Yes. Your outlandish hypothetical does not change our established jurisprudence.
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So?...If it Fidel Castro were elected and voted in by the Electoral College, the courts would have no say.
If a Marxist Congress refused to throw out the foreign born usurper Fidel Castro, we would have no recourse ( as common citizens) but to vote Fidel Castro out of office in the next election, or vote in representatives in 2010 would would impeach a man who was not natural born and could not be a president.
Yeah! I get it.
Yes you do. That is the way our Founders set it up. Moral of the story - make sure you stop Castro from being sworn in or be prepared to deal with it under the available alternatives.