn't want to be in contempt of court. Of course if he doesn't care about being in contempt, he can refuse to sign and force the Judge to make a decision.