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 {{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}} was discovered by Pavel Kropitz in May 2023, proving the lower bound:<math display="block">S(2,6) > \Sigma(2,6) > 10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}} > 10 \uparrow \uparrow \uparrow 3</math>

== Top Halters ==
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The 20 highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|10 ↑↑ 19892.08
|Peacemaker II
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB2LA5LB0RA1RA3LB_1LA4LA3LB3RB3RB1RZ|halt}}
|10 ↑↑ 23.9964
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|}
All decimal places are truncated.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in November 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~24B(illion) TMs of which ~2.278B were holdout TMs. This was reduced to ~22M holdout TMs (a 99.02% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
|style="text-align:left" |Terry Ligocki
|2,278,655,696
|2,109,114,609
|7.44%
|40.9
|1,150.90
|15,468.23
|style="text-align:left" |Reverse_Engineer_Filter.py
|style="text-align:left", rowspan="100" |[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|2,109,114,609
|683,067,538
|67.61%
|452.8
|874.77
|1,293.79
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|683,067,538
|210,993,434
|69.11%
|396.4
|330.85
|478.72
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|210,993,434
|141,680,232
|32.85%
|273.9
|70.29
|213.97
|style="text-align:left" |CPS_Filter.py --block-size=3 --max_steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|141,680,232
|66,029,536
|53.40%
|486.6
|43.18
|80.87
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|66,029,536
|46,119,004
|30.15%
|167.4
|33.05
|109.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|46,119,004
|39,034,142
|15.36%
|170.1
|11.57
|75.34
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|39,034,142
|29,109,512
|25.43%
|2,221.6
|1.24
|4.88
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|29,109,512
|24,536,819
|15.71%
|384.2
|3.31
|21.05
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=6 --recursive --no-steps --time=0.05 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|24,536,819
|22,302,296
|9.11%
|1,047.5
|0.59
|6.51
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=4 --recursive --no-steps --time=1.00 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|}

== Phase 2 ==
When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===
Andrew Ducharme ran another pass of "lr_enum_continue" with the maximum number of steps set to 10 million. The holdouts were reduced from ~22.3M TMs to ~20.4M TMs (a 8.72% reduction). The entry in the table below has a rather technical/arcane/cryptic description. This was an effort to capture enough information to rerun that filter in parallel with specific C++ code, lr_enum_continue, and a specific parallel queuing system, Slurm:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
|style="text-align:left" |Andrew Ducharme
|22,302,296
|20,358,011
|8.72%
|1,350.0
|0.40
|4.59
|style="text-align:left" |lr_enum_continue ${WORK_DIR}chunk_${SLURM_ARRAY_TASK_ID} 10000000 ${WORK_DIR}halt_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}inf_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}unknown_${SLURM_ARRAY_TASK_ID}.txt "" false
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|}

=== Stage 2 ===
Starting from the results of Stage 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 50 variations were run. The holdouts were reduced from ~20.4M TMs to ~907K TMs (a 95.5% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|20,358,011
|19,500,847
|4.21%
|22.0
|10.84
|257.42
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|19,500,847
|18,747,861
|3.86%
|86.0
|2.43
|63.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|18,747,861
|4,811,076
|74.34%
|47.0
|82.33
|110.75
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|4,811,076
|2,982,075
|38.02%
|17.1
|29.74
|78.22
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|2,982,075
|2,897,340
|2.84%
|15.2
|1.55
|54.64
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,897,340
|2,850,781
|1.61%
|16.7
|0.77
|48.17
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|2,850,781
|2,759,635
|3.20%
|13.7
|1.85
|58.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,759,635
|1,953,426
|29.21%
|13.6
|16.48
|56.42
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,953,426
|1,855,545
|5.01%
|2.4
|11.18
|223.14
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,855,545
|1,647,269
|11.22%
|6.6
|8.80
|78.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,647,269
|1,608,166
|2.37%
|3.4
|3.20
|134.96
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,608,166
|1,585,745
|1.39%
|9.6
|0.65
|46.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,585,745
|1,555,673
|1.90%
|5.7
|1.47
|77.73
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,555,673
|1,428,534
|8.17%
|9.3
|3.78
|46.31
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,428,534
|1,340,964
|6.13%
|0.8
|29.70
|484.55
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 10000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,340,964
|1,286,439
|4.07%
|0.8
|18.40
|452.56
|style="text-align:left" |chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,286,439
|1,273,911
|0.97%
|0.8
|4.20
|430.88
|style="text-align:left" |chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,273,911
|1,265,198
|0.68%
|0.8
|2.88
|420.73
|style="text-align:left" |chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,265,198
|1,258,925
|0.50%
|0.9
|1.99
|400.83
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,258,925
|1,242,136
|1.33%
|0.8
|5.51
|412.84
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,242,136
|1,231,731
|0.84%
|1.0
|2.78
|331.77
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,231,731
|1,216,646
|1.22%
|1.0
|4.15
|338.72
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,216,646
|1,214,294
|0.19%
|0.9
|0.76
|393.03
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,214,294
|1,213,431
|0.07%
|0.9
|0.28
|391.30
|style="text-align:left" |chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,213,431
|1,211,390
|0.17%
|1.1
|0.52
|307.13
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 1 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,211,390
|1,209,989
|0.12%
|1.1
|0.35
|306.09
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,989
|1,209,974
|0.00%
|0.9
|0.00
|381.42
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,974
|1,201,890
|0.67%
|2.5
|0.90
|134.19
|style="text-align:left" |chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,201,890
|1,200,086
|0.15%
|1.3
|0.37
|248.36
|style="text-align:left" |chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,200,086
|1,199,734
|0.03%
|1.2
|0.08
|270.32
|style="text-align:left" |chr_asth 0 chr_LRUH 3 chr_H 3 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,199,734
|1,198,893
|0.07%
|2.3
|0.10
|147.66
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 6 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,198,893
|1,165,493
|2.79%
|4.5
|2.05
|73.44
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 4 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,165,493
|1,153,863
|1.00%
|9.3
|0.35
|34.88
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 0 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,153,863
|1,144,711
|0.79%
|3.7
|0.69
|87.51
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 5 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,144,711
|1,127,789
|1.48%
|7.9
|0.60
|40.26
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 30000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,127,789
|1,124,762
|0.27%
|4.7
|0.18
|66.75
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|1,124,762
|1,117,226
|0.67%
|5.6
|0.37
|55.36
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,117,226
|1,109,057
|0.73%
|7.7
|0.30
|40.49
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,109,057
|1,083,097
|2.34%
|11.4
|0.63
|27.06
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 30000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,083,097
|1,077,833
|0.49%
|11.2
|0.13
|26.81
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,077,833
|1,066,795
|1.02%
|24.1
|0.13
|12.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 2 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,066,795
|1,039,229
|2.58%
|52.6
|0.15
|5.64
|style="text-align:left" |chr_LRUH 14 chr_H 6 MitM_CTL NG maxT 100000 NG_n 11 run
|-
|style="text-align:left" |Terry Ligocki
|1,039,229
|1,019,286
|1.92%
|43.5
|0.13
|6.63
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,019,286
|993,556
|2.52%
|66.8
|0.11
|4.24
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 8 H 2 tH 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|993,556
|985,718
|0.79%
|78.3
|0.03
|3.53
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|985,718
|981,095
|0.47%
|83.7
|0.02
|3.27
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 0 n 9 run
|-
|style="text-align:left" |Terry Ligocki
|981,095
|975,912
|0.53%
|79.4
|0.02
|3.43
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|975,912
|974,180
|0.18%
|84.6
|0.01
|3.20
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 4 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|974,180
|971,254
|0.30%
|96.9
|0.01
|2.79
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|971,254
|970,101
|0.12%
|105.6
|0.00
|2.56
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 0 tH 0 n 18 run
|}

=== Stage 3 ===
Starting from the results of Stage 2, Andrew Ducharme ran "lr_enum_continue" with the maximum number of steps set to 100 million, then "Enumerate.py" with various parameters. A total of 10 Enumerate variations were run. The holdouts were reduced from ~970K TMs to ~867K TMs (a 10.63% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Andrew Ducharme
|970,101
|939,447
|3.16%
| --
| --
| --
|style="text-align:left" |lr_enum_continue 100_000_000 steps
| rowspan="11" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|939,447
|903,224
|3.86%
|440.3
|0.03
|0.59
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=4 --no-ctl --lin-steps=0 --time=2 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|903,224
|895,813
|0.82%
|647.7
|0.00
|0.39
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=3 --no-ctl --lin-steps=0 --time=3 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|895,813
|889,838
|0.67%
|609.3
|0.00
|0.41
| style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=8 --no-ctl --lin-steps=0 --time=4 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|889,838
|880,278
|1.07%
|1,638.9
|0.00
|0.15
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=12 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|880,278
|877,485
|0.32%
|1,885.5
|0.00
|0.13
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=6 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|877,485
|875,062
|0.28%
|2,068.8
|0.00
|0.12
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=5 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|875,062
|873,469
|0.18%
|1,785.4
|0.00
|0.14
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=7 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|873,469
|870,085
|0.39%
|9,270.0
|0.00
|0.03
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=2 --tape-limit=500 --time=120 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|870,085
|869,001
|0.12%
|4,498.3
|0.00
|0.05
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=10_000_000 --block-mult=60 --tape-limit=5000 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|869,001
|867,008
|0.23%
|3997.4
|0.00
|0.06
|style="text-align:left"|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=9 --tape-limit=5000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl --force --save-freq=250
|}
The total time spent on the lr_enum_continue computation was not recorded.

=== Stage 4 ===
Following the release of @mxdys's implementation of FAR deciders in C++, these deciders were applied to the 2x6 holdouts by Andrew Ducharme. The details are given in this table, including links to the Google Drive with the holdouts and solved TMs per decider:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)
{| class="wikitable sortable" style="text-align: right"
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|867,008
|811,301
|6.43%
|0.043
|364.10
|5,666.72
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 1 tH 1 n 2
| rowspan="29" |[https://drive.google.com/drive/folders/18njhmOzRc67zCmVuLd0aDxl6ETBhL1gy?usp=sharing Google Drive]
|-
|811,301
|806,119
|0.64%
|0.159
|9.03
|1,413.42
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 3 H 1 tH 1 n 2
|-
|806,119
|736,690
|8.61%
|0.548
|35.21
|408.78
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 1 tH 1 n 2
|-
|736,690
|736,504
|0.03%
|0.009
|5.81
|23,021.56
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 1 H 1 tH 1 n 1
|-
|736,504
|735,317
|0.16%
|0.058
|5.71
|3,540.88
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
|-
|735,317
|733,717
|0.22%
|0.341
|1.30
|599.28
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 2
|-
|733,717
|673,920
|8.15%
|3.8
|4.43
|54.32
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 4
|-
|673,920
|652,828
|3.13%
|~10
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 6 H 2 tH 2 n 4
|-
|652,828
|645,264
|1.16%
|~12
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 8 H 2 tH 2 n 4
|-
|645,264
|641,388
|0.60%
|~15
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 100000 LRUH 10 H 2 tH 2 n 10
|-
|641,388
|635,505
|0.92%
|~200
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 10 H 1 tH 2 n 10
|-
|635,505
|616,639
|2.97%
| ---
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [3-10]
|-
|616,639
|592,039
|3.99%
|~700
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H 0 tH 0 n [1-10]
|-
|592,039
|576,938
|2.55%
|~800
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H [0-1] tH [0-1] n [1-10]
|-
|576,938
|572,963
|0.69%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 0 tH 0 n [1-10]
|-
|572,963
|567,971
|0.87%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 2 tH 0 n [1-10]
|-
|567,971
|566,096
|0.33%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 H 0 tH 0 n [1-10]
|-
|566,096
|564,290
|0.32%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 0 tH [0,2] n [1-10]
|-
|564,290
|559,553
|0.84%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 1 n [1-10]
|-
|559,553
|558,039
|0.27%
|~900
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 2 n [1-10]
|-
|558,039
|556,814
|0.22%
|~14,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [12,16] H [0-2] tH [0-2] n [1-10]
|-
|556,814
|554,479
|0.42%
|~3,600
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [1-3]
|-
|554,479
|551,586
|0.52%
|~5000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4
|-
|551,586
|548,993
|0.47%
|~13,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 5
|-
|548,993
|545,005
|0.73%
|~57,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 and 8
|-
|545,005
|542,325
|0.49%
|6851.2
|0.00
|0.022
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=96 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|542,325
|537,393
|0.91%
|9032.1
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=2 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|537,393
|536,112
|0.24%
|8969.4
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=3 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|536,112
|533,764
|0.43%
|8778.5
| ---
| ---
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=7 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|}

== References ==


[[Category: BB Domains]][[Category:BB(2,6)]]

TMBR: April 2026

2026-05-03T09:11:25Z

Polygon: /* Holdouts */ capitalsiation

TMBR: May 2026

2026-05-03T09:09:29Z

Polygon: Added BB(2,6) holdout reduction

{{TMBRnav|April 2026|June 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

== Holdouts ==

*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''533,764''' via Enumerate.py, a '''0.44%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 24K subtasks out of the 1 million subtasks (or '''24%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[TODO]</nowiki>]

[[Category:This Month in Beaver Research|2026-05]]

Holdouts lists

2026-05-03T08:40:21Z

Polygon: Updated BB(2,6) holdout count

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1104
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,049,358
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|533,764
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo April 30th, 2026]
|@Andrew Ducharme
|12,049,358
|[https://drive.google.com/file/d/1xCtI58doiCoI3lU_oQjK_21zsLFLF9g3/view 3x4_holdouts_12049358.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 16, 2025]
|
|970,101
|[https://drive.google.com/file/d/1Vjm3Qf-4a-Zr-E3EocjPWI5bjqH2YGil/view 2x6_holdouts_970101.txt.gz]
|Work done by Terry Ligocki
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 12, 2025]
|
|20,358,011
|[https://drive.google.com/file/d/1YOoh4Zh2Jn5zaq9-C638S6VR_4jzd6pa/view 2x6_holdouts_20358011.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

Main Page

2026-05-03T08:39:33Z

Polygon: /* This Month in Beaver Research (TMBR) */ linked May TMBR

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] ≥ <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] ≥ <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] ≥ <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] ≥ <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] ≥ <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] ≥ <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] ≥ <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] ≥ <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: March 2026]]
* [[TMBR: February 2026]]
* [[TMBR: January 2026]]

[[TMBR: April 2026]] and [[TMBR: May 2026]] are currently a work in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

TMBR: April 2026

2026-05-02T10:21:53Z

Polygon: /* BB Adjacent */ added BBµ(14) champion using min

TMBR: May 2026

2026-05-02T10:18:26Z

2026-04-30T17:40:44Z

Polygon: Added a Phase 3 Data section and fixed mess I created earlier

The Busy Beaver problem for 3 states and 4 symbols is unsolved, with the existence of [[Cryptids]] in the domain being given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current BB(3,4) [[Champions#3-Symbol TMs|champion]] is {{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}. It runs for <math>2 \uparrow^{15} 5</math> steps before halting. It was discovered by Pavel Kropitz in 2024 and was analysed by Shawn Ligocki in the same year.

== Top Halters ==
The top 20 known longest running halting BB(3,4) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
!Discoverer
|-
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|<math>2 \uparrow^{15} 5</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ1LA2RB_1RC3RC1LA2LB_2LB2RC1LC3RB|halt}}
|<math>2 \uparrow^{12} 4</math>
|Pavel Kropitz
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_3LA1RZ1LC2RA}}
|<math>2 \uparrow^{6} 5</math>
|Pavel Kropitz
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_1RA1RZ1LC2RA}}
|<math>2 \uparrow^{6} 4</math>
|Pavel Kropitz
|-
|{{TM|1RB0LB1RZ3LA_0LC3RB3RC1LB_2RB2LA3RA1LC|halt}}
|<math>10 \uparrow\uparrow {2\,048}</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ1LA2RB_2RC3RC3LB2RA_2LB3RB3LC1LA|halt}}
|<math>10 \uparrow\uparrow {189.05690}</math>
|[[User:XnoobSpeakable|XnoobSpeakable]]
|-
|{{TM|1RB3LA1RA2LB_2LC1RA2RC3RB_1RZ1LA2RA3RA|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|-
|{{TM|1RB1LB2LA3LC_1LA2RB3RA1LB_1RZ1RB2LB3LB|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|-
|{{TM|1RB1RA3LA3RB_2LC2RC1LB2RA_2RA2RB0LB1RZ|halt}}
|<math>10 \uparrow\uparrow {149.57147}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LB1RZ3LA_1LA3RC2LC1LC_2LA3RC0RA1LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LB1RZ3LA_1LA3RC3LB1LC_2RA3RC0RA1LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LC1RC2RA_1LA3RA2RB1RZ_3LB2LC1RA0LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|[[User:WarpedWartWars|Lúkos]]
|-
|{{TM|1RB1RA1LB3RA_1RC3LB1LC1RC_2LA2RA1RZ0LA|halt}}
|<math>10 \uparrow\uparrow {96.06390}</math>
|XnoobSpeakable
|-
|{{TM|1RB1RA1LB3RA_1RC3LB1LC1RC_2LA2RA1RZ1RB|halt}}
|<math>10 \uparrow\uparrow {96.06390}</math>
|XnoobSpeakable
|-
|{{TM|1RB3LC0RA1RC_2LA1RZ3LA2RB_3LA0RA1LA0RA|halt}}
|<math>10 \uparrow\uparrow {95.24590}</math>
|XnoobSpeakable
|-
|{{TM|1RB3LC0RA1RC_2LA1RZ3LA2RB_3LA0RA3LB0RA|halt}}
|<math>10 \uparrow\uparrow {95.24590}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LA1RA1RC_1LA3LC3RB1RC_3LA1RZ3LB2RC|halt}}
|<math>10 \uparrow\uparrow {84.19957}</math>
|XnoobSpeakable
|-
|{{TM|1RB0LB1RA2LB_2RC3LB3RB2LA_2LC1RZ1LA2RB|halt}}
|<math>10 \uparrow\uparrow {76.17604}</math>
|XnoobSpeakable
|-
|{{TM|1RB1LA1RZ0LB_2LC2LB1RC1RB_3LA3LC0LC0RA|halt}}
|<math>10 \uparrow\uparrow {74.18390}</math>
|XnoobSpeakable
|-
|{{TM|1RB2RA1LA1LB_2RC3RB1RZ3LC_2LA1RC1RB0RA|halt}}
|<math>10 \uparrow\uparrow {69.55341}</math>
|Lúkos
|}

== 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 600,253,156,787 TMs of which 42,391,877,333 were holdout TMs. Also found were 202,014,400,019 halting TMs and 355,846,879,435 infinite TMs. The number of holdouts was reduced to 434,787,751 TMs (a 98.97% reduction).

<div class="toccolours mw-collapsible mw-collapsed">'''Phase 1 Info'''<div class="mw-collapsible-content">
Two C++ programs were run before the filters in the table.
<pre>
lr_enum 3 4 8 /dev/null /dev/null 3x4.unk.txt false
00 <= XX < 47: lr_enum_continue 3x4.in.XX 1000 /dev/null /dev/null 3x4.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(3,4) of depth 8 and outputs the holdouts to 3x4.unk.txt. This file was then divided into 48 pieces, 3x4.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 3x4.unk.txt.XX.

For these runs the first command generated a total of 22,366,747 TMs: 932,198 halting, 235,446 infinite, and 21,199,103 holdouts. The second took the 22,366,747 holdout TMs and generated a total of 600,253,156,787 TMs: 202,014,400,019 halting, 355,846,879,435 infinite, and 42,391,877,333 holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.
</div></div>

The details are given in this table:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|42,391,877,333
|32,247,381,725
|23.93%
|788.0
|3,576.08
|14,943.74
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="8" style="text-align:left" |[https://drive.google.com/drive/folders/1EsFw5DOjjZ6Xzjc8QgRa2YkInrsmFnLX?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|32,247,381,725
|12,789,441,316
|60.34%
|6,185.2
|873.85
|1,448.22
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,789,441,316
|4,332,283,583
|66.13%
|4,771.0
|492.39
|744.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,332,283,583
|2,796,143,404
|35.46%
|7,873.1
|54.20
|152.85
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,796,143,404
|1,398,734,288
|49.98%
|10,239.4
|37.91
|75.85
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,398,734,288
|764,851,053
|45.32%
|3,203.1
|54.97
|121.30
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|764,851,053
|581,654,405
|23.95%
|1,925.3
|26.43
|110.35
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|581,654,405
|434,787,751
|25.25%
|9,042.5
|4.51
|17.87
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|}

== Phase 2 ==
XnoobSpeakable and Lúkos worked on applying mxdys' deciders/parameters to the remaining holdouts. The previous holdouts list by Terry Ligocki was split up into 8696 files containing 50,000 holdouts each* (*the last one actually has 37,751) and the deciders were applied to the files individually. These files can be called subtasks. Lúkos made a Python script to compute the subtasks in parallel, based off of XnoobSpeakable's earlier imperfect script.

Certain data such as TMs/sec/core is unavailable, because the amount of time the computation took was not kept track of.

Stages 3 and 8 are lr_enum_continue computations to 1 million and 10 million steps respectively. Lúkos did not contribute to computing these stages, but rather created a Python script to compute them in bulk, which XnoobSpeakable used.

For the "Contribution" column, the first number is the amount of subtasks solved by XnoobSpeakable, the 2nd is the amount solved by Lúkos. Note that the counts are unequal, because Lúkos has a slower CPU with a lower core count. According to XnoobSpeakable, the time contribution from both contributors is roughly equal. Lúkos also worked on the Python scripts used, while XnoobSpeakable worked on compiling holdouts lists and finding optimal deciders to apply.

[[User:Peacemaker II|Peacemaker II]] worked on verifying the validity of the results as well as providing solved TM counts per decider. Stages 1, 2, and 4 have been validated - no dropped TMs. XnoobSpeakable verified stage 6.

<nowiki>**</nowiki>Stage 9: The two subtasks are seperate sub-stages, applying Reverse-Engineer and Backtracking deciders.

Stage 10 is also seperated into substages A, B, and C for ease of computation.

[https://drive.google.com/drive/folders/1MamU8zivuqoe4j82vpb-jYBfc4wvXNRc All scripts used are available here.] The details will be given in this table:

{| class="wikitable sortable" style="text-align: left"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |% Reduced
! rowspan="2" |Description
! rowspan="2" |Subtasks
! rowspan="2" |Contribution
! rowspan="2" |Data
! rowspan="2" |Verified
|-
!Input
!Output
|-
| style="text-align:left" |XnoobSpeakable, Lúkos
|434,787,751
|64,777,377
|85.1%
| style="text-align:left" |Phase 2 stage 1
|8696
|7696 / 1000
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|64,777,377
|37,016,957
|42.9%
|Phase 2 stage 2
|1296
|1000 / 296
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable
|37,016,957
|26,266,261
|29.0%
|Phase 2 stage 3
|741
|741 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|26,266,261
|22,396,711
|14.7%
|Phase 2 stage 4
|526
|485 / 41
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|22,396,711
|17,983,810
|19.7%
|Phase 2 stage 5
|448
|390 / 58
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|17,983,810
|15,564,491
|13.5%
|Phase 2 stage 6
|360
|300 / 60
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|15,564,491
|15,136,283
|2.75%
|Phase 2 stage 7
|312
|260 / 52
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable
|15,136,283
|14,518,243
|4.08%
|Phase 2 stage 8
|303
|303 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable
|14,518,243
|14,413,460
|0.72%
|Phase 2 stage 9
|2**
|2 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|14,413,460
|13,733,388
|4.72%
|Phase 2 stage 10A
|289
|216 / 73
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|13,733,388
|13,334,244
|2.91%
|Phase 2 stage 10B
|275
|200 / 75
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|13,334,244
|12,435,284
|6.74%
|Phase 2 stage 10C
|267
|190 / 77
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|}

== Phase 2 Data ==
Here we will list the deciders applied during the various stages of Phase 2. Each decider within a stage was not applied in a specific order. Some stages have a number of TMs solved by every decider at the end of each row, provided by Peacemaker II.

<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
Phase 2 stage 1:
<pre>
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 1000 NG_n 2 run 86569103
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 1000 NG_n 1 run 26093656
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 13972946
MitM_CTL RWL_mod sim 1001 maxT 1000 H 6 mod 3 n 2 run 7190617
chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 1000 NG_n 2 run 90302784
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 1 n 4 run 4342619
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 12 H 0 tH 1 n 2 run 11764327
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 1000 NG_n 1 run 55465842
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 2 n 1 run 17805678
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 2 H 0 tH 0 n 1 run 12623067
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 3000 NG_n 2 run 2240499
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 1 tH 1 n 2 run 10223131
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 10000 NG_n 1 run 2901053
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 28515052
</pre>
Phase 2 stage 2:
<pre>
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 4 n 8 run 1030303
MitM_CTL RWL_mod sim 1001 maxT 3000 H 8 mod 2 n 2 run 1465523
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 5232902
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 1720994
MitM_CTL RWL_mod sim 1001 maxT 3000 H 8 mod 3 n 2 run 555883
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 3 H 1 tH 0 n 6 run 290547
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 1 run 67506
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 1577979
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 1604533
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 4 n 6 run 10012948
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 8 H 0 tH 0 n 4 run 526822
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 1 n 1 run 34821
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 3 run 356375
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 20 run 1970422
MitM_CTL RWL_mod sim 1001 maxT 1000 H 6 mod 3 n 1 run 443339
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 2 run 518824
MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 2 n 1 run 61865
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 1 tH 1 n 2 run 288834
</pre>
Phase 2 stage 3 (lr_enum_continue from [https://github.com/sligocki/busy-beaver busy-beaver] for 1 million steps):
<pre>
lr_enum_continue ./in.txt 1000000 ./halt.txt ./inf.txt ./unknown.txt "" false
</pre>
Phase 2 stage 4:
<pre>
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 3000 NG_n 2 run 202520
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 3000 NG_n 1 run 559013
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 4 n 1 run 1584
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 10000 NG_n 1 run 148544
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 1 run 812666
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 10000 NG_n 3 run 2043171
MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 2 n 1 run 102052
</pre>
Phase 2 stage 5:
<pre>
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 0 tH 1 n 10 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 16 H 1 tH 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 6 n 2 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 10000 NG_n 3 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 5 n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 3 H 1 tH 0 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 3 mod 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 2 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 5 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 4 n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 0 tH 1 n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 1 run
chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 3 run
chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 8 H 2 tH 1 n 2 run
chr_asth 0 chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 16 H 1 tH 0 n 6 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 0 tH 0 n 1 run
chr_LRUH 6 chr_H 2 MitM_CTL NG maxT 10000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 3 H 1 tH 0 n 2 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 3000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 3 n 1 run
chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 100000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 1 tH 0 n 4 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 4 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 2 tH 0 n 1 run
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000 NG_n 2 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 3 mod 6 n 1 run
chr_asth 0 chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 30000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 1 run
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run
chr_asth 0 chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 3000 NG_n 2 run
</pre>
Phase 2 stage 6:

XnoobSpeakable ran Enumerate.py with a 0.15s/TM time limit, Lúkos ran it with a 0.75s/TM time limit.
<pre>
python "+str(pl.Path("../../Code/Enumerate.py"))+" --infile="+str(pl.Path("./in.txt"))+" --outfile="+str(pl.Path("./out.pb"))+" --max-loops=990_000 --tape-limit=9_900 --no-steps --time="+str(time)+" --recursive --exp-linear-rules --lin-steps=0 --no-ctl --save-freq=10_000
</pre>
Phase 2 stage 7:
<pre>
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 1000 NG_n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 1 mod 5 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 3 H 0 tH 0 n 1 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 1000 NG_n 1 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 4 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 16 H 0 tH 1 n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 0 tH 1 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 0 tH 2 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 1 mod 2 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 2 H 0 tH 0 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 1 mod 3 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 1 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 2 H 0 tH 0 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 0 n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 3000 NG_n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 1000 NG_n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 1 mod 1 n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 2 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
</pre>Phase 2 stage 8 (lr_enum_continue from [https://github.com/sligocki/busy-beaver busy-beaver] for 10 million steps):<pre>lr_enum_continue ./in.txt 10000000 ./halt.txt ./inf.txt ./unknown.txt "" false</pre>Phase 2 stage 9a:<pre>python3 Code/Reverse_Engineer_Filter.py --infile='/home/xnoobspeakable/Desktop/Busy Beavers/BB(3,4)/resultsst8/3x4_holdouts_14518243.txt' --outfile='out.txt'</pre>Phase 2 stage 9b:<pre>python3 ./Code/Backtracking_Filter.py --infile='/home/xnoobspeakable/Desktop/Busy Beavers/BB(3,4)/resultsst9/unknown.txt' --outfile='./b1.pb' --steps=100</pre>Phase 2 stage 10a:<pre>chr_LRUH 28 chr_LRUn 2 MitM_CTL NG maxT 30000 NG_n 8 run
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 2 n 8 run</pre>Phase 2 stage 10b:<pre>MitM_CTL CPS_LRU sim 1,001 maxT 10000 LRUH 2 H 0 tH 0 n 20 run
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run</pre>Phase 2 stage 10c:<pre>chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 8 run
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 100000 NG_n 11 run</pre></div></div>

== Phase 3 ==

Phase 3 began with Andrew Ducharme applying a FAR decider by mxdys.

{| class="wikitable sortable" style="text-align: left"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |% Reduced
! rowspan="2" |Description
! rowspan="2" |Subtasks
! rowspan="2" |Contribution
! rowspan="2" |Data
! rowspan="2" |Verified
|-
!Input
!Output
|-
| style="text-align:left" |Andrew Ducharme
|12,435,284
|12,049,358
|3.10%
| style="text-align:left" |Phase 3 stage 1
|'''-'''
|'''-'''
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo Google Drive]
|No
|}

== Phase 3 Data ==
Here we will list the deciders applied during the various stages of Phase 3. Some stages have a number of TMs solved by every decider at the end of each row.

<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
Phase 2 stage 1:
<pre>
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2 368,121
FAR CPS_LRU maxT 100000 LRUH 1 H 0 tH 0 n [1-4] 17,660
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 1 145
</pre>

[[Category:BB Domains]][[Category:BB(3,4)]]

BB(3,4)

2026-04-30T17:36:54Z

Polygon: /* Phase 2 */ added a Phase 3 section

The Busy Beaver problem for 3 states and 4 symbols is unsolved, with the existence of [[Cryptids]] in the domain being given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current BB(3,4) [[Champions#3-Symbol TMs|champion]] is {{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}. It runs for <math>2 \uparrow^{15} 5</math> steps before halting. It was discovered by Pavel Kropitz in 2024 and was analysed by Shawn Ligocki in the same year.

== Top Halters ==
The top 20 known longest running halting BB(3,4) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
!Discoverer
|-
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|<math>2 \uparrow^{15} 5</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ1LA2RB_1RC3RC1LA2LB_2LB2RC1LC3RB|halt}}
|<math>2 \uparrow^{12} 4</math>
|Pavel Kropitz
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_3LA1RZ1LC2RA}}
|<math>2 \uparrow^{6} 5</math>
|Pavel Kropitz
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_1RA1RZ1LC2RA}}
|<math>2 \uparrow^{6} 4</math>
|Pavel Kropitz
|-
|{{TM|1RB0LB1RZ3LA_0LC3RB3RC1LB_2RB2LA3RA1LC|halt}}
|<math>10 \uparrow\uparrow {2\,048}</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ1LA2RB_2RC3RC3LB2RA_2LB3RB3LC1LA|halt}}
|<math>10 \uparrow\uparrow {189.05690}</math>
|[[User:XnoobSpeakable|XnoobSpeakable]]
|-
|{{TM|1RB3LA1RA2LB_2LC1RA2RC3RB_1RZ1LA2RA3RA|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|-
|{{TM|1RB1LB2LA3LC_1LA2RB3RA1LB_1RZ1RB2LB3LB|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|-
|{{TM|1RB1RA3LA3RB_2LC2RC1LB2RA_2RA2RB0LB1RZ|halt}}
|<math>10 \uparrow\uparrow {149.57147}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LB1RZ3LA_1LA3RC2LC1LC_2LA3RC0RA1LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LB1RZ3LA_1LA3RC3LB1LC_2RA3RC0RA1LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LC1RC2RA_1LA3RA2RB1RZ_3LB2LC1RA0LB|halt}}
|<math>10 \uparrow\uparrow {130.20142}</math>
|[[User:WarpedWartWars|Lúkos]]
|-
|{{TM|1RB1RA1LB3RA_1RC3LB1LC1RC_2LA2RA1RZ0LA|halt}}
|<math>10 \uparrow\uparrow {96.06390}</math>
|XnoobSpeakable
|-
|{{TM|1RB1RA1LB3RA_1RC3LB1LC1RC_2LA2RA1RZ1RB|halt}}
|<math>10 \uparrow\uparrow {96.06390}</math>
|XnoobSpeakable
|-
|{{TM|1RB3LC0RA1RC_2LA1RZ3LA2RB_3LA0RA1LA0RA|halt}}
|<math>10 \uparrow\uparrow {95.24590}</math>
|XnoobSpeakable
|-
|{{TM|1RB3LC0RA1RC_2LA1RZ3LA2RB_3LA0RA3LB0RA|halt}}
|<math>10 \uparrow\uparrow {95.24590}</math>
|XnoobSpeakable
|-
|{{TM|1RB2LA1RA1RC_1LA3LC3RB1RC_3LA1RZ3LB2RC|halt}}
|<math>10 \uparrow\uparrow {84.19957}</math>
|XnoobSpeakable
|-
|{{TM|1RB0LB1RA2LB_2RC3LB3RB2LA_2LC1RZ1LA2RB|halt}}
|<math>10 \uparrow\uparrow {76.17604}</math>
|XnoobSpeakable
|-
|{{TM|1RB1LA1RZ0LB_2LC2LB1RC1RB_3LA3LC0LC0RA|halt}}
|<math>10 \uparrow\uparrow {74.18390}</math>
|XnoobSpeakable
|-
|{{TM|1RB2RA1LA1LB_2RC3RB1RZ3LC_2LA1RC1RB0RA|halt}}
|<math>10 \uparrow\uparrow {69.55341}</math>
|Lúkos
|}

== 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 600,253,156,787 TMs of which 42,391,877,333 were holdout TMs. Also found were 202,014,400,019 halting TMs and 355,846,879,435 infinite TMs. The number of holdouts was reduced to 434,787,751 TMs (a 98.97% reduction).

<div class="toccolours mw-collapsible mw-collapsed">'''Phase 1 Info'''<div class="mw-collapsible-content">
Two C++ programs were run before the filters in the table.
<pre>
lr_enum 3 4 8 /dev/null /dev/null 3x4.unk.txt false
00 <= XX < 47: lr_enum_continue 3x4.in.XX 1000 /dev/null /dev/null 3x4.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(3,4) of depth 8 and outputs the holdouts to 3x4.unk.txt. This file was then divided into 48 pieces, 3x4.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 3x4.unk.txt.XX.

For these runs the first command generated a total of 22,366,747 TMs: 932,198 halting, 235,446 infinite, and 21,199,103 holdouts. The second took the 22,366,747 holdout TMs and generated a total of 600,253,156,787 TMs: 202,014,400,019 halting, 355,846,879,435 infinite, and 42,391,877,333 holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.
</div></div>

The details are given in this table:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|42,391,877,333
|32,247,381,725
|23.93%
|788.0
|3,576.08
|14,943.74
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="8" style="text-align:left" |[https://drive.google.com/drive/folders/1EsFw5DOjjZ6Xzjc8QgRa2YkInrsmFnLX?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|32,247,381,725
|12,789,441,316
|60.34%
|6,185.2
|873.85
|1,448.22
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,789,441,316
|4,332,283,583
|66.13%
|4,771.0
|492.39
|744.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,332,283,583
|2,796,143,404
|35.46%
|7,873.1
|54.20
|152.85
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,796,143,404
|1,398,734,288
|49.98%
|10,239.4
|37.91
|75.85
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,398,734,288
|764,851,053
|45.32%
|3,203.1
|54.97
|121.30
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|764,851,053
|581,654,405
|23.95%
|1,925.3
|26.43
|110.35
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|581,654,405
|434,787,751
|25.25%
|9,042.5
|4.51
|17.87
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|}

== Phase 2 ==
XnoobSpeakable and Lúkos worked on applying mxdys' deciders/parameters to the remaining holdouts. The previous holdouts list by Terry Ligocki was split up into 8696 files containing 50,000 holdouts each* (*the last one actually has 37,751) and the deciders were applied to the files individually. These files can be called subtasks. Lúkos made a Python script to compute the subtasks in parallel, based off of XnoobSpeakable's earlier imperfect script.

Certain data such as TMs/sec/core is unavailable, because the amount of time the computation took was not kept track of.

Stages 3 and 8 are lr_enum_continue computations to 1 million and 10 million steps respectively. Lúkos did not contribute to computing these stages, but rather created a Python script to compute them in bulk, which XnoobSpeakable used.

For the "Contribution" column, the first number is the amount of subtasks solved by XnoobSpeakable, the 2nd is the amount solved by Lúkos. Note that the counts are unequal, because Lúkos has a slower CPU with a lower core count. According to XnoobSpeakable, the time contribution from both contributors is roughly equal. Lúkos also worked on the Python scripts used, while XnoobSpeakable worked on compiling holdouts lists and finding optimal deciders to apply.

[[User:Peacemaker II|Peacemaker II]] worked on verifying the validity of the results as well as providing solved TM counts per decider. Stages 1, 2, and 4 have been validated - no dropped TMs. XnoobSpeakable verified stage 6.

<nowiki>**</nowiki>Stage 9: The two subtasks are seperate sub-stages, applying Reverse-Engineer and Backtracking deciders.

Stage 10 is also seperated into substages A, B, and C for ease of computation.

[https://drive.google.com/drive/folders/1MamU8zivuqoe4j82vpb-jYBfc4wvXNRc All scripts used are available here.] The details will be given in this table:

{| class="wikitable sortable" style="text-align: left"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |% Reduced
! rowspan="2" |Description
! rowspan="2" |Subtasks
! rowspan="2" |Contribution
! rowspan="2" |Data
! rowspan="2" |Verified
|-
!Input
!Output
|-
| style="text-align:left" |XnoobSpeakable, Lúkos
|434,787,751
|64,777,377
|85.1%
| style="text-align:left" |Phase 2 stage 1
|8696
|7696 / 1000
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|64,777,377
|37,016,957
|42.9%
|Phase 2 stage 2
|1296
|1000 / 296
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable
|37,016,957
|26,266,261
|29.0%
|Phase 2 stage 3
|741
|741 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|26,266,261
|22,396,711
|14.7%
|Phase 2 stage 4
|526
|485 / 41
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|22,396,711
|17,983,810
|19.7%
|Phase 2 stage 5
|448
|390 / 58
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|17,983,810
|15,564,491
|13.5%
|Phase 2 stage 6
|360
|300 / 60
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|Yes
|-
|XnoobSpeakable, Lúkos
|15,564,491
|15,136,283
|2.75%
|Phase 2 stage 7
|312
|260 / 52
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable
|15,136,283
|14,518,243
|4.08%
|Phase 2 stage 8
|303
|303 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable
|14,518,243
|14,413,460
|0.72%
|Phase 2 stage 9
|2**
|2 / 0
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|14,413,460
|13,733,388
|4.72%
|Phase 2 stage 10A
|289
|216 / 73
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|13,733,388
|13,334,244
|2.91%
|Phase 2 stage 10B
|275
|200 / 75
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|-
|XnoobSpeakable, Lúkos
|13,334,244
|12,435,284
|6.74%
|Phase 2 stage 10C
|267
|190 / 77
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i Google Drive]
|No
|}

== Phase 3 ==

Phase 3 began with Andrew Ducharme applying a FAR decider by mxdys.

{| class="wikitable sortable" style="text-align: left"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |% Reduced
! rowspan="2" |Description
! rowspan="2" |Subtasks
! rowspan="2" |Contribution
! rowspan="2" |Data
! rowspan="2" |Verified
|-
!Input
!Output
|-
| style="text-align:left" |Andrew Ducharme
|12,435,284
|12,049,358
|3.10%
| style="text-align:left" |Phase 3 stage 1
|'''-'''
|'''-'''
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo Google Drive]
|No
|-

== Phase 2 Data ==
Here we will list the deciders applied during the various stages of Phase 2. Each decider within a stage was not applied in a specific order. Some stages have a number of TMs solved by every decider at the end of each row, provided by Peacemaker II.

<div class="toccolours mw-collapsible mw-collapsed">'''Details'''<div class="mw-collapsible-content">
Phase 2 stage 1:
<pre>
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 1000 NG_n 2 run 86569103
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 1000 NG_n 1 run 26093656
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 13972946
MitM_CTL RWL_mod sim 1001 maxT 1000 H 6 mod 3 n 2 run 7190617
chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 1000 NG_n 2 run 90302784
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 1 n 4 run 4342619
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 12 H 0 tH 1 n 2 run 11764327
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 1000 NG_n 1 run 55465842
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 2 n 1 run 17805678
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 2 H 0 tH 0 n 1 run 12623067
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 3000 NG_n 2 run 2240499
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 1 tH 1 n 2 run 10223131
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 10000 NG_n 1 run 2901053
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 28515052
</pre>
Phase 2 stage 2:
<pre>
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 4 n 8 run 1030303
MitM_CTL RWL_mod sim 1001 maxT 3000 H 8 mod 2 n 2 run 1465523
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 5232902
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run 1720994
MitM_CTL RWL_mod sim 1001 maxT 3000 H 8 mod 3 n 2 run 555883
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 3 H 1 tH 0 n 6 run 290547
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 1 run 67506
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 1577979
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 1604533
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 4 n 6 run 10012948
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 8 H 0 tH 0 n 4 run 526822
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 1 n 1 run 34821
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 3 run 356375
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 20 run 1970422
MitM_CTL RWL_mod sim 1001 maxT 1000 H 6 mod 3 n 1 run 443339
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 2 run 518824
MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 2 n 1 run 61865
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 1 tH 1 n 2 run 288834
</pre>
Phase 2 stage 3 (lr_enum_continue from [https://github.com/sligocki/busy-beaver busy-beaver] for 1 million steps):
<pre>
lr_enum_continue ./in.txt 1000000 ./halt.txt ./inf.txt ./unknown.txt "" false
</pre>
Phase 2 stage 4:
<pre>
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 3000 NG_n 2 run 202520
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 3000 NG_n 1 run 559013
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 4 n 1 run 1584
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 10000 NG_n 1 run 148544
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 1 run 812666
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 10000 NG_n 3 run 2043171
MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 2 n 1 run 102052
</pre>
Phase 2 stage 5:
<pre>
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 0 tH 1 n 10 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 16 H 1 tH 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 6 n 2 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 10000 NG_n 3 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 5 n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 3 H 1 tH 0 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 3 mod 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 2 mod 3 n 2 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 5 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 4 n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 0 tH 1 n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 1 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 1 run
chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 3 run
chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 8 H 2 tH 1 n 2 run
chr_asth 0 chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 16 H 1 tH 0 n 6 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 0 tH 0 n 1 run
chr_LRUH 6 chr_H 2 MitM_CTL NG maxT 10000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 3 H 1 tH 0 n 2 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 3000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 3 n 1 run
chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 100000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 6 H 1 tH 0 n 4 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 4 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 2 tH 0 n 1 run
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000 NG_n 2 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 1 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 3 mod 6 n 1 run
chr_asth 0 chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 30000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 1 run
chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run
chr_asth 0 chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 3000 NG_n 2 run
</pre>
Phase 2 stage 6:

XnoobSpeakable ran Enumerate.py with a 0.15s/TM time limit, Lúkos ran it with a 0.75s/TM time limit.
<pre>
python "+str(pl.Path("../../Code/Enumerate.py"))+" --infile="+str(pl.Path("./in.txt"))+" --outfile="+str(pl.Path("./out.pb"))+" --max-loops=990_000 --tape-limit=9_900 --no-steps --time="+str(time)+" --recursive --exp-linear-rules --lin-steps=0 --no-ctl --save-freq=10_000
</pre>
Phase 2 stage 7:
<pre>
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 1000 NG_n 3 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 1 mod 5 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 3 H 0 tH 0 n 1 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 1000 NG_n 1 run
chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 4 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 16 H 0 tH 1 n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 4 H 0 tH 1 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 16 H 0 tH 2 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 30000 H 1 mod 2 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 2 H 0 tH 0 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 0 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 1 mod 3 n 1 run
MitM_CTL RWL_mod sim 1001 maxT 3000 H 1 mod 1 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 2 H 0 tH 0 n 1 run
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 0 n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 3000 NG_n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 1000 NG_n 1 run
chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 1 run
MitM_CTL RWL_mod sim 1001 maxT 10000 H 1 mod 1 n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 1 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 2 run
chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
</pre>Phase 2 stage 8 (lr_enum_continue from [https://github.com/sligocki/busy-beaver busy-beaver] for 10 million steps):<pre>lr_enum_continue ./in.txt 10000000 ./halt.txt ./inf.txt ./unknown.txt "" false</pre>Phase 2 stage 9a:<pre>python3 Code/Reverse_Engineer_Filter.py --infile='/home/xnoobspeakable/Desktop/Busy Beavers/BB(3,4)/resultsst8/3x4_holdouts_14518243.txt' --outfile='out.txt'</pre>Phase 2 stage 9b:<pre>python3 ./Code/Backtracking_Filter.py --infile='/home/xnoobspeakable/Desktop/Busy Beavers/BB(3,4)/resultsst9/unknown.txt' --outfile='./b1.pb' --steps=100</pre>Phase 2 stage 10a:<pre>chr_LRUH 28 chr_LRUn 2 MitM_CTL NG maxT 30000 NG_n 8 run
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 2 n 8 run</pre>Phase 2 stage 10b:<pre>MitM_CTL CPS_LRU sim 1,001 maxT 10000 LRUH 2 H 0 tH 0 n 20 run
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run</pre>Phase 2 stage 10c:<pre>chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 8 run
chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 100000 NG_n 11 run</pre></div></div>
[[Category:BB Domains]][[Category:BB(3,4)]]

Holdouts lists

2026-04-30T17:30:50Z

Polygon: Updated BB(3,4)

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1104
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,049,358
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|536,112
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo April 30th, 2026]
|@Andrew Ducharme
|12,049,358
|[https://drive.google.com/file/d/1xCtI58doiCoI3lU_oQjK_21zsLFLF9g3/view 3x4_holdouts_12049358.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 16, 2025]
|
|970,101
|[https://drive.google.com/file/d/1Vjm3Qf-4a-Zr-E3EocjPWI5bjqH2YGil/view 2x6_holdouts_970101.txt.gz]
|Work done by Terry Ligocki
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 12, 2025]
|
|20,358,011
|[https://drive.google.com/file/d/1YOoh4Zh2Jn5zaq9-C638S6VR_4jzd6pa/view 2x6_holdouts_20358011.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

Holdouts lists

2026-04-29T17:51:05Z

Polygon: /* Downloadable Holdouts Lists */ added two missing BB(2,6) holdouts lists

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1104
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|536,112
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 16, 2025]
|
|970,101
|[https://drive.google.com/file/d/1Vjm3Qf-4a-Zr-E3EocjPWI5bjqH2YGil/view 2x6_holdouts_970101.txt.gz]
|Work done by Terry Ligocki
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 12, 2025]
|
|20,358,011
|[https://drive.google.com/file/d/1YOoh4Zh2Jn5zaq9-C638S6VR_4jzd6pa/view 2x6_holdouts_20358011.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

BB(4,3)

2026-04-29T16:32:59Z

Polygon: /* Phase 1 */ fixed percent reduction

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 [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== 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 ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 99.93% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a [[TNF]] tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''99.93%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| class="wikitable sortable" style="text-align: right"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="28" style="text-align:left" |[https://drive.google.com/drive/folders/11FYe0CVDPczcgt4put3vsdGbeMWkM1S1?usp=sharing Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,641,006'''
|'''40.00%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

TMBR: April 2026

2026-04-29T15:48:00Z

Polygon: /* Holdouts */ Added new holdouts list

BB(6)

2026-04-29T15:46:09Z

Polygon: /* History */ updated holdout counts

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== History ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1104 machines up to equivalence and 2336 machines not considering equivalence as of April 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet]. The informal holdout count is 1205.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 to 1e15. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet].

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}, Space Needle
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}, similar to Space Needle
* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}, similar to Space Needle

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}
* {{TM|1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA|halt}}
|<math>10^{1\,400\,000\,000}</math>(runtime)
|Racheline<ref>https://discord.com/channels/960643023006490684/1345502880727040091/1345502880727040091</ref>
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub) upon initial application.

== References ==
<references />

[[Category:BB Domains]][[Category:BB(6)]]

Holdouts lists

2026-04-29T15:45:05Z

Polygon: Updated BB(6)

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1104
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|536,112
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

File:BB6 holdouts 1104.txt

2026-04-29T15:42:40Z

Polygon: Holdouts list shared on April 29 2026 by mxdys over Discord.

== Summary ==
Holdouts list shared on April 29 2026 by mxdys over Discord.

Consistent Collatz

2026-04-27T14:50:21Z

Polygon: Linked piecewise affine function

A '''consistent Collatz''' function is a [[Generalized Collatz Function]] (GCF) with the same coefficient in each [[Piecewise Affine Function|piecewise affine function]]. In other words, it is a function <math>g: \mathbb{N} \to \mathbb{N}</math> defined as:

<math display="block">g(x) = \begin{cases}
a x + b_0 & \text{if } x \equiv 0 \pmod{m} \\
a x + b_1 & \text{if } x \equiv 1 \pmod{m} \\
& \vdots \\
a x + b_{m-1} & \text{if } x \equiv m-1 \pmod{m} \\
\end{cases}</math>

A consistent Collatz sequence is a sequence of values obtained by iterating a consistent Collatz function: <math>x_{n+1} = g(x_n)</math> which can alternatively be defined by a recurrence relation of the following form:

<math>mx_{n+1} = rx_n + J[x_n\ {\operatorname{mod}}\ m]</math>

where the sequence itself is <math>x_0, x_1, \ldots</math>, <math>m</math> and <math>r</math> are positive integers, and <math>J</math> is a map from integers modulo <math>m</math> to (possibly negative) integers (with the values chosen to ensure that all <math>x_n</math> are integers). The sequence is entirely defined by <math>m</math>, <math>r</math>, <math>J</math>, and the value of its first element <math>x_0</math>.

Many small [[cryptids]] work by calculating the elements of a consistent Collatz sequence modulo <math>m</math>, and deciding whether or not to halt based on the remainders. For example, [[Hydra]] calculates the consistent Collatz sequence with <math>m=2</math>, <math>r=3</math>, <math>J=\{0 \mapsto 0, 1 \mapsto -1\}</math>, and <math>x_0=3</math>, halting only if the sequence has contained more than twice as many even elements as odd elements. <math>m=2, r=3</math> seems to be particularly common among small cryptids, due to creating the simplest consistent Collatz sequences that have nontrivial behaviour; however, other values have been observed, such as [[Bigfoot]]'s <math>m=81, r=256</math>.

== Efficiently calculating consistent Collatz sequences ==

For any given Collatz sequence, it is possible to calculate its sequence of remainders <math>x_n\ {\operatorname{mod}}\ m</math> in amortized [[wikipedia:quasilinear time|quasilinear time]]. This can be accomplished via the use of two helper sequences:

<math>w_n = x_n\ {\operatorname{mod}}\ m^{f(n)}</math>, except <math>w_0 = x_0</math> 
<math>j_n = m^{f(n)}x_n - r^{f(n)}x_{n-f(n)}</math>, with <math>j_0</math> undefined (the algorithm never uses it)

where <math>f(n)</math> is the largest power of 2 that divides into <math>n</math>.

=== Time complexity ===

The algorithm works by calculating <math>j_n</math> then <math>w_n</math> for each <math>n</math> in turn, with the only operations used being additions, subtractions, and multiplications of numbers whose number of digits is proportional to <math>f(n)</math>; division and modulo by powers of <math>m</math>; and calculation of <math>r^t</math> with <math>t \leq f(n)</math>. Because the values of <math>r^t</math> can be memoized, and it is possible to trivialise the division and modulus operations by storing the numbers in base <math>m</math>, this means that the only slow operations are additions and subtractions taking <math>O(f(n))</math> time, and multiplications taking <math>O(f(n) \log f(n) \log \log f(n))</math> time (and there are <math>O(\log f(n))</math> such operations performed), so the calculation of each <math>w_n, j_n</math> pair takes time quasilinear in <math>f(n)</math>. <math>\Sigma_{t=1}^n{f(t)}=O(n \log n)</math>: as such, the calculation of the entire sequences <math>w_0, \ldots, w_n</math> and <math>j_0, \ldots, j_n</math> takes time quasilinear in <math>n \log n</math>, thus quasilinear in <math>n</math>.

=== Details of the algorithm ===

The algorithm itself is:

<math>j_n = m^{f(n)-1} J[w_{n-1}\ {\operatorname{mod}}\ m] + \Sigma_{e=1}^{\log_2 f(n)}\left(m^{f(n)-2^e}r^{2^{e-1}}j_{n-2^{e-1}}\right)</math> 
<math>w'_n = j_n + r^{f(n)} w_{n-f(n)} \mod m^{2f(n)}</math> (i.e. all the calculations are done modulo <math>m^{2f(n)}</math>, saving time in cases where <math>w_{n-f(n)}</math> happens to be much larger than <math>m^{2f(n)}</math>) 
<math>w_n = w'_n \div m^{f(n)}</math> (which should always be an integer).

(If <math>n</math> is odd, then <math>f(n) = 1</math>, <math>\log_2 f(n) = 0</math>, and the sum in the calculation of <math>j_n</math> is a degenerate sum with no elements.)

=== Sketch proof of correctness ===

The proof that the algorithm is correct starts with the definition <math>j_n = m^{f(n)}x_n - r^{f(n)}x_{n-f(n)}</math>, adds a degenerate sum (that sums to zero) to produce the following expression:

<math>j_n = m^{f(n)}x_n + \Sigma_{e=1}^{\log_2 f(n)}\left(-m^{f(n)-2^{e-1}}r^{2^{e-1}}x_{n-2^{e-1}}+m^{f(n)-2^{e-1}}r^{2^{e-1}}x_{n-2^{e-1}}\right) - r^{f(n)}x_{n-f(n)}</math> (with each element of the sum being of the form <math>-q+q</math> and thus 0)

and then rebrackets by grouping the term before the sum with the first half of the first element of the sum, the second half of the first element of the sum with the first half of the second element of the sum, etc., and finally the second half of the last element of the sum with the term after the sum. From there, the proof is mostly just a matter of expanding definitions.

=== Possible tricks to optimize the implementation ===

Once the algorithm produces a value of <math>j_n</math> or <math>w_n</math>, it happens that it will never again read a value <math>j_{n'}</math> or <math>w_{n'}</math> where <math>f(n)=f(n')</math>. As such, it is possible to save memory by storing only one value of <math>j_n</math> and one value of <math>w_n</math> for each <math>f(n)</math>.

It is probably faster to switch to an alternative implementation (e.g. simple repeated multiplication) when the <math>f(n)</math> values are small, because at that point the numbers are small enough to fit into a machine register; this does not improve the asymptotic behaviour of the implementation but is likely to speed it up by a reasonably high constant factor.

Every <math>w'_n</math> is necessarily a multiple of <math>m^{f(n)}</math>. It seems like that might provide some sort of shortcut to calculate it faster, although the details are currently unclear.

=== Proof-of-concept implementation ===

Here's a proof-of-concept implementation, using Python 3 and gmpy2 (configured to calculate [[Hydra]], but it could easily be adapted for other consistent Collatz sequences with <math>m=2</math>):

<syntaxhighlight lang="python3">
import gmpy2
import time

# The Rules of Hydra:
# if x_n = 2y + 0, then x_{n+1} = 3y + 0, i.e. 2x_{n+1} = 3x_n + 0
# if x_n = 2y + 1, then x_{n+1} = 3y + 1, i.e. 2x_{n+1} = 3x_n - 1
m = 2 # modulus; denominator of the ratio between successive elements
r = 3 # numerator of the ratio between successive elements
J = [0, -1] # J[x_n % m] is the difference between m*x_{n+1} and r*x_n
x_0 = 3 # first term of the sequence

def f(n):
"""The largest power of 2 that divides into the argument"""
return n & ~(n - 1)

global mod_m_exp
if m == 2:
mod_m_exp = lambda v,e: gmpy2.f_mod_2exp(v, e)
else:
mod_m_exp = lambda v,e: gmpy2.f_mod(v, gmpy2.pow(m, e))

global div_m_exp
if m == 2:
div_m_exp = lambda v,e: gmpy2.f_div_2exp(v, e)
else:
div_m_exp = lambda v,e: gmpy2.f_div(v, gmpy2.pow(m, e))

global mul_m_exp
if m == 2:
mul_m_exp = lambda v,e: gmpy2.mpz(v) << e
else:
mul_m_exp = lambda v,e: gmpy2.mul(v, gmpy2.pow(m, e))

r_exp_cache = {1: gmpy2.mpz(r)}
def r_exp(e):
"""Returns r raised to the power of e. e must be a power of 2."""
global r_exp_cache
if not e in r_exp_cache:
r_exp_cache[e] = gmpy2.square(r_exp(e/2))
return r_exp_cache[e]

# The bulk of the calculation is to calculate w_n and j_n for each n.
# The definitions are:
# w_n = x_n mod m**f(n)
# j_n = m**f(n) * x_n - r**f(n) * x_{n-f(n)}
# The output from the program is the sequence of x_n mod m.
# This can be calculated by taking the values of w_n mod m.
w = {0: x_0} # most recently seen w for each f value
j = {} # most recently seen j for each f value

modulus_count = {0: 0, 1: 0}

n = 0
perf_counter_timestamp = time.perf_counter_ns()
while True:
last_f = f(n)
last_x_mod_m = mod_m_exp(w[last_f], 1)
# print(last_x_mod_m, end="")

modulus_count[last_x_mod_m] += 1
if n % 1000000 == 0:
last_timestamp = perf_counter_timestamp
perf_counter_timestamp = time.perf_counter_ns()
print("Reached n = ", n, "; modulus counts: ", modulus_count,
"; time for last 1000000 elements = ",
(perf_counter_timestamp - last_timestamp) // 1000000,
" ms", sep="")

n = n + 1
cur_f = f(n)

# j can be a sum of multiple terms; J[last_x_mod_m] is always present
# but if f > 1 there are other terms too
m_shift = cur_f - 1
r_shift = 1
new_j = mul_m_exp(J[last_x_mod_m], m_shift)
while m_shift >= r_shift:
m_shift -= r_shift
new_j += mul_m_exp(j[r_shift] * r_exp(r_shift), m_shift)
r_shift *= 2
j[cur_f] = new_j

# w can be calculated directly from the new j and the appropriate past w
wrap = lambda v: mod_m_exp(v, cur_f * 2)
past_w = w[f(n - f(n))]
new_w = wrap(wrap(new_j) + wrap(wrap(r_exp(cur_f)) * wrap(past_w)))
assert(mod_m_exp(new_w, cur_f) == 0)
new_w = div_m_exp(new_w, cur_f)
w[cur_f] = new_w
</syntaxhighlight>

This implementation is probably not suitable for serious use due to having poor constant factors: a faster implementation would use an alternative algorithm for low <math>f(n)</math> values. It is also intened only for <math>m=2</math>, because the complexity result is dependent on modulus by powers of <math>m</math> being fast regardless of the size of the dividend, but gmpy2 is only able to store numbers in binary and that operation is quick only if the modulus is a power of the base. As such, a full implementation of efficient consistent Collatz calculation would probably involve writing a new arbitrarily-large-integers library which is able to store numbers in arbitrary bases.

[[Category:Functions]]

Repeated Word List

2026-04-26T18:51:21Z

Polygon: /* RWLAcc */ underscore in Quick_Sim

{{Stub}}
[[File:RepWL 0RB0LC 1LA1RB 1RD0RE 1LC1LA ---0LD.png|thumb|500px|RepWL graph for non-halting TM {{TM|0RB0LC_1LA1RB_1RD0RE_1LC1LA_---0LD}} showing a closed graph.<ref>https://discord.com/channels/960643023006490684/1295401870298320977/1348277260221677641</ref>]]
'''Repeated Word List''' (short '''RepWL''', also called '''RepWL_ES''' where ES means ExecState) is a [[decider]] first introduced as part of [[Coq-BB5]]. It is a subset of the general [[CTL]] method and works by generating a graph of TM configurations, if the graph is shown to be closed, the TM is proven non-halting.

== Method ==
It works by splitting the tape contents into blocks ("words") of a given length <math>l</math>. Consecutive blocks grouped into powers by multiplicity. If there are more consecutive repeating blocks than a predefined repeat threshhold <math>T</math>, the exponent is given as <math>T+</math>. Consecutive blocks with no <math>+</math> in their exponent are called constant blocks. The simulation begins on the blank tape. When the TM head is facing a constant block, the TM is simulated until it either leaves the constant block, halts, or exceeds a predefined step limit <math>B</math>. This simulation inside a constant block is called block simulation. Once the TM has left the constant block, identical contiguous blocks are regrouped into powers. If the TM head is facing a group of blocks with a <math>+</math> in its exponent, the block directly faced by the TM head is separated from the group and the simulation splits into two branches: One where the original group's multiplicity is reduced to <math>T-1</math> and one where it stays at <math>T+</math>. This splitting is called regex branching. The decider constructs a graph of all instances of block simulation (called macro steps) and regex branching up to a predefined limit on the amount of nodes which are allowed to be visited, <math>N</math>. If the graph of a TM contains less than <math>N</math> nodes, the graph is closed. If the graph of a TM is closed and contains no halting configurations, the TM is proven non-halting.<ref name="bbchallenge 2025">https://arxiv.org/pdf/2509.12337 Determination of the fifth Busy Beaver value</ref>

== Variants ==

=== RWLAcc ===
RWLAcc is a block-based [[accelerated simulator]] similar to [[Quick_Sim]] capable of supporting L0 and L1 [[Inductive Proof System|inductive rules]].<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1333106708117196941</ref> It is part of mxdys's [[BB(6)]] Rocq decider set: https://github.com/ccz181078/busycoq/blob/BB6/verify/RWLAcc.v

== History ==
* In April 2024, RepWL is first introduced by mxdys as part of [[Coq-BB5]].<ref>https://github.com/ccz181078/Coq-BB5/blob/main/CoqBB5/BB5/Deciders/Decider_RepWL.v</ref>
* In May 2024, Repeated Word List is reproduced by savask.<ref>https://gist.github.com/savask/c7546bb6384984b2fb3cb90fc7925697</ref>
* In October 2024, [[User:Cosmo|Tristan Stérin]] reproduces Repeated Word List again.<ref>https://github.com/bbchallenge/bbchallenge-deciders/tree/main/decider-repeated-word-list-reproduction</ref>

== See also ==
* [[Halting Segment]]
* [[Finite Automata Reduction]]

== References ==

[[Category:Deciders]]

TMBR: April 2026

2026-04-26T18:42:54Z

Polygon: /* Holdouts */ Linked stage 3

User:Polygon

2026-04-26T18:28:49Z

Polygon: Added mxdys TM visualiser

List of user pages:
* [[User:Polygon/Collection of BB Champions]]
* [[User:Polygon/Better lower bound for BB(4,3)]]
* [[User:Polygon/Page for analyses]]
* [[User:Polygon/Page for testing]]

A list of interesting user pages:
* [[User:RobinCodes/Valuable user pages]]

Other sites:
* https://www.sligocki.com/
* https://groups.google.com/g/busy-beaver-discuss
* https://discuss.bbchallenge.org/
* https://bbchallenge.org/~pascal.michel/index.html
* https://nickdrozd.github.io/
* https://scottaaronson.blog/
* https://katelyndoucette.com/
* https://www.cse.unr.edu/~al/BusyBeaver.html
* https://cruzgodar.com/applets/lambda-calculus/?expression-textarea=%28%CE%BBa.aa%29%28%CE%BBa.a%28%CE%BBb.a%28%CE%BBc.ab%29%29a%29&animation-time-slider=1.095790162866908 (lambda calculus simulator)
* https://ccz181078.github.io/TM/LongAcc/index.html (Turing Machine Visualizer using longitudinal acceleration by mxdys)

Table of 30 longest running TMs across explored domains:
{| class="wikitable"
!Standard format
!Approximate sigma scores
!Discovered by
!Domain
|-
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|<math>(2 \uparrow^{15} 5) + 14</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1RZ1LA2RB_1RC3RC1LA2LB_2LB2RC1LC3RB|halt}}
|<math>(2 \uparrow^{12} 4) + 11</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|Pavel Kropitz
|[[BB(7)]]
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_3LA1RZ1LC2RA|halt}}
|<math>2 \uparrow^{6} 5</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_1RA1RZ1LC2RA|halt}}
|<math>2 \uparrow^{6} 4</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1})</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|<math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8.10237</math>
|mxdys
|[[BB(6)]]
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|<math>10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
|Pavel Kropitz
|[[BB(2,6)]]
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|<math>10 \uparrow\uparrow 11010000</math>
|mxdys
|[[BB(6)]]
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|<math>10 \uparrow\uparrow 19892.08</math>
|Peacemaker II
|[[BB(2,6)]]
|-
|{{TM|1RB0LB1RZ3LA_0LC3RB3RC1LB_2RB2LA3RA1LC|halt}}
|<math>10 \uparrow\uparrow 2048</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|<math>10 \uparrow\uparrow 1023.47221</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|<math>10 \uparrow\uparrow 619.07737</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|<math>10 \uparrow\uparrow 519.20</math>
|Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|<math>10 \uparrow\uparrow 512.10945</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|<math>10 \uparrow\uparrow 439.02781</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG|halt}}
|<math>10 \uparrow\uparrow 403.84</math>
|Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|<math>10 \uparrow\uparrow 286.17</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|<math>10 \uparrow\uparrow 246.32</math>
|lijil
|[[BB(7)]]
|-
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|<math>10 \uparrow\uparrow 243.88</math>
|lijil, Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|<math>10 \uparrow\uparrow 234.06408</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|<math>10 \uparrow\uparrow 228.78</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG|halt}}
|<math>10 \uparrow\uparrow 192.67</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|<math>10 \uparrow\uparrow 190.21359</math>
|Terry Ligocki
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ1LA2RB_2RC3RC3LB2RA_2LB3RB3LC1LA|halt}}
|<math>10 \uparrow\uparrow {189.05690}</math>
|[[User:XnoobSpeakable|XnoobSpeakable]]
|[[BB(3,4)]]
|-
|{{TM|1RB0LC_1LC1LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LA|halt}}
|<math>10 \uparrow\uparrow 188.28</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB3LA1RA2LB_2LC1RA2RC3RB_1RZ1LA2RA3RA|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|[[BB(3,4)]]
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|<math>10 \uparrow\uparrow 178.48320</math>
|Andrew Ducharme
|[[BB(4,3)]]
|}

BB(4,3)

2026-04-26T18:20:39Z

Polygon: /* Phase 1 */ linked TNF

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 [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== 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 ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a [[TNF]] tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| class="wikitable sortable" style="text-align: right"
! rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="28" style="text-align:left" |[https://drive.google.com/drive/folders/11FYe0CVDPczcgt4put3vsdGbeMWkM1S1?usp=sharing Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,641,006'''
|'''40.00%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(2,6)

2026-04-26T09:31:42Z

Polygon: /* Stage 4 */ fixed alignment

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 {{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}} was discovered by Pavel Kropitz in May 2023, proving the lower bound:<math display="block">S(2,6) > \Sigma(2,6) > 10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}} > 10 \uparrow \uparrow \uparrow 3</math>

== Top Halters ==
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The 20 highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|10 ↑↑ 19892.08
|Peacemaker II
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|}
All decimal places are truncated.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in November 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~24B(illion) TMs of which ~2.278B were holdout TMs. This was reduced to ~22M holdout TMs (a 99.02% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
|style="text-align:left" |Terry Ligocki
|2,278,655,696
|2,109,114,609
|7.44%
|40.9
|1,150.90
|15,468.23
|style="text-align:left" |Reverse_Engineer_Filter.py
|style="text-align:left", rowspan="100" |[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|2,109,114,609
|683,067,538
|67.61%
|452.8
|874.77
|1,293.79
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|683,067,538
|210,993,434
|69.11%
|396.4
|330.85
|478.72
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|210,993,434
|141,680,232
|32.85%
|273.9
|70.29
|213.97
|style="text-align:left" |CPS_Filter.py --block-size=3 --max_steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|141,680,232
|66,029,536
|53.40%
|486.6
|43.18
|80.87
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|66,029,536
|46,119,004
|30.15%
|167.4
|33.05
|109.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|46,119,004
|39,034,142
|15.36%
|170.1
|11.57
|75.34
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|39,034,142
|29,109,512
|25.43%
|2,221.6
|1.24
|4.88
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|29,109,512
|24,536,819
|15.71%
|384.2
|3.31
|21.05
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=6 --recursive --no-steps --time=0.05 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|24,536,819
|22,302,296
|9.11%
|1,047.5
|0.59
|6.51
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=4 --recursive --no-steps --time=1.00 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|}

== Phase 2 ==
When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===
Andrew Ducharme ran another pass of "lr_enum_continue" with the maximum number of steps set to 10 million. The holdouts were reduced from ~22.3M TMs to ~20.4M TMs (a 8.72% reduction). The entry in the table below has a rather technical/arcane/cryptic description. This was an effort to capture enough information to rerun that filter in parallel with specific C++ code, lr_enum_continue, and a specific parallel queuing system, Slurm:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
|style="text-align:left" |Andrew Ducharme
|22,302,296
|20,358,011
|8.72%
|1,350.0
|0.40
|4.59
|style="text-align:left" |lr_enum_continue ${WORK_DIR}chunk_${SLURM_ARRAY_TASK_ID} 10000000 ${WORK_DIR}halt_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}inf_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}unknown_${SLURM_ARRAY_TASK_ID}.txt "" false
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|}

=== Stage 2 ===
Starting from the results of Stage 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 50 variations were run. The holdouts were reduced from ~20.4M TMs to ~907K TMs (a 95.5% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|20,358,011
|19,500,847
|4.21%
|22.0
|10.84
|257.42
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|19,500,847
|18,747,861
|3.86%
|86.0
|2.43
|63.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|18,747,861
|4,811,076
|74.34%
|47.0
|82.33
|110.75
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|4,811,076
|2,982,075
|38.02%
|17.1
|29.74
|78.22
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|2,982,075
|2,897,340
|2.84%
|15.2
|1.55
|54.64
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,897,340
|2,850,781
|1.61%
|16.7
|0.77
|48.17
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|2,850,781
|2,759,635
|3.20%
|13.7
|1.85
|58.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,759,635
|1,953,426
|29.21%
|13.6
|16.48
|56.42
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,953,426
|1,855,545
|5.01%
|2.4
|11.18
|223.14
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,855,545
|1,647,269
|11.22%
|6.6
|8.80
|78.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,647,269
|1,608,166
|2.37%
|3.4
|3.20
|134.96
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,608,166
|1,585,745
|1.39%
|9.6
|0.65
|46.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,585,745
|1,555,673
|1.90%
|5.7
|1.47
|77.73
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,555,673
|1,428,534
|8.17%
|9.3
|3.78
|46.31
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,428,534
|1,340,964
|6.13%
|0.8
|29.70
|484.55
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 10000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,340,964
|1,286,439
|4.07%
|0.8
|18.40
|452.56
|style="text-align:left" |chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,286,439
|1,273,911
|0.97%
|0.8
|4.20
|430.88
|style="text-align:left" |chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,273,911
|1,265,198
|0.68%
|0.8
|2.88
|420.73
|style="text-align:left" |chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,265,198
|1,258,925
|0.50%
|0.9
|1.99
|400.83
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,258,925
|1,242,136
|1.33%
|0.8
|5.51
|412.84
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,242,136
|1,231,731
|0.84%
|1.0
|2.78
|331.77
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,231,731
|1,216,646
|1.22%
|1.0
|4.15
|338.72
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,216,646
|1,214,294
|0.19%
|0.9
|0.76
|393.03
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,214,294
|1,213,431
|0.07%
|0.9
|0.28
|391.30
|style="text-align:left" |chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,213,431
|1,211,390
|0.17%
|1.1
|0.52
|307.13
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 1 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,211,390
|1,209,989
|0.12%
|1.1
|0.35
|306.09
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,989
|1,209,974
|0.00%
|0.9
|0.00
|381.42
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,974
|1,201,890
|0.67%
|2.5
|0.90
|134.19
|style="text-align:left" |chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,201,890
|1,200,086
|0.15%
|1.3
|0.37
|248.36
|style="text-align:left" |chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,200,086
|1,199,734
|0.03%
|1.2
|0.08
|270.32
|style="text-align:left" |chr_asth 0 chr_LRUH 3 chr_H 3 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,199,734
|1,198,893
|0.07%
|2.3
|0.10
|147.66
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 6 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,198,893
|1,165,493
|2.79%
|4.5
|2.05
|73.44
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 4 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,165,493
|1,153,863
|1.00%
|9.3
|0.35
|34.88
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 0 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,153,863
|1,144,711
|0.79%
|3.7
|0.69
|87.51
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 5 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,144,711
|1,127,789
|1.48%
|7.9
|0.60
|40.26
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 30000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,127,789
|1,124,762
|0.27%
|4.7
|0.18
|66.75
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|1,124,762
|1,117,226
|0.67%
|5.6
|0.37
|55.36
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,117,226
|1,109,057
|0.73%
|7.7
|0.30
|40.49
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,109,057
|1,083,097
|2.34%
|11.4
|0.63
|27.06
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 30000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,083,097
|1,077,833
|0.49%
|11.2
|0.13
|26.81
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,077,833
|1,066,795
|1.02%
|24.1
|0.13
|12.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 2 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,066,795
|1,039,229
|2.58%
|52.6
|0.15
|5.64
|style="text-align:left" |chr_LRUH 14 chr_H 6 MitM_CTL NG maxT 100000 NG_n 11 run
|-
|style="text-align:left" |Terry Ligocki
|1,039,229
|1,019,286
|1.92%
|43.5
|0.13
|6.63
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,019,286
|993,556
|2.52%
|66.8
|0.11
|4.24
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 8 H 2 tH 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|993,556
|985,718
|0.79%
|78.3
|0.03
|3.53
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|985,718
|981,095
|0.47%
|83.7
|0.02
|3.27
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 0 n 9 run
|-
|style="text-align:left" |Terry Ligocki
|981,095
|975,912
|0.53%
|79.4
|0.02
|3.43
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|975,912
|974,180
|0.18%
|84.6
|0.01
|3.20
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 4 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|974,180
|971,254
|0.30%
|96.9
|0.01
|2.79
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|971,254
|970,101
|0.12%
|105.6
|0.00
|2.56
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 0 tH 0 n 18 run
|}

=== Stage 3 ===
Starting from the results of Stage 2, Andrew Ducharme ran "lr_enum_continue" with the maximum number of steps set to 100 million, then "Enumerate.py" with various parameters. A total of 10 Enumerate variations were run. The holdouts were reduced from ~970K TMs to ~867K TMs (a 10.63% 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:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Andrew Ducharme
|970,101
|939,447
|3.16%
| --
| --
| --
|style="text-align:left" |lr_enum_continue 100_000_000 steps
| rowspan="11" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|939,447
|903,224
|3.86%
|440.3
|0.03
|0.59
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=4 --no-ctl --lin-steps=0 --time=2 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|903,224
|895,813
|0.82%
|647.7
|0.00
|0.39
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=3 --no-ctl --lin-steps=0 --time=3 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|895,813
|889,838
|0.67%
|609.3
|0.00
|0.41
| style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=8 --no-ctl --lin-steps=0 --time=4 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|889,838
|880,278
|1.07%
|1,638.9
|0.00
|0.15
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=12 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|880,278
|877,485
|0.32%
|1,885.5
|0.00
|0.13
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=6 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|877,485
|875,062
|0.28%
|2,068.8
|0.00
|0.12
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=5 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|875,062
|873,469
|0.18%
|1,785.4
|0.00
|0.14
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=7 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|873,469
|870,085
|0.39%
|9,270.0
|0.00
|0.03
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=2 --tape-limit=500 --time=120 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|870,085
|869,001
|0.12%
|4,498.3
|0.00
|0.05
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=10_000_000 --block-mult=60 --tape-limit=5000 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|869,001
|867,008
|0.23%
|3997.4
|0.00
|0.06
|style="text-align:left"|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=9 --tape-limit=5000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl --force --save-freq=250
|}
The total time spent on the lr_enum_continue computation was not recorded.

=== Stage 4 ===
Following the release of @mxdys's implementation of FAR deciders in C++, these deciders were applied to the 2x6 holdouts by Andrew Ducharme. The details are given in this table, including links to the Google Drive with the holdouts and solved TMs per decider:

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)
{| class="wikitable sortable" style="text-align: right"
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|867,008
|811,301
|6.43%
|0.043
|364.10
|5,666.72
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 1 tH 1 n 2
| rowspan="28" |[https://drive.google.com/drive/folders/18njhmOzRc67zCmVuLd0aDxl6ETBhL1gy?usp=sharing Google Drive]
|-
|811,301
|806,119
|0.64%
|0.159
|9.03
|1,413.42
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 3 H 1 tH 1 n 2
|-
|806,119
|736,690
|8.61%
|0.548
|35.21
|408.78
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 1 tH 1 n 2
|-
|736,690
|736,504
|0.03%
|0.009
|5.81
|23,021.56
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 1 H 1 tH 1 n 1
|-
|736,504
|735,317
|0.16%
|0.058
|5.71
|3,540.88
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
|-
|735,317
|733,717
|0.22%
|0.341
|1.30
|599.28
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 2
|-
|733,717
|673,920
|8.15%
|3.8
|4.43
|54.32
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 4
|-
|673,920
|652,828
|3.13%
|~10
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 6 H 2 tH 2 n 4
|-
|652,828
|645,264
|1.16%
|~12
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 8 H 2 tH 2 n 4
|-
|645,264
|641,388
|0.60%
|~15
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 100000 LRUH 10 H 2 tH 2 n 10
|-
|641,388
|635,505
|0.92%
|~200
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 10 H 1 tH 2 n 10
|-
|635,505
|616,639
|2.97%
| ---
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [3-10]
|-
|616,639
|592,039
|3.99%
|~700
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H 0 tH 0 n [1-10]
|-
|592,039
|576,938
|2.55%
|~800
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H [0-1] tH [0-1] n [1-10]
|-
|576,938
|572,963
|0.69%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 0 tH 0 n [1-10]
|-
|572,963
|567,971
|0.87%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 2 tH 0 n [1-10]
|-
|567,971
|566,096
|0.33%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 H 0 tH 0 n [1-10]
|-
|566,096
|564,290
|0.32%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 0 tH [0,2] n [1-10]
|-
|564,290
|559,553
|0.84%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 1 n [1-10]
|-
|559,553
|558,039
|0.27%
|~900
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 2 n [1-10]
|-
|558,039
|556,814
|0.22%
|~14,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [12,16] H [0-2] tH [0-2] n [1-10]
|-
|556,814
|554,479
|0.42%
|~3,600
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [1-3]
|-
|554,479
|551,586
|0.52%
|~5000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4
|-
|551,586
|548,993
|0.47%
|~13,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 5
|-
|548,993
|545,005
|0.73%
|~57,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 and 8
|-
|545,005
|542,325
|0.49%
|6851.2
|0.00
|0.022
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=96 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|542,325
|537,393
|0.91%
|9032.1
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=2 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|537,393
|536,112
|0.24%
|8969.4
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=3 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|}

== References ==


[[Category: BB Domains]][[Category:BB(2,6)]]

TMBR: April 2026

2026-04-26T08:55:51Z

Polygon: /* Holdouts */ Added new BB(6) holdouts list

BB(6)

2026-04-26T08:47:50Z

Polygon: /* History */ updated holdout count

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== History ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1119 machines up to equivalence and 2337 machines not considering equivalence as of April 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet]. The informal holdout count is 1205.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 to 1e15. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet].

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}, Space Needle
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}, similar to Space Needle
* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}, similar to Space Needle

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}
* {{TM|1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA|halt}}
|<math>10^{1\,400\,000\,000}</math>(runtime)
|Racheline<ref>https://discord.com/channels/960643023006490684/1345502880727040091/1345502880727040091</ref>
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub) upon initial application.

== References ==
<references />

[[Category:BB Domains]][[Category:BB(6)]]

Holdouts lists

2026-04-26T08:45:55Z

Polygon: Updated BB(6) holdout count and added new holdouts list

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1119
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|536,112
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

File:BB6 holdouts 1119.txt

2026-04-26T08:43:04Z

Polygon: *April 25

== Summary ==
Holdouts list shared on April 25 2026 by mxdys over Discord.

File:BB6 holdouts 1119.txt

2026-04-26T08:42:37Z

Polygon: Holdouts list shared on April 26 2026 by mxdys over Discord.

== Summary ==
Holdouts list shared on April 26 2026 by mxdys over Discord.

User:Polygon/Collection of BB Champions

2026-04-26T08:40:16Z

Polygon: /* Regular Busy Beaver for Lambda Calculus (BBλ(n)) */ updated BBlambda(38) to proven

A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent [[:Category:Functions|functions]]. Note that for all functions with the input format f(n,m), n denotes the number of states and m denotes the number of symbols of the relevant Busy Beaver domain. Note that for functions with this input format f(n) = f(n,2).

Note: highest ref name in use: 4
= '''State-and-Symbol-Limited Busy Beaver functions''' =
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
== Original Busy Beaver Functions ==
=== Maximum Shifts Function ([[Busy Beaver Functions|S(n,m)]], also commonly called BB(n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Champions
|-
|[[BB(1)]]
|1
|{{TM|1RZ---|halt}}
|-
|[[BB(2)]]
|6
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|-
|[[BB(3)]]
|21
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|[[BB(4)]]
|107
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|[[BB(5)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|[[BB(6)]]
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(8)]]
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Runtime
!Champions
|-
|BB(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|[[BB(3,3)]]
|<math>\geq 119\,112\,334\,170\,342\,541 > 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|[[BB(4,3)]]
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Runtime
!Champions
|-
|BB(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|[[BB(3,4)]]
|<math>> (2 \uparrow^{15} 5) + 14</math><ref name=":0">S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Blog post, 2024. Accessed 15 August 2025.</ref>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
{| class="wikitable"
|+
!'''5 Symbols:'''
!Runtime
!Champions
|-
|BB(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|[[BB(2,5)]]
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|-
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|}
{| class="wikitable"
|+
!'''6 Symbols:'''
!Runtime
!Champions
|-
|BB(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1">S. Ligocki, "[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html BB(2,6) > 10↑↑10↑↑10↑↑3]". Blog post, 2023. Accessed 15 August 2025.</ref>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}

=== Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]]) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Score
!Champions
|-
|Σ(1)
|1
|{{TM|1RZ---|halt}}
|-
|Σ(2)
|4
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|Σ(3)
|6
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|-
|Σ(4)
|13
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|-
|Σ(5)
|4098
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|-
|Σ(6)
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|Σ(7)
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Score
!Champions
|-
|Σ(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|Σ(2,3)
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|Σ(3,3)
|≥ 374,676,383
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|Σ(4,3)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Score
!Champions
|-
|Σ(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|Σ(2,4)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|Σ(3,4)
|<math>\geq (2 \uparrow^{15} 5) + 14 </math><ref name=":0" />
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
{| class="wikitable"
|+
!'''5 Symbols:'''
!Score
!Champions
|-
|Σ(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|Σ(2,5)
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|}
{| class="wikitable"
|+
!'''6 Symbols:'''
!Score
!Champions
|-
|Σ(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|Σ(2,6)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1" />
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}

== Beeping Busy Beavers ==
=== Beeping Busy Beaver ([[BBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1)
|1
|
|-
|BBB(2)
|6
|{{TM|1RB1LB_1LB1LA}}
|-
|BBB(3)
|55
|{{TM|1RB0LB_1LA0RC_1LC1LA}}
|-
|BBB(4)
|≥ 32,779,478
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BBB(5)
|<math>\geq 10^{14006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,3)
|
|
|-
|BBB(2,3)
|59<ref name=":2">Nick Drozd. "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA}}
|-
|BBB(3,3)
|≥ 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,4)
|
|
|-
|BBB(2,4)
|<math>\geq 205\,770\,076\,433\,044\,242\,247\,859 > 2\times 10^{23}</math><ref name=":3" />
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
=== Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBBB(1)
|2
|
|-
|BBBB(2)
|17
|
|}

== Maximum Consecutive Ones Function ([[Num]](n,m)) ==
{| class="wikitable"
|+
!Domain
!Number of Ones
!Champions
|-
|num(1)
|1
|{{TM|1RZ---|halt}}
|-
|num(2)
|4
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|num(3)
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|num(4)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|-
|num(5)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|-
|num(2,3)
|6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|-
|num(3,3)
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|num(4,3)
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}

== Maximum Space Function ([[Maximum Space Function|BBspace]](n,m)) ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Cells visited
!Champions
|-
|BBspace(1,2)
|1
|{{TM|1RZ---|halt}}
|-
|BBspace(2,2)
|4
|{{TM|1RB1LB_1LA1RZ|halt}} and {{TM|1RB0LB_1LA1RZ|halt}}
|-
|BBspace(3,2)
|7
|{{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} and {{TM|1RB0RC_1LC1RZ_1RA0LB|halt}}
|-
|BBspace(4,2)
|16
|{{TM|1RB0RA_1LC0RD_0LD0LB_1RA1RZ|halt}}
|-
|BBspace(5,2)
|12289
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Cells visited
!Champions
|-
|BBspace(1,3)
|
|
|-
|BBspace(2,3)
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|BBspace(2,4)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|}

== Size of the runtime spectrum (R(n,m)) ==
There currently doesn't seem to be any available information about values of this function.

== Reversible Turing Machines ==
=== Maximum Shifts Function ([[Reversible Turing Machine|BBrev]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BBrev(1)
|
|
|-
|BBrev(2)
|6
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|BBrev(3)
|17
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|BBrev(4)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|BBrev(5)
|388
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|BBrev(6)
|≥ 537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|-
|BBrev(7)
|<math>> 10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|}

=== Maximum Score Function (Σrev(n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Score
!Champions
|-
|Σrev(1)
|
|
|-
|Σrev(2)
|≥ 2
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|Σrev(3)
|≥ 4
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|Σrev(4)
|≥ 6
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|Σrev(5)
|≥ 16
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|Σrev(6)
|≥ 1161
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|}

== Blanking Busy Beaver ([[Blanking Busy Beaver|BLB(n,m)]]) ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BLB(1)
|nonexistent
|nonexistent
|-
|BLB(2)
|8<ref name=":3">Nick Drozd. "[https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]". Accessed 15 August 2025.</ref>
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3)
|≥ 34<ref name=":4">Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(4)
|≥ 32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5)
|≥ 32,810,047<ref>Comment #71 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6)
|≥ 65,538,549<ref>Comment #62 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(7)
|> 10^183.76332<ref>https://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409</ref>
|{{TM|1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|nonexistent
|nonexistent
|-
|BLB(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|> 1042,745
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(1,4)
|nonexistent
|nonexistent
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== Lazy Beaver ==
=== Shifts Function ([[Lazy Beaver|LB]](n,m)) ===
{| class="wikitable"
|+
!
!1 State
!2 States
!3 States
!4 States
!5 States
!6 States
|-
|2 Symbols
|2
|7
|22
|72
|427
|8407
|-
|3 Symbols
|2
|23
|351
|189,270
|
|
|-
|4 Symbols
|2
|93
|242,789
|
|
|
|-
|5 Symbols
|2
|956
|
|
|
|
|-
|6 Symbols
|2
|33,851
|
|
|
|
|}

== Period-oriented Busy Beavers ==
=== Busy Preperiodic Beaver ([[BBS]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,2)
|0
|{{TM|1RA---}}
|-
|BBS(2,2)
|≥ 9
|{{TM|1RB0LB_1LA0RB}} proven winner?
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|≥ 119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(5,2)
|> 1014,006
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|0
|{{TM|1RA------}}
|-
|BBS(2,3)
|≥ 165<ref>Nick Drozd. [https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]</ref>
|{{TM|1RB0LA---_1LB2LA0RB}}
|-
|BBS(3,3)
|> 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|-
|BBS(4,3)
|> <math>10 \uparrow^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,4)
|0
|{{TM|1RA---------}}
|-
|BBS(2,4)
|≥ 205,770,076,433,044,242,247,860
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}

=== Busy Periodic Beaver ([[BBP]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Period
!Champions
|-
|BBP(1,2)
|1
|{{TM|1RA---}}
|-
|BBP(2,2)
|≥ 9
|{{TM|1RB0RB_1LB1RA}} proven winner?
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(4,2)
|≥ 212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Period
!Champions
|-
|BBP(1,3)
|1
|{{TM|1RA------}}
|-
|BBP(2,3)
|
|
|-
|BBP(3,3)
|≥ 1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|1
|{{TM|1RA---------}}
|-
|BBP(2,4)
|≥ 33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|}

= '''Instruction-Limited Busy Beaver''' =
== Instruction-Limited Classical Busy Beaver Functions ==
=== Instruction-Limited Maximum Shifts Function ([[BBi]](n)) ===
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BBi(1)
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|BBi(2)
|3
|{{TM|0RB---_1LA---|halt}}
|-
|BBi(3)
|5
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|16
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|37
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|123
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,963
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|BBi(8)
|<math>>6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}

=== Instruction-Limited Maximum Score Function ([[Σi]](n)) ===
{| class="wikitable"
|+
!
!Score
!Champions
|-
|Σi(1)
|1
|{{TM|1RH---|halt}}
|-
|Σi(2)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|Σi(3)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|Σi(4)
|5
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|Σi(5)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|Σi(6)
|14
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|Σi(7)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|Σi(8)
|<math>>1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|Σi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}

== Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BLBi(1)
|nonexistent
|nonexistent
|-
|BLBi(2)
|nonexistent
|nonexistent
|-
|BLBi(3)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLBi(4)
|12
|{{TM|1RB---_1RC---_1LC0RC}}
|-
|BLBi(5)
|30
|{{TM|1RB------_1RC------_2LC2RC0RC}}
|-
|BLBi(6)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|808
||{{TM|1RB------_1RC------_0RD2LC---_1LD2RD0RC}}
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|-
|BLBi(9)
|> 1042,745
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}

== Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|gBBi(1)
|1
|
|-
|gBBi(2)
|3
|
|-
|gBBi(3)
|5
|
|-
|gBBi(4)
|13
|
|-
|gBBi(5)
|19
|
|-
|gBBi(6)
|25
|
|-
|gBBi(7)
|41
|
|-
|gBBi(8)
|55
|
|-
|gBBi(9)
|238
|
|-
|gBBi(10)
|941
|
|-
|gBBi(11)
|1341
|
|-
|gBBi(12)
|10465
|
|-
|gBBi(13)
|10675
|
|-
|gBBi(14)
|≥ 9,874,580
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}

= '''Program-Limited Busy Beaver''' =
== Busy Beaver for Lambda Calculus ==
=== Regular Busy Beaver for Lambda Calculus ([[BBλ]](n)) ===
For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of <math> 20 \geq n </math> BBλ(n) = n.
{| class="wikitable"
|+
!
!BBλ(n)
!Champions
|-
|BBλ(4)
|4
|<math>\lambda 1</math>
|-
|BBλ(6)
|6
|<math>\lambda\lambda 1</math>
|-
|BBλ(7)
|7
|<math>\lambda\lambda 2</math>
|-
|BBλ(8)
|8
|<math>\lambda\lambda\lambda 1</math>
|-
|BBλ(9)
|9
|<math>\lambda\lambda\lambda 2</math>
|-
|BBλ(10)
|10
|<math>\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(11)
|11
|<math>\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(12)
|12
|<math>\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(13)
|13
|<math>\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(14)
|14
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(15)
|15
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(16)
|16
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(17)
|17
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(18)
|18
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(19)
|19
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(20)
|20
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(21)
|22
|<math>\lambda(1(\lambda 2))(1(\lambda 2))</math>
|-
|BBλ(22)
|24
|<math>\lambda(1 1) (1 1) (1 1)</math>
|-
|BBλ(23)
|26
|<math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>
|-
|BBλ(24)
|30
|<math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>
|-
|BBλ(25)
|42
|<math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|BBλ(26)
|52
|<math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>
|-
|BBλ(27)
|44
|<math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|BBλ(28)
|58
|<math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>
|-
|BBλ(29)
|223
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
|-
|BBλ(30)
|160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|-
|BBλ(31)
|267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|-
|BBλ(32)
|298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|-
|BBλ(33)
|1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|-
|BBλ(34)
|327,686
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(35)
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(36)
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
|-
|BBλ(37)
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|<math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(38)
|<math> 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|<math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(39)
|<math>\geq 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(40)
|<math>> (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(41)
|<math>\geq 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|<math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(42)
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(43)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|BBλ(44)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(45)
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|BBλ(46)
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(47)
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
|-
|BBλ(48)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(49)
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|BBλ(61)
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|BBλ(86)
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(90)
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(94)
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(95)
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(96)
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(100)
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(201)
|> q(5)
|<code>too large to show</code>
|-
|BBλ(331)
|lim(BMS)
|<code>too large to show</code>
|-
|BBλ(1850)
|<math>> \text{Loader's Number}</math>
|<code>too large to show</code>
|}

=== Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ1(n)]]) ===
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
|+
!
!BBλ1(n)
!Champions
|-
|BBλ1(1)
|0
|
|-
|BBλ1(2)
|1
|<math>1</math>
|-
|BBλ1(3)
|0
|
|-
|BBλ1(4)
|4
|<math>\lambda 1</math>
|-
|BBλ1(5)
|5
|<math>\lambda 2</math>
|-
|BBλ1(6)
|6
|<math>\lambda \lambda 1</math>
|-
|BBλ1(7)
|7
|<math>\lambda \lambda 2</math>
|-
|BBλ1(8)
|26
|<math>1 (\lambda 1)</math>
|-
|BBλ1(9)
|9
|<math>\lambda \lambda 2</math>
|-
|BBλ1(10)
|36
|<math>1 (\lambda \lambda 1)</math>
|-
|BBλ1(11)
|41
|<math>1 (\lambda \lambda 2)</math>
|-
|BBλ1(12)
|266
|<math>1 (1 (\lambda 1))</math>
|-
|BBλ1(13)
|51
|<math>1 (\lambda \lambda 2)</math>
|-
|BBλ1(14)
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<math>1 (1 (\lambda \lambda 1))</math>
|-
|BBλ1(15)
|<math>f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
|<math>1 (1 (\lambda \lambda 2))</math>
|-
|BBλ1(16)
|<math>f(266)</math>
|<math>1 (1 (1 (\lambda 1)))</math>
|-
|BBλ1(17)
|<math>f(51)</math>
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|-
|BBλ1(18)
|<math>f^{4}(4) = f(f(266))</math>
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(19)
|<math>f^{3}(7) = f(f(41))</math>
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|-
|BBλ1(20)
|<math>f^{6}(4) = f^{4}(266)</math>
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(21)
|<math>f^{7}(4) = f^{5}(266)</math>
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|-
|BBλ1(22)
|<math>f^{52}(4) = f^{50}(266)</math>
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|-
|BBλ1(28)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(29)
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|}

=== Busy Beaver for De Bruijn Lambda Calculus ===
{| class="wikitable"
!n
!Value
!Champion
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|10
|<math>> 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|11
|<math>> 10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|-
|15
|<math>> f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|}

== [[Busy Beaver for SKI calculus]] ==
=== Busy Beaver for SKI calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || 1 || S
|-
| 2 || 2 || SS
|-
| 3 || 3 || SSS
|-
| 4 || 4 || SSSS
|-
| 5 || 6 || SSS(SS)
|-
| 6 || 17 || SSS(SI)S
|-
|7
|≥ 18
|S(SSS(SI)S)
|-
|8
|≥ 19
|SS(SSS(SI)S)
|-
|9
|≥ 519
|SSI((S(SS)S)S)K
|-
|10
|≥ 1041
|SSI((S(SS)S)S)KS
|}

=== Busy Beaver for SK calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || = 1 || S
|-
| 2 || = 2 || SS
|-
| 3 || = 3 || SSS
|-
| 4 || = 4 || <code>SSSS</code>
|-
| 5 || = 6 || <code>SSS(SS)</code>
|-
| 6 || ≥ 8 || <code>SSS(SSS)</code>
|-
|7
|≥ 10
|<code>SSS(SSSS)</code>
|-
|8
|≥ 23
|<code>SSS(S(SKS))S</code>
|}

== Fractran ([[Fractran|BBf]](n)) ==
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|-
|20
| 746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
& -1 & -1 & +1 & \\
+1 & -1 & & & +1 \\
+1 & +1 & & -1 & -1 \\
-1 & & +1 & & \\
& +2 & -1 & &
\end{bmatrix}</math>
|-
|21
|≥ 31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|}

== Cyclic Tag ([[Cyclic Tag|CTBB(n)]]) ==
{| class="wikitable"
|+
!
!Runtime
!Champions
|-
|CTBB(2)
|5<ref>https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233</ref>
|
|-
|CTBB(3)
|> 38<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517</ref>
|
|-
|CTBB(4)
|≥ 672<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043</ref>
|
|-
|CTBB(5)
|≥ 2^2^2^2^182<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063</ref>
|
|-
|CTBB(6)
|> 2↑↑↑131<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479</ref>
|
|-
|CTBB(7)
|> 4↑↑↑↑(4↑↑↑3)<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281</ref> <ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217</ref>
|
|}

== [[Tiny Tag]] ==
{| class="wikitable"
|+
!n
!Runtime
!Champions
|-
|1
|1
|0
|-
|2
|5
|10
|-
|3
|9
|101
|-
|4
|13
|1001
|-
|5
|15
|10101
|-
|6
|334
|010111
|-
|7
|404
|1010111
|-
|8
|670
|11100101
|-
|9
|12584
|001101110
|-
|10
|2180995
|0100011110
|}

== Minsky Machines ([[Register machine|MBB(n)]]) ==

{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+Z</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|≥ 49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|}

= '''Turmites''' =
== Terminating Turmites ([[TT]](n,k), 1D Turmites) ==
Where n is the amount of states and k is the amount of symbols.
{| class="wikitable"
!Domain
!Runtime
!Champions
|-
|TT(2)
|≥ 13
|<code>1TB---_1PA0PB</code>
|-
|TT(3)
|≥ 82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4)
|≥ 48,186
|<code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>
|-
|TT(2,3)
|≥ 223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3)
|≥ 45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|-
|TT(2,4)
|> 3.467*1015
|<code>1TA2PB3TB---_3TA1PB1TA1PA</code>
|}

== 2D Turmites ([[Terminating Turmite|turNing machines]]) ==
There are currently no known/available Champions for this function.

= '''Doodle Function ([[Doodle function|doodle(c,n)]])''' =
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Champions
|-
|doodle(3,2)
|≥ 487
|
|}

= '''Bug Function''' =
Bug(2,2) = 2
<pre>
####
#S-#
#.F#
####
</pre>

Bug(3,3) = 8
<pre>
#####
#S..#
#.#.#
#.#F#
#####
</pre>

Bug(4,4) = 20
<pre>
######
#S...#
#..###
#....#
#..#F#
######
</pre>

Bug(5,5) = 42
<pre>
#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######
</pre>

Bug(6,6) = 96
<pre>
########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########
</pre>

Bug(7,7) = 218
<pre>
#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########
</pre>

Bug(8,8) = 506
<pre>
##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########
</pre>

= '''References''' =

User:Polygon/Collection of BB Champions

2026-04-26T08:39:30Z

Polygon: /* Fractran (BBf(n)) */ updated BBf(20) to proven

A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent [[:Category:Functions|functions]]. Note that for all functions with the input format f(n,m), n denotes the number of states and m denotes the number of symbols of the relevant Busy Beaver domain. Note that for functions with this input format f(n) = f(n,2).

Note: highest ref name in use: 4
= '''State-and-Symbol-Limited Busy Beaver functions''' =
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
== Original Busy Beaver Functions ==
=== Maximum Shifts Function ([[Busy Beaver Functions|S(n,m)]], also commonly called BB(n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Champions
|-
|[[BB(1)]]
|1
|{{TM|1RZ---|halt}}
|-
|[[BB(2)]]
|6
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|-
|[[BB(3)]]
|21
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|[[BB(4)]]
|107
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|[[BB(5)]]
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|[[BB(6)]]
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|-
|[[BB(8)]]
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Runtime
!Champions
|-
|BB(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|[[BB(2,3)]]
|38
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|[[BB(3,3)]]
|<math>\geq 119\,112\,334\,170\,342\,541 > 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|[[BB(4,3)]]
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Runtime
!Champions
|-
|BB(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|[[BB(2,4)]]
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|[[BB(3,4)]]
|<math>> (2 \uparrow^{15} 5) + 14</math><ref name=":0">S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Blog post, 2024. Accessed 15 August 2025.</ref>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
{| class="wikitable"
|+
!'''5 Symbols:'''
!Runtime
!Champions
|-
|BB(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|[[BB(2,5)]]
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|-
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|}
{| class="wikitable"
|+
!'''6 Symbols:'''
!Runtime
!Champions
|-
|BB(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1">S. Ligocki, "[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html BB(2,6) > 10↑↑10↑↑10↑↑3]". Blog post, 2023. Accessed 15 August 2025.</ref>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}

=== Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]]) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Score
!Champions
|-
|Σ(1)
|1
|{{TM|1RZ---|halt}}
|-
|Σ(2)
|4
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|Σ(3)
|6
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|-
|Σ(4)
|13
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|-
|Σ(5)
|4098
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|-
|Σ(6)
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|Σ(7)
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Score
!Champions
|-
|Σ(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|Σ(2,3)
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|Σ(3,3)
|≥ 374,676,383
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|Σ(4,3)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Score
!Champions
|-
|Σ(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|Σ(2,4)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|Σ(3,4)
|<math>\geq (2 \uparrow^{15} 5) + 14 </math><ref name=":0" />
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
{| class="wikitable"
|+
!'''5 Symbols:'''
!Score
!Champions
|-
|Σ(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|Σ(2,5)
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|}
{| class="wikitable"
|+
!'''6 Symbols:'''
!Score
!Champions
|-
|Σ(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|Σ(2,6)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1" />
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}

== Beeping Busy Beavers ==
=== Beeping Busy Beaver ([[BBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1)
|1
|
|-
|BBB(2)
|6
|{{TM|1RB1LB_1LB1LA}}
|-
|BBB(3)
|55
|{{TM|1RB0LB_1LA0RC_1LC1LA}}
|-
|BBB(4)
|≥ 32,779,478
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BBB(5)
|<math>\geq 10^{14006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,3)
|
|
|-
|BBB(2,3)
|59<ref name=":2">Nick Drozd. "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA}}
|-
|BBB(3,3)
|≥ 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,4)
|
|
|-
|BBB(2,4)
|<math>\geq 205\,770\,076\,433\,044\,242\,247\,859 > 2\times 10^{23}</math><ref name=":3" />
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
=== Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBBB(1)
|2
|
|-
|BBBB(2)
|17
|
|}

== Maximum Consecutive Ones Function ([[Num]](n,m)) ==
{| class="wikitable"
|+
!Domain
!Number of Ones
!Champions
|-
|num(1)
|1
|{{TM|1RZ---|halt}}
|-
|num(2)
|4
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|num(3)
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|num(4)
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|-
|num(5)
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|-
|num(2,3)
|6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|-
|num(3,3)
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|num(4,3)
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}

== Maximum Space Function ([[Maximum Space Function|BBspace]](n,m)) ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Cells visited
!Champions
|-
|BBspace(1,2)
|1
|{{TM|1RZ---|halt}}
|-
|BBspace(2,2)
|4
|{{TM|1RB1LB_1LA1RZ|halt}} and {{TM|1RB0LB_1LA1RZ|halt}}
|-
|BBspace(3,2)
|7
|{{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} and {{TM|1RB0RC_1LC1RZ_1RA0LB|halt}}
|-
|BBspace(4,2)
|16
|{{TM|1RB0RA_1LC0RD_0LD0LB_1RA1RZ|halt}}
|-
|BBspace(5,2)
|12289
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Cells visited
!Champions
|-
|BBspace(1,3)
|
|
|-
|BBspace(2,3)
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|BBspace(2,4)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|}

== Size of the runtime spectrum (R(n,m)) ==
There currently doesn't seem to be any available information about values of this function.

== Reversible Turing Machines ==
=== Maximum Shifts Function ([[Reversible Turing Machine|BBrev]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BBrev(1)
|
|
|-
|BBrev(2)
|6
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|BBrev(3)
|17
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|BBrev(4)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|BBrev(5)
|388
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|BBrev(6)
|≥ 537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|-
|BBrev(7)
|<math>> 10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|}

=== Maximum Score Function (Σrev(n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Score
!Champions
|-
|Σrev(1)
|
|
|-
|Σrev(2)
|≥ 2
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|Σrev(3)
|≥ 4
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|Σrev(4)
|≥ 6
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|Σrev(5)
|≥ 16
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|Σrev(6)
|≥ 1161
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|}

== Blanking Busy Beaver ([[Blanking Busy Beaver|BLB(n,m)]]) ==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BLB(1)
|nonexistent
|nonexistent
|-
|BLB(2)
|8<ref name=":3">Nick Drozd. "[https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]". Accessed 15 August 2025.</ref>
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3)
|≥ 34<ref name=":4">Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(4)
|≥ 32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5)
|≥ 32,810,047<ref>Comment #71 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6)
|≥ 65,538,549<ref>Comment #62 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|-
|BLB(7)
|> 10^183.76332<ref>https://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409</ref>
|{{TM|1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|nonexistent
|nonexistent
|-
|BLB(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|> 1042,745
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(1,4)
|nonexistent
|nonexistent
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}

== Lazy Beaver ==
=== Shifts Function ([[Lazy Beaver|LB]](n,m)) ===
{| class="wikitable"
|+
!
!1 State
!2 States
!3 States
!4 States
!5 States
!6 States
|-
|2 Symbols
|2
|7
|22
|72
|427
|8407
|-
|3 Symbols
|2
|23
|351
|189,270
|
|
|-
|4 Symbols
|2
|93
|242,789
|
|
|
|-
|5 Symbols
|2
|956
|
|
|
|
|-
|6 Symbols
|2
|33,851
|
|
|
|
|}

== Period-oriented Busy Beavers ==
=== Busy Preperiodic Beaver ([[BBS]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,2)
|0
|{{TM|1RA---}}
|-
|BBS(2,2)
|≥ 9
|{{TM|1RB0LB_1LA0RB}} proven winner?
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|≥ 119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(5,2)
|> 1014,006
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|0
|{{TM|1RA------}}
|-
|BBS(2,3)
|≥ 165<ref>Nick Drozd. [https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]</ref>
|{{TM|1RB0LA---_1LB2LA0RB}}
|-
|BBS(3,3)
|> 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|-
|BBS(4,3)
|> <math>10 \uparrow^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,4)
|0
|{{TM|1RA---------}}
|-
|BBS(2,4)
|≥ 205,770,076,433,044,242,247,860
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}

=== Busy Periodic Beaver ([[BBP]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Period
!Champions
|-
|BBP(1,2)
|1
|{{TM|1RA---}}
|-
|BBP(2,2)
|≥ 9
|{{TM|1RB0RB_1LB1RA}} proven winner?
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(4,2)
|≥ 212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Period
!Champions
|-
|BBP(1,3)
|1
|{{TM|1RA------}}
|-
|BBP(2,3)
|
|
|-
|BBP(3,3)
|≥ 1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|1
|{{TM|1RA---------}}
|-
|BBP(2,4)
|≥ 33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|}

= '''Instruction-Limited Busy Beaver''' =
== Instruction-Limited Classical Busy Beaver Functions ==
=== Instruction-Limited Maximum Shifts Function ([[BBi]](n)) ===
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BBi(1)
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|BBi(2)
|3
|{{TM|0RB---_1LA---|halt}}
|-
|BBi(3)
|5
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|16
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|37
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|123
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,963
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|BBi(8)
|<math>>6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}

=== Instruction-Limited Maximum Score Function ([[Σi]](n)) ===
{| class="wikitable"
|+
!
!Score
!Champions
|-
|Σi(1)
|1
|{{TM|1RH---|halt}}
|-
|Σi(2)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|Σi(3)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|Σi(4)
|5
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|Σi(5)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|Σi(6)
|14
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|Σi(7)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|Σi(8)
|<math>>1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|Σi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}

== Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BLBi(1)
|nonexistent
|nonexistent
|-
|BLBi(2)
|nonexistent
|nonexistent
|-
|BLBi(3)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLBi(4)
|12
|{{TM|1RB---_1RC---_1LC0RC}}
|-
|BLBi(5)
|30
|{{TM|1RB------_1RC------_2LC2RC0RC}}
|-
|BLBi(6)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|808
||{{TM|1RB------_1RC------_0RD2LC---_1LD2RD0RC}}
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|-
|BLBi(9)
|> 1042,745
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}

== Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|gBBi(1)
|1
|
|-
|gBBi(2)
|3
|
|-
|gBBi(3)
|5
|
|-
|gBBi(4)
|13
|
|-
|gBBi(5)
|19
|
|-
|gBBi(6)
|25
|
|-
|gBBi(7)
|41
|
|-
|gBBi(8)
|55
|
|-
|gBBi(9)
|238
|
|-
|gBBi(10)
|941
|
|-
|gBBi(11)
|1341
|
|-
|gBBi(12)
|10465
|
|-
|gBBi(13)
|10675
|
|-
|gBBi(14)
|≥ 9,874,580
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}

= '''Program-Limited Busy Beaver''' =
== Busy Beaver for Lambda Calculus ==
=== Regular Busy Beaver for Lambda Calculus ([[BBλ]](n)) ===
For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of <math> 20 \geq n </math> BBλ(n) = n.
{| class="wikitable"
|+
!
!BBλ(n)
!Champions
|-
|BBλ(4)
|4
|<math>\lambda 1</math>
|-
|BBλ(6)
|6
|<math>\lambda\lambda 1</math>
|-
|BBλ(7)
|7
|<math>\lambda\lambda 2</math>
|-
|BBλ(8)
|8
|<math>\lambda\lambda\lambda 1</math>
|-
|BBλ(9)
|9
|<math>\lambda\lambda\lambda 2</math>
|-
|BBλ(10)
|10
|<math>\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(11)
|11
|<math>\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(12)
|12
|<math>\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(13)
|13
|<math>\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(14)
|14
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(15)
|15
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(16)
|16
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(17)
|17
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(18)
|18
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(19)
|19
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|BBλ(20)
|20
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|BBλ(21)
|22
|<math>\lambda(1(\lambda 2))(1(\lambda 2))</math>
|-
|BBλ(22)
|24
|<math>\lambda(1 1) (1 1) (1 1)</math>
|-
|BBλ(23)
|26
|<math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>
|-
|BBλ(24)
|30
|<math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>
|-
|BBλ(25)
|42
|<math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|BBλ(26)
|52
|<math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>
|-
|BBλ(27)
|44
|<math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|BBλ(28)
|58
|<math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>
|-
|BBλ(29)
|223
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
|-
|BBλ(30)
|160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|-
|BBλ(31)
|267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|-
|BBλ(32)
|298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|-
|BBλ(33)
|1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|-
|BBλ(34)
|327,686
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(35)
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(36)
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
|-
|BBλ(37)
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|<math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(38)
|<math>\geq 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|<math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(39)
|<math>\geq 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(40)
|<math>> (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(41)
|<math>\geq 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|<math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(42)
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(43)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|BBλ(44)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(45)
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|BBλ(46)
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(47)
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
|-
|BBλ(48)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|BBλ(49)
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|BBλ(61)
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|BBλ(86)
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(90)
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(94)
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(95)
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|BBλ(96)
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(100)
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(201)
|> q(5)
|<code>too large to show</code>
|-
|BBλ(331)
|lim(BMS)
|<code>too large to show</code>
|-
|BBλ(1850)
|<math>> \text{Loader's Number}</math>
|<code>too large to show</code>
|}

=== Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ1(n)]]) ===
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
|+
!
!BBλ1(n)
!Champions
|-
|BBλ1(1)
|0
|
|-
|BBλ1(2)
|1
|<math>1</math>
|-
|BBλ1(3)
|0
|
|-
|BBλ1(4)
|4
|<math>\lambda 1</math>
|-
|BBλ1(5)
|5
|<math>\lambda 2</math>
|-
|BBλ1(6)
|6
|<math>\lambda \lambda 1</math>
|-
|BBλ1(7)
|7
|<math>\lambda \lambda 2</math>
|-
|BBλ1(8)
|26
|<math>1 (\lambda 1)</math>
|-
|BBλ1(9)
|9
|<math>\lambda \lambda 2</math>
|-
|BBλ1(10)
|36
|<math>1 (\lambda \lambda 1)</math>
|-
|BBλ1(11)
|41
|<math>1 (\lambda \lambda 2)</math>
|-
|BBλ1(12)
|266
|<math>1 (1 (\lambda 1))</math>
|-
|BBλ1(13)
|51
|<math>1 (\lambda \lambda 2)</math>
|-
|BBλ1(14)
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<math>1 (1 (\lambda \lambda 1))</math>
|-
|BBλ1(15)
|<math>f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
|<math>1 (1 (\lambda \lambda 2))</math>
|-
|BBλ1(16)
|<math>f(266)</math>
|<math>1 (1 (1 (\lambda 1)))</math>
|-
|BBλ1(17)
|<math>f(51)</math>
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|-
|BBλ1(18)
|<math>f^{4}(4) = f(f(266))</math>
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(19)
|<math>f^{3}(7) = f(f(41))</math>
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|-
|BBλ1(20)
|<math>f^{6}(4) = f^{4}(266)</math>
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(21)
|<math>f^{7}(4) = f^{5}(266)</math>
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|-
|BBλ1(22)
|<math>f^{52}(4) = f^{50}(266)</math>
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|-
|BBλ1(28)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|-
|BBλ1(29)
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|}

=== Busy Beaver for De Bruijn Lambda Calculus ===
{| class="wikitable"
!n
!Value
!Champion
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|10
|<math>> 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|11
|<math>> 10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|-
|15
|<math>> f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|}

== [[Busy Beaver for SKI calculus]] ==
=== Busy Beaver for SKI calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || 1 || S
|-
| 2 || 2 || SS
|-
| 3 || 3 || SSS
|-
| 4 || 4 || SSSS
|-
| 5 || 6 || SSS(SS)
|-
| 6 || 17 || SSS(SI)S
|-
|7
|≥ 18
|S(SSS(SI)S)
|-
|8
|≥ 19
|SS(SSS(SI)S)
|-
|9
|≥ 519
|SSI((S(SS)S)S)K
|-
|10
|≥ 1041
|SSI((S(SS)S)S)KS
|}

=== Busy Beaver for SK calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || = 1 || S
|-
| 2 || = 2 || SS
|-
| 3 || = 3 || SSS
|-
| 4 || = 4 || <code>SSSS</code>
|-
| 5 || = 6 || <code>SSS(SS)</code>
|-
| 6 || ≥ 8 || <code>SSS(SSS)</code>
|-
|7
|≥ 10
|<code>SSS(SSSS)</code>
|-
|8
|≥ 23
|<code>SSS(S(SKS))S</code>
|}

== Fractran ([[Fractran|BBf]](n)) ==
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|-
|20
| 746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
& -1 & -1 & +1 & \\
+1 & -1 & & & +1 \\
+1 & +1 & & -1 & -1 \\
-1 & & +1 & & \\
& +2 & -1 & &
\end{bmatrix}</math>
|-
|21
|≥ 31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|}

== Cyclic Tag ([[Cyclic Tag|CTBB(n)]]) ==
{| class="wikitable"
|+
!
!Runtime
!Champions
|-
|CTBB(2)
|5<ref>https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233</ref>
|
|-
|CTBB(3)
|> 38<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517</ref>
|
|-
|CTBB(4)
|≥ 672<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043</ref>
|
|-
|CTBB(5)
|≥ 2^2^2^2^182<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063</ref>
|
|-
|CTBB(6)
|> 2↑↑↑131<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479</ref>
|
|-
|CTBB(7)
|> 4↑↑↑↑(4↑↑↑3)<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281</ref> <ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217</ref>
|
|}

== [[Tiny Tag]] ==
{| class="wikitable"
|+
!n
!Runtime
!Champions
|-
|1
|1
|0
|-
|2
|5
|10
|-
|3
|9
|101
|-
|4
|13
|1001
|-
|5
|15
|10101
|-
|6
|334
|010111
|-
|7
|404
|1010111
|-
|8
|670
|11100101
|-
|9
|12584
|001101110
|-
|10
|2180995
|0100011110
|}

== Minsky Machines ([[Register machine|MBB(n)]]) ==

{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+Z</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|≥ 49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|}

= '''Turmites''' =
== Terminating Turmites ([[TT]](n,k), 1D Turmites) ==
Where n is the amount of states and k is the amount of symbols.
{| class="wikitable"
!Domain
!Runtime
!Champions
|-
|TT(2)
|≥ 13
|<code>1TB---_1PA0PB</code>
|-
|TT(3)
|≥ 82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4)
|≥ 48,186
|<code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>
|-
|TT(2,3)
|≥ 223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3)
|≥ 45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|-
|TT(2,4)
|> 3.467*1015
|<code>1TA2PB3TB---_3TA1PB1TA1PA</code>
|}

== 2D Turmites ([[Terminating Turmite|turNing machines]]) ==
There are currently no known/available Champions for this function.

= '''Doodle Function ([[Doodle function|doodle(c,n)]])''' =
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Champions
|-
|doodle(3,2)
|≥ 487
|
|}

= '''Bug Function''' =
Bug(2,2) = 2
<pre>
####
#S-#
#.F#
####
</pre>

Bug(3,3) = 8
<pre>
#####
#S..#
#.#.#
#.#F#
#####
</pre>

Bug(4,4) = 20
<pre>
######
#S...#
#..###
#....#
#..#F#
######
</pre>

Bug(5,5) = 42
<pre>
#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######
</pre>

Bug(6,6) = 96
<pre>
########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########
</pre>

Bug(7,7) = 218
<pre>
#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########
</pre>

Bug(8,8) = 506
<pre>
##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########
</pre>

= '''References''' =

User:Polygon

2026-04-26T08:36:58Z

Polygon: Added new BB(4,3) top halters

List of user pages:
* [[User:Polygon/Collection of BB Champions]]
* [[User:Polygon/Better lower bound for BB(4,3)]]
* [[User:Polygon/Page for analyses]]
* [[User:Polygon/Page for testing]]

A list of interesting user pages:
* [[User:RobinCodes/Valuable user pages]]

Other sites:
* https://www.sligocki.com/
* https://groups.google.com/g/busy-beaver-discuss
* https://discuss.bbchallenge.org/
* https://bbchallenge.org/~pascal.michel/index.html
* https://nickdrozd.github.io/
* https://scottaaronson.blog/
* https://katelyndoucette.com/
* https://www.cse.unr.edu/~al/BusyBeaver.html
* https://cruzgodar.com/applets/lambda-calculus/?expression-textarea=%28%CE%BBa.aa%29%28%CE%BBa.a%28%CE%BBb.a%28%CE%BBc.ab%29%29a%29&animation-time-slider=1.095790162866908 (lambda calculus simulator)

Table of 30 longest running TMs across explored domains:
{| class="wikitable"
!Standard format
!Approximate sigma scores
!Discovered by
!Domain
|-
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|<math>(2 \uparrow^{15} 5) + 14</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1RZ1LA2RB_1RC3RC1LA2LB_2LB2RC1LC3RB|halt}}
|<math>(2 \uparrow^{12} 4) + 11</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|Pavel Kropitz
|[[BB(7)]]
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_3LA1RZ1LC2RA|halt}}
|<math>2 \uparrow^{6} 5</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB3RB1LC2LA_2LA2RB1LB3RA_1RA1RZ1LC2RA|halt}}
|<math>2 \uparrow^{6} 4</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1})</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|Pavel Kropitz
|[[BB(4,3)]]
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|<math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8.10237</math>
|mxdys
|[[BB(6)]]
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|<math>10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
|Pavel Kropitz
|[[BB(2,6)]]
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|<math>10 \uparrow\uparrow 11010000</math>
|mxdys
|[[BB(6)]]
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|<math>10 \uparrow\uparrow 19892.08</math>
|Peacemaker II
|[[BB(2,6)]]
|-
|{{TM|1RB0LB1RZ3LA_0LC3RB3RC1LB_2RB2LA3RA1LC|halt}}
|<math>10 \uparrow\uparrow 2048</math>
|Pavel Kropitz
|[[BB(3,4)]]
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|<math>10 \uparrow\uparrow 1023.47221</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|<math>10 \uparrow\uparrow 619.07737</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|<math>10 \uparrow\uparrow 519.20</math>
|Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|<math>10 \uparrow\uparrow 512.10945</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|<math>10 \uparrow\uparrow 439.02781</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG|halt}}
|<math>10 \uparrow\uparrow 403.84</math>
|Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|<math>10 \uparrow\uparrow 286.17</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|<math>10 \uparrow\uparrow 246.32</math>
|lijil
|[[BB(7)]]
|-
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|<math>10 \uparrow\uparrow 243.88</math>
|lijil, Andrew Ducharme
|[[BB(7)]]
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|<math>10 \uparrow\uparrow 234.06408</math>
|Andrew Ducharme
|[[BB(4,3)]]
|-
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|<math>10 \uparrow\uparrow 228.78</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG|halt}}
|<math>10 \uparrow\uparrow 192.67</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|<math>10 \uparrow\uparrow 190.21359</math>
|Terry Ligocki
|[[BB(4,3)]]
|-
|{{TM|1RB1RZ1LA2RB_2RC3RC3LB2RA_2LB3RB3LC1LA|halt}}
|<math>10 \uparrow\uparrow {189.05690}</math>
|[[User:XnoobSpeakable|XnoobSpeakable]]
|[[BB(3,4)]]
|-
|{{TM|1RB0LC_1LC1LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LA|halt}}
|<math>10 \uparrow\uparrow 188.28</math>
|Terry Ligocki
|[[BB(7)]]
|-
|{{TM|1RB3LA1RA2LB_2LC1RA2RC3RB_1RZ1LA2RA3RA|halt}}
|<math>10 \uparrow\uparrow {188.16168}</math>
|XnoobSpeakable
|[[BB(3,4)]]
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|<math>10 \uparrow\uparrow 178.48320</math>
|Andrew Ducharme
|[[BB(4,3)]]
|}

Holdouts lists

2026-04-26T08:28:39Z

Polygon: Updated BB(4,3) holdout count and added new holdouts list

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1161
|17,823,260
|-
!3-symbol
|0
|4
|5,641,006
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|66
|
|
|
|
|
|-
!6-symbol
|536,112
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|Variable (49)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==