According to my HR definition, a dependent child must be financially dependent on an eligible employee for more than half of his or her support. And the child is listed as a dependent on the employee's federal tax return for the past two years.
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So if the child's mother agrees to file Form 8332 every year, saying that even she has physical custoday of the child, but I am the parent providing more than half of the child support. Then I can claim my child as dependent in my tax return, filed as head of household.
That case, I can establish the eligibility that I did provide more than half of the child support in my tax return. And the court has no rulling of who can claim a child as dependent.