Mike Bloomberg, New York Magazine Having Trouble Parsing 12th Amendment

There's a curious blurb today in New York Magazine's "Intelligencer".

Mayor Bloomberg’s political guru, Kevin Sheekey, continues his quest to get his boss in a position to run for president. According to a well-placed source, Sheekey’s latest project is to figure out if they could get California governor Arnold Schwarzenegger as a potential running mate. This summer, Bloomie and Arnie shared a few centrist photo ops. The East Coast–West Coast alliance would have its advantages in the Electoral College, certainly. But other parts of the Constitution are more problematic, like where it mentions that only a “natural-born Citizen … shall be eligible for the Office of President.” It doesn’t mention the veep, though. The source says Bloomberg has lawyers studying things like “succession plans” to see how it would work: If something happens, say, to President Bloomberg, does that mean Vice-President Schwarzenegger is overlooked and the Speaker of the House (Nancy Pelosi) gets to take over? A spokesman for Bloomberg declined to comment.

It's true that the Constitution as originally penned doesn't mention the Vice-Presidency in connection with the "natural-born citizen" requirement.  Specifically, in Article II, Section 1, it reads:

No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;

But the Twelfth Amendment makes it pretty clear:

But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.

The mayor's got a team of people at work on this?  Is one of them Lloyd Braun?  What am I missing?