The public can be informed of NHI, evidence can exist and the evidence can also be withheld from the public for national security reasons. All of this can be true.
Also doesn't really matter who is President. the administration would make the statement, and most likely in conjunction with the Joint Chiefs of Staff, the National Security Council and and the Homeland Security Council. The President would just be the mouthpiece. I doubt very much the President could get in front of the camera and state these kinds of things without unanimous agreement across the board on national security matters like this. The entire administration rapidly destabilizes if any one person or group went into business for themselves - hoax or otherwise.
I don't think the President would present evidence before making such a claim. It would be more of a diplomatic message, and probably in the form this being the "first step in the process". Evidence would eventually come, but not at the pace the nuts and bolts crowd would like (such as some intrepid posters here). I think that comes afterwards. Would be awesome if he trotted out the evidence in an SOTU though. Either way, I'd probably believe it.
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