1RB1LC 1LA1RE 0RD0LA 1RZ1LB 1LD0RF 0RD1RB

1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch)

Former BB(6) Champion. Discovered by mxdys on 16 June 2025.

It's in a family of 7 machines with the halting time and sigma score between 10↑↑11010000 and 10↑↑11011000:

1RB1LC_1LA1RE_0RD0LA_---1LB_1LD0RF_0RD1RB
1RB1LC_1LA1RE_0RD0LA_---1LB_1LE0RF_0RD1RB
1RB1LC_1LA1RD_1LA0LA_1LD0RE_0RF1RB_---1LB
1RB1LC_1LA1RD_1LA0LA_1LD0RE_0RF1RB_---1LC
1RB1LC_1LA1RD_1LA0LA_1LF0RE_0RF1RB_---1LB
1RB1LD_1LC1RE_---1LD_1LA0LA_1LE0RF_0RC1RB
1RB1LE_1LC0RA_1RB1LD_1LC0LC_1RF0LB_---1RE

Analysis

It's behavior can be described by the python code below:

def LBS(x):
    if x=='H1': return 'H1',1,1
    if x=='H2': return 'H4',2,2
    if x=='H3': print('Halted'); exit()
    if x=='H4': return 'H5',1,1
    if x=='H5': return 'H1',1,2
    w,k0,x = x
    if w=='W1':
        if k0==0 and x=='H1': return 'H5',1,2
        x0,n,k = LBS(x)
        k1=k0+k-2
        return ('W1',k1,x0),((n+1)*(1<<k1)-2)*4+5,2
    if w=='W2':
        if k0==0 and x=='H1': return 'H3',1,3
        if k0==1 and x=='H1': return 'H2',1,3
        x0,n,k = LBS(x)
        k1=k0+k-3
        return ('W2',k1,x0),((n+1)*(1<<k1)-2)*8+5,3
    assert 0

a,b,x = 57,5,'H1'
while 1:
    r,m2 = a%3,a//3
    x0,n,k = LBS(x)
    if r==2:
        m1 = b-3
        a,b,x = ((((n+1)*(1<<(m1+k))-2)*4+2)*(1<<(m2+1))+1), m2, ('W2',m1+k,x0)
    elif r==1:
        m1 = b-2
        a,b,x = ((((n+1)*(1<<(m1+k))-2)*2+2)*(1<<(m2+1))+1), m2, ('W1',m1+k,x0)
    elif r==0:
        m1 = b-1
        a,b,x = ((((n+1)*(1<<(m1+k))-1))*(1<<(m2+1))+1), (m2+m1+k+1), x0
    else: assert 0

Tape encoding (reversed):

H1 = 0^inf 1^7
H2 = 0^inf 1^5 101111 111111 1111
H3 = 0^inf 1^5 111111 1111
H4 = 0^inf 1^7 10 111111
H5 = 0^inf 1^7 111111
(W1,n,l) = l 111111^n 10 111111^2
(W2,n,l) = l 111111^n 1010 111111^3

(a,b,x) = x 111111^b <F0 11^1+a 0^inf

Execution process (shared by these 7 TMs):

(57,5,H1) -->
(66060289,25,H1) -->
(_,22020096,(W1,24,H1)) -->
(_,_,(W2,22020095,(W1,23,H1))) --> ...
(_,_,(...(W2,22020074,(W1,2,H1))...)) -->
(_,_,(...(W2,22020073,(W1,1,H1))...)) -->
(_,_,(...(W2,22020072,(W1,0,H1))...)) -->
(_,_,(...(W2,22020071,H5)...)) -->
(_,_,(...(W2,22020070,H1)...)) --> ...
(_,_,(...(W2,4,H1)...)) -->
(_,_,(...(W2,2,H1)...)) -->
(_,_,(...(W2,0,H1)...)) -->
(_,_,(...H3...)) -->
halt

In each iteration of (a,b,x) --> (a',b',x'), a'≈2^a. It halts after 11010064 iterations of (a,b,x) --> (a',b',x'). The omitted numbers in x and b are much smaller than a, and much larger than 66060289.

The halting config (reversed):

H3 = 0^inf 1 Z> 00110 110110^2
(W1,n,l) = l 110110^n 11 110110^2
(W2,n,l) = l 110110^n 1111 110110^3

(a,b,x) = x 110110^b 11^1+a 0^inf

where a is between 10↑↑11010000 and 10↑↑11011000.

The first 6 TMs in this family have exactly the same halting config, while 1RB1LE_1LC0RA_1RB1LD_1LC0LC_1RF0LB_---1RE has H3 = 0^inf 1 Z> 11110 110110^2, which results in a sigma score increase of 2.

We can see by looking at the space-time diagram that 1RB1LC_1LA1RE_0RD0LA_---1LB_1LD0RF_0RD1RB takes the longest time to halt.