User blog comment:Ikosarakt1/Inverse busy beaver function/@comment-1605058-20140311152606

\(\Sigma^{-1}(n)\) is obviously not strictly increasing - it has repeating values. But it's also (non-obviously) not nondecreasing: We have only (4*(6+1))^2*6=28^12<100^12=10^24 6-state Turing machines, so there is a number n<10^24 such that \(\Sigma^{-1}(n)>6\). On the other hand, if we take a 6-state machine which outputs m>10^24 (such as 6-state BB candidates) we will have \(\Sigma^{-1}(m)=6\). So we have \(\Sigma^{-1}(n)>\Sigma^{-1}(m)\) even though n<m, so function has decreased (Q.E.D.).