The question of whether Obama applied as a foreigner is critical. Why would anyone assume, if he did it, it was fraud? We can’t know for certain because we don’t know on what basis he made the claim. Did he present a foreign passport? Did he show some other form of foreign documentation? We just don’t know. I’m glad you would like to find out. It would certainly speak to his values, which you say are important, to find out (1) if he did it, and if so, (2) on what basis.
So if someone makes up some baseless rumor about a public official, the public official must continually try to placate that person by trying endlessly to meet an ever changing standard of proof?