Thank you for your question.
No, that isn't true. However, to enforce his rights in the event the mother refuses to allow access to the children, your son would need to obtain a court order for custody and/or visitation. It's the same as if the parents were married and one parent was refusing access. If your son's name is XXXXX XXXXX birth certificate and they've signed an acknowledgment of paternity, he is the legal father and could forgo paternity proceedings. If the mother is threatening to leave the state, emergency custody proceedings may be appropriate.
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