Evidence in nature, supporting the theory of evolution, for example.
Let's take one of Rand's biggies -- we can't sacrifice others to our own ends. By what logic can you prove that it's immoral to sacrifice others to my own ends? Evolutionary theory suggests that it can be just fine -- and we, as putative products of evolution, are subject to the same objective, "discoverable" rules as any other animal.
For example, if I'm a Pharaoh, who's to say that it's wrong to sacrice others to my own selfish ends? After all, it happens at all levels of nature, and has demonstrably good results for certain individuals who are, after all, "ends in themselves."
For objectivism to be valid, you have to prove that the Pharaoh is wrong, based on objective evidence. Knock yourself out.
At best, objectivist practice is merely one choice among many. Indeed, the arguments for it are often based on an alleged optimization of results. (Though the claims to optimality are themselves open to question.) But in that case, objectivism reduces to a merely relativist philosophy.
Tell you what. Give me a quote where Ayn rand said that, I'll send you a million bucks. She never said we can't sacrifice others to our own end, or any other end. She said it was morally wrong to sacrifice others for any reason. She is certainly on record as one who pointed out how this is a most common practice, and one of the essential evils of this world.
Those second-hand souls who have never discovered the requirements of their own nature to live, survive, and enjoy their lives as the fruit of their own effort, that this is what they must do to know they are of value, and are justified in their existense, and that they really do deserver to enjoy their own life, of course will never understand how there is a difference between creating your own wealth and life, or bludgeoning some else and taking theirs.
There is no point in reasoning with those who have intentionally dulled the source of their own moral values, be refusing to identify and recognize their own nature. If you do not know the difference between moral values and "optimization of results," you do not even understand what objectivism, even in it most primitive form (as originally explicated by Ayn Rand) is about.