As there is no other "known" parent, you would seem to have sole custody. If dad wants a say, including visitation, unless you and he acknowledge paternity, he will have to undergo a DNA test. Once done, he can contest the status of your sole custody. However, if you want to short circuit this, you would start the action against him, which you might consider doing only if he wants to walk away. Also, there is the child support issue and whether you want to actively seek it. Consult a local Family Law attorney to discuss this and other alternatives you may have.
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My response is for educational purposes and does not constitute legal advice nor creates an attorney client relationship which requires all the details and a personal conference.
You probably have custody just due to his absence on the certificate, however, you can never be sure unless a Family Court judge/magistrate awards it to you after you properly file and serve a Petition for Custody. It is not recommended to file petitions without an attorney. The father maintains the right to file and serve a Petition for Visitation, as well. You maintain the right to file and serve a Petition for Child Support.
If you know who the father is, then whether his name is on the birth certificate or not doesn't mean he's "not the father." You have a right to file a petition for a custody order in Family Court as well as a right to seek child support. You should further consult a Family Law attorney in your area.
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