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I found here some interesting notes on the record holder for \(\Sigma(6)\) (found by Kropitz). It also says, that when _11 is entered and the head is on the blank and the machine is in state A, it will run for over \(10^{10^{10^{10^{10^7}}}}\) steps. The following machine does this:

0 _ 1 l 4
1 _ 1 r 2
1 1 1 r 5
2 _ 1 l 3
2 1 _ r 1
3 _ 1 r 4
3 1 _ l 2
4 _ 1 l 6
4 1 _ r 3
5 _ 1 r halt
5 1 1 r 2
6 _ 1 r 1
6 1 1 l 4

(note: 6=old state A and 1=old state B, 2=old state C, 3=old state D, etc)

After the first step: _1, head on the blank, state 4

After the second step: _11, head on the blank, state 6, so

So, here are the bounds for \(\Sigma(7)\) and \(S(7)\)

\[S(7) > \Sigma(7) > 10^{10^{10^{10^{18705352}}}}\]

Acknowledgements

This machine is almost solely based on Pascal Michel's analysis and the machine provided by Kropitz. Therefore, they should get most credit for this discovery. Many thanks to Cloudy176 for providing the full proof (see comments).

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