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FYI: My son was born in England, and has a British father. I took him to the US Embassy in London when he was a month old to get him registered as a US Citizen. I was given a
The words "natural born" do not occur on any of the explanatory information that I was also given about his situation. My understanding has been that he could not run for POTUS, just as my husband could not play rugby for Wales. But that is not explicitly written anywhere on the official info given to me that day.
There's never been an official Supreme Court decision.
That said, after studying the matter for WAY longer than I should've ever put into it, I conclude that "natural born citizen" is, to all purposes, simply an equivalent term to "born a citizen." Although I confess it would've been a heck of a lot easier if they had just used the latter term instead of writing what was an old common-law term of art.
This is also the general consensus of legal scholars.
You son can run for President if he wants to. On meeting the other criteria (age and residency), he's eligible.