I don't have any personal experience with it (I got notarized permission from the father, even though he is not listed on the birth certificate) but the link that WebyJ provided for you explains exactly what is taken into consideration to prove sole responsibility. Your wife would need to make sure she can do that. Personally I would provide more than the birth certificate (perhaps a letter from the child's doctor and bank statements/receipts showing I was the only one financially supporting the child, along with a detailed letter explaining the situation with the father)...but I've never seen a clear breakdown of what is expected.