If the child has been living with you and you are the primary caretaker then you are likely to have primary physical custody and you and the father will get or share joint legal custody. Short of that you question is very confusing.
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First, it is important to note that there are two types of custody: legal custody and physical custody. Legal custody is about making the "big" decisions in a child's life, such as where a child will go to school, which physicians the child will see, and religious upbringing. Typically, parents share legal custody, unless there is some reason for the parents not to share legal custody (e.g. ongoing domestic violence).
Physical custody is about who physically has the child. There can be a primary physical custodian, or parents can have joint physical custody, where they share about equal time with the child.
The legal standard for determining both types of custody is "the best interests of the child." The court can look at many factors when considering this standard, such as the age of the child, who has primarily cared for the child in the past, how far apart the parents live from each other, etc.
For a child born out of wedlock, the law presumes that the mother has sole legal and sole physical custody absent an order from the court. However, the father still has the right to seek custody through the court. If/when a father files for custody in the court, the court will apply this "best interests" standard.
To answer your question, there is no straight answer without knowing all the details of your circumstances. You should consult with a knowledgeable Family Law attorney to discuss your circumstances in greater detail. Most attorneys, including my firm, offer free initial consultations.
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The Father of your child can file a Complaint to Establish Paternity if you are not willing to allow him to be put on the birth certificate as the child's father. This process requires that he take a DNA test to determine if he is in fact the father of your child. If after the DNA test the court determines him to be the father, then he will be put on the birth certificate, and if he chooses to pursue custody the issue will be dealt with at that point.
As was previously pointed out, generally both parents will share legal custody. The question will be what happens with physical custody? To determine physical custody the court will consider a number of factors, and will ultimately make its decision based upon what is in the best interest of your child. However, both you and the Father have the ability to craft an agreement/parenting schedule that works best for you. By working out an agreement with the Father, you have more control over the situation, than if you leave it up to a judge.
No matter which option you pursue, this is not something you should try to do on your own. If at all possible you should look into hiring an attorney to represent you in a case such as this.
The other answers provided great information. Just to add on, although you refer to sole custody, you may mean primary physical custody. Sole custody is usually reserved for situations where the other parent is a danger to the child and is not allowed to see the child ever, or only under supervised situations. If you truly mean primary physical custody (you have the child most of the time, and the father has the child part of the time) then based on the facts you provided you would probably be given primary physical custody. If the facts are different or not stated correctly, it could be a different situation.
As has been mentioned, based on the information provided, it would be wise to seek the assistance of an attorney to work out a shared parenting agreement and schedule with the father in order to avoid the difficult, and unpredictable situation that Court proceedings create.
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