What matters more are the constitution, Statutes, Treaties, administrative law decisions, and Court Precedents. The major authority on citizenship is the 14th Amendment, and any explicit "natural born citizen" definition used by the courts - likely 9-0, will use the birth citizenship as somebody born a citizen, or in its most restrictive form, born a citizen on US territory.
You're assuming there's a valid document out there confirming his eligibility. Well, the State of Hawaii refused to confirm his eligibility due to lack of documents. Nancy Pelosi took it upon herself to confirm the man.
And I shall not be surprised if they get it wrong. Any legal system that can produce Abortion out of the 14th amendment is fraudulent.