1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC

A probviously halting BB(6) Cryptid found by @mxdys on 30 Jun 2024.

Analysis by Shawn Ligocki:

1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC

C(a, b, c) = $ 1^2a+1 C> 0^2b 1^c 01 $

Level 1:
  C(a, b+2, c) -> C(a+3, b, c)
  C(a, 1, c+2) -> C(1, a+3, c)
  C(a, 0, c+1) -> C(1, a+1, c)

  C(a, 0, 0) -> C(1, 2, 2a+3)
  C(a, 1, 1) -> C(1, 2, 2a+7)
  C(a, 1, 0) -> Halt(2a+5)

Level 2:
  C(1, 2b,   c+1) -> C(1, 3b+2, c)
  C(1, 2b+1, c+2) -> C(1, 3b+4, c)

  C(1, 2b,   0) -> C(1, 2, 6b+5)
  C(1, 2b+1, 1) -> C(1, 2, 6b+9)
  C(1, 2b+1, 0) -> Halt(6b+7)


C(1, 0, 0)  @11
C(1, 2, 5)  @55
  ...
C(1, 17, 1)
C(1, 2, 57)
  ...
C(1, 70_091_065, 1)
C(1, 2, 210_273_201)
  ...

Rules validated in https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L1042

For my code I'm seeing O(n^2) runtime:

          0  C(1, 2, 5)  [0s]
          4  C(1, 2, 57)  [0s]
         45  C(1, 2, 210_273_201)  [0s]
    100_000  C(1, 10^17_602, 210_123_530)  [1s]
    200_000  C(1, 10^35_211, 209_973_408)  [4s]
    300_000  C(1, 10^52_820, 209_823_652)  [10s]
    400_000  C(1, 10^70_429, 209_673_652)  [17s]
    500_000  C(1, 10^88_039, 209_523_616)  [27s]
    600_000  C(1, 10^105_648, 209_373_548)  [38s]
    700_000  C(1, 10^123_257, 209_223_420)  [52s]
    800_000  C(1, 10^140_866, 209_073_477)  [68s]
    900_000  C(1, 10^158_475, 208_923_310)  [86s]
  1_000_000  C(1, 10^176_084, 208_773_236)  [107s]

ETA maybe 22 days for next "reset" (c <= 1).

Interestingly, both of the 2 reset's I've been able to simulate to both follow the C(1, 2b+1, 1) -> C(1, 2, 6b+9) path.

Equivalent TMs

This is equivalent to 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC (bbch) except slightly slower. Specifically: Specifically, 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC has 1 <E --(1)--> <D 1, while 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC has x1 <E --(3)--> x <D 1