The Twentieth Amendment, Section Three places the burden of proving eligibility squarely on the President Elect. If he/she fails to qualify, Congress must name an interim President. So, how does Cruz not qualify?
In your mind, what must he do to qualify?
First, he must have a valid birth certificate in order to prove age, place of birth and parentage. Since Obama apparently has no such thing except for the forged pdf, he was unable to prove anything and thus has never met the qualify portion of the Constitution. This means that he has never been a legal President because the eligibility requirements clearly forbid those who fail to qualify from holding the office of President. It doesn't matter how the failure to qualify occurred, it's still a failure to qualify.
In the case of Cruz, the question would be how those opposed to his taking office would interpret the natural born citizen question. If they take the traditional view of having to be born of citizen parents and born in a location under United States jurisdiction, he would not be allowed to serve. This is the central question concerning whether or not he is worth taking the chance to elect in the first place. My opinion, based upon the change made to the language of the Naturalization Act of 1790 by the Act of 1795, is that he would NOT be ruled as eligible.