An upper bound U for BB(6) implies that any program that runs longer than U never terminates. Thus the specific Collatz-type problems that can be encoded in 6 instructions can be run U+1 steps and if they don’t halt, they won’t halt.
The proof that BB(6) is relevant is that you can encode it in a 6 instruction program, which is what the link does.
The proof that BB(6) is relevant is that you can encode it in a 6 instruction program, which is what the link does.