1RB1LB2LC_1LA2RB1RB_---0LA2LA

1RB1LB2LC_1LA2RB1RB_---0LA2LA (bbch) is BB(3,3) holdout #397. It is similar to holdout #531, Wily Coyote.

Analysis by Legion

@Legion posted an analysis on Discord on the 17th of January, 2025, where he transformed Wily Coyote's analysis.

https://bbchallenge.org/1RB1LB2LC_1LA2RB1RB_---0LA2LA
https://bbchallenge.org/1LB1RB2RC_1RA2LB1LB_---0RA2RA
BB 0 => 2 AB: B0>1A,1B>B2

AB 1 => 2 BB: A1>1B,1B>B2
AB 2 => 1 CB: A2>2C,2B>B1
ABAB 1 => 1 BBCB: A1>1B,1B>B2,A2>2C,2B>B1
ABAB 2 => 2 CBBB: A2>2C,2B>B1,A1>1B,1B>B2
BB 1 => 2: B1>B2
BB 2 => 1: B2>B1
BBCB 1 => 1 AB: B1>B2,C2>2A,2B>B1
BBCB 2 => 2 ABAB: B2>B1,C1>0A,0B>1A,1B>B2
CB 1 => 2 ABAB: C1>0A,0B>1A,1B>B2
CB 2 => 1 AB: C2>2A,2B>B1
CBBB 1 => 1 ABAB: C1>0A,0B>1A,1B>B2,B2>B1
CBBB 2 => 2 AB: C2>2A,2B>B1,B1>B2

AB $ => 1$: A$>B1$
ABAB $ => 221$: A$>B1$,A1$>B221$
BB $ => 21$: B$>B21$
BBCB $ => 1221$: B$>B21$,C21$>1221$
CB $ => halt: C$>HH
CBBB $ => halt: C$>HH

a:AB
b:BB
c:CB
d:ABAB
e:CBBB
f:BBCB

a 0 => 1
a 1 => 2 b
a 2 => 1 c
b 0 => 2 a
b 1 => 2
b 2 => 1
c 0 => halt
c 1 => 2 d
c 2 => 1 a
d 0 => 2 b
d 1 => 1 f
d 2 => 2 e
e 0 => halt
e 1 => 1 d
e 2 => 2 a
f 0 => 1 d
f 1 => 1 a
f 2 => 2 d

cell 0: a(acde)
(Note that I've flipped the machine L ↔ R in this analysis, so that it cascades rightward)
Clearly, the only way for it to halt is from <C on the left side of the tape, but while that probviously can't occur (as I recall), it's not something that's trivial to prove, since the counter part is so chaotic.
Like the Wily Coyote machine, it seems to form chunks that grow in size