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= A(19, 2, 3)
= A(19, 2, 3)
</pre>
</pre>
A(19, 2, 3) --> <math>A(1, f^{9}(2), 3) --> A(1, 0, f^{10}(2)-3)</math>
A(19, 2, 3) --> <math>A(1, f^{9}(2), 3)</math> --> <math>A(1, 0, f^{10}(2)-3)</math>


Let m = <math>f^{10}(2)-3</math>
Let m = <math>f^{10}(2)-3</math>
Line 215: Line 215:


Or in tetration: 10^^9.873987 (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>
</div></div>
</div></div>


={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}=
={{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}=
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>.
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) and num(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">
<div class="toccolours mw-collapsible mw-collapsed">'''Analysis'''<div class="mw-collapsible-content">
{| class="wikitable"
{| class="wikitable"
Line 633: Line 655:


Rule 21 becomes:
Rule 21 becomes:
* <math>D(a, b, c) --> D(a, b-1, 2^{b+4} \times 3 -5)</math>
* <math>D(a, b, c) --> D(a, b-1, 2^{c+4} \times 3 -5) = D(a, b-1, f_1(c))</math>
* <math>D_1(a, b, c) --> D_1(a, b-1, 2^{b+4} \times 3 -5)</math>
* <math>D_1(a, b, c) --> D_1(a, b-1, 2^{c+4} \times 3 -5) = D_1(a, b-1, f_1(c))</math>
 
Rule 30 is constructed by repeating rule 21:
* <math>D(a, b, c) --> D(a, 0, f_1^{b}(c))</math>
* <math>D_1(a, b, c) --> D_1(a, 0, f_1^{b}(c))</math>


Rule 23 becomes:
Rule 23 becomes:
* <math>D(a, 0, c) --> D(a-3, 2c+11, 1)</math>
* <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>
* <math>D_1(a, 0, c) --> D_1(a-3, 2c+11, 1)</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> via rules 23 and 30, and
<math>D_1(a, 0, c) --> D_1(a-3, 2c+11, 1) --> D_1(a-3, 0, f_1^{2c+11}(1))</math> via rules 23 and 30
Thus rule 31 is:
* <math>D(a, 0, c) --> D(a-3, 0, f_1^{2c+11}(1)) = D(a-3, 0, f_2(1,c))</math> and
* <math>D_1(a, 0, c) --> D_1(a-3, 0, f_1^{2c+11}(1)) = D_1(a-3, 0, f_2(1,c))</math>
This rule can also be repeated to create rule 32, 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>


Rule 24 becomes:
Rule 24 becomes:
Line 644: Line 687:


Rule 25 becomes:
Rule 25 becomes:
* <math>D(2, 0, c) --> D(2c+8, 3, 1)</math>
* <math>D(2, 0, c) --> D_1(2c+8, 3, 1)</math>


Rule 26 becomes:
Rule 26 becomes:
Line 657: Line 700:
Rule 29 becomes:
Rule 29 becomes:
* <math>D_1(2, 0, c) --> D(2c+10, 2, 1)</math>
* <math>D_1(2, 0, c) --> D(2c+10, 2, 1)</math>
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>


==Trajectory==
==Trajectory==
Line 714: Line 743:
f_2(a,b) = f_1^(2*f_2(a-1,b)+11)(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)
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
Thus f_2(a,b) mod 3 = 1 for a ≠ 0.
</pre>
</pre>


Line 729: Line 758:
<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>
<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>


<math>D_1(1,0,e_3)</math>
<math>D_1(1,0,e_2)</math>


<math>e_2 mod 3 = 1</math>
<math>e_2 mod 3 = 1</math>
Line 735: Line 764:
--> <math>D_1(2e_2+7, 3, 1) --> D_1(2e_2+7, 0, f_1^{2}(91))</math>
--> <math>D_1(2e_2+7, 3, 1) --> D_1(2e_2+7, 0, f_1^{2}(91))</math>


2e_3 + 7
2e_2 + 7


Modulus: 2 + 7 --> 9 mod 3 = 0
Modulus: 2 + 7 --> 9 mod 3 = 0
Line 766: Line 795:
<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>
<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>


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


<math>e_1 = 2^{95} \times 3 -5</math>
<math>e_1 = 2^{95} \times 3 -5</math>
Line 777: Line 806:
<math>2^{n+5} < f_1(n) < 2^{n+6}</math>
<math>2^{n+5} < f_1(n) < 2^{n+6}</math>


<math>(2 \uparrow)^{a} n+5 < f_1^{a}(n) < (2 \uparrow)^{a} n+7</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} 8</math>
<math>(2 \uparrow)^{a} 5 < f_1^{a}(1) < (2 \uparrow)^{a} 8</math>
Line 803: Line 832:


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>
== Reset behavior ==
Let's restate the rules:
Rule 30:
* <math>D(a, b, c) --> D(a, 0, f_1^{b}(c))</math>
* <math>D_1(a, b, c) --> D_1(a, 0, f_1^{b}(c))</math>
Rule 23:
* <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 31 for a greater than or equal to 3:
* <math>D(a, 0, c) --> D(a-3, 0, f_1^{2c+11}(1)) = D(a-3, 0, f_2(1,c))</math> and
* <math>D_1(a, 0, c) --> D_1(a-3, 0, f_1^{2c+11}(1)) = D_1(a-3, 0, f_2(1,c))</math>
Rule 32:
* <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>
Rule 24:
* <math>D(0, 0, c) --> D(c+1, 3, 1)</math>
Rule 28:
* D(1, 0, c) --> halt with score 4c + 18
Rule 25:
* <math>D(2, 0, c) --> D_1(2c+8, 3, 1)</math>
Rule 27:
* <math>D_1(0, 0, c) --> D(2c+5, 3, 1)</math>
Rule 26:
* <math>D_1(1, 0, c) --> D_1(2c+7, 3, 1)</math>
Rule 29:
* <math>D_1(2, 0, c) --> D(2c+10, 2, 1)</math>
Let's explore what happens after the remainder rules:
AR1: <math>D(0, 0, c) --> D(c+1, 3, 1) --> D(c+1, 0, f_1^{3}(1))</math> by rules 24 and 30.
AR2: D(1, 0, c) --> halt with score 4c + 18 by rule 28.
AR3: <math>D(2, 0, c) --> D_1(2c+8, 3, 1) --> D(2c+8, 0, f_1^{3}(1))</math> by rules 25 and 30.
AR4: <math>D_1(0, 0, c) --> D(2c+5, 3, 1) --> D(2c+5, 0, f_1^{3}(1))</math> by rules 27 and 30.
AR5: <math>D_1(1, 0, c) --> D_1(2c+7, 3, 1) --> D_1(2c+7, 0, f_1^{3}(1))</math> by rules 26 and 30.
AR6: <math>D_1(2, 0, c) --> D(2c+10, 2, 1) --> D(2c+10, 0, f_1^{2}(1))</math> by rules 29 and 30.
Together with:
AR7 by rule 30:
* <math>D(a, b, c) --> D(a, 0, f_1^{b}(c))</math>
* <math>D_1(a, b, c) --> D_1(a, 0, f_1^{b}(c))</math>
AR8 by rule 32 for k ≠ 0:
* <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>
Let's also consider the following which was shown earlier:
<pre>
f_1(n) mod 3 = 1
f_1^a(n) mod 3 = 1
f_2(a,b) mod 3 = 1 for a ≠ 0
</pre>
Every one of these rules will set c to either f_1(n), f_1^a(n) or f_2(a,b) with a ≠ 0, thus c mod 3 = 1 after the first rule application.
Thus, c can be considered to be of the form 3m+1:
AR1-2: <math>D(0, 0, 3m+1) --> D(3m+2, 0, f_1^{3}(1)) --> D(2, 0, f_2(m,f_1^3(1)))</math> by rules AR1 and AR8. AR1-2 leads to AR3-2.
AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2-2 halts.
AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR5-2.
AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2
AR5-2: <math>D_1(1, 0, 3m+1) --> D_1(6m+9, 0, f_1^{3}(1)) = D_1(3(2m+3), 0, f_1^{3}(1)) --> D_1(0, 0, f_2(2m+3,f_1^3(1)))</math> by rules AR5 and AR8. AR5-2 leads to AR4-2.
AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.
The maximum trajectory following these rules is: AR6-2 --> AR1-2 --> AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt
This TM follows the trajectory: AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt
</div></div>
</div></div>


Line 1,236: Line 1,377:
3^2k+1 mod 4 = 3
3^2k+1 mod 4 = 3


g_2(4k+1) = 3^(3*(4k+1)+7)-2 = 3^(12k+10)-2 = 3^2m-2
g_2(4k+1) = 3^(3*(4k+1)+7)-2 = 3^(12k+10)-2 = 3^2m-2 where m = 6k+5
3^2m mod 4 = 1 --> -2
3^2m mod 4 = 1 --> -2
3^2m - 2 mod 4 = 3
3^2m - 2 mod 4 = 3
Line 1,242: Line 1,383:
--> g_2(4k+1) = 4m+3
--> g_2(4k+1) = 4m+3


g_2(4k+3) = 3^(3*(4k+3)+7)-2 = 3^(12k+16)-2 = 3^2m-2
g_2(4k+3) = 3^(3*(4k+3)+7)-2 = 3^(12k+16)-2 = 3^2m-2 where m = 6k+8
3^2m-2 mod 4 = 3
3^2m-2 mod 4 = 3
g_2(4k+1) mod 4 = 3
g_2(4k+1) mod 4 = 3
Line 1,329: Line 1,470:


<math>3 \uparrow\uparrow\uparrow 88574 < \sigma < S < 3 \uparrow\uparrow\uparrow 88575</math>
<math>3 \uparrow\uparrow\uparrow 88574 < \sigma < S < 3 \uparrow\uparrow\uparrow 88575</math>
</div>
 
==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 ≥ 5: 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 ≥ 3: 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]
 
a4: A(0, 5, c, ...) --> A(0, c+8, ...) [+3c +39 steps]
by:
A(0, 5, c, ...)
--> A(2, 1, c+3, ...) [+19] by rule 17
--> A(0, c+8, ...) [+3c +39] by rule 21
</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, ...) rule A1 [+8k^2 +4vk +k steps] by rule 8
  b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) rule A2 [+3c +13 steps] by rule 9
  b = 0:
      a mod 2 = 0:
        a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) rule A3 [+4a -6 steps] by rule 10
        a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) rule A4 [+3c +17 steps] by rule 11
        a = 0: unreachable
      a mod 2 = 1:
        a ≥ 5: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) rule A5 [+4a steps] by rule 13
        a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... rule A6 [+16 steps] by rule 14
        a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) rule A7 [+3c +15 steps] 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, ...) rule A8 [+b +4a +2 steps] by rule 16
        a = 0:
            b ≥ 7: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) rule A9 [+8k^2 +36k +3c -65 steps] by rule a3
            b = 5: A(0, 5, c, ...) --> A(0, c+8, ...) rule A10 [+3c +39 steps] by rule a4
            b = 3: unreachable
      a mod 2 = 1:
        a ≥ 3: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) rule A11 [+b +4a +8 steps] by rule 19
        a = 1: unreachable
  b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) rule A12 [+3c +11 steps] 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>

Latest revision as of 12:51, 18 October 2026

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) and num(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−1,2c+4×3−5)=D(a,b−1,f1(c))
  • D1(a,b,c)−−>D1(a,b−1,2c+4×3−5)=D1(a,b−1,f1(c))

Rule 30 is constructed by repeating rule 21:

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

Rule 23 becomes:

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


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)) via rules 23 and 30, and

D1(a,0,c)−−>D1(a−3,2c+11,1)−−>D1(a−3,0,f12c+11(1)) via rules 23 and 30

Thus rule 31 is:

  • D(a,0,c)−−>D(a−3,0,f12c+11(1))=D(a−3,0,f2(1,c)) and
  • D1(a,0,c)−−>D1(a−3,0,f12c+11(1))=D1(a−3,0,f2(1,c))


This rule can also be repeated to create rule 32, 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))

Rule 24 becomes:

  • D(0,0,c)−−>D(c+1,3,1)

Rule 25 becomes:

  • D(2,0,c)−−>D1(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)

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 for a ≠ 0.

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,e2)

e2mod3=1

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

2e_2 + 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(2e1+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↑)a(n+5)<f1a(n)<(2↑)a(n+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

Reset behavior

Let's restate the rules:

Rule 30:

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

Rule 23:

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

Rule 31 for a greater than or equal to 3:

  • D(a,0,c)−−>D(a−3,0,f12c+11(1))=D(a−3,0,f2(1,c)) and
  • D1(a,0,c)−−>D1(a−3,0,f12c+11(1))=D1(a−3,0,f2(1,c))

Rule 32:

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

Rule 24:

  • D(0,0,c)−−>D(c+1,3,1)

Rule 28:

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

Rule 25:

  • D(2,0,c)−−>D1(2c+8,3,1)

Rule 27:

  • D1(0,0,c)−−>D(2c+5,3,1)

Rule 26:

  • D1(1,0,c)−−>D1(2c+7,3,1)

Rule 29:

  • D1(2,0,c)−−>D(2c+10,2,1)


Let's explore what happens after the remainder rules:

AR1: D(0,0,c)−−>D(c+1,3,1)−−>D(c+1,0,f13(1)) by rules 24 and 30.

AR2: D(1, 0, c) --> halt with score 4c + 18 by rule 28.

AR3: D(2,0,c)−−>D1(2c+8,3,1)−−>D(2c+8,0,f13(1)) by rules 25 and 30.

AR4: D1(0,0,c)−−>D(2c+5,3,1)−−>D(2c+5,0,f13(1)) by rules 27 and 30.

AR5: D1(1,0,c)−−>D1(2c+7,3,1)−−>D1(2c+7,0,f13(1)) by rules 26 and 30.

AR6: D1(2,0,c)−−>D(2c+10,2,1)−−>D(2c+10,0,f12(1)) by rules 29 and 30.

Together with:

AR7 by rule 30:

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

AR8 by rule 32 for k ≠ 0:

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

Let's also consider the following which was shown earlier: 

f_1(n) mod 3 = 1
f_1^a(n) mod 3 = 1
f_2(a,b) mod 3 = 1 for a ≠ 0

Every one of these rules will set c to either f_1(n), f_1^a(n) or f_2(a,b) with a ≠ 0, thus c mod 3 = 1 after the first rule application.

Thus, c can be considered to be of the form 3m+1:


AR1-2: D(0,0,3m+1)−−>D(3m+2,0,f13(1))−−>D(2,0,f2(m,f13(1))) by rules AR1 and AR8. AR1-2 leads to AR3-2.

AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2-2 halts.

AR3-2: D(2,0,3m+1)−−>D1(6m+10,0,f13(1))=D1(3(2m+3)+1,0,f13(1))−−>D1(1,0,f2(2m+3,f13(1))) by AR3 and AR8. AR3-2 leads to AR5-2.

AR4-2: D1(0,0,3m+1)−−>D(6m+7,0,f13(1))=D(3(2m+2)+1,0,f13(1))−−>D(1,0,f2(2m+2,f13(1))) by rules AR4 and AR8. AR4-2 leads to AR2-2

AR5-2: D1(1,0,3m+1)−−>D1(6m+9,0,f13(1))=D1(3(2m+3),0,f13(1))−−>D1(0,0,f2(2m+3,f13(1))) by rules AR5 and AR8. AR5-2 leads to AR4-2.

AR6-2: D1(2,0,3m+1)−−>D(6m+12,0,f12(1))=D(3(2m+4),0,f12(1))−−>D(0,0,f2(2m+4,f12(1))) by rules AR6 and AR8. AR6-2 leads to AR1-2.

The maximum trajectory following these rules is: AR6-2 --> AR1-2 --> AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

This TM follows the trajectory: AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

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 where m = 6k+5
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 where m = 6k+8
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 ≥ 5: 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 ≥ 3: 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]

a4: A(0, 5, c, ...) --> A(0, c+8, ...) [+3c +39 steps]
by:
A(0, 5, c, ...)
--> A(2, 1, c+3, ...) [+19] by rule 17
--> A(0, c+8, ...) [+3c +39] by rule 21

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, ...) rule A1 [+8k^2 +4vk +k steps] by rule 8
   b = 2: A(a, 2, c, d, ...) --> A(a+2, c+1, d, ...) rule A2 [+3c +13 steps] by rule 9
   b = 0:
      a mod 2 = 0:
         a ≥ 4: A(a, 0, c, ...) --> A(1, a-4, c+4, ...) rule A3 [+4a -6 steps] by rule 10
         a = 2: A(2, 0, c, d, ...) --> A(2, c+2, d, ...) rule A4 [+3c +17 steps] by rule 11
         a = 0: unreachable
      a mod 2 = 1:
         a ≥ 5: A(a, 0, c, ...) --> A(2, a-4, c+4, ...) rule A5 [+4a steps] by rule 13
         a = 3: A(3, 0, c, ...) --> halt with 0^inf 1^2 0 1 Z> 1^c+4 ... rule A6 [+16 steps] by rule 14
         a = 1: A(1, 0, c, d, ...) --> A(0, c+4, d, ...) rule A7 [+3c +15 steps] 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, ...) rule A8 [+b +4a +2 steps] by rule 16
         a = 0:
            b ≥ 7: A(0, 2k+1, c, ...) --> A(0, c+6k-4, ...) rule A9 [+8k^2 +36k +3c -65 steps] by rule a3
            b = 5: A(0, 5, c, ...) --> A(0, c+8, ...) rule A10 [+3c +39 steps] by rule a4
            b = 3: unreachable
      a mod 2 = 1:
         a ≥ 3: A(a, b, c, ...) --> A(2, a-2, b-2, c+3, ...) rule A11 [+b +4a +8 steps] by rule 19
         a = 1: unreachable
   b = 1: A(a, 1, c, d, ...) --> A(0, a+c+3, d, ...) rule A12 [+3c +11 steps] 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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