No.
Nothing about the number of months the child lived with you is sent to the IRS. That question is only asked so that TurboTax can determine what tax attributes you are allowed to claim. As the non-custodial parent, your answer must be something less than 6. It doesn't matter whether you enter 0 or 5 or something in between. It is OK for the other parent to answer 12, since he is the custodial parent for tax purposes.
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For others reading this, the custodial parent does have to show on IRS Schedule EIC, when claiming the Earned Income Credit, how many months the child lived with him. Entering 7, 8, 9, 10, 11, or 12 will qualify him for EIC. In this situation, the child alternated between parents for the first 8 months of the year. So, the child lived with him all year. The custodial parent would not be lying to claim 12 months.
