1RB1RD_0LC1RA_0RA1LC_1RA1LE_1RF0LD_---0RE

1RB1RD_0LC1RA_0RA1LC_1RA1LE_1RF0LD_---0RE (bbch) is a BB(6) potential Cryptid found by Marco Dos Santos using Gemini and reported in his paper At the Limits of Computability: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier. The probvious behavior of these TMs has not yet been analyzed.

Analysis by Marco Dos Santos

Consider:
P_1(c) = 0^inf C> 0 1^c 0^inf
P_2(a, c) = 0^inf 10^a 1 C> 0 1^c 0^inf
D(a, b, w) = 0^inf 10^a 1^b D> 1^{w+1} 0^inf
where c >= 1

P_1(1) -> P_1(3)
P_1(w+2) -> D(0, 1, w)
P_2(a, 1) -> P_2(a, 3)
P_2(a, w+2) -> D(a+1, 1, w)

D(0, 2k+1, 0) -> P_1(2k+4)
D(0, 2k+1, 1) -> P_1(2k+6)
D(0, 2k+1, w+2) -> D(0, 2k+4, w)

D(0, 2k+2, 0) -> Halt
D(0, 2k+2, 1) -> P_2(k, 3)
D(0, 2k+2, 2) -> P_2(k, 5)
D(0, 2k+2, 3) -> P_2(k, 7)
D(0, 2k+2, w+4) -> D(k+1, 5, w)

D(1, 2k+1, 0) -> P_1(2k+5)
D(1, 2k+1, 1) -> P_1(2k+7)
D(a+2, 2k+1, 0) -> P_2(a, 2k+5)
D(a+2, 2k+1, 1) -> P_2(a, 2k+7)
D(a+1, 2k+1, w+2) -> D(a, 2k+5, w)

Start: P_1(1), reached after 3 raw TM steps

This can be simplified into:

Consider:
D(a, b, w) = 0^inf 10^a 1^b D> 1^{w+1} 0^inf where b >= 1, and b is odd if a >= 1
D(0, 2k+1, w) -> D(0, 1, 2k+2w+2) if w <= 1
D(0, 2k+1, w+2) -> D(0, 2k+4, w)
D(0, 2k+2, 0) -> Halt
D(0, 2k+2, w) -> D(k+1, 1, 2w-1) if 1 <= w <= 3
D(0, 2k+2, w+4) -> D(k+1, 5, w)
D(a+1, 2k+1, w) -> D(a, 1, 2k+2w+3) if w <= 1
D(a+1, 2k+1, w+2) -> D(a, 2k+5, w)
Start: D(0, 1, 1), reached after 13 raw TM steps