1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB
1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB (bbch) is a probviously halting BB(6) Cryptid analyzed by Racheline on 9 February 2025.
Analysis by Racheline
https://discord.com/channels/960643023006490684/1239205785913790465/1337994337790853181
if it halts, it seems to do so after at least around 1.24^1.24^530 steps (around 10^(3*10^48)). it's a shift-overflow counter but the counting is chaotic, like in the two chaotic counter 3x3 holdouts. here's my python code for figuring out the sequence of states hitting a given bit, in case anyone wants to see if they can do something with it (0 is E, 1 is A and 2 is D):
| Click to expand code |
|---|
<syntaxhighlight lang="python" line="1">
a = [0]
ats = [1]
atlen = 1
for _ in range(77):
b = []
bts = []
btlen = 0
t = 0
bit = 0
starting = 1
while starting or bit:
starting = 0
for i in range(len(a)):
s = a[i]
t += ats[i]
btlen += ats[i]
if s == 0:
bit = 1 - bit
b.append(bit)
bts.append(t)
t = 0
elif s == 1 and bit:
b.append(2)
bts.append(t)
t = 0
elif s == 2:
b.append(bit)
bts.append(t)
t = 0
t += atlen - ats[-1]
a = b
ats = bts
|
Analysis by Opus 4.7 / DrDisentangle
Halting transition: F,0 → ---. F is reached only via D,0 → 0LF, so halting requires D's left-sweep to end on a 0 whose L cell is also 0.
Let [b₁, b₂, ..., bₙ, (k)] denote the tape at an E-turnaround:
0^inf 1^b₁ 0 1^b₂ 0 ... 0 1^bₙ 0 1^k E> 0 0^inf
where E> means state E reading the cell immediately right of the rightmost 1.
Block sizes are read left-to-right (b₁ = leftmost block, bₙ = second-rightmost,
k = rightmost block). Reading right-to-left from the head, the Side expression is:
left = ones k *> [false] *> ones bₙ *> [false] *> ... *> [false] *> ones b₁ *> blank∞
right = blank∞
Invariant carried by all proven rules:
- every bᵢ ≥ 2 and k ≥ 2;
- all observed block sizes satisfy bᵢ ≡ 2 (mod 3);
In each rule, "..." on the left matches any (possibly empty) block prefix.
Parameters: j ≥ 0 ranges over ℕ; L : Side is an arbitrary leftward Side.
Block sizes in [..] positions can be any ≥ 2 value ≡ 2 (mod 3).
Init52: blank tape -> [5, (2)]
in 59 steps. [init_to_M_52]
Init55: blank tape -> [5, (5)]
in 76 steps. (= Init52 + Bump0) [init_to_M_55]
--- Simple bump (rightmost block ≡ 2 mod 6) ---
General: [..., (6j+2)] -> [..., (6j+5)]
in 16j+17 steps. [simple_bump]
Bump0: [..., (2)] -> [..., (5)]
in 17 steps. [simple_bump_base]
Bump1: [..., (8)] -> [..., (11)]
in 33 steps. [simple_bump_j1]
Carry0: [..., 2, 2, (5)] -> [..., 5, (8)]
in 45 steps. [carry_22_5]
Carry1: [..., 2, 2, (11)] -> [..., 5, (14)]
in 61 steps. [carry_22_11]
(Base cases of the family [..., 2, 2, k] → [..., 5, k+3]
with closed form dt = (8k+95)/3 for k = 6j+5.)
--- Swap rule: [..., 2, 5, k] → [..., 5, 2, k+3] ---
Swap11: [..., 2, 5, (11)] -> [..., 5, 2, (14)]
in 73 steps. [swap_25_11]
(Base case of the family [..., 2, 5, k] → [..., 5, 2, k+3]
with closed form dt = (8k+131)/3 for k = 6j+5.)
--- Carry rule: [..., 8, 5, k] → [..., 11, 2, k+3] ---
Carry8: [..., 8, 5, (35)] -> [..., 11, 2, (38)]
in 153 steps. [carry_85_35]
(Base case of the family [..., 8, 5, k] → [..., 11, 2, k+3]
with closed form dt = (8k+179)/3 for k = 6j+5.)
Rule families observed (500M steps):
- 335+ distinct macro-rule families
- All match closed form dt = (8k +
C_family) / 3, Δk = +3 C_familydepends only on the few rightmost blocks- Families above (
simple_bump,carry_22,swap_25,carry_85) are the four most frequent.
Shape-schema enumeration (2M configs):
- 12 "core" active-region schemata across 6 states (see machine.lean)
- ~20–25 total including edge/transient schemata
- Foundation for a potential ClosedSet-style nonhalt proof
See [1] for full proofs and the ClosedSet / regularity / schemata-enumeration discussion.