There is no clearly defined or motivated problem of "proving Friendliness". We need to understand what goals are, what humane goals are, what process can be used to access their formal definition, and what kinds of things can be done with them how to what end. We need to understand these things well, which (on psychological level) triggers association with mathematical proofs, and will probably actually involve some mathematics suitable to the task. Whether the answers take the form of something describable as "provable Friendliness" seems to me an unclear/unmotivated consideration. Unpacking that label might make it possible to provide a more useful response to the question.