Totally irrelevant. Cruz did not apply for naturalization. He was not naturalized. He was a citizen at birth, which is synonymous with natural born citzen (unless you can show otherwise, which I doubt).
Did or did not Cruz go before a judge in 2014 and formally renounce his Canadian citizenship that he had from birth?
The law states that one who must renounce a foreign citizenship because they were born of a foreign parent in the country of the foreign parent, that person is a naturalized citizen. I have read the Canadian citizenship laws of 1970 and in 1970, Canada did not grant citizenship to babies born in the country unless, 1) one of the parents was a Canadian, 2) both of the parents were Canadian & 3) the parents had applied and were granted permanent legal resident status because they had begun the process of becoming Canadian citizens.
Cruz Sr. was a Canadian citizen when the family relocated back to the United States. He remained a Canadian citizen until 2005. So for Constitutional purposes, Rafael Edward (Ted) Cruz was a natural born Canadian citizen at birth. One can not be natural born of 2 countries and Cruz legally admitted he was a natural born Canadian when he formally & legally renounced his foreign citizenship in 2014.