But “natural born” is very vaguely defined.
The fact that he was born to a US Citizen mother and was never subsequently naturalized (as an immigrant has to be), would probably suffice to fit the definition if it were ever seriously contested.
I too would like to see his BC, because there is doubtless something of great interest there, of a rather different significance.
No. It was very well defined at the time of the Constitution.
Clearing the Smoke on Obamas Eligibility: An Intelligence Investigators June 10 Report
William Blackstone, Commentaries 1:354, 35758, 36162 Year 1765
Natural-born subjects are such as are born within the dominions of the crown of England, that is, within the ligeance, or as it is generally called, the allegiance of the king; and aliens, such as are born out of it.. . .
To encourage also foreign commerce, it was enacted by statute 25 Edw. III. st. 2. that all children born abroad, provided both their parents were at the time of the birth in allegiance to the king,
might inherit as if born in England: