\(S(6,3) > \Sigma(6,3) > 10^{10^{10^{10^{18705352}}}}\)
0 _ 2 l 4 0 1 1 r 2 0 2 1 l 1 1 _ 1 r 2 1 1 1 r 0 2 _ 1 l 3 2 1 _ r 1 3 _ 1 r 4 3 1 _ l 2 3 2 1 r halt 4 _ 1 l 6 4 1 _ r 3 6 _ 1 r 1 6 1 1 l 4
Again a bound using Pavel Kropitz' \(\Sigma(6)\) machine and the note by Pascal Michel.
The begin state is the original state 5. It now writes two where it used to halt. Because tape looks like
#_11(H_)# before halting, we can make sure that it halts as it did.
If there is a space before the 2, then we'll end up with 111 where the the head is on second one in state one, which is the same as the asked position (_11 with head on space and state 6) when one step is preformed.
Therefore, \(S(6,3) > \Sigma(6,3) > 10^{10^{10^{10^{18705352}}}}\)