20,363
pages

I decided to create some hydra machines.

Note: current Kirby Paris results can be found in my subpages, I wrote this post to get attention.

## Kirby Paris hydra

### 79 states, 2 colors

Input: Multiplier in the form of 111111...111111, where 2n 1s is a multiplier of n. Then two spaces. Then hydra without rightmost )'s, where a ( is changed to _1 and a ) is changed to 11.

0 1 1 r 0
0 _ _ 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 0
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 0
77 _ _ l 78
78 _ 1 l 26
78 1 1 l 26


### 34 states, 3 colors

2 states improvement over Peng's machine, yay!

Note that although I built this machine from scratch, it uses Peng's idea to eliminate the last  )'s.

Input: Multiplier in the form of (((...(((, then a space, then a hydra without rightmost )'s.

0 ( ( r 0
0 _ _ l 24
1 _ _ r halt
1 ( ( r 2
2 ( ( r 2
2 ) ) r 2
2 _ _ l 13
3 ) _ r 4
3 _ _ l 25
3 ( _ r 8
4 ( ( r 4
4 ) ) r 4
4 _ _ r 5
5 ( ( r 5
5 _ ( l 6
6 ( ( l 6
6 _ _ l 7
7 ( ( l 7
7 ) ) l 7
7 _ ) l 3
8 ( ( r 8
8 ) ) r 8
8 _ _ r 9
9 ( ( r 9
9 _ _ l 10
10 ( _ l 11
10 _ ) l 14
11 ( ( l 11
11 _ _ l 12
12 ( ( l 12
12 ) ) l 12
12 _ ( l 3
13 ) _ l 13
13 ( _ l 3
14 ( ( l 14
14 ) ) l 14
14 _ ( r 15
15 ( _ r 16
15 ) _ r 20
15 _ _ l 26
16 ( ( r 16
16 ) ) r 16
16 _ _ r 17
17 ( ( r 17
17 ) ) r 17
17 _ ( l 18
18 ( ( l 18
18 ) ) l 18
18 _ _ l 19
19 ( ( l 19
19 ) ) l 19
19 _ ( r 15
20 ) ) r 20
20 ( ( r 20
20 _ _ r 21
21 ( ( r 21
21 ) ) r 21
21 _ ) l 22
22 ( ( l 22
22 ) ) l 22
22 _ _ l 23
23 ( ( l 23
23 ) ) l 23
23 _ ) r 15
24 ( ) r 24
24 _ _ r 1
25 ( ( l 25
25 ) ( l 25
25 _ ( r 0
26 ( ( l 26
26 ) ) l 26
26 _ _ l 27
27 ( ( l 27
27 ) ( l 28
28 ( ) r 29
28 _ _ r 31
29 ( ( r 29
29 _ _ r 30
30 ( ( r 30
30 ) ) r 30
30 _ ( r 15
31 ( ( r 31
31 _ _ r 32
32 ( ( r 32
32 ) ) r 32
32 _ ( l 33
33 ( ( l 33
33 ) ) l 33
33 _ _ l 25


### 28 states, 4 colors

No states 9, 10, 11, 12, 19, 23 thanks to a smart tric with a fourth color.

Input: Multiplier in the form of (((...(((, then a space, then a hydra without rightmost )'s.

0 ( ( r 0
0 _ _ l 24
1 _ _ r halt
1 ( ( r 2
2 ( ( r 2
2 ) ) r 2
2 _ _ l 13
3 ) _ r 4
3 _ _ l 25
3 ( _ r 6
4 ( ( r 4
4 ) ) r 4
4 x x r 4
4 _ x l 5
5 x x l 5
5 ( ( l 5
5 ) ) l 5
5 _ ) l 3
6 ( ( r 6
6 ) ) r 6
6 x x r 6
6 _ _ l 7
7 x _ l 8
7 _ _ r 13
7 ) ) r 13
8 x x l 8
8 ( ( l 8
8 ) ) l 8
8 _ ( l 3
13 _ ) l 14
13 ) _ l 13
13 ( _ l 3
14 ( ( l 14
14 ) ) l 14
14 _ ( r 15
15 ( x r 16
15 ) x r 20
15 _ _ l 26
16 ( ( r 16
16 ) ) r 16
16 _ _ r 17
17 ( ( r 17
17 ) ) r 17
17 _ ( l 18
18 ( ( l 18
18 ) ) l 18
18 _ _ l 18
18 x ( r 15
20 ) ) r 20
20 ( ( r 20
20 _ _ r 21
21 ( ( r 21
21 ) ) r 21
21 _ ) l 22
22 ( ( l 22
22 ) ) l 22
22 _ _ l 22
22 x ) r 15
24 ( ) r 24
24 _ _ r 1
25 ( ( l 25
25 ) ( l 25
25 _ ( r 0
26 ( ( l 26
26 ) ) l 26
26 _ _ l 27
27 ( ( l 27
27 ) ( l 28
28 ( ) r 29
28 _ _ r 31
29 ( ( r 29
29 _ _ r 30
30 ( ( r 30
30 ) ) r 30
30 _ ( r 15
31 ( ( r 31
31 _ _ r 32
32 ( ( r 32
32 ) ) r 32
32 _ ( l 33
33 ( ( l 33
33 ) ) l 33
33 _ _ l 25


### 17 states, 5 colors

Unbelievable! I was able to reduce 11 states more: 1, 15, 17, 21, 22, 24, 25, 27, 30, 31, 32!

Input: Multiplier in the form of xxxx...xxxx, then a hydra without rightmost )'s.

0 x x r 0
0 ( ( l 2
0 _ _ r halt
2 x y r 2
2 ( ( r 2
2 ) ) r 2
2 y ( r 2
2 _ _ l 13
3 ) _ r 4
3 y x l 33
3 ( _ r 6
4 ( ( r 4
4 ) ) r 4
4 x x r 4
4 _ x l 5
5 x x l 5
5 ( ( l 5
5 ) ) l 5
5 _ ) l 3
6 ( ( r 6
6 ) ) r 6
6 x x r 6
6 _ _ l 7
7 x _ l 8
7 _ _ r 13
7 ) ) r 13
8 x x l 8
8 ( ( l 8
8 ) ) l 8
8 _ ( l 3
13 _ ) r 26
13 ) _ l 13
13 ( _ l 3
14 ( ( l 14
14 ) ) l 14
14 _ ( r 28
16 ( ( r 16
16 ) ) r 16
16 y y r 16
16 _ ( l 18
18 ( ( l 18
18 ) ) l 18
18 y y l 18
18 x ( r 28
18 _ ) r 28
20 ) ) r 20
20 ( ( r 20
20 y y r 20
20 _ ) l 18
26 ( ( l 26
26 ) ) l 26
26 x x l 26
26 y x l 28
26 _ y l 14
28 x y r 29
28 _ x r 0
28 ( x r 16
28 ) _ r 20
28 y y l 26
29 x x r 29
29 ( ( r 29
29 ) ) r 29
29 y _ r 29
29 _ y l 14
33 ( ( l 33
33 ) ) l 33
33 x x l 33
33 y x l 33
33 _ x r 0


### 16 states, 7 colors

Reduced state 5 beacause of state reduction using temporary color change.

0 x x r 0
0 ( ( l 2
0 _ _ r halt
2 x y r 2
2 ( ( r 2
2 ) ) r 2
2 y ( r 2
2 _ _ l 13
3 ) _ r 4
3 y x l 33
3 ( _ r 6
3 x x l 3
3 [ ( l 3
3 ] ) l 3
3 _ ) l 3
4 ( [ r 4
4 ) ] r 4
4 x x r 4
4 _ x l 3
6 ( ( r 6
6 ) ) r 6
6 x x r 6
6 _ _ l 7
7 x _ l 8
7 _ _ r 13
7 ) ) r 13
8 x x l 8
8 ( ( l 8
8 ) ) l 8
8 _ ( l 3
13 _ ) r 26
13 ) _ l 13
13 ( _ l 3
14 ( ( l 14
14 ) ) l 14
14 _ ( r 28
16 ( ( r 16
16 ) ) r 16
16 y y r 16
16 _ ( l 18
18 ( ( l 18
18 ) ) l 18
18 y y l 18
18 x ( r 28
18 _ ) r 28
20 ) ) r 20
20 ( ( r 20
20 y y r 20
20 _ ) l 18
26 ( ( l 26
26 ) ) l 26
26 x x l 26
26 y x l 28
26 _ y l 14
28 x y r 29
28 _ x r 0
28 ( x r 16
28 ) _ r 20
28 y y l 26
29 x x r 29
29 ( ( r 29
29 ) ) r 29
29 y _ r 29
29 _ y l 14
33 ( ( l 33
33 ) ) l 33
33 x x l 33
33 y x l 33
33 _ x r 0


### 15 states, 8 colors

Reduced state 18 beacause of state reduction using temporary color change.

0 x x r 0
0 ( ( l 2
0 _ _ r halt
2 x y r 2
2 ( ( r 2
2 ) ) r 2
2 y ( r 2
2 _ _ l 13
3 ) _ r 4
3 y x l 33
3 ( _ r 6
3 x x l 3
3 [ ( l 3
3 ] ) l 3
3 _ ) l 3
4 ( [ r 4
4 ) ] r 4
4 x x r 4
4 _ x l 3
6 ( ( r 6
6 ) ) r 6
6 x x r 6
6 _ _ l 7
7 x _ l 8
7 _ _ r 13
7 ) ) r 13
8 x x l 8
8 ( ( l 8
8 ) ) l 8
8 _ ( l 3
13 _ ) r 26
13 ) _ l 13
13 ( _ l 3
14 ( ( l 14
14 ) ) l 14
14 _ ( r 28
14 [ ( l 14
14 ] ) l 14
14 y y l 14
14 x ( r 28
14 c ) r 28
16 ( [ r 16
16 ) ] r 16
16 y y r 16
16 _ ( l 14
20 ) ] r 20
20 ( [ r 20
20 y y r 20
20 _ ) l 14
26 ( ( l 26
26 ) ) l 26
26 x x l 26
26 y x l 28
26 _ y l 14
28 x y r 29
28 _ x r 0
28 ( x r 16
28 ) c r 20
28 y y l 26
29 x x r 29
29 ( ( r 29
29 ) ) r 29
29 y _ r 29
29 _ y l 14
33 ( ( l 33
33 ) ) l 33
33 x x l 33
33 y x l 33
33 _ x r 0


## Bounds using Kirby-Paris machines

### $$\Sigma(85,2)$$

The needed input is 11___1(_1)n for some n to create a big 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

5 1 _ l 5
5 _ _ l 6
6 1 _ l 7
7 1 1 l 7
7 _ 1 r M0


This will be transformed to 1111___1(_1)1909

State 6 can be reduced into the main machine.

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

### $$\Sigma(37,3)$$

Construction is similiar to Cloudy's in here.

$$\Sigma(37,3) > f_{\varepsilon_0}(373676378)$$

### $$\Sigma(36,3)$$

Using 2 state 3 Busy Beaver.

$$\Sigma(36,3) > f_{\varepsilon_0}(7)$$

### $$\Sigma(31,4)$$

Construction is similiar to Cloudy's in here.

$$\Sigma(31,4) > f_{\varepsilon_0}(373676378)$$

### $$\Sigma(19,5)$$

0 _ 1 r 1
0 1 2 l 0
0 2 1 r 0
0 3 2 l 1
0 4 2 l 0
1 _ _ l 0
1 1 2 r 1
1 2 3 r 1
1 3 4 r 0
1 4 1 l M28


The best known 2 state 5 color machine. It outputs

_122222...2222 (~1.7*10^352 2's)
^


Let 2 be ( and 1 be x, it outputs what we want.

$$\Sigma(19,5) > f_{\varepsilon_0}(1.7\cdot10^{352})$$

### $$\Sigma(18,7)$$

$$\Sigma(18,7) > f_{\varepsilon_0}(1.7\cdot10^{352})$$

### $$\Sigma(17,8)$$

$$\Sigma(17,8) > f_{\varepsilon_0}(1.7\cdot10^{352})$$

## (0,1) game

### 92 states, 5 colors

Note that although I built this machine from scratch, it uses Peng's idea to eliminate the last  )'s.

Input: Multiplier in the form of (((...(((, then a space, then a hydra without rightmost )'s and ]'s.

Note: no states 49, 50, 54, 55.

0 ( ( r 0
0 _ _ l 3
1 _ _ r halt
1 ( ( r 2
2 ( ( r 2
2 ) ) r 2
2 [ [ r 2
2 ] ] r 2
2 _ _ l 4
3 ( ) r 3
3 _ _ r 1
4 ) _ l 4
4 ( _ l 5
4 [ _ l 58
4 ] _ l 4
4 _ [ r 26
5 ( _ r 6
5 [ _ r 11
5 ) _ r 16
5 ] _ r 20
5 _ _ l 57
6 ( ( r 6
6 [ [ r 6
6 ) ) r 6
6 ] ] r 6
6 _ _ r 7
7 ( ( r 7
7 _ _ l 8
8 ( _ l 9
8 _ ) l 24
9 ( ( l 9
9 _ _ l 10
10 ( ( l 10
10 ) ) l 10
10 [ [ l 10
10 ] ] l 10
10 _ ( l 5
11 ( ( r 11
11 [ [ r 11
11 ) ) r 11
11 ] ] r 11
11 _ _ r 12
12 ( ( r 12
12 _ _ l 13
13 ( _ l 14
13 _ ] l 25
14 ( ( l 14
14 _ _ l 15
15 ( ( l 15
15 ) ) l 15
15 [ [ l 15
15 ] ] l 15
15 _ [ l 5
16 ( ( r 16
16 [ [ r 16
16 ) ) r 16
16 ] ] r 16
16 _ _ r 17
17 ( ( r 17
17 _ ( l 18
18 ( ( l 18
18 _ _ l 19
19 ( ( l 19
19 ) ) l 19
19 [ [ l 19
19 ] ] l 19
19 _ ) l 5
20 ( ( r 20
20 [ [ r 20
20 ) ) r 20
20 ] ] r 20
20 _ _ r 21
21 ( ( r 21
21 _ ( l 22
22 ( ( l 22
22 _ _ l 23
23 ( ( l 23
23 ) ) l 23
23 [ [ l 23
23 ] ] l 23
23 _ ] l 5
24 ( ( l 24
24 ) ) l 24
24 [ [ l 24
24 ] ] l 24
24 _ ( r 26
25 ( ( l 25
25 ) ) l 25
25 [ [ l 25
25 ] ] l 25
25 _ [ r 26
26 ( _ r 27
26 ) _ r 31
26 [ _ r 35
26 ] _ r 39
26 _ _ l 43
27 ( ( r 27
27 ) ) r 27
27 [ [ r 27
27 ] ] r 27
27 _ _ r 28
28 ( ( r 28
28 ) ) r 28
28 [ [ r 28
28 ] ] r 28
28 _ ( l 29
29 ( ( l 29
29 ) ) l 29
29 [ [ l 29
29 ] ] l 29
29 _ _ l 30
30 ( ( l 30
30 ) ) l 30
30 [ [ l 30
30 ] ] l 30
30 _ ( r 26
31 ( ( r 31
31 ) ) r 31
31 [ [ r 31
31 ] ] r 31
31 _ _ r 32
32 ( ( r 32
32 ) ) r 32
32 [ [ r 32
32 ] ] r 32
32 _ ) l 33
33 ( ( l 33
33 ) ) l 33
33 [ [ l 33
33 ] ] l 33
33 _ _ l 34
34 ( ( l 34
34 ) ) l 34
34 [ [ l 34
34 ] ] l 34
34 _ ) r 26
35 ( ( r 35
35 ) ) r 35
35 [ [ r 35
35 ] ] r 35
35 _ _ r 36
36 ( ( r 36
36 ) ) r 36
36 [ [ r 36
36 ] ] r 36
36 _ [ l 37
37 ( ( l 37
37 ) ) l 37
37 [ [ l 37
37 ] ] l 37
37 _ _ l 38
38 ( ( l 38
38 ) ) l 38
38 [ [ l 38
38 ] ] l 38
38 _ [ r 26
39 ( ( r 39
39 ) ) r 39
39 [ [ r 39
39 ] ] r 39
39 _ _ r 40
40 ( ( r 40
40 ) ) r 40
40 [ [ r 40
40 ] ] r 40
40 _ ] l 41
41 ( ( l 41
41 ) ) l 41
41 [ [ l 41
41 ] ] l 41
41 _ _ l 42
42 ( ( l 42
42 ) ) l 42
42 [ [ l 42
42 ] ] l 42
42 _ ] r 26
43 ( ( l 43
43 ) ) l 43
43 [ [ l 43
43 ] ] l 43
43 _ _ l 44
44 ( ( l 44
44 ) ( l 45
44 _ [ l 56
45 _ _ r 51
45 ( ) r 46
46 ( ( r 46
46 _ _ r 47
47 ( ( r 47
47 ) ) r 47
47 [ [ r 47
47 ] ] r 47
47 _ _ l 48
48 ) ) r 48
48 ] ] r 4
48 _ ( r 26
51 ( ( r 51
51 _ _ r 52
52 ( ( r 52
52 ) ) r 52
52 [ [ r 52
52 ] ] r 52
52 _ _ l 53
53 ) ) r 53
53 ] ] r 44
53 _ ( l 56
56 ( ( l 56
56 ) ) l 56
56 [ [ l 56
56 ] ] l 56
56 _ _ l 57
57 ( ( l 57
57 ) ( l 57
57 _ ( r 0
58 ( _ r 59
58 ) _ r 64
58 [ _ r 68
58 ] _ r 73
59 ( ( r 59
59 [ [ r 59
59 ) ) r 59
59 ] ] r 59
59 _ _ r 60
60 ( ( r 60
60 _ _ l 61
61 ( _ l 62
61 _ _ l 77
62 ( ( l 62
62 _ _ l 63
63 ( ( l 63
63 ) ) l 63
63 ] ] l 63
63 [ [ l 63
63 _ ( l 58
64 ( ( r 64
64 [ [ r 64
64 ) ) r 64
64 ] ] r 64
64 _ _ r 65
65 ( ( r 65
65 _ ( l 66
66 ( ( l 66
66 _ _ l 67
67 ( ( l 67
67 ) ) l 67
67 ] ] l 67
67 [ [ l 67
67 _ ) l 58
68 ( ( r 68
68 [ [ r 68
68 ) ) r 68
68 ] ] r 68
68 _ _ r 69
69 ( ( r 69
69 _ _ l 70
70 ( _ l 71
70 _ _ l 72
71 ( ( l 71
71 _ _ l 72
72 ( ( l 72
72 ) ) l 72
72 ] ] l 72
72 [ [ l 72
72 _ [ l 58
73 ( ( r 73
73 [ [ r 73
73 ) ) r 73
73 ] ] r 73
73 _ _ r 74
74 ( ( r 74
74 _ ( l 75
75 ( ( l 75
75 _ _ l 76
76 ( ( l 76
76 ) ) l 76
76 ] ] l 76
76 [ [ l 76
76 _ ] l 58
77 ( ( l 77
77 ) ) l 77
77 [ [ l 77
77 ] ] l 77
77 _ ( r 78
78 ( _ r 79
78 ) _ r 83
78 [ _ r 87
78 ] _ r 91
78 _ ( r 95
79 ( ( r 79
79 ) ) r 79
79 [ [ r 79
79 ] ] r 79
79 _ _ r 81
80 ( ( l 80
80 ) ) l 80
80 [ [ l 80
80 ] ] l 80
80 _ ( r 78
81 ( ( r 81
81 ) ) r 81
81 [ [ r 81
81 ] ] r 81
81 _ ( l 82
82 ( ( l 82
82 ) ) l 82
82 [ [ l 82
82 ] ] l 82
82 _ _ l 80
83 ( ( r 83
83 ) ) r 83
83 [ [ r 83
83 ] ] r 83
83 _ _ r 85
84 ( ( l 84
84 ) ) l 84
84 [ [ l 84
84 ] ] l 84
84 _ ) r 78
85 ( ( r 85
85 ) ) r 85
85 [ [ r 85
85 ] ] r 85
85 _ ) l 86
86 ( ( l 86
86 ) ) l 86
86 [ [ l 86
86 ] ] l 86
86 _ _ l 84
87 ( ( r 87
87 ) ) r 87
87 [ [ r 87
87 ] ] r 87
87 _ _ r 89
88 ( ( l 88
88 ) ) l 88
88 [ [ l 88
88 ] ] l 88
88 _ [ r 78
89 ( ( r 89
89 ) ) r 89
89 [ [ r 89
89 ] ] r 89
89 _ [ l 90
90 ( ( l 90
90 ) ) l 90
90 [ [ l 90
90 ] ] l 90
90 _ _ l 88
91 ( ( r 91
91 ) ) r 91
91 [ [ r 91
91 ] ] r 91
91 _ _ r 93
92 ( ( l 92
92 ) ) l 92
92 [ [ l 92
92 ] ] l 92
92 _ ] r 78
93 ( ( r 93
93 ) ) r 93
93 [ [ r 93
93 ] ] r 93
93 _ ] l 94
94 ( ( l 94
94 ) ) l 94
94 [ [ l 94
94 ] ] l 94
94 _ _ l 92
95 ( ( r 95
95 ) ) r 95
95 [ [ r 95
95 ] ] r 95
95 _ ( l 56


### 59 states, 6 colors

33 states reduced: 7, 9, 12, 14, 17, 18, 21, 22, 28, 30, 32, 34, 36, 38, 40, 42, 46, 60, 62, 65, 66, 69, 71, 74, 75, 81, 82, 85, 86, 89, 90, 93, 94.

Input: Multiplier in the form of (((...(((, then a space, then a hydra without rightmost )'s and ]'s.

0 ( ( r 0
0 _ _ l 3
1 _ _ r halt
1 ( ( r 2
2 ( ( r 2
2 ) ) r 2
2 [ [ r 2
2 x _ l 4
2 ] ] r 2
2 _ _ l 4
3 ( ) r 3
3 _ _ r 1
4 ) _ l 4
4 ( _ l 5
4 [ _ l 58
4 ] _ l 4
4 _ [ r 26
5 ( _ r 6
5 [ _ r 11
5 ) _ r 16
5 ] _ r 20
5 _ _ l 57
6 ( ( r 6
6 [ [ r 6
6 ) ) r 6
6 ] ] r 6
6 x x r 6
6 _ _ l 8
8 x _ l 10
8 ( ( r 26
8 ) ) r 26
8 [ [ r 26
8 ] ] r 26
8 _ _ r 26
10 x x l 10
10 ( ( l 10
10 ) ) l 10
10 [ [ l 10
10 ] ] l 10
10 _ ( l 5
11 ( ( r 11
11 [ [ r 11
11 ) ) r 11
11 ] ] r 11
11 x x r 11
11 _ _ l 13
13 _ x l 25
13 x _ l 15
13 ) ) r 78
13 [ [ r 78
13 ] ] r 78
13 ( ( r 78
15 ( ( l 15
15 ) ) l 15
15 [ [ l 15
15 ] ] l 15
15 _ [ l 5
16 ( ( r 16
16 [ [ r 16
16 ) ) r 16
16 ] ] r 16
16 x x r 16
16 _ x l 19
19 x x l 19
19 ( ( l 19
19 ) ) l 19
19 [ [ l 19
19 ] ] l 19
19 _ ) l 5
20 ( ( r 20
20 [ [ r 20
20 ) ) r 20
20 ] ] r 20
20 _ x l 22
20 x x l 23
23 x x l 23
23 ( ( l 23
23 ) ) l 23
23 [ [ l 23
23 ] ] l 23
23 _ ] l 5
24 ( ( l 24
24 ) ) l 24
24 [ [ l 24
24 ] ] l 24
24 _ ( r 26
25 ( ( l 25
25 ) ) l 25
25 [ [ l 25
25 ] ] l 25
25 _ [ r 26
26 ( _ r 27
26 ) _ r 31
26 [ _ r 35
26 ] _ r 39
26 x x r 51
26 _ ) r 48
27 ( ( r 27
27 ) ) r 27
27 [ [ r 27
27 ] ] r 27
27 x x r 27
27 _ ( l 29
29 ( ( l 29
29 ) ) l 29
29 [ [ l 29
29 ] ] l 29
29 x x l 29
29 _ ( r 26
31 ( ( r 31
31 ) ) r 31
31 [ [ r 31
31 ] ] r 31
31 x x r 31
31 _ ) l 33
33 ( ( l 33
33 ) ) l 33
33 [ [ l 33
33 ] ] l 33
33 x x l 33
33 _ ) r 26
35 ( ( r 35
35 ) ) r 35
35 [ [ r 35
35 ] ] r 35
35 x x r 35
35 _ [ l 37
37 ( ( l 37
37 ) ) l 37
37 [ [ l 37
37 ] ] l 37
37 x x l 37
37 _ [ r 26
39 ( ( r 39
39 ) ) r 39
39 [ [ r 39
39 ] ] r 39
39 x x r 39
39 _ ] l 41
41 ( ( l 41
41 ) ) l 41
41 [ [ l 41
41 ] ] l 41
41 x x l 41
41 _ ] r 26
43 ( ( l 43
43 ) ) l 43
43 [ [ l 43
43 ] ] l 43
43 x x l 43
43 _ _ l 44
44 ( ( l 44
44 ) ( l 45
44 _ [ l 56
45 _ _ r 52
45 ( ) r 47
47 ( ( r 47
47 ) ) r 47
47 [ [ r 47
47 ] ] r 47
47 _ _ r 47
47 x x l 48
48 ) ) r 48
48 ] ] r 4
48 x ( r 26
48 _ x l 24
51 ( ( r 51
51 ) ) r 51
51 [ [ r 51
51 ] ] r 51
51 _ x l 43
52 _ _ r 52
52 ( ( r 52
52 ) ) r 52
52 [ [ r 52
52 ] ] r 52
52 x _ l 53
53 ) ) r 53
53 ] ] r 44
53 _ ( l 56
56 ( ( l 56
56 ) ) l 56
56 [ [ l 56
56 ] ] l 56
56 _ _ l 57
57 ( ( l 57
57 ) ( l 57
57 _ ( r 0
58 ( _ r 59
58 ) _ r 64
58 [ _ r 68
58 ] _ r 73
58 _ x l 77
59 ( ( r 59
59 [ [ r 59
59 ) ) r 59
59 ] ] r 59
59 x x r 59
59 _ _ l 61
61 x _ l 63
61 ( ( r 58
61 ) ) r 58
61 [ [ r 58
61 ] ] r 58
61 _ _ r 58
63 x x l 63
63 ( ( l 63
63 ) ) l 63
63 ] ] l 63
63 [ [ l 63
63 _ ( l 58
64 ( ( r 64
64 [ [ r 64
64 ) ) r 64
64 ] ] r 64
64 x x r 64
64 _ x l 67
67 x x l 67
67 ( ( l 67
67 ) ) l 67
67 ] ] l 67
67 [ [ l 67
67 _ ) l 58
68 ( ( r 68
68 [ [ r 68
68 ) ) r 68
68 ] ] r 68
68 x x r 68
68 _ _ l 70
70 x _ l 72
70 _ _ l 72
72 x x l 72
72 ( ( l 72
72 ) ) l 72
72 ] ] l 72
72 [ [ l 72
72 _ [ l 58
73 ( ( r 73
73 [ [ r 73
73 ) ) r 73
73 ] ] r 73
73 x x r 73
73 _ x l 76
76 x x l 76
76 ( ( l 76
76 ) ) l 76
76 ] ] l 76
76 [ [ l 76
76 _ ] l 58
77 ( ( l 77
77 ) ) l 77
77 [ [ l 77
77 ] ] l 77
77 _ ( r 78
78 ( _ r 79
78 ) _ r 83
78 [ _ r 87
78 ] _ r 91
78 x ( r 95
78 _ ] r 13
79 ( ( r 79
79 ) ) r 79
79 [ [ r 79
79 ] ] r 79
79 x x r 79
79 _ ( l 80
80 ( ( l 80
80 ) ) l 80
80 [ [ l 80
80 ] ] l 80
80 x x l 80
80 _ ( r 78
83 ( ( r 83
83 ) ) r 83
83 [ [ r 83
83 ] ] r 83
83 x x r 83
83 _ ) l 84
84 ( ( l 84
84 ) ) l 84
84 x x l 84
84 [ [ l 84
84 ] ] l 84
84 _ ) r 78
87 ( ( r 87
87 ) ) r 87
87 x x r 87
87 [ [ r 87
87 ] ] r 87
87 _ [ l 88
88 ( ( l 88
88 ) ) l 88
88 [ [ l 88
88 x x l 88
88 ] ] l 88
88 _ [ r 78
91 ( ( r 91
91 ) ) r 91
91 x x r 91
91 [ [ r 91
91 ] ] r 91
91 _ ] l 92
92 ( ( l 92
92 ) ) l 92
92 x x l 92
92 [ [ l 92
92 ] ] l 92
92 _ ] r 78
95 ( ( r 95
95 ) ) r 95
95 [ [ r 95
95 ] ] r 95
95 _ ( l 56


### 44 states, 7 colors

15 states reduced: 1, 3, 10, 20, 23, 25, 33, 41, 45, 57, 72, 73, 76, 84, 92.

Input: Multiplier in the form of yyy...yyy, then a hydra without rightmost )'s and ]'s.

0 y y r 0
0 ( ( l 2
0 _ _ r halt
2 ( ( r 2
2 ) ) r 2
2 [ [ r 2
2 x _ l 4
2 ] ] r 2
2 _ _ l 4
2 y x r 2
4 ) _ l 4
4 ( _ l 5
4 [ _ l 58
4 ] _ l 4
4 _ [ r 26
5 ( _ r 6
5 [ y r 11
5 ) _ r 16
5 ] y r 16
5 x y l 56
6 ( ( r 6
6 [ [ r 6
6 ) ) r 6
6 ] ] r 6
6 x x r 6
6 _ _ l 8
8 x _ l 15
8 ( ( r 26
8 ) ) r 26
8 [ [ r 26
8 ] ] r 26
8 _ _ r 26
11 ( ( r 11
11 [ [ r 11
11 ) ) r 11
11 ] ] r 11
11 x x r 11
11 _ _ l 13
13 _ x l 24
13 x _ l 15
13 ) ) r 78
13 [ [ r 78
13 ] ] r 78
13 ( ( r 78
15 ( ( l 15
15 ) ) l 15
15 x x l 15
15 [ [ l 15
15 ] ] l 15
15 y [ l 5
15 _ ( l 5
16 ( ( r 16
16 [ [ r 16
16 ) ) r 16
16 ] ] r 16
16 x x r 16
16 _ x l 19
19 x x l 19
19 ( ( l 19
19 ) ) l 19
19 [ [ l 19
19 ] ] l 19
19 y ] l 5
19 _ ) l 5
24 ( ( l 24
24 ) ) l 24
24 [ [ l 24
24 ] ] l 24
24 _ ( r 26
24 y [ r 26
26 ( _ r 27
26 ) y r 31
26 [ _ r 35
26 ] y r 39
26 x x r 51
26 _ ) r 48
27 ( ( r 27
27 ) ) r 27
27 [ [ r 27
27 ] ] r 27
27 x x r 27
27 _ ( l 29
29 ( ( l 29
29 ) ) l 29
29 [ [ l 29
29 ] ] l 29
29 x x l 29
29 _ ( r 26
29 y ) r 26
31 ( ( r 31
31 ) ) r 31
31 [ [ r 31
31 ] ] r 31
31 x x r 31
31 _ ) l 29
35 ( ( r 35
35 ) ) r 35
35 [ [ r 35
35 ] ] r 35
35 x x r 35
35 _ [ l 37
37 ( ( l 37
37 ) ) l 37
37 [ [ l 37
37 ] ] l 37
37 x x l 37
37 _ [ r 26
37 y ] r 26
39 ( ( r 39
39 ) ) r 39
39 [ [ r 39
39 ] ] r 39
39 x x r 39
39 _ ] l 37
43 ( ( l 43
43 ) ) l 43
43 [ [ l 43
43 ] ] l 43
43 x x l 44
43 _ _ r 52
43 y x r 47
44 ( ( l 44
44 ) ) l 44
44 [ [ l 44
44 ] ] l 44
44 y y l 44
44 x y l 43
44 _ [ l 56
47 y y r 47
47 ( ( r 47
47 ) ) r 47
47 [ [ r 47
47 ] ] r 47
47 x x l 48
48 ) ) r 48
48 ] ] r 4
48 x ( r 26
48 _ x l 24
51 ( ( r 51
51 ) ) r 51
51 [ [ r 51
51 ] ] r 51
51 _ x l 43
52 y y r 52
52 ( ( r 52
52 ) ) r 52
52 [ [ r 52
52 ] ] r 52
52 x _ l 53
53 ) ) r 53
53 ] ] r 44
53 _ ( l 56
56 ( ( l 56
56 ) ) l 56
56 [ [ l 56
56 ] ] l 56
56 y y l 56
56 x y l 56
56 _ y r 0
58 ( _ r 59
58 ) _ r 64
58 [ y r 68
58 ] y r 64
58 _ x l 77
59 ( ( r 59
59 [ [ r 59
59 ) ) r 59
59 ] ] r 59
59 x x r 59
59 _ _ l 61
61 x _ l 63
61 ( ( r 58
61 ) ) r 58
61 [ [ r 58
61 ] ] r 58
61 _ _ r 58
63 x x l 63
63 ( ( l 63
63 ) ) l 63
63 ] ] l 63
63 [ [ l 63
63 _ ( l 58
63 y [ l 58
64 ( ( r 64
64 [ [ r 64
64 ) ) r 64
64 ] ] r 64
64 x x r 64
64 _ x l 67
67 x x l 67
67 ( ( l 67
67 ) ) l 67
67 ] ] l 67
67 [ [ l 67
67 _ ) l 58
67 y ] l 58
68 ( ( r 68
68 [ [ r 68
68 ) ) r 68
68 ] ] r 68
68 x x r 68
68 _ _ l 70
70 x _ l 63
70 _ _ l 63
77 ( ( l 77
77 ) ) l 77
77 [ [ l 77
77 ] ] l 77
77 _ ( r 78
78 ( _ r 79
78 ) y r 83
78 [ _ r 87
78 ] y r 91
78 x ( r 95
78 _ ] r 13
79 ( ( r 79
79 ) ) r 79
79 [ [ r 79
79 ] ] r 79
79 x x r 79
79 _ ( l 80
80 ( ( l 80
80 ) ) l 80
80 [ [ l 80
80 ] ] l 80
80 x x l 80
80 _ ( r 78
80 y ) r 78
83 ( ( r 83
83 ) ) r 83
83 [ [ r 83
83 ] ] r 83
83 x x r 83
83 _ ) l 80
87 ( ( r 87
87 ) ) r 87
87 x x r 87
87 [ [ r 87
87 ] ] r 87
87 _ [ l 88
88 ( ( l 88
88 ) ) l 88
88 [ [ l 88
88 x x l 88
88 ] ] l 88
88 _ [ r 78
88 y ] r 78
91 ( ( r 91
91 ) ) r 91
91 x x r 91
91 [ [ r 91
91 ] ] r 91
91 _ ] l 88
95 ( ( r 95
95 ) ) r 95
95 [ [ r 95
95 ] ] r 95
95 _ ( l 56


### 40 states, 9 colors

4 states reduced: 15, 37, 67, 88.

This is raw version, meaning that there will be probably more improvements later. But the improvement I have in mind might need even more colors.

Input: Multiplier in the form of yyy...yyy, then a hydra without rightmost )'s and ]'s.

0 y y r 0
0 ( ( l 2
0 _ _ r halt
2 ( ( r 2
2 ) ) r 2
2 [ [ r 2
2 x _ l 4
2 ] ] r 2
2 _ _ l 4
2 y x r 2
4 ) _ l 4
4 ( _ l 5
4 [ _ l 58
4 ] _ l 4
4 _ [ r 26
5 ( _ r 6
5 [ y r 11
5 ) } r 16
5 ] > r 16
5 x y l 56
6 ( ( r 6
6 [ [ r 6
6 ) ) r 6
6 ] ] r 6
6 x x r 6
6 _ _ l 8
8 x _ l 15
8 ( ( r 26
8 ) ) r 26
8 [ [ r 26
8 ] ] r 26
8 _ _ r 26
11 ( ( r 11
11 [ [ r 11
11 ) ) r 11
11 ] ] r 11
11 x x r 11
11 _ _ l 13
13 _ x l 24
13 x _ l 15
13 ) ) r 78
13 [ [ r 78
13 ] ] r 78
13 ( ( r 78
15 ( ( l 15
15 ) ) l 15
15 x x l 15
15 [ [ l 15
15 ] ] l 15
15 } ) l 5
15 > ] l 5
15 y [ l 5
15 _ ( l 5
16 ( ( r 16
16 [ [ r 16
16 ) ) r 16
16 ] ] r 16
16 x x r 16
16 _ x l 15
24 ( ( l 24
24 ) ) l 24
24 [ [ l 24
24 ] ] l 24
24 _ ( r 26
24 y [ r 26
26 ( _ r 27
26 ) y r 31
26 [ } r 35
26 ] > r 39
26 x x r 51
26 _ ) r 48
27 ( ( r 27
27 ) ) r 27
27 [ [ r 27
27 ] ] r 27
27 x x r 27
27 _ ( l 29
29 ( ( l 29
29 ) ) l 29
29 [ [ l 29
29 ] ] l 29
29 x x l 29
29 _ ( r 26
29 y ) r 26
29 > ] r 26
29 } [ r 26
31 ( ( r 31
31 ) ) r 31
31 [ [ r 31
31 ] ] r 31
31 x x r 31
31 _ ) l 29
35 ( ( r 35
35 ) ) r 35
35 [ [ r 35
35 ] ] r 35
35 x x r 35
35 _ [ l 29
39 ( ( r 39
39 ) ) r 39
39 [ [ r 39
39 ] ] r 39
39 x x r 39
39 _ ] l 29
43 ( ( l 43
43 ) ) l 43
43 [ [ l 43
43 ] ] l 43
43 x x l 44
43 _ _ r 52
43 y x r 47
44 ( ( l 44
44 ) ) l 44
44 [ [ l 44
44 ] ] l 44
44 y y l 44
44 x y l 43
44 _ [ l 56
47 y y r 47
47 ( ( r 47
47 ) ) r 47
47 [ [ r 47
47 ] ] r 47
47 x x l 48
48 ) ) r 48
48 ] ] r 4
48 x ( r 26
48 _ x l 24
51 ( ( r 51
51 ) ) r 51
51 [ [ r 51
51 ] ] r 51
51 _ x l 43
52 y y r 52
52 ( ( r 52
52 ) ) r 52
52 [ [ r 52
52 ] ] r 52
52 x _ l 53
53 ) ) r 53
53 ] ] r 44
53 _ ( l 56
56 ( ( l 56
56 ) ) l 56
56 [ [ l 56
56 ] ] l 56
56 y y l 56
56 x y l 56
56 _ y r 0
58 ( _ r 59
58 ) } r 64
58 [ y r 68
58 ] > r 64
58 _ x l 77
59 ( ( r 59
59 [ [ r 59
59 ) ) r 59
59 ] ] r 59
59 x x r 59
59 _ _ l 61
61 x _ l 63
61 ( ( r 58
61 ) ) r 58
61 [ [ r 58
61 ] ] r 58
61 _ _ r 58
63 x x l 63
63 ( ( l 63
63 ) ) l 63
63 ] ] l 63
63 [ [ l 63
63 _ ( l 58
63 y [ l 58
63 } ) l 58
63 > ] l 58
64 ( ( r 64
64 [ [ r 64
64 ) ) r 64
64 ] ] r 64
64 x x r 64
64 _ x l 63
68 ( ( r 68
68 [ [ r 68
68 ) ) r 68
68 ] ] r 68
68 x x r 68
68 _ _ l 70
70 x _ l 63
70 _ _ l 63
77 ( ( l 77
77 ) ) l 77
77 [ [ l 77
77 ] ] l 77
77 _ ( r 78
78 ( _ r 79
78 ) y r 83
78 [ } r 87
78 ] > r 91
78 x ( r 95
78 _ ] r 13
79 ( ( r 79
79 ) ) r 79
79 [ [ r 79
79 ] ] r 79
79 x x r 79
79 _ ( l 80
80 ( ( l 80
80 ) ) l 80
80 [ [ l 80
80 ] ] l 80
80 x x l 80
80 _ ( r 78
80 y ) r 78
80 } [ r 78
80 > ] r 78
83 ( ( r 83
83 ) ) r 83
83 [ [ r 83
83 ] ] r 83
83 x x r 83
83 _ ) l 80
87 ( ( r 87
87 ) ) r 87
87 x x r 87
87 [ [ r 87
87 ] ] r 87
87 _ [ l 80
91 ( ( r 91
91 ) ) r 91
91 x x r 91
91 [ [ r 91
91 ] ] r 91
91 _ ] l 80
95 ( ( r 95
95 ) ) r 95
95 [ [ r 95
95 ] ] r 95
95 _ ( l 56


## Bounds using (0,1) game machines

They can be made using the 5 color 2 state record holder.

$$\Sigma(94,5) > f_{\vartheta(\varepsilon_{\Omega+1})}(1.7 \cdot 10^{352})$$

$$\Sigma(61,6) > f_{\vartheta(\varepsilon_{\Omega+1})}(1.7 \cdot 10^{352})$$

$$\Sigma(46,7) > f_{\vartheta(\varepsilon_{\Omega+1})}(1.7 \cdot 10^{352})$$

$$\Sigma(42,9) > f_{\vartheta(\varepsilon_{\Omega+1})}(1.7 \cdot 10^{352})$$