1RB0RA 1LC1LF 1RD0LB 1RA1LE 1RZ0LC 1RG1LD 0RG0RF

1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch) is a halting BB(7) TM which runs for over $2\uparrow ^{11}2\uparrow ^{11}3$ (or $2\uparrow ^{11}2\uparrow ^{10}4$) steps. It was discovered by Pavel Kropitz on 10 May 2025 (Discord link) and analyzed by Shawn Ligocki (here) on 13 May 2025.

Analysis by Shawn Ligocki

Consider general configurations matching the regex:

$$0^{\infty }\;11\;(1\;(01)^{*})^{*}\;0011100\;{\textrm {A>}}\;0^{\infty }$$

Low level rules

              01 1 01^n 0011100 A> 00   -->                 1 01^n+2 0011100 A>
           01^3 11 01^n 0011100 A> 0^6  -->            1 01^n+5 1 01 0011100 A>
01^3 (1 01)^k+1 11 01^n 0011100 A> 0^6  -->  1 01^n+6 (1 01)^k 11 01 0011100 A>
 011 (1 01)^k   11 01^n 0011100 A> 0^2  -->  1 Z> 111 01^n+1 00 101^k+2

Mid level rules

Let

$$B(a;b,c,...,z)=0^{\infty }\;111\;(01)^{3z+1}\;1\;\cdots \;1(01)^{3c+1}\;1\;(01)^{3b+1}\;1\;(01)^{0}\;1\;(01)^{3a+1}\;0011100\;{\textrm {A>}}\;0^{\infty }$$

and let B(a; [x]*k, y, ...) = $B(a;\underbrace {x,\cdots ,x} _{k},y,...)$ (In other words, [x]*k represents k repeats of the value x in a config).

then

B(a; b+1, ...) -> B(2a+4; b, ...)
B(a; [0]*k, 0, n+1, ...) -> B(0; [0]*k, a+2, n, ...)
B(a; [0]*k) -> Halt(3a + 2k + 9)

Start at step 8178: B(2, [1]*12)

High level rule

Let

$${\begin{array}{l}g_{0}(x)&=&2x+4\\g_{k+1}(x)&=&g_{k}^{x+2}(0)\\\end{array}}$$

then

$$B(a;\underbrace {0,\cdots ,0} _{k},n,...)\to B(g_{k}^{n}(a);\underbrace {0,\cdots ,0} _{k},0,...)$$

Bound

Let

$${\begin{array}{l}a_{0}&=&2\\a_{k+1}&=&g_{k}(a_{k})\\\end{array}}$$

then

$$B(a_{0};\underbrace {1,\cdots ,1} _{k})\to B(a_{k},\underbrace {0,\cdots ,0} _{k})\to {\text{Halt}}(3a_{k}+2k+9)$$

and

$${\textrm {Start}}\to B(a_{0};\underbrace {1,\cdots ,1} _{12})$$

and so this TM halts with a sigma score of $\sigma =3a_{12}+33$

Note that $g_{k}(x)=(2\uparrow ^{k}(x+4))-4$ and so for $k\geq 2$,

$$a_{k+1}+4>2\uparrow ^{k}2\uparrow ^{k}3$$

and so this TM halts with sigma score $\sigma >2\uparrow ^{11}2\uparrow ^{11}3$.

This bound is pretty tight: $\sigma <2\uparrow ^{11}2\uparrow ^{11}4=2\uparrow ^{12}4$.