1RB1LF_0RC1RD_1LA1LD_1LE0RA_---1LC_0LA1LA

1RB1LF_0RC1RD_1LA1LD_1LE0RA_---1LC_0LA1LA (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:
C(a, b) = 0^inf 110 A> 11^a 0 1^b 0^inf with b >= 6
M_3(a, c, d) = 0^inf 110 A> 11^a 01 11^c 0 1^d 0^inf
M_4(a, b, c, d) = 0^inf 11^{a+1} 110 A> 11^b 01 11^c 0 1^d 0^inf

Notice that C(a, 2c+1) = M_3(a, c, 0)

C(0, 2k+6) -> C(k+1, 9)
C(a+1, 2k+6) -> M_4(a, k+1, 4, 0)

M_3(0, 0, d) -> Halt
M_3(1, 0, d) -> Halt
M_3(a+2, 0, d) -> C(a+1, d+9)
M_3(0, c+1, d) -> C(c+1, d+6)
M_3(a+1, c+1, d) -> M_3(a, c, d+6)

M_4(a, 0, c, d) -> C(a+c+1, d+6)
M_4(0, 1, 0, d) -> Halt
M_4(a+1, 1, 0, d) -> C(a+1, d+12)
M_4(a, b+2, 0, d) -> M_3(a, b, d+9)
M_4(a, b+1, c+1, d) -> M_4(a, b, c, d+6)

Start: C(0, 9), reached after 74 raw TM steps

This can be simplified to:

Consider:
C(a, b) = 0^inf 110 A> 11^a 0 1^b 0^inf with b >= 6

C(0, 2k+6) -> C(k+1, 9)
C(a+1, 2k+6) -> C(a-k+4, 6k+12) if k <= a+3
C(a+1, 2k+6) -> Halt if k = a+4 or k = a+5
C(a+1, 2k+6) -> C(k-a-5, 6a+39) if k >= a+6

C(a, 2k+1) -> C(k-a, 6a+6) if k > a
C(a, 2k+1) -> Halt if a = k or a = k+1
C(a, 2k+1) -> C(a-k-1, 6k+9) if a >= k+2

Start: C(0, 9), reached after 74 raw TM steps