This is a good point that a lot of people have missed.
The US Constitution is an "enabling" document. Think of it as a "requirements statement" -- if you are familiar with contracting.
Laws must be separately enacted to achieve the requirements set out in the Constitution. And the judicial branch is suppose to determine the Constitutionality of a particular law by whether it meets or conflicts with the requirements in the US Constitution.
The US Code doesn't say that the President has to present evidence of being a natural-born citizen, mostly because it's never been an issue until now. There are some efforts to rectify that before the next Presidential election.
It's a bit complicated because the Electoral College effectively creates 51 different elections, and the results are combined. So, each individual state has to enforce this requirement. Theoretically, a few states could enact the requirement into law and be ignored by an ineligible candidate -- because he doesn't need their electoral votes to win.
Here's where you can find reference to this "requirement" in U.S. Code. I've made the relevant portion easier to read by bolding the text. In my opinion, we currently have a "vacancy" at the office of president.
CITE: 3USC19
TITLE 3--THE PRESIDENT
CHAPTER 1--PRESIDENTIAL ELECTIONS AND VACANCIES
Sec. 19. Vacancy in offices of both President and Vice President; officers eligible to act
(a)(1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.