Request a letter from Social Security Administration stating that your child has not been issued a SSN. It's not all that uncommon for children.
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I had a vasectomy 27 years ago, can I give the state a letter that says I can't have any kids eligible for child support? I never give my SSN on any 4473's. Wait until the Demonrats find out you can buy a gun without a SSN but need one to go fishing.