BB(2,6)

The 2-state, 6-symbol Busy Beaver problem, BB(2,6), is unsolved. With cryptids like Hydra in the preceding domain BB(2,5), we know that we must solve a Collatz-like problem in order to solve BB(2,6).

The current BB(2,6) champion 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch) was discovered by Pavel Kropitz in May 2023, proving the lower bound:

$$S(2,6)>\Sigma (2,6)>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}>10\uparrow \uparrow \uparrow 3$$

Top Halters

The highest known scoring machines are:

TM	Approximate sigma score	Discoverer
1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)	10 ↑↑↑ 3	Pavel Kropitz
1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ (bbch)	10 ↑↑ 91	Pavel Kropitz
1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA (bbch)	10 ↑↑ 70	Shawn Ligocki
1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA (bbch)	10 ↑↑ 69.68	Andrew Ducharme
1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ (bbch)	10 ↑↑ 54.90	Andrew Ducharme*
1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB (bbch)	10 ↑↑ 42.17	Andrew Ducharme*
1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB (bbch)	10 ↑↑ 40.07	Andrew Ducharme*
1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ (bbch)	10 ↑↑ 21.54	Andrew Ducharme*
1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA (bbch)	10 ↑↑ 20.58	Shawn Ligocki
1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ (bbch)	10 ↑↑ 17.53	Andrew Ducharme*

All decimal places are truncated. Discoverers are asterisked where it is unclear if the TM had been found but unreported by someone previously (namely Shawn Ligocki).

Filtering

Starting from Terry Ligocki's holdouts list of 22,302,296 TMs, additional filtering has been performed:

(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			$*^{3}$	$*^{4}$
Andrew Ducharme	22,302,296	20,778,101	6.8%	274.6		22.56	Enumerate.py with --no-sim and --lin-steps=10_000	Google Drive
Andrew Ducharme	20,778,101	19,257,876	7.3%	200.0		28.00	lr_enum_continue 1M steps
Andrew Ducharme	19,280,508	19,004,377	1.4%	2,174.6		2.46	Enumerate.py with --block-multiple=1, max-loops=20_000, and --time=1
Andrew Ducharme	19,005,529	18,952,159	0.3%	1,952.7		2.70	Enumerate.py with --block-multiple=2, max-loops=20_000, and --time=120
Andrew Ducharme	18,952,159	18,054,938	4.7%	4,168.4		1.26	CPS_Filter with --block-size=7
Andrew Ducharme	18,068,066	17,996,475	0.4%	1,100.0		4.50	lr_enum_continue 10M steps
Andrew Ducharme	17,999,451	17,629,828	2.1%	1,610.6		0.31	Enumerate.py with --block-multiple=8, max-loops=100_000, and --time=0.45
Terry Ligocki	17,629,828	17,224,474	2.3%	18.7	6.01	261.45	MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
Terry Ligocki	17,224,474	16,824,222	2.3%	71.1	1.56	67.28	MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
Terry Ligocki	16,824,222	4,434,034	73.6%	41.1	83.71	113.67	chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
Terry Ligocki	4434034	2,715,631	38.8%	13.3	35.92	92.67	chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
Terry Ligocki	2,715,631	2,642,604	2.7%	12.7	1.59	59.22	MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
Terry Ligocki	2,642,604	2,606,842	1.4%	12.8	0.78	57.29	chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
Terry Ligocki	2,606,842	2,527,197	3.1%	11.0	2.01	65.89	MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
Terry Ligocki	2,527,197	1,785,192	29.4%	10.6	19.51	66.45	chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
Terry Ligocki	1,785,192	1,690,054	5.3%	7.4	3.55	66.67	MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
Terry Ligocki	1,690,054	1,505,268	10.9%	7.4	6.90	63.09	MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
Terry Ligocki	1,505,268	1,460,289	3.0%	7.4	1.68	56.24	chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
Terry Ligocki	1,460,289	1,442,538	1.2%	8.5	0.58	47.78	MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
Terry Ligocki	1,442,538	1,416,921	1.8%	7.4	0.96	53.83	chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
Terry Ligocki	1,416,921	1,300,334	8.2%	8.2	3.96	48.11	MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run

A far more efficient pipeline would immediately apply lr_enum_continue out to 1M steps to Terry Ligocki's holdout list. lr_enum_continue, written in C++, is about 400x faster than Enumerate.py at checking for Lin Recursion. Using Enumerate.py meant its Reverse Engineering decider was applied to all holdouts, and solved 74,089 TMs (0.33% of holdouts)...at the cost of roughly 274.1 hours of compute.