BB(4,3)

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of Cryptids in the domain is given by the discovery of Bigfoot in BB(3,3). The current champion appears to be 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch) which was discovered by Pavel Kropitz in May 2024 and analyzed by Racheline in Feb 2025, demonstrating the lower bounds:

$$S(4,3)>\Sigma (4,3)>2\uparrow \uparrow \uparrow 2^{2^{32}}$$

The exact status of championship currently remains unclear because of this list of "Potential Champions" below which has not yet been fully investigated.

Top Halters

The top 20 longest running halting BB(4,3) TMs are:

Standard format	(approximate) runtime
0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)	$2\uparrow \uparrow \uparrow 2^{2^{32}}$

Potential Champions

In May 2024, Pavel Kropitz found 7 halting TMs that run for a large number of steps, but have not been analyzed in detail:

https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120

Current champion and equivalent TMs:

• The current champion 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch): 1 2^((80*2^((<(8*2^((8*2^(29) - 2)) - 5); (<(80*2^((b - 10)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4); (<(80*2^((<(80*2^((8*2^((8*2^(29) - 2)) - 3)) - 13)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> - 6)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4)> - 10)/5) - 3)) 1 0 1 2 1^2 Z> 1 2^2 1
• 0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch): 1 2^((80*2^((<(8*2^((8*2^(29) - 2)) - 5); (<(80*2^((b - 10)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4); (<(80*2^((<(80*2^((8*2^((8*2^(29) - 2)) - 3)) - 13)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> - 6)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4)> - 10)/5) - 3)) 1 0 1 2 1^2 Z> 1 2^2 1
• 1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA (bbch): 1 2^((80*2^((<(8*2^((8*2^(29) - 2)) - 5); (<(80*2^((b - 10)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4); (<(80*2^((<(80*2^((8*2^((8*2^(29) - 2)) - 3)) - 13)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> - 6)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4)> - 10)/5) - 3)) 1 0 1 2 1^2 Z> 1 2^2 1
• 1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB (bbch): 1 2^((80*2^((<(8*2^((8*2^(29) - 2)) - 5); (<(80*2^((b - 10)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4); (<(80*2^((<(80*2^((8*2^((8*2^(29) - 2)) - 3)) - 13)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> - 6)/5) - 17)/9; (40*2^((8*2^((a - 11)/5) - 2)) - 4); (40*2^(2) - 4)> + 4)> - 10)/5) - 3)) 1 0 1 2 1^2 Z> 1 2^2 1

Others:

• 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD (bbch): 1 Z> 1^((8*<7; (6*2^((4b + 14)) - 4); (6*2^((48*2^(21) - 2)) - 4)> + 33)) 2
• 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch): 1 Z> 1^((2*<(<(<(16*2^(92) - 3); (24*2^((24*2^(<(b + 10); (24*2^(b) - 4); 2>) - 3)) - 11); (24*2^((24*2^(<(24*2^((24*2^(<(24*2^((24*2^(92) - 3)) - 2); (24*2^(b) - 4); 92>) - 3)) - 1); (24*2^(b) - 4); 2>) - 3)) - 11)> + 8)/3; (24*2^((24*2^(<(b + 10); (24*2^(b) - 4); 2>) - 3)) - 11); (24*2^((24*2^(<1; (24*2^(b) - 4); 2>) - 3)) - 11)> + 5)/3; (24*2^((24*2^(<(b + 10); (24*2^(b) - 4); 2>) - 3)) - 11); (24*2^((24*2^(<1; (24*2^(b) - 4); 2>) - 3)) - 11)> + 19))
• 1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD (bbch): 1 Z> 1^(162*3^((3*<(243*3^(6) - 5)/2; (<(54*3^((3b + 11)/2) - 2); (54*3^((3b + 14)/2) - 6); (54*3^(7) - 6)> + 1); (<(54*3^((3*<(54*3^(7) - 3); (54*3^((3b + 14)/2) - 6); (54*3^((81*3^(7) - 2)) - 6)> + 14)/2) - 2); (54*3^((3b + 14)/2) - 6); (54*3^(7) - 6)> + 1)> + 11)/2)) 2