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A page for analyses of individual machines.
={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}=
={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}=
<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
1. One of the seven potential [[BB(4,3)]] champions discovered by Pavel Kropitz in May 2024. Analysed in September 2025. This TM runs the shortest of Pavel's potential champions, achieving a score of about <math>10 \uparrow\uparrow 9.873987</math>.
<div class="toccolours mw-collapsible mw-collapsed">'''Analysis'''<div class="mw-collapsible-content">
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 42: Line 44:
by:
by:
S (12)^a 2 (12)^b A> 0^2 S
S (12)^a 2 (12)^b A> 0^2 S
--> S (12)^a 2 (12)^b 1 B> 0 S
--> S (12)^a 2 (12)^b <A (11) S
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
Line 51: Line 51:


8. S (12)^a 2 (12)^b A> 0^inf --> S 2 (12)^b+2a A> 0^inf [+4a^2 +8a +4ba steps]
8. S (12)^a 2 (12)^b A> 0^inf --> S 2 (12)^b+2a A> 0^inf [+4a^2 +8a +4ba steps]
Obtained by repeating rule 8.
Obtained by repeating rule 7.


9. S (12)^a <D (11)^b 0^inf --> S (12)^a-1 <D (11)^2b+3 0^inf [+4b^2 +22b +22 steps]
9. S (12)^a <D (11)^b 0^inf --> S (12)^a-1 <D (11)^2b+3 0^inf [+4b^2 +22b +22 steps]
Line 105: Line 105:
--> S (11)^a-2 (12)^b+3 2^2 A> (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 A> (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 A> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 A> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 1 B> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 <A (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 1 <C 1 (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 <A (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 2^2 <A (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 2 <C 1 (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 2 <C 1 (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf
--> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf
12. S 1^a <A (11)^b 0^inf --> 1^a-1 <A (11)^b+1 2 0^inf [+4b +5 steps]
by:
S 1^a <A (11)^b 0^inf
--> S 1^a D> (11)^b 0^inf
--> S 1^a (21)^b D> 0^inf
--> S 1^a (21)^b 2 B> 0^inf
--> S 1^a (21)^b 2 <B 2 0^inf
--> S 1^a (21)^b <C 1 2 0^inf
--> S 1^a <C (11)^b 1 2 0^inf
--> 1^a-1 <A (11)^b+1 2 0^inf
</pre>
</pre>
==Functions==
Let A(a,b,c) = S (11)^a (12)^b <D (11)^c 0^inf
Let A(a,b,c) = S (11)^a (12)^b <D (11)^c 0^inf
* Rule 9: A(a, b, c) --> A(a, b - 1, 2c + 3)
* Rule 9: A(a, b, c) --> A(a, b - 1, 2c + 3)
Line 118: Line 127:


Further: let <math>f(n) = 2^{n+1} \times 3</math>
Further: let <math>f(n) = 2^{n+1} \times 3</math>
* If c = 3: A(a, b, 3) --> A(a, 0, f(c) - 3) --> A(a - 2, f(c), 3)
* If c = 3: A(a, b, 3) --> A(a, 0, f(b) - 3) --> A(a - 2, f(b), 3)


* A(2k + d, 0, c) --> <math>A(d, f^{k-1}(c+3), 3)</math>
* A(a, 0, c) --> <math>A(a-2, c+3, 3)</math> --> <math>A(a-2, 0, f(c+3)-3)</math>
 
* A(2k + d, b, 3) --> <math>A(d, f^{k}(b), 3)</math>
 
==Trajectory==
<pre>
<pre>
The TM enters configuration A(19, 2, 3) after 799 steps with tape:
S=0: 0^inf A> 0^inf
0^inf 2 1 (11)^19 (12)^2 <D (11)^3 0^inf
S=1: 0^inf 1 B> 0^inf
S=5: 0^inf <A (11)^1 0^inf
S=6: 0^inf 1 B> (11)^1 0^inf
S=8: 0^inf 1 (11)^1 B> 0^inf
S=9: 0^inf 1 (11)^1 <B 2 0^inf
S=10: 0^inf 1 (11)^1 B> 2 0^inf
S=11: 0^inf 1 (11)^1 <C 1 0^inf
S=12: 0^inf (11)^1 <A (11)^1 0^inf
S=21: 0^inf 1 <A (11)^2 2 0^inf by rule 12
S=22: 0^inf 1 D> (11)^2 2 0^inf
S=26: 0^inf 1 (21)^2 D> 2 0^inf
S=27: 0^inf 1 (21)^2 2 D> 0^inf
S=28: 0^inf 1 (21)^2 2^2 B> 0^inf
S=29: 0^inf 1 (21)^2 2^2 <B 2 0^inf
S=30: 0^inf 1 (21)^2 2 <C 1 2 0^inf
S=31: 0^inf 1 (21)^2 <D (11)^1 2 0^inf
S=32: 0^inf 1 (21)^1 2^2 A> (11)^1 2 0^inf
S=34: 0^inf 1 (21)^1 2^2 (12)^1 A> 2 0^inf
S=35: 0^inf 1 (21)^1 2^2 (12)^1 <C 1 0^inf
S=36: 0^inf 1 (21)^1 2^2 1 <D (11)^1 0^inf
S=37: 0^inf 1 (21)^1 2^3 A> (11)^1 0^inf
S=39: 0^inf (12)^2 2^2 (12)^1 A> 0^inf
S=46: 0^inf (12)^2 2^2 <A (11)^2 0^inf
S=47: 0^inf (12)^2 2 <C 1 (11)^2 0^inf
S=48: 0^inf (12)^2 <D (11)^3 0^inf
S=172: 0^inf (12)^1 <D (11)^9 0^inf by rule 9
S=716: 0^inf <D (11)^21 0^inf by rule 9
S=717: 0^inf 2 B> (11)^21 0^inf
S=759: 0^inf 2 (11)^21 B> 0^inf
S=760: 0^inf 2 (11)^21 <B 2 0^inf
S=761: 0^inf 2 (11)^21 B> 2 0^inf
S=762: 0^inf 2 (11)^21 <C 1 0^inf
S=763: 0^inf 2 (11)^20 1 <A (11)^1 0^inf
S=772: 0^inf 2 (11)^20 <A (11)^2 2 0^inf
S=773: 0^inf 2 (11)^20 D> (11)^2 2 0^inf
S=777: 0^inf 2 (11)^20 (21)^2 D> 2 0^inf
S=778: 0^inf 2 (11)^20 (21)^2 2 D> 0^inf
S=779: 0^inf 2 1 (11)^19 1 (21)^2 2^2 B> 0^inf
S=780: 0^inf 2 1 (11)^19 (12)^3 2 <B 2 0^inf
S=781: 0^inf 2 1 (11)^19 (12)^3 <C 1 2 0^inf
S=782: 0^inf 2 1 (11)^19 (12)^2 1 <D (11)^1 2 0^inf
S=783: 0^inf 2 1 (11)^19 (12)^2 2 A> (11)^1 2 0^inf
S=785: 0^inf 2 1 (11)^19 (12)^2 2 (12)^1 A> 2 0^inf
S=786: 0^inf 2 1 (11)^19 (12)^2 2 (12)^1 <C 1 0^inf
S=787: 0^inf 2 1 (11)^19 (12)^2 2 1 <D (11)^1 0^inf
S=788: 0^inf 2 1 (11)^19 (12)^2 2^2 A> (11)^1 0^inf
S=790: 0^inf 2 1 (11)^19 (12)^2 2^2 (12)^1 A> 0^inf
S=797: 0^inf 2 1 (11)^19 (12)^2 2^2 <A (11)^2 0^inf
S=798: 0^inf 2 1 (11)^19 (12)^2 2 <C 1 (11)^2 0^inf
S=799: 0^inf 2 1 (11)^19 (12)^2 <D (11)^3 0^inf
= A(19, 2, 3)
</pre>
</pre>
==Trajectory==
A(19, 2, 3) --> <math>A(1, f^{9}(2), 3)</math> --> <math>A(1, 0, f^{10}(2)-3)</math>
A(19, 2, 3) --> A(19, 0, 21) --> <math>A(1, f^{8}(24), 3)</math>


--> <math>A(1, 0, f^{9}(24) - 3)</math>
Let m = <math>f^{10}(2)-3</math>


Let's have <math> f^{9}(24) - 3</math> = m.
--> 0^inf 2 1 (11)^1 <D (11)^m 0^inf
<pre>
<pre>
Final trajectory:
Final trajectory:
Line 145: Line 207:
--> 0^inf 1 Z> (11)^m+3 (12)^1 0^inf
--> 0^inf 1 Z> (11)^m+3 (12)^1 0^inf
Score = 2m + 9
Score = 2m + 9
</pre>


==Approximate Score==
Score calculated in HyperCalc:
Score calculated in HyperCalc:
(10^)^8 30,302,671.815163
(10^)^8 30,302,671.815163
Or in tetration: 10^^10.873987 (truncated)
 
Or in tetration: 10^^9.873987 (truncated)
 
==Permutations==
'''Starting in state B'''
<pre>
0^inf <B 0^inf
--> translated cycler
</pre>
 
'''Starting in state C'''
<pre>
0^inf <C 0^inf
--> 0^inf 1 Z> 0^inf
</pre>
 
'''Starting in state D'''
<pre>
0^inf <D 0^inf
S=1: 0^inf 2 B> 0^inf
S=2: 0^inf 2 <B 2 0^inf
S=3: 0^inf <C 1 2 0^inf
S=4: 0^inf 1 Z> 1 2 0^inf
</pre>
</pre>
</div></div>
</div></div>


=1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD=
={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}=
<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
2. One of the seven potential [[BB(4,3)]] champions discovered by Pavel Kropitz in May 2024. Analysed in October 2025. This TM runs the longest of Pavel's potential champions and is - as of October 20th 2025 - the highest scoring BB(4,3) TM with score > <math>2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 7.92 \times 10^{28}</math>.
<div class="toccolours mw-collapsible mw-collapsed">'''Analysis'''<div class="mw-collapsible-content">
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 191: Line 279:
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]
by:
S (12)^a A> 0^2 S
--> S (12)^a <A (11)^1 S
--> S <A (11)^a+1 S


7. S A> (11)^1 2^b S --> S 2 A> (11)^1 2^b-1 S [+5 steps]
7. S A> (11)^1 2^b S --> S 2 A> (11)^1 2^b-1 S [+5 steps]
Line 260: Line 352:
by repetition of rule 14
by repetition of rule 14


16. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 (12)^b+1 1 <D (11)^1 0^inf [+10b +26 steps]
16. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf [+10b +28 steps]
by:
by:
S (12)^a 2 1 <D (11)^b 0^inf
S (12)^a 2 1 <D (11)^b 0^inf
Line 268: Line 360:
--> S (12)^a 2 <C 1 (11)^b+1 0^inf
--> S (12)^a 2 <C 1 (11)^b+1 0^inf
--> S (12)^a <D (11)^b+2 0^inf
--> S (12)^a <D (11)^b+2 0^inf
--> S (12)^a-1 1 <D (11)^b+2 2 0^inf by rule 10
--> S (12)^a-1 1 <D (11)^b+3 2 0^inf by rule 10
--> S (12)^a-1 2 A> (11)^b+2 2 0^inf
--> S (12)^a-1 2 A> (11)^b+3 2 0^inf
--> S (12)^a-1 2 (12)^b+2 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+3 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+2 <C 1 0^inf
--> S (12)^a-1 2 (12)^b+3 <C 1 0^inf
--> S (12)^a-1 2 (12)^b+1 1 <D (11)^1 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf


17. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 1 <D (11)^2b+3 0^inf [+4a^2 +4ab +6b +22 steps]
17. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 1 <D (11)^2b+5 0^inf
by:
by:
S (12)^a 2 1 <D (11)^b 0^inf
S (12)^a 2 1 <D (11)^b 0^inf
--> S (12)^a-1 2 (12)^b+1 1 <D (11)^1 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf by rule 16
--> S (12)^a-1 2 1 <D (11)^2b+3 0^inf
--> S (12)^a-1 2 1 <D (11)^2b+5 0^inf by rule 15


18. S (12)^a 2 1 <D (11)^b 0^inf --> S 2 1 <D (11)^(2^a)*b+(2^a)*3-3 0^inf
18. S (12)^a 2 1 <D (11)^b 0^inf --> S 2 1 <D (11)^(2^a)*b+(2^a)*5-5 0^inf
by repetition of rule 17
by repetition of rule 17


---
---
19. S (12)^a 2 1 <D (11)^b 2 0^inf --> S (12)^a 2 1 <D (11)^2b-1 0^inf
19. S (12)^a 2 1 <D (11)^b 2 0^inf --> S (12)^a 2^2 1 <D (11)^2b-1 0^inf
by:
by:
S (12)^a 2 1 <D (11)^b 2 0^inf
S (12)^a 2 1 <D (11)^b 2 0^inf
Line 302: Line 394:
--> S (12)^a 2 1 <D (11)^2b-1 0^inf by rule 15
--> S (12)^a 2 1 <D (11)^2b-1 0^inf by rule 15


21. S (12)^a 2^2 1 <D (11)^b 0^inf --> S (12)^a-1 2^2 1 <D (11)^2^(b+5)-3 0^inf
21. S (12)^a 2^2 1 <D (11)^b 0^inf --> S (12)^a-1 2^2 1 <D (11)^2^(b+4)*3-5 0^inf
by:
by:
S (12)^a 2^2 1 <D (11)^b 0^inf
S (12)^a 2^2 1 <D (11)^b 0^inf
Line 317: Line 409:
--> S (12)^a-1 2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 2 1 <D (11)^1 0^inf by rule 20
--> S (12)^a-1 2 (12)^b+3 2 1 <D (11)^1 0^inf by rule 20
--> S (12)^a-1 2^2 1 <D (11)^(2^(b+3)*1)+(2^(b+3)*3)-3 0^inf by rule 18
--> S (12)^a-1 2^2 1 <D (11)^(2^(b+3)*1)+(2^(b+3)*5)-5 0^inf by rule 18
= S (12)^a-1 2^2 1 <D (11)^(2^(b+5)-3) 0^inf
= S (12)^a-1 2^2 1 <D (11)^(2^(b+4)*3-5) 0^inf


22. S 1 <D (11)^b 2^2 0^inf --> S 2 (12)^b-1 2 1 <D (11)^1 0^inf
22. S 1 <D (11)^b 2^2 0^inf --> S 2 (12)^b-1 2 1 <D (11)^1 0^inf
Line 357: Line 449:
--> S (11)^a-3 1 (21)^2b+11 <D (11)^1 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 <D (11)^1 2^2 0^inf
= S (11)^a-3 (12)^2b+11 1 <D (11)^1 2^2 0^inf
= S (11)^a-3 (12)^2b+11 1 <D (11)^1 2^2 0^inf
--> S (11)^a-3 (12)^2b+11 2 (12)^0 2 1 <D (11)^1 0^inf
--> S (11)^a-3 (12)^2b+11 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf
= S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf


Line 381: Line 473:
--> 0^inf (11)^c+1 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
--> 0^inf (11)^c+1 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf
= 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf


Line 392: Line 484:
--> 0^inf (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf (11)^2 <D (11)^c+3 2 0^inf
--> 0^inf (11)^2 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^1 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 (12)^c+4 A> 2 0^inf
--> 0^inf (11)^1 (12)^c+4 A> 2 0^inf
Line 417: Line 509:
--> 0^inf 1 (11)^2c+8 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
--> 0^inf 1 (11)^2c+8 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
--> 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf
= 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf


26. 0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf
26. 0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf
Line 455: Line 547:
--> 0^inf 1 (11)^2c+7 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf
= 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf


Line 471: Line 563:
--> 0^inf 2 (12)^c+3 <C 1 0^inf
--> 0^inf 2 (12)^c+3 <C 1 0^inf
--> 0^inf 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf 2 1 <D (11)^2c+5 0^inf
--> 0^inf 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf 2^2 A> (11)^2c+5 0^inf
--> 0^inf 2^2 A> (11)^2c+5 0^inf
--> 0^inf 2^2 (12)^2c+5 A> 0^inf
--> 0^inf 2^2 (12)^2c+5 A> 0^inf
Line 490: Line 582:
--> 0^inf (11)^2c+5 (12)^3 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 <D (11)^1 2^2 0^inf by rule 22
--> 0^inf (11)^2c+5 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
--> 0^inf (11)^2c+5 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf
= 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf


Line 507: Line 599:
--> 0^inf (12)^c+4 <C 1 0^inf
--> 0^inf (12)^c+4 <C 1 0^inf
--> 0^inf (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf 1 <D (11)^2c+7 0^inf
--> 0^inf 1 <D (11)^2c+7 0^inf by rule 15
--> 0^inf 2 A> (11)^2c+7 0^inf
--> 0^inf 2 A> (11)^2c+7 0^inf
--> 0^inf 2 (12)^2c+7 A> 0^inf
--> 0^inf 2 (12)^2c+7 A> 0^inf
Line 513: Line 605:
--> 0^inf <C 1 (11)^2c+8 0^inf
--> 0^inf <C 1 (11)^2c+8 0^inf
--> 0^inf 1 Z> 1 (11)^2c+8 0^inf
--> 0^inf 1 Z> 1 (11)^2c+8 0^inf
Note: Rule 29 is not relevant to this TMs trajectory.
29. 0^inf 1 (11)^2 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^2c+10 (12)^2 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 (11)^2 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 (11)^2 2^3 A> (11)^c 0^inf
--> 0^inf 1 (11)^2 2^3 (12)^c A> 0^inf
--> 0^inf 1 (11)^2 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf 1 (11)^2 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^2 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^2 2 (12)^c+3 A> 2 0^inf
--> 0^inf (11)^2 2 (12)^c+3 <C 1 0^inf
--> 0^inf (11)^2 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf (11)^2 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf (11)^2 2^2 A> (11)^2c+5 0^inf
--> 0^inf (11)^2 2^2 (12)^2c+5 A> 0^inf
--> 0^inf (11)^2 2^2 <A (11)^2c+6 0^inf
--> 0^inf (11)^2 2 <C 1 (11)^2c+6 0^inf
--> 0^inf (11)^2 <D (11)^2c+7 0^inf
--> 0^inf (11)^1 1 2 A> (11)^2c+7 0^inf
--> 0^inf (11)^1 (12)^2c+8 A> 0^inf
--> 0^inf (11)^1 <A (11)^2c+9 0^inf
--> 0^inf 1 <A (11)^2c+10 2 0^inf by rule 12
--> 0^inf <A (11)^2c+11 2^2 0^inf by rule 13
--> 0^inf 1 B> (11)^2c+11 2^2 0^inf
--> 0^inf 1 (11)^2c+11 B> 2^2 0^inf
--> 0^inf 1 (11)^2c+11 <C 1 2 0^inf
--> 0^inf (11)^2c+11 <A (11)^1 2 0^inf
--> 0^inf (11)^2c+10 1 <A (11)^2 2^2 0^inf by rule 13
--> 0^inf (11)^2c+10 1 D> (11)^2 2^2 0^inf
--> 0^inf (11)^2c+10 1 (21)^2 D> 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2^2 D> 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^2c+10 (12)^2 2^2 1 <D (11)^1 0^inf
</pre>
</pre>
==Functions==
==Functions==
Line 519: Line 650:
Let D_1(a, b, c) = 0^inf 1 (11)^a (12)^b 2^2 1 <D (11)^c 0^inf
Let D_1(a, b, c) = 0^inf 1 (11)^a (12)^b 2^2 1 <D (11)^c 0^inf


Let <math>f_1(n) = 2^{n+5}-3</math>
Let <math>f_1(n) = 2^{n+4} \times 3 -5</math>


Let <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>
Let <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>


Rule 21 becomes:
Rule 21 becomes:
* <math>D(a, b, c) --> D(a, b-1, 2^{b+5}-3)</math>
* <math>D(a, b, c) --> D(a, b-1, 2^{b+4} \times 3 -5)</math>
* <math>D_1(a, b, c) --> D_1(a, b-1, 2^{b+5}-3)</math>
* <math>D_1(a, b, c) --> D_1(a, b-1, 2^{b+4} \times 3 -5)</math>


Rule 23 becomes:
Rule 23 becomes:
Line 545: Line 676:
Rule 28 becomes:
Rule 28 becomes:
* D(1, 0, c) --> halt with score 4c + 18
* D(1, 0, c) --> halt with score 4c + 18
Rule 29 becomes:
* <math>D_1(2, 0, c) --> D(2c+10, 2, 1)</math>


By repeating rule 21, a stronger rule can be constructed:
By repeating rule 21, a stronger rule can be constructed:
Line 559: Line 693:
* <math>D(3k+d, 0, c) --> D(d, 0, f_2(k, c))</math>
* <math>D(3k+d, 0, c) --> D(d, 0, f_2(k, c))</math>
* <math>D_1(3k+d, 0, c) --> D_1(d, 0, f_2(k, c))</math>
* <math>D_1(3k+d, 0, c) --> D_1(d, 0, f_2(k, c))</math>
The trajectory of D_1(2, 0, c) is currently unexplored.


==Trajectory==
==Trajectory==
Line 592: Line 724:
D(2, 2, 1) -->
D(2, 2, 1) -->


<math>D(2, 0, f_1^{2}(1)) = D(2, 0, f_1(61))</math>
<math>D(2, 0, f_1^{2}(1)) = D(2, 0, f_1(91))</math>
 
<math>e_1 = f_1(61) = 2^{66}-3</math>


--><math>D_1(2e_1+8, 3, 1) --> D_1(2e_1+8, 0, f_1^{2}(61))</math>
<math>e_1 = f_1(91) = 2^{95} \times 3 -5</math>


<math>e_2 = 2^{67} +2</math>
<pre>
<pre>
2^67 +2 mod 3 = ?
f_1(n) = 2^(n+4)*3 - 5
2^2k mod 3 = 1
Note that the times three means that this expression of of the form 3k - 5 which can be rewritten as 3(k-1)-2 which can again be rewritten as 3(k-2)+1.
2^2k+1 mod 3 = 2
Next, 3k+1 mod 3 = 1
2^67 mod 3 = 2 --> +2 = 4; 4 mod 3 = 1
So f_1(n) mod 3 = 1
Thus f_1^a(n) mod 3 = 1
f_2(a,b) = f_1^(2*f_2(a-1,b)+11)(1)
Note that f_1^(2*f_2(a-1,b)+11)(1) is also of the form f_1^a(n)
Thus f_2(a,b) mod 3 = 1
</pre>
</pre>
<math>D_1(e_2, 0, f_1^{2}(61))</math>


--> <math>D_1(1,0,f_2(\frac{e_2-1} 3, f_1^{2}(61)))</math>
<math>D(2, 0, e_1)</math>


<math>e_3 = f_2(\frac{e_2-1} 3, f_1^{2}(61))</math>
--><math>D_1(2e_1+8, 3, 1) --> D_1(2e_1+8, 0, f_1^{2}(91))</math>
<pre>
 
f_1(2k+1) = 2^2k+6-3
e_1 mod 3 = 1; 2*1 + 8 = 10 --> 10 mod 3 = 1
Note that 2k+6 is of the form 2k, and that the whole expression is of the form 2k+1.
 
This means that any f_1^a(2k+1) = some f_1^a-1(2k+1)
<math>D_1(2e_1+8, 0, f_1^{2}(91))</math>
Now consider that 2^2k mod 3 = 1 and 2^2k+1 mod 3 = 2
 
f_1(2k+1) = 2^2k+6-3
--> <math>D_1(1,0,f_2(\frac{2e_1+7} 3, f_1^{2}(91)))</math>
Note that 2k + 6 is of the form 2k, meaning that 2^2k+6 mod 3 = 1
 
The -3 will not affect the Modulus here, as it is a full cycle back to 1.
<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>
Thus f_1(2k+1) mod 3 = 1, and, by earlier f_1^a(2k+1) mod 3 = 1
f_2(a,b) = f_1^(2*f_2(a-1,b)+11)(1)
Note that f_1^(2*f_2(a-1,b)+11)(1) is also of the form f_1^a(2k+1)
Thus f_2(a,b) mod 3 = 1
</pre>


<math>D_1(1,0,e_3)</math>
<math>D_1(1,0,e_3)</math>


<math>e_3 mod 3 = 1</math>
<math>e_2 mod 3 = 1</math>


--> <math>D_1(2e_3+7, 3, 1) --> D_1(2e_3+7, 0, f_1^{2}(61))</math>
--> <math>D_1(2e_2+7, 3, 1) --> D_1(2e_2+7, 0, f_1^{2}(91))</math>


2e_3 + 7
2e_3 + 7
Line 634: Line 761:
Modulus: 2 + 7 --> 9 mod 3 = 0
Modulus: 2 + 7 --> 9 mod 3 = 0


--> <math>D_1(0, 0, f_2(\frac{2e_3+7} 3, f_1^{2}(61)))</math>
--> <math>D_1(0, 0, f_2(\frac{2e_2+7} 3, f_1^{2}(91)))</math>


<math>e_4 = f_2(\frac{2e_3+7} 3, f_1^2{2}(61))</math>
<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>




<math>D_1(0, 0, e_4)</math>
<math>D_1(0, 0, e_3)</math>


--> <math>D(2e_4+5, 3, 1) --> D(2e_4+5, 0, f_1^{2}(61))</math>
--> <math>D(2e_3+5, 3, 1) --> D(2e_3+5, 0, f_1^{2}(91))</math>


e_4 mod 3 = 1; 2*1+5 = 7 --> 7 mod 3 = 1
e_3 mod 3 = 1; 2*1+5 = 7 --> 7 mod 3 = 1


--> <math>D(1, 0, f_2(\frac{2e_4+4} 3, f_1^{2}(61)))</math>
--> <math>D(1, 0, f_2(\frac{2e_3+4} 3, f_1^{2}(91)))</math>


<math>e_5 = f_2(\frac{2e_4+4} 3, f_1^{2}(61))</math>
<math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>




<math>D(1, 0, e_5)</math>
<math>D(1, 0, e_4)</math>


--> halts with score <math>4e_5 + 18</math>.
--> halts with score <math>4e_4 + 18</math>.


==Approximate Score==
==Approximate Score==
<math>4e_5 + 18</math>
<math>4e_4 + 18</math>


<math>e_5 = f_2(\frac{2e_4+4} 3, f_1^{2}(61))</math>
<math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>


<math>e_4 = f_2(\frac{2e_3+7} 3, f_1^{2}(61))</math>
<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>


<math>e_3 = f_2(\frac{e_2-1} 3, f_1^{2}(61))</math>
<math>e_2 = f_2(\frac{e_1+7} 3, f_1^{2}(91))</math>


<math>e_2 = 2^{67} +2</math>
<math>e_1 = 2^{95} \times 3 -5</math>


<math>f_1(n) = 2^{n+5}-3</math>
<math>f_1(n) = 2^{n+4} \times 3 -5</math>


<math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where <math>f_2(0, b) = b</math>
<math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where <math>f_2(0, b) = b</math>


* <math>f_1(n):</math>
* <math>f_1(n):</math>
<math>2^{n+4} < f_1(n) < 2^{n+5}</math>
<math>2^{n+5} < f_1(n) < 2^{n+6}</math>


<math>(2 \uparrow)^{a} n+4 < f_1^{a}(n) < (2 \uparrow)^{a} n+6</math>
<math>(2 \uparrow)^{a} n+5 < f_1^{a}(n) < (2 \uparrow)^{a} n+7</math>


<math>(2 \uparrow)^{a} 5 < f_1^{a}(1) < (2 \uparrow)^{a} 7</math>
<math>(2 \uparrow)^{a} 5 < f_1^{a}(1) < (2 \uparrow)^{a} 8</math>


<math>2 \uparrow\uparrow (a+2) < f_1^{a}(1) < 2 \uparrow\uparrow (a+3)</math>
<math>2 \uparrow\uparrow (a+2) < f_1^{a}(1) < 2 \uparrow\uparrow (a+3)</math>
Line 685: Line 812:
<math>(2 \uparrow\uparrow)^{a} b < f_2(a,b) < (2 \uparrow\uparrow)^{a+1} b</math>
<math>(2 \uparrow\uparrow)^{a} b < f_2(a,b) < (2 \uparrow\uparrow)^{a+1} b</math>


<math>2 \uparrow\uparrow 2 \uparrow\uparrow 2 < e_2 < f_1^{2}(61) < 2 \uparrow\uparrow 2 \uparrow\uparrow 2 \uparrow\uparrow 2</math>
<math>2 \uparrow\uparrow 2 \uparrow\uparrow 2 < e_1 < f_1^{2}(91) < 2 \uparrow\uparrow 2 \uparrow\uparrow 2 \uparrow\uparrow 2</math>


<math>(2 \uparrow\uparrow)^{a} (2 \uparrow\uparrow 2 \uparrow\uparrow 2) < f_2(a, f_1^{2}(61)) < (2 \uparrow\uparrow)^{a} (2 \uparrow\uparrow 2 \uparrow\uparrow 2 \uparrow\uparrow 2)</math>
<math>(2 \uparrow\uparrow)^{a} (2 \uparrow\uparrow 2 \uparrow\uparrow 2) < f_2(a, f_1^{2}(91)) < (2 \uparrow\uparrow)^{a} (2 \uparrow\uparrow 2 \uparrow\uparrow 2 \uparrow\uparrow 2)</math>


<math>2 \uparrow\uparrow\uparrow (a+3) < f_2(a, f_1^{2}(61)) < 2 \uparrow\uparrow\uparrow (a+4)</math>
<math>2 \uparrow\uparrow\uparrow (a+3) < f_2(a, f_1^{2}(91)) < 2 \uparrow\uparrow\uparrow (a+4)</math>


<math>2 \uparrow\uparrow\uparrow (4.91 \times 10^{19}) < e_3 < 2 \uparrow\uparrow\uparrow (4.92 \times 10^{19})</math>
<math>2 \uparrow\uparrow\uparrow (7.92 \times 10^{28}) < e_2 < 2 \uparrow\uparrow\uparrow (7.93 \times 10^{28})</math>


<math>2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (4.91 \times 10^{19}) < e_4 < 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (4.92 \times 10^{19})</math>
<math>2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28}) < e_3 < 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.93 \times 10^{28})</math>


<math>2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (4.91 \times 10^{19}) < e_5 < \sigma < S < 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (4.92 \times 10^{19})</math>
<math>2 \uparrow^{4} 5 < 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28}) < e_4 < \sigma < S < 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.93 \times 10^{28})</math>


This score would make {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} the new [[BB(4,3)]] champion.
This score would make {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} the new [[BB(4,3)]] champion.
==Permutations==
'''Starting in state B'''
<pre>
0^inf <B 0^inf
--> 0^inf <B 2^k 0^inf
--> translated cycler
</pre>
'''Starting in state C'''
<pre>
0^inf <C 0^inf
-->0^inf 1 Z> 0^inf
</pre>
'''Starting in state D'''
<pre>
0^inf <D 0^inf
--> 0^inf <B 0^inf
--> translated cycler
</pre>
</div></div>
={{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}=
3. One of the seven potential [[BB(4,3)]] champions discovered by Pavel Kropitz in May 2024. Analysed in October 2025. This TM has rules which are based on the remainder of some value modulo 4, although it is quite unlucky that three of the four possible remainders lead to halting. The TM achieves a score of around <math>3 \uparrow\uparrow\uparrow 88574</math>.
<div class="toccolours mw-collapsible mw-collapsed">'''Analysis'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!
!0
!1
!2
|-
|A
|1RB
|2LB
|0LB
|-
|B
|2LC
|2LA
|0LA
|-
|C
|2RD
|1LC
|1RZ
|-
|D
|1RA
|2LD
|1RD
|}
<pre>
S is any tape configuration
1. S 1^a <C S --> S <C 1^a S [+a steps]
2. S 1^a <D S --> S <D 2^a S [+a steps]
3. S D> 2^a S --> S 1^a D> S [+a steps]
4. S (11)^a <A S --> S <A (22)^a S [+2a steps]
  S (11)^a <B S --> S <B (22)^a S [+2a steps]
5. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3 A> (22)^b-1 S [+8a +24 steps]
by:
0^inf 2 (11)^a A> (22)^b S
--> 0^inf 2 (11)^a <B 0 2 (22)^b-1 S [+1]
--> 0^inf 2 <B (22)^a 0 2 (22)^b-1 S [+2a +1]
--> 0^inf <A 0 (22)^a 0 2 (22)^b-1 S [+2a +2]
--> 0^inf 1 B> 0 (22)^a 0 2 (22)^b-1 S [+2a +3]
--> 0^inf 1 <C 2 (22)^a 0 2 (22)^b-1 S [+2a +4]
--> 0^inf <C 1 2 (22)^a 0 2 (22)^b-1 S [+2a +5]
--> 0^inf 2 D> 1 2 (22)^a 0 2 (22)^b-1 S [+2a +6]
--> 0^inf 2 <D (22)^a+1 0 2 (22)^b-1 S [+2a +7]
--> 0^inf 1 D> (22)^a+1 0 2 (22)^b-1 S [+2a +8]
--> 0^inf 1 (11)^a+1 D> 0 2 (22)^b-1 S [+4a +10]
--> 0^inf (11)^a+2 A> 2 (22)^b-1 S [+4a +11]
--> 0^inf (11)^a+2 <B 0 (22)^b-1 S [+4a +12]
--> 0^inf <B (22)^a+2 0 (22)^b-1 S [+6a +16]
--> 0^inf <C 2 (22)^a+2 0 (22)^b-1 S [+6a +17]
--> 0^inf 2 D> 2 (22)^a+2 0 (22)^b-1 S [+6a +18]
--> 0^inf 2 1 (11)^a+2 D> 0 (22)^b-1 S [+8a +23]
--> 0^inf 2 (11)^a+3 A> (22)^b-1 S [+8a +24]
6. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3b A> S
by repetition of rule 5
7. 0^inf 2 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^a+2 0 (22)^b-1 S [+6a +28 steps]
by:
0^inf 2 (11)^a A> 0 (22)^b S
--> 0^inf 2 (11)^a 1 B> (22)^b S [+1]
--> 0^inf 2 1 (11)^a <A 0 2 (22)^b-1 S [+2]
--> 0^inf 2 1 <A (22)^a 0 2 (22)^b-1 S [+2a +2]
--> 0^inf 2 <B 2 (22)^a 0 2 (22)^b-1 S [+2a +3]
--> 0^inf <A 0 2 (22)^a 0 2 (22)^b-1 S [+2a +4]
--> 0^inf 1 B> 0 2 (22)^a 0 2 (22)^b-1 S [+2a +5]
--> 0^inf 1 <C (22)^a+1 0 2 (22)^b-1 S [+2a +6]
--> 0^inf <C 1 (22)^a+1 0 2 (22)^b-1 S [+2a +7]
--> 0^inf 2 D> 1 (22)^a+1 0 2 (22)^b-1 S [+2a +8]
--> 0^inf 2 <D 2 (22)^a+1 0 2 (22)^b-1 S [+2a +9]
--> 0^inf 1 D> 2 (22)^a+1 0 2 (22)^b-1 S [+2a +10]
--> 0^inf (11)^a+2 D> 0 2 (22)^b-1 S [+4a +13]
--> 0^inf (11)^a+2 1 A> 2 (22)^b-1 S [+4a +14]
--> 0^inf (11)^a+2 1 <B 0 (22)^b-1 S [+4a +15]
--> 0^inf 1 <B (22)^a+2 0 (22)^b-1 S [+6a +19]
--> 0^inf <A 2 (22)^a+2 0 (22)^b-1 S [+6a +20]
--> 0^inf 1 B> 2 (22)^a+2 0 (22)^b-1 S [+6a +21]
--> 0^inf 1 <A 0 (22)^a+2 0 (22)^b-1 S [+6a +22]
--> 0^inf <B 2 0 (22)^a+2 0 (22)^b-1 S [+6a +23]
--> 0^inf <C 2 2 0 (22)^a+2 0 (22)^b-1 S [+6a +24]
--> 0^inf 2 D> (22)^1 0 (22)^a+2 0 (22)^b-1 S [+6a +25]
--> 0^inf 2 1 1 D> 0 (22)^a+2 0 (22)^b-1 S [+6a +27]
--> 0^inf 2 1 (11)^1 A> (22)^a+2 0 (22)^b-1 S [+6a +28]
8. 0^inf 2 (11)^a A> 2 0 2 S --> 0^inf 2 1 (11)^a+3 A> S [+8a +27 steps]
by:
0^inf 2 (11)^a A> 2 0 2 S
--> 0^inf 2 (11)^a <B 0^2 2 S [+1]
--> 0^inf 2 <B (22)^a 0^2 2 S [+2a +1]
--> 0^inf <A 0 (22)^a 0^2 2 S [+2a +2]
--> 0^inf 1 B> 0 (22)^a 0^2 2 S [+2a +3]
--> 0^inf 1 <C 2 (22)^a 0^2 2 S [+2a +4]
--> 0^inf <C 1 2 (22)^a 0^2 2 S [+2a +5]
--> 0^inf 2 D> 1 2 (22)^a 0^2 2 S [+2a +6]
--> 0^inf 2 <D (22)^a+1 0^2 2 S [+2a +7]
--> 0^inf 1 D> (22)^a+1 0^2 2 S [+2a +8]
--> 0^inf 1 (11)^a+1 D> 0^2 2 S [+4a +10]
--> 0^inf (11)^a+2 A> 0 2 S [+4a +11]
--> 0^inf (11)^a+2 1 B> 2 S [+4a +12]
--> 0^inf (11)^a+2 1 <A 0 S [+4a +13]
--> 0^inf 1 <A (22)^a+2 0 S [+6a +17]
--> 0^inf <B 2 (22)^a+2 0 S [+6a +18]
--> 0^inf <C (22)^a+3 0 S [+6a +19]
--> 0^inf 2 D> (22)^a+3 0 S [+6a +20]
--> 0^inf 2 (11)^a+3 D> 0 S [+8a +26]
--> 0^inf 2 1 (11)^a+3 A> S [+8a +27]
9. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^3a+4 A> (22)^b-1 S
by:
0^inf 2 1 (11)^a A> (22)^b S
--> 0^inf 2 1 (11)^a <B 0 2 (22)^b-1 S
--> 0^inf 2 1 <B (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 <A 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf <B 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf <C 2 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 D> 2 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 1 D> 0 2 (22)^a 0 (22)^b-1 2 S
--> 0^inf 2 (11)^1 A> (22)^a 2 0 (22)^b-1 2 S
--> 0^inf 2 (11)^3a+1 A> 2 0 2 (22)^b-1 S by rule 6
--> 0^inf 2 1 (11)^3a+4 A> (22)^b-1 S by rule 8
= 0^inf 2 1 (11)^g_1(a) A> (22)^b-1 S
10. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^g_1^b(a) A> S
by repetition of rule 9
g_1(n) = 3n + 4
11. 0^inf 2 1 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 (22)^b-2 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^b S
--> 0^inf 2 (11)^a+1 B> (22)^b S
--> 0^inf 2 (11)^a+1 <A 0 2 (22)^b-1 S
--> 0^inf 2 <A (22)^a+1 0 2 (22)^b-1 S
--> 0^inf <B 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf <C 2 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 D> 2 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 1 D> 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 (11)^1 A> (22)^a+1 0 (22)^b-1 2 S
--> 0^inf 2 (11)^3a+4 A> 0 (22)^b-1 2 S by rule 6
Call this rule 11-1
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 (22)^b-2 2 S by rule 7
Call this rule 11-2
12. 0^inf 2 (11)^a A> 0 11 S --> 0^inf 2 1 (11)^1 A> 0 (22)^a+2 2 S [+6a +31 steps]
by:
0^inf 2 (11)^a A> 0 11 S
--> 0^inf 2 (11)^a 1 B> 11 S [+1]
--> 0^inf 2 (11)^a 1 <A 2 1 S [+2]
--> 0^inf 2 1 <A (22)^a 2 1 S [+2a +2]
--> 0^inf 2 <B (22)^a+1 1 S [+2a +3]
--> 0^inf <A 0 (22)^a+1 1 S [+2a +4]
--> 0^inf 1 B> 0 (22)^a+1 1 S [+2a +5]
--> 0^inf 1 <C 2 (22)^a+1 1 S [+2a +6]
--> 0^inf <C 1 2 (22)^a+1 1 S [+2a +7]
--> 0^inf 2 D> 1 2 (22)^a+1 1 S [+2a +8]
--> 0^inf 2 <D (22)^a+2 1 S [+2a +9]
--> 0^inf 1 D> (22)^a+2 1 S [+2a +10]
--> 0^inf 1 (11)^a+2 D> 1 S [+4a +14]
--> 0^inf 1 (11)^a+2 <D 2 S [+4a +15]
--> 0^inf <D (22)^a+3 S [+6a +20]
--> 0^inf 1 A> (22)^a+3 S [+6a +21]
--> 0^inf 1 <B 0 2 (22)^a+2 S [+6a +22]
--> 0^inf <A 2 0 2 (22)^a+2 S [+6a +23]
--> 0^inf 1 B> 2 0 2 (22)^a+2 S [+6a +24]
--> 0^inf 1 <A 0 0 2 (22)^a+2 S [+6a +25]
--> 0^inf <B 2 0 0 2 (22)^a+2 S [+6a +26]
--> 0^inf <C (22)^1 0 0 2 (22)^a+2 S [+6a +27]
--> 0^inf 2 D> (22)^1 0 0 2 (22)^a+2 S [+6a +28]
--> 0^inf 2 (11)^1 D> 0 0 2 (22)^a+2 S [+6a +30]
--> 0^inf 2 1 (11)^1 A> 0 (22)^a+2 2 S [+6a +31]
13. 0^inf 2 1 (11)^a A> 0 2^b S --> 0^inf 2 1 (11)^g_2(a) A> 0 2^b-3 S
by:
0^inf 2 1 (11)^a A> 0 2^b S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 2^b-3 S by rule 11-2
--> 0^inf 2 1 (11)^g_1^(3a+6)(1) A> 0 2^b-3 S by rule 10
= 0^inf 2 1 (11)^(3^(3a+7)-2) A> 0 2^b-3 S
= 0^inf 2 1 (11)^g_2(a) A> 0 2^b-3 S
14. 0^inf 2 1 (11)^a A> 0 2^3k+v S --> 0^inf 2 1 (11)^(g_2)^k(a) A> 0 2^v S
by repetition of rule 13
15. 0^inf 2 1 <A S --> 0^inf 1 D> 2^3 S [+8 steps]
by:
0^inf 2 1 <A S
--> 0^inf 2 <B 2 S
--> 0^inf <A 0 2 S
--> 0^inf 1 B> 0 2 S
--> 0^inf 1 <C 2 2 S
--> 0^inf <C 1 2^2 S
--> 0^inf 2 D> 1 2^2 S
--> 0^inf 2 <D 2^3 S
--> 0^inf 1 D> 2^3 S
16. 0^inf 2 1 (11)^a A> 0 2 1 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3a+7 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 2 1 2 0^inf
--> 0^inf 2 (11)^a+1 B> 2 1 2 0^inf
--> 0^inf 2 (11)^a+1 <A 0 1 2 0^inf
--> 0^inf 2 <A (22)^a+1 0 1 2 0^inf
--> 0^inf <B 0 (22)^a+1 0 1 2 0^inf
--> 0^inf <C 2 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 D> 2 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 1 D> 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 (11)^1 A> (22)^a+1 0 1 2 0^inf
--> 0^inf 2 (11)^3a+4 A> 0 1 2 0^inf by rule 6
--> 0^inf 2 (11)^3a+4 1 B> 1 2 0^inf
--> 0^inf 2 (11)^3a+4 1 <A (22)^1 0^inf
--> 0^inf 2 1 <A (22)^3a+5 0^inf
--> 0^inf 1 D> 2 (22)^3a+6 0^inf by rule 15
--> 0^inf (11)^3a+7 D> 0^inf
--> 0^inf (11)^3a+7 1 A> 0^inf
--> 0^inf (11)^3a+8 B> 0^inf
--> 0^inf (11)^3a+8 <C 2 0^inf
--> 0^inf <C (11)^3a+8 2 0^inf
--> 0^inf 2 D> (11)^3a+8 2 0^inf
--> 0^inf 2 <D 2 1 (11)^3a+7 2 0^inf
--> 0^inf 1 D> 2 1 (11)^3a+7 2 0^inf
--> 0^inf (11)^1 D> 1 (11)^3a+7 2 0^inf
--> 0^inf (11)^1 <D 2 (11)^3a+7 2 0^inf
--> 0^inf <D 2^3 (11)^3a+7 2 0^inf
--> 0^inf 1 A> (22)^1 2 (11)^3a+7 2 0^inf
--> 0^inf 1 <B 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <A 2 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 1 B> 2 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 1 <A 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <B 2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <C 2 2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 D> 2^2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 (11)^1 D> 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3a+7 2 0^inf
17. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1 S
--> 0^inf 2 (11)^3a+4 A> 0 2 1 S by rule 11-1
--> 0^inf 2 (11)^3a+4 1 B> 2 1 S
--> 0^inf 2 1 (11)^3a+4 <A 0 1 S
--> 0^inf 2 1 <A (22)^3a+4 0 1 S
--> 0^inf 1 D> 2 (22)^3a+5 0 1 S by rule 15
--> 0^inf (11)^3a+6 D> 0 1 S
--> 0^inf (11)^3a+6 1 A> 1 S
--> 0^inf (11)^3a+6 1 <B 2 S
--> 0^inf 1 <B (22)^3a+6 2 S
--> 0^inf <A (22)^3a+7 S
--> 0^inf 1 B> (22)^3a+7 S
--> 0^inf 1 <A 0 2 (22)^3a+6 S
--> 0^inf <B 2 0 2 (22)^3a+6 S
--> 0^inf <C 2 2 0 2 (22)^3a+6 S
--> 0^inf 2 D> 2 2 0 (22)^3a+6 2 S
--> 0^inf 2 (11)^1 D> 0 (22)^3a+6 2 S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S
18. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^g_2(a) A> 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1 S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S by rule 17
--> 0^inf 2 1 (11)^g_1^(3a+6)(1) A> 2 S by rule 10
= 0^inf 2 1 (11)^(3^(3a+7)-2) A> 2 S
= 0^inf 2 1 (11)^g_2(a) A> 2 S
19. 0^inf 2 1 (11)^a A> 2 1^3 S --> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 S
by:
0^inf 2 1 (11)^a A> 2 1^3 S
--> 0^inf 2 1 (11)^a <B 0 1^3 S
--> 0^inf 2 1 <B (22)^a 0 1^3 S
--> 0^inf 2 <A 2 (22)^a 0 1^3 S
--> 0^inf <B 0 2 (22)^a 0 1^3 S
--> 0^inf <C 2 0 2 (22)^a 0 1^3 S
--> 0^inf 2 D> 2 0 2 (22)^a 0 1^3 S
--> 0^inf 2 1 D> 0 (22)^a 2 0 1^3 S
--> 0^inf 2 (11)^1 A> (22)^a 2 0 1^3 S
--> 0^inf 2 (11)^3a+1 A> 2 0 1^3 S by rule 6
--> 0^inf 2 (11)^3a+1 <B 0 0 1^3 S
--> 0^inf 2 <B (22)^3a+1 0 0 1^3 S
--> 0^inf <A 0 (22)^3a+1 0 0 1^3 S
--> 0^inf 1 B> 0 (22)^3a+1 0 0 1^3 S
--> 0^inf 1 <C 2 (22)^3a+1 0 0 1^3 S
--> 0^inf <C 1 2 (22)^3a+1 0 0 1^3 S
--> 0^inf 2 D> 1 2 (22)^3a+1 0 0 1^3 S
--> 0^inf 2 <D (22)^3a+2 0 0 1^3 S
--> 0^inf 1 D> (22)^3a+2 0 0 1^3 S
--> 0^inf 1 (11)^3a+2 D> 0 0 1^3 S
--> 0^inf (11)^3a+3 A> 0 1^3 S
--> 0^inf (11)^3a+3 1 B> 1^3 S
Call this rule 19***
with:
0^inf 2 1 (11)^a A> 2 S --> 0^inf (11)^3a+3 1 B> S
Continuing:
--> 0^inf (11)^3a+3 1 <A 2 1^2 S
--> 0^inf 1 <A (22)^3a+3 2 1^2 S
--> 0^inf <B (22)^3a+4 1^2 S
Call this rule 19**
with:
0^inf 2 1 (11)^a A> 2 1 S --> 0^inf <B (22)^3a+4 S
Continuing:
--> 0^inf <C 2 (22)^3a+4 1^2 S
--> 0^inf 2 D> 2 (22)^3a+4 1^2 S
--> 0^inf 2 1 (11)^3a+4 D> 1^2 S
--> 0^inf 2 1 (11)^3a+4 <D 2 1 S
--> 0^inf 2 <D (22)^3a+5 1 S
--> 0^inf 1 D> (22)^3a+5 1 S
--> 0^inf 1 (11)^3a+5 D> 1 S
Call this rule 19*
with:
0^inf 2 1 (11)^a A> 2 1^2 S --> 0^inf 1 (11)^3a+5 D> S
Continuing:
--> 0^inf 1 (11)^3a+5 <D 2 S
--> 0^inf <D (22)^3a+6 S
--> 0^inf 1 A> (22)^3a+6 S
--> 0^inf 1 <B 0 2 (22)^3a+5 S
--> 0^inf <A 2 0 2 (22)^3a+5 S
--> 0^inf 1 B> 2 0 2 (22)^3a+5 S
--> 0^inf 1 <A 0 0 2 (22)^3a+5 S
--> 0^inf <B 2 0 0 (22)^3a+5 2 S
--> 0^inf <C (22)^1 0 0 (22)^3a+5 2 S
--> 0^inf 2 D> (22)^1 0 0 (22)^3a+5 2 S
--> 0^inf 2 (11)^1 D> 0 0 (22)^3a+5 2 S
--> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 S
This rule can be rewritten as:
0^inf 2 1 (11)^a A> 2 1^b S --> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 1^b-3 S
20. 0^inf 2 1 (11)^a A> 0 (22)^1 1^b S --> 0^inf 2 1 (11)^g_3(a) A> 0 (22)^1 1^b-4 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^b S
--> 0^inf 2 1 (11)^g_2(a) A> 2 1^b-1 S by rule 18
--> 0^inf 2 1 (11)^1 A> 0 (22)^3*g_2(a)+5 2 1^b-4 S by rule 19
= 0^inf 2 1 (11)^1 A> 0 2^6*g_2(a)+11 1^b-4 S
Modulus for rule 14:
6*g_2(a)+11 = 3(2*g_2(a)+3)+2 = 3k+2
3k+2 mod 3 = 2
--> 0^inf 2 1 (11)^(g_2)^(2*g_2(a)+3)(1) A> 0 (22)^1 1^b-4 S
= 0^inf 2 1 (11)^g_3(a) A> 0 (22)^1 1^b-4 S
21. 0^inf 2 1 (11)^a A> 0 (22)^1 1^4k+v S --> 0^inf 2 1 (11)^g_3^k(a) A> 0 (22)^1 1^v S
by repetition of rule 20
22. 0^inf 2 1 (11)^a A> 0 (22)^1 1^3 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 1^6*g_2(a)+12 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^3 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 1^2 2 0^inf by rule 18
--> 0^inf 1 (11)^3*g_2(a)+5 D> 2 0^inf by rule 19*
--> 0^inf (11)^3*g_2(a)+6 D> 0^inf
--> 0^inf (11)^3*g_2(a)+6 1 A> 0^inf
--> 0^inf (11)^3*g_2(a)+7 B> 0^inf
--> 0^inf (11)^3*g_2(a)+7 <C 2 0^inf
--> 0^inf <C (11)^3*g_2(a)+7 2 0^inf
--> 0^inf 2 D> (11)^3*g_2(a)+7 2 0^inf
--> 0^inf 2 <D 2 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 D> 2 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf (11)^1 D> 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf (11)^1 <D 2 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <D 2^3 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 A> 2^3 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 <B 0 2^2 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <A 2 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 B> 2 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 <A 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <B 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <C 2 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 D> 2 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 (11)^1 D> 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
= 0^inf 2 1 (11)^1 A> 0 (22)^1 1^6*g_2(a)+12 2 0^inf
23. 0^inf 2 1 (11)^a A> 0 (22)^1 1^2 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^2 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 1 2 0^inf by rule 18
--> 0^inf <B (22)^3*g_2(a)+4 2 0^inf by rule 19**
--> 0^inf <C (22)^3*g_2(a)+5 0^inf
--> 0^inf 2 D> (22)^3*g_2(a)+5 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 D> 0^inf
--> 0^inf 2 1 (11)^3*g_2(a)+5 A> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+6 B> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+6 <C 2 0^inf
--> 0^inf 2 <C (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf
24. 0^inf (11)^a A> 0 (22)^1 1 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf
by:
0^inf (11)^a A> 0 (22)^1 1 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 2 0^inf by rule 18
--> 0^inf (11)^3*g_2(a)+3 1 B> 2 0^inf by rule 19***
--> 0^inf (11)^3*g_2(a)+3 1 <A 0^inf
--> 0^inf 1 <A (22)^3*g_2(a)+3 0^inf
--> 0^inf <B 2 (22)^3*g_2(a)+3 0^inf
--> 0^inf <C (22)^3*g_2(a)+4 0^inf
--> 0^inf 2 D> (22)^3*g_2(a)+4 0^inf
--> 0^inf 2 (11)^3*g_2(a)+4 D> 0^inf
--> 0^inf 2 1 (11)^3*g_2(a)+4 A> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 B> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 <C 2 0^inf
--> 0^inf 2 <C (11)^3*g_2(a)+5 2 0^inf
--> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf
25. 0^inf 2 1 (11)^a A> 0 (22)^1 2 0^inf --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 0^inf by rule 14
--> 0^inf 2 (11)^g_2(a)+1 B> 0^inf
--> 0^inf 2 (11)^g_2(a)+1 <C 2 0^inf
--> 0^inf 2 <C (11)^g_2(a)+1 2 0^inf
--> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf
</pre>
==Functions==
<math>g_1(n) = 3n + 4</math>
Note that <math>(3^{k}-2) \times 3 + 4 = 3^{k+1} - 2</math>
And <math>1 = 3^1 - 2</math>
It follows that <math>g_1^{n}(1) = 3^{n+1}-2</math>
<math>g_2(n) = 3^{3n + 7}-2</math>
<math>g_3(n) = g_2^{2 \times (g_2(n)+3)}(1)</math>
<pre>
Modulus of g_2^a(1):
1 is of the form 4k+1
g_2(n) = 3^(3n+7)-2
3^2k mod 4 = 1
3^2k+1 mod 4 = 3
g_2(4k+1) = 3^(3*(4k+1)+7)-2 = 3^(12k+10)-2 = 3^2m-2
3^2m mod 4 = 1 --> -2
3^2m - 2 mod 4 = 3
g_2(4k+1) mod 4 = 3
--> g_2(4k+1) = 4m+3
g_2(4k+3) = 3^(3*(4k+3)+7)-2 = 3^(12k+16)-2 = 3^2m-2
3^2m-2 mod 4 = 3
g_2(4k+1) mod 4 = 3
--> g_2(4k+3) = 4m+3
-->g_2^k(1) = 4m+3
-->g_2^k(1) mod 4 = 3
</pre>
<pre>
Let's have L(a, b) = 0^inf 2 1 (11)^a A> 0 (22)^1 1^b 2 0^inf
* L(a, 4k+v) --> L(g_3^k(a), v) by rule 21
* L(a, 0) --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf by rule 25
* L(a, 1) --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf by rule 24
* L(a, 2) --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf by rule 23
* L(a, 3) --> L(1, 6*g_2(a) + 12) by rule 22
</pre>
==Trajectory==
<pre>
S=0: 0^inf <A 0^inf
S=1: 0^inf 1 B> 0^inf
S=2: 0^inf 1 <C 2 0^inf
S=3: 0^inf <C 1 2 0^inf
S=4: 0^inf 2 D> 1 2 0^inf
S=5: 0^inf 2 <D 2^2 0^inf
S=6: 0^inf 1 D> 2^2 0^inf
S=8: 0^inf 1^3 D> 0^inf
S=9: 0^inf (11)^2 A> 0^inf
S=10: 0^inf (11)^2 1 B> 0^inf
S=11: 0^inf (11)^2 1 <C 2 0^inf
S=16: 0^inf <C (11)^2 1 2 0^inf
S=17: 0^inf 2 D> (11)^2 1 2 0^inf
S=18: 0^inf 2 <D 2 (11)^2 2 0^inf
S=19: 0^inf 1 D> 2 (11)^2 2 0^inf
S=20: 0^inf (11)^1 D> (11)^2 2 0^inf
S=21: 0^inf (11)^1 <D 2 1 (11)^1 2 0^inf
S=23: 0^inf <D 2^3 1^3 2 0^inf
S=24: 0^inf 1 A> 2^3 1^3 2 0^inf
S=25: 0^inf 1 <B 0 2^2 1^3 2 0^inf
S=26: 0^inf <A 2 0 2^2 1^3 2 0^inf
S=27: 0^inf 1 B> 2 0 2^2 1^3 2 0^inf
S=28: 0^inf 1 <A 0 0 2^2 1^3 2 0^inf
S=29: 0^inf <B 2 0 0 2^2 1^3 2 0^inf
S=30: 0^inf <C 2 2 0 0 2^2 1^3 2 0^inf
S=31: 0^inf 2 D> 2^2 0 0 2^2 1^3 2 0^inf
S=33: 0^inf 2 (11)^1 D> 0^2 2^2 1^3 2 0^inf
S=34: 0^inf 2 1 (11)^1 A> 0 (22)^1 1^3 2 0^inf
= L(1, 3) after 34 steps
</pre>
L(1, 3) --> <math>L(1, 6*g_2(1) + 12)</math> by rule 22, which can be simplified to L(1, 354294)
--> <math>L(g_3^{88573}(1), 2)</math>
<math>L(g_3^{88573}(1), 2)</math> --> 0^inf 1 Z> <math>(11)^{3 \times (g_2(g_3^{88573}(1)) +6}</math> 2 0^inf
<math>\sigma = 6 \times g_2(g_3^{88573}(1)) + 14</math>
==Approximate Score==
<math>g_2(n) = 3^{3n+7}-2</math>
<math>3^{n} < g_2(n) < 3^{3^{n}}</math>
<math>g_3(n) = g_2^{2 \times (g_2(n)) +3}(1)</math>
<math>(3 \uparrow)^{k} n < g_2^{k}(n) < (3 \uparrow)^{2k} n</math>
<math>(3 \uparrow)^{k} 1 < g_2^{k}(1) < (3 \uparrow)^{2k} 1</math>
<math>3 \uparrow\uparrow k < g_2^{k}(1) < 3 \uparrow\uparrow 2k</math>
<math>3 \uparrow\uparrow 2 \times g_2(n) +6 < g_3(n) < 3 \uparrow\uparrow 4 \times g_2(n) +12</math>
<math>3 \uparrow\uparrow 2 \times 3^{3 \uparrow\uparrow 2 \times g_2(n) +6} +6 < g_3^{2}(n)</math>
<math>3 \uparrow\uparrow 3 \uparrow\uparrow (2 \times g_2(n) +7) < g_3^{2}(n) < 3 \uparrow\uparrow 3 \uparrow\uparrow (4 \times g_2(n) +13)</math>
<math>(3 \uparrow\uparrow)^{k} (2 \times g_2(n) + 7) < g_3^{k}(n) < (3 \uparrow\uparrow)^{k} (5 \times g_2(n) + 13)</math>
<math>(3 \uparrow\uparrow)^{k} 118101 < g_3^{k}(1) < (3 \uparrow\uparrow)^{k} 295248</math>
<math>3 \uparrow\uparrow\uparrow k+1 < g_3^{k}(1) < 3 \uparrow\uparrow\uparrow k+2</math>
<math>\sigma = 6 \times g_2(g_3^{88573}(1)) + 14</math>
<math>3 \uparrow\uparrow\uparrow 88574 < \sigma < S < 3 \uparrow\uparrow\uparrow 88575</math>
==Permutations==
'''Starting in state B'''
<pre>
S=0: 0^inf <B 0^inf
S=1: 0^inf <C 2 0^inf
S=2: 0^inf 2 D> 2 0^inf
S=3: 0^inf 2 1 D> 0^inf
S=4: 0^inf 2 1^2 A> 0^inf
S=5: 0^inf 2 1^3 B> 0^inf
S=6: 0^inf 2 1^3 <C 2 0^inf
S=9: 0^inf 2 <C 1^3 2 0^inf
S=10: 0^inf 1 Z> 1^3 2 0^inf
</pre>
'''Starting in state C'''
<pre>
S=0: 0^inf <C 0^inf
S=1: 0^inf 2 D> 0^inf
S=2: 0^inf 2 1 A> 0^inf
S=3: 0^inf 2 1^2 B> 0^inf
S=4: 0^inf 2 1^2 <C 2 0^inf
S=6: 0^inf 2 <C 1^2 2 0^inf
S=7: 0^inf 1 Z> 1^2 2 0^inf
</pre>
'''Starting in state D'''
<pre>
Enters configuration L(1,0) after 23 steps.
Halting tape: 0^inf 1 Z> (11)^10 1 2 0^inf
</pre>
</div></div>
={{TM|1RB1LE_1LB1LC_1RD0LE_---0RB_1RF1LA_0RA0RD}}=
4. A [[BB(6)]] [[holdout]] TM. Analysed in October 2025. Its fate is currently unknown.
<div class="toccolours mw-collapsible mw-collapsed">'''Analysis'''<div class="mw-collapsible-content">
For this analysis the undefined D0 transition will be set to 1RZ.
{| class="wikitable"
|+
!
!0
!1
|-
|A
|1RB
|1LE
|-
|B
|1LB
|1LC
|-
|C
|1RD
|0LE
|-
|D
|1RZ
|0RB
|-
|E
|1RF
|1LA
|-
|F
|0RA
|0RD
|}
<pre>
S is any tape configuration
1. S 0^a <B S --> S <B 1^a S [+a steps]
2. S D> 1^2 S --> S 1 D> 1 S [+3 steps]
by:
S D> 1^2 S
--> S 0 B> 1 S [+1]
--> S 0 <C 1 S [+2]
--> S 1 D> 1 S [+3]
3. S D> 1^a 1 S --> S 1^a D> 1 S |for a > 0 [+3a steps]
by repetition of rule 2
4. S (11)^a <E S --> S <E (11)^a S [+2a steps]
  S (11)^a <A S --> S <A (11)^a S [+2a steps]
5. S 1^2 D> 1 0 S --> S <E 0 1^3 S [+5 steps]
by:
S 1^2 D> 1 0 S
--> S 1^2 0 B> 0 S [+1]
--> S 1^2 0 <B 1 S [+2]
--> S 1^2 <B 1^2 S [+3]
--> S 1 <C 1^3 S [+4]
--> S <E 0 1^3 S [+5]
6. S 0 1^a <E S --> S 1 0 1^a-2 D> 1 S [+4a -4 steps] (if a mod 2 = 0)
by:
S 0 1^a <E S
--> S 0 <E 1^a S [+a]
--> S 1 F> 1^a S [+a +1]
--> S 1 0 D> 1^a-1 S [+a +2]
= S 1 0 D> 1^a-2 1 S
--> S 1 0 1^a-2 D> 1 S [+4a -4]
7. S 0 1^a D> 1 0 S --> S 1 0 1^a-4 D> 1 0 1^3 S [+4a -7 steps] (if a mod 2 = 0 and a >= 4)
by:
S 0 1^a D> 1 0 S
--> S 0 1^a 0 B> 0 S [+1]
--> S 0 1^a 0 <B 1 S [+2]
--> S 0 1^a <B 1^2 S [+3]
--> S 0 1^a-1 <C 1^3 S [+4]
--> S 0 1^a-2 <E 0 1^3 S [+5]
--> S 1 0 1^a-4 D> 1 0 1^3 S [+4a -7] by rule 6
8. S 0 1^4k+v D> 1 0 1^b S --> S 1^k 0 1^v D> 1 0 1^b+3k S (if v mod 2 = 0 and k ≥ 1)
[+4bk -8k^2 +k steps where b = 4k + v]
by repetition of rule 7
9. S 0 1^2 D> 1 0 1^c S --> S 1^2 0 1^c+1 D> 1 S [+3c +13 steps]
by:
S 0 1^2 D> 1 0 1^c S
--> S 0 1^2 0 B> 0 1^c S [+1]
--> S 0 1^2 0 <B 1^c+1 S [+2]
--> S 0 1^2 <B 1^c+2 S [+3]
--> S 0 1 <C 1^c+3 S [+4]
--> S 0 <E 0 1^c+3 S [+5]
--> S 1 F> 0 1^c+3 S [+6]
--> S 1 0 A> 1^c+3 S [+7]
--> S 1 0 <E 1^c+3 S [+8]
--> S 1^2 F> 1^c+3 S [+9]
--> S 1^2 0 D> 1^c+2 S [+10]
= S 1^2 0 D> 1^c+1 1 S [+10]
--> S 1^2 0 1^c+1 D> 1 S [+3c +13]
10. S 0 1^a 0 D> 1 0 S --> S 1 0 1^a-4 D> 1 0 1^4 S [+4a -6 steps] (if a mod 2 = 0 and a ≥ 4)
by:
S 0 1^a 0 D> 1 0 S
--> S 0 1^a 0^2 B> 0 S [+1]
--> S 0 1^a 0^2 <B 1 S [+2]
--> S 0 1^a <B 1^3 S [+4]
--> S 0 1^a-1 <C 1^4 S [+5]
--> S 0 1^a-2 <E 0 1^4 S [+6]
--> S 0 <E 1^a-2 0 1^4 S [+a +4]
--> S 1 F> 1^a-2 0 1^4 S [+a +5]
--> S 1 0 D> 1^a-3 0 1^4 S [+a +6]
= S 1 0 D> 1^a-4 1 0 1^4 S [+a +6]
--> S 1 0 1^a-4 D> 1 0 1^4 S [+4a -6]
11. S 0 1^2 0 D> 1 0 1^c S --> S 1^2 0 1^c+2 D> 1 S [+3c +17 steps]
by:
S 0 1^2 0 D> 1 0 1^c S
--> S 0 1^2 0^2 B> 0 1^c S [+1]
--> S 0 1^2 0^2 <B 1^c+1 S [+2]
--> S 0 1^2 <B 1^c+3 S [+4]
--> S 0 1 <C 1^c+4 S [+5]
--> S 0 <E 0 1^c+4 S [+6]
--> S 1 F> 0 1^c+4 S [+7]
--> S 1 0 A> 1^c+4 S [+8]
--> S 1 0 <E 1^c+4 S [+9]
--> S 1^2 F> 1^c+4 S [+10]
--> S 1^2 0 D> 1^c+3 S [+11]
= S 1^2 0 D> 1^c+2 1 S
--> S 1^2 0 1^c+2 D> 1 S [+3c +17]
12. 0^inf D> 1 0 S --> non-halting translated cycler
by:
0^inf D> 1 0 S
--> 0^inf B> 0 S
--> 0^inf <B 1 S
--> spin out
13. S 0^2 1^a 0 D> 1 0 S --> S 1^2 0 1^a-4 D> 1 0 1^4 S [+4a steps] (if a mod 2 = 1 and a ≥ 4)
by:
S 0^2 1^a 0 D> 1 0 S
--> S 0^2 1^a 0^2 B> 0 S [+1]
--> S 0^2 1^a 0^2 <B 1 S [+2]
--> S 0^2 1^a <B 1^3 S [+4]
--> S 0^2 1^a-1 <C 1^4 S [+5]
--> S 0^2 1^a-2 <E 0 1^4 S [+6]
= S 0^2 1 1^a-3 <E 0 1^4 S
--> S 0^2 1 <E 1^a-3 0 1^4 S [+a +3]
--> S 0^2 <A 1^a-2 0 1^4 S [+a +4]
--> S 0 1 B> 1^a-2 0 1^4 S [+a +5]
--> S 0 1 <C 1^a-2 0 1^4 S [+a +6]
--> S 0 <E 0 1^a-2 0 1^4 S [+a +7]
--> S 1 F> 0 1^a-2 0 1^4 S [+a +8]
--> S 1 0 A> 1^a-2 0 1^4 S [+a +9]
--> S 1 0 <E 1^a-2 0 1^4 S [+a +10]
--> S 1^2 F> 1^a-2 0 1^4 S [+a +11]
--> S 1^2 0 D> 1^a-3 0 1^4 S [+a +12]
= S 1^2 0 D> 1^a-4 1 0 1^4 S
--> S 1^2 0 1^a-4 D> 1 0 1^4 S [+4a]
14. S 0^2 1^3 0 D> 1 0 S --> S 1^2 0 1 Z> 1^4 S [+16 steps]
by:
S 0^2 1^3 0 D> 1 0 S
--> S 0^2 1^3 0^2 B> 0 S [+1]
--> S 0^2 1^3 0^2 <B 1 S [+2]
--> S 0^2 1^3 <B 1^3 S [+4]
--> S 0^2 1^2 <C 1^4 S [+5]
--> S 0^2 1 <E 0 1^4 S [+6]
--> S 0^2 <A 1 0 1^4 S [+7]
--> S 0 1 B> 1 0 1^4 S [+8]
--> S 0 1 <C 1 0 1^4 S [+9]
--> S 0 <E 0 1 0 1^4 S [+10]
--> S 1 F> 0 1 0 1^4 S [+11]
--> S 1 0 A> 1 0 1^4 S [+12]
--> S 1 0 <E 1 0 1^4 S [+13]
--> S 1^2 F> 1 0 1^4 S [+14]
--> S 1^2 0 D> 0 1^4 S [+15]
--> S 1^2 0 1 Z> 1^4 S [+16]
15. S 0 1 0 D> 1 0 1^c S --> S 1^c+4 D> 1 S [+3c +15 steps]
by:
S 0 1 0 D> 1 0 1^c S
--> S 0 1 0^2 B> 0 1^c S [+1]
--> S 0 1 0^2 <B 1^c+1 S [+2]
--> S 0 1 <B 1^c+3 S [+4]
--> S 0 <C 1^c+4 S [+5]
--> S 1 D> 1^c+4 S [+6]
= S 1 D> 1^c+3 1 S [+6]
--> S 1^c+4 D> 1 S [+3c +15]
16. S 0 1^a 0 1^b D> 1 0 S --> S 1 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +2 steps] (if b mod 2 = 1 and a mod 2 = 0 and a ≥ 2)
by:
S 0 1^a 0 1^b D> 1 0 S
--> S 0 1^a 0 1^b 0 B> 0 S [+1]
--> S 0 1^a 0 1^b 0 <B 1 S [+2]
--> S 0 1^a 0 1^b <B 1^2 S [+3]
--> S 0 1^a 0 1^b-1 <C 1^3 S [+4]
--> S 0 1^a 0 1^b-2 <E 0 1^3 S [+5]
--> S 0 1^a 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0 1^a 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0 1^a+1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0 1^a+1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0 1^a <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0 <E 1^a 0 1^b-2 0 1^3 S [+b +a +6]
--> S 1 F> 1^a 0 1^b-2 0 1^3 S [+b +a +7]
--> S 1 0 D> 1^a-1 0 1^b-2 0 1^3 S [+b +a +8]
= S 1 0 D> 1^a-2 1 0 1^b-2 0 1^3 S
--> S 1 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +2]
17. S 0^2 1^b D> 1 0 S --> S 1^2 0 1^b-4 D> 1 0 1^3 S [+4b -1 steps] (if b mod 2 = 1 and b ≥ 5)
by:
S 0^2 1^b D> 1 0 S
--> S 0^2 1^b 0 B> 0 S [+1]
--> S 0^2 1^b 0 <B 1 S [+2]
--> S 0^2 1^b <B 1^2 S [+3]
--> S 0^2 1^b-1 <C 1^3 S [+4]
--> S 0^2 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 <A 1^b-2 0 1^3 S [+b +3]
--> S 0 1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0 1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0 <E 0 1^b-2 0 1^3 S [+b +6]
--> S 1 F> 0 1^b-2 0 1^3 S [+b +7]
--> S 1 0 A> 1^b-2 0 1^3 S [+b +8]
--> S 1 0 <E 1^b-2 0 1^3 S [+b +9]
--> S 1^2 F> 1^b-2 0 1^3 S [+b +10]
--> S 1^2 0 D> 1^b-3 0 1^3 S [+b +11]
= S 1^2 0 D> 1^b-4 1 0 1^3 S
--> S 1^2 0 1^b-4 D> 1 0 1^3 S [+4b -1]
18. S 0^2 1^3 D> 1 0 S --> S 1^2 0 1 Z> 1^3 S [+15 steps]
by:
S 0^2 1^3 D> 1 0 S
--> S 0^2 1^3 0 B> 0 S [+1]
--> S 0^2 1^3 0 <B 1 S [+2]
--> S 0^2 1^3 <B 1^2 S [+3]
--> S 0^2 1^2 <C 1^3 S [+4]
--> S 0^2 1 <E 0 1^3 S [+5]
--> S 0^2 <A 1 0 1^3 S [+6]
--> S 0 1 B> 1 0 1^3 S [+7]
--> S 0 1 <C 1 0 1^3 S [+8]
--> S 0 <E 0 1 0 1^3 S [+9]
--> S 1 F> 0 1 0 1^3 S [+10]
--> S 1 0 A> 1 0 1^3 S [+11]
--> S 1 0 <E 1 0 1^3 S [+12]
--> S 1^2 F> 1 0 1^3 S [+13]
--> S 1^2 0 D> 0 1^3 S [+14]
--> S 1^2 0 1 Z> 1^3 S [+15]
19. S 0^2 1^a 0 1^b D> 1 0 S --> S 1^2 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +8 steps] (if b mod 2 = 1 and a mod 2 = 1 and a ≥ 2)
by:
S 0^2 1^a 0 1^b D> 1 0 S
--> S 0^2 1^a 0 1^b 0 B> 0 S [+1]
--> S 0^2 1^a 0 1^b 0 <B 1 S [+2]
--> S 0^2 1^a 0 1^b <B 1^2 S [+3]
--> S 0^2 1^a 0 1^b-1 <C 1^3 S [+4]
--> S 0^2 1^a 0 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1^a 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 1^a 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0^2 1^a+1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0^2 1^a+1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0^2 1^a <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0^2 1 <E 1^a-1 0 1^b-2 0 1^3 S [b +a +5]
--> S 0^2 <A 1^a 0 1^b-2 0 1^3 S [+b +a +6]
--> S 0 1 B> 1^a 0 1^b-2 0 1^3 S [+b +a +7]
--> S 0 1 <C 1^a 0 1^b-2 0 1^3 S [+b +a +8]
--> S 0 <E 0 1^a 0 1^b-2 0 1^3 S [+b +a +9]
--> S 1 F> 0 1^a 0 1^b-2 0 1^3 S [+b +a +10]
--> S 1 0 A> 1^a 0 1^b-2 0 1^3 S [+b +a +11]
--> S 1 0 <E 1^a 0 1^b-2 0 1^3 S [+b +a +12]
--> S 1^2 F> 1^a-1 1 0 1^b-2 0 1^3 S [+b +a +13]
--> S 1^2 0 D> 1^a-2 1 0 1^b-2 0 1^3 S [+b +a +14]
--> S 1^2 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +8]
20. S 0^2 1 0 1^b D> 1 0 S --> S 1^2 0 1 Z> 1^b-2 0 1^3 S [+b +16 steps] (if b mod 2 = 1)
by:
S 0^2 1 0 1^b D> 1 0 S
--> S 0^2 1 0 1^b 0 B> 0 S [+1]
--> S 0^2 1 0 1^b 0 <B 1 S [+2]
--> S 0^2 1 0 1^b <B 1^2 S [+3]
--> S 0^2 1 0 1^b-1 <C 1^3 S [+4]
--> S 0^2 1 0 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 1 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0^2 1^2 B> 1^b-2 0 1^3 S [+b +4]
--> S 0^2 1^2 <C 1^b-2 0 1^3 S [+b +5]
--> S 0^2 1 <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0^2 <A 1 0 1^b-2 0 1^3 S [+b +7]
--> S 0 1 B> 1 0 1^b-2 0 1^3 S [+b +8]
--> S 0 1 <C 1 0 1^b-2 0 1^3 S [+b +9]
--> S 0 <E 0 1 0 1^b-2 0 1^3 S [+b +10]
--> S 1 F> 0 1 0 1^b-2 0 1^3 S [+b +11]
--> S 1 0 A> 1 0 1^b-2 0 1^3 S [+b +12]
--> S 1 0 <E 1 0 1^b-2 0 1^3 S [+b +13]
--> S 1^2 F> 1 0 1^b-2 0 1^3 S [+b +14]
--> S 1^2 0 D> 0 1^b-2 0 1^3 S [+b +15]
--> S 1^2 0 1 Z> 1^b-2 0 1^3 S [+b +16]
21. S 0 1^a 0 1 D> 1 0 1^c S --> S 0 1^a+c+3 D> 1 S [+3c +11 steps]
by:
S 0 1^a 0 1 D> 1 0 1^c S
--> S 0 1^a 0 1 0 B> 0 1^c S [+1]
--> S 0 1^a 0 1 0 <B 1^c+1 S [+2]
--> S 0 1^a 0 1 <B 1^c+2 S [+3]
--> S 0 1^a 0 <C 1^c+3 S [+4]
--> S 0 1^a+1 D> 1^c+2 1 S [+5]
--> S 0 1^a+c+3 D> 1 S [+3c +11]
</pre>
==Functions==
<pre>
Let A(a, b, c, d, e, f, ..., k) = 0^inf 1^a 0 1^b D> 1 0 1^c 0 1^d 0 1^e 0 1^f ... 0 1^k 0^inf
b mod 2 = 0:
  b ≥ 4: A(a, b, c, ...) --> A(a+1, b-4, c+3, ...) by rule 7
  b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) by rule 9
  b = 0:
      a mod 2 = 0:
        a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) by rule 10
        a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) by rule 11
        a = 0: A(0, 0, c, ...) --> spin out by rule 12
      a mod 2 = 1:
        a ≥ 4: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) by rule 13
        a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... by rule 14
        a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) by rule 15
b mod 2 = 1:
  b ≥ 3:
      a mod 2 = 0:
        a ≥ 2: A(a, b, c, ...) --> A(1, a-2, b-2, c+3, ...) by rule 16
        a = 0:
            b ≥ 5: A(0, b, c, ...) --> A(2, b-4, c+3, ...) by rule 17
            b = 3: A(0, 3, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+3 ... by rule 18
      a mod 2 = 1:
        a ≥ 2: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) by rule 19
        a = 1: A(1, b, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^b-2 0 1^c+3 ... by rule 20
  b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) by rule 21
</pre>
Rules 12, 18 and 20 are not reachable by any of these rules (reaching them would require negative entries), meaning that they can only be triggered if they are the TMs starting configurations.
'''Accelerated rules:'''
<pre>
R8: A(a, 4k+v, c, ...) --> A(a+k, v, c+3k, ...) [+4bk -8k^2 +k steps] (if v mod 2 = 0 and k ≥ 1)
A1: A(0, 2k+1, c, ...) --> A(0, 2(k-1)+1, c+6, ...) [+16k +10 steps] (if k ≥ 3)
by:
A(0, 2k+1, c, ...)
--> A(2, 2(k-2)+1, c+3, ...) by rule 17 [+8k +3]
--> A(1, 0, 2(k-3)+1, c+6, ...) by rule 16 [+10k +10]
--> A(0, 2(k-1)+1, c+6, ...) by rule 15 [+16k +10]
A2: A(0, 2k+1, c, ...) --> A(0, 5, c+6k-12) [+8k^2 +18k -68 steps] (if k ≥ 3)
by repetition of rule A1
A3: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) [+8k^2 +36k +3c -65 steps] (if k ≥ 3)
by:
A(0, 2k+1, c, ...)
--> A(0, 5, c+6k-12, ...) by rule A2 [+8k^2 +18k -68]
--> A(2, 1, c+6k-9, ...) by rule 17 [8k^2 +18k -49]
--> A(0, c+6k-4, ...) by rule 21 [+8k^2 +36k + 3c -65]
</pre>
'''Using the accelerated rules:'''
<pre>
Let A(a, b, c, d, e, f, ..., k) = 0^inf 1^a 0 1^b D> 1 0 1^c 0 1^d 0 1^e 0 1^f ... 0 1^k 0^inf
b mod 2 = 0:
  b ≥ 4: A(a, 4k+v, c, ...) --> A(a+k, v, c+3k, ...) by rule 8
  b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) by rule 9
  b = 0:
      a mod 2 = 0:
        a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) by rule 10
        a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) by rule 11
        a = 0: unreachable
      a mod 2 = 1:
        a ≥ 4: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) by rule 13
        a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... by rule 14
        a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) by rule 15
b mod 2 = 1:
  b ≥ 3:
      a mod 2 = 0:
        a ≥ 2: A(a, b, c, ...) --> A(1, a-2, b-2, c+3, ...) by rule 16
        a = 0:
            b ≥ 7: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) by rule A3
            b = 5: A(0, b, c, ...) --> A(2, b-4, c+3, ...) by rule 17
            b = 3: unreachable
      a mod 2 = 1:
        a ≥ 2: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) by rule 19
        a = 1: unreachable
  b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) by rule 21
</pre>
==Trajectory==
<pre>
S=0: 0^inf <A 0^inf
S=1: 0^inf 1 B> 0^inf
S=2: 0^inf 1 <B 1 0^inf
S=3: 0^inf <C 1^2 0^inf
S=4: 0^inf 1 D> 1^2 0^inf
S=7: 0^inf 1^2 D> 1 0^inf = A(0, 2, 0)
So this TM reaches configuration A(0, 2, 0) after 7 steps.
</pre>
==Permutations==
'''Starting in state B'''
<pre>
S=0: 0^inf <B 0^inf
--> spin out
</pre>
'''Starting in state C'''
<pre>
S=0: 0^inf <C 0^inf
S=1: 0^inf 1 D> 0^inf
S=2: 0^inf 1^2 Z> 0^inf
</pre>
'''Starting in state D'''
<pre>
S=0: 0^inf <D 0^inf
S=1: 0^inf 1 Z> 0^inf
</pre>
'''Starting in state E'''
<pre>
S=0: 0^inf <E 0^inf
S=1: 0^inf 1 F> 0^inf
S=2: 0^inf 1 0 A> 0^inf
S=3: 0^inf 1 0 1 B> 0^inf
S=4: 0^inf 1 0 1 <B 1 0^inf
S=5: 0^inf 1 0 <C 1^2 0^inf
S=6: 0^inf 1^2 D> 1^2 0^inf
S=9: 0^inf 1^3 D> 1 0^inf = A(0, 3, 0)
--> Reaches configuration A(0, 3, 0) after 9 steps.
S=24: 0^inf 1^2 0 1 Z> 1^3 0^inf
</pre>
'''Starting in state F'''
<pre>
S=0: 0^inf <F 0^inf
S=1: 0^inf A> 0^inf
--> Equivalent to starting in state A, but started one step earlier.
</pre>
</div></div>
</div></div>

Latest revision as of 18:40, 10 November 2025

A page for analyses of individual machines.

1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD (bbch)

1. One of the seven potential BB(4,3) champions discovered by Pavel Kropitz in May 2024. Analysed in September 2025. This TM runs the shortest of Pavel's potential champions, achieving a score of about 10↑↑9.873987.

Analysis
0 1 2
A 1RB 1RD 1LC
B 2LB 1RB 1LC
C 1RZ 1LA 1LD
D 2RB 2RA 2RD 
S is any tape configuration
1. S D> 2^a S --> S 2^a D> S [+a steps]
2. S B> 1^a S --> S 1^a B> S [+a steps]
3. S 1 B> 0 S --> S <A 1^2 S [+4 steps]
4. S D> (11)^a S --> S (21)^a D> S [+2a steps]
   S A> (11)^a S --> S (12)^a A> S [+2a steps]
5. S (21)^a <C S --> S <C (11)^a S [+2a steps]
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]

7. S (12)^a 2 (12)^b A> 0^2 S --> S (12)^a-1 2 (12)^b+2 A> S [+4b +7 steps]
by:
S (12)^a 2 (12)^b A> 0^2 S
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
--> S (12)^a-1 1 <D (11)^b+2 S
--> S (12)^a-1 2 A> (11)^b+2 S
--> S (12)^a-1 2 (12)^b+2 A> S

8. S (12)^a 2 (12)^b A> 0^inf --> S 2 (12)^b+2a A> 0^inf [+4a^2 +8a +4ba steps]
Obtained by repeating rule 7.

9. S (12)^a <D (11)^b 0^inf --> S (12)^a-1 <D (11)^2b+3 0^inf [+4b^2 +22b +22 steps]
by:
S (12)^a <D (11)^b 0^inf
--> S (12)^a D> (11)^b 0^inf
--> S (12)^a (21)^b D> 0^inf
--> S (12)^a (21)^b 2 B> 0^inf
--> S (12)^a (21)^b 2 <B 2 0^inf
--> S (12)^a (21)^b <C 1 2 0^inf
--> S (12)^a <C (11)^b 1 2 0^inf
--> S (12)^a-1 1 <D (11)^b+1 2 0^inf
--> S (12)^a-1 2 A> (11)^b+1 2 0^inf
--> S (12)^a-1 2 (12)^b+1 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+1 <C 1 0^inf
--> S (12)^a-1 2 (12)^b 1 <D 11 0^inf
--> S (12)^a-1 2 (12)^b 2 A> (11)^1 0^inf
--> S (12)^a-1 2 (12)^b 2 (12)^1 A> 0^inf
--> S (12)^a-1 2 2 (12)^2b+1 A> 0^inf
--> S (12)^a-1 2^2 <A (11)^2b+2 0^inf
--> S (12)^a-1 2 <C 1 (11)^2b+2 0^inf
--> S (12)^a-1 <D (11)^2b+3 0^inf

10. S (12)^a <D (11)^b 0^inf --> S <D (11)^((2^(a))*b+(2^(a))*3-3) 0^inf
Obtained by repeating rule 9.

11. S (11)^a <D (11)^b 0^inf --> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf [+10b +50 steps]
by:
S (11)^a <D (11)^b 0^inf
--> S (11)^a-1 1 2 A> (11)^b 0^inf
--> S (11)^a-1 (12)^b+1 A> 0^inf
--> S (11)^a-1 <A (11)^b+2 0^inf
--> S (11)^a-1 D> (11)^b+2 0^inf
--> S (11)^a-1 (21)^b+2 D> 0^inf
--> S (11)^a-1 (21)^b+2 2 B> 0^inf
--> S (11)^a-1 (21)^b+2 2 <B 2 0^inf
--> S (11)^a-1 (21)^b+2 <C (12)^1 0^inf
--> S (11)^a-1 <C (11)^b+2 1 2 0^inf
--> S (11)^a-2 1 <A (11)^b+3 2 0^inf
--> S (11)^a-2 1 D> (11)^b+3 2 0^inf
--> S (11)^a-2 1 (21)^b+3 D> 2 0^inf
--> S (11)^a-2 1 (21)^b+3 2 D> 0^inf
--> S (11)^a-2 1 (21)^b+3 2^2 B> 0^inf
--> S (11)^a-2 1 (21)^b+3 2^2 <B 2 0^inf
--> S (11)^a-2 1 (21)^b+3 2 <C (12)^1 0^inf
--> S (11)^a-2 1 (21)^b+3 <D 1 1 2 0^inf
Note that 1 (21)^k = (12)^k 1
= S (11)^a-2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (11)^a-2 (12)^b+3 2 A> (11)^1 2 0^inf
--> S (11)^a-2 (12)^b+3 2 (12)^1 A> 2 0^inf
--> S (11)^a-2 (12)^b+3 2 (12)^1 <C 1 0^inf
--> S (11)^a-2 (12)^b+3 2 1<D (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 A> (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 A> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 <A (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 2 <C 1 (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf

12. S 1^a <A (11)^b 0^inf --> 1^a-1 <A (11)^b+1 2 0^inf [+4b +5 steps]
by:
S 1^a <A (11)^b 0^inf
--> S 1^a D> (11)^b 0^inf
--> S 1^a (21)^b D> 0^inf
--> S 1^a (21)^b 2 B> 0^inf
--> S 1^a (21)^b 2 <B 2 0^inf
--> S 1^a (21)^b <C 1 2 0^inf
--> S 1^a <C (11)^b 1 2 0^inf
--> 1^a-1 <A (11)^b+1 2 0^inf

Functions

Let A(a,b,c) = S (11)^a (12)^b <D (11)^c 0^inf

  • Rule 9: A(a, b, c) --> A(a, b - 1, 2c + 3)
  • Rule 10: A(a, b, c) --> A(a,0,2b×c+2b×3−3) which becomes A(a,0,2b+1×3−3) if c = 3.
  • Rule 11: A(a, 0, c) --> A(a - 2, c + 3, 3)

Further: let f(n)=2n+1×3

  • If c = 3: A(a, b, 3) --> A(a, 0, f(b) - 3) --> A(a - 2, f(b), 3)
  • A(a, 0, c) --> A(a−2,c+3,3) --> A(a−2,0,f(c+3)−3)
  • A(2k + d, b, 3) --> A(d,fk(b),3)

Trajectory

S=0: 0^inf A> 0^inf
S=1: 0^inf 1 B> 0^inf
S=5: 0^inf <A (11)^1 0^inf
S=6: 0^inf 1 B> (11)^1 0^inf
S=8: 0^inf 1 (11)^1 B> 0^inf
S=9: 0^inf 1 (11)^1 <B 2 0^inf
S=10: 0^inf 1 (11)^1 B> 2 0^inf
S=11: 0^inf 1 (11)^1 <C 1 0^inf
S=12: 0^inf (11)^1 <A (11)^1 0^inf
S=21: 0^inf 1 <A (11)^2 2 0^inf by rule 12
S=22: 0^inf 1 D> (11)^2 2 0^inf
S=26: 0^inf 1 (21)^2 D> 2 0^inf
S=27: 0^inf 1 (21)^2 2 D> 0^inf
S=28: 0^inf 1 (21)^2 2^2 B> 0^inf
S=29: 0^inf 1 (21)^2 2^2 <B 2 0^inf
S=30: 0^inf 1 (21)^2 2 <C 1 2 0^inf
S=31: 0^inf 1 (21)^2 <D (11)^1 2 0^inf
S=32: 0^inf 1 (21)^1 2^2 A> (11)^1 2 0^inf
S=34: 0^inf 1 (21)^1 2^2 (12)^1 A> 2 0^inf
S=35: 0^inf 1 (21)^1 2^2 (12)^1 <C 1 0^inf
S=36: 0^inf 1 (21)^1 2^2 1 <D (11)^1 0^inf
S=37: 0^inf 1 (21)^1 2^3 A> (11)^1 0^inf
S=39: 0^inf (12)^2 2^2 (12)^1 A> 0^inf
S=46: 0^inf (12)^2 2^2 <A (11)^2 0^inf
S=47: 0^inf (12)^2 2 <C 1 (11)^2 0^inf
S=48: 0^inf (12)^2 <D (11)^3 0^inf
S=172: 0^inf (12)^1 <D (11)^9 0^inf by rule 9
S=716: 0^inf <D (11)^21 0^inf by rule 9
S=717: 0^inf 2 B> (11)^21 0^inf
S=759: 0^inf 2 (11)^21 B> 0^inf
S=760: 0^inf 2 (11)^21 <B 2 0^inf
S=761: 0^inf 2 (11)^21 B> 2 0^inf
S=762: 0^inf 2 (11)^21 <C 1 0^inf
S=763: 0^inf 2 (11)^20 1 <A (11)^1 0^inf
S=772: 0^inf 2 (11)^20 <A (11)^2 2 0^inf
S=773: 0^inf 2 (11)^20 D> (11)^2 2 0^inf
S=777: 0^inf 2 (11)^20 (21)^2 D> 2 0^inf
S=778: 0^inf 2 (11)^20 (21)^2 2 D> 0^inf
S=779: 0^inf 2 1 (11)^19 1 (21)^2 2^2 B> 0^inf
S=780: 0^inf 2 1 (11)^19 (12)^3 2 <B 2 0^inf
S=781: 0^inf 2 1 (11)^19 (12)^3 <C 1 2 0^inf
S=782: 0^inf 2 1 (11)^19 (12)^2 1 <D (11)^1 2 0^inf
S=783: 0^inf 2 1 (11)^19 (12)^2 2 A> (11)^1 2 0^inf
S=785: 0^inf 2 1 (11)^19 (12)^2 2 (12)^1 A> 2 0^inf
S=786: 0^inf 2 1 (11)^19 (12)^2 2 (12)^1 <C 1 0^inf
S=787: 0^inf 2 1 (11)^19 (12)^2 2 1 <D (11)^1 0^inf
S=788: 0^inf 2 1 (11)^19 (12)^2 2^2 A> (11)^1 0^inf
S=790: 0^inf 2 1 (11)^19 (12)^2 2^2 (12)^1 A> 0^inf
S=797: 0^inf 2 1 (11)^19 (12)^2 2^2 <A (11)^2 0^inf
S=798: 0^inf 2 1 (11)^19 (12)^2 2 <C 1 (11)^2 0^inf
S=799: 0^inf 2 1 (11)^19 (12)^2 <D (11)^3 0^inf
= A(19, 2, 3)

A(19, 2, 3) --> A(1,f9(2),3) --> A(1,0,f10(2)−3)

Let m = f10(2)−3

--> 0^inf 2 1 (11)^1 <D (11)^m 0^inf 

Final trajectory:
0^inf 2 1 (11)^1 <D (11)^m 0^inf
--> 0^inf 2 1 1 2 A> (11)^m 0^inf
--> 0^inf 2 1 (12)^m+1 A> 0^inf
--> 0^inf 2 1 <A (11)^m+2 0^inf
--> 0^inf 2 1 D> (11)^m+2 0^inf
--> 0^inf (21)^m+3 D> 0^inf
--> 0^inf (21)^m+3 2 B> 0^inf
--> 0^inf (21)^m+3 2 <B 2 0^inf
--> 0^inf (21)^m+3 <C (12)^1 0^inf
--> 0^inf <C (11)^m+3 (12)^1 0^inf
--> 0^inf 1 Z> (11)^m+3 (12)^1 0^inf
Score = 2m + 9

Approximate Score

Score calculated in HyperCalc:

(10^)^8 30,302,671.815163

Or in tetration: 10^^9.873987 (truncated)

Permutations

Starting in state B 

0^inf <B 0^inf
--> translated cycler

Starting in state C 

0^inf <C 0^inf
--> 0^inf 1 Z> 0^inf

Starting in state D 

0^inf <D 0^inf
S=1: 0^inf 2 B> 0^inf
S=2: 0^inf 2 <B 2 0^inf
S=3: 0^inf <C 1 2 0^inf
S=4: 0^inf 1 Z> 1 2 0^inf

1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)

2. One of the seven potential BB(4,3) champions discovered by Pavel Kropitz in May 2024. Analysed in October 2025. This TM runs the longest of Pavel's potential champions and is - as of October 20th 2025 - the highest scoring BB(4,3) TM with score > 2↑↑↑2↑↑↑2↑↑↑7.92×1028.

Analysis
0 1 2
A 1RB 1RD 1LC
B 2LB 1RB 1LC
C 1RZ 1LA 1LD
D 0RB 2RA 2RD 
S is any tape configuration
1. S D> 2^a S --> S 2^a D> S [+a steps]
2. S B> 1^a S --> S 1^a B> S [+a steps]
3. S A> 0^2 S --> S <A 1^2 S [+5 steps]
4. S D> (11)^a S --> S (21)^a D> S [+2a steps]
   S A> (11)^a S --> S (12)^a A> S [+2a steps]
5. S (21)^a <C S --> S <C (11)^a S [+2a steps]
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]
by:
S (12)^a A> 0^2 S
--> S (12)^a <A (11)^1 S
--> S <A (11)^a+1 S

7. S A> (11)^1 2^b S --> S 2 A> (11)^1 2^b-1 S [+5 steps]
by:
S A> (11)^1 2^b S
--> S (12)^1 A> 2^b S
--> S (12)^1 <C 1 2^b-1 S
--> S 1 <D (11)^1 2^b-1 S
--> S 2 A> (11)^1 2^b-1 S
8. S A> (11)^1 2^b S --> S 2^b A> (11)^1 S [+5b steps]
by repetition of rule 7

9. S D> 0^2 S --> S <B 2^2 S [+3 steps]

10. S 2 <D (11)^a 0^2 S --> S <D (11)^a+1 2 S [+4a +7 steps]
by:
S 2 <D (11)^a 0^2 S
--> S 2 D> (11)^a 0^2 S
--> S 2 (21)^a D> 0^2 S
--> S 2 (21)^a <B 2^2 S
--> S 2 (21)^a B> 2^2 S
--> S 2 (21)^a <C 1 2 S
--> S 2 <C (11)^a 1 2 S
--> S <D (11)^a+1 2 S

11. S 2 <D (11)^a 2 0^2 S --> S <D (11)^a+1 2^2 S [+4a +7 steps]
by:
S 2 <D (11)^a 2 0^2 S
--> S 2 D> (11)^a 2 0^2 S
--> S 2 (21)^a D> 2 0^2 S
--> S 2 (21)^a 2 D> 0^2 S
--> S 2 (21)^a 2 <B 2^2 S
--> S 2 (21)^a <C 1 2^2 S
--> S 2 <C (11)^a 1 2^2 S
--> S <D (11)^a+1 2^2 S

12. S  1^a <A (11)^b 0^2 S --> S 1^a-1 <A (11)^b+1 2 S [+4b +7 steps]
by:
S 1^a <A (11)^b 0^2 S
--> S 1^a D> (11)^b 0^2 S
--> S 1^a (21)^b D> 0^2 S
--> S 1^a (21)^b <B 2^2 S
--> S 1^a (21)^b B> 2^2 S
--> S 1^a (21)^b <C 1 2 S
--> S 1^a <C (11)^b 1 2 S
--> S 1^a-1 <A (11)^b+1 2 S

13. S 1^a <A (11)^b 2 0^2 S --> S 1^a-1 <A (11)^b+1 2^2 S [+4b +7 steps]
by:
S 1^a <A (11)^b 2 0^2 S
--> S 1^a D> (11)^b 2 0^2 S
--> S 1^a (21)^b D> 2 0^2 S
--> S 1^a (21)^b 2 D> 0^2 S
--> S 1^a (21)^b 2 <B 2^2 S
--> S 1^a (21)^b <C 1 2^2 S
--> S 1^a <C (11)^b 1 2^2 S
--> S 1^a-1 <A (11)^b+1 2^2 S

14. S (12)^a 1 <D (11)^b 0^2 S --> S (12)^a-1 1 <D (11)^b+2 [+4b +8 steps]
by:
S (12)^a 1 <D (11)^b 0^2 S
--> S (12)^a 2 A> (11)^b 0^2 S
--> S (12)^a 2 (12)^b A> 0^2 S
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
--> S (12)^a-1 1 <D (11)^b+2 S

15. S (12)^a 1 <D (11)^b 0^inf --> S 1 <D (11)^b+2a 0^inf [+4a^2 +4ba + 4a steps]
by repetition of rule 14

16. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf [+10b +28 steps]
by:
S (12)^a 2 1 <D (11)^b 0^inf
--> S (12)^a 2^2 A> (11)^b 0^inf
--> S (12)^a 2^2 (12)^b A> 0^inf
--> S (12)^a 2^2 <A (11)^b+1 0^inf
--> S (12)^a 2 <C 1 (11)^b+1 0^inf
--> S (12)^a <D (11)^b+2 0^inf
--> S (12)^a-1 1 <D (11)^b+3 2 0^inf by rule 10
--> S (12)^a-1 2 A> (11)^b+3 2 0^inf
--> S (12)^a-1 2 (12)^b+3 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+3 <C 1 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf

17. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 1 <D (11)^2b+5 0^inf
by:
S (12)^a 2 1 <D (11)^b 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf by rule 16
--> S (12)^a-1 2 1 <D (11)^2b+5 0^inf by rule 15

18. S (12)^a 2 1 <D (11)^b 0^inf --> S 2 1 <D (11)^(2^a)*b+(2^a)*5-5 0^inf
by repetition of rule 17

---
19. S (12)^a 2 1 <D (11)^b 2 0^inf --> S (12)^a 2^2 1 <D (11)^2b-1 0^inf
by:
S (12)^a 2 1 <D (11)^b 2 0^inf
--> S (12)^a 2^2 A> (11)^b 2 0^inf
--> S (12)^a 2^2 (12)^b A> 2 0^inf
--> S (12)^a 2^2 (12)^b <C 1 0^inf
--> S (12)^a 2^2 (12)^b-1 1 <D (11)^1 0^inf
--> S (12)^a 2^2 1 <D (11)^2b-1 0^inf by rule 15

20. S (12)^a 1 <D (11)^b 2 0^inf --> S (12)^a 2 1 <D (11)^2b-1 0^inf
by:
S (12)^a 1 <D (11)^b 2 0^inf
--> S (12)^a 2 A> (11)^b 2 0^inf
--> S (12)^a 2 (12)^b A> 2 0^inf
--> S (12)^a 2 (12)^b <C 1 0^inf
--> S (12)^a 2 (12)^b-1 1 <D (11)^1 0^inf
--> S (12)^a 2 1 <D (11)^2b-1 0^inf by rule 15

21. S (12)^a 2^2 1 <D (11)^b 0^inf --> S (12)^a-1 2^2 1 <D (11)^2^(b+4)*3-5 0^inf
by:
S (12)^a 2^2 1 <D (11)^b 0^inf
--> S (12)^a 2^3 A> (11)^b 0^inf
--> S (12)^a 2^3 (12)^b A> 0^inf
--> S (12)^a 2^3 <A (11)^b+1 0^inf
--> S (12)^a 2^2 <C 1 (11)^b+1 0^inf
--> S (12)^a 2 <D (11)^b+2 0^inf
--> S (12)^a <D (11)^b+3 2 0^inf by rule 10
--> S (12)^a-1 1 <D (11)^b+4 2^2 0^inf by rule 11
--> S (12)^a-1 2 A> (11)^b+4 2^2 0^inf
--> S (12)^a-1 2 (12)^b+4 A> 2^2 0^inf
--> S (12)^a-1 2 (12)^b+4 <C 1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 2 1 <D (11)^1 0^inf by rule 20
--> S (12)^a-1 2^2 1 <D (11)^(2^(b+3)*1)+(2^(b+3)*5)-5 0^inf by rule 18
= S (12)^a-1 2^2 1 <D (11)^(2^(b+4)*3-5) 0^inf

22. S 1 <D (11)^b 2^2 0^inf --> S 2 (12)^b-1 2 1 <D (11)^1 0^inf
by:
S 1 <D (11)^b 2^2 0^inf
--> S 2 A> (11)^b 2^2 0^inf
--> S 2 (12)^b A> 2^2 0^inf
--> S 2 (12)^b <C 1 2 0^inf
--> S 2 (12)^b-1 1 <D (11)^1 2 0^inf
--> S 2 (12)^b-1 2 1 <D (11)^1 0^inf by rule 20

23. S (11)^a 2^2 1 <D (11)^b 0^inf --> S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf
by:
S (11)^a 2^2 1 <D (11)^b 0^inf
--> S (11)^a 2^3 A> (11)^b 0^inf
--> S (11)^a 2^3 (12)^b A> 0^inf
--> S (11)^a 2^3 <A (11)^b+1 0^inf
--> S (11)^a 2^2 <C 1 (11)^b+1 0^inf
--> S (11)^a 2 <D (11)^b+2 0^inf
--> S (11)^a <D (11)^b+3 2 0^inf by rule 10
--> S (11)^a-1 1 2 A> (11)^b+3 2 0^inf
--> S (11)^a-1 (12)^b+4 A> 2 0^inf
--> S (11)^a-1 (12)^b+4 <C 1 0^inf
--> S (11)^a-1 (12)^b+3 1 <D (11)^1 0^inf
--> S (11)^a-1 1 <D (11)^2b+7 0^inf by rule 15
--> S (11)^a-1 2 A> (11)^2b+7 0^inf
--> S (11)^a-1 2 (12)^2b+7 A> 0^inf
--> S (11)^a-1 2 <A (11)^2b+8 0^inf
--> S (11)^a-1 <C 1 (11)^2b+8 0^inf
--> S (11)^a-2 1 <A (11)^2b+9 0^inf
--> S (11)^a-2 <A (11)^2b+10 2 0^inf by rule 12
--> S (11)^a-3 1 <A (11)^2b+11 2^2 0^inf by rule 13
--> S (11)^a-3 1 D> (11)^2b+11 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 D> 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 2^2 D> 0^inf
--> S (11)^a-3 1 (21)^2b+11 2^2 <B 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 2 <C 1 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 <D (11)^1 2^2 0^inf
= S (11)^a-3 (12)^2b+11 1 <D (11)^1 2^2 0^inf
--> S (11)^a-3 (12)^2b+11 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf

24. 0^inf 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 2^2 1 <D (11)^c 0^inf
--> 0^inf 2^3 A> (11)^c 0^inf
--> 0^inf 2^3 (12)^c A> 0^inf
--> 0^inf 2^3 <A (11)^c+1 0^inf
--> 0^inf 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 2 <D (11)^c+2 0^inf
--> 0^inf <D (11)^c+3 2 0^inf by rule 10
--> 0^inf B> (11)^c+3 2 0^inf
--> 0^inf (11)^c+3 B> 2 0^inf
--> 0^inf (11)^c+3 <C 1 0^inf
--> 0^inf (11)^c+2 1 <A (11)^1 0^inf
--> 0^inf (11)^c+2 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^c+1 1 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^c+1 1 D> (11)^3 2^2 0^inf
--> 0^inf (11)^c+1 1 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^c+1 1 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^c+1 1 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf

25. 0^inf (11)^2 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf (11)^2 2^2 1 <D (11)^c 0^inf
--> 0^inf (11)^2 2^3 A> (11)^c 0^inf
--> 0^inf (11)^2 2^3 (12)^c A> 0^inf
--> 0^inf (11)^2 2^3 <A (11)^c+1 0^inf
--> 0^inf (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf (11)^2 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^1 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 (12)^c+4 A> 2 0^inf
--> 0^inf (11)^1 (12)^c+4 <C 1 0^inf
--> 0^inf (11)^1 (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf (11)^1 1 <D (11)^2c+7 0^inf by rule 15
--> 0^inf (11)^1 2 A> (11)^2c+7 0^inf
--> 0^inf (11)^1 2 (12)^2c+7 A> 0^inf
--> 0^inf (11)^1 2 <A (11)^2c+8 0^inf
--> 0^inf (11)^1 <C 1 (11)^2c+8 0^inf
--> 0^inf 1 <A (11)^2c+9 0^inf
--> 0^inf <A (11)^2c+10 2 0^inf by rule 12
--> 0^inf 1 B> (11)^2c+10 2 0^inf
--> 0^inf 1 (11)^2c+10 B> 2 0^inf
--> 0^inf 1 (11)^2c+10 <C 1 0^inf
--> 0^inf (11)^2c+10 <A (11)^1 0^inf
--> 0^inf (11)^2c+9 1 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+9 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+9 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^2c+9 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf

26. 0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 (11)^1 2^3 A> (11)^c 0^inf
--> 0^inf 1 (11)^1 2^3 (12)^c A> 0^inf
--> 0^inf 1 (11)^1 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 (11)^1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 (11)^1 2 <D (11)^c+2 0^inf
--> 0^inf 1 (11)^1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 2 (12)^c+3 A> 2 0^inf
--> 0^inf (11)^1 2 (12)^c+3 <C 1 0^inf
--> 0^inf (11)^1 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf (11)^1 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf (11)^1 2^2 A> (11)^2c+5 0^inf
--> 0^inf (11)^1 2^2 (12)^2c+5 A> 0^inf
--> 0^inf (11)^1 2^2 <A (11)^2c+6 0^inf
--> 0^inf (11)^1 2 <C 1 (11)^2c+6 0^inf
--> 0^inf (11)^1 <D (11)^2c+7 0^inf
--> 0^inf 1 2 A> (11)^2c+7 0^inf
--> 0^inf (12)^2c+8 A> 0^inf
--> 0^inf <A (11)^2c+9 0^inf
--> 0^inf 1 B> (11)^2c+9 0^inf
--> 0^inf 1 (11)^2c+9 B> 0^inf
--> 0^inf 1 (11)^2c+9 <B 2 0^inf
--> 0^inf 1 (11)^2c+9 B> 2 0^inf
--> 0^inf 1 (11)^2c+9 <C 1 0^inf
--> 0^inf 1 (11)^2c+8 1 <A (11)^1 0^inf
--> 0^inf 1 (11)^2c+8 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+8 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+8 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+8 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+8 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^2c+8 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf

27. 0^inf 1 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 2^3 A> (11)^c 0^inf
--> 0^inf 1 2^3 (12)^c A> 0^inf
--> 0^inf 1 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 2 <D (11)^c+2 0^inf
--> 0^inf 1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf 2 A> (11)^c+3 2 0^inf
--> 0^inf 2 (12)^c+3 A> 2 0^inf
--> 0^inf 2 (12)^c+3 <C 1 0^inf
--> 0^inf 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf 2^2 A> (11)^2c+5 0^inf
--> 0^inf 2^2 (12)^2c+5 A> 0^inf
--> 0^inf 2^2 <A (11)^2c+6 0^inf
--> 0^inf 2 <C 1 (11)^2c+6 0^inf
--> 0^inf <D (11)^2c+7 0^inf
--> 0^inf B> (11)^2c+7 0^inf
--> 0^inf (11)^2c+7 B> 0^inf
--> 0^inf (11)^2c+7 <B 2 0^inf
--> 0^inf (11)^2c+7 B> 2 0^inf
--> 0^inf (11)^2c+7 <C 1 0^inf
--> 0^inf (11)^2c+6 1 <A (11)^1 0^inf
--> 0^inf (11)^2c+6 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+5 1 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+5 1 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+5 1 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2^2 D> 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf

28. 0^inf (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 Z> 1 (11)^2c+8 0^inf
by:
0^inf (11)^1 2^2 1 <D (11)^c 0^inf
--> 0^inf (11)^1 2^3 A> (11)^c 0^inf
--> 0^inf (11)^1 2^3 (12)^c A> 0^inf
--> 0^inf (11)^1 2^3 <A (11)^c+1 0^inf
--> 0^inf (11)^1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^1 2 <D (11)^c+2 0^inf
--> 0^inf (11)^1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (12)^c+4 A> 2 0^inf
--> 0^inf (12)^c+4 <C 1 0^inf
--> 0^inf (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf 1 <D (11)^2c+7 0^inf by rule 15
--> 0^inf 2 A> (11)^2c+7 0^inf
--> 0^inf 2 (12)^2c+7 A> 0^inf
--> 0^inf 2 <A (11)^2c+8 0^inf
--> 0^inf <C 1 (11)^2c+8 0^inf
--> 0^inf 1 Z> 1 (11)^2c+8 0^inf

Note: Rule 29 is not relevant to this TMs trajectory.
29. 0^inf 1 (11)^2 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^2c+10 (12)^2 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 (11)^2 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 (11)^2 2^3 A> (11)^c 0^inf
--> 0^inf 1 (11)^2 2^3 (12)^c A> 0^inf
--> 0^inf 1 (11)^2 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf 1 (11)^2 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^2 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^2 2 (12)^c+3 A> 2 0^inf
--> 0^inf (11)^2 2 (12)^c+3 <C 1 0^inf
--> 0^inf (11)^2 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf (11)^2 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf (11)^2 2^2 A> (11)^2c+5 0^inf
--> 0^inf (11)^2 2^2 (12)^2c+5 A> 0^inf
--> 0^inf (11)^2 2^2 <A (11)^2c+6 0^inf
--> 0^inf (11)^2 2 <C 1 (11)^2c+6 0^inf
--> 0^inf (11)^2 <D (11)^2c+7 0^inf
--> 0^inf (11)^1 1 2 A> (11)^2c+7 0^inf
--> 0^inf (11)^1 (12)^2c+8 A> 0^inf
--> 0^inf (11)^1 <A (11)^2c+9 0^inf
--> 0^inf 1 <A (11)^2c+10 2 0^inf by rule 12
--> 0^inf <A (11)^2c+11 2^2 0^inf by rule 13
--> 0^inf 1 B> (11)^2c+11 2^2 0^inf
--> 0^inf 1 (11)^2c+11 B> 2^2 0^inf
--> 0^inf 1 (11)^2c+11 <C 1 2 0^inf
--> 0^inf (11)^2c+11 <A (11)^1 2 0^inf
--> 0^inf (11)^2c+10 1 <A (11)^2 2^2 0^inf by rule 13
--> 0^inf (11)^2c+10 1 D> (11)^2 2^2 0^inf
--> 0^inf (11)^2c+10 1 (21)^2 D> 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2^2 D> 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2c+10 (12)^2 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
= 0^inf (11)^2c+10 (12)^2 2^2 1 <D (11)^1 0^inf

Functions

Let D(a, b, c) = 0^inf (11)^a (12)^b 2^2 1 <D (11)^c 0^inf

Let D_1(a, b, c) = 0^inf 1 (11)^a (12)^b 2^2 1 <D (11)^c 0^inf

Let f1(n)=2n+4×3−5

Let f2(a,b)=f12×f2(a−1,b)+11(1), wheref2(0,b)=b

Rule 21 becomes:

  • D(a,b,c)−−>D(a,b−1,2b+4×3−5)
  • D1(a,b,c)−−>D1(a,b−1,2b+4×3−5)

Rule 23 becomes:

  • D(a,0,c)−−>D(a−3,2c+11,1)
  • D1(a,0,c)−−>D1(a−3,2c+11,1)

Rule 24 becomes:

  • D(0,0,c)−−>D(c+1,3,1)

Rule 25 becomes:

  • D(2,0,c)−−>D(2c+8,3,1)

Rule 26 becomes:

  • D1(1,0,c)−−>D1(2c+7,3,1)

Rule 27 becomes:

  • D1(0,0,c)−−>D(2c+5,3,1)

Rule 28 becomes:

  • D(1, 0, c) --> halt with score 4c + 18

Rule 29 becomes:

  • D1(2,0,c)−−>D(2c+10,2,1)

By repeating rule 21, a stronger rule can be constructed:

  • D(a,b,c)−−>D(a,0,f1b(c))
  • D1(a,b,c)−−>D1(a,0,f1b(c))

If a is greater than or equal to 3: D(a,0,c)−−>D(a−3,2c+11,1)−−>D(a−3,0,f12c+11(1)) =D(a−3,0,f2(1,c))

  • D(a,0,c)−−>D(a−3,0,f12c+11(1))

This rule can also be repeated, also note that f12c+11(1)=f2(1,c) and f12×f2(a,b)+11(1)=f2(a+1,b):

  • D(3k+d,0,c)−−>D(d,0,f2(k,c))
  • D1(3k+d,0,c)−−>D1(d,0,f2(k,c))

Trajectory

S=0: 0^inf A> 0^inf
S=5: 0^inf <A (11)^1 0^inf
S=6: 0^inf 1 B> (11)^1 0^inf
S=8: 0^inf 1 (11)^1 B> 0^inf
S=9: 0^inf 1 (11)^1 <B 2 0^inf
S=10: 0^inf 1 (11)^1 B> 2 0^inf
S=11: 0^inf 1 (11)^1 <C 1 0^inf
S=12: 0^inf (11)^1 <A (11)^1 0^inf
S=23: 0^inf 1 <A (11)^2 2 0^inf
S=38: 0^inf <A (11)^3 2^2 0^inf
S=39: 0^inf 1 B> (11)^3 2^2 0^inf
S=45: 0^inf 1 (11)^3 B> 2^2 0^inf
S=46: 0^inf 1 (11)^3 <C 1 2 0^inf
S=47: 0^inf (11)^3 <A (11)^1 2 0^inf
S=58: 0^inf 1 (11)^2 <A (11)^2 2^2 0^inf
S=59: 0^inf 1 (11)^2 D> (11)^2 2^2 0^inf
S=63: 0^inf 1 (11)^2 (21)^2 D> 2^2 0^inf
S=65: 0^inf 1 (11)^2 (21)^2 2^2 D> 0^inf
S=68: 0^inf 1 (11)^2 (21)^2 2^2 <B 2^2 0^inf
S=69: 0^inf 1 (11)^2 (21)^2 2 <C 1 2^2 0^inf
S=70: 0^inf (11)^2 (12)^2 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2 (12)^2 2 (12)^0 2 1 <D (11)^1 0^inf
=0^inf (11)^2 (12)^2 2^2 1 <D (11)^1 0^inf
= D(2, 2, 1)
So, the TM starts in configuration D(2, 2, 1).

D(2, 2, 1) -->

D(2,0,f12(1))=D(2,0,f1(91))

e1=f1(91)=295×3−5 

f_1(n) = 2^(n+4)*3 - 5
Note that the times three means that this expression of of the form 3k - 5 which can be rewritten as 3(k-1)-2 which can again be rewritten as 3(k-2)+1.
Next, 3k+1 mod 3 = 1
So f_1(n) mod 3 = 1
Thus f_1^a(n) mod 3 = 1
f_2(a,b) = f_1^(2*f_2(a-1,b)+11)(1)
Note that f_1^(2*f_2(a-1,b)+11)(1) is also of the form f_1^a(n)
Thus f_2(a,b) mod 3 = 1

D(2,0,e1)

-->D1(2e1+8,3,1)−−>D1(2e1+8,0,f12(91))

e_1 mod 3 = 1; 2*1 + 8 = 10 --> 10 mod 3 = 1

D1(2e1+8,0,f12(91))

--> D1(1,0,f2(2e1+73,f12(91)))

e2=f2(2e1+73,f12(91))

D1(1,0,e3)

e2mod3=1

--> D1(2e2+7,3,1)−−>D1(2e2+7,0,f12(91))

2e_3 + 7

Modulus: 2 + 7 --> 9 mod 3 = 0

--> D1(0,0,f2(2e2+73,f12(91)))

e3=f2(2e2+73,f12(91))


D1(0,0,e3)

--> D(2e3+5,3,1)−−>D(2e3+5,0,f12(91))

e_3 mod 3 = 1; 2*1+5 = 7 --> 7 mod 3 = 1

--> D(1,0,f2(2e3+43,f12(91)))

e4=f2(2e3+43,f12(91))


D(1,0,e4)

--> halts with score 4e4+18.

Approximate Score

4e4+18

e4=f2(2e3+43,f12(91))

e3=f2(2e2+73,f12(91))

e2=f2(e1+73,f12(91))

e1=295×3−5

f1(n)=2n+4×3−5

f2(a,b)=f12×f2(a−1,b)+11(1), where f2(0,b)=b

  • f1(n):

2n+5<f1(n)<2n+6

(2↑)an+5<f1a(n)<(2↑)an+7

(2↑)a5<f1a(1)<(2↑)a8

2↑↑(a+2)<f1a(1)<2↑↑(a+3)

  • f2(a,b):

f2(a,b)=f12×f2(a−1,b)+11(1), where f2(0,b)=b

2↑↑(2×f2(a−1,b)+13)<f2(a,b)<2↑↑(2×f2(a−1,b)+14)

(2↑↑)ab<f2(a,b)<(2↑↑)a+1b

2↑↑2↑↑2<e1<f12(91)<2↑↑2↑↑2↑↑2

(2↑↑)a(2↑↑2↑↑2)<f2(a,f12(91))<(2↑↑)a(2↑↑2↑↑2↑↑2)

2↑↑↑(a+3)<f2(a,f12(91))<2↑↑↑(a+4)

2↑↑↑(7.92×1028)<e2<2↑↑↑(7.93×1028)

2↑↑↑2↑↑↑(7.92×1028)<e3<2↑↑↑2↑↑↑(7.93×1028)

2↑45<2↑↑↑2↑↑↑2↑↑↑(7.92×1028)<e4<σ<S<2↑↑↑2↑↑↑2↑↑↑(7.93×1028)

This score would make 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) the new BB(4,3) champion.

Permutations

Starting in state B 

0^inf <B 0^inf
--> 0^inf <B 2^k 0^inf
--> translated cycler

Starting in state C 

0^inf <C 0^inf
-->0^inf 1 Z> 0^inf

Starting in state D 

0^inf <D 0^inf
--> 0^inf <B 0^inf
--> translated cycler

1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD (bbch)

3. One of the seven potential BB(4,3) champions discovered by Pavel Kropitz in May 2024. Analysed in October 2025. This TM has rules which are based on the remainder of some value modulo 4, although it is quite unlucky that three of the four possible remainders lead to halting. The TM achieves a score of around 3↑↑↑88574.

Analysis
0 1 2
A 1RB 2LB 0LB
B 2LC 2LA 0LA
C 2RD 1LC 1RZ
D 1RA 2LD 1RD 
S is any tape configuration

1. S 1^a <C S --> S <C 1^a S [+a steps]
2. S 1^a <D S --> S <D 2^a S [+a steps]
3. S D> 2^a S --> S 1^a D> S [+a steps]

4. S (11)^a <A S --> S <A (22)^a S [+2a steps]
   S (11)^a <B S --> S <B (22)^a S [+2a steps]

5. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3 A> (22)^b-1 S [+8a +24 steps]
by:
0^inf 2 (11)^a A> (22)^b S
--> 0^inf 2 (11)^a <B 0 2 (22)^b-1 S [+1]
--> 0^inf 2 <B (22)^a 0 2 (22)^b-1 S [+2a +1]
--> 0^inf <A 0 (22)^a 0 2 (22)^b-1 S [+2a +2]
--> 0^inf 1 B> 0 (22)^a 0 2 (22)^b-1 S [+2a +3]
--> 0^inf 1 <C 2 (22)^a 0 2 (22)^b-1 S [+2a +4]
--> 0^inf <C 1 2 (22)^a 0 2 (22)^b-1 S [+2a +5]
--> 0^inf 2 D> 1 2 (22)^a 0 2 (22)^b-1 S [+2a +6]
--> 0^inf 2 <D (22)^a+1 0 2 (22)^b-1 S [+2a +7]
--> 0^inf 1 D> (22)^a+1 0 2 (22)^b-1 S [+2a +8]
--> 0^inf 1 (11)^a+1 D> 0 2 (22)^b-1 S [+4a +10]
--> 0^inf (11)^a+2 A> 2 (22)^b-1 S [+4a +11]
--> 0^inf (11)^a+2 <B 0 (22)^b-1 S [+4a +12]
--> 0^inf <B (22)^a+2 0 (22)^b-1 S [+6a +16]
--> 0^inf <C 2 (22)^a+2 0 (22)^b-1 S [+6a +17]
--> 0^inf 2 D> 2 (22)^a+2 0 (22)^b-1 S [+6a +18]
--> 0^inf 2 1 (11)^a+2 D> 0 (22)^b-1 S [+8a +23]
--> 0^inf 2 (11)^a+3 A> (22)^b-1 S [+8a +24]

6. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3b A> S
by repetition of rule 5

7. 0^inf 2 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^a+2 0 (22)^b-1 S [+6a +28 steps]
by:
0^inf 2 (11)^a A> 0 (22)^b S
--> 0^inf 2 (11)^a 1 B> (22)^b S [+1]
--> 0^inf 2 1 (11)^a <A 0 2 (22)^b-1 S [+2]
--> 0^inf 2 1 <A (22)^a 0 2 (22)^b-1 S [+2a +2]
--> 0^inf 2 <B 2 (22)^a 0 2 (22)^b-1 S [+2a +3]
--> 0^inf <A 0 2 (22)^a 0 2 (22)^b-1 S [+2a +4]
--> 0^inf 1 B> 0 2 (22)^a 0 2 (22)^b-1 S [+2a +5]
--> 0^inf 1 <C (22)^a+1 0 2 (22)^b-1 S [+2a +6]
--> 0^inf <C 1 (22)^a+1 0 2 (22)^b-1 S [+2a +7]
--> 0^inf 2 D> 1 (22)^a+1 0 2 (22)^b-1 S [+2a +8]
--> 0^inf 2 <D 2 (22)^a+1 0 2 (22)^b-1 S [+2a +9]
--> 0^inf 1 D> 2 (22)^a+1 0 2 (22)^b-1 S [+2a +10]
--> 0^inf (11)^a+2 D> 0 2 (22)^b-1 S [+4a +13]
--> 0^inf (11)^a+2 1 A> 2 (22)^b-1 S [+4a +14]
--> 0^inf (11)^a+2 1 <B 0 (22)^b-1 S [+4a +15]
--> 0^inf 1 <B (22)^a+2 0 (22)^b-1 S [+6a +19]
--> 0^inf <A 2 (22)^a+2 0 (22)^b-1 S [+6a +20]
--> 0^inf 1 B> 2 (22)^a+2 0 (22)^b-1 S [+6a +21]
--> 0^inf 1 <A 0 (22)^a+2 0 (22)^b-1 S [+6a +22]
--> 0^inf <B 2 0 (22)^a+2 0 (22)^b-1 S [+6a +23]
--> 0^inf <C 2 2 0 (22)^a+2 0 (22)^b-1 S [+6a +24]
--> 0^inf 2 D> (22)^1 0 (22)^a+2 0 (22)^b-1 S [+6a +25]
--> 0^inf 2 1 1 D> 0 (22)^a+2 0 (22)^b-1 S [+6a +27]
--> 0^inf 2 1 (11)^1 A> (22)^a+2 0 (22)^b-1 S [+6a +28]

8. 0^inf 2 (11)^a A> 2 0 2 S --> 0^inf 2 1 (11)^a+3 A> S [+8a +27 steps]
by:
0^inf 2 (11)^a A> 2 0 2 S
--> 0^inf 2 (11)^a <B 0^2 2 S [+1]
--> 0^inf 2 <B (22)^a 0^2 2 S [+2a +1]
--> 0^inf <A 0 (22)^a 0^2 2 S [+2a +2]
--> 0^inf 1 B> 0 (22)^a 0^2 2 S [+2a +3]
--> 0^inf 1 <C 2 (22)^a 0^2 2 S [+2a +4]
--> 0^inf <C 1 2 (22)^a 0^2 2 S [+2a +5]
--> 0^inf 2 D> 1 2 (22)^a 0^2 2 S [+2a +6]
--> 0^inf 2 <D (22)^a+1 0^2 2 S [+2a +7]
--> 0^inf 1 D> (22)^a+1 0^2 2 S [+2a +8]
--> 0^inf 1 (11)^a+1 D> 0^2 2 S [+4a +10]
--> 0^inf (11)^a+2 A> 0 2 S [+4a +11]
--> 0^inf (11)^a+2 1 B> 2 S [+4a +12]
--> 0^inf (11)^a+2 1 <A 0 S [+4a +13]
--> 0^inf 1 <A (22)^a+2 0 S [+6a +17]
--> 0^inf <B 2 (22)^a+2 0 S [+6a +18]
--> 0^inf <C (22)^a+3 0 S [+6a +19]
--> 0^inf 2 D> (22)^a+3 0 S [+6a +20]
--> 0^inf 2 (11)^a+3 D> 0 S [+8a +26]
--> 0^inf 2 1 (11)^a+3 A> S [+8a +27]

9. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^3a+4 A> (22)^b-1 S
by:
0^inf 2 1 (11)^a A> (22)^b S
--> 0^inf 2 1 (11)^a <B 0 2 (22)^b-1 S
--> 0^inf 2 1 <B (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 <A 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf <B 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf <C 2 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 D> 2 0 2 (22)^a 0 2 (22)^b-1 S
--> 0^inf 2 1 D> 0 2 (22)^a 0 (22)^b-1 2 S
--> 0^inf 2 (11)^1 A> (22)^a 2 0 (22)^b-1 2 S
--> 0^inf 2 (11)^3a+1 A> 2 0 2 (22)^b-1 S by rule 6
--> 0^inf 2 1 (11)^3a+4 A> (22)^b-1 S by rule 8
= 0^inf 2 1 (11)^g_1(a) A> (22)^b-1 S

10. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^g_1^b(a) A> S
by repetition of rule 9
g_1(n) = 3n + 4 

11. 0^inf 2 1 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 (22)^b-2 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^b S
--> 0^inf 2 (11)^a+1 B> (22)^b S
--> 0^inf 2 (11)^a+1 <A 0 2 (22)^b-1 S
--> 0^inf 2 <A (22)^a+1 0 2 (22)^b-1 S
--> 0^inf <B 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf <C 2 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 D> 2 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 1 D> 0 (22)^a+1 0 2 (22)^b-1 S
--> 0^inf 2 (11)^1 A> (22)^a+1 0 (22)^b-1 2 S
--> 0^inf 2 (11)^3a+4 A> 0 (22)^b-1 2 S by rule 6
Call this rule 11-1
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 (22)^b-2 2 S by rule 7
Call this rule 11-2

12. 0^inf 2 (11)^a A> 0 11 S --> 0^inf 2 1 (11)^1 A> 0 (22)^a+2 2 S [+6a +31 steps]
by:
0^inf 2 (11)^a A> 0 11 S
--> 0^inf 2 (11)^a 1 B> 11 S [+1]
--> 0^inf 2 (11)^a 1 <A 2 1 S [+2]
--> 0^inf 2 1 <A (22)^a 2 1 S [+2a +2]
--> 0^inf 2 <B (22)^a+1 1 S [+2a +3]
--> 0^inf <A 0 (22)^a+1 1 S [+2a +4]
--> 0^inf 1 B> 0 (22)^a+1 1 S [+2a +5]
--> 0^inf 1 <C 2 (22)^a+1 1 S [+2a +6]
--> 0^inf <C 1 2 (22)^a+1 1 S [+2a +7]
--> 0^inf 2 D> 1 2 (22)^a+1 1 S [+2a +8]
--> 0^inf 2 <D (22)^a+2 1 S [+2a +9]
--> 0^inf 1 D> (22)^a+2 1 S [+2a +10]
--> 0^inf 1 (11)^a+2 D> 1 S [+4a +14]
--> 0^inf 1 (11)^a+2 <D 2 S [+4a +15]
--> 0^inf <D (22)^a+3 S [+6a +20]
--> 0^inf 1 A> (22)^a+3 S [+6a +21]
--> 0^inf 1 <B 0 2 (22)^a+2 S [+6a +22]
--> 0^inf <A 2 0 2 (22)^a+2 S [+6a +23]
--> 0^inf 1 B> 2 0 2 (22)^a+2 S [+6a +24]
--> 0^inf 1 <A 0 0 2 (22)^a+2 S [+6a +25]
--> 0^inf <B 2 0 0 2 (22)^a+2 S [+6a +26]
--> 0^inf <C (22)^1 0 0 2 (22)^a+2 S [+6a +27]
--> 0^inf 2 D> (22)^1 0 0 2 (22)^a+2 S [+6a +28]
--> 0^inf 2 (11)^1 D> 0 0 2 (22)^a+2 S [+6a +30]
--> 0^inf 2 1 (11)^1 A> 0 (22)^a+2 2 S [+6a +31]

13. 0^inf 2 1 (11)^a A> 0 2^b S --> 0^inf 2 1 (11)^g_2(a) A> 0 2^b-3 S
by:
0^inf 2 1 (11)^a A> 0 2^b S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 2^b-3 S by rule 11-2
--> 0^inf 2 1 (11)^g_1^(3a+6)(1) A> 0 2^b-3 S by rule 10
= 0^inf 2 1 (11)^(3^(3a+7)-2) A> 0 2^b-3 S
= 0^inf 2 1 (11)^g_2(a) A> 0 2^b-3 S

14. 0^inf 2 1 (11)^a A> 0 2^3k+v S --> 0^inf 2 1 (11)^(g_2)^k(a) A> 0 2^v S
by repetition of rule 13

15. 0^inf 2 1 <A S --> 0^inf 1 D> 2^3 S [+8 steps]
by:
0^inf 2 1 <A S
--> 0^inf 2 <B 2 S
--> 0^inf <A 0 2 S
--> 0^inf 1 B> 0 2 S
--> 0^inf 1 <C 2 2 S
--> 0^inf <C 1 2^2 S
--> 0^inf 2 D> 1 2^2 S
--> 0^inf 2 <D 2^3 S
--> 0^inf 1 D> 2^3 S

16. 0^inf 2 1 (11)^a A> 0 2 1 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3a+7 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 2 1 2 0^inf
--> 0^inf 2 (11)^a+1 B> 2 1 2 0^inf
--> 0^inf 2 (11)^a+1 <A 0 1 2 0^inf
--> 0^inf 2 <A (22)^a+1 0 1 2 0^inf
--> 0^inf <B 0 (22)^a+1 0 1 2 0^inf
--> 0^inf <C 2 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 D> 2 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 1 D> 0 (22)^a+1 0 1 2 0^inf
--> 0^inf 2 (11)^1 A> (22)^a+1 0 1 2 0^inf
--> 0^inf 2 (11)^3a+4 A> 0 1 2 0^inf by rule 6
--> 0^inf 2 (11)^3a+4 1 B> 1 2 0^inf
--> 0^inf 2 (11)^3a+4 1 <A (22)^1 0^inf
--> 0^inf 2 1 <A (22)^3a+5 0^inf
--> 0^inf 1 D> 2 (22)^3a+6 0^inf by rule 15
--> 0^inf (11)^3a+7 D> 0^inf
--> 0^inf (11)^3a+7 1 A> 0^inf
--> 0^inf (11)^3a+8 B> 0^inf
--> 0^inf (11)^3a+8 <C 2 0^inf
--> 0^inf <C (11)^3a+8 2 0^inf
--> 0^inf 2 D> (11)^3a+8 2 0^inf
--> 0^inf 2 <D 2 1 (11)^3a+7 2 0^inf
--> 0^inf 1 D> 2 1 (11)^3a+7 2 0^inf
--> 0^inf (11)^1 D> 1 (11)^3a+7 2 0^inf
--> 0^inf (11)^1 <D 2 (11)^3a+7 2 0^inf
--> 0^inf <D 2^3 (11)^3a+7 2 0^inf
--> 0^inf 1 A> (22)^1 2 (11)^3a+7 2 0^inf
--> 0^inf 1 <B 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <A 2 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 1 B> 2 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 1 <A 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <B 2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf <C 2 2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 D> 2^2 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 (11)^1 D> 0 0 (22)^1 (11)^3a+7 2 0^inf
--> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3a+7 2 0^inf

17. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1 S
--> 0^inf 2 (11)^3a+4 A> 0 2 1 S by rule 11-1
--> 0^inf 2 (11)^3a+4 1 B> 2 1 S
--> 0^inf 2 1 (11)^3a+4 <A 0 1 S
--> 0^inf 2 1 <A (22)^3a+4 0 1 S
--> 0^inf 1 D> 2 (22)^3a+5 0 1 S by rule 15
--> 0^inf (11)^3a+6 D> 0 1 S
--> 0^inf (11)^3a+6 1 A> 1 S
--> 0^inf (11)^3a+6 1 <B 2 S
--> 0^inf 1 <B (22)^3a+6 2 S
--> 0^inf <A (22)^3a+7 S
--> 0^inf 1 B> (22)^3a+7 S
--> 0^inf 1 <A 0 2 (22)^3a+6 S
--> 0^inf <B 2 0 2 (22)^3a+6 S
--> 0^inf <C 2 2 0 2 (22)^3a+6 S
--> 0^inf 2 D> 2 2 0 (22)^3a+6 2 S
--> 0^inf 2 (11)^1 D> 0 (22)^3a+6 2 S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S

18. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^g_2(a) A> 2 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1 S
--> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S by rule 17
--> 0^inf 2 1 (11)^g_1^(3a+6)(1) A> 2 S by rule 10
= 0^inf 2 1 (11)^(3^(3a+7)-2) A> 2 S
= 0^inf 2 1 (11)^g_2(a) A> 2 S

19. 0^inf 2 1 (11)^a A> 2 1^3 S --> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 S
by:
0^inf 2 1 (11)^a A> 2 1^3 S
--> 0^inf 2 1 (11)^a <B 0 1^3 S
--> 0^inf 2 1 <B (22)^a 0 1^3 S
--> 0^inf 2 <A 2 (22)^a 0 1^3 S
--> 0^inf <B 0 2 (22)^a 0 1^3 S
--> 0^inf <C 2 0 2 (22)^a 0 1^3 S
--> 0^inf 2 D> 2 0 2 (22)^a 0 1^3 S
--> 0^inf 2 1 D> 0 (22)^a 2 0 1^3 S
--> 0^inf 2 (11)^1 A> (22)^a 2 0 1^3 S
--> 0^inf 2 (11)^3a+1 A> 2 0 1^3 S by rule 6
--> 0^inf 2 (11)^3a+1 <B 0 0 1^3 S
--> 0^inf 2 <B (22)^3a+1 0 0 1^3 S
--> 0^inf <A 0 (22)^3a+1 0 0 1^3 S
--> 0^inf 1 B> 0 (22)^3a+1 0 0 1^3 S
--> 0^inf 1 <C 2 (22)^3a+1 0 0 1^3 S
--> 0^inf <C 1 2 (22)^3a+1 0 0 1^3 S
--> 0^inf 2 D> 1 2 (22)^3a+1 0 0 1^3 S
--> 0^inf 2 <D (22)^3a+2 0 0 1^3 S
--> 0^inf 1 D> (22)^3a+2 0 0 1^3 S
--> 0^inf 1 (11)^3a+2 D> 0 0 1^3 S
--> 0^inf (11)^3a+3 A> 0 1^3 S
--> 0^inf (11)^3a+3 1 B> 1^3 S
Call this rule 19***
with:
0^inf 2 1 (11)^a A> 2 S --> 0^inf (11)^3a+3 1 B> S
Continuing:
--> 0^inf (11)^3a+3 1 <A 2 1^2 S
--> 0^inf 1 <A (22)^3a+3 2 1^2 S
--> 0^inf <B (22)^3a+4 1^2 S
Call this rule 19**
with:
0^inf 2 1 (11)^a A> 2 1 S --> 0^inf <B (22)^3a+4 S
Continuing:
--> 0^inf <C 2 (22)^3a+4 1^2 S
--> 0^inf 2 D> 2 (22)^3a+4 1^2 S
--> 0^inf 2 1 (11)^3a+4 D> 1^2 S
--> 0^inf 2 1 (11)^3a+4 <D 2 1 S
--> 0^inf 2 <D (22)^3a+5 1 S
--> 0^inf 1 D> (22)^3a+5 1 S
--> 0^inf 1 (11)^3a+5 D> 1 S
Call this rule 19*
with:
0^inf 2 1 (11)^a A> 2 1^2 S --> 0^inf 1 (11)^3a+5 D> S
Continuing:
--> 0^inf 1 (11)^3a+5 <D 2 S
--> 0^inf <D (22)^3a+6 S
--> 0^inf 1 A> (22)^3a+6 S
--> 0^inf 1 <B 0 2 (22)^3a+5 S
--> 0^inf <A 2 0 2 (22)^3a+5 S
--> 0^inf 1 B> 2 0 2 (22)^3a+5 S
--> 0^inf 1 <A 0 0 2 (22)^3a+5 S
--> 0^inf <B 2 0 0 (22)^3a+5 2 S
--> 0^inf <C (22)^1 0 0 (22)^3a+5 2 S
--> 0^inf 2 D> (22)^1 0 0 (22)^3a+5 2 S
--> 0^inf 2 (11)^1 D> 0 0 (22)^3a+5 2 S
--> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 S
This rule can be rewritten as:
0^inf 2 1 (11)^a A> 2 1^b S --> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 1^b-3 S

20. 0^inf 2 1 (11)^a A> 0 (22)^1 1^b S --> 0^inf 2 1 (11)^g_3(a) A> 0 (22)^1 1^b-4 S
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^b S
--> 0^inf 2 1 (11)^g_2(a) A> 2 1^b-1 S by rule 18
--> 0^inf 2 1 (11)^1 A> 0 (22)^3*g_2(a)+5 2 1^b-4 S by rule 19
= 0^inf 2 1 (11)^1 A> 0 2^6*g_2(a)+11 1^b-4 S
Modulus for rule 14:
6*g_2(a)+11 = 3(2*g_2(a)+3)+2 = 3k+2
3k+2 mod 3 = 2
--> 0^inf 2 1 (11)^(g_2)^(2*g_2(a)+3)(1) A> 0 (22)^1 1^b-4 S
= 0^inf 2 1 (11)^g_3(a) A> 0 (22)^1 1^b-4 S

21. 0^inf 2 1 (11)^a A> 0 (22)^1 1^4k+v S --> 0^inf 2 1 (11)^g_3^k(a) A> 0 (22)^1 1^v S
by repetition of rule 20

22. 0^inf 2 1 (11)^a A> 0 (22)^1 1^3 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 1^6*g_2(a)+12 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^3 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 1^2 2 0^inf by rule 18
--> 0^inf 1 (11)^3*g_2(a)+5 D> 2 0^inf by rule 19*
--> 0^inf (11)^3*g_2(a)+6 D> 0^inf
--> 0^inf (11)^3*g_2(a)+6 1 A> 0^inf
--> 0^inf (11)^3*g_2(a)+7 B> 0^inf
--> 0^inf (11)^3*g_2(a)+7 <C 2 0^inf
--> 0^inf <C (11)^3*g_2(a)+7 2 0^inf
--> 0^inf 2 D> (11)^3*g_2(a)+7 2 0^inf
--> 0^inf 2 <D 2 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 D> 2 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf (11)^1 D> 1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf (11)^1 <D 2 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <D 2^3 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 A> 2^3 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 <B 0 2^2 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <A 2 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 B> 2 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 <A 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <B 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf <C 2 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 D> 2 2 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 (11)^1 D> 0 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3*g_2(a)+6 2 0^inf
= 0^inf 2 1 (11)^1 A> 0 (22)^1 1^6*g_2(a)+12 2 0^inf

23. 0^inf 2 1 (11)^a A> 0 (22)^1 1^2 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 1^2 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 1 2 0^inf by rule 18
--> 0^inf <B (22)^3*g_2(a)+4 2 0^inf by rule 19**
--> 0^inf <C (22)^3*g_2(a)+5 0^inf
--> 0^inf 2 D> (22)^3*g_2(a)+5 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 D> 0^inf
--> 0^inf 2 1 (11)^3*g_2(a)+5 A> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+6 B> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+6 <C 2 0^inf
--> 0^inf 2 <C (11)^3*g_2(a)+6 2 0^inf
--> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf

24. 0^inf (11)^a A> 0 (22)^1 1 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf
by:
0^inf (11)^a A> 0 (22)^1 1 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 2 2 0^inf by rule 18
--> 0^inf (11)^3*g_2(a)+3 1 B> 2 0^inf by rule 19***
--> 0^inf (11)^3*g_2(a)+3 1 <A 0^inf
--> 0^inf 1 <A (22)^3*g_2(a)+3 0^inf
--> 0^inf <B 2 (22)^3*g_2(a)+3 0^inf
--> 0^inf <C (22)^3*g_2(a)+4 0^inf
--> 0^inf 2 D> (22)^3*g_2(a)+4 0^inf
--> 0^inf 2 (11)^3*g_2(a)+4 D> 0^inf
--> 0^inf 2 1 (11)^3*g_2(a)+4 A> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 B> 0^inf
--> 0^inf 2 (11)^3*g_2(a)+5 <C 2 0^inf
--> 0^inf 2 <C (11)^3*g_2(a)+5 2 0^inf
--> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf

25. 0^inf 2 1 (11)^a A> 0 (22)^1 2 0^inf --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf
by:
0^inf 2 1 (11)^a A> 0 (22)^1 2 0^inf
--> 0^inf 2 1 (11)^g_2(a) A> 0^inf by rule 14
--> 0^inf 2 (11)^g_2(a)+1 B> 0^inf
--> 0^inf 2 (11)^g_2(a)+1 <C 2 0^inf
--> 0^inf 2 <C (11)^g_2(a)+1 2 0^inf
--> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf

Functions

g1(n)=3n+4

Note that (3k−2)×3+4=3k+1−2

And 1=31−2

It follows that g1n(1)=3n+1−2

g2(n)=33n+7−2

g3(n)=g22×(g2(n)+3)(1) 

Modulus of g_2^a(1):

1 is of the form 4k+1
g_2(n) = 3^(3n+7)-2
3^2k mod 4 = 1
3^2k+1 mod 4 = 3

g_2(4k+1) = 3^(3*(4k+1)+7)-2 = 3^(12k+10)-2 = 3^2m-2
3^2m mod 4 = 1 --> -2
3^2m - 2 mod 4 = 3
g_2(4k+1) mod 4 = 3
--> g_2(4k+1) = 4m+3

g_2(4k+3) = 3^(3*(4k+3)+7)-2 = 3^(12k+16)-2 = 3^2m-2
3^2m-2 mod 4 = 3
g_2(4k+1) mod 4 = 3
--> g_2(4k+3) = 4m+3

-->g_2^k(1) = 4m+3
-->g_2^k(1) mod 4 = 3

Let's have L(a, b) = 0^inf 2 1 (11)^a A> 0 (22)^1 1^b 2 0^inf

* L(a, 4k+v) --> L(g_3^k(a), v) by rule 21
* L(a, 0) --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf by rule 25
* L(a, 1) --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf by rule 24
* L(a, 2) --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf by rule 23
* L(a, 3) --> L(1, 6*g_2(a) + 12) by rule 22

Trajectory

S=0: 0^inf <A 0^inf
S=1: 0^inf 1 B> 0^inf
S=2: 0^inf 1 <C 2 0^inf
S=3: 0^inf <C 1 2 0^inf
S=4: 0^inf 2 D> 1 2 0^inf
S=5: 0^inf 2 <D 2^2 0^inf
S=6: 0^inf 1 D> 2^2 0^inf
S=8: 0^inf 1^3 D> 0^inf
S=9: 0^inf (11)^2 A> 0^inf
S=10: 0^inf (11)^2 1 B> 0^inf
S=11: 0^inf (11)^2 1 <C 2 0^inf
S=16: 0^inf <C (11)^2 1 2 0^inf
S=17: 0^inf 2 D> (11)^2 1 2 0^inf
S=18: 0^inf 2 <D 2 (11)^2 2 0^inf
S=19: 0^inf 1 D> 2 (11)^2 2 0^inf
S=20: 0^inf (11)^1 D> (11)^2 2 0^inf
S=21: 0^inf (11)^1 <D 2 1 (11)^1 2 0^inf
S=23: 0^inf <D 2^3 1^3 2 0^inf
S=24: 0^inf 1 A> 2^3 1^3 2 0^inf
S=25: 0^inf 1 <B 0 2^2 1^3 2 0^inf
S=26: 0^inf <A 2 0 2^2 1^3 2 0^inf
S=27: 0^inf 1 B> 2 0 2^2 1^3 2 0^inf
S=28: 0^inf 1 <A 0 0 2^2 1^3 2 0^inf
S=29: 0^inf <B 2 0 0 2^2 1^3 2 0^inf
S=30: 0^inf <C 2 2 0 0 2^2 1^3 2 0^inf
S=31: 0^inf 2 D> 2^2 0 0 2^2 1^3 2 0^inf
S=33: 0^inf 2 (11)^1 D> 0^2 2^2 1^3 2 0^inf
S=34: 0^inf 2 1 (11)^1 A> 0 (22)^1 1^3 2 0^inf
= L(1, 3) after 34 steps

L(1, 3) --> L(1,6*g2(1)+12) by rule 22, which can be simplified to L(1, 354294)

--> L(g388573(1),2)

L(g388573(1),2) --> 0^inf 1 Z> (11)3×(g2(g388573(1))+6 2 0^inf

σ=6×g2(g388573(1))+14

Approximate Score

g2(n)=33n+7−2

3n<g2(n)<33n

g3(n)=g22×(g2(n))+3(1)

(3↑)kn<g2k(n)<(3↑)2kn

(3↑)k1<g2k(1)<(3↑)2k1

3↑↑k<g2k(1)<3↑↑2k

3↑↑2×g2(n)+6<g3(n)<3↑↑4×g2(n)+12

3↑↑2×33↑↑2×g2(n)+6+6<g32(n)

3↑↑3↑↑(2×g2(n)+7)<g32(n)<3↑↑3↑↑(4×g2(n)+13)

(3↑↑)k(2×g2(n)+7)<g3k(n)<(3↑↑)k(5×g2(n)+13)

(3↑↑)k118101<g3k(1)<(3↑↑)k295248

3↑↑↑k+1<g3k(1)<3↑↑↑k+2


σ=6×g2(g388573(1))+14

3↑↑↑88574<σ<S<3↑↑↑88575

Permutations

Starting in state B 

S=0: 0^inf <B 0^inf
S=1: 0^inf <C 2 0^inf
S=2: 0^inf 2 D> 2 0^inf
S=3: 0^inf 2 1 D> 0^inf
S=4: 0^inf 2 1^2 A> 0^inf
S=5: 0^inf 2 1^3 B> 0^inf
S=6: 0^inf 2 1^3 <C 2 0^inf
S=9: 0^inf 2 <C 1^3 2 0^inf
S=10: 0^inf 1 Z> 1^3 2 0^inf

Starting in state C 

S=0: 0^inf <C 0^inf
S=1: 0^inf 2 D> 0^inf
S=2: 0^inf 2 1 A> 0^inf
S=3: 0^inf 2 1^2 B> 0^inf
S=4: 0^inf 2 1^2 <C 2 0^inf
S=6: 0^inf 2 <C 1^2 2 0^inf
S=7: 0^inf 1 Z> 1^2 2 0^inf

Starting in state D 

Enters configuration L(1,0) after 23 steps.
Halting tape: 0^inf 1 Z> (11)^10 1 2 0^inf

1RB1LE_1LB1LC_1RD0LE_---0RB_1RF1LA_0RA0RD (bbch)

4. A BB(6) holdout TM. Analysed in October 2025. Its fate is currently unknown.

Analysis

For this analysis the undefined D0 transition will be set to 1RZ.

0 1
A 1RB 1LE
B 1LB 1LC
C 1RD 0LE
D 1RZ 0RB
E 1RF 1LA
F 0RA 0RD 
S is any tape configuration

1. S 0^a <B S --> S <B 1^a S [+a steps]

2. S D> 1^2 S --> S 1 D> 1 S [+3 steps]
by:
S D> 1^2 S
--> S 0 B> 1 S [+1]
--> S 0 <C 1 S [+2]
--> S 1 D> 1 S [+3]

3. S D> 1^a 1 S --> S 1^a D> 1 S |for a > 0 [+3a steps]
by repetition of rule 2

4. S (11)^a <E S --> S <E (11)^a S [+2a steps]
   S (11)^a <A S --> S <A (11)^a S [+2a steps]

5. S 1^2 D> 1 0 S --> S <E 0 1^3 S [+5 steps]
by:
S 1^2 D> 1 0 S
--> S 1^2 0 B> 0 S [+1]
--> S 1^2 0 <B 1 S [+2]
--> S 1^2 <B 1^2 S [+3]
--> S 1 <C 1^3 S [+4]
--> S <E 0 1^3 S [+5]

6. S 0 1^a <E S --> S 1 0 1^a-2 D> 1 S [+4a -4 steps] (if a mod 2 = 0)
by:
S 0 1^a <E S
--> S 0 <E 1^a S [+a]
--> S 1 F> 1^a S [+a +1]
--> S 1 0 D> 1^a-1 S [+a +2]
= S 1 0 D> 1^a-2 1 S
--> S 1 0 1^a-2 D> 1 S [+4a -4]

7. S 0 1^a D> 1 0 S --> S 1 0 1^a-4 D> 1 0 1^3 S [+4a -7 steps] (if a mod 2 = 0 and a >= 4)
by:
S 0 1^a D> 1 0 S
--> S 0 1^a 0 B> 0 S [+1]
--> S 0 1^a 0 <B 1 S [+2]
--> S 0 1^a <B 1^2 S [+3]
--> S 0 1^a-1 <C 1^3 S [+4]
--> S 0 1^a-2 <E 0 1^3 S [+5]
--> S 1 0 1^a-4 D> 1 0 1^3 S [+4a -7] by rule 6

8. S 0 1^4k+v D> 1 0 1^b S --> S 1^k 0 1^v D> 1 0 1^b+3k S (if v mod 2 = 0 and k ≥ 1)
[+4bk -8k^2 +k steps where b = 4k + v]
by repetition of rule 7

9. S 0 1^2 D> 1 0 1^c S --> S 1^2 0 1^c+1 D> 1 S [+3c +13 steps]
by:
S 0 1^2 D> 1 0 1^c S
--> S 0 1^2 0 B> 0 1^c S [+1]
--> S 0 1^2 0 <B 1^c+1 S [+2]
--> S 0 1^2 <B 1^c+2 S [+3]
--> S 0 1 <C 1^c+3 S [+4]
--> S 0 <E 0 1^c+3 S [+5]
--> S 1 F> 0 1^c+3 S [+6]
--> S 1 0 A> 1^c+3 S [+7]
--> S 1 0 <E 1^c+3 S [+8]
--> S 1^2 F> 1^c+3 S [+9]
--> S 1^2 0 D> 1^c+2 S [+10]
= S 1^2 0 D> 1^c+1 1 S [+10]
--> S 1^2 0 1^c+1 D> 1 S [+3c +13]

10. S 0 1^a 0 D> 1 0 S --> S 1 0 1^a-4 D> 1 0 1^4 S [+4a -6 steps] (if a mod 2 = 0 and a ≥ 4)
by:
S 0 1^a 0 D> 1 0 S
--> S 0 1^a 0^2 B> 0 S [+1]
--> S 0 1^a 0^2 <B 1 S [+2]
--> S 0 1^a <B 1^3 S [+4]
--> S 0 1^a-1 <C 1^4 S [+5]
--> S 0 1^a-2 <E 0 1^4 S [+6]
--> S 0 <E 1^a-2 0 1^4 S [+a +4]
--> S 1 F> 1^a-2 0 1^4 S [+a +5]
--> S 1 0 D> 1^a-3 0 1^4 S [+a +6]
= S 1 0 D> 1^a-4 1 0 1^4 S [+a +6]
--> S 1 0 1^a-4 D> 1 0 1^4 S [+4a -6]

11. S 0 1^2 0 D> 1 0 1^c S --> S 1^2 0 1^c+2 D> 1 S [+3c +17 steps]
by:
S 0 1^2 0 D> 1 0 1^c S
--> S 0 1^2 0^2 B> 0 1^c S [+1]
--> S 0 1^2 0^2 <B 1^c+1 S [+2]
--> S 0 1^2 <B 1^c+3 S [+4]
--> S 0 1 <C 1^c+4 S [+5]
--> S 0 <E 0 1^c+4 S [+6]
--> S 1 F> 0 1^c+4 S [+7]
--> S 1 0 A> 1^c+4 S [+8]
--> S 1 0 <E 1^c+4 S [+9]
--> S 1^2 F> 1^c+4 S [+10]
--> S 1^2 0 D> 1^c+3 S [+11]
= S 1^2 0 D> 1^c+2 1 S
--> S 1^2 0 1^c+2 D> 1 S [+3c +17]

12. 0^inf D> 1 0 S --> non-halting translated cycler
by:
0^inf D> 1 0 S
--> 0^inf B> 0 S
--> 0^inf <B 1 S
--> spin out

13. S 0^2 1^a 0 D> 1 0 S --> S 1^2 0 1^a-4 D> 1 0 1^4 S [+4a steps] (if a mod 2 = 1 and a ≥ 4)
by:
S 0^2 1^a 0 D> 1 0 S
--> S 0^2 1^a 0^2 B> 0 S [+1]
--> S 0^2 1^a 0^2 <B 1 S [+2]
--> S 0^2 1^a <B 1^3 S [+4]
--> S 0^2 1^a-1 <C 1^4 S [+5]
--> S 0^2 1^a-2 <E 0 1^4 S [+6]
= S 0^2 1 1^a-3 <E 0 1^4 S
--> S 0^2 1 <E 1^a-3 0 1^4 S [+a +3]
--> S 0^2 <A 1^a-2 0 1^4 S [+a +4]
--> S 0 1 B> 1^a-2 0 1^4 S [+a +5]
--> S 0 1 <C 1^a-2 0 1^4 S [+a +6]
--> S 0 <E 0 1^a-2 0 1^4 S [+a +7]
--> S 1 F> 0 1^a-2 0 1^4 S [+a +8]
--> S 1 0 A> 1^a-2 0 1^4 S [+a +9]
--> S 1 0 <E 1^a-2 0 1^4 S [+a +10]
--> S 1^2 F> 1^a-2 0 1^4 S [+a +11]
--> S 1^2 0 D> 1^a-3 0 1^4 S [+a +12]
= S 1^2 0 D> 1^a-4 1 0 1^4 S
--> S 1^2 0 1^a-4 D> 1 0 1^4 S [+4a]

14. S 0^2 1^3 0 D> 1 0 S --> S 1^2 0 1 Z> 1^4 S [+16 steps]
by:
S 0^2 1^3 0 D> 1 0 S
--> S 0^2 1^3 0^2 B> 0 S [+1]
--> S 0^2 1^3 0^2 <B 1 S [+2]
--> S 0^2 1^3 <B 1^3 S [+4]
--> S 0^2 1^2 <C 1^4 S [+5]
--> S 0^2 1 <E 0 1^4 S [+6]
--> S 0^2 <A 1 0 1^4 S [+7]
--> S 0 1 B> 1 0 1^4 S [+8]
--> S 0 1 <C 1 0 1^4 S [+9]
--> S 0 <E 0 1 0 1^4 S [+10]
--> S 1 F> 0 1 0 1^4 S [+11]
--> S 1 0 A> 1 0 1^4 S [+12]
--> S 1 0 <E 1 0 1^4 S [+13]
--> S 1^2 F> 1 0 1^4 S [+14]
--> S 1^2 0 D> 0 1^4 S [+15]
--> S 1^2 0 1 Z> 1^4 S [+16]

15. S 0 1 0 D> 1 0 1^c S --> S 1^c+4 D> 1 S [+3c +15 steps]
by:
S 0 1 0 D> 1 0 1^c S
--> S 0 1 0^2 B> 0 1^c S [+1]
--> S 0 1 0^2 <B 1^c+1 S [+2]
--> S 0 1 <B 1^c+3 S [+4]
--> S 0 <C 1^c+4 S [+5]
--> S 1 D> 1^c+4 S [+6]
= S 1 D> 1^c+3 1 S [+6]
--> S 1^c+4 D> 1 S [+3c +15]

16. S 0 1^a 0 1^b D> 1 0 S --> S 1 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +2 steps] (if b mod 2 = 1 and a mod 2 = 0 and a ≥ 2)
by:
S 0 1^a 0 1^b D> 1 0 S
--> S 0 1^a 0 1^b 0 B> 0 S [+1]
--> S 0 1^a 0 1^b 0 <B 1 S [+2]
--> S 0 1^a 0 1^b <B 1^2 S [+3]
--> S 0 1^a 0 1^b-1 <C 1^3 S [+4]
--> S 0 1^a 0 1^b-2 <E 0 1^3 S [+5]
--> S 0 1^a 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0 1^a 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0 1^a+1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0 1^a+1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0 1^a <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0 <E 1^a 0 1^b-2 0 1^3 S [+b +a +6]
--> S 1 F> 1^a 0 1^b-2 0 1^3 S [+b +a +7]
--> S 1 0 D> 1^a-1 0 1^b-2 0 1^3 S [+b +a +8]
= S 1 0 D> 1^a-2 1 0 1^b-2 0 1^3 S
--> S 1 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +2]

17. S 0^2 1^b D> 1 0 S --> S 1^2 0 1^b-4 D> 1 0 1^3 S [+4b -1 steps] (if b mod 2 = 1 and b ≥ 5)
by:
S 0^2 1^b D> 1 0 S
--> S 0^2 1^b 0 B> 0 S [+1]
--> S 0^2 1^b 0 <B 1 S [+2]
--> S 0^2 1^b <B 1^2 S [+3]
--> S 0^2 1^b-1 <C 1^3 S [+4]
--> S 0^2 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 <A 1^b-2 0 1^3 S [+b +3]
--> S 0 1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0 1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0 <E 0 1^b-2 0 1^3 S [+b +6]
--> S 1 F> 0 1^b-2 0 1^3 S [+b +7]
--> S 1 0 A> 1^b-2 0 1^3 S [+b +8]
--> S 1 0 <E 1^b-2 0 1^3 S [+b +9]
--> S 1^2 F> 1^b-2 0 1^3 S [+b +10]
--> S 1^2 0 D> 1^b-3 0 1^3 S [+b +11]
= S 1^2 0 D> 1^b-4 1 0 1^3 S
--> S 1^2 0 1^b-4 D> 1 0 1^3 S [+4b -1]

18. S 0^2 1^3 D> 1 0 S --> S 1^2 0 1 Z> 1^3 S [+15 steps]
by:
S 0^2 1^3 D> 1 0 S
--> S 0^2 1^3 0 B> 0 S [+1]
--> S 0^2 1^3 0 <B 1 S [+2]
--> S 0^2 1^3 <B 1^2 S [+3]
--> S 0^2 1^2 <C 1^3 S [+4]
--> S 0^2 1 <E 0 1^3 S [+5]
--> S 0^2 <A 1 0 1^3 S [+6]
--> S 0 1 B> 1 0 1^3 S [+7]
--> S 0 1 <C 1 0 1^3 S [+8]
--> S 0 <E 0 1 0 1^3 S [+9]
--> S 1 F> 0 1 0 1^3 S [+10]
--> S 1 0 A> 1 0 1^3 S [+11]
--> S 1 0 <E 1 0 1^3 S [+12]
--> S 1^2 F> 1 0 1^3 S [+13]
--> S 1^2 0 D> 0 1^3 S [+14]
--> S 1^2 0 1 Z> 1^3 S [+15]

19. S 0^2 1^a 0 1^b D> 1 0 S --> S 1^2 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +8 steps] (if b mod 2 = 1 and a mod 2 = 1 and a ≥ 2)
by:
S 0^2 1^a 0 1^b D> 1 0 S
--> S 0^2 1^a 0 1^b 0 B> 0 S [+1]
--> S 0^2 1^a 0 1^b 0 <B 1 S [+2]
--> S 0^2 1^a 0 1^b <B 1^2 S [+3]
--> S 0^2 1^a 0 1^b-1 <C 1^3 S [+4]
--> S 0^2 1^a 0 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1^a 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 1^a 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0^2 1^a+1 B> 1^b-2 0 1^3 S [+b +4]
--> S 0^2 1^a+1 <C 1^b-2 0 1^3 S [+b +5]
--> S 0^2 1^a <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0^2 1 <E 1^a-1 0 1^b-2 0 1^3 S [b +a +5]
--> S 0^2 <A 1^a 0 1^b-2 0 1^3 S [+b +a +6]
--> S 0 1 B> 1^a 0 1^b-2 0 1^3 S [+b +a +7]
--> S 0 1 <C 1^a 0 1^b-2 0 1^3 S [+b +a +8] 
--> S 0 <E 0 1^a 0 1^b-2 0 1^3 S [+b +a +9]
--> S 1 F> 0 1^a 0 1^b-2 0 1^3 S [+b +a +10]
--> S 1 0 A> 1^a 0 1^b-2 0 1^3 S [+b +a +11]
--> S 1 0 <E 1^a 0 1^b-2 0 1^3 S [+b +a +12]
--> S 1^2 F> 1^a-1 1 0 1^b-2 0 1^3 S [+b +a +13]
--> S 1^2 0 D> 1^a-2 1 0 1^b-2 0 1^3 S [+b +a +14]
--> S 1^2 0 1^a-2 D> 1 0 1^b-2 0 1^3 S [+b +4a +8]

20. S 0^2 1 0 1^b D> 1 0 S --> S 1^2 0 1 Z> 1^b-2 0 1^3 S [+b +16 steps] (if b mod 2 = 1)
by:
S 0^2 1 0 1^b D> 1 0 S
--> S 0^2 1 0 1^b 0 B> 0 S [+1]
--> S 0^2 1 0 1^b 0 <B 1 S [+2]
--> S 0^2 1 0 1^b <B 1^2 S [+3]
--> S 0^2 1 0 1^b-1 <C 1^3 S [+4]
--> S 0^2 1 0 1^b-2 <E 0 1^3 S [+5]
--> S 0^2 1 0 1 <E 1^b-3 0 1^3 S [+b +2]
--> S 0^2 1 0 <A 1^b-2 0 1^3 S [+b +3]
--> S 0^2 1^2 B> 1^b-2 0 1^3 S [+b +4]
--> S 0^2 1^2 <C 1^b-2 0 1^3 S [+b +5]
--> S 0^2 1 <E 0 1^b-2 0 1^3 S [+b +6]
--> S 0^2 <A 1 0 1^b-2 0 1^3 S [+b +7]
--> S 0 1 B> 1 0 1^b-2 0 1^3 S [+b +8]
--> S 0 1 <C 1 0 1^b-2 0 1^3 S [+b +9]
--> S 0 <E 0 1 0 1^b-2 0 1^3 S [+b +10]
--> S 1 F> 0 1 0 1^b-2 0 1^3 S [+b +11]
--> S 1 0 A> 1 0 1^b-2 0 1^3 S [+b +12]
--> S 1 0 <E 1 0 1^b-2 0 1^3 S [+b +13]
--> S 1^2 F> 1 0 1^b-2 0 1^3 S [+b +14]
--> S 1^2 0 D> 0 1^b-2 0 1^3 S [+b +15]
--> S 1^2 0 1 Z> 1^b-2 0 1^3 S [+b +16]

21. S 0 1^a 0 1 D> 1 0 1^c S --> S 0 1^a+c+3 D> 1 S [+3c +11 steps]
by:
S 0 1^a 0 1 D> 1 0 1^c S
--> S 0 1^a 0 1 0 B> 0 1^c S [+1]
--> S 0 1^a 0 1 0 <B 1^c+1 S [+2]
--> S 0 1^a 0 1 <B 1^c+2 S [+3]
--> S 0 1^a 0 <C 1^c+3 S [+4]
--> S 0 1^a+1 D> 1^c+2 1 S [+5]
--> S 0 1^a+c+3 D> 1 S [+3c +11]

Functions

Let A(a, b, c, d, e, f, ..., k) = 0^inf 1^a 0 1^b D> 1 0 1^c 0 1^d 0 1^e 0 1^f ... 0 1^k 0^inf

b mod 2 = 0:
   b ≥ 4: A(a, b, c, ...) --> A(a+1, b-4, c+3, ...) by rule 7
   b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) by rule 9
   b = 0:
      a mod 2 = 0:
         a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) by rule 10
         a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) by rule 11
         a = 0: A(0, 0, c, ...) --> spin out by rule 12
      a mod 2 = 1:
         a ≥ 4: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) by rule 13
         a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... by rule 14
         a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) by rule 15
b mod 2 = 1:
   b ≥ 3:
      a mod 2 = 0:
         a ≥ 2: A(a, b, c, ...) --> A(1, a-2, b-2, c+3, ...) by rule 16
         a = 0:
            b ≥ 5: A(0, b, c, ...) --> A(2, b-4, c+3, ...) by rule 17
            b = 3: A(0, 3, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+3 ... by rule 18
      a mod 2 = 1:
         a ≥ 2: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) by rule 19
         a = 1: A(1, b, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^b-2 0 1^c+3 ... by rule 20
   b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) by rule 21

Rules 12, 18 and 20 are not reachable by any of these rules (reaching them would require negative entries), meaning that they can only be triggered if they are the TMs starting configurations.

Accelerated rules: 

R8: A(a, 4k+v, c, ...) --> A(a+k, v, c+3k, ...) [+4bk -8k^2 +k steps] (if v mod 2 = 0 and k ≥ 1)

A1: A(0, 2k+1, c, ...) --> A(0, 2(k-1)+1, c+6, ...) [+16k +10 steps] (if k ≥ 3)
by:
A(0, 2k+1, c, ...)
--> A(2, 2(k-2)+1, c+3, ...) by rule 17 [+8k +3]
--> A(1, 0, 2(k-3)+1, c+6, ...) by rule 16 [+10k +10]
--> A(0, 2(k-1)+1, c+6, ...) by rule 15 [+16k +10]

A2: A(0, 2k+1, c, ...) --> A(0, 5, c+6k-12) [+8k^2 +18k -68 steps] (if k ≥ 3)
by repetition of rule A1

A3: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) [+8k^2 +36k +3c -65 steps] (if k ≥ 3)
by:
A(0, 2k+1, c, ...)
--> A(0, 5, c+6k-12, ...) by rule A2 [+8k^2 +18k -68]
--> A(2, 1, c+6k-9, ...) by rule 17 [8k^2 +18k -49]
--> A(0, c+6k-4, ...) by rule 21 [+8k^2 +36k + 3c -65]

Using the accelerated rules: 

Let A(a, b, c, d, e, f, ..., k) = 0^inf 1^a 0 1^b D> 1 0 1^c 0 1^d 0 1^e 0 1^f ... 0 1^k 0^inf

b mod 2 = 0:
   b ≥ 4: A(a, 4k+v, c, ...) --> A(a+k, v, c+3k, ...) by rule 8
   b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) by rule 9
   b = 0:
      a mod 2 = 0:
         a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) by rule 10
         a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) by rule 11
         a = 0: unreachable
      a mod 2 = 1:
         a ≥ 4: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) by rule 13
         a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... by rule 14
         a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) by rule 15
b mod 2 = 1:
   b ≥ 3:
      a mod 2 = 0:
         a ≥ 2: A(a, b, c, ...) --> A(1, a-2, b-2, c+3, ...) by rule 16
         a = 0:
            b ≥ 7: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) by rule A3
            b = 5: A(0, b, c, ...) --> A(2, b-4, c+3, ...) by rule 17
            b = 3: unreachable
      a mod 2 = 1:
         a ≥ 2: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) by rule 19
         a = 1: unreachable
   b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) by rule 21

Trajectory

S=0: 0^inf <A 0^inf
S=1: 0^inf 1 B> 0^inf
S=2: 0^inf 1 <B 1 0^inf
S=3: 0^inf <C 1^2 0^inf
S=4: 0^inf 1 D> 1^2 0^inf
S=7: 0^inf 1^2 D> 1 0^inf = A(0, 2, 0)
So this TM reaches configuration A(0, 2, 0) after 7 steps.

Permutations

Starting in state B 

S=0: 0^inf <B 0^inf
--> spin out

Starting in state C 

S=0: 0^inf <C 0^inf
S=1: 0^inf 1 D> 0^inf
S=2: 0^inf 1^2 Z> 0^inf

Starting in state D 

S=0: 0^inf <D 0^inf
S=1: 0^inf 1 Z> 0^inf

Starting in state E 

S=0: 0^inf <E 0^inf
S=1: 0^inf 1 F> 0^inf
S=2: 0^inf 1 0 A> 0^inf
S=3: 0^inf 1 0 1 B> 0^inf
S=4: 0^inf 1 0 1 <B 1 0^inf
S=5: 0^inf 1 0 <C 1^2 0^inf
S=6: 0^inf 1^2 D> 1^2 0^inf
S=9: 0^inf 1^3 D> 1 0^inf = A(0, 3, 0)
--> Reaches configuration A(0, 3, 0) after 9 steps.
S=24: 0^inf 1^2 0 1 Z> 1^3 0^inf

Starting in state F 

S=0: 0^inf <F 0^inf
S=1: 0^inf A> 0^inf
--> Equivalent to starting in state A, but started one step earlier.
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