Show me ... in your own words ... how Rand's premises equate to mysticism.
For reference, here is the relevant portion of Rand's definition of mysticism:
What is mysticism? Mysticism is the acceptance of allegations without evidence or proof, either apart from or against the evidence of one's senses and one's reason. Mysticism is the claim to some non-sensory, non-rational, non-definable, non-identifiable means of knowledge, such as "instinct," "revelation," or any form of "just knowing."
With that in hand, let's dissect Rand's third premise (from the link):
A. "Man — every man — is an end in himself, not the means to the ends of others."
Says who? If I apply Rand's premises 1 and 2, then I can plausibly and scientifically suggest that man -- every man -- is merely a transitional piece in the evolutionary chain, and that my striving is merely a means to the genetic good of successive generations. For me to accept Rand's demand that her premise is absolute, requires me to reject what my senses tell me about evolution and DNA -- i.e., Rand has engaged in mysticism.
He must exist for his own sake, neither sacrificing himself to others nor sacrificing others to himself.
Why? Who says? Aside from the obvious corollaries drawn from the evolutionary argument above, it is not obvious that there is something intrinsically, objectively wrong with somebody like a Pharaoh sacrificing others to himself. After all, according to Rand I must exist for my own sake: If I can live like a pharaoh, and die old, rich, and happy, why would that be wrong? Rand never really says -- she's merely making a "claim to some non-sensory, non-rational, non-definable, non-identifiable means of knowledge" that says Pharaoh shouldn't do that because it's wrong. I.e., once again she's engaging in mysticism.
The pursuit of his own rational self-interest and of his own happiness is the highest moral purpose of his life.
What, precisely, is my own happiness? Is it the same as yours? The truth is, "happiness" is pretty damned subjective, and knowing what it is requires me to resort to "'instinct,' 'revelation,' or any form of 'just knowing.'" Mysticism, again.
As for "rational self-interest," why is it not in my rational self-interest merely to limit my crimes to those things I know I can get away with? (And a Pharaoh can get away with a lot of things.) Well, because Ayn Rand says so, apparently -- we must accept her allegations without proof, which is (again) mysticism.
The reason Ayn Rand can get away with these errors at all, is because many of her underlying premises had already been derived from a theological perspective, and people had already accepted them on that alternative basis. That is why I called her a moral looter.
Rand's philosophy appears to be founded on her atheism, and her objectivism is best seen as an attempt to "work backwards" toward a non-theistic basis for certain desirable philosophical principles. Unfortunately, in order to do that, Rand requires a us to accept as absolute a very specific set of conditions, many of which do not withstand logical scrutiny. Yet she claims to have done it all very rationally. That is why I called her a philosophical fraud.