1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD

1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD (bbch) is a tetrational halting BB(4,3) TM. It was discovered in May 2024 by Pavel Kropitz as one of seven long running TMs and achieves a score of around 10 ↑↑ 9.873987. Polygon analysed the TM by hand in September 2025, providing its score.

Pavel listed the halting tape as:

1 Z> 1^((8*<7; (6*2^((4b + 14)) - 4); (6*2^((48*2^(21) - 2)) - 4)> + 33)) 2

Analysis by Polygon

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

9. S (12)^a <D (11)^b 0^inf --> S (12)^a-1 <D (11)^2b+3 0^inf [+4b^2 +22b +22 steps]
10. S (12)^a <D (11)^b 0^inf --> S <D (11)^((2^(a))*b+(2^(a))*3-3) 0^inf

11. S (11)^a <D (11)^b 0^inf --> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf [+10b +50 steps]

12. S 1^a <A (11)^b 0^inf --> 1^a-1 <A (11)^b+1 2 0^inf [+4b +5 steps]

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

• Rule 9: $A(a,b,c)\to A(a,b-1,2c+3)$
• Rule 10: $A(a,b,c)\to A(a,0,2^b\times c+2^b\times 3-3)$ which becomes $A(a,0,2^{b+1}\times 3-3)$ if c = 3.
• Rule 11: $A(a,0,c)\to A(a-2,c+3,3)$

Further: let $f(n)=2^{n+1}\times 3$

• If c = 3: $A(a,b,3)\to A(a,0,f(b)-3)\to A(a-2,f(b),3)$

• $A(a,0,c)\to A(a-2,c+3,3)\to A(a-2,0,f(c+3)-3)$

• $A(2k+d,b,3)\to A(d,f^k(b),3)$

Trajectory:

The TM enters configuration $A(19,2,3)$ with S = 2 1 after 799 steps.

$A(19,2,3)\to A(1,f^9(2),3)\to A(1,0,f^{10}(2)-3)$

Let m = $f^{10}(2)-3$

--> 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

Score calculated in HyperCalc:

$(10\uparrow {)}^{8}30302671.815163$

Or in tetration: $10\uparrow \uparrow 9.873987$ (truncated)