1RB1RF 0LC1RC 1RD1LC 1RZ0RE 1RA1LF 1RA0LE

1RB1RF_0LC1RC_1RD1LC_1RZ0RE_1RA1LF_1RA0LE (bbch) is a halting BB(6) TM. It halts with sigma score 4,419,340,317.

Analysis by Shawn Ligocki

https://discord.com/channels/960643023006490684/1239205785913790465/1430590536825442384

1RB1RF_0LC1RC_1RD1LC_---0RE_1RA1LF_1RA0LE

A> 10 -> 11 A>
0 1^n A> 00 -> 11 A> 1^n 0   for n >= 1
0 1^2k+3 A> 11 -> 1^4 0 1^2k+1 A>
0 1 A> 1^2 0 -> 1^5 Z>  (Halt)
0 1 A> 1^3 0 -> 1^4 0 1 A>
0 1 A> 1^4   -> 1^5 A> 1
0 1^2k A> 11 -> 1^2k+3 A>


A(a,b,c) = 0^inf 1^a 0 1^b A> 1^c 0^inf

A(a,b,1) -> A(a,b+2,0)
A(a,b,0) -> A(0,a+2,b)
A(a,2k+3,c+2) -> A(a+4,2k+1,c)
A(a,1,2) -> Halt(a+5)
A(a,1,3) -> A(a+4,1,0)
A(a,1,c+4) -> A(0,a+5,c+1)
A(a,2k,c+2) -> A(0,a+2k+3,c)


if b >= c:
  A(0, 2n+1, 2m)   -> A(0, 4m+5, 2(n-m)-1)   if n >= m + 1
  A(0, 2n+1, 2m+1) -> A(0, 4m+5, 2(n-m)+1)   if n >= m
  A(0, 2n+1, 2n)   -> A(0, 5, 4n+2)
if b < c:
  A(0, 2n+1, c) -> A(0, 4n+5, c-2n-3)   if c >= 2n + 4
  A(0, 2n+1, 2n+3) -> A(0, 5, 4n+6)
  A(0, 2n+1, 2n+2) -> Halt(4n+5)

Start @step 16: A(0, 5, 0)

@Bricks noticed that this halts after 1.8B rule steps with score 4,419,340,317: https://discord.com/channels/960643023006490684/1239205785913790465/1430668239158907085

B(n, c) = A(0, 2n+1, c)

            0  :  B(             2,              0)  (0.03s)
  100_000_000  :  B(   104_768_252,     23_779_441)  (16.28s)
  200_000_000  :  B(    20_853_246,    424_931_081)  (32.53s)
  300_000_000  :  B(    73_587_498,    552_796_212)  (48.77s)
  400_000_000  :  B(   339_264_902,    254_785_766)  (65.03s)
  500_000_000  :  B(    35_411_234,  1_095_798_412)  (81.49s)
  600_000_000  :  B(   146_681_726,  1_106_587_498)  (98.76s)
  700_000_000  :  B(   225_344_990,  1_182_601_470)  (116.50s)
  800_000_000  :  B(   731_823_582,    402_997_335)  (134.49s)
  900_000_000  :  B(   726_356_142,    647_251_151)  (152.53s)
1_000_000_000  :  B(   125_259_918,  2_082_755_014)  (170.74s)
1_100_000_000  :  B( 1_249_366_982,     67_881_740)  (189.27s)
1_200_000_000  :  B(   569_963_118,  1_660_048_591)  (208.10s)
1_300_000_000  :  B( 1_441_422_382,    150_484_951)  (226.80s)
1_400_000_000  :  B(     5_449_630,  3_255_763_599)  (245.36s)
1_500_000_000  :  B(   302_635_372,  2_894_710_243)  (264.06s)
1_600_000_000  :  B(   384_944_862,  2_963_428_001)  (283.06s)
1_700_000_000  :  B(   558_908_586,  2_848_805_949)  (302.31s)
1_800_000_000  :  B( 1_914_732_464,    370_483_439)  (321.60s)

Halted after 1_894_017_994 rule steps with score 4_419_340_317

Quick_Sim confirms that it halts (after over 24hr!):

Code/Quick_Sim.py 1RB1RF_0LC1RC_1RD1LC_---0RE_1RA1LF_1RA0LE --no-steps --freeze-prover --print-loops=1_000_000
...

        Steps:                     Times Applied:

Total:   0                             35039235619
Macro:   N/A                           23359504647
Chain:   N/A                            9785713060
Rule:    N/A                            1894017912
Rules proven: 4
Collatz rules: 0
Failed proofs: 0
Prover num past configs: 0
Tape copies: 4261543974
Elapsed time: 105504.40982294083
00^inf 01^1 11^2_209_670_157 (11) Z> 00^inf
Num Nonzeros: 4_419_340_317  =  4_419_340_317
Turing Machine reached Undefined transition
State:  3
Symbol: 0
Nonzeros: 4_419_340_317  =  4_419_340_317