1RB0LF 1RC1RB 1LD0RA 1LB0LE 1RZ0LC 1LA1LF

1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF (bbch) is a long running Halting BB(6) TM first analyzed by Katelyn Doucette on 1 Jun 2025 (Discord Link). It is a shift overflow counter that halts with a final sigma score of roughly $10\uparrow \uparrow 7.52390$. It is currently the second longest known running halting BB(6) TM.

Analysis by Katelyn Doucette

(These rules were automatically generated by https://github.com/Laturas/shift_overflow_subset_decider)

START = A(5, 2^2 - 1, 0) at step 57
A(3k, 2^n - 1, 0) -> HALT
A(3k + 1, 2^n - 1, 0) -> A(5, 2^2 - 1, n + 2k + 0)
A(3k + 2, 2^n - 1, 0) -> A(5, 2^2 - 1, n + 2k + 1)
A(a, 2^n - 1, c) -> A(a + 2^n (4(4^c - 1)/3) - 3c, 2^{n+2c} - 1, 0)

---------- RULE EVALUATION ----------
START = A(5, 2^2 - 1, 0)
A(5, 2^2 - 1, 0) -> A(5, 2^2 - 1, 5) -> A(5446, 2^12 - 1, 0)    (mod 6377292)
A(5446, 2^12 - 1, 0) -> A(5, 2^2 - 1, 3642) -> A(195563, 2^7286 - 1, 0)    (mod 708588)
A(195563, 2^7286 - 1, 0) -> A(5, 2^2 - 1, 137661) -> A(72002, 2^39128 - 1, 0)    (mod 78732)
A(72002, 2^39128 - 1, 0) -> A(5, 2^2 - 1, 8397) -> A(7430, 2^8048 - 1, 0)    (mod 8748)
A(7430, 2^8048 - 1, 0) -> A(5, 2^2 - 1, 1337) -> A(838, 2^732 - 1, 0)    (mod 972)
A(838, 2^732 - 1, 0) -> A(5, 2^2 - 1, 318) -> A(107, 2^98 - 1, 0)    (mod 108)
A(107, 2^98 - 1, 0) -> A(5, 2^2 - 1, 25) -> A(6, 2^4 - 1, 0)    (mod 12)
 -> HALT

Analysis by Shawn Ligocki

https://discord.com/channels/960643023006490684/1375512968569028648/1378925307544997898

1RB0LF_1RC1RB_1LD0RA_1LB0LE_---0LC_1LA1LF
   0 (5, 2, 5, 2, 0, 1)
   1 (((-46 + 2^14)/3), 12, 5, 2, 0, 1)
   2 (((-11 + 2^((56 + 2^16)/9) + -1 * 2^15)/3), ((38 + 2^16)/9), 5, 2, 0, 1)
   3 (((-14 + 2^((62 + 2^((74 + 2^16)/9))/9) + -1 * 2^((65 + 2^16)/9))/3), ((44 + 2^((74 + 2^16)/9))/9), 5, 2, 0, 1)
   4 (((-14 + 2^((62 + 2^((80 + 2^((74 + 2^16)/9))/9))/9) + -1 * 2^((71 + 2^((74 + 2^16)/9))/9))/3), ((44 + 2^((80 + 2^((74 + 2^16)/9))/9))/9), 5, 2, 0, 1)
   5 (((-14 + 2^((62 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9) + -1 * 2^((71 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/3), ((44 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9), 5, 2, 0, 1)
   6 (((-11 + 2^((56 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9) + -1 * 2^((71 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/3), ((38 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9), 5, 2, 0, 1)
   7 (((-14 + 2^((62 + 2^((74 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9))/9) + -1 * 2^((65 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9))/3), ((44 + 2^((74 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9))/9), 5, 2, 0, 1)
Halted with score: ~10↑↑7.52390  =  ((34 + 2^((71 + 2^((74 + 2^((80 + 2^((80 + 2^((80 + 2^((74 + 2^16)/9))/9))/9))/9))/9))/9))/9)

Equivalent TMs

The following TMs have near identical behavior, producing the same values but do so slightly more efficiently:

1RB0LF_1RC1RB_1LD0RA_1LF0LE_---0LC_1LA1LF
1RB0LF_1RC1RB_1LD0RA_1RF0LE_---0LC_1LA1LF

Related TMs

This TM is part of a broader family of 15 turing machines with similar rules. The family is made up of the following:

1RB1RA_1LC0RF_---0LD_0LE0LB_1LF1LE_1RA0LE
1RB0RE_1LC0RF_1LD1LC_1RA0LB_---0RD_1RB1RF
1RB0LF_1RC1RB_1LD0RA_0RF0LE_---0LC_1LA1LF
1RB0LF_1RC1RB_1LD0RA_0RA0LE_---0LC_1LA1LF
1RB0LF_1LC1LB_1RD0LB_1RE1RD_1LA0RC_---0LE
1RB0LF_1RC1RB_1LD0RA_1RA0LE_---0LC_1LA1LF
1RB0LE_1LC0RD_1LA1LC_---0RA_1LC0RF_1RE1RF
1RB0LF_1RC1RB_1LD0RA_1LA0LE_---0LC_1LA1LF
1RB0LD_1RC0RF_1RD1RC_1LE0RC_1LA1LE_---0RA
1RB0LF_1RC1RB_1LD0RA_1RF0LE_---0LC_1LA1LF
1RB0LD_1RC0RF_1LA1LC_1LC0RE_1RD1RE_---0RA
1RB1RA_1LC0RA_1LD1LC_1RE0LB_0LA0RF_---0RD
1RB0LC_0LC0RF_1LE0RD_1RC1RD_1LA1LE_---0RA
1RB0LF_1RC1RB_1LD0RA_1LB0LE_---0LC_1LA1LF
1RB0LF_1RC1RB_1LD0RA_1LF0LE_---0LC_1LA1LF

All of these turing machines have been shown to halt. Interestingly, 1RB1RA_1LC0RF_---0LD_0LE0LB_1LF1LE_1RA0LE is the only one to halt when the unary counter is equal to 1 mod 3 or 2 mod 3 instead of 0 mod 3.

All of these machines are fundamentally characterized by the rule A(a, 2^n - 1, c) -> A(a + 2^n (4(4^c - 1)/3) - 3c, 2^{n+2c} - 1, 0) and only differ in starting conditions and reset behavior.

1RB1LE_1RC1RC_1LD0LF_1LE1LD_1RF0LC_1RZ0RA (bbch) exhibits similar behavior but never halts. See proof at: (Discord Link).