1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC

1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch)

Potential BB(6) Cryptid found by @mxdys on 18 Aug 2024. Andrew Ducharme forward simulated the combined map (P(x), Q(x,y)) for 10^8 iterations. The TM did not yet halt. After 10^7 iterations, the TM reached a value (x',1) where x' ~ 10^604100, and after 10^8 iterations, it reached a value x ~ 10^(6.04305 x 10^6).

start: P(2)

P(2a)   -> P(3a+4)
P(2a+1) -> Q(a+2,1)

Q(2a+3,b) -> P(b+5a+6)
Q(2a+2,b) -> Q(a,b+2a+5)

Q(1,2b+1) -> P(3b+8)
Q(1,2b)   -> Q(b+2,1)

Q(0,b)    -> halt

P(a) := 0^inf 1^a 011 <D 0^inf
Q(a,b) := 0^inf 1^(2a+1) <D 0 1^b 0^inf

The TM will halt after N iterations of the map Q if it ever takes the form ${\textstyle Q(2^{N+1}-2,y)}$.

On the trajectory starting from P(2), the Q(1,y) rules, at least through 10^5 steps, are never triggered. Thus, the implementation of the forward simulation can be made slightly faster (roughly 10% in practice) by, instead of nesting the execution of Q(x,y) within two if statements predicated on x=0 and x=1, only checking if x > 2, applying Q(x,y) if true, and throwing an exception if false.

CoSearch: https://cosearch.bbchallenge.org/contricution/tpxh8d8d