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

Phase 1

The initial phase of enumeration and reduction of holdouts took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~663B(illion) TMs of which ~34B were holdout TMs. This was reduced to ~461M holdout TMs (a 98.66% reduction). The details will be given in this table:

(done to reduce column size: $*^{1}$= % Reduced, $*^{2}$= Runtime (hours), $*^{3}$= Decided, $*^{4}$= Processed)

Done by	Holdout TMs		$*^{1}$	$*^{2}$	TMs/sec/core		Description	Data
	Input	Output			$*^{3}$	$*^{4}$
---	---	---	---	---	---	---	---

Phase 2

In this phase, Terry Ligocki used mxdys' C++ code, main.cpp, to find and apply deciders/parameters to the holdouts from Phase 1. This took place in September 2025. The initial ~461M holdout TMs were reduced to ~98M (a 78.8% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(done to reduce column size: $*^{1}$= % Reduced, $*^{2}$= Compute Time (core-hours), $*^{3}$= Decided, $*^{4}$= Processed)

Done by	Holdout TMs		$*^{1}$	$*^{2}$	TMs/sec/core		Description	Data
	Input	Output			$*^{3}$	$*^{4}$
Terry Ligocki	460,916,384	114,288,393	75.20%	30.3	3,181.42	4,230.39	chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 1000 NG_n 8 run	Google Drive
Terry Ligocki	114,288,393	105,922,759	7.32%	7.4	315.37	4,308.43	MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 5 n 3 run
Terry Ligocki	105,922,759	98,061,985	7.42%	7.4	296.09	3,989.74	MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 0 tH 2 n 2 run
Terry Ligocki	98,061,985	97,770,372	0.30%	1.5	54.14	18,206.56	chr_asth 0 chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 1000 NG_n 6 run
Terry Ligocki	97,770,372	97,705,106	0.07%	1.7	10.45	15,652.68	MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 2 tH 0 n 6 run
Terry Ligocki	97,705,106	97,701,052	0.00%	1.0	1.09	26,353.22	MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 2 tH 0 n 5 run