We need to see his Consular Report of Foreign Birth (CRFB) to be able to answer that question. Technically without a CRFB Cruz is not even a citizen of the USA. But he has been here long enough that I am OK with not deporting him back to Canada.
Is your theory that he was denied citizenship and then applied for naturalization and went through the naturalization process?
Apparently his mother got it for him when he was 16. Before that, he was on paper, by law, technically a Canadian.