"If random can not be proven, perhaps it is because there is no such phenomenon."
No, the reason randomness cannot be proven is due to the limitations of logic and computation; it has nothing to do with whether there exist random phenomenon in the physical universe or not. It is possible that nothing is random, but this is completely unprovable one way or the other. This is precisely why ID can never be made a rigorous scientific theory...unless one is willing to put limitations on the complexity of God's mind. If you put explicit restrictions on the resources of God's mind (space and time resources) then it becomes possible to test whether or not a particular sequence is beyond the capacity of God's mind or not. But are you willing to admit such restrictions? If so, what would those restrictions be?
There absolutely are limits on the Mind of God, or, at least, on His practices. He chooses to limit Himself from evil, chaos and absurdity. Keep in mind that there is a natural struggle between chaos and order. Religious thought typically refers to this as the struggle between good and evil.
Is there no place in science for values?