User:Wythagoras/Rado's sigma function/BB(85)

\(\Sigma(85) > f_{\varepsilon_0}(1907)\) 0 _ 1 l bb1 0 1 1 r bb2 bb1 _ _ r 0 bb1 1 _ r bb3 bb2 _ 1 r 0 bb2 1 1 l C1 bb3 _ 1 r bb1 bb3 1 1 l bb4 bb4 _ _ l bb3 bb4 1 _ l bb1 C1 1 _ l C1 C1 _ _ l 2 2 1 _ l C7 C7 1 1 l C7 C7 _ 1 r 0 0a 1 1 r 0a 0a _ _ l 1 1 1 _ r 1 1 _ _ r 3 2 _ 1 r 17 3 _ _ r 4 4 _ _ r 60 4 1 1 r halt 5 1 1 r 61 5 _ _ l 6 6 _ _ l 7 6 1 1 l 8 7 1 _ l 7 7 _ 1 r 32 8 1 _ l 9 8 _ _ l 77 9 1 _ r 10 9 _ _ r 18 10 _ _ r 11 11 1 1 r 11 11 _ _ r 12 12 1 1 r 11 12 _ _ r 13 13 1 1 r 13 13 _ 1 l 14 14 1 1 l 14 14 _ _ l 27 15 1 1 l 27 15 _ _ l 16 16 1 1 l 27 16 _ 1 r 32 17 1 1 r 17 17 _ _ r 72 18 _ _ r 19 19 1 1 r 19 19 _ _ r 20 20 1 1 r 19 20 _ _ r 21 21 1 1 r 21 21 _ _ l 22 22 1 _ l 23 22 _ _ l 34 23 1 1 l 23 23 _ _ l 24 24 _ _ l 25 25 1 1 l 28 25 _ _ l 29 26 1 1 l 26 26 _ 1 l 31 27 _ _ l 15 27 1 1 l 15 28 1 1 l 25 28 _ _ l 25 29 _ _ r 30 29 1 1 l 25 30 _ 1 l 33 31 _ 1 r 0a 32 _ 1 l 6 33 _ _ l 8 34 _ _ l 35 34 1 1 l 35 35 1 1 l 34 35 _ _ l 36 36 _ _ r 37 36 1 1 l 35 37 _ 1 r 40 37 1 _ r 49 38 1 1 r 38 38 _ _ r 39 39 _ _ r 42 39 1 1 r 38 40 _ _ r 41 40 1 _ r 37 41 1 _ r 38 41 _ _ l 63 42 1 1 r 42 42 _ _ r 43 43 1 1 r 42 43 _ 1 l 44 44 _ _ l 45 44 1 1 l 44 45 _ _ l 47 45 1 1 l 44 46 1 1 l 47 46 _ _ l 48 47 1 1 l 46 47 _ _ l 46 48 1 1 l 46 48 _ _ r 37 49 1 1 r 49 49 _ _ r 50 50 _ _ r 52 50 1 1 r 49 51 _ _ r 66 51 1 1 r 51 52 1 1 r 52 52 _ _ r 53 53 1 1 r 52 53 _ 1 l 59 54 _ _ l 55 54 1 1 l 54 55 _ _ l 57 55 1 1 l 54 56 1 1 l 57 56 _ _ l 58 57 1 1 l 56 57 _ _ l 56 58 1 1 l 56 58 _ 1 r 37 59 _ 1 l 55 59 1 1 l 67 60 _ _ r halt 60 1 1 r 61 61 1 1 r 61 61 _ _ r 5 62 1 1 l 63 62 _ _ l 64 63 1 1 l 62 63 _ _ l 62 64 _ _ l 65 64 1 1 l 62 65 1 1 l 65 65 _ 1 l 59 66 _ _ * 73 66 1 1 r 51 67 1 _ r 68 67 _ 1 l 2 68 1 1 r 68 68 _ _ r 71 69 1 1 r 69 69 _ _ r 70 70 1 1 r 69 70 _ _ * 37 71 _ _ r 69 72 _ _ r 51 73 _ 1 l 74 74 _ _ l 75 74 1 1 l 74 75 _ _ l 76 75 1 1 l 74 76 1 1 l 74 76 _ _ l 0a 77 _ _ l 78 78 _ 1 l 26 78 1 1 l 26

Explanation
The needed input is 11___1(_1)n for some moderately large n to create a huge output. The head must be on one of the first ones.

The following machine outputs 1111_11(_1)1909 where the head is on the fifth one. 0 _ 1 l 1 0 1 1 r 2 1 _ _ r 0 1 1 _ r 3 2 _ 1 r 0 2 1 1 l halt 3 _ 1 r 1 3 1 1 l 4 4 _ _ l 3 4 1 _ l 1 Using C5 1 _ l C5 C5 _ _ l C6 C6 1 _ l C7 C7 1 1 l C7 C7 _ 1 r M0 This will be transformed to 1111___1(_1)1909

State C6 can be reduced into the main machine.

\(\Sigma(85,2) > f_{\varepsilon_0}(1907)\)