Just in case you missed my point in #336, the "authority" I referenced was Mr. Benjamin Franklin, in his letter of 1775 to Mr. Charles Dumas, thanking him for the three copies of Vattel's work. It's a little difficult to speak with Ben right now. Fortunately for us, he left many of his thoughts in writing so that we could review them hundreds of years later.
I agree that the "Laws of Nations" as capitalized in the Constitution is insufficient evidence of Vattel's influence (every noun seems to be capitalized), but Franklin's own words support it's importance.
Nobody’s denying Ben said what he said. But Vattel loses the contest against English common law when it comes to the meaning of natural born citizenship.
Not one single reputable lawyer or scholar agrees with you. Doesn’t that mean anything to you?
http://avalon.law.yale.edu/subject_menus/17th.asp