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={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}=
<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!
!0
!1
!2
|-
|A
|1RB
|1RD
|1LC
|-
|B
|2LB
|1RB
|1LC
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|C
|1RZ
|1LA
|1LD
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|D
|2RB
|2RA
|2RD
|}
<pre>
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 (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 <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 8.
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 (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 <C 1 (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf
</pre>
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) --> <math>A(a,0,2^{b} \times c + 2^{b} \times 3 - 3)</math> which becomes <math>A(a,0,2^{b+1} \times 3 - 3)</math> if c = 3.
* Rule 11: A(a, 0, c) --> A(a - 2, c + 3, 3)
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)
* A(2k + d, 0, c) --> <math>A(d, f^{k-1}(c+3), 3)</math>
<pre>
The TM enters configuration A(19, 2, 3) after 799 steps with tape:
0^inf 2 1 (11)^19 (12)^2 <D (11)^3 0^inf
</pre>
==Trajectory==
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's have <math> f^{9}(24) - 3</math> = m.
<pre>
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
Score calculated in HyperCalc:
(10^)^8 30,302,671.815163
Or in tetration: 10^^10.873987 (truncated)
</pre>
</div></div>
=1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD=
<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!
!0
!1
!2
|-
|A
|1RB
|1RD
|1LC
|-
|B
|2LB
|1RB
|1LC
|-
|C
|1RZ
|1LA
|1LD
|-
|D
|0RB
|2RA
|2RD
|}
<pre>
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]
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+1 1 <D (11)^1 0^inf [+10b +26 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+2 2 0^inf by rule 10
--> S (12)^a-1 2 A> (11)^b+2 2 0^inf
--> S (12)^a-1 2 (12)^b+2 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+2 <C 1 0^inf
--> S (12)^a-1 2 (12)^b+1 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]
by:
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 1 <D (11)^2b+3 0^inf
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
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
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+5)-3 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)*3)-3 0^inf by rule 18
= S (12)^a-1 2^2 1 <D (11)^(2^(b+5)-3) 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
= 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
= 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
--> 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
= 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
--> 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 by rule 22
--> 0^inf (11)^2c+5 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
= 0^inf (11)^2c+5 (12)^3 2^2 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
--> 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
</pre>
==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 <math>f_1(n) = 2^{n+5}-3</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:
* <math>D(a, b, c) --> D(a, b-1, 2^{b+5}-3)</math>
* <math>D_1(a, b, c) --> D_1(a, b-1, 2^{b+5}-3)</math>
Rule 23 becomes:
* <math>D(a, 0, c) --> D(a-3, 2c+11, 1)</math>
* <math>D_1(a, 0, c) --> D_1(a-3, 2c+11, 1)</math>
Rule 24 becomes:
* <math>D(0, 0, c) --> D(c+1, 3, 1)</math>
Rule 25 becomes:
* <math>D(2, 0, c) --> D(2c+8, 3, 1)</math>
Rule 26 becomes:
* <math>D_1(1, 0, c) --> D_1(2c+7, 3, 1)</math>
Rule 27 becomes:
* <math>D_1(0, 0, c) --> D(2c+5, 3, 1)</math>
Rule 28 becomes:
* D(1, 0, c) --> halt with score 4c + 18
By repeating rule 21, a stronger rule can be constructed:
* <math>D(a, b, c) --> D(a, 0, f_1^{b}(c))</math>
* <math>D_1(a, b, c) --> D1(a, 0, f_1^{b}(c))</math>
If a is greater than or equal to 3:
<math>D(a, 0, c) --> D(a-3, 2c+11, 1) --> D(a-3, 0, f_1^{2c+11}(1))</math>
=<math>D(a-3, 0, f_2(1,c))</math>
* <math>D(a, 0, c) --> D(a-3, 0, f_1^{2c+11}(1))</math>
This rule can also be repeated, also note that <math>f_1^{2c+11}(1) = f_2(1,c)</math> and <math>f_1^{2 \times f_2(a,b) + 11}(1) = f_2(a+1,b)</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>
The trajectory of D_1(2, 0, c) is currently unexplored.
==Trajectory==
<pre>
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).
</pre>
D(2, 2, 1) -->
<math>D(2, 0, f_1^{2}(1)) = D(2, 0, f_1(61))</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_2 = 2^{67} +2</math>
<pre>
2^67 +2 mod 3 = ?
2^2k mod 3 = 1
2^2k+1 mod 3 = 2
2^67 mod 3 = 2 --> +2 = 4; 4 mod 3 = 1
</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>e_3 = f_2(\frac{e_2-1} 3, f_1^{2}(61))</math>
<pre>
f_1(2k+1) = 2^2k+6-3
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)
Now consider that 2^2k mod 3 = 1 and 2^2k+1 mod 3 = 2
f_1(2k+1) = 2^2k+6-3
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.
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>e_3 mod 3 = 1</math>
--> <math>D_1(2e_3+7, 3, 1) --> D_1(2e_3+7, 0, f_1^{2}(61))</math>
2e_3 + 7
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>e_4 = f_2(\frac{2e_3+7} 3, f_1^2{2}(61))</math>
<math>D_1(0, 0, e_4)</math>
--> <math>D(2e_4+5, 3, 1) --> D(2e_4+5, 0, f_1^{2}(61))</math>
e_4 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>e_5 = f_2(\frac{2e_4+4} 3, f_1^{2}(61))</math>
<math>D(1, 0, e_5)</math>
--> halts with score <math>4e_5 + 18</math>.
==Approximate Score==
<math>4e_5 + 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+7} 3, f_1^{2}(61))</math>
<math>e_3 = f_2(\frac{e_2-1} 3, f_1^{2}(61))</math>
<math>e_2 = 2^{67} +2</math>
<math>f_1(n) = 2^{n+5}-3</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>2^{n+4} < f_1(n) < 2^{n+5}</math>
<math>(2 \uparrow)^{a} n+4 < f_1^{a}(n) < (2 \uparrow)^{a} n+6</math>
<math>(2 \uparrow)^{a} 5 < f_1^{a}(1) < (2 \uparrow)^{a} 7</math>
<math>2 \uparrow\uparrow (a+2) < f_1^{a}(1) < 2 \uparrow\uparrow (a+3)</math>
*<math>f_2(a,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>2 \uparrow\uparrow (2 \times f_2(a-1,b) + 13) < f_2(a,b) < 2 \uparrow\uparrow (2 \times f_2(a-1,b) + 14)</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)^{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\uparrow (a+3) < f_2(a, f_1^{2}(61)) < 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 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 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>
This score would make {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} the new [[BB(4,3)]] champion.
</div></div>

Revision as of 11:13, 4 October 2025

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