[John Valentine]

There has to be some kind of documentation of his American citizenship.

There is. His birth certificate and his mother's.

For things in the U.S. that require citizenship, what would he present? His mother's birth certificate?

His and his mother's. There is also a State Department form called "Certificate of US Citizen Born Abroad", but requesting one is not a requirement, and Cruz has not produced one to the public. Still, he may have one; I don't have any way of knowing. If his mother was diligent, she would have requested one. But if she didn't, it's not dispositive of anything.

[Behind the Blue Wall]

His mother’s birth certificate by itself doesn’t prove that she was a citizen at the time of his birth.