BB(111) is not computable by "just computation". The issue is you have to prove that the programs that don't halt, don't halt. There is no general way to do this (Halting Problem). In fact once you get to a large enough n, BB(n) will be undecidable in whatever axioms you're using. We already have a simple proof of this by taking any known undecidable statement and turning it into a Turing machine halting problem.
