BB(7)

The 7-state, 2-symbol Busy Beaver problem, BB(7), refers to the unsolved 7^th value of the Busy Beaver function. With the compilation of the Cryptid machine Bigfoot into a 7-state, 2-symbol machine in May 2024, we now know that we must solve a Collatz-like problem in order to solve BB(7).

The current BB(7) champion 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch) was discovered by Pavel Kropitz in May 2025, proving the lower bound:

$$S(7)>\Sigma (7)>2\uparrow ^{11}2\uparrow ^{11}3$$

History

Before 2025, the only known BB(7) champions were produced by hand, not by search. In 1964, Milton Green designed a machine that had sigma score 22,961. In 2014, Wythagoras modified a BB(6) champion to produce a machine that had sigma score $>10\uparrow \uparrow 5$.

In May 2025, mxdys shared C++ code that breaks up the BB(7) enumeration into 1 million subtasks which each run for about 2 minutes and leave ~100 holdouts each. Various folks on Discord have been investigating different sections of this domain to search for champions.

Within three days of the code's release, the Ligockis found three champions after applying their deciders to enumerator output. Shawn Ligocki found the first two, 1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB (bbch) and 1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC (bbch), with sigma scores of approximately 10 ↑↑ 22 and 10 ↑↑ 35. That evening, Terry Ligocki found 1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB (bbch), with sigma score ~10 ↑↑ 46. A few days later, Pavel found a TM that outpaces all of them.

Pavel's champion is enumerated in subtask 243308.

Cryptids

BB(7) has not been seriously investigated by hand, so no native BB(7) Cryptids have yet been discovered.

Probviously non-halting Cryptids:

0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB (bbch), Bigfoot (a BB(3,3) Cryptid) compiled into a 2-symbol TM by Iijil in 2024.

Top Halters

The top 20 scoring known machines are:

TM	Approximate sigma score	Discoverer
1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)	$2\uparrow ^{11}2\uparrow ^{11}3$	Pavel Kropitz
1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE (bbch)	10 ↑↑ 519.20	Andrew Ducharme
1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE (bbch)	10 ↑↑ 519.20	Andrew Ducharme
1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE (bbch)	10 ↑↑ 519.20	@gerbil5709, Terry Ligocki
1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ (bbch)	10 ↑↑ 286.17	Terry Ligocki
1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD (bbch)	10 ↑↑ 246.32	@Iijil
1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA (bbch)	10 ↑↑ 246.32	@star, Terry Ligocki
1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG (bbch)	10 ↑↑ 243.88	@Iijil, Andrew Ducharme
1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ (bbch)	10 ↑↑ 228.78	Terry Ligocki
1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_0RG0LA_1RB1RZ (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_1RG0LA_0LE1RZ (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RG0LG_1RB1LG (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB1RZ_1LC0RE_0LD1LB_1RE0LG_1RF0RG_0RA0LG_1RB1LG (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RC0LD (bbch)	10 ↑↑ 192.67	Andrew Ducharme
1RB1LA_1LC0RE_0LD1LB_1RE1LG_1RF0LG_0RA0LA_0RF1RZ (bbch)	10 ↑↑ 192.67	Andrew Ducharme
1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA (bbch)	10 ↑↑ 192.67	@C7X
1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ (bbch)	10 ↑↑ 192.67	@Iijil, Terry Ligocki
1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_1RG0LA_0LE1RZ (bbch)	10 ↑↑ 192.67	@Iijil, Terry Ligocki

The top 20 known halters with unique scores are:

TM	Approximate sigma score	Discoverer
1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)	$2\uparrow ^{11}2\uparrow ^{11}3$	Pavel Kropitz
1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE (bbch)	10 ↑↑ 519.20	Andrew Ducharme
1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ (bbch)	10 ↑↑ 286.17	Terry Ligocki
1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD (bbch)	10 ↑↑ 246.32	@Iijil
1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG (bbch)	10 ↑↑ 243.88	@Iijil, Andrew Ducharme
1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ (bbch)	10 ↑↑ 228.78	Terry Ligocki
1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG (bbch)	10 ↑↑ 192.67	Terry Ligocki
1RB0LC_1LC1LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LA (bbch)	10 ↑↑ 188.28	Terry Ligocki
1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC (bbch)	10 ↑↑ 140.28	@stokastic
1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA (bbch)	10 ↑↑ 136.64	Katelyn Doucette, Andrew Ducharme
1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG (bbch)	10 ↑↑ 133.85	@poppuncher
1RB1RZ_1RC0RF_1LD1RB_1RG0LE_1LD0RA_1RE0LD_0RC1LF (bbch)	10 ↑↑ 129.24	@Iijil
1RB0RG_1LC0RE_1LF1LD_0LE1LC_1RA1RB_1LD0LF_1RZ0RF (bbch)	10 ↑↑ 127.52	Andrew Ducharme
1RB0LC_1RC0RG_1RD0LF_1RE0RF_1LA1RG_1LE1LF_1RZ1RD (bbch)	10 ↑↑ 126.20	@stokastic
1RB0LD_1RC1RA_0RD1RG_1LE1LF_0LF1LE_0RA0LD_0RA1RZ (bbch)	10 ↑↑ 124.86	Terry Ligocki
1RB1LF_1RC1RA_1LD0LD_1LA1LE_0LA0LD_1LG0RF_0LE1RZ (bbch)	10 ↑↑ 116.98	Terry Ligocki
1RB0RD_1RC0LA_0LA0LE_1RE1RZ_1RF0RA_1LG0LE_1LC0LG (bbch)	10 ↑↑ 116.05
1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ (bbch)	10 ↑↑ 115.52	@prurq
1RB0RG_1LC0LE_1LD0LB_0LE1RE_0RA1RF_0RD1RC_1RD1RZ (bbch)	10 ↑↑ 114.83	Andrew Ducharme
1RB0RE_1LC0LA_1LD0LC_0LE0LA_1RF0RG_1RD0LE_1RA1RZ (bbch)	10 ↑↑ 114.60	Shawn Ligocki

Phase 1

Phase 1's first two stages were carried out from May 2025 to July 2025. The table below summaries some of that activity and was used to coordinate the effort of doing the raw computation. Fourteen people (see below) contributed directly and the overall bbchallenge group participated in discord discussions, etc.

Stage 1, "enumeration" in the table, involved running 1 million subtasks using mxdys's code in 100 batches of 10 thousand subtasks.

Stage 2, "linear rule" in the table, processing the output of Stage 1 using Shawn Ligocki's linear rule code which also used a few other deciders, e.g., a version of CTL.

Stage 3, the final stage of Phase 1, is the compilation, verification, and presentation of the final results from Phase 1. It was completed during August 2025.

(done to reduce column size: $*^{1}$= enumeration, $*^{2}$= linear rule)

Task range	Done by	Completed		# holdouts	Maximum Score TM	~Sigma	Source
		$*^{1}$	$*^{2}$
00-01xxxx	@Iijil	Yes	Yes	1,545,673	1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD (bbch)	10 ↑↑ 246.32	Google Drive folder
02-04xxxx	@Iijil Terry Ligocki	Yes	Yes	2,279,734	1RB0LF_1RC1RA_1RD0RG_1LE1RZ_1LA0LF_1RA1LE_0RE1RG (bbch)	10 ↑↑ 93.81	@Iijil Terry Ligocki
05-09xxxx	@Iijil Andrew Ducharme	Yes	Yes	3,889,955	1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG (bbch)	10 ↑↑ 243.88	@Iijil Andrew
10-12xxxx	Andrew Ducharme	Yes	Yes	2,708,888	1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE (bbch)	10 ↑↑ 519.20	Google Drive folder
13xxxx	Shawn Ligocki	Yes	Yes	1,192,442	1RB0RE_1LC0LA_1LD0LC_0LE0LA_1RF0RG_1RD0LE_1RA1RZ (bbch)	10 ↑↑ 114.60	Google Drive folder
14-16xxxx	Andrew Ducharme	Yes	Yes	2,701,637	1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0LD_0RA1RE_1RZ1LA (bbch)	10 ↑↑ 188.28	Google Drive folder
17-18xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	1,898,156	1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ (bbch)	10 ↑↑ 192.67	@gerbil5709 Terry Ligocki
19xxxx	Katelyn Doucette Andrew Ducharme	Yes	Yes	1,099,752	1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA (bbch)	10 ↑↑ 136.64	Google Drive folder
20-23xxxx	@C7X	Yes	Yes	4,528,827	1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA (bbch)	10 ↑↑ 192.67	Google Drive folder
24xxxx	Andrew Ducharme	Yes	Yes	712,356	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)*	$2\uparrow ^{11}2\uparrow ^{11}3^{*}$	Google Drive folder
25-34xxxx	@stokastic	Yes	Yes	10,339,816	1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC (bbch)	10 ↑↑ 140.28	Google Drive folder
35-39xxxx	Terry Ligocki	Yes	Yes	4,894,047	1RB1RZ_1LC0RF_0LD1LB_1RD0LE_1RB1LE_1RG0RE_0RA0LE (bbch)	10 ↑↑ 192.67	Google Drive folder
40-47xxxx	Andrew Ducharme	Yes	Yes	6,181,327	1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE (bbch)	10 ↑↑ 519.20	Google Drive folder
48xxxx	@star Terry Ligocki	Yes	Yes	727,875	1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA (bbch)	10 ↑↑ 246.32	@star Terry Ligocki
49xxxx	Tobiáš Brichta Terry Ligocki	Yes	Yes	804,722	1RB0LG_1RC0RG_0LD1RE_1RD0RE_1LF1RB_0LA1RZ_1LC1LG (bbch)	10 ↑↑ 126.20	Tobiáš Brichta Terry Ligocki
50xxxx	@prurq Andrew Ducharme	Yes	Yes	797,224	1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ (bbch)	10 ↑↑ 115.52	Google Drive folder
51-53xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	3,016,175	1RB0LC_1LC0LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LC (bbch)	10 ↑↑ 140.28	@gerbil5709 Terry Ligocki
54-59xxxx	Terry Ligocki	Yes	Yes	5,689,850	1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LA (bbch)	10 ↑↑ 188.28	Google Drive folder
60-64xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	3,817,876	1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE (bbch)	10 ↑↑ 519.20	@gerbil5709 Terry Ligocki
65-68xxxx	Terry Ligocki	Yes	Yes	3,076,778	1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD (bbch)	10 ↑↑ 192.67	Google Drive folder
69xxxx	@poppuncher	Yes	Yes	1,053,119	1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG (bbch)	10 ↑↑ 133.85	Google Drive folder
70-71xxxx	@hipparcos	Yes	Yes	1,899,094	1RB1RZ_1LC1RD_0LD0LC_1LE1RA_1LF0LE_1RF0RG_1RG0RD (bbch)	10 ↑↑ 77.50	Github release
72-79xxxx	Terry Ligocki	Yes	Yes	7,627,514	1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ (bbch)	10 ↑↑ 228.78	Google Drive folder
80-81xxxx	XnoobSpeakable	Yes	Yes	1,537,533	1RB0LA_0RC1RZ_0RD0RG_1LE1RA_1LF1LD_1RG0RG_1RD1RC (bbch)	10 ↑↑ 74.85	Google Drive folder
82-99xxxx	Terry Ligocki	Yes	Yes	15,673,786	1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ (bbch)	10 ↑↑ 286.17	Google Drive folder

*The current BB(7) champion TM 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)* was discovered by Pavel Kropitz in the enumeration of subtask 243308. The remaining subtasks in the 24xxxx range were enumerated and filtered by Andrew Ducharme.

Exploration after Phase 1

People are now looking at reducing the number of holdouts (~85M TMs) after Phase 1. They are trying different deciders not used in Phase 1 and different parameters with the deciders used in Phase 1. This table is somewhere they can put the results of these explorations so that there is a record of what is being done. It is hoped that this will inspire everyone to contribute when they can, a reference point for discord discussions, and something that can be questioned and/or verified.

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

Done by	Holdout TMs		$*^{1}$	$*^{2}$	TMs/sec/core		Description	Source
	Input	Output			$*^{3}$	$*^{4}$
Shawn Ligocki	858,538	733,830	14.5%	5.0	6.9282	47.6966	Translated Cycler and CPS on 7x2_p01_s02_holdouts_rand_13.txt	Discord link
Andrew Ducharme	872,041	784,099	10.1%	456.4	0.0535	0.5034	Enumerate.py w/ 250k max-loops and block-mult=3 on 7x2_p01_s02_holdouts_rand_72.txt	Discord link
Terry Ligocki	1,000	982	1.8%	3.3	0.0015	0.0830	MITMWFAR Code with options -n=10 -m=1 -pm=1 on the first 1,000 TMs in 7x2_p01_s03_holdouts_rand_47.txt
Andrew Ducharme	800,507	693,348	13.3%	2818.7	0.0106	0.0789	Enumerate.py w/ 1M max-loops and block-mult=4 on holdouts from above Enumerate run on7x2_p01_s02_holdouts_rand_72.txt
Andrew Ducharme	70,000	65,615	6.3%	6.6	0.1842	2.9412	Enumerate.py with --no-sim and --lin-steps=100000 on 7x2_p01_s02_holdouts_rand_65.txt
Terry Ligocki	10,000	N/A	2% - 32%	N/A	N/A	N/A	See the "Source" discussion on the discord channel which includes graphs with quantitative data. A parameter study of random sample of 10,000 TMs from the Phase 1, Stage 3 holdouts	Discord link

Phase 2

Phase 2 has now begun with Andrew Ducharme's initial decider runs (see table below). The goal of Phase 2 is to run deciders on the current holdout list (the last in the table) and produce an accessible new holdout list which can then be added to the table. Input Holdout TMs may be unequal to the next row's Output Holdout TMs because more TMs are TNF-enumerated in the process of executing the next row's filtering. Accessing the holdout list can be done through the link in the "Source" column. If there isn't a link there it means the data is still being put somewhere everyone can access it.

(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	Source	Data
	Input	Output			$*^{3}$	$*^{4}$
Andrew Ducharme	86,129,304	82,226,951	4.5%	119.6	9.06	200.04	CPS_Filter with --max-block-size=4	discord	Google Drive
Andrew Ducharme	82,226,951	73,751,624	10.3%	120.0	19.62	190.34	lr_enum_continue 1M steps	discord
Andrew Ducharme	73,751,624	72,470,054	1.7%	1040.0	0.34	19.70	lr_enum_continue 3M steps	discord
Andrew Ducharme	72,470,054	69,347,610	4.3%	1113.4	0.78	18.08	CPS_Filter with --min-block-size=5, --max-block-size=6	discord
Andrew Ducharme	69,347,610	68,695,205	0.9%	1491.8	0.12	12.92	CPS_Filter with --block-size=7	discord
Andrew Ducharme	69,180,361	61,875,401	10.6%	1726.7	1.17	11.11	Enumerate.py with --block-multiple=12, max-loops=100_000, and --time=0.1	discord
Andrew Ducharme	61,891,137	60,986,231	1.5%	1660.7	0.15	10.35	Enumerate.py with --block-multiple=8, max-loops=100_000, and --time=0.1	discord
Andrew Ducharme	60,988,359	60,765,943	0.4%	1619.4	0.04	10.46	Enumerate.py with --block-multiple=16, max-loops=100_000, and --time=0.1	discord
Andrew Ducharme	60,765,943	59,727,905	1.7%	2329.4	0.12	7.25	CPS_Filter with --block-size=8	discord
Andrew Ducharme	59,727,905	57,452,672	3.9%	2472.2	0.26	6.71	Enumerate.py with --block-multiple=5, max-loops=200_000, and time=0.2	discord
Cumulative	86,129,304	57,452,672	33.3%	---	---	---	---	---
Andrew Ducharme	57,452,672	52,605,872	8.4%	150.0	8.98	106.39	chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3	discord
Andrew Ducharme	52,605,872	50,268,427	4.4%	100.0	6.49	146.13	chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3	discord
Andrew Ducharme	50,268,427	45,980,438	8.5%	250.0			MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8	discord
Andrew Ducharme	45,980,438	43,870,806	4.5%	220.0			MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6	discord
Andrew Ducharme	43,870,806	42,700,370	2.6%	80.0			chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2	discord
Andrew Ducharme	42,700,370	41,926,200	1.8%	20.0			MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1	discord
Andrew Ducharme	41,926,200	41,590,605	0.8%	50.0			MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6	discord
Andrew Ducharme	41,590,605	40,481,477	2.6%	75.0			chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2	discord
Andrew Ducharme	40,481,477	38,641,627	4.5%	200.0			MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12	discord
Andrew Ducharme	38,641,627	37,514,197	2.9%	80.0			chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5
Cumulative	86,129,304	37,514,197	56.4%	---	---	---	---	---