[[By way of example, NOTHING in the law speaks to “gives up custody to the father.” You are deliberately fogging things up.]]
1409 talks about a child living with a father, and explains what the father’s responsibilities are concerning whether the child needs to be naturalized or not- Where the mother is a Citizen, the father does NOT have to meet the requirements, where them other is NOT a citizen, the father would have to meet the requirements, in order for child to be citizen- where the father has sole custody, he would need to meet the requirements etc 1409 is speaking to unwed parents, if the child is with the father, he needs to meet requirements in order to child to be citizen- this infers that them other is not a citizen nor the one in custody because 1401 covers this case with no need for the father to meet any requirement-