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Why do you want to screw mess with your sister or sister-in-law?

Your nephew can be your dependent if he lived in your home more than half the nights of the year.  It is not required that you prove you provided support, just where the child lived.

However, the child is a dependent of his mother if he lived with his mother more than half the nights of the year, including "temporary absences."  A temporary absence is just what it says, presuming the reason for the child living away is temporary and it is reasonable to expect that the child will return home after the absence.  The child is considered to live where they would have lived if not for the reason for the temporary absence.  This concept usually applies to sleepovers, summers with grandparents, and college.  But it may apply here.

If you claim your nephew and are blocked from e-filing, you can print your return and mail it in.  The IRS will investigate. Your argument would be that the child lived in your home more than half the nights of the year.  The mother's argument would be that the child lived in her home more than half the nights of the year because the time in your home should be considered a "temporary absence."  Can you prove otherwise?

This situation seems like a judgement call, you would be relying on the auditor to see it your way.