1RB3RB---3LA1RA_2LA3RA4LB0LB1LB

1RB3RB---3LA1RA_2LA3RA4LB0LB1LB (bbch) sometimes called "Bonus Cryptid" is a BB(2,5) found by Daniel Yuan in May 2024. It appeared to be a Cryptid, but in Aug 2024, Andrew Ducharme showed that it reached an undefined transition, so more investigation is necessary.

See https://www.sligocki.com/2024/05/10/bb-2-5-is-hard.html#a-bonus-cryptid

Behavior

1RB3RB---3LA1RA_2LA3RA4LB0LB1LB

Define
A(a, b) = 0^inf <B 0^a 3^b 2 0^inf

The rules are
A(3n, 0) -> Halt
A(3n, b+1) -> A(4n+3, b)
A(3n+1, b) -> A(4n+3, b+3)
A(3n+2, 0) -> ?
A(3n+2, b+1) -> A(4n+5, b)

Starts: A(3, 1)

Analysis by Andrew Ducharme

https://discord.com/channels/960643023006490684/1259770421046411285/1336973758182981634I found in August that it hits A(3n+2,0) after 113 A rule steps, but never seriously looked into what happened from there. The tape of the actual TM at that point is ginormous, so I looked at cases A(3n+2,0) -> 0<A 2 0^(3n+2) 2 0. From observations of the cases n=[0,1,2,...,10], the tape A(3n+2, 0) goes to (16/3 n + 7 , 1) if n = 0 (mod 3), (16/3n + 25/3, 2) if n = 2 (mod 3). God knows what happens if n is congruent to 1. Using these rules, I was able to go from 113 rule steps to........118 before the congruent 1 case was triggered.

In the first two cases, the lone 2 on the right hand side of the tape becomes a lone 4, and then is subsumed into the normal A(a,b) framework. In the last case, the lone 2 becomes a lone 4, then becomes a lone 1, but from there, I don't see a pattern. Sometimes it fixes itself, like when n = 1, but sometimes we enter a whole new phase like with n = 4.