“Naturalization only occurs after birth.”
A child who is granted the right to claim U.S. citizenship retroactively by automatic naturalization at birth is in fact naturalized after birth. This can be observed for example in such cases where a child is born with a U.S. citizen parent and an alien parent outside the jurisdiction of the United States and who never comes forward to perfect the right to claim naturalized U.S. citizenship at some point in time after the birth. Instead the child is born with a foreign citizenship, the parents never come forward to claim naturalized U.S. citizenship at birth for the child, and the child reaches the age of majority and does not claim naturalized U.S. citizenship. A person who is an actual natural born citizen is born in the jurisdiction of the citizen parents’ sovereign and without any foreign citizenship. A person who is an actual natural born citizen cannot be a foreign citizen at birth,, because the parents are U.S. citizens and the birth occurred within the sole jurisdiction of the sovereign United States Article III courts. In the case of Ted Cruz, he may or may not have acquired U.S. citizenship, but if he did so it could only have been under the authority of the U.S. Immigration and Naturalization Act of 1952 which authorized the parents to file a claim after the birth claiming naturalized citizenship retroactively effective at birth.
Incorrect.
A person’s birth circumstances can not change, thus their citizenship is fixed at birth. Naturalization changes that status after birth via a legal process. That is why Title 8 Section 1101 subsection A part (23) (https://www.law.cornell.edu/uscode/text/8/1101) defines naturalization as:
The term “naturalization” means the conferring of nationality of a state upon a person after birth, by any means whatsoever.
Sen Cruz was born a US Citizen, has always been a citizen, has never needed to be naturalized and has never been naturalized. See USC Title 8 Section 1401 - Nationals and Citizens of the United States at birth. https://www.law.cornell.edu/uscode/text/8/1401