If the 3 of you live together all year, the IRS doesn't care who claims the child, especially if you both file as single. Only the person who both pays more than half of the expenses of keeping up the home AND claims the child can file as HOH. If you do not live together at all, it's impossible for you to truthfully say the child lives with each of you half the year because the test is MORE than half of the year and only one of you would qualify as the custodial parent.
A non-custodial parent can only claim the child if they meet all the tests and the custodial parent gives them a signed form 8332. The form does not convey HOH or EIC rights.
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